Emulsion-Assisted Nanoparticle Precipitation: Time Scale Analysis

ARTICLE pubs.acs.org/IECR

Emulsion-Assisted Nanoparticle Precipitation: Time Scale Analysis and Dynamic Simulation Michael Fricke† and Kai Sundmacher*,†,‡ † ‡

Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany Process Systems Engineering, Otto-von-Guericke University Magdeburg, Universit€atsplatz 2, 39106 Magdeburg, Germany ABSTRACT: This contribution provides a model-based approach of a nonspecific emulsion-assisted precipitation process in order to analyze the influence of process parameters on the particle size and size distribution of primary nanoparticles. The here presented process route comprises mass transfer of reactant molecules from the continuous phase of the emulsion into the dispersed droplets in a stirred tank reactor. In a time scale analysis, the penetration of the reactant through the surfactant-laden interface can be identified as the controlling process mechanism. With regard to the rate-controlling process step, a mathematical model based on the population balance approach is derived. A high resolution discretization scheme extended by a flux limiter function is applied to solve the population balance equations. The dynamics of the particle formation process and the final particle size distribution are investigated in numerical simulations by variations of the droplet size, the initial concentration level of the dissolved reactant in the droplet, and the feeding rate into the reactor. It has been figured out that an increase in the feeding rate and a decrease in the emulsion droplet diameter results in a reduction of the mean particle size. In contrast, low feeding rates and large droplets are favorable to obtaining homogeneous reaction conditions in the stirred tank reactor.

1. INTRODUCTION Several sophisticated technical applications require particulate precursors on the nanometer scale. The special optical, electrical, magnetic, and catalytic properties of inorganic nanomaterials depend primarily on the size and shape of these crystals.1 Therefore, it is of major importance to identify a suitable particle synthesis route on an industrial scale, which enables the adjustment of a desired particle size with a narrow size distribution. An established bottom-up approach in wet chemistry is the classical bulk precipitation in stirred tank reactors.2 Due to the ionic character of the two dissolved reactants A and B, a very fast and irreversible chemical reaction of the type A+B f C f P V

ð1Þ

competes with mixing of the reactants on the molecular level. If the chemical reaction is much faster than micromixing, concentration gradients in the solution appear, which lead to local inhomogeneities of the supersaturation level.3 Since the subsequent elementary subprocesses of precipitation, namely, nucleation and growth, are directly influenced by supersaturation inhomogeneities, broad particle size distributions are obtained. Furthermore, secondary processes such as Ostwald ripening and aggregation in highly concentrated reactors may occur4 and shift the mean particle size to undesired bigger particles. A promising technique to overcome the drawbacks of the bulk phase system is the precipitation in the dispersed phase of an emulsion. Hence, emulsion droplets act as microreaction containers that comprise a confined number of molecules which can form particles. Furthermore, the use of surfactants prevents particle aggregation by means of steric and/or electrostatic barriers. Two major routes realizing precipitation reactions in emulsions are discussed in the literature, which are the one-emulsion technique5,6 and the two-emulsion technique.7,8 A schematic r 2011 American Chemical Society

Figure 1. Illustration of emulsion-based precipitation, (a) one-emulsion technique and (b) two-emulsion technique.

illustration of the mechanisms of both emulsion-assisted techniques is depicted in Figure 1. The one-emulsion technique is based on the mass transfer of reactant A from the continuous phase across the surfactant-laden interface into the droplets. The second reactant B is dissolved in the droplet, where the precipitation reaction takes place. In the case of water-soluble substances, the reactants A and B are dissolved in droplets of two separate water-in-oil (w/o) emulsions. These two emulsions are subsequently mixed together by means of agitation. The interdroplet exchange and the particle synthesis occur due to coalescence events of droplets of dissimilar composition. This synthesis route is called the two-emulsion technique. The two-emulsion technique has been tested successfully for the synthesis of a multitude of different nanomaterials in thermodynamically stable microemulsion droplets on the laboratory Special Issue: Nigam Issue Received: March 28, 2011 Accepted: June 10, 2011 Revised: June 10, 2011 Published: June 10, 2011 1579

dx.doi.org/10.1021/ie2006164 | Ind. Eng. Chem. Res. 2012, 51, 1579–1591

Industrial & Engineering Chemistry Research scale.7,9,10 Also, kinetically stabilized emulsions with much higher dispersed phase fractions compared to microemulsions were recently used for the synthesis of inorganic nanoparticles.6 Sager et al.8 were the first to prepare colloidal ceramic precursors in kinetically stabilized emulsion droplets. FeOOH particles having a particle diameter of 2030 nm were prepared using the twoemulsion technique. Shi and Verweij11 synthesized ZrO2 and Fe2O3 particles with an average diameter of 4 nm also using the two-emulsion technique. SiO2 and TiO2 particles with a mean size ranging from 30 to 230 nm were obtained by alkoxide hydrolysis using the one-emulsion technique.6 Lu and Saha12 also applied the one-emulsion technique for the precipitation of SrBi2Ta2O9 nanoparticles with an average size of 60 nm. It is believed that the size of the synthesized nanoparticles depends basically on the droplet diameter and on the number of molecules, which are applied for the particle formation in the droplets.6 Well-defined aqueous reactant solutions can easily be prepared prior to the emulsification process. In the case of kinetically stabilized emulsion systems, the size of the droplets can be adjusted by the physicochemical constitution and the preparation method of the emulsion.13 Different names and size ranges for these kinds of emulsions can be found in the literature.14,15 In this article, kinetically stabilized emulsions with a mean diameter on the submicrometer scale are termed miniemulsions. Another important aspect of the emulsion-assisted particle synthesis is the method of initialization of the chemical reaction. In this context, the one-emulsion technique reveals a better controllability compared to the two-emulsion technique, since spontaneous coalescence of miniemulsion droplets is prevented. Thus, the presented work is focused on the oneemulsion technique. Its aim is to improve the understanding of the particle formation phenomena within one single droplet by means of a detailed model-based analysis. A variety of model-based approaches for particle synthesis processes using microemulsions can be found in the literature.16 For crystallization processes in kinetically stabilized emulsions, only a few publications are available in the literature.17,18 In order to identify the process-controlling mixing and mass transport mechanisms of the one-emulsion technique, a time scale analysis has to be accomplished. On the basis of the results of the time scale analysis, a population balance equation model, which accounts for nucleation and growth on the crystal length coordinate, has been derived. In numerical simulations, the influence of the droplet size, the initial concentration level of reactant B in the droplets, and the feeding rate of reactant A into the reactor on the particle size distribution are investigated for typical kinetic parameters of precipitation reactions.

