Electrokinetic Lift Effects Observed in the Transport of Submicrometer

Langmuir 1996, 12, 613-623

613

Electrokinetic Lift Effects Observed in the Transport of Submicrometer Particles through Microcapillary Tubes Andrew D. Hollingsworth and Cesar A. Silebi* Department of Chemical Engineering and Emulsion Polymers Institute, Lehigh University, Bethelehem, Pennsylvania 18015 Received May 26, 1995. In Final Form: September 14, 1995X Experimental evidence for the electrokinetic lifting of submicrometer colloidal particles in laminar flow through microcapillary tubes is presented. Electrokinetic lift is a new colloidal force originating when a charged particle and the diffuse part of the electrical double layer surrounding it are made to move relative to each other in the presence of another surface. This force occurs in the absence of an applied external field and is comparable to double-layer repulsion for low conductivity fluids. Our measurements of the average residence time of monodisperse polystyrene latexes being pumped under laminar flow through microcapillaries have shown a strong flow rate dependence using eluants of less than 10-3 M ionic strength. This behavior was most noticeable at the lowest electrolyte concentration (3 × 10-6 M) using a surfactantfree eluant. Deviation from theoretically-predicted separation factors (a parameter equal to the ratio of the mean displacement velocities of particle and eluant) became more pronounced with increasing particle size and decreasing tube diameter. The mathematical model which was developed from first principles predicts that particles in this size range will not be influenced significantly by hydrodynamic forces, thus precluding any dependence of the average particle velocity on the eluant flow rate. A repulsive force of electrokinetic origin may be responsible for the anomalous data which show good qualitative agreement with theoretical results when we incorporate a recently derived lift force expression into the model.

Introduction The transport of colloidal particles suspended in laminar tube flow is of central importance to the determination of particle size and size distribution using capillary hydrodynamic fractionation (CHDF), a recently developed analytical method for particle size characterization in the submicrometer size range.1,2 It consists of a hydrodynamic flow fractionation process in which colloidal dispersions are pumped under laminar flow conditions through an open-bore microcapillary tube, eluting in order of decreasing particle diameter. Various turbidimetric techniques can be used to monitor the optical densities of the fractionated species which, together with the elution time data, are analyzed to calculate average particle size and particle size distribution.3,4 The fractionation step results in much higher resolution than would otherwise be possible in determining submicrometer particle size distributions, particularly if the sample is multimodal or has a broad size distribution. This technique has also been used to obtain analytical separations of macromolecules in solution.5,6 Bos et al.7 determined the dissociation temperature of organic micelles using capillary hydrodynamic fractionation with an on-column detector. Re* To whom correspondence should be addressed at the Department of Chemical Engineering, Iacocca Hall, Lehigh University, Bethelehem, PA 18015. X Abstract published in Advance ACS Abstracts, December 15, 1995. (1) Silebi, C. A.; DosRamos, J. G. J. Colloid Interface Sci. 1989, 130, 14. (2) DosRamos, J. G.; Silebi, C. A. J. Colloid Interface Sci. 1989, 133, 302. (3) DosRamos, J. G.; Silebi, C. A. J. Colloid Interface Sci. 1990, 135, 165. (4) DosRamos, J. G.; Silebi, C. A. In Particle Size Distribution Assessment and Characterization II; Provder, T., Ed.; ACS Symp. Series No. 472; American Chemical Society: Washington, DC, 1991; p 292. (5) Tijssen, R.; Bos, J.; van Kreveld, M. E. Anal. Chem. 1986, 58, 3036. (6) Sugarman, J. H.; Prud’homme, R. K.; Langshorst, M. A.; Stanley, F. W. J. Appl. Polym. Sci. 1987, 33, 693. (7) Bos, J.; Tijssen, R.; van Kreveld, M. E. Anal. Chem. 1989, 61, 1318.

0743-7463/96/2412-0613$12.00/0

cently, Chun et al.8 developed a theoretical model to predict the concentration depletion profile of rodlike polymers in a CHDF system. Relative elution times of particles are dependent on particle size, tube internal diameter, surfactant species and concentration, and the average velocity and ionic strength of the eluant being pumped through the capillary.1 The mechanism of size separation is due to the laminar velocity profile of the fluid inside the conduit and the steric exclusion of the particles from the slower velocity streamlines near the capillary wall. In addition to these two effects, CHDF performance is also affected by the radial forces acting on the particles due to the fluid inertial effect9,10 and the electrostatic repulsion and van der Waals attraction between particles and the wall of the capillary. The extent to which one or more of these forces is dominant depends upon the operating conditions of CHDF. A mathematical model which has been developed from first principles without the use of any adjustable parameters is capable of predicting both the average particle velocity as well as the degree of axial dispersion of colloidal particles, i.e. the efficiency of separation of different sized particles.2,11 At low eluant ionic strength, however, poor prediction of the separation behavior has been reported by Venkatesan.12 This result was observed when nonionic polymeric surfactants were dissolved in the eluant. The adsorption of long chain molecules on the capillary wall and particle surfaces reduces the tube internal diameter while increasing the effective particle size. Larger separation factors (a dimensionless parameter representing the relative velocity of different sized particles with respect to the eluant) are typically observed under these conditions.1,13 Similar results have been reported in flow-through packed column hydrodynamic (8) Chun, M.-S.; Park, O. O.; Yang, S.-M. J. Colloid Interface Sci. 1993, 161, 247. (9) Segre´, G.; Silberberg, A. Nature (London) 1961, 189, 209. (10) Segre´, G.; Silberberg, A., J. Fluid Mech. 1962, 14, 115, 136. (11) Silebi, C. A.; DosRamos, J. G. AIChE J. 1989, 35, 1351. (12) Venkatesan, J. Ph.D. Dissertation, Lehigh University, 1992. (13) deJaeger, N. C.; Trappers, J. L.; Lardon, P. Part. Charact. 1986, 3, 187.

