Article pubs.acs.org/IECR
Role of In-Situ Generated CO2 Bubbles in Heterogeneous Nucleation of Solid Solutes in the Precipitation by Pressure Reduction of Gas-Expanded Liquid (PPRGEL) Process Mriganka Mondal, Sandip Roy,* and Mamata Mukhopadhyay Department of Chemical Engineering, IIT Bombay, Mumbai 400076, India ABSTRACT: Precipitation by pressure reduction of a gas-expanded liquid (PPRGEL) entails dissolution of subcritical CO2 in the solution of an organic solvent and a solid solute resulting in a gas-expanded liquid (GEL). Depressurization of a GEL leads to evolution of CO2 bubbles along with a large decrease in temperature facilitating solid precipitation. An illustrative system comprising CO2 (1), acetone (2), and cholesterol (3) is considered. It is proposed that CO2 bubbles generated from the depressurized GEL provide a substrate for heterogeneous nucleation of cholesterol. The size of nucleated CO2 bubbles and their rate of generation with time are predicted. The effective size of the CO2 bubbles ranges over 200−500 μm and their generation rate is ∼106−107 #/s. Comparing these with the corresponding solute nuclei size and nucleation rates, and considering the GEL−solid interfacial energetics, we conclude that the CO2 bubbles provide the required surface area for solid nucleation.
1. INTRODUCTION Production of micro/nanoparticles of uniform size is a necessity for a large section of the food and pharmaceutical industry. In recent years processes using either subcritical or supercritical fluids, especially CO2 have been accorded preference in the production of such particles over other conventional processes like crystallization, spray drying, and liquid antisolvent precipitation. These novel processes use CO2 as a solvent, cosolvent, or antisolvent and are based on the principles of depressurization crystallization, antisolvent crystallization, and thermally induced crystallization. Examples include rapid expansion of supercritical solutions (RESS), particles from gas saturated solutions (PGSS), gas anti-solvent (GAS), supercritical anti-solvent (SAS), solution enhanced dispersion by supercritical fluids (SEDS), and aerosol solvent extraction system (ASES).1 These processes differ in their modes of attainment of very high and rapid supersaturation of the solid solute, which is required for formation micro/nanoparticles.2 Precipitation by pressure reduction of a gas-expanded liquid (PPRGEL) is a more recently developed process for production of ultrafine particles.3 In this process subcritical CO2 is first dissolved in an organic solvent with solid solute under pressure to form a CO2-gas-expanded liquid (GEL) solution. Subsequently, the GEL is depressurized rapidly, which leads to large systemic temperature drop. Consequently, high, rapid, and uniform supersaturation of the solute is attained, resulting in its nucleation, which is followed by crystalline growth. In our preceding work on PPRGEL,4 involving a system composed of CO2 (1), acetone (2), and cholesterol (3), a composite model was proposed to predict the solute particle size and its distribution. The model was developed on the basis of the assumption that a sufficiently large number of CO2 bubbles are present at all times during the depressurization process, and they are large © 2017 American Chemical Society
enough to provide planar surface for solid nucleation. The present work aims to validate these assumptions through an analysis of CO2 bubble dynamics, which involves prediction of timedependent variations in their supersaturation, nucleation rates, growth and eventual collapse. Nucleation processes, in general, may be either homogeneous or heterogeneous. It has been shown that nucleation of gas bubbles in aqueous solutions requires supersaturation of the order of 100 or more.5−7 According to Blatteau et al.8 also, homogeneous bubble nucleation in pure water requires very high supersaturation (∼1400), and bubbles are likely to form at lower supersaturation by means of pre-existing gas microbubbles.9,10 Other authors11,12 have also considered bubble nucleation from pre-existing gas cavities; i.e., when a bubble is detached, a fraction of it remains at the cavity, which promotes the formation of the next bubble. The existence of such gas filled cavities or microbubbles (∼1 μm), however, is unlikely if the system pressure is significantly above atmospheric.8 Because the initial pressure in PPRGEL process is 4.0−7.0 MPa, it is reasonable to assume pre-existing microbubbles are not responsible for generation of CO2 bubbles. Thus, heterogeneous nucleation of CO2 bubbles may be the only possibility at lower values of supersaturation (i.e., S1 < 10). Lubetkin and Blackwell13 have experimentally determined the number of CO2 bubbles generated per unit time for CO2−water aqueous system in a polished stainless steel vessel. Reporting an estimated supersaturation of ∼8 for the system under study, they have proposed that under such low supersaturation Received: Revised: Accepted: Published: 9331