2. TIME SCALE ANALYSIS This section provides insight into the prevailing mixing and mass transport mechanisms of the one-emulsion technique. In contrast to processes in homogeneous media, the emulsionassisted precipitation demands a distinction between mass transport phenomena in the organic continuous phase of the emulsion and subprocesses within the dispersed droplet phase. 2.1. Mass Transport and Mixing in the Continuous Phase. It is well-known that mixing affects the course of very fast homogeneous chemical reactions, and for this reason mixing influences the particle property distribution in bulk precipitation of sparingly soluble materials.1921 In particular, molecular level processes such as the chemical reaction, nucleation, and growth

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Figure 2. Multiscale mixing approach in emulsion-assisted precipitation (adopted from ref 22).

of particles are directly influenced by mixing on the molecular scale.20 Furthermore, homogenization on coarser scales determines the environment for mixing on the molecular level.22 Therefore, depending on the process parameters, micromixing and mesomixing are the rate-determining steps in bulk precipitation. In contrast, a direct contact between both reactants A and B in emulsion-assisted precipitation is prevented, since reactant B is dissolved in the dispersed phase of the w/o emulsion. The chemical reaction is initiated when molecules of reactant A penetrate from the oil phase into the droplets. The presence of the surfactant-laden interface between the organic continuous phase and the aqueous droplets can retard the interfacial mass transfer.23 The influence of the additional transport resistance can be ranked by comparing the mixing time constants in the vessel with the time needed to overcome the mass transport barrier at the interface. For sufficiently small miniemulsion droplets (d32 < λk) and a relatively low dispersed phase fraction ϕw, interactions between emulsion droplets and turbulent eddies do not appear, and hence the emulsion may be regarded as a quasi-homogeneous liquid. Thus, the analysis of mixing in homogeneous systems on different scales in the semibatch operation of a stirred tank20,24,25 can also be applied for the one-emulsion technique. The one-emulsion technique can be considered as a semibatch operation. In a first process step, the well-prepared w/o emulsion with defined properties is filled into the reactor, where reactant B is dissolved in the dispersed phase. It has to be noted that the droplet size of the miniemulsion is not affected by stirring in the reactor. In a second process step, reactant A is dosed as a concentrated solution to the agitated reaction vessel. The molecules of reactant A undergo mixing on different time and 1580

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length scales in the continuous phase of the emulsion in the turbulent flow field of the stirred tank until they reach the water droplets. A schematic presentation of the multiscale approach of the one-emulsion technique is depicted in Figure 2. The process of mixing on the scale of the whole reactor determines the macroscopic inhomogeneities in the vessel. A good approximation of the macromixing time constant is given by the circulation time τc:20 τc ¼

Vr Qc

ð2Þ

with Vr being the liquid volume in the reactor and Qc being the circulation capacity, which depends on the geometry of the impeller. The circulation capacity can be estimated as doubled pumping capacity: Qc ¼ 2Qp

ð3Þ

For a disk-like turbine, the pumping capacity is given by20 Qp ¼ 0:76nds3

ð4Þ

where n and ds stand for the stirring rate and the stirrer diameter, respectively. The turbulent exchange between the freshly entered material and its surroundings proceeds on the next smaller scale compared to the reactor scale but is of coarse scale relative to mixing on the molecular level.26 The dispersion of the highly concentrated domains in the vicinity of the feed pipe, which is termed mesomixing, is brought about by turbulent diffusion and the disintegration of large eddies, as shown in Figure 2b. The time scale for turbulent diffusion is given by26 q0 τtd ¼ ð5Þ uD ̅ t with the volumetric flow rate of the feed q0, the turbulent diffusivity Dt, and u being the local average fluid velocity in the surrounding of the feed pipe. In order to estimate u, the reactor flow field is simulated by a plug flow loop with the circulation capacity Qc. Assuming that the length of the plug flow loop is 3-fold larger than the reactor diameter, the mean fluid velocity in the stirred tank is given by Qc u̅ ¼ 3dr Vr

