Parametric analysis of homogeneous and heterogeneous nucleation in subcritical CO2-mediated antisolvent crystallization

chemical engineering research and design 1 0 6 ( 2 0 1 6 ) 283–297

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Parametric analysis of homogeneous and heterogeneous nucleation in subcritical CO2 -mediated antisolvent crystallization Shital D. Bachchhav, Sandip Roy ∗ , Mamata Mukhopadhyay Department of Chemical Engineering, IIT Bombay, Mumbai 400076, Maharashtra, India

a r t i c l e

i n f o

a b s t r a c t

Article history:

The present work investigates the tunability of antisolvency effect of subcritical carbon

Received 15 September 2015

dioxide (CO2 ), in a solution of cholesterol in acetone, which facilitates high and rapid

Received in revised form

supersaturation needed for producing cholesterol micro-particles. A thermodynamic anal-

11 December 2015

ysis is proposed for selection of operating conditions that result in high solid solute

Accepted 16 December 2015

supersaturation. This is further coupled to a computational analysis of homogeneous and

Available online 23 December 2015

heterogeneous nucleation, and crystal growth kinetics. A numerical strategy has been evolved and for verifying its consistency the predicted particle size has been compared

Keywords:

with that obtained experimentally. In addition, the effects of pressure (60–70 bar), tem-

Subcritical CO2

perature (291–303 K), initial solute concentration (90–100%), specific dissolution rate of CO2

Antisolvent crystallization

(0.095–6.0 min−1 ), and nuclei-substrate contact angle (30–50◦ ) on average particle size have

Secondary nucleation

been ascertained. In the case of homogeneous nucleation, the particle size increases with

Heterogeneous nucleation

temperature, while pressure has a negligible effect. Further, the particle size decreases

Particle size

when the antisolvent dissolution rate and initial solute loading in solvent are increased. For heterogeneous nucleation, an enhancement in contact angle increases the particle size. These trends are in agreement with the experimental observations reported in the literature. The computational method thus elucidates a generalized approach for engineering desired particle size in subcritical CO2 -mediated antisolvent crystallization. © 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

Production of ultra-fine particles of controlled size has been a challenge to the chemical and pharmaceutical industries for many years (Dodds et al., 2007; Elvassore et al., 2003; Jung and Perrut, 2001; Subramaniam et al., 1997; Tabernero et al., 2012). Conventional techniques employed for synthesizing ultra-fine particles include jet milling, freezedrying, spray-drying, liquid antisolvent process and thermal cooling (Elvassore et al., 2003; Martín and Cocero, 2008; Subramaniam et al., 1997). However, one of the relatively recent techniques, with considerable potential for use in pharmaceutical industries, is crystallization employing supercritical or subcritical carbon dioxide (CO2 ) as an antisolvent.

∗

CO2 is used as an antisolvent because it is inert, safe, nonflammable, environment-friendly green gas, and highly soluble in many organic solvents (Esfandiari and Ghoreishi, 2013; Subramaniam et al., 1997; Tabernero et al., 2012). Processes that use supercritical CO2 as an antisolvent have been explored widely (Subramaniam et al., 1997; Tabernero et al., 2012). Examples of such processes include precipitation with compressed antisolvent (PCA), supercritical antisolvent (SAS), and solution enhanced dispersion with supercritical fluids (SEDS) (Elvassore et al., 2003; Tabernero et al., 2012). Those employing subcritical CO2 include precipitation by pressure reduction of gas expanded liquid (PPRGEL) (Dalvi and Mukhopadhyay, 2009) and gas antisolvent (GAS) process (Elvassore et al., 2003).

Corresponding author. Tel.: +91 22 25767249; fax: +91 22 2572 6895. E-mail address: [email protected] (S. Roy). http://dx.doi.org/10.1016/j.cherd.2015.12.016 0263-8762/© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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Nomenclature Ap C3 C∗3 Dp Dp D32 h J kB kg MW3 NA nn ng Np P R rc r0 rp S Sab T vL V VL v2 vm Vc Vt Vp rb x1 max x1 * Yf

surface area of the particle [m2 ] instantaneous solid concentration [mol/m3 ] equilibrium solid concentration [mol/m3 ] diameter of the particle [m] average diameter of the particle [m] diffusion coefficient of the solid solute [m2 /s] width of the cap-shaped particle [m] nucleation rate [#.m3 /s] Boltzman’s constant [J/K] mass transfer coefficient [m/s] molecular weight of solid solute [g/mol] Avogadro’s number [mol−1 ] number of moles of solid precipitated due to nucleation [mol] number of moles of solid precipitated due to growth [mol] number of nuclei formed [–] pressure [bar] universal gas constant [J/mol/K] critical radius of the nuclei [m] molecular radius of the solid solute [m] radius of the particle [m] solid solute supersaturation [–] absolute solid solute supersaturation [–] temperature [K] molar volume of the liquid [m3 /mol] total volume of the vessel [m3 ] initial volume of the solvent [m3 ] partial molar volume of the solvent [m3 /mol] molar volume of GEL solution [m3 /mol] critical volume of the nuclei [m3 ] total volume of GEL [m3 ] volume of the particle [m3 ] base radius of the cap-shaped particle [m] maximum CO2 mole fraction when vessel is completely filled [–] equilibrium CO2 mole fraction [–] pre-defined solid solute yield [%]

Greek symbols three-phase contact angle  solute density [kg/m3 ] 3  mixture viscosity [Pa s] solid solute-GEL interfacial tension [N/m1 ]  cl solid solute-substrate interfacial tension  cs [N/m1 ] substrate–GEL interfacial tension [N/m1 ]  sl Abbreviations GAS gas antisolvent process GEL gas-expanded liquid initial solute concentration ISC rapid expansion of supercritical solutions RESS SDR specific dissolution rate SC CO2 supercritical carbon dioxide

The present paper focuses on a parametric analysis of subcritical CO2 -mediated antisolvent crystallization of cholesterol particles from acetone solution. In this process the solid solute to be precipitated is pre-dissolved in a suitable organic