© 1996 American Chemical Society

614

Langmuir, Vol. 12, No. 3, 1996

chromatography.14 Venkatesan’s systematic investigation of eluant composition effects on the separation factor in CHDF indicated that nonionic surfactants with both a low critical micelle concentration (cmc) and low solution conductivity yielded optimal separation performance and that the separation factor could be maximized near the cmc. The latter observation was attributed to the presence of an osmotic repulsive force which together with steric interaction was believed to account for anomalous hydrodynamic measurements of the adsorbed layer thickness.15 The proposed osmotic pressure mechanism is related to the depletion stabilization of colloids by free polymer;16,17 to date, no experimental evidence for these relatively short-range interaction forces has been reported. Recent experimental and theoretical work suggests that a repulsive force of electrokinetic origin may be operative in low Reynolds number flow. The force, termed electrokinetic lift by Prieve and Bike,18 may explain the anomalous experimental results of Prieve and Alexander19 and Alexander and Prieve20 which were obtained when attempting to measure the potential interaction energy between a single colloid particle and a flat plate. The apparent velocity-dependent lift force was present only in the more viscous fluids making it inconsistent with an inertial lift mechanism. It is comparable to double-layer repulsion for low conductivity fluids and is predicted to increase with decreasing fluid conductivity and increasing particle velocity. In the CHDF system, an electrokinetic lift force would be radially directed toward the tube axis, pushing the charged particles away from the fused silica surface in the microcapillary. Larger separation factors and reduced axial dispersion would result under such conditions. In the absence of an applied external field, electrokinetic phenomena occur when a charged surface and the diffuse part of the electrical double layer (EDL) are made to move relative to each other.21 Here, electrical effects arise solely from the deformation of the EDL by the primary fluid motion. For simple shear flow, electrokinetic lift of colloidal particles may be due to an induced streaming potential in the particle-wall gap18,22,23 or the asymmetry of the electric field caused by the presence of the wall.24 Lift force expressions were derived for translational particle motion in both cases. A general electrokinetic lift theory for charged particles in simple shear flow has been published by Bike;25 however, the applicability of the equilibrium double layer assumption (Pećlet number , 1) made in her analysis may not be valid for the low conductivity conditions in CHDF. At low electrolyte concentrations, the Debye double-layer thickness is comparable to the particle radius and the EDL is likely to deform under the influence of shear flow.26 Additionally, the assumption of linear shear flow may be inappropriate (14) Small, H.; Saunders, F. L.; Solc, J. Adv. Colloid Interface Sci. 1976, 6, 237. (15) Venkatesan, J.; DosRamos, J. G.; Silebi, C. A. In Particle Size Distribution Assessment and Characterization II; Provder, T., Ed.; ACS Symp. Series No. 472; American Chemical Society: Washington, DC, 1991; p 279. (16) Feigin, R. I.; Napper, D. H. J. Colloid Interface Sci. 1980, 75, 525. (17) Clark, A. T.; Lal, M. J. Chem. Soc., Faraday Trans. 1981, 77, 981. (18) Prieve, D. C.; Bike, S. G. Chem. Eng. Commun. 1987, 55, 149. (19) Prieve, D. C.; Alexander, B. M. Science 1986, 231, 1269. (20) Alexander, B. M.; Prieve, D. C. Langmuir 1987, 3, 788. (21) Saville, D. A. Ann. Rev. Fluid Mech. 1977, 9, 321. (22) Bike, S. G.; Prieve, D. C. J. Colloid Interface Sci. 1990, 136, 95. (23) Bike, S. G.; Prieve, D. C. J. Colloid Interface Sci. 1992, 154, 87. (24) van de Ven, T. G. M.; Warszynski, P.; Dukhin, S. S. J. Colloid Interface Sci. 1993, 157, 328. (25) Bike, S. G. Ph.D. Dissertation, Carnegie-Mellon University, 1988. (26) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: London, 1988; Chapter 3.

Hollingsworth and Silebi

for particles in CHDF traveling near the tube axis where the parabolic nature of the flow field becomes important. Velocity-induced lift forces have been postulated to account for anomalous retention in steric field flow fractionation of micrometer size latexes.27,28 Williams et al.29 examined data collected from sedimentation fieldflow fractionation (SFFF) experiments in their attempt to characterize hydrodynamic lift forces occurring in the vicinity of the channel wall. Multiple linear regression analysis resulted in an expression for the lift force which is inconsistent with fluid inertial effects and also electrokinetic lifting. Their observation that lift force increased with viscosity using modified carrier fluids in the SFFF experiment is consistent with Alexander and Prieve’s20 observation of greater particle-wall separation with higher fluid viscosity; however, Williams et al.29 propose that the observed lift force in SFFF is related to lubrication phenomena. The purpose of our investigation was to examine the effects of low ionic strength eluant on particle transport in capillary hydrodynamic fractionation and to provide an explanation for the anomalies that have characterized recent work using nonionic surfactants dissolved in the eluant. In this contribution we present what we believe to be the first experimental evidence for electrokinetic lifting of colloidal particles in tube flow. Related to this study are particle-particle interaction effects which are expected to become significant as particles are forced toward the tube axis and hence closer together. The results of this work are of potential importance in other applications such as field-flow fractionation, the rheology of dispersions, tribology, and particle capture and interaction. Experimental Section The CHDF system used in this study is an experimental unit built in our laboratory.1,30 Eluant was pumped from a reservoir using a positive displacement type metering pump equipped with a pulse dampener into a low dispersion injector fitted with a position-sensing switch, a fused silica capillary tube, and a multiwavelength detector with a 15 µL fluid cell. The average internal diameter of the capillary tubes was calculated hydrodynamically using the Hagen-Poiseuille law31 which is justified by the low Reynolds number conditions in CHDF.1 The capillary tube length was determined from the minimum residence time required for full development of the radial concentration profile.32 In order to minimize dead volume effects associated with the low flow rates used, sample splitting at the entrance and eluant makeup at the exit of the capillary were used. A constant flow rate pump was used to deliver the make-up stream of distilled deionized (DDI) water to the detector cell. The detector output (fractogram) was monitored at various wavelengths in the UV range of interfaced with a microcomputer and chart recorder for data acquisition. The detector wavelength was chosen such that the strongest peak would be obtained at a minimum noise to signal ratio; monitoring several wavelengths was useful for turbidimetric analysis. Eluant solutions of a nonionic water-soluble surfactant belonging to the poly(ethylene oxide)/alkyl ether class of compounds (BRIJ35 SP, ICI Americas, Inc.) were used at concentrations above the critical micelle concentration. To remove the (27) Caldwell, K. D.; Nguyen, T. T.; Myers, M. N.; Giddings, J. C. Sep. Sci. Technol. 1979, 14, 935. (28) Giddings, J. C.; Myers, M. N.; Caldwell, K. D.; Pav, J. W. J. Chromatogr. 1979, 185, 261. (29) Williams, P. S.; Koch, T.; Giddings, J. C. Chem. Eng. Commun. 1992, 111, 121. (30) DosRamos, J. G. Ph.D. Dissertation, Lehigh University, 1988. (31) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960; Chapter 2. (32) DosRamos, J. G.; Jenkins, R. D.; Silebi, C. A. In Particle Size Distribution Assessment and Characterization II; Provder, T., Ed.; ACS Symp. Series No. 472; American Chemical Society: Washington, DC, 1991; p 264.

Electrokinetic Lift of Submicrometer Particles

Langmuir, Vol. 12, No. 3, 1996 615 situation never occurs in practice due to the retarding effect of the capillary wall on the particle velocity.

