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Dielectrophoretic behavior of single polystyrene-carboxylate microparticles and polystyrene .... In the plot of Re[Ke] vs log f, there is a point of i...
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Langmuir 2000, 16, 3866-3872

Positive Dielectrophoretic Mobilities of Single Microparticles Enhanced by the Dynamic Diffusion Cloud of Ions Satoshi Tsukahara, Takashi Sakamoto, and Hitoshi Watarai* Department of Chemistry, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043 Japan Received April 17, 1998. In Final Form: January 7, 2000 Dielectrophoretic behavior of single polystyrene-carboxylate microparticles and polystyrene microparticles was studied with a quadrupole microelectrode. For positive dielectrophoresis (DEP) of polystyrenecarboxylate particle observed in the lower frequency region (e10 kHz), a DEP mobility coefficient (R) was almost proportional to the square of ac voltage (Urms, root mean square), as was expected from standard theories of DEP. The R values did not depend on the particle radius (re), but depended on the kind of electrolyte in the order HCl . KCl ≈ KOH g tetrabutylammonium chloride, showing a specific enhancement by H+. Since this observation was against the standard theories, we introduced a DEP radius (rDEP) of a Debye-type function instead of the particle radius to fit the observed R values in the lower frequency region. The positive DEP behaviors were reproduced by a function of rDEP and the surface conductivity of the negatively charged particles. As for negative DEP observed in the higher frequency region (g56 kHz), proportional relationships between R and Urms2 and ones between R and re2 were obtained as predicted by the standard theories. The obtained rDEP values increased with a decrease in the ac frequency, and the rDEP value in the HCl system was larger than those for the other systems. These suggested that the rDEP was controlled by the moving distance of cations swung by the applied ac electric field in the dynamic diffusion cloud around the particles.

Introduction Many separation and characterization methods for molecules and ions, including high-performance liquid chromatography, capillary zone electrophoresis, and solvent extraction, have been widely used in many fields. However, for relatively larger particles (0.1-10 µm in diameter), such as colloids, liposomes, and biological cells, there are a few methods available for separation and characterization; for example, zonal centrifugation, ultrafiltration, field-flow fractionation (FFF)1 and electrophoresis. By these techniques, only differences in electric charge, density, or size of particle are measurable. Recently, photophoresis was proposed as a novel separation technique for microparticles; photophoresis can distinguish the difference in refractivity.2-4 To develop new methods for colloidal characterization and separation, it is required to invent a technique to measure other properties of particles, such as conductivity and permittivity. The motion of particles, with a dielectric permittivity different from that of a surrounding liquid medium, in a nonuniform electric field, was termed dielectrophoresis (DEP).5 The strength and direction of the DEP force depend on the dielectric properties of both the particles and the medium. “Positive DEP” implies that a particle is attracted to a region of stronger electric field and “negative DEP” is the opposite phenomenon. Using DEP, it has been * To whom correspondence may be addressed. E-mail: watarai@ chem.sci.osaka-u.ac.jp. (1) Giddings, J. C. Unified Separation Science; John Wiley & Sons: New York, 1991. (2) Hirai, A.; Monjushiro, H.; Watarai, H. Langmuir 1996, 12, 5570. (3) Kaneta, T.; Ishidzu, Y.; Mishima, N.; Imasaka, T. Anal. Chem. 1997, 69, 2701. (4) Hatano, T.; Kaneta, T.; Imasaka, T. Anal. Chem. 1997, 69, 2711. (5) Pohl, H. A. Dielectrophoresis; Combridge University Press: Cambridge, 1978.

demonstrated that dead and living cells6-8 or ill and healthy cells9 can be distinguished, and that mineralpowder mixtures can be separated into their component parts.10 It has been recently reported that single cells can be manipulated with a DEP microelectrode11 and that a molybdenum tip is coated with diamond powder by using DEP force.12 The DEP technique is applicable to all kinds of particles whether charged or not, and this technique provides an attractive noninvasive method for investigating the electrical properties (permittivity and conductivity) of individual particles, colloids, and cells. The DEP behavior of cells or liposomes has been discussed with a model consisting of a conductive sphere enclosed by a thin insulating layer.13,14 However, an anomalous positive DEP has been observed at the ac frequency e1 kHz,15-19 where standard theories predict only a negative DEP.15,16 Inorganic and organic samples, (6) Ting, I. I.; Jolley, K.; Beasley, C. A.; Pohl, H. A. Biochim. Biophys. Acta 1971, 234, 324. (7) Pohl, H. A.; Crane, J. S. Biophys. J. 1971, 11, 711. (8) Iglesias, F. J.; Lopez, M. C.; Santamarı´a, C.; Dominguez, A. Biochim. Biophys. Acta 1984, 804, 221. (9) Gascoyne, P. R. C.; Huang, Y.; Pethig, R.; Vykoukal, J.; Becker, F. F. Meas. Sci. Technol. 1992, 3, 439. (10) Verschure, R. H.; Ijlst, L. Nature 1966, 211, 619. (11) Matsue, T.; Matsumoto, N.; Koike, S.; Uchida, I. Biochim. Biophys. Acta 1993, 1157, 332. (12) Choi, W. B.; Cuomo, J. J.; Zhirnov, V. V.; Myers, A. F.; Hren, J. J. Appl. Phys. Lett. 1996, 68, 720. (13) Kaler, K. V. I. S.; Jones, T. B. Biophys. J. 1990, 57, 173. (14) Stoicheva, N. G.; Hui, S. W. Biochim. Biophys. Acta 1994, 1195, 39. (15) Paul, R.; Kaler, K. V. I. S.; Jones, T. B. J. Phys. Chem. 1993, 97, 4745. (16) Kaler, K. V. I. S.; Jones, T. B.; Paul, R. Biophys. J. 1992, 63, 58. (17) Paddison, S. J.; Paul, R.; Kaler, K. V. I. S. Bioelectrochem. Bioenerg. 1995, 38, 321. (18) Barrie, G.; Paul, R.; Kaler, K. V. I. S.; Jones, T. B. J. Colloid Interface Sci. 1995, 175, 97. (19) Kaler, K. V. I. S.; Jones, T. B.; Paul, R. J. Colloid Interface Sci. 1995, 175, 108.