solvent and subcritical CO2 is added to it. The dissolution of CO2 progressively expands the solution forming a gas expanded liquid (GEL), which is accompanied by substantial reduction in the solvent power for the solid solute, which in turn enhances the solute supersaturation and then precipitates the solute (Bakhbakhi et al., 2005; Dodds et al., 2007; Esfandiari and Ghoreishi, 2013; Muhrer et al., 2002). Several researchers have studied the effects of operating parameters on the characteristics of the cholesterol particles precipitated using different supercritical carbon dioxide (SCCO2 ) processes. Subra et al. (2004) investigated cholesterol particle formation using rapid expansion of supercritical solutions (RESS) and supercritical antisolvent (SAS) processes. For the RESS process the particle morphology was needle-like and the average size ranged from 1–15 ␮m. However, in the SAS process the particle morphology and sizes were influenced by flow rates and concentration. The morphology varied from needle-shaped particles to curved or flat elongated structures. In another work, Liu et al. (2002) produced cholesterol particles from acetone solution via the GAS process. The morphology of cholesterol particles varied depending on the CO2 injection rate. Low injection rate produced needle-like crystals whereas rapid injection rate resulted in flattened tabular crystals. Dalvi and Mukhopadhyay (2009) studied the effects of various process parameters on the particle size and yield of cholesterol particles when precipitated from its acetone or ethanol solutions using subcritical CO2 by the PPRGEL process. Particle size ranged from 200 nm to 7 ␮m, and the morphologies were near-spherical to needle-like depending on the operating conditions. In general, formation of ultra-fine particles requires attainment of very high and rapid supersaturation of the solid solute in the solution, followed by its nucleation. Subsequent growth of the nuclei occurs due to diffusion of solute molecules from solution to the nuclei surface followed by their surface integration (Myerson, 2002). The nucleation process may be either homogeneous or heterogeneous. In the existent literature, the antisolvent crystallization has largely been addressed by considering homogeneous nucleation which may involve primary and secondary nucleation (Bakhbakhi et al., 2005; Dodds et al., 2007; Erriguible et al., 2015; Esfandiari and Ghoreishi, 2013; Muhrer et al., 2002). For example, Erriguible et al. (2015), developed a mathematical model for the estimation of nucleation and growth parameters of naproxen + nicotinamide cocrystals precipitated from acetone solution using the GAS process. This was achieved by fitting the predicted particle size distribution data to that obtained experimentally. In general, however, the nature of nucleation occurring during crystallization has not been rigorously justified in the literature. As it is well-known, the homogeneous nucleation can occur only when the solution is absolutely free of foreign particles (i.e., dust, impurities), which typically is not obtained in most practical crystallization processes (Kashchiev and ´ van Rosmalen, 2003; Mersmann, 2001; Mullin, 2001; Nyvlt, 1984). This is because impurities that may be usually present in the system can always provide sites for heterogeneous nucleation. In their study on heterogeneous nucleation, Liu (2000) investigated the effect of foreign particles on nucleation of paracetamol from aqueous solution, both in the absence and presence of additives. It was concluded that the greater is the affinity between the foreign particle and the crystalline phase, the lower is the supersaturation required for the

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onset of nucleation. Another theoretical investigation by the same authors for heterogeneous nucleation on surface which contains cavities, concluded that at lower levels of supersaturation, the rate of heterogeneous nucleation is higher than that of the homogeneous nucleation (Liu et al., 2008). Further, a modeling and simulation study by Kumar et al. (2013), reported particle size estimation in the PPRGEL process for CO2 –acetone–cholesterol system in which solid precipitation is induced by co-generated CO2 gas bubbles, the latter providing heterogeneous nucleation sites for cholesterol particles. The authors predicted that the values of nucleation rates obtained are higher for heterogeneous nucleation as compared to homogeneous nucleation. The foregoing considerations suggest that the nature of nucleation is a critical determinant in crystallization including its sensitivity on process parameters. The present work thus attempts to address the subject through a systematic modeling and simulation for the system comprising subcritical CO2 (1), acetone (2), and cholesterol (3) considering both homogeneous and heterogeneous nucleation phenomena, separately. The main objective of this work is to present a theoretical approach that provides a rationale for selecting the various process parameters and the nature of substrate necessary for engineering ultra-fine solid particles for antisolvent crystallization. The effects of various process parameters on the nucleation and growth kinetics induced by the antisolvency effect of CO2 and in turn, on the final particle size are also elucidated.

2.

Computational methodology

Precipitation of solids from GEL is governed by phase equilibrium thermodynamics as well as nucleation and growth kinetics. The computational methodology adopted in this work is executed in two consecutive steps namely, (i) the selection of operating conditions such as pressure, P, temperature, T, initial solute concentration (ISC), specific dissolution rate (SDR) of CO2 , and contact angle between the nuclei and substrate, ; (ii) estimation of the average particle size considering homogeneous and heterogeneous nucleation phenomena. The thermodynamic variables P, T, and ISC are selected for obtaining a very high supersaturation of the solid solute in GEL. On the other hand SDR of CO2 is selected for attaining rapid supersaturation. The parameter SDR governs the nucleation and growth kinetics and hence determines the average size and size distribution of the precipitating particles. In addition, the effect of the energetics of a foreign, heterogeneous substrate on particle size is demonstrated by systematic variation of the initial three phase contact angle, , that may exist at the substrate–solid solute–GEL contact point.

2.1. Equilibrium solid solute solubility and supersaturation 2.1.1.

equations of state (EOS) have been used, they do not result in accurate values of v2 at high CO2 mole fractions (x1 > 0.80), which is of interest in the present work. Mukhopadhyay and Dalvi (2004) have shown that x3 * is very sensitive to the variation of x1 at a high value of x1 (i.e., x1 > 0.80) at any T and even a small uncertainty in the computed values of v2 leads to a large error in the estimated values of x3 * and supersaturation. Therefore, in the present work, x3 * of cholesterol in CO2 –acetone mixture has been regressed from the experimental data at 308 K and 318 K reported in the literature (Liu et al., 2002), by fitting an appropriate polynomial function and extrapolating to lower temperatures, i.e., 291 K, 298 K, and 303 K.

2.1.2.

Solid solute supersaturation

Supersaturation, S, is the driving force for solute crystallization, which, in this work, is defined as: S=

x3 x3∗

(1)

where x3 is the mole fraction of the solid solute in GEL and is calculated as: x3 =

n3 n3 = n1 + n2 + n3 nt

(2)

where n1 is the time-dependent number of moles of CO2 , n2 is the number of moles of acetone, n3 is the instantaneous number of moles of solid solute remaining in the solution during nucleation and growth, and nt is the total number of moles. These are in turn used to calculate the instantaneous x3 and hence S. The time-dependent supersaturation is further used for the calculation of nucleation rate.

2.2.

Nucleation rate

This section provides a brief overview of the procedures adopted to separately estimate the homogeneous and heterogeneous nucleation rates needed for the particle size estimation.

2.2.1.

Primary nucleation

In homogeneous nucleation particle formation is attributed to both primary nucleation, with rate Jp , and to secondary nucleation, with rate Js . The overall homogeneous nucleation rate amounts to the sum of these two contributions. The time-dependent primary nucleation rate, Jp , is estimated by the following equation (Bakhbakhi, 2009; Bakhbakhi et al., 2005; Dodds et al., 2007; Erriguible et al., 2015; Esfandiari and Ghoreishi, 2013; Lindenberg and Mazzotti, 2009; Mersmann, 2001; Muhrer et al., 2002; Mullin, 2001): Jp = 1.5D32 (C∗3 SNA )7/3



cl w exp kT

 −

16 3





cl3 (kB T)

3

w2



2 (ln S)

(3)

Equilibrium solid solute solubility

Knowledge of equilibrium solid solubility, x3 * , in GEL at high CO2 mole fraction is necessary for estimating supersaturation, nucleation and growth rates. Methods available in the literature based on the equations of state (EOS) approach can be used to predict x3 * in CO2 -expanded solvent (Dodds et al., 2007; Muhrer et al., 2002; Mukhopadhyay and Dalvi, 2004). However, these methods require an accurate knowledge of liquid phase partial molar volume of the solvent, v2 , and the corresponding molar volume of the solution, vm . Though several cubic

The critical radius of nuclei that are formed due to homogeneous nucleation is expressed as (Mullin, 2001): rc =