Table 1. Polystyrene Latex Standards latex designation

diameter, µm

std dev, µm

LS-1132-B LS-1044-E LS-1045-E LS-1047-E LS-1010-E LS-1115-E LS-1117-B LS-1166-B

0.091 0.109 0.176 0.234 0.357 0.610 0.794 1.101

0.0058 0.0027 0.0023 0.0026 0.0056 0.0090 0.0044 0.0055

CHDF Model The most important result of the Dos Ramos and Silebi2 analysis of particle transport by CHDF is an expression for the average particle velocity in the axial direction

∫0R -R vpz(r)e-E(r)r dr 〈vpz〉 ) ∫0R -R e-E(r)r dr o

residual ionic contamination from this surfactant, a concentrated solution was treated by a mixed-bed ion exchange resin. The ionic strength was then adjusted with the addition of a small quantity of sodium chloride. Dilute NaCl solutions were also used as eluants. Sodium lauryl sulfate (anionic surfactant) solutions were prepared at 5 × 10-3, 1 × 10-3, and 1 × 10-4 M concentrations. The water used in this study was distilled and then cleaned by passing it through ion exchange columns and particulate filters. The measured specific conductance of the DDI water ranged from 0.3 to 0.4 µS‚cm-1. The ionic strength of the high-purity water and all solutions at or below 10-4 M was calculated from the measured solution conductivity and pH assuming the electrolyte to be composed of H+, OH-, Na+, and Cl- ions. All eluants were filtered through a 0.22 µm membrane and an in-line filter installed upstream of the injection valve. A series of uniform polystyrene latexes manufactured by the Dow Chemical Co. were used in this investigation. Average particle size and standard deviation as determined by electron microscopy appear in Table 1. All latex standards were cleaned using an ion-exchange procedure to remove excess surfactant and ionic contaminants. When uncleaned latexes were injected into the low ionic strength eluants, we routinely observed longer particle residence times than those measured using the cleaned standards. We speculated that particle transport may have been retarded by the high conductivity fluid in which the latex particles were dispersed. For the smaller particle sizes, two fractogram peaks were frequently associated with an uncleaned standard. The two peaks were found to have the same ratios of turbidity at two wavelengths of UV light, implying that they represent the same size particle. Experiments were conducted at the lowest possible particle concentrations to avoid the influence of particle-particle interaction on the separation factor. This procedure was also followed by Okubo33 to minimize the influence of particle concentration effects in the measurement of sedimentation velocities in completely deionized suspensions. Prior to injection into the CHDF system, the samples were diluted to the appropriate weight fraction using a nonionic surfactant solution (0.05% (w/w) BRIJ35 SP). For the lowest ionic strength eluants, a solids content of 0.001% was necessary for the smallest particle sizes. Samples were sonicated before their injection in order to break up any aggregates that may have formed during the dilution. Particle transport through the microcapillary tube was quantified experimentally by the ratio of the capillary length to the particles’ average residence time in the eluant stream. For a symmetrical fractogram, the average residence time corresponds to the peak elution time. Asymmetrical fractograms required the calculation of the centroid of the distribution which was necessary in some cases.34 The average eluant velocity was measured using a UV-absorbing marker species (sodium benzoate). The dimensionless parameter referred to as the separation factor, Rf, represents the relative velocity of different sized particles with respect to the eluant:

Rf ) tm/tp

(1)

where tm and tp represent the average residence times of the marker and the colloidal particles, respectively. Separation factors are always greater than unity indicating that on an average basis, the particles travel faster than the eluant. The maximum value of the separation factor is 2, which occurs when particles travel at the centerline position in the tube. This (33) Okubo, T. J. Phys. Chem. 1994, 98, 1472. (34) Dyson, N. A. Chromatographic Integration Methods; Royal Society of Chemistry: Cambridge, 1992; Chapter 1.

p

o

(2)

p

where E(r) represents the total interaction potential normalized with respect to thermal energy kT

E(r) )

1 [Φ(r) kT

∫0rF(r) dr]

(3)

and Φ(r) and F(r) represent the colloidal interaction potential and the sum of fluid inertial and electrokinetic lift forces, respectively. Here, we have included an expression for electrokinetic lift which does not appear in the original CHDF model. The average particle velocity is calculated by weighting the local particle velocity at a given radial position with the concentration at that position (as determined by a probability density function) and integrating over the cross section of the tube accessible to the particle. The upper integration limit in eq 2 represents the difference of the tube and particle radii, accounting for the size exclusion layer adjacent to the wall. Using experimentally measured parameters (eluant ionic strength, particle diameter, and eluant flow rate) together with specific material parameters, eq 2 is numerically integrated to obtain the theoretical value of Rf. The local particle velocity is a function of radial position in the tube and is given by

( ( )) r Ro

vpz ) 2vm 1 -

2

- vps

(4)

where vm is the mean fluid axial velocity and vps is the particle slip velocity. Here, r/Ro represents the ratio of the radial position of the particle’s center of mass to the capillary radius. The slip velocity of a particle suspended in tube flow depends on its radial position, the ratio of the particle to tube radii, κ (tRp/Ro), and the mean fluid velocity. The expression for this velocity term is given by

vps )

( )[(

4 r v κ2 + 5vmκ3 3 m Ro

1-

)(

r Ro

2

1+

)]

r Ro

-1

(5)

The colloidal interaction potential is a result of doublelayer repulsion and van der Waals attraction between the particle and the capillary wall (Φ(r) ) ΦDL + ΦvW). Due to the small particle to capillary ratios, it is calculated assuming a sphere-plane interaction. The appropriate expressions are35,36 (35) Bell, G. M.; Levine, S.; McCartney, L. N. J. Colloid Interface Sci. 1970, 33, 335. (36) Oshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 11.

616


( )

32 ΦDL )

( )


( ) ( ) ( )]

eψ1 eψ2 a tanh R exp 4kT 4kT p κD eψ2 0.5 2Rp/κD + 1 2 1+ 1tanh 4kT (Rp/κD + 1)2 (6)

kT e

2

tanh

[

and37

ΦvW ) -

AHRp 6a

1 14a 1+ λ

( ) kT 8πe2ci

0.5

(8)

The use of eq 6 in our model requires knowledge of the particle and capillary surface potentials. Although the potential at the shear surfaces differs from that on the respective surfaces due to layers of adsorbed ions and molecules, we have identified the zeta potential with the surface potential. This assumption is justified by the fact that the colloidal interaction potential (and hence Rf) is insensitive to the values of ψi chosen at low electrolyte concentrations. The radial inertial force, FH, is responsible for driving the particles toward an eccentric equilibrium position in the capillary tube. This term can be calculated using Stoke’s drag force equation with a correction of O(κ) due to the presence of the wall,

(

(

9 r FH ) 6πRp µvpr 1 + κ 1 8 Ro

)) -1

(9)

where µ is the eluant viscosity and vpr represents the particle’s radial migration velocity. For a freely rotating, neutrally buoyant particle in a tube flow, the radial velocity is given by38,39

vpr )

[ ( )

