Influence of Agitation and Fluid Shear on Primary Nucleation in Solution

Article pubs.acs.org/crystal

Influence of Agitation and Fluid Shear on Primary Nucleation in Solution Jin Liu† and Åke C. Rasmuson*,†,‡ †

Department of Chemical Engineering and Technology, KTH Royal Institute of Technology, Stockholm, Sweden Department of Chemical and Environmental Science, Solid State Pharmaceutical Cluster, Materials and Surface Science Institute, University of Limerick, Limerick, Ireland

‡

ABSTRACT: The influence of mechanical energy on primary nucleation of butyl paraben has been investigated through 1320 cooling crystallization experiments. The induction time has been measured at different supersaturations, temperatures, and levels of mechanical energy input, in two different flow systems. There is an overall tendency in the experiments that primary nucleation is promoted by increased input of mechanical energy. In small vials agitated by magnetic stir bars, the induction time was found to decrease with increasing agitation power input raised to 0.2 in the low agitation region. However, further increase in agitation leads to an increase again in the induction time. In a concentric cylinder apparatus of Taylor−Couette flow type, the induction time is inversely related to the shear rate. By fitting the parameters of the classical nucleation theory to experimental data, it is shown that the results can be explained as an influence on the pre-exponential factor. The treatment behind the pre-exponential factor is extended to account for the contribution of forced convection in a solution exposed to agitation and fluid shear. However, the analysis cannot verify that increased rate of mass transfer can explain the results. Alternative mechanisms are discussed based on a comprehensive review of the relevant literature. Shear-induced molecular alignment and in particular agitation-enhanced cluster aggregation are mechanisms that appear to deserve further attention.

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INTRODUCTION Crystallization processes are of significant importance to our society in industrial production of metallurgic and polymeric materials, and of inorganic and organic compounds; in the formation of shells and bone structures in nature; and in diseases such as the appearance of kidney stones and precipitates of amyloid proteins. In the industrial production of pharmaceutical compounds, crystallization is repeatedly used to separate and purify intermediates and is of major importance for the purification of the final product. Primary nucleation denotes the formation of a new particle in the solution, having a size sufficient for it to be thermodynamically stable at the prevailing conditions. Crystal nucleation has a governing influence on product properties and robustness of the process but is the mechanism of crystallization that is the least understood. This leads to significant problems in the design, operation, and control of industrial processes. Nucleation behavior is known to be unreliable and case sensitive. Because of this, industrial processes are developed by trial and error, and they often lack sufficient robustness. Sometimes lack of reproducibility requires rework or even disposal of the batch. The slow development of the fundamental understanding of primary nucleation is due to the nanoscale size range, the very strong nonlinear dependence on the supersaturation, and the significant stochastic component. The earliest evidence on the effect of mechanical stimulus on primary nucleation can be traced to the beginning of the 20th century. Young1−3 and Berkeley4 carried out experiments where an anvil placed in a supersaturated solution was stroke by a hammer. Their results showed that the critical supercooling © XXXX American Chemical Society

was reduced by the mechanical energy and an empirical correlation was proposed (Young3). The main focus of the work was on the essence of metastable limit, and no further comments were given on the mechanisms involved in the influence of the mechanical energy. In 1960s, Mullin and Raven5,6 and Nyvlt et al.7 more systematically studied the influence of mechanical agitation on nucleation. Mullin and Raven performed metastable zone width experiments with NH3H2PO4, MgSO4, and NaNO3 in aqueous solution5,6 and found that increasing agitation initially reduces the supercooling required for nucleation, but further increase in agitation actually leads to an increase again. They explained these results by a combined effect of diffusion to and attrition of clusters/small crystals. Growth of the cluster benefits from agitation-enhanced diffusion, while attrition of the clusters hampered nucleation. Nyvlt et al.7 found the maximum supersaturation of NaNO3 decreased monotonically with increasing stirring rate. Their interpretation is based on a detailed analysis of the influence of fluid shear on mass transfer and a model combining stirring enhanced cluster formation with probability of cluster disappearance. Lev and Joseph8 developed a model over clusters in centrifugal fields of free eddies generated in turbulent flows. By the centrifugal force, the clusters will move to the periphery of the eddy, where they concentrate and aggregate to form critical nuclei. A correlation between nucleation rate and stirrer speed was established, which can Received: May 17, 2013 Revised: August 6, 2013

