Magnetophoresis of Nonmagnetic, Submicrometer Particles in Magnetic Fluids

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1 Magnetohoresis of Nonmagnetic, Submicrometer Particles in Magnetic Fluids Lino Gonzalez, Seif Fateen, Kenneth Smith and T. Alan Hatton Deartment of Chemical Engineering Massachusetts Institute of Technology, Cambridge, Massachusetts Present Address: Nestle R&D Center Marrysville, Ohio Abstract We studied the migration of nonmagnetic, submicrometer olystyrene beads submerged in a magnetic fluid in the resence of nonuniform magnetic fields as a otential method for size-based searation of submicrometer, nonmagnetic secies. Since the olystyrene beads are much larger than the magnetic fluid nanoarticles, the magnetic fluid was treated as a onecomonent continuum with resect to the beads. We found that the olystyrene beads will migrate in the direction of decreasing magnetic fields and will focus over a region where the magnetic field or its gradient vanishes, as redicted by our model. The concentration rofiles redicted by our model, which has no adjustable or fitted arameters, agree reasonably well with the exerimental data both qualitatively and quantitatively. I. INTRODUCTION Searations is an active research field in chemical engineering. With increasing advances in technology, the focus has shifted in recent years from macro-scale to micro-scale searations, esecially in biological alications. It is very imortant to be able to searate cells, roteins, DNA, and viruses based on both their hysical and chemical/biological roerties. Deending on the article size-range of interest, there are several searation methods that have been used traditionally to searate articles based on size, secific gravity, and chemical roerties. Some of the hysical methods are summarized in Table 1, where the size 1 µm was chosen arbitrarily to searate small from large articles. Magnetic fluids have been used in flotation methods to searate articles based on density differences, not size difference, and only on rather large articles (> 1 µm). It was discovered in 1969 that magnetically neutral minerals would exerience different magnetic buoyancy forces deending on their relative densities [1]. Patented rocesses were designed where minerals of different densities would reach different equilibrium heights above external magnets, making continuous mineral searation ossible [2]. However, these methods are alicable only to materials of different secific gravities and, because of Brownian motion, are not alicable to submicrometer searations. In recent years there has been extensive work in biological searations using magnetic articles. In most of these rocesses, small magnetic articles are chemically or hysically attached to the nonmagnetic secies [3]-[6]. Once the secies of interest have been effectively magnetized, magnetic fields are used to move them around the carrier fluid. With this rocedure, biological secies can be searated because the small magnetic articles would only attach to the secies of interest. However, this method is not effective in searating nonmagnetic secies that have similar chemical and biological roerties. Also, the secies would need to be treated at the end of the searation rocess to remove the magnetic articles. The resent work differs from revious submicrometer searation techniques in that the magnetic articles are not chemically or hysically attached to the secies of interest. Instead, a magnetic fluid continuum is used to generate a magnetic ressure that can be used to move nonmagnetic bodies. Since the articles we are interested in searating are smaller than a micrometer in diameter, buoyancy effects are negligible, so equilibrium searation in sace is not ossible. Searation is achieved based on a balance between magnetic and drag forces. The nonmagnetic articles reach equilibrium velocities when subjected to external, nonuniform magnetic fields, as oosed to equilibrium heights above a magnet. This work is based on the exerimental findings of Fateen [7], who studied the forced diffusion of submicrometer, aroximately nonmagnetic olystyrene beads in magnetic fluids. Watarai et al. [8] indeendently studied a similar rocess, where migration of micrometer-sized olystyrene beads in a aramagnetic MnCl 2 solution was described by considering a force balance between viscous drag and forces of magnetic origin. However, water-based magnetic fluids are referred over aramagnetic salts since they are environmentally and biologically friendly, can achieve much higher magnetic suscetibilities, and can be easily recovered and recycled using High Gradient Magnetic Searation (HGMS). We can also maniulate the magnetization and the ionic strength of the carrier fluid indeendently in magnetic fluids, which allows for high magnetization without having any of the nonmagnetic articles agglomerate and reciitate out of solution due to charge screening. Our work deals with smaller articles than those studied by [8] resent at much larger concentrations. Thus, we model our rocess as a forced-diffusion rocess and kee track of concentration rofiles, instead of keeing track of individual olystyrene beads. Our model is also able to account for article-article interactions, such as electrostatic reulsion between olystyrene beads, which are significant at the concentration levels of interest to us.