(

vpr ) 0.7 vm

(7)

where is the eluant dielectric constant, e is the protonic charge, a is the minimum separation distance between particle and the capillary surface, Ro - r - Rp, ψi is the surface potential for the capillary, i ) 1, or particle, i ) 2, κD is the Debye double-layer thickness, AH is the Hamaker’s constant, and λ is the characteristic interaction wavelength. Born and steric repulsive interactions represent very short range forces and are not included in the model. The double layer thickness may be determined in terms of the ionic concentration in the eluant phase, ci, by

κD )

functions which were substituted into eq 10 to determine vpr. The use of eq 10 requires Rep , Rp/Ro , 1, with the particle Reynolds number given by Rep ) Rpvps/ν. This condition is easily satisfied in our experiments using particles in the 0.09-0.36 µm diameter size range. For one set of data, we also used the following empirically derived equation40 for the radial particle velocity in our CHDF model

( ) ( ( ) ( ))]

6πRp r r U∞2h - U∞vmg + ν Ro Ro

r r 5 2 2 + f2 v κ f1 9 m Ro Ro

(10)

where ν is the kinematic viscosity of the fluid and U∞ is the sedimentation velocity of the particle in an unbounded fluid. Following Dos Ramos and Silebi,2 we have substituted the axial slip velocity when wall effects are not negligible for U∞. The functions h, g, f1, and f2 represent volume integrals containing the Green’s function for creeping flow in a circular tube; we have used eqs 32, 33, and 34 in Dos Ramos and Silebi2 to represent these (37) Gregory, J. J. Colloid Interface Sci. 1981, 83, 138. (38) Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1968, 23, 147. (39) Ishii, K.; Hasimoto, H. J. Phys. Soc. Jpn. 1980, 48, 2144.

) ( )(

2Rovm ν

0.5

)

r r r* Ro Ro

κ2

(11)

where r* represents the equilibrium radial position given by41 r* ) 0.67(1 - κ). The electrokinetic lift force expression for translational motion of a spherical particle near a planar surface has been derived by Bike and Prieve.23 For a cylindrical geometry, the appropriate expression is

Fek )

( ) ( )( 4π

3

27π vpzκ 16 KRo

2

1-

)

r Ro

-4

ζ2(ζ2 + 2ζ1)

(12)

where Fek is the electrokinetic lift force, ζ1 and ζ2 are the zeta potentials of the wall and particle respectively, and K is the fluid conductivity. This theoretical result is based on the following assumptions: 1. Both the particle and wall surfaces bear EDL’s whose thicknesses are small compared to both the radius of the particle and the separation distance between the surfaces, i.e., κD , Rp , l where l is the distance from the wall to the sphere center. 2. The distribution of ions and charge within the EDL are not perturbed significantly by flow, i.e., Pe , 1. The Pećlet number, Pe, is defined as the ratio of ion convection by bulk flow to the Brownian (thermal) diffusion of ions in the EDL: Pe ) vp κD(ωkT)-1, where ω ) (6πµai) is the average ion mobility based on ai, the mean ion radius, and vp is a typical particle velocity. Note that for thin double layers, the appropriate length scale is κD, not Rp.42-44 Values of the zeta potential used were based on Hlatshwayo’s45 measurements of the electrophoretic mobilities of the polystyrene latex standards in the presence of 10-5 M SLS surfactant in a 25 µm i.d. capillary. Electroosmotic velocities were also measured for various SLS and NaCl concentrations using a 25 µm i.d. capillary and used to estimate the zeta potential of the capillary wall.45 Results and Discussion Separation factors were measured as a function of particle diameter, eluant average velocity, and eluant ionic strength using two microcapillary tubes of 9.0 and 25.6 µm average internal diameter. All experimental data were compared with theoretical separation factors using the mathematical model described in the previous section. The dimensionless product of the particle Reynolds number (Rep ) 2Rpvps(0)/ν) and the Pećlet number (Pe ) 2Rovm/D) was used to establish the conditions under which fluid inertial forces become significant in CHDF.2 Our calculations show that the separation factor will increase 1% above that value obtained in the absence of inertial forces when RepPe ) 3, 9, and 10 for electrolyte concentrations of 10-3, 10-4, and 10-5 M, respectively. Table 2 (40) Jeffrey, R. C.; Pearson, J. R. A. J. Fluid Mech. 1965, 22, 721. (41) Walz, D.; Grun, F. J. Colloid Interface Sci. 1973, 45, 467. (42) Russel, W. B. J. Fluid Mech. 1978, 85, 673. (43) Lever, D. A. J. Fluid Mech. 1979, 92, 421. (44) Sherwood, J. D. J. Fluid Mech. 1980, 101, 609. (45) Hlatchwayo, A. B. Ph.D. Dissertation, Lehigh University, 1994.


Langmuir, Vol. 12, No. 3, 1996 617

Table 2. Critical Velocities for 9.0 and 25.6 µm Diameter Capillaries Dp ) 0.091 µm

Dp ) 0.109 µm

Dp ) 0.176 µm

Dp ) 0.234 µm

Dp ) 0.357 µm

κ ) 0.010 κ ) 0.0036 κ ) 0.012 κ ) 0.0043 κ ) 0.020 κ ) 0.0069 κ ) 0.026 κ ) 0.0091 κ ) 0.040 κ ) 0.0140 v*, cm/s (10-3 M) v*, cm/s (10-4 M) v*, cm/s (10-5 M)

35.8 62.0 65.3

60.4 104.5 110.2

24.9 43.2 45.5

42.1 72.9 76.8

Figure 1. Theoretical calculation showing the effect of eluant average velocity, 〈ve〉, on the separation factor at high and low eluant ionic strengths: capillary radius, 4.5 µm; Fek ) 0.

presents the critical eluant velocity, v*, for ten ratios of particle to tube radius at these ionic strengths. Most of our experiments were performed at average eluant velocities smaller than the corresponding critical values, above which separation factors would be expected to show flow rate dependence. When the colloidal interaction force is large enough to push particles beyond the equilibrium radial position, it will be opposed by hydrodynamic forces directed toward the tube wall. Therefore, increasing the eluant flow rate could, under certain conditions, result in smaller separation factors as shown in Figure 1. Here, all lines represent simulated separation factors as a function of particle size for a 9 µm i.d. capillary. The upper and lower set of curves correspond to eluants typical of those used in our experiments; also shown is the case when the hydrodynamic force is not accounted for in the CHDF model. As the average eluant velocity increases from 2.5 to 10 cm/s, the theoretical separation factor increases for the 50 mM eluant, while just the opposite trend is predicted for the low ionic strength fluid. Electrokinetic lift was not included in this simulation. Figure 2 compares the colloidal interaction force, FC, with the hydrodynamic lateral force, FH, and electrokinetic lift force, Fek, as functions of radial position in the capillary tube. The ratio of particle to tube radius specified corresponds to one of the largest particle sizes and the smaller capillary inner diameter used in our experiments. This combination can result in a relatively large hydrodynamic force at high eluant flow rates. The low electrolyte concentration represents the experimental condition at which we observed a large deviation from theoretically-predicted separation factors (setting Fek ) 0 in the model). Assuming the energy of interaction is related to the interaction force by