ð6Þ

with dr being the diameter of the reactor. The other important mechanism of mesomixing is the disintegration of large eddies. In this inertial convective process, the scale of the concentration fluctuations is reduced from an initial scale toward the Kolmogorov scale λk.20,27 The time constant characterizing the disintegration is given by 3 5 2=3 Λ2=3 c ð7Þ τed ¼ 2 π ε 1=3 with Λc being the scale of the turbulent eddies and ε̅ being the mean energy dissipation rate. In the case of the feeding of the fresh material by a small feed pipe into a larger turbulent environment, the length scale of the turbulent eddies Λc can be approximated by21,24 1=2 q0 Λc ¼ ð8Þ πu̅ The mean energy dissipation rate in an agitated vessel depends on the type and geometry of the impeller and on the stirring rate. With

the definition of the power number, Po: Vr ε ð9Þ n3 ds5 the mean energy dissipation rate can be estimated.20 Mixing on the molecular level or micromixing in a turbulent flow field is realized by the incorporation of fluid elements from the environment, forming a short-lived laminated structure within a vortex tube (see Figure 2c). The remaining concentration inhomogeneities between adjacent layers of dissimilar composition are equalized by unsteady molecular diffusion.3 Since the diffusion process is accelerated by the deformation of the laminated structure, which results in thinning laminae, the half-life time for molecular diffusion is given by20 1=2 ν τmd ¼ 2 arcsin hð0:05ScÞ ð10Þ ε where ν stands for the kinematic viscosity and Sc is the Schmidt number. The characteristic time scale of engulfment depends on the engulfment rate constant E:26 1=2 1 ν τe ¼ 17 ð11Þ E ε In contrast to the precipitation in homogeneous media, where a direct contact of both reactants is brought about by micromixing, in the emulsion-assisted process using the one-emulsion technique, micromixing is the preceding step for the mass transfer of reactant A into the droplets. The course of the chemical reaction within the dispersed emulsion droplets depends therefore on mixing in the reactor but also on the interfacial mass transfer. In order to estimate the time scale of the mass transfer across the liquidliquid interface, we have to differentiate between the mass transfer resistance in the film around the droplets in the continuous phase of the emulsion and the resistance of the penetration through the surfactant-laden interface. The mass transfer phenomena coupled with the precipitation subprocesses inside the emulsion droplets are discussed in the next section. Within the turbulent vortices, the time constant of mass transport in the film around the droplets depends on the diffusivity of reactant A in the oil phase DA,o, on the film thickness δ, and on the organic phase mass transfer coefficient in the film kmf:24 Po

δ2 δ DA, o ¼ ¼ 2 ð12Þ kmf DA, o kmf The continuous phase mass transfer is mainly influenced by the fluid motion around the emulsion droplets. The dominant transport mechanism can be evaluated by the use of the Peclet number, Pe, which is generally defined as the product of the Reynolds number, Re, and the Schmidt number, Sc: τmf ¼

d32 u ð13Þ DA, o where u is the fluid velocity relative to the droplet velocity in the flow field and d32 is the Sauter mean diameter of the miniemulsion droplets. An equivalent expression of the Peclet number as a function of the mean dissipation rate was given by Batchelor:28 Pe ReSc ¼

1=2

2 d32 ε ð14Þ DA, o ν1=2 3 3 For a typical reactor setup on the laboratory scale (Vr = 10 m ) with a reactor-to-stirrer diameter ratio of dr/ds = 3 and with a

Pe ¼

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filling level equal to the reactor diameter, h = dr, the mean energy dissipation rate as a function of the stirrer speed for Po = 0.55 is presented in Figure 3. For mean energy dissipation rates on the order of ε̅ ≈ 101 W/kg and droplets smaller than 1 μm in size (miniemulsion), the Peclet number is Pe , 1000. According to eq 13, the Reynolds number is (Re < 1) for Schmidt numbers of Sc ≈ 1000. Therefore, the following relationship to estimate the Sherwood number Sh in the Stokes regime (Re < 1) can be used:29 Sh ¼ 1 + ð1 + PeÞ1=3

ð15Þ

The mass transfer coefficient can be derived from the definition of the Sherwood number: Sh

kmf d32 DA, o

ð16Þ

The characteristic time constant for the penetration through the interface is given by τmi ¼

1

ð17Þ kmi ai where kmi is the interfacial mass transfer coefficient. The interfacial area of all droplets in the emulsion related to the volume of the organic phase ai is given by the following relation: ai ¼

ϕw 6 1 ϕw d32

ð18Þ

It is known that the presence of surfactant molecules at liquidliquid interfaces reduces the interfacial mass transfer rate markedly.23,30 On

the basis of mass transfer experiments of benzoic acid in colloidal aphron systems,31,32 the order of magnitude of the interfacial mass transfer resistance can be estimated. According to the literature, kmi ≈ 5 107 m/s is a good approximation of the interfacial mass transfer coefficient in emulsion systems. A comparison of mixing and mass transfer time scales of the one-emulsion technique is presented in Figure 4 as function of the mean energy dissipation rate and of the Sauter mean diameter. The information about the reactor geometry and the operating conditions employed for the calculations of the time scales are summarized in Table 1. It is obvious from both diagrams in Figure 4 that the mass transfer in the continuous phase around the droplets is the fastest subprocess. With an increasing energy dissipation rate in the reactor, the mixing times on all scales are decreasing, in which the macromixing is the slowest mixing mechanism. The penetration of reactant A through the interface proceeds on a time scale of a few seconds and is therefore slower than mixing on the meso- and microscale. Since this time scale depends on the relative interfacial area of the emulsion droplets, an increase in size of the emulsion droplets results in a decreasing mass transfer rate. For high energy dissipation rates and large Sauter mean droplet diameters, the time scale of the interfacial mass transfer exceeds the circulation time. Under these conditions, mixing on all scales in the continuous phase is completed before molecules of reactant A penetrate into the droplets. This means homogeneous conditions for the precipitation of particles are available in the dispersed droplet phase. Since well-prepared miniemulsions possess Table 1. Parameter Set for the Calculation of Mixing and Mass Transport Time Scales in the Continuous Phase parameter

Figure 3. Mean energy dissipation as function of the stirrer speed for different Power numbers.