2 TABLE I SOME WIDELY USED SEPARATIONS PROCESSES FOR DIFFERENT PAICLE SIZES. > 1 µm < 1 µm Filtration Cycloning Sedimentation Sieving Centrifugation Flotation Fig. 1. Two hase aqueous artitioning Ultracentrifugation Ultrafiltration Entroic traing Vector chromatograhy Field flow fractionation Magnetic fluid structure II. MAGNETIC FLUID STRUCTURE The rincial tye of ferrofluid used in ractice is a colloidal susension of magnetic, single-domain articles in a liquid carrier, stabilized against agglomeration by a molecular layer of disersant. Thermal agitation kees the articles susended because of Brownian motion, and the coatings revent the articles from sticking to each other. Most colloidal ferrofluids are synthesized, for they are not commonly found in nature (certain bacteria roduce colloidal magnetite, but not in sufficient amounts for commercial use). The two most common methods for rearing a magnetic colloid are size reduction and reciitation [9]. A tyical ferrofluid contains 1023 articles er cubic meter and is oaque to visible light. The most common magnetic material used to make ferrofluids is magnetite (FeO-Fe 2 O 3 ). The average magnetite article diameter is about 10 nm, with the surfactant or olymer layer making the total diameter about nm. Some of the most common carrier fluids include diesters, hydrocarbons, esters, fluorocarbons, and water. The most common carrier for biological alications is water, as it is most comatible with biological materials and the resulting ferrofluid would not be hazardous if trace amounts were left behind in searation rocesses. In this research, we use a magnetic fluid reared by reciitation, consisting of a susension aroximately 10 nm magnetite nanoarticles coated with a olymer layer of olyethylene oxide for colloidal stability [10]. A schematic diagram of a coated magnetite article is shown in Fig. 1. The olymer layer stabilizes our magnetic nanoarticles by a combination of steric hindrance and electrostatic reulsion. III. THEORY We will start by erforming a force balance on a single article (such as a olystyrene bead) submerged in a ferrofluid in the resence of a satially nonuniform magnetic field. Since the articles of interest are much bigger than the magnetic fluid nanoarticles, as shown in Fig. 2, the ferrofluid will be treated as a one-comonent continuum (Fig. 2 is drawn to scale to emhasize the difference in size between the olystyrene beads used in this research and the coated magnetic fluid nanoarticles). This force balance will allow us to study the relationshi between the size of the article and its migration velocity in a ferrofluid continuum. We will then include diffusion effects and some article-article interactions to obtain a more comlete model describing the forced diffusion of small articles in a magnetic fluid continuum. Since the magnetic nanoarticles that make u the ferrofluid are so small in diameter, the alied magnetic fields will not be strong enough to comete with the Brownian motion of these nanoarticles. The comosition of the ferrofluid will therefore remain aroximately uniform over time and sace. Thus, the density of the ferrofluid continuum will remain aroximately constant in time and sace. A. Force Balance on a Single Particle The force of magnetic origin on a article submerged in a ferrofluid f is given by, to leading order, F m µ 0 V (M M f ) H, (1) where M f and M reresent the magnetization of the fluid and the article, resectively, V the volume of the article, H the magnetic field strength, and µ 0 the ermeability of free sace [9]. This exression assumes that the ferrofluid behaves as a continuum around the article of interest, meaning that the magnetic nanoarticles that make u the ferrofluid must be much smaller than the article. The magnetization of the fluid and the articles are assumed to be collinear with the magnetic field, meaning that the above-given force exression is not valid for high-frequency, alternating magnetic fields. Other assumtions include negligible electrical conductivity of the ferrofluid, nearly constant temerature or temeratures much smaller than the Currie temerature of the ferrofluid, constant density, constant comosition of the magnetic fluid, small volume fraction of the articles (φ 1), nearly constant H and H over the article volume V, and that the secies magnetizations satisfy the conditions Fig. 2. Polystyrene bead in magnetic fluid continuum M H, M f H 1. (2)