March 16, 2017 July 26, 2017 July 26, 2017 July 26, 2017 DOI: 10.1021/acs.iecr.7b01105 Ind. Eng. Chem. Res. 2017, 56, 9331−9340
Article
Industrial & Engineering Chemistry Research
induce a drop in temperature of the GEL solution, leading to lowering of solubility of the solid solute, which in turn nucleates on the existent CO2 bubbles. To the best of our knowledge a mathematical model for such simultaneous and related gas and solid nucleation has not been reported in the existing literature. The objective of this work is to verify this mechanism (presented in Figure 1) by calculating both the size of CO2 bubbles and their rate of generation as a function of time. A critical parameter that determines the nucleation rate of CO2 bubbles is the CO2 bubble−GEL contact angle. However, because during the depressurization process the system temperature varies continuously, this contact angle is also expected to vary. The variation of the contact angle is regressed by equating the fundamental nucleation rate expression to the experimentally deduced rate of CO2 depletion from the GEL. The range of contact angle so obtained is compared with values reported in the literature. Lastly, the rate of generation of CO2 bubble is compared to that necessary for solute nucleation.
nucleation of CO2 bubbles is essentially heterogeneous in nature, and that this occurs on the vessel surface. Nucleation of gas bubbles on solid surfaces also depends on substrate characteristics. For example, formation of CO2 bubbles at low supersaturation (S1 < 10) is not feasible on very smooth surfaces, such as glass.14 However, for steel surfaces, which are typically less smooth, gas nucleation may occur on microcavities. In their work, Lubetkin and Blackwell13 have studied gas bubble nucleation on conical cavities (diameter ∼ 1.5 μm) present on the vessel surface. Blander’s15 work has also highlighted that bubble nucleation generally takes place at irregularities and cavities present on vessel surface. Wilt6 has shown that the production of gas bubbles is possible only on conical cavities on surfaces at a low range of supersaturation ( r. Finally, eq 5 (i.e., with Fnet = 0) is used to estimate the value of rd. Numerically, this is achieved by using a criterion Fnet/(Fs + Fd) < ε. The value of ε is set at 0.1 for the ease of computation; i.e., on the average, the total force acting to detach the bubble from the surface is greater than 90% of the adhesion force. Subsequently, the corresponding time of detachment of CO2 bubbles termed as τd can also be simultaneously calculated by eqs 5 and 6. Once the first set of calculations is completed by starting from time t = 0, it yields the bubble size for the first set, which leaves the vessel surface at a time τd(1).. Following this, a new set of bubbles nucleate on the same sites as the previous ones and the entire set of calculations (as in Figure 2) are repeated to obtain the detachment time τd(2) for this second set and so on. This calculation procedure is repeated until the end of depressurization process (t = τ). 2.3. Calculation of Number of CO2 Bubbles (Nn) Generated and Contact Angle (ω). The CO2 nucleation rate may be calculated with the classical heterogeneous rate equation (eq 13) using relevant physical properties:6
where N is the number density of nucleation site per unit volume (molecules/m3), m is the mass of CO2 molecule (kg), k is the Boltzmann constant (J/K), and B* denotes chemical and mechanical equilibrium states of the bubble. For chemical equilibrium, B* = 1 and for mechanical equilibrium it is given by eq 14:6 B* ≃ 1 −
N
f (ω ,ψ ) =
⎪
⎪
⎪
1 − sin(ω − ψ ) 2
(15)
f1 (ω ,ψ ) = 2 − 2 sin(ω − ψ ) + cos(ω) cos3(ω − ψ )/sin(ψ ) 4 (16)
Here ω is the contact angle between the CO2 bubble and vessel surface at the point of nucleation (Figure 3) and ψ is the conical cavity half-angle (Figure 1). Equation 13 yields the nucleation rate per unit surface area; this may be converted to per unit volume by multiplying it with the surface to volume ratio (A/Vt) for the vessel (A is the surface area of the vessel and Vt is the total volume of the solution at any instant of time). The ratio A/Vt is held constant during all above computations. The reason for this assumption is that as gas bubbles are formed rapidly during depressurization of the GEL,