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dx.doi.org/10.1021/cg4007636 | Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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Figure 1. A schematic representation of the different mechanisms.

pseudoplastic, also called shear-thinning, fluids the viscosity decreases with increasing shear rate. This behavior is found in certain complex solutions, such as blood, paint, polymer solutions, and molten polymers. The molecular basis is described as breaking up of aggregates and alignment of molecules with the streamlines. By molecular simulation (most commonly nonequilibrium molecular dynamics), the behavior of molecules under shear has been explored. It is reported19−24 that molecules tend to align with the flow direction as the shear rate increases. For atoms and small molecules, very high shear rates (>1010 s−1) are required to cause a prominent alignment. Gray22 estimated the magnitude of thermally induced shear fluctuations (1011 s−1) from the statistics of the fluid velocity field and explained that the reason for the high shear rates was to surpass the random shear fluctuations. Even though it is perceived as being too simplistic to describe real nucleation processes, simulations by the Ising model of nucleation under shear25,26 have provided additional perspectives suggesting the interplay of three different mechanisms: shear enhanced cluster growth, coalescence, and breakup.

explain that the nucleation rate rapidly increases with increasing agitation and then levels out. During the following decades, additional studies showed similar effects in cooling crystallization9−11 and precipitation,12−14 but not much progress was made in the actual understanding. Some authors9,15−18 argue that the influence of agitation on induction time should be explained by secondary nucleation. In order for the nucleation to become visible to the naked eye, there has to be a certain concentration of crystals. Only the first nucleus is actually due to primary nucleation, and all nuclei following the first would strictly speaking be due to secondary nucleation. A higher agitation rate would induce a faster secondary nucleation after the first crystal is born and thus lead to a faster detectability of the nucleation event. However, often the induction time is much longer than the time required to go from first observation to total cloudiness, which suggests that this is not likely to be a dominating explanation. For the understanding of rheological behavior of nonNewtonian fluids, e.g., in lubrication, it is important to understand how the molecules react to fluid shear. For B



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In Figure 1, we provide a schematic representation of the more interesting mechanisms suggested. Unlike the slow pace of development on the nucleation of small molecules, significant progress has been made with respect to shear-induced nucleation of polymers. In 1965, Pennings and Kiel27 found ribbon-like crystals in stirred polyethylene solutions using microscopy, later known as Shish-kebab structures. These structures can be found in elongation and shear flow.28−30 Keller et al.31 observed that when the strain rate reached a critical value, an abrupt change occurred in the birefringence, which indicated a fully extended chain conformation. Lagasse and Maxwell32 showed that shear flow decreased the induction time of nucleation in molten polyethylene, a result that was later supported by further experiments using a rotational rheometer,33−35 and with fiber stretching experiments.36,37 These results were explained by orientation of the polymer chains under fluid shear, which decreases the system entropy and makes nucleation more favorable. On the basis of this mechanism, Acierno et al.35 proposed a microrheological crystallization model for polymers. The model correlated induction time to the shear rate in a steady-state shear flow, which agreed very well with experimental data of polypropylene. In the area of crystallization of low to medium molecular weight organic molecules from solution, very little systematic work has been done on the influence of hydrodynamics and fluid shear on primary nucleation. In the present study, our ambition is to provide experimental data that clearly reveal whether there is a substantial influence or not in such systems, to reveal the character of that influence and to review some of the theories that can provide explanations for this behavior. Two experimental systems have been used: (i) a 20 mL vial with a magnetic stir bar operated in a multivial system by which a large number of experiments could be conducted, and (ii) a rotating concentric tube system, a system sometimes called a Taylor−Couette flow system, which allowed generating a more uniform shear stress in the solution. In each set up, the induction time for nucleation of butyl paraben dissolved in ethanol was investigated at different levels of supersaturation and agitation/shear.

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Figure 2. Multivial experiment setup.