FC ) -dΦ/dr

(13)

and using eqs 6 and 7, we calculated FC for an eluant ionic strength of 10-5 M. Electrostatic repulsion dominates the van der Waals attraction force for this eluant ionic strength. Equations 9, 10, and 12 were used to calculate the hydrodynamic and electrokinetic lift forces at two

9.6 16.6 17.5

16.1 27.9 29.5

5.4 9.4 9.9

9.1 15.8 16.7

2.3 4.1 4.2

3.9 6.8 7.2

Figure 2. Theoretical calculation comparing the absolute value of the colloidal interaction (Fc), fluid inertial (Fh), and gravitational (Fg) forces with translational electrokinetic lift force (Fek) in CHDF. The scaling is such that the sphere’s net weight equals unity. Relevant parameters are as follows: ) 78.5; ζwall ) 100 mV; ζparticle ) 100 mV; Ro ) 4.5 µm; Rp ) 0.179 µm; Hamaker constant, 5 × 10-21 J; T ) 298 K; characteristic interaction wavelength, 100 nm; specific conductance, 1.5 µS/ cm; µ ) 0.001 Pa‚s.

eluant average velocities representative of our experiments. The cusp observed in three of these curves represents the radial position at which the force changes sign; to the right of this point, forces are negatively-valued and directed away from the capillary wall. In the case of FH, this particle radial position represents the familiar “tubular pinch point”. At the very low electrolyte concentration and higher flow rate, |Fek| . |FC| > |FH| for radial positions from r* to 0.85; below the equilibrium position, FH changes sign and is directed away from the tube axis. This behavior is consistent with our experimental observations. Effect of Eluant Composition. The effect of eluant composition on the separation factor is shown in Figures 3-7 for various particle sizes and two capillary inner diameters. The eluants studied contained either an anionic surfactant (Figure 3), a nonionic surfactant (Figures 4 and 5), or no surfactant species (Figures 6 and 7). Separation factors obtained using a 50 mM SLS eluant show good agreement with the theoretically-predicted values. At the lower SLS concentrations, we observed a significant deviation from the simulated separation factors for the larger particle sizes. When an empirical relation for radial particle velocity (eq 11), together with the Walz and Grun41 expression for the equilibrium radial position, were incorporated into the CHDF model, we were able to fit the data in the 0-0.40 µm size range; however, the prediction of separation factors for larger particle sizes was not good, and almost none of the 50 mM SLS data agreed with the simulated values. From this result, we concluded that these empirical expressions result in the overestimation of the fluid inertial forces in our system and that the Ishii-Hasimoto39 radial velocity equation (eq 10) is valid for these experimental conditions. The anomalous measurements appear to be due to the electrolyte concentration rather than hydrodynamic effects since the deviation from the corresponding theoretical values grows stronger with decreasing ionic strength and

618


Figure 3. Comparison between predicted and measured separation factor-particle diameter behavior for three sodium lauryl sulfate (SLS) surfactant concentrations: solid line, Rf simulation using the particle radial velocity given by eq 10; - - -, Rf simulation using eq 11 and the Walz and Grun41 expression for equilibrium radial position, r*. Eluant average velocity, 2.0 cm/s; capillary radius, 4.8 µm; Fek ) 0.


Figure 5. Comparison between predicted and measured separation factor-particle diameter behavior for 0.02% (w/w) BRIJ35 SP eluant: solid line: Rf simulation with Fek ) 0; - - -, Fek term included in Rf simulation using ζwall ) 100 mV, ζparticle ) 50 mV; ‚‚‚, Fek term included in Rf simulation using ζwall ) 100 mV, ζparticle ) 100 mV. Eluant average velocity, 1.6 cm/s; capillary radius, 4.5 µm; estimated ionic strength, 7.0 × 10-6 M; specific conductance, 1.0 µS/cm.

Figure 4. Comparison between predicted and measured separation factor-particle diameter behavior for 0.01% (w/w) BRIJ35 SP and NaCl eluant: solid line, Rf simulation with Fek ) 0; - - -, Fek term included in Rf simulation using ζwall ) 100 mV, ζparticle ) 50 mV; ‚‚‚, Fek term included in Rf simulation using ζwall ) 100 mV, ζparticle ) 100 mV. Eluant average velocity, 2.1 cm/s; capillary radius, 4.5 µm; estimated ionic strength, 2.3 × 10-5 M; specific conductance, 3.0 µS/cm.

Figure 6. Comparison between predicted and measured separation factor-particle diameter behavior for DDI water eluant: solid line, Rf simulation with Fek ) 0; - - -, Fek term included in Rf simulation using ζwall ) 100 mV, ζparticle ) 50 mV; ‚‚‚, Fek term included in Rf simulation using ζwall ) 100 mV, ζparticle ) 100 mV. Eluant average velocity, 4.2 cm/s; capillary radius, 12.8 µm; estimated ionic strength, 4.0 × 10-6 M; specific conductance, 0.44 µS/cm.

because all data were measured at about the same flow rate. To ensure that fluid inertial forces would be minimal, we restricted subsequent experiments to the 0-0.40 µm particle size range. From the results presented in Figures 4-7, polymeric molecules dissolved in the eluant at these electrolyte concentrations have no significant effect on Rf. Rather, the eluant ionic strength and/or fluid conductivity appear to strongly influence particle transport through the microcapillary tube. As the eluant conductivity was reduced from 3.0 to 0.44 µS‚cm-1, the deviation between theoretical and measured separation factors grew progressively stronger when we did not use an electrokinetic lift force expression to predict Rf. Separation factors as large as 1.7 to 1.8 for particle to tube diameter ratios of less than 0.01 are indeed remarkable, and this result cannot be explained by fluid inertia (RepPe < 10 for all experiments) or double layer repulsion forces alone. Figures 6 and 7 reveal that the measured separation factors are also flow rate dependent. When 〈ve〉 was raised 50% using the lowest conductivity eluant, average particle residence times decreased dramatically. This effect was more pronounced for the larger particles but appeared to

diminish for Rp g 0.15 µm. We believe that the particles’ near-centerline average radial position may be responsible for this limiting behavior. The use of surfactant in flow separation systems is necessitated by the possibility of particle aggregation. This is the reason why CHDF is typically operated using 10-4 to 10-3 M sodium lauryl sulfate eluants. To verify that the fractogram signals we obtained using surfactant-free eluants represented single particles and not agglomerated species, turbidity ratios were measured and compared with theoretical values. By taking the ratio of the measured optical density at two wavelengths of light in the UV range, we calculated turbidity ratios as a function of particle elution time. The theoretical turbidity ratios presented in Table 3 were determined using correlated values for the refractive indices of these polystyrene latex particles at the wavelength ratios indicated.12 The variation associated with the experimental measurements ranged from (0.6% for the largest latex standard to about (5% for the smallest size. This result may reflect the fractionation of the latex standard since the experimental turbidity ratios always increased with elution time. More importantly, the range we observed precludes the pos-