2wcl kB T ln S

(4)

where w is the molecular volume of the solid solute (=MW3 /3 NA ), MW3 is the molecular weight of solid solute, 3 is the density of solid solute, NA is the Avogadro’s number, kB

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is the Boltzmann constant, T is the process temperature, D32 is the diffusion coefficient of solid solute in GEL,  cl is the interfacial tension between solid solute and GEL, and C3 * (=x3 * /vm ) is the equilibrium concentration of solid solute in GEL. The value of D32 in GEL is estimated using the Stokes–Einstein equation (Dodds et al., 2007; Kumar et al., 2013): D32 =

kB T 6r0 

(5)

where r0 is the molecular radius of solid solute, which can be obtained by equating the molecular volume of the solid solute to the equivalent volume of a sphere, as: r0 =

 3MW 1/3 3

(6)

43 NA

The viscosity, , of the medium, i.e., GEL is estimated by the following equation (Kumar et al., 2013):

=

2

(7)

xi i

basis of an empirical equation (Dodds et al., 2007; Erriguible et al., 2015): Js = ASN Cm s (S − 1)

n

(11)

where m and n are the empirical constants, generally regressed from experimental data. For the present computations the values of m and n are chosen to be 1.95 and 0.99, respectively as reported (Erriguible et al., 2015) for a system in which the antisolvent and solvent are the same as in the present system, i.e., CO2 and acetone, respectively. The parameter Cs is the instantaneous concentration of the suspended solid in GEL and is calculated as: Cs =

n30 − n3 Vt

(12)

where n30 is the initial number of moles of solid solute. The value of Cs changes with the progression of nucleation and growth of the particles. The total volume of CO2 -expanded solvent (i.e., GEL), Vt , is calculated as:

i=1

Vt = nt vm where xi and i are the mole fraction and viscosity of component ‘i’ respectively. Since the initial cholesterol mole fraction in GEL is small in the present work (i.e., x30 ∼10−3 ), the latter is approximated as a binary solution of CO2 and acetone. The viscosity of each of these components can be calculated from the empirical expressions presented in the literature (Kumar et al., 2013) as:



1 = exp 18.775 −



402.92 − 4.6854 log T − 6.917 × 10−26 T 10 T (8)



2 = exp −14.918 +

1023.4 + 0.5461 log T T

 (9)

The solid solute-GEL interfacial tension,  cl , is a critical system property for crystallization. Mersmann (1990) developed a theoretical approach to calculate the interfacial tension of binary solid–liquid systems based on a free energy balance over the interface. The interfacial tension,  cl , between the solid solute-GEL is given by: cl = 0.414kB T

 N  2/3 A 3 MW3

ln

3 MW3 C3

 (10)

where C3 (=x3 /vm ) is the instantaneous concentration of solid solute in GEL. It may be emphasized that Eq. (10) is used to compute the progressively changing interfacial tension corresponding to the instantaneous solute concentration in GEL as used by others (Dodds et al., 2007; Esfandiari and Ghoreishi, 2013), rather than a constant interfacial tension at an equilibrium solute concentration (Bakhbakhi, 2009; Bakhbakhi et al., 2005; Muhrer et al., 2002).

2.2.2.

Secondary nucleation

Secondary nucleation occurs subsequently in the vicinity of the nuclei formed due to primary nucleation (Myerson, 2002). The rate of secondary nucleation, Js , may be calculated on the

(13)

In the present study, the molar volume of GEL has been estimated using the generalized correlation developed by Dalvi and Mukhopadhyay (2007). The molar volume of GEL, vm , is defined as vm =

ZL RT P

(14)

where ZL = APBr and Pr = P/(Pc,1 x1 + Pc,2 x2 ). Here, A and B are the system-specific coefficients for the relevant ranges of P and T, and Pc is the critical pressure. In this study, A and B for CO2 –acetone system are obtained from the literature (Dalvi and Mukhopadhyay, 2007). The empirical constant, ASN in Eq. (11) is dependent on the physical characteristics of the system (Dodds et al., 2007) and is reported to be proportional to D32 and inversely proportional to d4m , i.e., ASN ∝

D32 d4m

(15)

where dm [=2r0 ] is the molecular diameter of the solid solute. ASN is also dependent on temperature as it is a function of diffusivity (Myerson, 2002). The value of ASN usually lies in the range of 109 –1011 (Dodds et al., 2007; Erriguible et al., 2015). In the present work, the values of ASN reported by Erriguible et al. (2015) have been appropriately corrected for cholesterol based on its functional dependence on T and dm as expressed by Eq. (15). Combining Eqs. (3) and (11), one can determine the overall homogeneous nucleation rate (i.e., Jhom = Jp + Js ) for the selected operating condition.

2.2.3.

Heterogeneous nucleation

Heterogeneous nucleation is principally governed by the affinity of the solid solute for the foreign (e.g. metallic, polymeric, or ceramic) substrate, which is typically characterized in terms of the three-phase contact angle. The following equation has been employed for computing heterogeneous nucleation rate

chemical engineering research and design 1 0 6 ( 2 0 1 6 ) 283–297

Fig. 1 – Three-phase contact angle for nuclei formation on a flat substrate. (a) Initial spherical cap geometry of nuclei; (b) geometries for different contact angles (Â  90◦ , Â ≈ 90◦ , Â > 90◦ , Â ≈ 180◦ ). considering that the nucleation occurs on a flat substrate as (Löffelmann, 2002):

∗

Jhet = 1.5D32 (C SNA )

 −

exp

16 3



f ()cl w kT

7/3



cl3 (kB T)

3

w2 (ln S)

 (16)

where f() is the geometric correction factor which depends on three-phase contact angle, , and is given by: f () =

2 − 3 cos  + cos3  4

(17)

The critical radius of the nuclei formed due to the heterogeneous nucleation is the same as that given by Eq. (4). The critical volume, Vc , of the nuclei formed due to heterogeneous nucleation, however, is a function of  and is less than that of nuclei formed by homogeneous nucleation. The expression for Vc of the spherical nuclei for homogeneous nucleation is: Vc(hom) =

4 3 r 3 c

(18)

whereas Vc of the cap-shaped nuclei (Fig. 1a) in the case of heterogeneous nucleation is (Mullin, 2001): Vc(het) =

4 3 r f () 3 c

2.3.

Three-phase contact angle

(19)

As shown in Fig. 1a the three-phase contact angle corresponds to that which forms between the substrate (s), solid solute (c), and GEL (l). From a force balance criterion at the three-phase contact line, one can obtain (Mullin, 2001): cos  =

sl − cs cl

To obtain the solid solute-substrate contact angle, it is necessary to estimate all three interfacial tensions ( sl ,  cs , and  cl ). The expression for the calculation of solid solute-GEL interfacial tension is already given in Eq. (10). In principle, the substrate–GEL interfacial tension,  sl , can be calculated based on the individual surface energies of the substrate and liquid using the Van Oss–Chaudhury–Good (VCG) theory (Rieke, 1997). The method to calculate solid solute-substrate interfacial tension,  cs , has also been discussed in the literature and is based on the individual surface energies of substrate and solid solute (Rieke, 1997). However, in the present work the latter two interfacial tensions have not been specifically computed. This is because the objective of the present work is to directly analyze the sensitivity of the final particle size to a variation in the three-phase contact angle. Accordingly a range of  between 30–50◦ has been selected to analyze the effect of the nature of substrate.