Figure 3. Flow diagram of induction time experiment (30 parallel vials). The vials filled with solution were initially put in a 20 °C water bath for 2 h to ensure complete dissolution. Then the entire rack holding the vials was moved to a 12 or 11 °C water bath to create two different levels of supersaturation. The temperature of the water bath was controlled by a Julabo FP50 refrigerated circulators. The time for cooling the solution in the vials is within 1 min, and the temperature stability is at least ±0.1 °C. A Sony HDR-XR200 high-resolution digital camcorder was used to record the nucleation event in each vial. Immediately after the nucleation, the solution in the whole vial turned turbid within a few seconds. Therefore, the identification of the moment of nucleation could be done with sufficient precision from the video recordings, and the uncertainty is much less than the variation in induction time measurements. After the set time for recording vial nucleation, the vial rack was returned to the 20 °C water bath, and the series of steps shown in Figure 3 were repeated up to 10 times with different rotation rates. For each rotation rate, 90 induction time measurements were collected (i.e., 3 times 30 parallel vials), except for in a few cases where up to 180 (6 times 30 parallel vials) values were obtained. The rate of agitation was changed in a random manner from experiment (30 parallel vials) to experiment. Seven levels of rotation rate were compared: 100, 150, 200, 300, 500, 800 r/min, and additional experiments were performed with intermittent agitation. When each vial was filled, the solution was filtered through a 0.2 μm syringe filter. Each vial was filled with the same amount of solution. For redissolution, the vials were placed into the higher-temperature water bath for the same time period (2 h) to avoid variations in the history of solution.41,46,47 The experiments were conducted at low supersaturation to make sure the induction time is long enough that the time spent on cooling the vials to the target temperature can be neglected. So the induction time can be considered as the time elapsed from the moment the vials were put into the low temperature water bath until the nuclei appeared.

EXPERIMENTAL SECTION

Nucleation experiments recording the induction time for primary nucleation have been performed in two different apparatus: a rotating concentric cylinder device and magnetically stirred 30 mL vials. All work has been done on butyl paraben in ethanol. Material. Butyl paraben (CAS No: 94-26-8) of >99.0% purity was purchased from Aldrich and was used without further purification. Ethanol of 99.7% purity was purchased from Solveco chemicals. Solubility data of butyl paraben in ethanol has been reported previously.38 Magnetically Agitated Vials. In small scale, nucleation experiments often show a significant variation, which is assumed to reflect the stochastic nature of nucleation itself.39−45 In order to capture this variation and to obtain statistically valid data, a multivial system was designed and operated, Figure 2. In each experiment, 30 vials are operated in parallel held in a specially designed multivial rack. Each vial (D = 25 mm, H = 60 mm) was filled with 20 mL solution and furnished with a magnetic stir bar (L = 20 mm, D = 6 mm, with a ring in the middle to reduce friction with the vial bottom). The vials are placed on a submersible multipole magnetic driver unit produced by 2mag AG. The solution was prepared with concentration of 1.76 g of butyl paraben/g of ethanol, corresponding to a saturation temperature of 15 °C. Figure 3 shows the flow diagram of induction time experiment. C



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In some introductory experiments, the influence of the supersaturation on the induction time is established, as well as the influence of the length of the stir bar and the solution volume. In the latter experiments, two levels of rotation rate: 100 and 200 r/min were used for three groups of vials: long stir bar (20 mm) with full volume of solution (20 mL), long stir bar (20 mm) with half volume of solution (10 mL), and short stir bar (10 mm) with full volume of solution (20 mL). Taylor−Couette Flow System. Although the multivial experiments can be performed in high numbers, the flow conditions are complex. The hydrodynamic conditions and shear rates are very nonuniform, and the shearing conditions between the stir bar and the bottom of the vial are difficult to characterize. In order to create a simpler hydrodynamic situation, an apparatus with two concentric cylinders were designed. As shown in Figure 4, the shell (D = 50 mm,

variation were undertaken similar to those of the vial experiments described above. Even though the rotating cylinder is hollow, the total mass of this device is of course much larger than that of the individual vials. Figure 5 shows curves over the temperature change in going from

Figure 5. Cooling profile of solution in Taylor−Couette system. the undersaturated to the supersaturated state. The time to reach 95% of the required temperature change is less than 6 min, and in 9 min the target temperature is reached within 0.2 °C.