10.1021/la980441k CCC: $19.00 © 2000 American Chemical Society Published on Web 03/14/2000

Mobilities of Single Microparticles

for example, silica,5 clay,20,21 and poly(vinyl chloride) (PVC) particles,22 also showed similar anomaly. Many researchers were interested in this anomalous positive DEP and attempted some theoretical explanations. The majority of them discussed the effect of the surface conductivity of a charged particle,15,20,22 which resulted from the motion of ions existing in the Stern and Gouy layers of the particles.23,24 Others postulated a nonlinear polarization at the interface between protoplasts and a medium,17 and a production of a large dipole induced by small-scale motion of particles in the lower frequency region (e100 Hz).18,19 They improved only the Clausius-Mossotti factor, but it was pointed out that other factors must be considered, such as motion of ions around the particles at the low frequency.16,19 Recently, we have reported a two-dimensional quadrupole microelectrode method25 to observe and to analyze the DEP migration of a single particle in a nonuniform electric field. Since the photolithographic technique allows us to make two-dimensional microelectrodes of complicated structure with excellent precision, this type of microelectrode has been widely applied for the DEP separation in recent years.9,26-29 Therefore, we have intended to make the two-dimensional quadrupole microelectrode25 instead of the classical three-dimensional electrode.5,30 Furthermore, there are very few studies on the quantitative analysis of DEP behavior. In the present study, we have found out that the migration velocity of polystyrene particles is enhanced for the positive DEP under acidic conditions. Some factors governing the anomalous positive DEP have been determined by the quantitative analysis of behaviors of the microparticles. Experimental Section Chemicals and Apparatus. Polystyrene-carboxylate microparticle (Polybead Carboxylate Sampler Kit-I, 0.702, 0.731, 1.917, and 3.001 µm diameter) and polystyrene microparticle (Polybead Polystyrene Sampler Kit-I, 2.836 µm diameter) certified as particle size standards were purchased from Funakoshi Co. (Tokyo). Water was distilled and purified with a Milli-Q system (Milli-Q Sp. Toc., Millipore). All other reagents were of analytical reagent grade. Sample aqueous suspensions containing 0.005% (w/w) (7.6 × 108-3.5 × 106 particles per mL) polystyrene-carboxylate particles, 0-5.0 × 10-3 M KCl, and 2.6 × 10-7 M Rhodamine B (Wako, Kyoto) were prepared. Rhodamine B, a fluorescent dye, was adsorbed to the surface of the microparticle, and behaviors of the particles were clearly observed. In some cases, 1 × 10-3 M HCl, KOH, or tetrabutylammonium chloride, (TBACl, (C4H9)4NCl) was added instead of KCl. Aqueous suspensions containing 0.005% (w/w) (4.4 × 106 particles per mL) polystyrene particles and 1.0 × 10-3 M HCl, KCl, KOH, or (C4H9)4NCl were also prepared. Conductivity of these suspensions was measured with a conductometer (CM-40V, TOA Electronics, Tokyo). (20) Shang, J. Q.; Inculet, I. I.; Lo, K. Y. J. Electrostat. 1994, 33, 229. (21) Lockhart, N. C. J. Electrostat. 1983, 10, 17. (22) Santamarı´a, C.; Iglesias, F. J.; Domı´nguez, A. J. Colloid Interface Sci. 1985, 103, 508. (23) Schwan, H. P.; Schwarz, G.; Maczuk, J.; Pauly, H. J. Phys. Chem. 1962, 66, 2626. (24) O’Konski, C. T. J. Phys. Chem. 1960, 64, 605. (25) Watarai, H.; Sakamoto, T.; Tsukahara, S. Langmuir 1997, 13, 2417. (26) Rousselet, J.; Salome, L.; Ajdari, A.; Prost, J. Nature 1994, 370, 446. (27) Matsumoto, N.; Matsue, T.; Uchida, I. Bioelectrochem. Bioenerg. 1994, 34, 199. (28) Wang, X.-B.; Huang, Y.; Burt, J. P. H.; Markx, G. H.; Pethig, R. J. Phys. D: Appl. Phys. 1993, 26, 1278. (29) Schnelle, T.; Mu¨ller, T.; Voigt, A.; Reimer, K.; Wagner, B.; Fuhr, G. Langmuir 1996, 12, 801. (30) Foster, K. R.; Sauer, F. A.; Schwan, H. P. Biophys. J. 1992, 63, 180.