⎧ ⎫ ⎧ 2γ B* ⎫1/2 16πγ33f1 (ω ,ψ ) ⎪ ⎪ 3 ⎬ exp⎪ ⎨− ⎬ f (ω ,ψ )⎨ 2⎪ ⎩ πmf1 (ω ,ψ ) ⎭ ⎩ 3kT ((S1 − 1)P) ⎭ ⎪
(14)
where PL and PG are pressure in the liquid and gas pressure inside the critical bubble.8 In the present work, chemical equilibrium between the gas phase (CO2) and liquid phase (GEL) is assumed. The parameters f(ω,ψ) and f1(ω,ψ) are defined by eqs 15 and 16:6
Jhetero = 2/3
P ⎞ 1⎛ 2 ⎜1 − L ⎟ ≃ 3⎝ PG ⎠ 3
(13) 9334
DOI: 10.1021/acs.iecr.7b01105 Ind. Eng. Chem. Res. 2017, 56, 9331−9340
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Industrial & Engineering Chemistry Research
and temperature are chosen to be 5.5 MPa (i.e., Pi) and 303 K (i.e., Ti) respectively, while the total depressurization time is fixed at 60 s. The experimental values of pressure and temperature were simultaneously measured and recorded by using data logger during depressurization.23 Then with the entropy balance method mentioned in earlier work,4 both P vs t (Figure 4a) and T vs t (Figure 4b) data were predicted and compared with the experimental data shown in Figure 4. The average absolute relative deviations (%AARD) resulting from such comparisons for pressure and temperature were 8.8 and 0.1, respectively (for the case of Pi = 5.5 MPa and Ti = 303 K). In addition, the values of these two process parameters are also predicted by the methodology developed in our earlier work,4 based on the system entropy balance considerations. On the basis of the temporal values of P and T, the variation of x1 is computed. Concomitantly, the equilibrium CO2 mole fraction x1* at the same times is calculated with Aspen-ONE V-7 employing the Peng−Robinson EOS. The calculated values of x1* at several temperatures have also been verified with the experimental data reported in the literature.24 These values are displayed in Figure 5a. It may be seen that even though the pressure falls rapidly during the initial stage (Figure 4a), the onset of decrease of temperature (Figure 4b) “lags” behind. This is because it requires time for CO2 to attain supersaturation and the bubbles to commence nucleating, which is responsible for the GEL temperature drop. Using the instantaneous values of x1 and x1*, we compute the variation of S1 (Figure 5b), which goes through a maximum. This may be explained as follows. As shown in Figure 5a, during the earlier phase of depressurization, the rate of decrease in equilibrium solubility of CO2 (x1*) is faster compared to the rate of decrease in its actual mole fraction (x1), which causes S1 to increase in time. However, toward the later part of the depressurization process the ratio of x1 and x1* decreases as the two converge at the final equilibrium condition characterized by a system pressure of 0.1 MPa (i.e., approximately atmospheric pressure). This implies that S1 must eventually fall to a value of unity. 3.2. Variation of CO2 Bubble Size (rd) and Detachment Time (τd) with Depressurization Time. To calculate CO2 bubble detachment size (rd) and detachment time (τd) by solving eqs 5 and 6, the required physical properties, namely, (a) the interfacial tension between the GEL and CO2 (γ3) and (b) the diffusion coefficient (D12) have been calculated by eqs 27 and 28, respectively, of the previous work.4 However, the density of the GEL and gaseous CO2 have been calculated by using the ZL−Pr correlation developed by Dalvi and Mukhopadhyay25 and virial EOS.21 Table 1 shows the variation of these physical properties with time for the entire depressurization process. However, to solve eqs 5 and 6 simultaneously, one needs to make appropriate an assumption of the possible dimension of cavities on the vessel surface on which, as discussed earlier, CO2 bubbles are likely to nucleate. The field emission gun-scanning electron microscopic (FEG-SEM) images of the representative steel (SS316) vessel surface used in the present study are shown in Figure 6. The images appear to suggest existence of cavities in the form of microchannels of an average width of 5 μm. In their analysis of surface characteristics of stainless steel and brass surfaces Qi26 has reported engineered conical cavities of ∼2.7 μm diameter. On the basis of these observations, in the present work the width of the microchannel has been assumed to be equivalent to conical cavity with a mouth diameter of
Figure 3. CO2−GEL contact angle and interfacial energy.