value

d32

0.110

dr/ds

3

dr/h

1

DA,o

1 10o 7

unit μm

m2/s

kmi

5 10

Po

5

q0

10

mL/min

Vr ε̅

1 103 1 103 to 1 10 1

m3 W/kg

ν

1 106

m2/s

ϕw

0.1

m/s

Figure 4. Time scales of the one-emulsion technique as function of the energy dissipation rate for (a) a fixed Sauter mean droplet diameter of d32 = 1 μm and (b) as function of the Sauter mean droplet diameter for a fixed mean dissipation rate ε̅ = 102 W/kg (macromixing τc, turbulent diffusion τtd, eddy disintegration τed, molecular diffusion τmd, engulfment τe, mass transfer in the film τmf, mass transfer across the interface τmi). 1582



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Figure 5. Feeding time and circulation times for different mean energy dissipation rates as function of the droplet diameter.

droplet sizes below 1 μm, macromixing might play a role in this process. However, for a sufficiently slow addition of reactant A (τf > τc), a uniform bulk composition can be assumed.20 The feeding time constant τf depends on the volumetric flow rate q0 and on the concentration relation between reactant B in the aqueous droplets and reactant A in the feeding stream. For a stoichiometric operation mode, the feeding time constant is given by τf ¼

ϕw Vr c0B, w q0 c0A, o

ð19Þ

A comparison between feeding and macromixing time scales as a function of the feeding rate is presented in Figure 5 for a droplet size of 1 μm, and the initial concentration relation c0B,w/c0A,o = 0.01. The condition (τf > τc) is fulfilled for very low volumetric flow rates, as depicted in Figure 5. In contrast to precipitation in homogeneous media, where mixing on the meso- and microscale are the controlling mechanisms, the mass transfer across the surfactant-laden interface is the ratedetermining step of the one-emulsion technique. Consequently, the very fast chemical reaction is decelerated by the mass transfer resistance, and homogeneous reaction conditions in the dispersed droplet phase throughout the reactor are provided. 2.2. Mass Transport and Precipitation in the Droplet. The chemical reaction of the reactants A and B is the basis for the subsequent particle formation process within the droplet. Due to the ionic character of these reactants, the chemical reaction can be considered instantaneous compared to the time scale of the reactant mass transfer within the droplet. Therefore, reactants A and B cannot coexist inside the droplet. As emerged from the study presented in the preceding section, the penetration of reactant A proceeds on the order of seconds. Thus, only small amounts of A encounter a large excess of B relative to A. Since the diffusive transport rate of B to the reaction front is large compared to the interfacial mass transfer rate, the reaction plane is localized at the droplet boundary,24 as sketched in Figure 6. The question of whether spatial inhomogeneities within the droplet appear can be answered by considering the time scales of the following subprocesses in the droplet: the chemical reaction, the diffusive transport, and the consumption of product C by nucleation. The production of C at the droplet boundary can be characterized by the time scale of penetration of component A through the interphase. Therefore, the time scale of the chemical reaction reads as follows: τcr ¼ τmi

ð20Þ

Figure 6. Concentration profiles of reactants A and B in the vicinity of the droplet boundary.

The characteristic time of the diffusive mass transport within the droplet is given by τd ¼

dd2 4DC, w

ð21Þ

with the diffusivity of C in the droplet being DC,w ≈ DA,o. The characteristic time constant for the nucleation process can be estimated by20 nn τn ¼ ð22Þ Vd Bn where nn stands for the number of nuclei, Vd for the droplet volume, and Bn for the nucleation rate. The nuclei number can be approximated by the assumption that a certain proportion N of the maximum possible number of nuclei in the droplet nmax is generated. Hence, nn can be estimated by nn ¼ Nnmax

ð23Þ

If we suppose that the existing material in the droplet is exclusively consumed by nucleation, the resulting maximum possible number of nuclei of size xn is given by nmax ¼

Vd MC c0B, w kv x3n FC

ð24Þ

with kv being the volume shape factor of the particles and FC and MC being the density and the molar mass of pure C, respectively. The nucleation rate Bn can be expressed by33 A Bn ¼ kn exp 2 ð25Þ ln S where kn and A are kinetic parameters and S is the supersaturation in the droplet. In order to evaluate the appearance of spatial dependencies within the droplet during the particle formation, the time constants of the intradroplet subprocesses have to be compared. The parameter set, which was used to calculate the time constants, is summarized in Tables 1 and 2. The time scales of the chemical reaction and the mass transfer within the droplet are show in Figure 7a as function of the droplet size. The dependency of the characteristic time of nucleation on the supersaturation level in the droplet is presented in Figure 7b. In the droplet size range of miniemulsions (d32 < 1 μm), the diffusive mass transfer is much faster than the production of C at the droplet interface. Furthermore, due to the slow production of C, the supersaturation is slowly increasing. As shown in Figure 7b, 1583



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nucleation at low supersaturation levels is extremely slow. Therefore, nucleation begins when a sufficiently high supersaturation inside the droplet is attained. A comparison of the time scales reveals that nucleation starts when the product C is already equally distributed inside the droplet. Hence, the droplets can be considered as ideally mixed for the particle formation process.