Langmuir, Vol. 12, No. 3, 1996 619 Table 4. Summary of Experiments Associated with Anomalous Results CHDF (high K)

Bike25

Physical Parameters 0.013 66.8 5.0 0.18 50 50-100 0.022 1.0

1.0 0.18 50-100 0.007

Particle Velocity and Elevation vp (cm/s) 0.0013-0.0053 1.1-3.0 [Rp + δ] (cm × 104) 5.7-8.0 1.5-1.6

0.3-3.2 2.2-3.0

K (µS/cm) Rp (µm) ζ (mV) I (mM)

Pe Rp/κD δ/Rp

Figure 7. Comparison between predicted and measured separation factor-particle diameter behavior for DDI water eluant: solid line, Rf simulation with Fek ) 0; - - -, Fek term included in Rf simulation using ζwall ) 100 mV, ζparticle ) 50 mV; ‚‚‚, Fek term included in Rf simulation using ζwall ) 100 mV, ζparticle ) 100 mV. Eluant average velocity, 6.4 cm/s; capillary radius, 12.8 µm; estimated ionic strength, 4.0 × 10-6 M; specific conductance, 0.44 µS/cm. Table 3. Measured Turbidity Ratios Using a Surfactant-free Eluant (4 × 10-6 M) particle diameter, µm

wavelength ratio, nm:nm

av exptl turbidity ratio

theoretical turbidity ratio

0.091 0.109 0.176 0.234 0.357

200:235 200:235 220:255 220:255 220:255

3.88 2.34 1.88 1.22 0.683

3.110 2.350 1.755 1.210 0.672

sibility of aggregate formation. No evidence of particle agglomeration was observed in any experiments involving surfactant-free eluants. Existence of Electrokinetic Lift in CHDF. When an electrokinetic lift force expression (eq 12) was used to predict separation factors in our low eluant conductivity experiments, reasonably good qualitative results were obtained (cf. Figures 4-7). Considering that the general theory of electrokinetic lift25 is unable to explain the anomalous results of Prieve and Alexander,19,20 it is natural to question the applicability of eq 12 to this work. Bike’s analysis found the predicted total lift force to be at least 4 orders of magnitude less than double layer and gravity forces at the smallest separation distances.25 Furthermore, the expression which we chose represents only the translational variant of the total lift force and is valid for large particle-wall separation distances, i.e., δ/Rp > 1, where δ is the gap thickness. The effects of particle rotation and shear flow, which are expected to increase the magnitude of the electrokinetic lift predicted by eq 12, are not included in this expression. Therefore, we would like to know if our experiments are in fact a validation of this theoretical result, or whether the qualitative agreement is fortuitous due to the use of a result outside the regime where it is presumed to be valid. The discrepancy between the general theory and Bike’s experimental results25 was attributed to the presence nonequilibrium EDL’s and emphasized the need to account for ion diffusion within the double layers. To form a more complete analysis, the ion conservation equation must be solved in addition to Stokes’ equation with charge effects, the equation of continuity, and Poisson’s equation. Recent theoretical work based on the lubrication approximation for motion of a long cylinder near a solid wall (both surfaces bearing thin double layers) in creeping flow indicates that an alteration in the double layer structure will enhance

CHDF (low K)

Dimensionless Groups 0.62-2.5 0.11-0.29 104 19 0.14-0.6 7.3-7.9

0.35-3.7 1.6 11.2-15.7

electrokinetic lift by at least 2 orders of magnitude.46 To our knowledge, a completely general analysis of electrokinetic lift which accounts for both nonequilibrium and thick double layers at arbitrary particle-wall separation distances has not yet been developed. Table 4 summarizes some of the physical parameters and dimensionless groups associated with the anomalous retention time of particles in a 9 µm i.d. microcapillary tube, together with the results of Bike25 which have been attributed to electrokinetic lift. In order to estimate the particle-wall separation distances in the capillary tube, we have assumed that particle and fluid velocities are equivalent (vps ) 0) and that the average particle elevation may be related to the separation factor by

( [

h ) Ro 1 - 1 -

])

Rf 2

0.5

(14)

where h (dδ + Rp) is measured from the sphere center to the inner wall surface. Although separation distances in the CHDF experiments satisfy the condition δ/Rp > 1, the use of eq 12 is questionable since both the Pećlet number and the electrokinetic radius (Rp/κD) are comparable to unity in many of our experiments. That our Pećlet numbers are nearly the same as those for the glycerol experiments is due to the magnitude of the ion diffusion coefficient (10-5 and 10-8 cm2/s for water and pure glycerol, respectively). The electrokinetic radius is, however, 1 to 2 orders of magnitude smaller than Rp/κD in Bike’s work. Clearly, the assumption of thin, equilibrium double layers is not applicable in our experiments. As 〈ve〉 approaches zero, however, electrokinetic and hydrodynamic lift forces are expected to be negligible, and the colloidal interaction potential alone will determine Rf. At very low eluant flow rates, we find that the CHDF model predicts the measured separation factors fairly well for all ionic strengths. If we make the thin double layer approximation and omit the correction for particle curvature from eq 6, Φ(r) decreases 10% and our model tends to underpredict Rf only slightly at the lowest ionic strength. In all cases, the ratio of the effective particle diameter (Rp + κD) to the effective capillary inner diameter (Ro - κD) is small enough to assume a sphere-infinite plate interaction. The influence of double layer polarization in the related problem of particle electrophoresis has been reviewed thoroughly by Dukhin.47 Comparison of the electrophoretic mobility computed using the thin double layer approximation of Smoluchowski with numerical calculations48 shows that the criteria for the two to agree will (46) Personal communication with Professor T. G. M. van de Ven. (47) Dukhin, S. S. Adv. Colloid Interface Sci. 1993, 44, 1. (48) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1978, 74, 1607.

620



Figure 8. Comparison between predicted and measured separation factor-eluant average velocity behavior at a relatively high eluant ionic strength: eluant, 1 mM sodium lauryl sulfate (SLS); ζwall ) 100 mV, ζparticle ) 100 mV; capillary radius, 4.5 µm; particle radius, 0.179 µm; specific conductance, 66.8 µS/cm. Fek is based on tabulated results25 for total electrokinetic lift.

Figure 9. Comparison between predicted and measured separation factor-eluant average velocity behavior for three particle sizes: eluant, 0.02% (w/w) BRIJ35 SP; capillary radius, 4.5 µm; estimated ionic strength, 7.0 × 10-6 M; specific conductance, 1.0 µS/cm; Fek ) 0.