2.4.

f () 2

(20)

where  sl = substrate–GEL interfacial tension,  cs = solid solute-substrate interfacial tension, and  cl = solid solute-GEL interfacial tension. Fig. 1b shows different geometric of nuclei for the possible range of variation of . The value of f() ranges between 0 and 1. The condition,  = 0◦ , wherein f() = 0, is obtained when the nuclei fully ‘wets’ the substrate, i.e., there is no energy barrier for nucleation on the substrate. Conversely, when  = 180◦ , and f() = 1, the nuclei have no affinity for the substrate. It, therefore, plays no role in the nucleation process, which then is homogeneous in nature.

287

Crystal growth rate

The growth of the crystal may occur sequentially by solute diffusion in the bulk of the solution followed by surface integration. Diffusion controlled and surface integration controlled resistances can be combined in the form of an empirical growth rate equation depending on the process condition and is given as (Erriguible et al., 2015): G = kg C∗3 (S − 1)

g

(21)

where kg is the mass transfer coefficient and g is the growth parameter. These parameters are generally regressed from experimental data and may vary substantially depending upon the system conditions. In the present work, kg and g have been taken as 2.48 × 10−5 m/s and 1.0, respectively corresponding to the values reported in the literature for a similar system (i.e., containing CO2 and acetone) (Erriguible et al., 2015).

2.5.

Particle size estimation

This section presents the sequential steps employed in the present work to determine the final average particle size as well as the size distribution. As a numerical strategy, the total time over which particles nucleate and grow in GEL is discretized into a large number of small intervals, t (∼ 0.1 s), and all relevant parameters are updated progressively with time. The process conditions such as P, T, and SDR are first selected. The selected SDR decides the rate of change of x1 which in turn is used for the estimation of n1 . Next the value of ISC is calculated for the selected T and then it is further used for calculating the initial solid solute supersaturation, S, using Eq. (1). Next, the nucleation rate is calculated for a given value of initial supersaturation using Eq. (3) or Eq. (16) depending on the type of nucleation. For any subsequent ith time interval the number of nuclei formed due to nucleation is calculated as Np(i) = J(i)hom/het Vt(i) t

(22)

The value of Np(i) is rounded to an integer value. The total volume of GEL, Vt(i) is calculated using x1 at the beginning of the ith time interval by Eq. (13).

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Further, the number of moles of solute consumed due to nucleation, nn , for the (j–1)th interval is calculated using the following expression: nn(j−1) =

(Jhom/het Vt )j−1 tVc 3

(23)

MW3

Once the nuclei are formed, the growth in their size over time is computed using Eq. (21) and the number of moles of solute consumed due to crystal growth during the (j–1)th time interval is calculated using the following equation: ng(j−1) =



g

Np(i) kg Ap(i,j−1)hom/het C∗3 (S − 1)j−1 t

(24)

i

where Ap is the surface area of the particle. As is known, the particle morphology is determined by the level of supersaturation prevailing during the crystallization process (Subra et al., 2004). However the objective of this work being the study of the effects of key operating parameters (P, T, ISC, and SDR) on the resultant particle size, for homogeneous nucleation the nuclei and the growing crystals are both assumed to be spherical in shape, whereas, for heterogeneous nucleation it is assumed to be spherical cap-shaped as it is initially formed on a flat substrate. The cap-shaped geometry is assumed based on the configuration illustrated in Fig. 1a. The respective surface areas of the particle formed due to homogeneous or heterogeneous nucleation (used in Eq. (24)) are given, as: 2 Ap(i,j)hom = 4rp(i,j)

(25)

2 Ap(i,j)het = 2rp(i,j) (1 − cos )

(26)

where rp(i,j) is the radius of the spherical or cap-shaped particles either by homogeneous or heterogeneous nucleation. Removal of solute from GEL phase occurs not only due to nucleation, but also due to the subsequent growth of the nuclei. The number of moles of solute present in GEL at the jth time interval is computed by subtracting the total number of moles of solute consumed due to both nucleation and growth during the (j–1)th time interval from the moles of solute present in the beginning of the (j–1)th time interval and is given as: n3(j) = n3(j−1) − nn(j−1) − ng(j−1)

(27)

Using the prevalent values of n1 and n3 , that of S is updated using Eq. (1) for the instantaneous time corresponding to the beginning of the jth time interval. The radius of the particle, rp , at the beginning of the jth time interval is estimated using the following balance equation: 4 3 4 3 g f () = rp(i,j−1) f () + (Ap )(i,j−1)hom/het kg C∗3 (S − 1)(j−1) t r 3 p(i,j) 3



MW3 3

 (28)

where ith is the time interval when the particle is formed and jth is the time interval when its size is calculated. It may be noted that i ≤ j. In addition, rp(i,i) = rc(i,i) for homogeneous nucleation, whereas rp(i,i) = rc(i,i) (f())1/3 for heterogeneous nucleation. For a pre-defined solid solute yield, Yf , all the above computational steps from Eqs. (22)–(28) are

performed for the prediction of final average particle size, as higher is the yield, higher is the particle size. The yield of the solid solute at the jth time interval, Y(j) , can be calculated as: Y(j) =

n30 − n3(j) n30

× 100

(29)

For the special case of f() = 1 in Eq. (28), the particle size obtained is due to homogeneous nucleation. For simulations of the heterogeneous nucleation and growth of all particles formed, it is assumed that the nucleating solid particle is nondeformable, and hence the three-phase contact angle remains invariant during the growth of the nuclei into full particles. In other words, the initial spherical cap geometry of the nuclei defined by the contact angle in Eq. (20) remains unaltered during growth. Thus, for a cap-shaped particle on a flat substrate the base radius of the particle, rb , the maximum ‘width’ of the particle, h, and volume, Vcap , as illustrated in Fig. 1a are calculated at any point of time during nucleation and subsequent growth. The final average size of the particle formed either due to homogeneous or heterogeneous nucleation is calculated using the following expression:

Dp =

n N D i=1 p(i) p(i,)

n i=1

Np(i)

(30)

where th is the last time interval when pre-defined yield is attained and Dp(i,␶) [=2rp(i,␶) ] is the diameter of the particle. It is to be noted that in the case of heterogeneous nucleation the final average equivalent diameter of the cap-shaped particle is obtained from the diameter of the spherical particle having the same volume as of the cap-shaped particle. The overall algorithm for computing the final average particle size employed in this work is shown in Fig. 2.

3.

Results and discussion

The selection of various process conditions is aimed at attaining high and rapid supersaturation, an essential condition for formation of ultra-fine particles. A thermodynamic analysis is performed to select the suitable operating conditions that would yield high actual solute supersaturation, S, based on the criterion that the same would yield the highest absolute solute supersaturation, Sab . In other words, Sab is the value of solute supersaturation that would be obtained at the highest value of x1 max in GEL, without considering depletion of solute by nucleation and growth. Once the appropriate conditions (of P, T, and ISC) are selected, based on high Sab , the particle size estimation is performed by considering continuous solute precipitation. Progressive solute depletion from GEL by nucleation and growth determines the actual instantaneous solute supersaturation, S. The nucleation and growth kinetics are then pursued up to a pre-defined solute yield in accordance with the algorithm presented in Fig. 2. The analysis is extended to examine the effects of various process parameters on the average particle size.