3. MODEL DERIVATION In this section, a mathematical model of the formation of primary nanocrystals using the one-emulsion technique will be presented. The one-emulsion technique is a semibatch process, where an A-rich organic solution is dosed into the stirred w/o emulsion. In the course of the process, reactant A penetrates into the droplets. For an adequate description of this process, a distinction between phenomena in the reaction vessel that contains the w/o emulsion and phenomena taking place within the droplets during particle formation is necessary. Since the mass transfer of reactant A across the liquidliquid interface is the rate determining step, the continuous phase can be considered as ideally mixed. Therefore, the treatment of particle population dynamics in one representative single droplet is sufficient, if a monodispersed emulsion with a constant droplet diameter dd is assumed. 3.1. Solid Particle Formation in an Emulsion Droplet. A well-established approach for the characterization of dispersed solid particles in a surrounding liquid is the concept of population balance modeling, where the collection of single crystals is considered a population. Population balances typically describe phenomena in terms of internal and external coordinates of a dispersed system. Due to the fact that the droplet can be treated as ideally mixed, it is sufficient to describe the dynamics of the Table 2. Parameter Set for the Calculation of Intradroplet Time Scales parameter

value

unit

A

2500

c0B,w

0.01

mol/L

kn kv

1 1036 π/6

1/(m3 s)

MC

100

g/mol

N

0.01

S

1 102 to 1 104

xn

1

nm

FC

500

kg/m3

particle formation along the internal coordinate. Since the crystal size seems to be an adequate property of the population, the internal state space is defined by the characteristic crystal length x. This restriction implicates a fixed crystal shape with a constant volume shape factor kv. The number of particles per unit length of crystal size between x and x + dx is given by the number density function F(x,t) dx. The population balance accounting for primary nucleation in the droplet water bulk and growth is given by ∂F ∂F ¼ G + Vd Bn δðx xn Þ ð26Þ ∂t ∂x with the initial condition F(x,t = 0) = 0, where G stands for the growth rate, Bn for the nucleation rate, and Vd is the droplet volume. The δ function δ(x xn) represents the uniform distribution of nuclei of size xn. Secondary changes like agglomeration or Ostwald ripening are disregarded by our model. Breakage is also excluded due to the small size of the synthesized primary particles. Since the birth term of nucleation is expressed explicitly on the right-hand side of eq 26, the following boundary value on the crystal size coordinate is given: Fjx ¼ 0 ¼ 0

ð27Þ

With regard to classical nucleation theory, the rate of nucleation Bn is calculated as a function of the supersaturation S = cC,w/ceq C,w, where cC,w is the concentration of the dissolved component C in the aqueous phase and ceq C,w is the solubility of C in water. According to Wei and Garside,33 the kinetics of nucleation are 8 > < kn exp A , S g 1 ln2 S ð28Þ Bn ¼ > : 0, S τf dt Vo with the initial condition nA,o(t = 0) = 0, nA,o being the amount of reactant A in the continuous phase and Ai being the interfacial area between the aqueous and the organic phases of the emulsion. The feeding stream stops when the amount of A in the emulsion obeys the stoichiometric condition of the chemical reaction equation. During the dosing period of the A-rich solution (t e τf), the volume of the oil phase Vo increases: Vo ¼ q0 t + Vo0

ð31Þ

where V0o is the initial volume of the organic phase. Since the chemical reaction can be considered instantaneous and the reaction plane is located at the droplet boundary, reactant A does not exist inside the droplet. Thus, the production rate of component C is given by the mass transfer rate of reactant A across the interface.24 The dynamics of the dissolved component C are closely linked to the particle population dynamics. Assuming that the volume of the precipitated particles is negligible compared to the droplet volume, the material balance of component C reads as follows: dnC, w Ad F ¼ km, i nA, o 3 C Gkv dt Vo MC

Z 0

∞

x2 F dx

FC kv x3 Vd Bn ð32Þ MC n

with the initial condition nC,w(t = 0) = 0. In eq 32, nC,w is the amount of the dissolved component C in the droplet, and Ad is the surface area of the droplet. 3.3. Dimensionless Formulation of Model Equations. For the numerical treatment, it is useful to reformulate the model equation into dimensionless quantities. The dimensionless number density is defined as ψ = Fxmax with xmax being the maximum

Vd MC c0B, w k v FC

!1=3 ð33Þ

The kinetics of nucleation, growth, interfacial mass transfer, and the feeding rate are expressed in terms of time constants. The characteristic time of nucleation is given by eq 22. For the nondimensionalization, we relate the maximum possible number of nuclei nmax to the nucleation rate at maximum possible supersaturation Bn,max. Assuming that the chemical reaction is finished but the consumption of C by nucleation and growth of particles has not been initiated, the maximum possible supersaturation level in the droplet is given by Smax ¼

c0B, w eq cC, w

ð34Þ

For the particle growth, the time constant can be expressed by τg ¼

xmax Gmax

ð35Þ

where Gmax stands for the growth rate at maximum possible supersaturation. The time constants for the interfacial mass transfer and the feeding flow rate are given by eqs 17 and 19. Finally, the time is scaled to characteristic time of the feeding flow rate τ = t/τf. The reformulated model equations for S g 1 read as follows: The dimensionless population balance equation is ∂ψ τf Smax xC, w 1 g ∂ψ ¼ ∂τ Smax 1 ∂ξ τg ! τf A A + nmax exp 2 ðξ ξn Þ ð36Þ τn ln Smax ln2 Smax xC, w with the initial condition ψ(ξ,τ = 0) = 0 and the boundary condition ψ|ξ=0 = 0. The reformulated material balance of A in the organic phase is given by ( 1, τ e 1 dxA, o τf ϕ0o ¼ xA, o + ð37Þ 0, τ > 1 dτ τmi ϕo with the initial condition xA,o(τ = 0) = 0. The increase of the oil phase volume fraction ϕo can be calculated by ! 0 c B, w ϕo ¼ ϕ0w 0 τ 1 + 1 ð38Þ cA, o where ϕ0w is the initial volume fraction of the dispersed phase. Finally, the reformulated material balance of the dissolved 1585



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Table 3. Kinetic Parameters parameter

Table 4. Specified Process and Material Parameters value

unit

parameter

value

unit

A

2500

c0A,o

1

g

1

c0B,w

0.0010.1

mol/L

kg

1 10 12

m/s

ceq C,w

1 10 7

mol/L

km,i

5 10 7

m/s

dd

0.12

μm

kn

1 1036

1/(m3 s)