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RESULTS AND DISCUSSION Magnetically Agitated Vial Experiments. The influence of the supersaturation on the induction time for nucleation is presented in Figure 6, evaluated in accordance with the classical

Figure 4. Taylor−Couette flow system. H = 150 mm) is made by glass. The rotating cylinder inside (D = 40 mm, H = 100 mm) is made by stainless steel. The cylinder is hollow to reduce its heat capacity and hence the time for adjustment to the nucleation temperature. The wall thickness is 2 mm, and the cylinder is closed to retain no liquid. The gap between the inner rotating and the outer stationary cylinders is designed to be 5 mm. In early preliminary work, it was found that a bearing inside the supersaturated liquid could induce nucleation when the axis rotates. Hence, in the final design the axis is only supported in the upper end as shown in Figure 4. Because of this, even though the axis is of 6 mm stainless steel, the cylinder is “swaying” somewhat in particular at higher rotation rates. Accordingly, 5 mm should be considered as the average gap. The inner cylinder is rotated by an agitation driver unit, providing a rotation rate from 100 to 400 r/min in this study. The vessel is completely filled with solution (approximately 150 mL) and is then connected to the driver, both of which were held on an iron stand. The unit is then submersed into a water bath for temperature control. The same steps as shown in Figure 3 were followed for determination of the induction time. The solution was prepared with concentration of 1.76 g of butyl paraben/g of ethanol, corresponding to a saturation temperature of 15 °C. After proper dissolution, the unit is moved into another water bath and cooled down to 9, 10, 11, or 12 °C, corresponding to the supersaturation ratios (C/C*) of 1.16, 1.13, 1.10, and 1.08, respectively. Four levels of rotation rate were investigated at each supersaturation: 100, 200, 300, and 400 r/min. For each condition, five repetition experiments were performed. Precautions to ensure reproducibility within the stochastic

Figure 6. Linear regression of ln t versus T−3(ln S)−2 in butyl parabenethanol solution.

nucleation theory, where primary nucleation of a spherical nucleus is described by48 ⎛ 16πσ 3v 2 ⎞ J = A n exp⎜ − 3 3 m 2 ⎟ ⎝ 3k T (ln S) ⎠

(1)

The exponential term represents a free energy barrier for formation of the nucleus, while the pre-exponential factor An can be considered as a kinetic parameter. The induction time is assumed to be inversely proportional to the nucleation rate, by which eq 1 can be rewritten as B ln t ind = −ln A + 3 T (ln S)2 (2) where A is proportional to the pre-exponential factor An. In Figure 6, the induction time distribution of 30 experiments at each driving force is represented by the median value. The D



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Since the number of vials that did not nucleate varies from experiment to experiment, median values of the induction time distributions are used in the evaluation, Figure 8. As the

surface free energy can be obtained from the slope of the line and becomes: σ = 1.16 mJ/m2, which is very close to the value of σ = 1.13 mJ/m2 obtained in previous work.49 The results of the introductory experiments on volume and stir bar length are shown in Table 1. The induction time is Table 1. Median Induction Time of Experiments with Long Stir Bar (20 mm) in Full Volume of Solution (20 mL), Long Stir Bar (20 mm) in Half Volume of Solution (10 mL), and Short Stir Bar (10 mm) in Full Volume of Solution (20 mL) at Rotation Rate of 100 and 200 r/mina rotation rate (r/min)

length of the stir bar (mm)

volume of the solution (mL)

induction time (min)

200

20 20 10 20 20 10

10 20 20 10 20 20

67.3 40.8 45.9 131.5 101.9 92.8

100

a

Figure 8. Influence of rotation rate on the median induction time (Ts = 15 °C, ΔT = 3 °C).