3.1.

Selection of process conditions

3.1.1.

Pressure and temperature

As indicated earlier, the process explored in this work involves progressive dissolution of antisolvent (CO2 ) in the solvent (acetone) in which solid solute (cholesterol) is pre-dissolved. At

chemical engineering research and design 1 0 6 ( 2 0 1 6 ) 283–297

289

Fig. 2 – Algorithm for the estimation of final average particle size.

the end of the dissolution process, the vessel may be either (i) partially filled or (ii) completely filled by GEL depending on the values of P, T, and VL (i.e., the initial volume of the solvent). For example, for a partially filled vessel at the operating pressure less than the saturated vapor pressure of CO2 (i.e., at P ≤ P1 S ), the maximum CO2 mole fraction that can be attained in GEL is x1 max , where x1 max ≤ x1 * , the latter being the equilibrium CO2 mole fraction. However, a still higher value of CO2 mole fraction (i.e., x1 max ≥ x1 * ) can be attained in the single phase region when P ≥ P1 S and the vessel is completely filled. Since the solute is typically present in a relatively very small amount (i.e., x30 ≤ 10−3 ), the final volume of GEL and x1 max in it are determined by the binary VLE data for a partially filled vessel and the molar volume of GEL for a completely filled vessel. From VLE phase diagram for CO2 –acetone system (Chang et al., 1997), it is known that x1 * increases with an increase in P at a constant T and decreases with an increase in T at a constant P. Using such data the appropriate ranges of P and T are selected to be 60–70 bar and 291–303 K, respectively.

3.1.2.

Initial solute concentration

The loading of solid solute or ISC is also a determinant to the supersaturation that may be attained due to antisolvency. It can be changed in two ways, namely by changing (i) percent saturation of solid solute at a fixed T or (ii) the solid solubility in the solvent by changing T. In the present work, ISC has been selected for a given T based on 100% and 95% saturation of the solid solute in the initial volume of the pure solvent.

3.1.3.

Specific dissolution rate

The rate at which CO2 dissolves in the acetone-cholesterol solution during nucleation and growth, is also a significant parameter as it determines the rate at which supersaturation is attained. This parameter is termed as specific dissolution rate (SDR) which is defined as:

SDR =

Molar rate of dissolution of CO2 in the solvent Initial moles of solvent (31)

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Table 1 – Regressed coefficients of cubic polynomials for x3 * vs. x1 at different temperatures. From experimental data (Liu et al., 2002) Coefficients

T = 308 K

B0 B1 B2 B3

0.00635 −0.00714 −4.2099E − 4 6.9699E − 4

From calculated data (Eq. (32)) T = 318 K 0.01035 −0.01036 −0.00512 0.00447

In a stirred batch crystallizer, charged with the solution in which CO2 is added (typically through a dip tube), the initial value of SDR remains high, but falls progressively as x1 in GEL rises. The value of SDR tails off with time, t, to a low and reasonably constant value at high x1 (i.e., x1 ≥ 0.80) (Lin et al., 2003). For homogeneous nucleation, this range of x1 is of interest in the present work as such nucleation rate is negligible for x1 < 0.80. Accordingly, for all corresponding computations, time has been initialized (i.e., t = 0) at x1 = 0.80. In other words, the actual time for attaining a value of x1 > 0.80 is the sum of the initial time required for attaining x1 = 0.80 and the subsequent time required beyond x1 = 0.80. On the other hand, for heterogeneous nucleation, time has been initialized (i.e., t = 0) at x1 = 0.50 for the present computations, as nucleation occurs at a relatively lower x1 . For illustrating the effect of SDR on nucleation and growth rates, and in turn on the average particle size via homogeneous nucleation, three values namely, 0.095 min−1 (equivalent to the reported (Erriguible et al., 2015) addition rate of 3 g/min to 40 g of solvent), 0.632 min−1 (equivalent to the reported (Erriguible et al., 2015) addition rate of 20 g/min to 40 g of solvent), and 2 min−1 (based on our experimental work) are chosen for x1 in the range of 0.80 < x1 ≤ 0.95. However, for heterogeneous nucleation SDR value of 6 min−1 has been considered for x1 in the range of 0.50 < x1 ≤ 0.80. It is to be noted that SDR is the decisive factor that determines the instantaneous value of n1 in GEL required for calculating x3 by Eq. (2) and then instantaneous S.

3.2.

Estimation of equilibrium solid solubility

As mentioned earlier, the reported (Liu et al., 2002) experimental data of equilibrium solid solubility, x3 * , as a function of x1 at 308 K and 318 K have been fitted to a polynomial function of following form: x3∗ (T) = B0 + B1 x1 + B2 x12 + B3 x13

T = 291 K

T = 298 K

T = 303 K

0.00254 0.00347 0.00197 0.00137

0.00375 −0.00475 0.00157 −9.3703E − 4

0.00491 −0.00585 7.9763E − 4 −3.0492E − 4

Table 2 – Coefficients A and B in Eq. (33) for different x1 values. CO2 mole fraction (x1 ) 0.0 0.2 0.4 0.6 0.8 0.9

A

B

4.561 4.633 4.347 3.646 2.396 2.594

−2082.7 −2139.4 −2097.7 −1951.2 −1706.7 −2053.0

regression of the polynomial coefficients in Eq. (32). Table 1 also lists the values of the polynomial coefficients at these temperatures in the last three columns. The solubility of cholesterol in pure acetone, as presented in Fig. 3, illustrates the validity of Eq. (33) for the entire range of T from 291 to 318 K. The predicted data of x3 * vs. x1 at different values of T are shown in Fig. 4. It can be seen that x3 * decreases sharply at x1 > 0.80 due to the high antisolvency effect which leads to the desired high supersaturation.

3.3. Effects of process parameters on absolute solid solute supersaturation 3.3.1.

Effect of x1 max

For analyzing the effect of x1 max on Sab computations have been performed at appropriate operating conditions (P = 60 bar, T = 291 K, and V = 500 mL). The value of x1 max is calculated by equating the vessel volume, V, to the volume of GEL, Vt , i.e., when the vessel is completely filled and is given as x1max = 1 −

VL vm V vL

(34)

(32)

The regressed values of the coefficients in Eq. (32) at these two temperatures are reported in the first two columns of Table 1. For the purpose of extrapolation of x3 * to lower T i.e., 291, 298, and 303 K, the following linear relationship has been employed based on the van’t Hoff equation (for a constant x1 ) and is given as: log (x3∗ ) = A +

B T

(33)

The coefficients A and B in Eq. (33) have been obtained from x3 * calculated using Eq. (32) at different values of x1 and are listed in Table 2. Further, these value of A and B have been employed to estimate x3 * vs. x1 data at the required T = 291, 298, and 303 K for

Fig. 3 – The solubility of cholesterol in pure acetone with temperature.

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solute saturation) from 13.4 to 26.0 mg/mL. This leads to an increase in Sab from 1.17 to 1.22.

3.3.3.