the latter is likely to be kept vigorously agitated and expanded as in a foaming situation. Under such a condition the vessel surface wetted by the GEL may remain close to the original extent, i.e., prior to depressurization. To calculate the contact angle, ω, one needs to know the interfacial tensions between the bubble and solid substrate, γ1, and between the GEL and solid substrate, γ2. These, however, are difficult to estimate or measure. Hence, a direct application of eq 13 to derive the bubble nucleation rate is not feasible. An alternative approach is therefore employed. The experimental P, T data obtained during the depressurization phase of the PPRGEL process is used to estimate the change in number of moles of CO2 with time.4 Accordingly, the number of bubbles that leave the vessel surface (and eventually coalesce with the GEL−vapor interface to exit the system) after a given detachment time τd is computed by the following equation: Jhetero =
Δn1
( Mρ )
VtτdVp
1
(17)
where Δn1 is the change in number of moles of CO2 after τd (computed on the basis of the change in x1 over time from P and T data), Vp is the volume of a single bubble (m3) at the point of detachment, ρ1 is density (kg/m3) of CO2 at P and T, and M is the molecular weight of CO2 (kg/mol). Equation 17 may be reorganized as Nm = Jhetero Vt =
Δn1
( Mρ )
τdVp
1
(18)
where Nm (#/s) is the number of bubble generated per unit time The nucleation rate of CO2 bubbles (Jhetero) calculated from eq 17 at a given point of time is equated to that with eq 13, so as to regress the contact angle ω over the same period during which it is assumed to remain constant. These calculations are repeated so as to obtain the progressive change in the value of ω over the entire depressurization time. The continuous drop in GEL temperature during depressurization alters all three interfacial energies shown in Figure 3 as a result of which the contact angle ω also undergoes a concomitant change.
3. RESULTS AND DISCUSSION In this section, results of the computation of variation of CO2 supersaturation (S1), bubble detachment size (rd), number of bubbles generated (Nm), and the contact angle (ω) with time are presented sequentially. These results are utilized to verify the proposed mechanism that CO2 bubbles provide the requisite surface area for heterogeneous nucleation of the solid solute. 3.1. Variation of CO2 Supersaturation (S1) with Time. For the purpose of illustration of the above-mentioned computational results, the predepressurization GEL pressure 9335
DOI: 10.1021/acs.iecr.7b01105 Ind. Eng. Chem. Res. 2017, 56, 9331−9340
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Figure 4. Variation of experimental and predicted (a) pressure (P) and (b) temperature (T) vs depressurization time (t)4 for Pi = 5.5 MPa and τ = 60 s.