Table 5. Comparison of Predicted Lift Force per Unit Area Using Equation 12

Fek/2πRp2 (µdyn/cm2) rel force/areaa a

Bike25

CHDF (high K)

CHDF (low K)

2.2-9.3 0.24-1.0

59-410 2.3-44

5100 to 1.7 × 105 550 to 1.8 × 104

Normalized with respect to 9.3 µdyn/cm2.

depend upon the zeta potential of the particle. For ζ ) 50 mV (typical of our experiments), acceptable accuracy (agreement to within 10 %) is obtained for Rp/κD > 30. For the high-conductivity CHDF experiments, Pe ≈ 0.3 and Rp/κD ≈ 20, and it seems likely that these conditions are not too far outside the regime where eq 12 is supposed to be valid. Although the Rf values are underpredicted significantly as shown in Figure 8, Fek is large enough to influence our numerical simulation and it appears that eq 12 shows qualitative agreement with the experimental results. At the low conductivity conditions, the predicted electrokinetic lift force is large enough to influence the simulated separation factors even more. Table 5 shows the lift force per hemispherical surface area as predicted by eq 12 for both the CHDF experiments and those of Bike.25 The results presented correspond to the particle elevations appearing in the previous table and are normalized with respect to the largest electrokinetic lift force predicted for the glycerol experiment. On the basis of a comparison of the magnitude of these theoretical lift forces, we expect that an electrokinetic lift effect would be amplified greatly for the transport of submicrometer particles in capillary flow. Our calculations using eq 12 have also shown that Rf is sensitive to the choice of zeta potentials and that accurate measurements of the eluant electroosmotic velocity and electrophoretic mobilities of the polystyrene latex standards are necessary. Particle-particle interaction at the lowest eluant conductivity is also significant and may have influenced some of our measurements. In summary, an electrokinetic lift force, consistent with that which is predicted by eq 12, is thought to be responsible for the observed effects in CHDF. In order to observe when electrokinetic effects become significant in particle transport through the capillary tube, we have set Fek ) 0 for all subsequent analyses. Effect of Eluant Velocity. Figures 9-11 show the effect of eluant velocity on the separation factor for a small capillary internal diameter. These results indicate that

Figure 10. Comparison between predicted and measured separation factor-eluant average velocity behavior for three eluants of various composition: eluant, 10-4 M SLS (open circle), 10-4 M NaCl (inverted open triangle), 0.02% (w/w) BRIJ35 SP/ NaCl (open triangle); capillary radius, 4.5 µm; particle radius, 0.179 µm; Fek ) 0.

Figure 11. Comparison between predicted and measured separation factor-eluant average velocity behavior for high and low ionic strength eluants: specific conductance, 0.44 µS/ cm (open circle, filled inverted triangle), and 66.8 µS/cm (filled triangle); capillary radius, 4.5 µm; particle radius, 0.179 µm; Fek ) 0.

above a certain velocity, the average residence time of particles as small as 0.1 µm in diameter depends on the eluant flow rate. This velocity dependence is not predicted by the model and may depend upon the eluant conductivity, not ionic strength, as revealed in Figure 10. It appears


to be unrelated to fluid inertia (RepPe < 3 for most experiments performed) and becomes stronger with increasing particle size at a fixed ionic strength. The fluid conductivity dependence may be explained in terms of ionic mobility. On the basis of the limiting equivalent conductance of lauryl sulfate,49 the mobility of these anions is 31/2 times smaller than that of chloride ions. Thus for a given ionic strength, the conductivity of NaCl solution would be greater than that corresponding to a solution of sodium lauryl sulfate. Additional experiments are needed to verify this behavior in CHDF. The velocity dependence also appears to grow stronger with decreasing ionic strength for a given particle size. Flow rate dependent separation factors were observed at the 10-3 M concentration, occurring at an eluant average velocity of about 1.8 cm/s. It is not clear whether a limiting separation factor exists at the high eluant flow rates since the effect of particleparticle interaction appears to interfere with this measurement. Negatively-skewed fractograms observed at high eluant average velocities suggest a column overloading condition, in which case the particles interfere with one another’s migration across fluid streamlines. As the sample solids content is reduced, the signal to noise ratio diminishes, requiring increased detector sensitivity. This limitation may be eliminated by using an on-column detection system. At the lowest eluant ionic strength, the strongest sample concentration dependence is observed. The electrostatic repulsive force together with a noninertial lifting force appear to be opposed by doublelayer repulsion between particles, resulting in lower separation factors when the solids content is increased. The effective size of the particles due to the expanded double layers surrounding them at the very low ion concentrations is probably not responsible for the large separation factors. In the case of DDI water eluant, the Debye thickness is about 0.15 µm and the diameter of the 0.109 µm particles would effectively increase by a factor of 3.8. The corresponding theoretical separation factor (Fek ) 0) for this diameter is still much lower than that which is measured. Increasing the eluant velocity will intensify the shear field, tending to shorten the axis of the Debye cloud in the direction normal to flow. This situation implies that separation factors might decrease with increasing shear rate. As the actual particle size increases, the effective size factor decreases and we would also expect any deviation to decrease. Just the opposite trends are observed for the measured separation factors. For the larger capillary inner diameter, a similar velocity-dependent separation factor is observed for all particle sizes (cf. Figure 12). These data were compared with separation factor-particle diameter behavior in the 9.0 µm diameter tube using the same low conductivity eluant; here we have scaled particle diameter with the tube diameter. Although the exclusion layer thickness due to electrostatic repulsion remains essentially constant with increasing tube radius, the core region in the capillary grows as the tube radius squared. Thus, when the average eluant velocity, fluid ionic strength, and κ are held constant, we expect the separation factor to become smaller as the tube radius is increased. At an average eluant velocity of 2 cm/s, we observed that, for particle sizes below about 0.18 µm diameter, the CHDF model accurately predicted separation factors for both capillaries. Deviation from the theoretical values grows at about the same rate with increasing κ for both tube inner diameters as indicated by the parallel dotted lines. (49) Mukerjee, P.; Mysels, K. J.; Dulin, C. I. J. Phys. Chem. 1958, 62, 1390.

Langmuir, Vol. 12, No. 3, 1996 621

Figure 12. Comparison between predicted and measured separation factor-particle diameter behavior for various eluant average velocities. Particle diameter has been scaled with the capillary diameter and the equivalent shear rate condition for the two capillary tubes is represented by ∆p/∆p* ) 1.0. Capillary radii: 4.5, 12.8 µm; eluant: 10-5 M NaCl; K ) 1.6 µS/cm; Fek ) 0.

Figure 13. Comparison between predicted and measured separation factor-eluant ionic strength behavior for two particle diameters: capillary radius, 4.5 µm; eluant average velocity, 2.2 cm/s; Fek ) 0.