MC

100

g/mol

q0

0.11000

mL/min

V0r

1

L

xn FC

1 500

nm kg/m3

ϕ0w

0.1

m3/m3

product C reads Z dxC, w τf ϕ0o τf Smax xC, w 1 g ∞ 2 ¼ xA, o 3 ξ ψ dξ Smax 1 dτ τmi ϕo τg 0 ! τf A A exp ð39Þ τn ln2 Smax ln2 Smax xC, w with the initial condition xC,w(τ = 0) = 0. 3.4. Numerical Simulation of Population Balance Equations. In order to simulate the dynamics of particle formation in emulsion droplets, accurate and robust numerical solution algorithms of partial differential equations have to be used. A multitude of different numerical methods can be found in the literature.3436 For the solution of the derived dimensionless population balance equations, a finite volume scheme was applied, since this technique exhibits the exact conservation of mass. Therefore, a semidiscretization of the population balance with respect to the property coordinate x is required. This means the desired crystal size range is subdivided into a finite number of grid cells. The cell average value of the dimensionless number density function ψi is determined by integration over each grid cell. As result of the semidiscretization, the partial differential equation is approximated by a finite number of ordinary differential equations. This system of ordinary differential equations can be calculated by standard ODE solvers. The accuracy of the finite volume discretization scheme of population dynamics is primarily influenced by the approximation of the cell-face flux on the crystal size coordinate. A computationally efficient way to approximate this flux over the edges of the cells is the application of the high-resolution finite volume method extended by a flux limiter function. This discretization technique was derived by Koren37 for a two-dimensional advectiondiffusion equation. Motz et al.38 and Qamar et al.39 rederived the semidiscrete highresolution scheme with a flux limiter for one-dimensional population balance equations. In the cited references, a detailed consideration of the discretization scheme is presented.

4. SIMULATION RESULTS Simulated particle size distributions are attained by implementing the semidiscrete model equations in MatLab 7.5. The considered crystal size domain is subdivided into 500 equally sized finite grid cells for the high-resolution discretization scheme. The resulting discretized ordinary differential equations are calculated using the standard ODE solver ODE15s. In order to investigate the influence of process parameters on the particle size distribution, reliable kinetics are needed. Nucleation and growth kinetics were estimated by typical constants for precipitation processes. The preexponential nucleation parameter kn and parameter A were adopted on the basis of the work of Wei and Garside33 for the precipitation of BaSO4. The magnitude of the growth kinetic constant kg and the kinetic order

mol/L

g were evaluated on the basis of the literature data.33,40,41 The value for the interfacial mass transfer coefficient was adopted on the basis of the work of He and co-workers.31,32 All kinetic parameters are summarized in Table 3. The variety of process parameters used for the simulations was restricted by the model assumptions and the process practicability. For material parameters, typical values of precipitation processes were used. A summary of these parameters is given in Table 4. Two limiting cases for the particle precipitation in emulsion droplets are possible. In the first case, we assume that only one particle is generated inside the droplet and the dissolved component C is depleted by growth. The size of this maximum possible particle xmax can be calculated by eq 33. In the second case, it is supposed that the existing material in the droplet is exclusively consumed by nucleation. The resulting maximum possible number of nuclei of size xn = 1 nm is given by eq 24. Both the maximum particle size and the maximum number of particles are depicted in Figure 9 for three different initial concentration levels of reactant B in the droplet. For the maximum particle size, a linear dependency on the droplet diameter exists. If only one particle of x = 10 nm should be generated, the droplet size has to be reduced to the order of dd = 0.1 μm. In the limiting case of pure nucleation, up to 1 million nuclei can be born in a 1 μm sized droplet. These limiting values can be used to normalize the absolute particle size and number for the presentation of the simulation results. In order to survey simulation results but also to describe the dynamic behavior of the particle formation, moments of the size distribution are employed. The kth moment is defined as follows: Z

∞

Mk ¼

xk F dx, k ¼ 0, 1, 2

ð40Þ

0

The zeroth moment represents the number of particles. The mean particle size xm is the ratio of the first to the zeroth moment: xm ¼

M1 M0

ð41Þ

A common choice to describe the breadth of a distribution is the polydispersity p, which is defined as p¼ 1586

M2 M0 M12

ð42Þ



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Figure 9. Maximum possible size of one single particle in one emulsion droplet (a) and maximum possible number of nuclei in one emulsion droplet (b) as a function of the droplet diameter.

Figure 10. Dynamic evolution of the concentration of reactant A in the oil phase (a), of the supersaturation (b), of the normalized particle number and of the normalized mean particle size (c), and the final particle size distribution (d).

In order to compare particle size distributions, the normalized number density q is introduced: q¼

F M0

ð43Þ

4.1. Dynamics of Particle Formation. In this section, a qualitative description of the particle formation dynamics for a set of representative process parameters (dd = 1 μm, c0B,w = 0.01 mol/L, q0 = 10 mL/min) is presented. Dynamic evolutions of the concentration level of reactant A in the organic phase, of the supersaturation level within the droplet, of the dimensionless mean particle size, and of the number of particles inside the droplet are depicted in Figure 10. Due to the small volumetric flow rate of reactant A into the reactor (τf < τmi), only small

amounts of reactant A accumulate in the organic phase of the emulsion. The termination of the feeding period can be identified by the kink in the course of the time-dependent concentration data. When reactant A penetrates into the droplets, the product C is instantaneously produced, whose concentration is expressed as supersaturation level. The time range around the maximum in the dynamic evolution of the supersaturation level in Figure 10b indicates the period of nucleation. When a critical supersaturation level is exceeded, the formation of nuclei in the droplet water bulk sets in until the supersaturation level falls below this critical value in the course of the process. The time scale of nucleation can be deduced from the dynamic evolution of the normalized particle number in the droplet, as shown in Figure 10c. Since nucleation inside the droplet proceeds on the order of seconds 1587



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Figure 11. Normalized and absolute mean particle size (a) and polydispersity (b) as a function of the droplet diameter dd for c0B,w = 0.01 mol/L and q0 = 10 mL/min.