Figure 5. Variation of (a) CO2 mole fraction (x1), equilibrium CO2 mole fraction (x1*), and (b) CO2 supersaturation with time (t).4
Table 1. Physical Properties of the System for Pi = 5.5 MPa and Ti = 303 K t (s) 0 4 5 10 15 20 30 40 60
γ3 (N/m) 0.001102 0.001352 0.001953 0.005671 0.010360 0.018662 0.031026 0.038581 0.050363
D12 (m2/s) 5.59 5.45 5.28 3.00 1.90 1.10 5.83 4.17 2.60
× × × × × × × × ×
vm (m3/mol)
−09
5.42 5.41 5.43 5.50 5.41 5.36 5.31 5.35 5.87
10 10−09 10−09 10−09 10−09 10−09 10−10 10−10 10−10
5 μm (i.e., R = 2.5 μm). As the cavity size is relatively small, numerical application of eq 4 shows that CO2 bubbles at the vessel bottom horizontal surface cannot detach as the surface anchoring force dominates over the other force elements. However, computations using eq 5 show that bubbles can detach from vertical surface of the vessel. As mentioned earlier, once a set of bubbles nucleate on the vessel surface, they grow until they attain a critical radius (rd) for detachment at a corresponding time designated as “detachment time” (τd). Once they detach, a new set of bubbles are nucleated at the same surface cavities and the process repeats. However, the detachment bubble radius and time for each set are not the same as the physical conditions
× × × × × × × × ×
−05
10 10−05 10−05 10−05 10−05 10−05 10−05 10−05 10−05
ρGEL (kg/m3)
ρ1 (kg/m3)
854.824 856.340 857.448 872.402 903.848 942.579 987.356 1002.560 972.555
130.792 125.806 124.823 55.7984 29.7843 16.374 12.130 6.354 3.714
(P, T, x1, etc.) of the system changes continuously in time. The absolute time is the cumulative sum of progressive detachment times for consecutive bubble sets. Diagrammatically, this phenomenon may be represented by a series of bars, each one extending over the associated detachment time. However, the number of such bubble sets forming over the total depressurization time is very large; the actual detachment times are of the order of 0.021−7.75 s, whereas the total depressurization time is ∼60 s. Thus, the detachment radii and time for the consecutive bubble sets are displayed as a continuous function of (absolute) time in Figure 7a,b, respectively. It can be observed that rd increases with time throughout the process. As the depressurization proceeds, the 9336
DOI: 10.1021/acs.iecr.7b01105 Ind. Eng. Chem. Res. 2017, 56, 9331−9340
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Industrial & Engineering Chemistry Research
Figure 6. FEG-SEM images of a stainless steel (SS-316) surface of (a) sample 1 and (b) sample 2.
Figure 7. Variation of (a) CO2 bubble size (rd, m) and (b) time of detachment (τd, s) with depressurization time (t).
process (t > 40 s), τd again increases with time. As S1 decreases after 40 s (Figure 5b), the value of the parameter β decreases concurrently. Thus, as evident from application of eq 6, a larger time is required to for the bubble to attain a detachment size. This results in an increase in time of detachment, τd. 3.3. Variation of Rate of Number of CO2 Bubbles (Nm) with Time (t). The number of bubbles generated per interval of time (Nm) is computed by eq 17 and are shown in Figure 8.
large and rapid temperature drop (Figure 4b) and the attendant reduction in CO2 mole fraction leads to a continuous increase in the GEL−CO2 interfacial energy (γ3).4 It follows that the interfacial force component in eq 6 increases with time, thus requiring progressively a larger detachment bubble size for buoyancy and pressure force together to overcome the former. Moreover, as described by Keshock and Siegel (1962), the drag force is negligible in realistic applications (Fd),20 and also the order of the order of magnitude of inertial force (Fi) is ∼10−10 N compared to other forces having an order of magnitude of 10−8−10−7 N during the time of depressurization. Thus, the effect of these two forces have been neglected for the present system to calculate the predetachment size of CO2 bubbles. As evident from Figure 7b, the bubble detachment time goes through some form of minimum, which may be explained as follows. During the initial stages of depressurization, S1 (Figure 5a) is low leading to low values of β (calculated by combining eq 7 and 12), which results in slower growth rates of CO2 bubbles. As a consequence, the detachment times are relatively higher during the initial phase of the process. Because φ ∝ S1 (=C1/C1*) and φ ∝ β (from eq 7 and 12), it follows that when the supersaturation of CO2 increases (as in Figure 5b), the dimensionless growth parameter, β, also increases, leading to higher growth rates. Thus, smaller time is required for the bubble to detach from the vessel surface, leading to a decrease in τd. Toward the latter part of depressurization