3 The urose of using a ferrofluid is to be able to searate articles that are almost nonmagnetic, since magnetic articles can be easily searated without having to immerse them in a ferrofluid. Thus, we will generally also have that M 1. (3) M f We will therefore not include the magnetization of the submicrometer articles in our exressions. For cases where the article magnetization is significant, the magnetization of the fluid aearing in our exressions can simly be relaced by the magnetization difference between the magnetic fluid and the articles of interest. The magnetization is related to the magnetic field by Fig. 3. Magnetization of 1% magnetic fluid versus alied magnetic field M = χh, (4) where χ denotes magnetic suscetibility. In eq. (1), we have allowed for the article and the ferrofluid to be nonlinear, meaning that χ = χ(h). This is very imortant since magnetic fluids are not magnetically linear, as seen in Fig. 3. Although more rigorous equations are available to relate χ to H, we decided to use the simle, emirical relationshi χ = M s H T + H, (5) where M s is the saturation magnetization of the fluid (the magnetization at infinite alied magnetic field) and H T is the magnetic field strength at which M = M s /2. As shown in Fig. 3, this emirical relationshi fits the exerimental data very well and its simlicity allows for fast numerical solutions to our model equations. In Fig. 3, a negative value of the magnetic field is used to indicate that the field was alied in the oosite direction. The resulting magnetization is symmetric, as exected. When using eq. (5), the value of H is the magnitude of the alied field. The values used to generate the curve are M s = 665 A/m and H T = A/m. Finally, eq. (1) is valid whether the article is stationary or migrating at a velocity u (t), as long as the assumtions described above are satisfied and the article is small enough so that inertial effects are negligible. Equation (1) tells us that the force of magnetic origin acting on articles submerged in a ferrofluid is roortional to their volume, the difference in magnetization between the articles and the ferrofluid continuum, and to the magnetic field gradient. This means that nonmagnetic articles will be ushed in the direction of decreasing magnetic fields until either the magnetic field or its gradient vanishes (the force of magnetic origin will vanish when the magnetic field is zero, since the fluid magnetization will also be zero at that oint). All sufficiently large nonmagnetic articles will migrate in the direction in which the magnetic field is decreasing and eventually focus in a region where the magnetic field or its gradient vanishes (since gravitational effects are negligible), regardless of their relative sizes. Thus, size searation is not ossible when the articles reach their equilibrium osition. However, since this force of magnetic origin is also roortional to the article volume, different-sized articles will migrate toward the equilibrium ositions at different velocities, meaning that dynamic searation in sace is ossible even if static searation is not. B. Forced Diffusion Since the articles of interest are about a micrometer in diameter, Brownian motion and article-article interactions are significant and must be accounted for. To achieve this, we will introduce a diffusion force, given as the negative of the gradient of the chemical otential of the article at constant temerature. A balance between this diffusive force, the viscous drag force on a article, the gravitational force, and the force of magnetic origin gives (to leading order) 1 N a ( ζ ) T + 6πηa ( v -u ) + ρ V g µ 0 V M f H = 0, where N a is Avogadro s number, ζ is the chemical otential of the article, T is temerature, η is the viscosity of the fluid, a is the hydrodynamic radius of the article, u is the velocity of the article relative to a fixed reference frame, v is the volume-average velocity of the article, ρ is the density of the article, and g is the gravitational constant vector. Here the chemical otential is not a function of H since we have introduced the force of magnetic origin as an external force on the article, not as a force density on the system. The above exression can be rearranged to give u v = 1 N a ( ζ ) T + ρ V g µ 0 V M f H 6πηa (6). (7) Using the Stokes-Einstein equation for the diffusion coefficient results in u v = D ( ζ ) T ρ V g, (8) +µ 0 V M f H