Ts = 15°C, ΔT = 3 °C.

given as the median of 30 parallel experiments for each condition. Obviously, at both rotation rates the induction time becomes shorter when the volume increases, while the stir bar length does not show a clear influence. The influence of the volume reflects that the larger the volume the greater is the likelihood for nucleation somewhere in the vial volume, and this effect is obviously strong enough to counteract the reduced specific power input of the agitation (i.e., P/V). In the following experiments presented below, the stir bar length and the vial liquid volume were kept constant as given in the main description of the experiments. The cumulative induction time distributions in the vial experiments are shown in Figure 7, covering six different levels of rotation rate. Obviously, and as expected,39,44 fairly broad distributions are obtained, significantly tailoring toward longer induction times. Furthermore, less than 80% of the vials nucleated within the time period of each recording (200 min). Not even after 24 h can all vials be expected to have nucleated. Detailed analysis shows that individual vials behave essentially as randomly as the whole set of experiments. This agrees with our findings in previous work.

rotation rate increases, the induction time at first decreases clearly, then reaches a minimum after which follows an increase into a plateau. The shape of this curve resembles the curve described by Mullin and Raven.5,6 The agitation plate used cannot reduce the rate below 100 rpm. In order to extend the work into the low agitation region, intermittent agitation also was used in a second set of experiments at a higher supersaturation. In Table 1 the induction time median values of 90 parallel experiments at each level of stirring intensity (8/52 denotes 8 s of stirring followed by 52 s of nonstirring, and then repeating the cycle again throughout the experiment) are shown. Obviously the induction time increases as the intensity of agitation is reduced. Often different modes of input energy into a solution such as impact,3 ultrasound,50 and agitation14 are characterized by the corresponding power input. For the intermittent experiments, we have calculated a corrected stirring power to compare with the continuous agitation experiments as P = (tr /t tot)NpρN3DS5

(3)

Figure 7. Influence of agitation rate on the cumulative induction time distributions. Saturation temperature Ts = 15 °C, supercooling ΔT = 3 °C. The volume of the solution in each vial is 20 mL. E



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where tr and ttot are the rotational time and total time in one period; ρ is density of solution; N is rotation rate; DS is the impeller diameter, which equal the length of the stir bar in this case. The stirring power number Np can be estimated using Nagata’s correlation for an unbaffled vessel:51 ⎛ 103 + 1.2Re 0.66 ⎞C ⎛ h ⎞(0.35 + b / DT) A t ⎟ ⎜ ⎟ Np = + B⎜⎜ 3 0.66 ⎟ Ret ⎝ 10 + 3.2Ret ⎠ ⎝ DT ⎠ (sin θ)1.2

(4)

where Ret = NDS ρ/μ is the Reynolds number; h is the height of liquid level; DT is the diameter of the vial; b is the width of the impeller blade, which equals the diameter of the stir bar; θ is the angle between agitator blade surface and horizontal surface, taken as 90 deg for the stir bar in this case; A, B, and C are parameters that are related to the system geometry. By using the above method, the power input for each case was calculated and is given in Table 2 together with the agitation Reynolds number. The induction time results are plotted against input power shown in Figure 9. 2

Figure 10. Correlation of ln P versus ln t. Diamond: continuous stirring. Blue: N ≤ 200 r/min; red: N > 200 r/min). Triangle: intermittent stirring. Ts = 15 °C, ΔT = 4 °C.

The diagram also shows with perfect clarity that as the power input is further increased the behavior changes. It is worth noting that the transition from the first region of decreasing induction time with increasing agitation to that of increasing occurs at approximately the same agitation rate (200 rpm continuous) for the two different levels of supercooling. Taylor−Couette Flow Experiments. The Couette flow in the concentric gap between the rotating inner and stationary outer cylinders becomes unstable as the rotation speed of the inner cylinder increases. This results in so-called Taylor− Couette flow, i.e., the appearance of pairs of counter rotating vortices − Taylor vortices, that are superimposed on the laminar Couette flow.52 The critical value of the Reynolds number (Re = NπDidρ/μ, Di is the diameter of the inner cylinder, d is the gap between the cylinders) to form Taylor vortices is about 110.53 In the present work, the Taylor− Couette flow region is reached already at the lowest rotation rate (Re = 483). However, the influence of the Taylor vortices in the range of the experiments is quite limited. The highest radial and axial velocities are less than 15% of the tangential velocity,54 and the shear stress is mainly determined by the Couette flow.55 Accordingly, for the purpose of estimation of the shear rate, the velocity profile in the gap is approximated by the solution to the Navier−Stokes equation for incompressible tangential Newtonian flow:

Table 2. Induction Time and Power Input at Different Levels of Stirring Intensitya rotation rate (r/min) 100