Effect of pressure

As mentioned earlier, at any P and T, homogeneous nucleation occurs at high value of x1 , i.e., in the range of 0.80 < x1 ≤ 0.95, as the antisolvency effect is generally pronounced in this range. As can be seen from Table 3, for this range, the higher is the value of P, the higher is the value of x1 max and so the higher is the value of Sab . Accordingly a higher pressure is preferred to avail of a higher supersaturation.

3.3.4.

Fig. 4 – Variation of equilibrium solubility of cholesterol (x3 * ) in CO2 -expanded acetone solution with CO2 mole fraction (x1 ) at different temperatures.

Table 3 shows the results of detailed parametric study performed at different operating conditions. In case of variation of Sab with x1 max , it can be seen that when x1 max is 0.611 (i.e., x1 max < x1 * ), Sab is 1.17. If x1 max is increased from 0.611 to 0.911, Sab also increases from 1.17 to 136. Table 3 additionally shows that for x1 max = 0.97 (i.e., x1 max = x1 * ), Sab becomes very high >105 . This is due to the high antisolvency effect of CO2 (i.e., at such a high value of x1 , x3 * < 10−5 ). As x1 is the key parameter for the antisolvency effect, P and T are accordingly selected for attaining the desired high value of x1 , which in turn yield a high value of supersaturation (Sab > 5).

3.3.2.

Effect of ISC

The effect of ISC on Sab at T = 291 K for a fixed value of P is also given in Table 3. As the initial solid solute saturation is increased from 95% (ISC = 12.7 mg/mL) to 100% (ISC = 13.4 mg/mL), Sab marginally increases from 1.11 to 1.17. Further, due to the effect of T on the solid solubility, an increase in T from 291 to 303 K, results an increase in ISC (at 100% solid

Effect of temperature

Temperature affects Sab in two ways: i.e., due to changes in x1 and x3 * as both are sensitive to a variation in T. However, at a constant x1 , as can be seen from Table 3, with an increase in T from 291 K to 303 K, there is a decrease in Sab from 6.97 to 3.93. This is because an increase in T increases x3 * (as seen in Fig. 4) which in turn lowers Sab . The effect of T on the variation of Sab with x1 is depicted in Fig. 5. It can be seen that, at a constant T, Sab increases significantly at higher values of x1 , but at any x1 it decreases with increase in T. Hence, a lower value of T is preferred for availing of a higher supersaturation.

3.4. Effects of process parameters on average particle size for homogeneous nucleation The effects of various process parameters (as listed in Section 3.1) on the time-dependent variation of supersaturation, homogeneous nucleation (primary and secondary), growth, and hence on the predicted average particle size, Dp , are described in this section. The computational methodology has been used to predict the particle size (Dp ) which in turn, has exp been compared with that obtained experimentally (Dp ) for verifying its consistency.

3.4.1.

Comparison with experimental particle size

The experimental set-up and procedure used in the present work are similar to those reported earlier (Mukhopadhyay and Patel, 2009; Mukhopadhyay and Singh, 2004). A high pressure,

Table 3 – Effects of x1 max , ISC, P, and T on Sab . P (bar)

T (K)

x1 *

Effect of x1 max 60

291

0.97

250 125 83.1 64.4 22.3

0.611 0.822 0.884 0.911 0.970

13.4a

1.17 1.66 3.41 136 >105

Effect of ISC 60

291

0.97

250

0.611

303

240

0.611

13.4a 12.7b 26.0a

1.17 1.11 1.22

303

69.4 66.0 62.7

0.900 0.905 0.910

26.0a

7.63 11.7 34.2

291 298 303

64.1

0.900 0.900c 0.900c

13.4a

6.97 4.85 3.93

60 Effect of P 60 65 70 Effect of T 60

a b c

100% solid solute saturation. 95% solid solute saturation. Less than x1 max .

VL (mL)

x1 max

ISC (mg/mL)

Sab

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Table 4 – Empirical constant ASN for secondary nucleation rate used in the present work. SDR (min−1 )

5.473 × 109 * 2.532 × 1010 * 9.5 × 1010

0.095 0.632 2.0 ∗

Fig. 5 – Effect of temperature on the variation of absolute supersaturation (Sab ) as a function of CO2 mole fraction (x1 ) at P = 65 bar, and ISC = 13.4 mg/mL.

Corrected values for CO2 -acetone-cholesterol system at T = 303 K.

are mostly needle-shaped. For simplicity, in the present work, the particles have been assumed to be spherical in shape and exp the average equivalent diameter (Dp ) has been computed as follows. The length of each needle-shaped particle was measured using image analysis software ImageJ. The total volume of each particle was calculated in terms of its length, L, according to the scheme used by Erriguible et al. (2015) which is as follows: Vp = eiL =

well-stirred SS316 vessel with an internal volume of 500 mL (diameter = 4 cm; height = 67 cm) was used as the precipitator to perform the antisolvent crystallization experiments. The inlet of the vessel was connected to the CO2 cylinder from which CO2 was passed though the dip tube and then into the vessel, initially charged with a known volume of solution containing the solute to be precipitated. At the bottom of the vessel, a metal frit (pore size = 0.5 ␮m) was placed in order to retain the particles. During an experimental run about 5 mL of the gas expanded liquid (GEL) was sampled periodically and depressurized through water displacement method to obtain the amount of carbon dioxide in the sampled GEL. The data were also utilized to compute SDR (using Eq. (31)) taking into account the corresponding GEL volume and the initial solvent volume. This enables estimation of rate of change of CO2 mole fraction in GEL during crystallization. Finally, the precipitator was depressurized and particles formed were collected for optical microscopy analysis. The experiment on crystallization of cholesterol from acetone solution was performed at P = 65.9 bar, T = 302.3 K, ISC = 26.0 mg/mL, and SDR = 2 min−1 . The microscopy images of this crystallized cholesterol particles are shown in Fig. 6. It can be observed that the particles

Fig. 6 – Microscopy image of cholesterol particles crystallized from CO2 -acetone solution at P = 65.9 bar, T = 302.3 K, ISC = 26.0 mg/mL, and SDR = 2 min−1 .

ASN

L3 125

(35)

where e is the thickness, i is the width, L is the length of the particle; for needle shaped particles as shown in Fig. 6, e and exp i are expressed in terms of L as e = L/25 and i = L/5. Dp is the equivalent diameter of the spherical particle having Vp as its exp volume, from which it may be shown that: Dp = L/4. Based on the population of the particles given in Fig. 6, the number exp average particle size (using Eq. (30)) Dp has been calculated to be ∼30 ␮m. For comparison with this experimental data, the predicted average particle size, Dp , of cholesterol was estimated at the same process condition (P = 65 bar, T = 303 K, ISC = 26.0 mg/mL, and SDR = 2 min−1 ) and it is found to be ∼36 ␮m, for which ASN has been regressed and is found to be 9.5 × 101◦ . This is in the range of values of ASN reported in the literature (Erriguible et al., 2015) for a similar system. The corrected values of ASN for the present computations based on the reported values of SDR are given in Table 4.

3.4.2.

Evolution of primary and secondary nucleation

Fig. 7 shows the time-dependent evolution of primary nucleation rate and secondary nucleation rate, for P = 65 bar, T = 291 K,

Fig. 7 – Variation of primary and secondary nucleation rates with CO2 mole fraction at P = 65 bar, T = 291 K, ISC = 13.4 mg/mL, SDR = 0.632 min−1 .