4 where D denotes the diffusion coefficient of article in the ferrofluid f, R is the gas constant, and V is the molar volume of the article. We can now obtain the diffusive flux relative to the molar-average velocity, given by J C ( u v ) (9) = C D ( ζ ) T ρ V g +µ 0 V M f H, (10) where C is the molar concentration of the articles. This can be rearranged to give J = D C ( ζ ) T φ ρ g, (11) +φ µ 0 M f H where φ is the volume fraction of articles in the mixture. To interret the result obtained, let s examine (11) in the case were the chemical otential and gravity effects are negligible. For a small, fixed number of sherical articles n, we have that the volume fraction is roortional to a 3, the diffusion coefficient is roortional to the radius a, so the flux of articles relative to the fluid is roortional to a 2, as exected from a balance between the force of magnetic origin (roortional to a 3 ) and the drag force (roortional to a). It is this flux deendence on article size that makes dynamic, size-based searation of articles in sace ossible. In (11), a large ortion of the hysics involved in this forceddiffusion rocess are contained in the chemical otential ζ, which can be written as ζ = ζ 0 (T, P ) + ln x + ζ ex (T, P, x ). (12) Here ζ 0 is the chemical otential is a reference chemical otential at the temerature T and ressure P of interest, x is the mole fraction of articles, and ζ ex is the excess chemical otential. We can therefore write ( ζ ) T = V P + x x + ζ ex (T, P, x ). (13) For the rocess of interest to us, ressure dros due to fluid flow are negligible in comarison to hydrostatic ressure gradients, so we will therefore have P ρg. (14) Inserting these result back into (10) gives C x + C ζ ex (T, P, x ) J = D +(φ w )ρg +φ µ 0 M f H, (15) where C is the total concentration of the fluid. For our system of interest, the density of the olystyrene beads and the dilute magnetic fluid carrier are equal, so that the gravitational buoyancy term aearing in (15) (φ w )ρg is comletely negligible. Even when substantial density differences are resent, the gravitational buoyancy term in this exression is usually much smaller than the molecular diffusion term x for small articles. We thus have that, for our case of interest, J = D C x + C ζ ex (T, P, x ) +φ µ 0 M f H. (16) For an ideal system in the absence of magnetic fields, we see that J = CD x, (17) which is Fick s law for the molar flux relative to the molaraverage velocity [11]. The other terms in (16) account for flux due to nonidealities (such as article-article interactions) and magnetic effects. Finally, for a binary system, the molar flux relative to the mass-average velocity is given by J = W f C ρ J (18) = W f CD C x + C µ ex ρ, (19) +φ µ 0 M f H where W f is the molecular weight of the magnetic fluid solvent. Now that we have an exression for the molar flux, we can obtain the governing equation for the evolution of the concentration rofile of the articles. At constant density and in the absence of convection (no mass-average velocity), the concentration rofile is governed by C t + J = 0, (20) where the molar concentration and the mole fraction are related by C = x C and J is given by eq. (19). C. Nonidealities In our derivation, we have allowed for an excess chemical otential to describe effects that cannot be catured by ordinary diffusion. We have found exerimentally that ordinary diffusion is too small to comete with the magnetic effects, so there must be other mechanisms that are affecting the evolution of the concentration rofiles for the olystyrene beads. The ideal chemical otential assumes that the individual articles occuy zero volume and that there in no interaction between articles. Since we know that the olystyrene beads have a finite volume and that there is electrostatic reulsion between the beads, we assumed that the excess chemical otential should consist of mostly excluded-volume effects and electrostatic-reulsion effects. Furthermore, since volumeexclusion effects are so much smaller than electrostatic effects at the conditions of interest to us, we neglected volume exclusion and only focused on electrostatic effects. The total excess chemical otential of secies in solvent s was calculated to be ζ ex 16π 2 εψ 2 0a 2 κ 2 [1 + 2κa] N 2 acx, (21)

5 where we have used the fact that φ 1 and that the volume exclusion contribution is considerably smaller than the electrostatic contribution [7]. In this exression, ε is the electric ermittivity of the fluid, Ψ 0 is the electric otential at the surface of the articles, κ 1 is the Debye length, and N a is Avogadro s number.thus, we have that, at constant density, ζ ex = 16π2 εψ 2 0a 2 κ 2 [1 + 2κa] NaC 2 2 W s x. (22) ρ In deriving these exressions, we have assumed that the double-layer theory, which is strictly and electro-quasi-static concet, is still valid for our system. Since magnetic fluids are non-conducting, our magnetic fields do not vary in time, and the migration of the olystyrene beads is slow enough not induce any significant fields, the equations used and assumtions made in the double-layer theory are indeed valid. By introducing a constant defined as Fig. 4. Exerimental setu Ψ 2 0 = 16π2 εψ 2 0a 2 κ 2 [1 + 2κa] N 2 acw s ρ, (23) we can rewrite the molar diffusive flux