Figure 12. Normalized and absolute mean particle size (a) and polydispersity (b) as a function of the initial concentration of component B, c0B,w, for dd = 1 μm and q0 = 10 mL/min.

and is therefore much slower than the diffusive mass transport of the dissolved component C, the assumption of the ideally mixed droplet for the particle formation process is reasonable. With further decreasing supersaturation, the remaining dissolved material is consumed by growth. The course of the particle growth is illustrated by the dynamic evolution of the normalized mean particle size in Figure 10c. After a period of time of about 20 s, the particle formation within the droplet is completed. The resulting final particle size distribution is depicted in Figure 10d. For this set of process parameters, the mean particle size is approximately 7% of the maximum possible particle size in the droplet. 4.2. Variation of Process Parameters. Process parameters, which can be easily varied in the real particle formation process using the one-emulsion technique are the droplet size dd, the initial concentration level of component B within the droplets c0B,w, and the volumetric flow rate of the feeding stream q0. Investigations on the influence of these parameters on the particle size and size distribution were accomplished by means of numerical simulations and are presented in this section. In order to compare the impact of the particular parameters on the product properties, the relative and absolute mean particle size and the polydispersity of the size distribution are depicted over the selected parameter range. The effect of the droplet diameter for c0B,w = 0.01mol/L and q0 = 10 mL/min is shown in Figure 11. In the typical droplet size range of miniemulsions, the absolute particle size slightly increases with increasing droplet diameters. As shown in the time scale analysis, the interfacial mass transfer is improved for decreasing emulsion droplet sizes. Thus, with

increasing droplet diameters, the mass transfer across the surfactant-laden interface and the production of product C are decelerated. This leads to a slight reduction of the nucleation rate. Consequently, more material inside the droplets is consumed by growth. For this reason, slightly larger absolute particle sizes are obtained for increasing droplet diameters. Since the increase of the maximum possible particle size for growing droplet sizes is larger than the slight increase of the absolute particle size, the normalized particle size reveals a decreasing trend for the droplet size increase. Figure 11b shows also only a marginal increase of the polydispersity for increasing droplet sizes. Basically, the effect of the droplet size on both the particle size and the polydispersity is weak, because the time constant of the feeding rate (τf = 6 s) is larger than the dropletsize-dependent time constant of the interfacial mass transfer (τmi = 0.36 s). The interfacial mass transfer is not the controlling process step in the droplet size range of miniemulsions for the given feeding rate. The dependencies of the initial concentration level of reactant B on the particle size and polydispersity for dd = 1 μm and q0 = 10 mL/min are shown in Figure 12. The reduction of the initial concentration level of B entails a decrease of the mean absolute particle size until a minimum is reached. A further decrease of the concentration level, however, results in a slight increase of the absolute particle size. This behavior is due to the fact that less material is dissolved in the droplet, and hence, at some concentration level an insufficient supersaturation level for nucleation appears. Consequently, less particles are created in the droplet, and the remaining dissolved component C is consumed by 1588



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Figure 13. Normalized and absolute mean particle size (a) and polydispersity (b) as a function of the feeding rate q0 for dd = 1 μm and c0B,w = 0.01 mol/L.

growth, which results in the increase of the mean particle size. In the presentation of the normalized mean particle size in Figure 12a, this effect is more pronounced. For a further increase of the initial concentration of B, the rate of increase of both the absolute mean particle size and the maximum possible particle size is equal. Therefore, the normalized particle size remains constant. The increase of the polydispersity for decreasing concentration levels indicates a more pronounced superposition of nucleation and growth compared to high initial concentration levels. By varying the initial concentration level of B the particle size can be influenced to a certain degree. In Figure 13, the impact of the feeding rate on the particle size and polydispersity for dd = 1 μm and c0B,w = 0.01 mol/L is depicted. In the case of the variation of the feeding rate, the absolute and the normalized mean particle size show the same behavior, since xmax is independent of the feeding rate. For low volumetric flow rates, the largest particles are obtained. With increasing feeding rates, the mean particle size is reduced until a constant value is obtained for q0 > 100 mL/min. In the slow feeding regime, the process is controlled by the low feeding rate. Therefore, product C is also slowly produced, which results in a reduction of the nucleation rate. The remaining material is consumed by growth, and larger particles are obtained. With an increase of the feeding rate, the production rate of the sparingly soluble product is accelerated as well, and hence more particles are born inside the droplet. Consequently, less material can be used for growth, and a reduction in the mean particle size appears. In the fast feeding regime, the interfacial mass transfer is the controlling mechanism of the particle formation process. Thus, a further increase of the feeding rate has no influence on the mean particle size. The polydispersity only slightly increases for growing feeding rates. It has to be noted that for q0 > 10 mL/min (see Figure 5), macromixing effects might also influence the course of the particle formation in the droplets. Compared to the variation of the droplet diameter and the initial concentration of B, the feeding rate reveals the highest potential to control the particle size in the emulsion-assisted precipitation process using the one-emulsion technique.