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ISC = 13.4 mg/mL (corresponding to 100% saturation of cholesterol at T = 291 K), and SDR = 0.632 min−1 . As discussed in Section 3.1, the appropriate range of x1 for primary homogeneous nucleation to take place is x1 ≥ 0.80 which is also evident from Fig. 7 where it is observed that the primary nucleation rate begins to rise at x1 > 0.903 (at this x1 , S > 10). The onset of primary nucleation and secondary nucleation are decided by respective values of x1 . As can be seen from Fig. 7, beyond x1 = 0.903 the primary nucleation increases very sharply, attains a maximum value and then decreases very sharply. There are two opposing effects that determine such a behavior. The increase in dissolution of CO2 in GEL enhances the antisolvency effect (i.e., x3 * decreases) and hence the solute supersaturation, which in turn is responsible for the increase in nucleation rate. At the same time x3 decreases as the solute is continuously removed from GEL by nucleation and growth and thereby decreases supersaturation and nucleation. As a consequence of the interplay of these two opposing phenomena the nucleation rate passes through a maximum value with time. The present computations further indicate that around 90% of the solid solute depleted from GEL is due to its consumption by growth of nuclei. It is further observed that the growth dominates over nucleation as it persists even after nucleation becomes insignificant. This contributes to the relatively sharper declining behavior of nucleation rate with x1 . As is evident from Fig. 7, that the secondary nucleation rate becomes significant after primary nucleation diminishes. The primary nucleation takes place in the range of x1 between 0.9030 and 0.9045 whereas the secondary nucleation occurs between 0.9044 and 0.9050. According to Eq. (11), the rise in secondary nucleation is prompted by the increase in the solid suspension density resulting from primary nucleation. Conversely, the decrease in secondary nucleation is due to eventual decrease in supersaturation owing to progressive solute precipitation. As is evident from Fig. 7 the secondary nucleation dominates the overall homogeneous nucleation, since the order of magnitude for secondary nucleation is at least four times higher than that for primary nucleation.

3.4.3.

Effect of temperature on particle size

To analyze the effect of T on average particle size, simulations have been carried out at T = 291, 298, and 303 K at P = 65 bar and SDR = 0.632 min−1 . The amount of cholesterol dissolved in acetone was assumed to have ISC corresponding to 100% saturation of cholesterol (i.e., ISC = 13.4 mg/mL) at T = 291 K. The effects of T along with other process parameters on the average particle size are summarized in Table 5. It can be seen from Table 5 that when T is varied from 291 to 303 K the average particle size increases from 79.7 ␮m to 86.4 ␮m. The effect of T on the variation of the effective homogeneous nucleation rate with time is presented in Fig. 8. As x3 * increases with T at a constant x1 , which in turn lowers supersaturation, and hence nucleation rate. Therefore, at a lower T, due to higher nucleation rate, relatively more number of nuclei would form, resulting in smaller final size of the particles. It can also be seen that for a lower T, the onset of nucleation is earlier and less time is required for completion of nucleation and growth.

3.4.4.

Table 5 – Effects of process parameters on average particle size for homogeneous nucleation. P (bar)

T (K)

Effect of T 65

Effect of P 60 65 70 Effect of ISC 65

ISC (mg/mL) SDR (min−1 )

Dp (␮m)

291 298 303

13.4

0.632

79.7 85.9 86.4

303

26.0

0.632

54.4 54.5 54.6

303

26.0 24.7 23.4

0.632

54.5 56.6 58.8

26.0

0.095 0.632 2.0

88.5 54.5 35.7

Effect of SDR 303 65

For these three simulations the final average particle size are shown in Table 5. The average particle size increases marginally from 54.4 ␮m to 54.6 ␮m, with an increase in P from 60 to 70 bar. It may be noted that P does not affect x3 * , hence supersaturation and nucleation rate are expected to be invariant with P. This is also evident from Fig. 9. The marginal increase in particle size is due to the relatively minor effect of P on the relevant physical properties of the system. A similar trend on the variation of particle size with pressure was reported by Liu et al. (2002) in their experimental study on gas antisolvent crystallization of cholesterol particles performed at 45 bar and 60 bar. It was observed that the particle size remained almost invariant with P.

3.4.5.

Effect of ISC on particle size

The effect of ISC on the average particle size has been investigated for three different values of ISC corresponding to three different values of percent solid solute saturation, namely, 90%, 95%, and 100% at T = 303 K and is reported in Table 5. The other parameters were fixed as P = 65 bar, and SDR = 0.632 min−1 . The behavior of the nucleation rate with ISC is illustrated in Fig. 10.

Effect of pressure on particle size

The effect of P on the final average particle size has been analyzed at P = 60, 65, and 70 bar (i.e., subcritical conditions of CO2 ), with T = 303 K, ISC = 26.0 mg/mL, SDR = 0.632 min−1 .

Fig. 8 – Effect of temperature on variation of homogeneous nucleation rate with time (initialized at x1 = 0.8) at P = 65 bar, ISC = 13.4 mg/mL, and SDR = 0.632 min−1 .

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Fig. 9 – Effect of the pressure on variation of homogeneous nucleation rate with time (initialized at x1 = 0.8) at T = 303 K, ISC = 26.0 mg/mL, and SDR = 0.632 min−1 . As can be seen from Fig. 10, the maximum nucleation rate for ISC = 26.0 mg/mL is ∼1.4 × 1011 (#/m3 s) whereas in the case of ISC = 23.4 mg/mL, it is 1.2 × 1011 (#/m3 s). Therefore, a higher ISC leads to a higher nucleation rate and in turn, a larger number of nuclei. These subsequently grow to a smaller average particle size in spite of a higher initial amount of solid solute being present in the solution. It can be seen from Table 5 that with a decrease in ISC (from 26.0 to 23.4 mg/mL), the particle size increases from 54.5 ␮m to 58.8 ␮m. Muhrer et al. (2002) in their theoretical study and Fusaro et al. (2004) in their experimental work on GAS crystallization, observed a similar trend in the variation in particle size with ISC.

3.4.6.

Effect of SDR on particle size

The effect of SDR on the average particle size has been studied for three values of SDR namely, 0.095, 0.632, and 2 min−1 at P = 65 bar, T = 303 K, and ISC = 26.0 mg/mL (corresponding to 100% saturation of cholesterol at T = 303 K). Fig. 11 shows the evolution of homogeneous nucleation rates for three different

Fig. 10 – Effect of the initial solid solute saturation (ISC) on variation of homogeneous nucleation rate with time (initialized at x1 = 0.8) at P = 65 bar, T = 303 K and SDR = 0.632 min −1 .