relative to the massaverage velocity as ( ) J = W f CD C x 1 + Cx Ψ2 0 ρ. (24) +φ µ 0 M f H IV. EXPERIMENTAL SETUP The exerimental arrangement used in this research is shown schematically in Fig. 4. Fluorescently-tagged latex beads were used as the nonmagnetic articles. These beads emit fluorescent light when excited by light at the correct frequency. The excitation source was a 200-W mercury lam, and the desired excitation frequency was selected using aroriate filters. A digital camera was used to cature images of the emitted light, whose intensity is roortional to the concentration of beads. Thus, we were able to monitor concentration rofiles by measuring the intensity of the emitted light. The concentration of the magnetic fluid used in our exeriments was a dilute 1% by mass of magnetite, since a more concentrated magnetic fluid would be too oaque and block out olystyrene fluorescence. Fig. 5 shows a schematic reresentation of the exeriments erformed. A caillary tube with an inner diameter of 350 µm and an outer diameter of 500 µm was filled with a mixture of 1% by mass magnetite (stabilized by a olymer layer) and 0.4% by mass of fluorescently-tagged olystyrene beads. The remaining 98.6% of the mass consisted of deionized water and the olymer layer surrounding the magnetite articles. At time t = 0, two 12.7 mm diameter magnets where laced at a osition z = 0, with a 7 mm ga between the magnets. Two different sizes of olystyrene beads where used: 510 nm and 840 nm in diameter. The setu described in Fig. 4 was used to monitor the concentration rofile of the olystyrene beads Fig. 5. Magnetohoresis exeriment as a function of time. All of the exeriments were erformed at room temerature. V. RESULTS Exeriments were erformed using different bead sizes. Here we show and analyze the results obtained for 840 nm beads as well as 510 nm beads with the exerimental setu described in Fig. 5. The magnetic field rofile generated by the two magnets is given in Fig. 6, where z is the distance along the center of the caillary tube and z = 0 is the center of the two magnets (the lane of symmetry). A. Exerimental Results The exerimental data for the 840 nm and the 510 nm beads have been lotted in Fig. (7) and (8), resectively. The lots are normalized by the initial concentration of the olystyrene beads. The results are lotted at four different times: 100 s, 490 s, 1080 s, and 4630 s. As exected, the latex beads migrated away from the region of high magnetic fields and settled in the region where the magnetic field and its gradient vanished. Also as exected, the 840 nm beads migrated faster and reached a Fig. 6. Magnetic field rofile

6 TABLE II PARAMETERS USED IN MODEL Fig. 7. Exerimental concentration rofile for 840 nm articles PARAMETER R T VALUE J/(mol K) 298 K µ 0 4π 10 7 H/m ε Ψ 0 κ F/m V m M s 665 A/m H T A/m ρ 1050 Kg/m 3 η Kg/(m s) C mol/m 3 W s Kg/mol W Kg/mol W Kg/mol D m 2 /s D m 2 /s x x Fig. 8. Exerimental concentration rofile for 510 nm articles higher eak concentration, since the magnetic effects become more significant than the diffusive and electrostatic effects with increasing article size. If the exeriments were erformed over a larger eriod of time, the eak that reaches a maximum at about 4000 seconds would eventually decay toward the ositive z direction, since there is no magnetic force in this region and the only active terms in the molar flux are diffusion and electrostatic reulsion. However, due to the rather large size of the articles, this rocess would take days to comlete. B. Comarison with Model Equation (20) was solved in one satial dimension and time to redict the concentration rofiles for the exerimental conditions described above. The comuter rogram MATLAB was used to erform the numerical integration. The initial condition used was a constant concentration rofile for all z. The boundary conditions used were no flux at z = 0 and x = x (t = 0) at z = 15 mm. The latter condition was used since the exerimental results suggest that the article concentration will not vary significantly at this oint. A list of the arameters used is given in Table 2. Since the volume fraction of latex beads never exceeds 0.02 (5 times the original concentration) and since the magnetic fluid only contains 0.01 weight fraction of magnetite, we assumed that the concentration and viscosity of the mixture can be aroximated by the concentration and viscosity of ure water, resectively. The densities of the magnetic fluid and the latex beads are both 1050 Kg/m 3, so the density of the mixture is also 1050 Kg/m 3. The molecular weight of the olystyrene beads was calculated from the weight of beads of the given diameter, and the diffusion coefficients were