Figure 8. Variation of number of bubbles (#/s) with depressurization time (t) for Pi = 5.5 MPa. 9337
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values of ω in Figure 9 corresponds to a similar range of contact angle data reported (77−90°) for gas bubbles nucleating on the solid substrate from an aqueous solution.5,10,26 To ascertain the validity of our model, calculations of ω have also been carried out for the case of an initial GEL pressure of 6.2 MPa. As is evident, the variation of ω occurs over a range similar to that for the previous case (Figure 9). The value of f1 (eq 16) normally varies between 0 and 1 but it can also be negative16 if [ω − ψ] > 90°. Under such a condition the CO2 gas bubble surface becomes concave toward the GEL, restricting the growth of bubbles10 and so eq 13 no longer remains meaningful. Hence, the limiting condition for eq 13 to be valid is [ω − ψ] < 90°. As shown in Figure 9, the contact angle for CO2 nucleation; i.e., ω does not exceed 95.0°. Because the vessel surface conical cavity half-angle has been taken to be ∼5°, it follows that, at all times during depressurization, the condition [ω − ψ] < 90° remains valid. 3.5. Comparison of Nucleation Rates of CO2 Bubbles and of Cholesterol Particles. Figure 10 shows the predicted
As explained in the previous section, bubbles are formed in distinct sets. However, the time spent by a bubble on the vessel surface prior to its detachment, which is equivalent to the time for its generation, is much lower compared to the depressurization time. Thus, as an approximation, the number of bubbles formed over a given time interval is shown against the median time over the interval. Because the number of bubble sets over the entire depressurization time is large (∼2600), the variable Nm is displayed as a continuous function of (absolute) time. It may be observed that during the initial phase of depressurization time (∼5 s) there is no generation of CO2 bubbles; this is due to the insufficient level of CO2 supersaturation prevailing at these times. This is also evident from the negligible amount of temperature drop that occurs during first 5 s in Figure 4b. Although S1 rises as the depressurization continues (Figure 5b), the bubble generation rate (Nm) decreases. This occurs as there is an increase in the GEL−CO2 interfacial energy (γ3) with time and hence in the free energy barrier to nucleation (i.e., the exponential term in eq 13). 3.4. Variation of ω vs t. As explained earlier, the contact angle ω between the CO2 bubble and the vessel surface (Figure 3), embodied in eq 13, is difficult to estimate as it depends on the interfacial tension between the vessel surface and CO2 (γ1), and between the vessel surface and GEL (γ2), the values of which are generally not easily available. As a result, it is not feasible to compute the nucleation rate by eq 13. Several authors27,11 have assumed a constant contact angle of ∼90° in the foaming process (for particle stabilized aqueous foams) to calculate the heterogeneous nucleation rate of gas bubbles. For the PPRGEL process, however, ω is expected to vary considerably over time due to large change in γ3 (∼0.01−0.053 N/m) brought about by large temperature drop during depressurization. The contact angle, ω, at any time has been calculated through trial and error by combining eqs 13 and 17. As shown in Figure 9, for a pre-depressurization GEL
Figure 10. Variation of solid (cholesterol) nucleation rate with time in the GEL solution for Pi = 5.5 MPa, Ti = 303 K, and τ = 60 s.4
variation of the nucleation rate of cholesterol particles for the present system for the case of an initial GEL pressure of 5.5 MPa. This is sourced from our earlier modeling work,4 which was premised on the assumption that CO2 bubbles provide a planar heterogeneous surface for the nucleation of cholesterol particles. As had been shown,4 the majority of solid precipitation takes place toward the later part of the depressurization process (t > 40 s). Drawing on the data presented in Figure 7, it may be seen that the detachment size of the bubbles at ∼50 s is ≈225 μm. The critical radius of cholesterol nuclei is ∼0.002 μm.4 The effect of surface curvature of a substrate on solid nucleation rates becomes important if the substrate diameter is