200

400 800 a

interval

induction time (min)

continuous 18/12 8/52 8/112 8/232 continuous 18/12 8/52 continuous continuous

65. 9 80.1 96 117.5 148.5 48.4 50.8 70.2 73.6 70.8

Re

Np

308

1.79

616

1.56

1232 2464

1.36 1.17

P (*10−5 watt) 1.79 1.07 0.24 0.12 0.06 12.39 7.43 1.65 86.4 594.6

Ts = 15 °C, ΔT = 4 °C.

uθ = Ar + B /r , ur = 0, uz = 0

(6)

where uθ, ur, and uz are the azimuthal, radial, and axial components of velocity. A and B depend on the radius ratio, η = ri/ro of the inner cylinder radius ri and the outer cylinder radius ro, and the rotational speed of the inner cylinder, Ωi, as Figure 9. Median induction time of different levels of power input. Diamond: continuous stirring. Triangle: intermittent stirring (data from Table 1). Ts = 15 °C, ΔT = 4 °C.

A = −Ωi

η2 1 − η2 ,

B = Ωi

ri 2 1 − η2

(7)

The shear rate in the gap can be obtained from the derivative of eq 6, and using eq 7 In Figure 10, all results are presented in a log−log diagram. Obviously, in the low agitation region there is a fairly clear power law relation between the induction time and the average agitation power input: t ind ∝

1 P 0.20

γ (r ) =

ri 2 ⎞ du θ −Ω i ⎛ 2 B ⎜ ⎟ =A− 2 = η + dr r 1 − η2 ⎝ r2 ⎠

(8)

In the present work, η = 0.8 and ri = 0.002m, and the calculated average shear rate as well as the highest (r = ri) and lowest (r = ro) values are shown in Table 3. The relative difference is within 14%, which means that the shear rate in the Taylor−Couette

(5) F



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Table 3. Shear Rate in Taylor−Couette Flow System rotation rate (r/ min)

average shear rate (s−1)

highest shear rate (s−1)

lowest shear rate (s−1)

100 200 300 400

41.9 83.8 125 167

47.7 95.4 143 190

37.2 74.5 111 148

flow is reasonably uniform. Therefore, we can relate the induction time directly to the average shear rate as shown in Figure 11. Each point in the figure is an average of five repetition experiments, and the bars indicate the 95% confidence intervals. Figure 13. Determination of parameters A and B (eq 2) at different rotation rates.

given in Table 4. Notably, the variation in the slope parameter B is quite small, while the parameter A systematically increases Table 4. Nucleation Parameters for Different Rotation Rates rotation rate (r/min)

A (*10−3 s−1)

B

100 200 300 400

0.75 1.931 3.441 4.534

216280 248294 224959 244207

Figure 11. Induction time at different flow shear rates in the Taylor− Couette system.

Figure 12 presents how the induction time depends on the shear rate in the Taylor−Couette system for different

Figure 14. Influence of the rotation rate on the pre-exponential factor A.

with increasing rotation rate. In Figure 14, A is plotted versus rotation rate, showing a very good linear relationship. Hence, a reasonable interpretation is that increasing the rotation rate leads to an increase in the pre-exponential factor, while as would be expected there is no clear influence on the thermodynamic barrier for nucleation. In a deeper analysis of the basis for eq 7, the nucleation rate is described as56

Figure 12. Induction time versus shear rate at different levels of supersaturation in the Taylor−Couette system.

nucleation driving forces: RT ln S. Each point in Figure 12 represents the average of five repetition experiments. As can be seen, a similar trend was found for all levels of supersaturation, and the induction time is shorter at a higher supersaturation as expected.

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J = f *ZC*

DISCUSSION AND ANALYSIS In Figure 13, the nucleation data of the Taylor−Couette flow system is plotted in accordance with eq 2, for each rotation rate. The lines are drawn using least-squares fits, and the corresponding values of A and B for each rotation rate are

(9)

where C* is the equilibrium concentration of critical nuclei, Z is the so-called Zeldovich factor, which compensates for deviation from the equilibrium state, and f * is the attachment frequency factor describing the rate by which monomers attach to the nucleus. f * can be expressed as the product of a sticking G