calculated using the Stokes- Einstein equation, D = 6πηaN a. (25) The results of the numerical integrations are lotted at the times of interest for the 840 nm and the 510 nm articles in Fig. (9) and (10), resectively. C. Discussion As we can see from comaring the results obtained from our diffusion model to the exerimental data, the model redicts the correct trends and the correct height of the concentration eak. This is very encouraging since the model does not contain any adjustable or fitted arameters. The exressions used for the magnetic force and the electrostatic reulsion are only leading-order aroximations. Several of the discreancies observed between the exeriments and the model can be exlained by examining the aroximations made in deriving the model. First, we assumed that the concentration of the magnetic fluid would not vary at all during the exeriment. This aroximation is not strictly valid in the neighborhood of z = 0 due to the high magnetic forces resent there. We also neglected any interactions between the magnetic nanoarticles and the latex beads. Since the magnetic nanoarticles are stabilized with a olymer that will have a net negative charge (the latex beads also have a negative charge when susended in water), these interactions may become significant. Finally, the

7 a better fit with the exerimental data, some of our simlifications need to be relaxed, and the exerimental setu needs to be modified in order to obtain more accurate exerimental data. ACKNOWLEDGMENT The authors would like to thank the Singaore-MIT Alliance (SMA) for artial financial suort of this research. Fig. 9. Fig. 10. Predicted concentration rofile for 840 nm articles Predicted concentration rofile for 510 nm articles magnetic fluid nanoarticles are not comletely monodiserse, so in ractice the larger magnetic nanoarticles will tend to concentrate in the vicinity of z = 0. The combination of these effects may exlain the discreancies observed in the neighborhood of z = 0. We can clearly see form our results that our model overredicts the thickness of the concentration eak. Our model only accounts for diffusion and electrostatic effects, and only to leading order. We will need to account for other tyes of interactions to obtain a better fit between the exerimental and the redicted results. The exerimental data also shows a di in the concentration rofile to the right of the concentration eak. This is not redicted by the model and we do not know if this is a real effect or an error in exerimental measurements. In our exerimental setu, the light from the mercury lam shines on the samle at an angle, and it is ossible that the high concentration of beads at the eak will block the light and revent it from reaching the fluorescently-tagged latex beads just to the right of the eak. We are develoing a new setu that uses a dichroic mirror to deflect the light coming from the mercury lam so that it shines on our samle at a right angle. In summary, our model is able to redict the correct trends in the concentration rofile of olystyrene beads and the correct numerical value for the concentration eaks. To obtain REFERENCES [1] R. E. Rosensweig, Material searation using ferromagnetic liquid techniques, U.S. atent , [2] J. Shimoiizaka, K. Nakatsuka, T. Fujita, and A. Kounosu, Sink-float searators using ermanent magnets and water based magnetic fluid, IEEE Trans. Magnetics, vol. MAG-16 (2), , [3] M. Berger, J. Castelino, R. Huang, M. Shah, and R. H. Austin, Design of a microfabricated magnetic searator, Electrohoresis, vol. 22, , Oct [4] J. Choi, T. M. Liakooulos, and C. H. Ahn, An on-chi magnetic bead searator using siral electromagnets with semi-encasulated ermalloy, Biosensors and Bioelectronics, vol. 16, , Aug [5] P. A. Liberti and B. P. Feeley, Microtitre well cell-searation with biorecetor ferrofluids in high-gradient magnetic searations, FASEB Journal, vol. 5 (6), a1690 art 3, [6] S. Roath, R. Smith, and J. H. P. Watson, High-gradient magnetic searation in blood and bone-marrow rocessing, J. Magn. Magn. Mater., vol. 85 (1-3), , Ar [7] S. Fateen, Magnetohoretic focusing of submicron articles disersed in a olymer-based magnetic fluid, Ph.D. dissertation, Det. Chem. Eng., Massachusetts Institute of Technology, Cambridge, MA [8] H. Watari and N. Makoto, Magnetohoretic Behavior of Single Polystyrene Particles in Aqueous Manganese(II) Chloride, Anal. Sci., vol. 17, , Oct [9] R. E. Rosensweig, Ferrohydrodynamics. New York, NY: Cambridge University Press, 1985, [10] G. Moeser, K. A. Roach, W. H. Green, P. E. Laibinis, and T.A. Hatton, Ind. Eng. Chem. Res., vol. 41 (19), , Se [11] W. M. Deen, Analysis of Transort Phenomena. New York, NY: Oxford University Press, 1998,. 4-5.