Because the ratio of particle radius to the Debye length was approximately unity for these experiments, we expected that the expanded double layer surrounding each latex particle would be deformed by the shear flow in much the same way that a liquid drop is distorted in Poiseuille flow. In the latter case, the observation of axial migration toward regions of lower shear has been explained on the basis of minimum dissipation of energy.50 If this condition were to apply here, we might expect that Rf-κ behavior could be superimposed for equivalent shear rate conditions in different capillary inner diameters. The highest eluant average velocity indicated in Figure 12 corresponds to that flow rate which would be required to match the shear field present in the smaller capillary. Hence, ∆p* represents the calculated pressure gradient necessary to achieve this condition as specified by the shear force expression for laminar flow of fluids in a circular tube.31 At ∆p/∆p* ) 1.0, our results indicate that the particle transport mechanism is not purely shear-induced since the measured separation factors for the two capillaries do not follow the same trend. Separation Factor as a Function of Ionic Strength. Figures 13-17 show the effect of eluant ionic strength on the separation factor for two capillary diameters. A theoretical calculation reveals that the largest flow rate (50) Jeffery, G. B. Proc. R. Soc. London 1922, A102, 161.

622



Figure 14. Comparison between predicted and measured separation factor-eluant ionic strength behavior for two average eluant velocities: capillary radius, 4.5 µm; particle radius, 0.179 µm; Fek ) 0.

Figure 17. Comparison between predicted and measured separation factor-eluant ionic strength behavior for various average eluant velocities: capillary radius, 12.8 µm; particle radius, 0.179 µm; Fek ) 0.


ionic strengths below this critical value, the separation factor is expected to decrease with increasing eluant velocity since hydrodynamic forces will cause particles to migrate away from the tube center toward the tubular pinch point. In our experiments, we always find the strongest deviation from theory at the lowest ionic strengths. In the smaller capillary, the measured separation factors for the 0.176 µm particles agree with predicted values when the ionic strength is above 10-5 M. Generally, as particle size increases this deviation begins to occur at higher electrolyte concentrations and is more pronounced at higher eluant average velocities. A slight flow rate dependence is predicted for the 0.357 µm particles for relatively high ionic strength eluants; in all other cases, hydrodynamic lift forces are expected to be insignificant. A similar trend was observed in experiments conducted with a 25.6 µm diameter capillary. At the 10-5 M concentration, our model predicts that the separation factor will be independent of flow rate for the 0.357 µm particles. Instead, we find a significant velocity dependence which tends to diminish with decreasing particle size and increasing ionic strength. In the case of the 0.357 µm particles at the lowest ionic strength eluant, the separation factor appears to become less flow rate dependent at the high eluant average velocities. This result may be due to particle interaction effects and/or the particles’ near-centerline position in the capillary tube. Equilibrium Radial Position. The measurement of separation factors for monodisperse latex standards can be used to determine an average radial position as suggested by Noel et al.51 Assuming that particle and fluid velocities are equivalent (zero axial slip velocity), then this position may be expressed as


dependence is expected at the highest ionic strengths. This result is due to the relative magnitude of electrostatic double-layer and hydrodynamic forces. Although the fluid inertial force increases with increasing eluant velocity and increasing particle to tube diameter ratios, a particular ionic strength exists for which the separation factor is independent of flow rate. At this concentration, electrostatic repulsion between the particles and the capillary wall causes particles to travel, on the average, at this equilibrium streamline or tubular pinch point. At eluant

[

]

Rf r ) 1Ro 2

0.5

(15)

Theoretical equilibrium radial positions were determined with eqs 2 and 3 using the following expression:

[

]

〈vpz〉 r ) 1Ro 2〈ve〉

0.5

(16)

where 〈ve〉 is the average eluant velocity. (51) Noel, R. J.; Gooding, K. M.; Regnier, F. E.; Ball, D. M.; Orr, C.; Mullins, M. E. J. Chromatogr. 1978, 166, 373.


Figure 18. Comparison between predicted and measured average equilibrium positions, r/Ro, at various eluant ionic strengths and average velocities: capillary radius, 12.8 µm; Fek ) 0.

Karnis et al.52 and Walz and Grun41 found that for neutrally buoyant particles, r/Ro varies with κ when the tube Reynolds number is less than or equal to unity, as was the case for all data analyzed here. We observed that r/Ro varies inversely with particle diameter at a constant Ro which is consistent with their analysis; however, the effective radial position we calculate decreases significantly with increasing eluant flow rate and decreasing electrolyte concentration. Our calculated values range from 0.29 to 0.63 as shown in Figure 18. Again, this behavior is inconsistent with the tubular pinch effect, leading us to conclude that the apparent lifting mechanism cannot be of inertial origin. Conclusions We have observed that the average residence time of submicrometer-size polystyrene spheres being pumped under laminar flow through a microcapillary tube is strongly dependent upon the fluid average velocity when eluants of low electrolyte concentration are used. The severe depletion of particles from the region near the surface of the capillary is enhanced by decreasing the eluant ionic strength, increasing particle size, or by decreasing the tube diameter. The apparent lifting force responsible for these anomalous results does not appear to be of inertial origin. Additionally, our results are not (52) Karnis, A.; Goldsmith, H. L.; Mason, S. G. Can. J. Chem. Eng. 1966, 44, 181.

Langmuir, Vol. 12, No. 3, 1996 623

predicted by a CHDF model which takes into account only the effects of fluid inertia and colloidal interaction forces. Qualitative agreement with recently derived lift force expressions suggests that our results may be due to an electroviscous effect called electrokinetic lifting. A comparison of our results when the recent work of others in which electrokinetic lifting was observed revealed several important factors. Most important was the fact that the relatively high shear and low viscosity conditions at which our experiments were performed result in Pećlet numbers which are close to unity. That is, the condition of nonequilibrium double layers is not severe and does not appear to invalidate our use of a recently derived theoretical result to establish the existence of electrokinetic lift in CHDF. The influence of thick double layer polarization on this lift force is not well understood and emphasizes the need for a more general theory of electrokinetic lift which is valid for arbitrary Pe and electrokinetic radius. Evidence for the existence of electrokinetic lift was found in almost all of our experiments. Particle interaction significantly affects particle transport in CHDF under the low ionic strength conditions and requires the use of low solids content samples. We speculate that the lifting force which causes particles to migrate toward the tube axis is opposed by double layer repulsive forces which result when the particles are forced together. A paper detailing our investigation of sample concentration effects using low ionic strength eluants in CHDF is in preparation. Future studies will attempt to investigate the effect of eluant viscosity on the particle residence time. These results should provide a more complete analysis of electrokinetic effects in CHDF. Particle separation experiments will be conducted to validate the modified CHDF model. The CHDF process involves the measurement of statistical averages rather than direct observation of individual particle response to shear flow. Despite this limitation, we believe that our technique may provide the definitive experiment for the electrokinetic lifting of small particles. These results also indicate that the use of low ionic strength eluants in capillary hydrodynamic fractionation may provide better resolution of colloidal particles than that presently obtained using higher electrolyte concentrations. Acknowledgment. This research was supported by the Emulsion Polymers Institute Liaison Program at Lehigh University. LA950407V

Electrokinetic Lift Effects Observed in the Transport of Submicrometer

Recommend Documents