Fig. 11 – Effect of the specific dissolution rate (SDR) on the homogeneous nucleation rate as a function of time (initialized at x1 = 0.8) at P = 65 bar, T = 303 K, and ISC = 26.0 mg/mL. values of SDR. It is observed that the higher is SDR, the earlier is the onset as well as the attainment of maximum nucleation rate. This is attributed to an earlier attainment of the higher level of supersaturation at a higher SDR. In addition, a higher nucleation rate in general, yields more number of smaller sized particles. As evident from Table 5, the final average particle size corresponding to a higher SDR of 2 min−1 is smaller compared to that for a lower SDR of 0.095 min−1 , i.e., 35.7 ␮m and 88.5 ␮m, respectively. In other words, the average particle size decreases with increase in SDR. This also conforms to the observation on the effect of SDR on Dp reported in the literature (Bakhbakhi et al., 2006, 2005; Dodds et al., 2007; Erriguible et al., 2015; Fusaro et al., 2004; Muhrer et al., 2002). Fig. 12 shows the predicted particle size distribution at three different values of SDR namely, 0.095, 0.632, and 2 min−1 including the experimentally obtained particle size distribution at SDR of 2 min−1 . The particle size follows a skewed distribution due to a larger number of smaller sized particles formed by secondary nucleation at the later stages of the nucleation process (as shown in Fig. 7). In general, a higher SDR

Fig. 12 – Comparison of effects of specific dissolution rate (SDR) on predicted and experimental particle size distributions at P = 65 bar, T = 303 K, and ISC = 26.0 mg/mL.

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Table 6 – Effect of contact angle on characteristic spherical cap dimensions, width (h), base radius (rb) , and volume (Vcap ) at P = 65 bar, T = 303 K, ISC = 26.0 mg/mL, and SDR = 6 min−1 . h (␮m)

 ◦

30 40◦ 50◦

3.86 12.8 28.2

Vcap (␮m3 )

rb (␮m)

1.29 × 10 2.62 × 104 1.73 × 105 3

14.4 35.3 60.4

Fig. 13 – Effect of contact angle on variation of supersaturation with CO2 mole fraction and onset of heterogeneous nucleation at P = 65 bar, T = 303 K, ISC = 26.0 mg/mL, and SDR = 6 min−1 . results in a narrower particle size distribution. It can also be seen that higher number of smaller sized particles are formed during antisolvent crystallization experiments.

3.5. Effect of contact angle on particle size for heterogeneous nucleation In the case of heterogeneous nucleation, a change in the nature of the substrate entails a change in three phase contact angle. Three different values of the solid-substrate contact angles, namely,  = 30◦ , 40◦ , and 50◦ have been investigated to analyze their effects on the average particle size at P = 65 bar, T = 303 K, ISC = 26.0 mg/mL, and SDR = 6.0 min−1 .

13.5 36.8 69.2

Fig. 13 shows the variation of S with x1 after the onset of nucleation for different contact angles. The onset of nucleation occurs at relatively higher values of x1 (shown as the starting point of S vs. x1 profile) as the contact angle increases. This indicates that the threshold value of x1 (when a sufficient number of nuclei are formed), occurs only at x1 ∼0.5 for the case of  = 30◦ ; whereas the corresponding values of x1 are ∼0.6746 and ∼0.7711 for  = 40◦ and 50◦ , respectively. This is consistent with the fact that the solid-substrate affinity decreases with an increase in  and thus a higher supersaturation is needed for nucleation, which can be obtained only at increased values of x1 . The effect of contact angle on the variation of heterogeneous nucleation rate with CO2 mole fraction is depicted in Fig. 14. It can be observed that decrease in the affinity of the substrate toward nuclei, i.e., increase in,  significantly decreases nucleation rate despite of a higher S (Fig. 13). For  = 30◦ the maximum nucleation rate is of order of ∼1015 (#/m3 s) whereas in the case of  = 50◦ it reduces to ∼1013 (#/m3 s). This implies that it is easier for cholesterol particle to nucleate on the surface of the substrate with a lower contact angle. A comparison of Fig. 7 and 14 shows that heterogeneous nucleation occurs at a lower x1 compared to homogeneous nucleation in agreement with the reported observation (Kumar et al., 2013). The average particle size has also been computed considering heterogeneous nucleation employing the algorithm described in Fig. 3. To reiterate in the case of heterogeneous nucleation the final average equivalent diameter of the capshaped particle is obtained from the diameter of the spherical particle having the same volume as of the cap-shaped particle. Table 6 reports the representative computed dimensions of the cap-shaped particle along with the final average particle size for different contact angles. With an increase in  from 30◦ to 50◦ , the average diameter of equivalent spherical particles also increases from 13.5 ␮m to 69.2 ␮m.

4.

Fig. 14 – Effect of contact angle on the variation of heterogeneous nucleation rate with CO2 mole fraction at P = 65 bar, T = 303 K, ISC = 26.0 mg/mL, and SDR = 6 min−1 .

Dp (␮m)

Conclusions

The present work illustrates a strategy for screening the process conditions for producing cholesterol micro-particles of a specific size range from acetone solution using subcritical CO2 -assisted antisolvent crystallization process. Formation of ultra-fine solid particles is enabled by high and rapid attainment of supersaturation. Overall, such a process is dictated by thermodynamic considerations as well as nucleation and growth kinetics. The former enables identification of appropriate process conditions such as P, T, and ISC. These in turn determine the maximum attainable mole fraction of dissolved CO2 and the resultant solid solute supersaturation, S. Thermodynamic analysis indicates that lower T, higher P and ISC favor higher absolute solid solute supersaturation. Kinetic analysis of both homogeneous and heterogeneous nucleation, and growth of particles has been carried out to

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elucidate the effects of various operating parameters on the final particle size which, in addition to P, T, and ISC, include specific dissolution rate of CO2 (SDR) in GEL, and three-phase contact angle, . For this purpose, a numerical strategy has been developed and verified by comparing the predicted and experimental particle sizes. For a specific set of conditions P = 65.9 bar, T = 302.3 K, and SDR = 2.0 min−1 the experimental particle size is determined to be ∼30 ␮m, while the predicted size is ∼36 ␮m. Simulations for homogeneous nucleation show that the particle size increases with temperature, while pressure exerts a minor effect. Also the particle size decreases as the initial solute loading in solvent is enhanced. With an increase in the antisolvent dissolution rate particle size also decreases; additionally, a narrower particle size distribution is obtained. Simulation results further suggest that while the heterogeneous nucleation sets in at a lower CO2 mole fraction (x1 ∼0.58 for  = 30◦ ) and hence at lower solute supersaturation, compared to the homogeneous nucleation which requires a higher degree of supersaturation, its onset occurring at x1 > 0.80. This is attributed to the comparatively lower Gibbs free energy barrier for heterogeneous nucleation arising out of specific solute affinity for a foreign substrate. The heterogeneous nucleation effects are elucidated by varying . A lower contact angle, which signifies higher solute nucleisubstrate affinity, is shown to result in finer particles. Our work substantiates the general observation that higher supersaturation results in smaller particles due to higher nucleation rates and consequent lesser growth of larger number of individual particles. However, both homogeneous and heterogeneous nucleation rates depend on supersaturation as well as other physical properties. For the case of homogeneous nucleation, nucleation rate increases with increase in supersaturation. For the case of heterogeneous nucleation, our study shows that higher supersaturation does not necessarily entail high nucleation rate, as it is also dependent on the contact angle, which represents the nature of external substrate. These computational results derived for both homogeneous and heterogeneous nucleation phenomena are validated by trends observed in experimental studies reported by other authors. It is therefore concluded that the present work illustrates a generalized computational methodology for selecting the operating conditions as well as for identifying the type of external substrate, as characterized by three-phase contact angle, that are conducive to engineering of ultra-fine particles of a desired size.

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