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rate of mass transfer as was suggested, e.g., by Nyvlt et al.7 However, we do recognize that when agitation is provided by a magnetic stir bar rotating at the bottom of the vial, the highest shear rates will occur between the stir bar and the bottom. If the tip speed of the stir bar is divided by δ0, the shear rate becomes much higher than given by the equation above. Of course, the maximum shear rate will arise when the rotating stir bar touches the glass surface and the distance approaches zero. However, there is no corresponding behavior in the Taylor− Couette flow, and hence it cannot explain in general our results on the influence of agitation and shear on the nucleation. It should be recognized that there is a significant difference between nucleation and standard mass transfer processes. Both of the above correlations are based on an averaged effect on mass transfer, while for nucleation it would be sufficient that a nucleus is formed somewhere in the system in order for the entire system to nucleate like an “avalanche” (Mersmann61). On the microscale, correlations such as eq 11 are based on Kolmogoroff’s theory of isotropic homogeneous turbulence,62 in which the statistical properties of turbulence are assumed to be spatially homogeneous and isotropic. However, the variations of velocity and pressure may have a considerable influence on the possibility of formation of a nucleus somewhere in the solution. These variations increase as the Reynold’s number increases.63 Therefore, mechanisms that enhance the mass transfer can still contribute to an explanation for the influence of fluid shear on primary nucleation. Mullin and Raven5,6 proposed that the behavior shown in Figure 8 could be explained as increased mass transfer gradually being counteracted by increased particle attrition. However, the basis for the latter is that attrition of sub-micrometer particles follows the same mechanisms as large crystals. Large crystal attrition at the hydrodynamic conditions of the present work would primarily be due to particles colliding with the revolving agitator blades or vessel walls, and would depend on the inertia of the particles decreasing to the third power with particle size. A particle of the size of a critical nucleus [0.01−0.001] μm will follow the streamlines very closely and is very unlikely to collide with the agitator blades. Effects of shear-induced molecular alignment are very difficult to quantify without more elaborated simulations, even though we can assume that the significance would be more important the more anisotropic the molecule shape and would increase with increasing shear rate. In polymer nucleation, the stretching and ordering of molecules were observed at shear rates of 0.5−10 s−1.30,64 However, for simple diatomic molecules, very high shear rates (more than 1010 s−1)22 are required for molecular ordering. In our experiments, the average shear rate is about 2−26 s−1 (eq 13) in the vials and 42−168 s−1 in the cylinder experiments. The butyl paraben molecule is obviously quite anisotropic and big compared to the diatomic molecules in the dynamic simulations but fairly short compared to polymer molecules. The actual existence of small molecular clusters has been established more convincingly in recent work.65,66 In supersaturated solutions of calcium carbonate, Gebauer et al.65 found a few fairly distinct cluster sizes, which was interpreted as suggesting that nucleation takes place by cluster aggregation. Jawor-Baczynska et al.66 showed in experiments on glycine that the mechanical action of a tumbling stir bar significantly shortened the time to formation of observable crystals. Since this mechanical action also appeared to lead to coalescence of 250 nm nanodroplets into larger nanodroplets, the latter were

coefficient accounting for that only a fraction of molecules colliding with the nucleus surface leads to a successful attachment, the diffusion flux, j, of solute molecules to the nucleus surface and the nucleus surface area a. In homogeneous nucleation, for a spherical nucleus of radius r* = (3υmn*/4π)1/3, j* is obtained as j* = DC/r* by solving the unsteady state pure diffusion problem, and the resulting formula for f* can be written as57 f * = κDC 48π 2υmn*

(10)

where κ is the sticking coefficient, vm is the mole volume, n* is the number of molecules contained in a critical nucleus, C is the solute concentration in the bulk solvent, and D is the diffusion coefficient. In the case of agitation, the mass transfer rate is faster than in a quiescent liquid. For mass transfer from the liquid to fine solid particles in an agitated suspension, the Sherwood number is often calculated as58,59 Shp =

k SLd p D

Sc1/3 = 2 + 0.52Reo 0.52 ̀

(11)

where kSL is the solid particle-liquid mass transfer coefficient, dp is particle diameter, and Reε = (εdp4/ν3)1/3 is the Reynold’s number based on Kolmogoroff’s theory of isotropic turbulence. Sc = μ/ρD is the Schmidt number. The value 2 on the righthand side accounts for the molecular diffusion contribution in the case of a spherical particle. As the particle size decreases, the Reynold’s number decreases, and the second term on the righthand side of eq 11 becomes negligible for particles smaller than a few micrometers. Accordingly, within the theory behind eq 11, we cannot explain the influence of rotation on nucleation. The laminar shear rate in turbulent flow enhances the mass transfer within a fluid layer. Although the agitation in the vials is far from being strictly turbulent, this model has also been examined. During shearing, the fluid layer becomes thinner, which corresponds to a convective transport in the direction of reducing layer thickness. This convective velocity depends on the shear rate60 uc =