5. CONCLUSIONS This contribution provides a time scale analysis of the emulsion-assisted nanoparticle formation in a stirred tank reactor. On the basis of the identified controlling process steps a population balance model-based approach was used to estimate the influence of process parameters on the product properties of synthesized nanoparticles. The preseneted model-based analysis may be

regarded as a starting point for further theoretical and experimental investigations. On the assumption that the miniemulsion system behaves as quasi-homogeneous liquid, the characteristic times of mixing in the continuous phase on different length scales can be estimated on the basis of studies for homogeneous media. The following rate-controlling steps of the one-emulsion technique were identified: feeding of reactant A into the stirred vessel, mixing on the scale of the whole reactor, and penetration of reactant A through the surfactant-laden interface. For sufficiently low feeding rates and high energy dissipation, the influence of macromixing in the reactor can be neglected. With the goal of achieving homogeneous reaction conditions in the dispersed droplet phase of the w/o miniemulsion, an increase of the droplet diameter is favorable. In this case, mixing of reactant A in the continuous phase is completed before A penetrates into the droplets. The instantaneous chemical reaction inside the droplets is retarded by the interfacial mass transfer resistance. Therefore, the diffusive mass transfer in submicrometer miniemulsion droplets is much faster than the time scale of nucleation at small supersaturation levels. As an outcome of the time scale analysis, both the continuous phase of the emulsion and the droplets can be considered ideally mixed in our particle formation model. Solely the mass transfer across the liquidliquid interface has to be accounted for by the model. In order to describe the synthesis of nanoparticles, a onedimensional population balance model in one representative droplet was derived. In numerical simulations, the influence of the droplet size, the initial concentration level of reactant B, and the feeding rate of reactant A into the stirred tank reactor on the particle size were analyzed on the basis of typical material parameters and kinetic data of precipitation reactions. It was figured out that the influence of the droplet diameter and the initial concentration of B on the mean particle size is weak. In contrast, the influence of the feeding rate on the mean particle size is much more pronounced. A slow addition of reactant A into the reactor results in the production of only a few particles with a relatively large mean particle size. For increasing feeding rates, a reduction of the particle size was obtained. In conclusion, we can state that high feeding rates and small droplets are advantageous for the production of small nanoparticles. Contrary to the synthesis of small nanoparticles, low feeding rates and large miniemulsion droplets are favorable for obtaining homogeneous reaction conditions in the emulsion-assisted precipitation process. In future work, the time constant of the interfacial mass transfer will be experimentally investigated. Furthermore, for an adequate precipitation process, an experimental validation of our model is necessary. 1589



’ AUTHOR INFORMATION Corresponding Author

*Tel. +49 391 6110-350. Fax: +49 391 6110-353. E-mail: [email protected].

’ NOTATION Latin Letters

ai = relative interfacial area, m2/m3 A = nucleation rate parameter Ai = interfacial area, m2 Ad = surface area of one droplet, m2 Bn = nucelation rate, 1/(m3s) Bn,max = maximum nucelation rate, 1/(m3s) cA,o = concentration of A in the oil phase, mol/L c0A,o = concentration of A in the feed, mol/L cB,w = concentration of B in the aqueous phase, mol/L c0B,w = initial concentration of B in the aqueous phase, mol/L cC,w = concentration of C in the aqueous phase, mol/L ceq C,w = solubility of C in water, mol/L d32 = Sauter mean diameter, m dd = droplet diameter, m dr = reactor diameter, m ds = stirrer diameter, m Dt = turbulent diffusivity, m2/s DA,o = diffusivity of A in the oil phase, m2/s DC,w = diffusivity of C in the aqueous phase, m2/s E = engulfment rate, 1/s F = number density, 1/m g = order of growth rate G = growth rate, m/s Gmax = maximum growth rate, m/s h = filling level, m kmf = mass transfer coefficient in the film, m/s kmi = interfacial mass transfer coefficient, m/s kn = nucleation rate parameter, 1/(m3s) kv = volume shape factor MC = molar mass of component C, g/mol Mi = moment of order i, mi n = stirring rate, 1/s nA,o = amount of substance A in the oil phase, mol nA,o,max = maximum possible amount of substance A in the oil phase, mol nC,w = amount of substance C in the aqueous phase, mol nn = number of nuclei nmax = maximum number of particles N = normalized number of particles p = polydispersity Pe = Peclet number Po = power number q = normalized number density, 1/m q0 = feedinhg rate, m3/s Qc = circulation capacity, m3/s Qp = pumping capacity, m3/s Re = Reynolds number S = supersaturation Smax = maximum possible supersaturation Sc = Schmidt number t = time, s u = local average fluid velocity, m/s u = relative fluid velocity, m/s

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Vd = droplet volume, m3 Vo = volume of the oil phase, m3 V0o = initial volume of the oil phase, m3 Vr = liquid volume in the reactor, m3 x = particle size, m xA,o = dimensionless amount of substance A in the oil phase xC,w = dimensionless amount of substance C in the aqueous phase xm = mean particle size, m xmax = maximum possible particle size, m xn = size of the nuclei, m Greek Letters

δ = film thickness, m ε̅ = mean energy dissipation rate, W/kg λk = Kolmogorov scale, m Λc = length scale of turbulent eddies, m ν = kinematic viscosity, m2/s ξ = dimensionless particle size FC = density of component C, kg/m3 τ = dimensionless time τc = circulation time, s τcr = time scale of the chemical reaction, s τd = time scale of molecular diffusion in the droplet, s τe = time scale of engulfment, s τed = time scale of disintegration of large eddies, s τf = time scale of the feeding rate, s τmd = time scale of molecular diffusion in laminated structure, s τm = time scale of mass transfer in the film, s τmi = time scale of interfacial mass transfer, s τn = time scale of nucleation, s τtd = time scale of turbulent diffusion, s ϕo = volume fraction of the oil phase, m3/m3 ϕ0o = initial volume fraction of the oil phase, m3/m3 ϕw = volume fraction of the dispersed phase, m3/m3 ψ = dimensionless number density

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Emulsion-Assisted Nanoparticle Precipitation: Time Scale Analysis

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