δ0γ 2t (1 + γ 2t 2)3/2

(12)

The laminar shear rate γ in turbulent flow can be estimated as γ = 0.5

ε v

(13)

where ε is the energy dissipation rate per unit mass, ν is the kinematic viscosity, and the initial thickness δ0 can be taken as half of the Kolmogoroff microscale: ⎛ v 3 ⎞1/4 δ0 = 0.5⎜ ⎟ ⎝ε⎠

(14)

Thus, the transport from the bulk to the cluster surface should be replaced by

j = ucC +

DC r*

(15)

However, it appears as if this convective contribution is negligible at the level of energy dissipation rates of the experiments of this work and even for values more close to industrial conditions (∼1 W/kg). Hence we are not able to verify that the influence on nucleation is related to an increased H



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kSL l m N Np n* P R Ret Reε

assumed to be the prime source for crystal nucleation. Both these studies thus lend support to the relevance of the “agitation enhanced cluster aggregation” theory.

solid particle-liquid mass transfer coefficient length mass rotation rate power number molecular number contained in a critical nucleus stirring power gas constant Renault number based on terminal velocity Renault number based on Kolmogoroff’s theory of isotropic turbulence r radius S supersaturation ratio, C/C* Sc Schmidt number Shp Sherwood number based on particle diameter T temperature Ts saturated temperature ΔT supercooling t time uc convective velocity uθ Azimuthal velocity ur radial velocity uz axial velocity vm molecular volume Z Zeldovich factor γ shear rate ε power input (or energy dissipation) per unit mass η radius ratio of inner cylinder and outer cylinder θ angle between agitator blade surface and horizontal surface κ sticking coefficient μ dynamic viscosity ν Kkinematic viscosity ρ density σ interfacial energy Ω rotation speed

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CONCLUSION The results in this paper clearly reveal that there is a substantial influence of fluid shear on primary nucleation of butyl paraben. The results from the multivial system show that the induction time is first reduced as the rotation rate increases and then increases after a minimum value. In an extended study in the low agitation rate region, an exponential relationship between power input and induction time is obtained. In the Taylor− Couette flow system, the induction time is inversely related to the flow shear rate. The results clearly reveal that the influence of mechanical energy is not limited to agitation by a magnetic stir bar lying at the bottom bumping up and down and rubbing the bottom surface. There is a clear influence of pure fluid shear. By fitting the parameters of the classical nucleation theory to the Taylor−Couette flow experimental data, it is shown that the results can be explained as an influence on the pre-exponential factor in the nucleation rate equation. Expansion of the pre-exponential factor to account for convective mass transfer does not provide a reasonable explanation to the experimental results. Shear-induced molecular alignment and in particular agitation-enhanced cluster aggregation are mechanisms that appear to deserve further attention.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: +46-8-7908227. Fax: +46-8-105228. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J.L. gratefully acknowledges the CSC scholarship from the Chinese Government and a scholarship from the Industrial Association of Crystallization Research and Development, and Å.C.R. acknowledges the support of the Science Foundation Ireland (10/IN.1/B3038).

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A a An B b C C* C0 D d dp f* G H h J j k

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NOTATION constant area exponential coefficient in nucleation rate equation constant width of the impeller blade (i) concentration; (ii) constant saturated concentration concentration of nucleation sites in a system at equilibrium (i) molar diffusion coefficient; (ii) diameter distance of the gap between inner cylinder and outer wall particle diameter molecular attachment frequency Gibbs free energy height height of liquid level steady-state nucleation rate diffusion flux Boltzmann constant I



Article

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Influence of Agitation and Fluid Shear on Primary Nucleation in Solution

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