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Figure 1. Schematic illustration of hyperbolic quadrupole microelectrode around the working area and the wiring system. Black regions are microelectrodes on a glass substratum, which are made of chromium (lower layer, 100 nm thickness) and gold (upper layer, 50 nm), and the inscribed circle shows the working area. A DEP cell with a hyperbolic quadrupole microelectrode shown in Figure 1 was made by ordinary photolithographic technique on a glass plate. An electrode and its opposite one were wired to be the same polarity of alternating current (ac). Excitation and emission wavelengths of a fluorescence microscope BX60 (OLYMPUS) were set to 520-550 and 580-800 nm, respectively, for the observation of the particles adsorbing Rhodamine B. Measurement of Migration of Single Particles. The migration velocity of single particles was measured as previously.25 An aliquot (3 µL) of the sample suspension was dropped in the working area of the DEP cell, and a small cover glass plate was placed on it. An applied voltage (Urms, root mean square) and frequency (f) of ac were changed in the region of 2.5-5.0 V and 1 kHz-1 MHz, respectively, by a function generator (FG273, Kenwood). All experiments were carried out in a thermostated room at 25 ( 1 °C. The distance (R) from the center of the quadrupole electrode to a migrating particle was measured as a function of time (t). Migration of particles in the working area of the DEP cell was observed by the fluorescence microscope with a CCD camera (ImagePoint, Photometrix) and the image was recorded on a videotape. The image was then transferred as digital pictures to a personal computer (Power Macintosh 8100/100AV, Apple) at certain time intervals. The distance (R) was determined from the digital pictures. Analysis of DEP migration. The theoretical expression of migration of a single particle in the quadrupole microelectrode cell was described in detail in ref 25. Briefly, when an ac voltage, Up cos(ωt), is applied to the quadrupole microelectrode, the timeaveraged motive force 〈FDEP〉 of the particle caused by DEP is expressed as25,31,32

4πre3mRUrms2Re[Ke]

〈FDEP〉 ) β

d4

(1)

where ω is the angular frequency ()2πf) of ac, β the efficiency of DEP force ()0.39), re the radius of the spherical particle, m the permittivity of the medium, Re[Ke] the real part of the Clausius-Mossotti factor Ke, and d the radius of the inscribed circle in the working area of the microelectrode (see Figure 1). The underline means a complex property and Urms2 is equal to Up2/2. The frictional force that a migrating particle will undergo from a medium is given by the Stokes equation:

FSt ) 6πηre

dR dt

(2)

(31) Benguigui, L.; Lin, I. J. J. Appl. Phys. 1982, 53, 1141. (32) Watarai, H.; Sakamoto, T.; Tsukahara, S. Chem. Lett. 1998, 279.

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where η is the viscosity of the medium. Consequently, R can be expressed as a function of time (t) as ln R ) Rt + ln R0,25 where R and R0 are the DEP mobility coefficient and the initial R value, respectively. R was defined as

2re2mUrms2Re[Ke]

R)β

3ηd4

(3)

The Clausius-Mossotti factor is expressed as

Ke )

 p - m p + 2m

(4)

where the subscripts p and m are particle and medium, respectively. The complex, , is  - σ/(jω), where  is the permittivity, σ the conductivity, and j x-1. In the plot of Re[Ke] vs log f, there is a point of inflection at the Maxwell-Wagner relaxation frequency (fMW) that can be written as5,31

1 (σp + 2σm) fMW ) 2π (p + 2m)

(5)

The conductivity, σp, of the polystyrene-carboxylate and polystyrene particles is much less than that of the aqueous media, σm. By substituting 78.3o and 2.55o33 for m and p, respectively, fMW values were calculated for all cases. Log fMW was 5.7 in the case of [KCl] ) 2.3 × 10-4 M (σm ) 2.39 × 10-3 S‚m-1), but their values were larger than 6.3 (higher than the upper limit of our experiments) in the other cases. This means that the real part of the Clausius-Mossotti factor is controlled by conductivities of particle and medium, and it can be reduced to

Re[Ke] ≈

σp - σm σp + 2σm

(6)

under our experimental conditions except for one instance.

Results and Discussion Plots of ln R against the DEP time showed linear relationships with high correlation for all instances. The slopes of the plots were obtained as the DEP mobility coefficient, R, by the least-squares method. The conductivity, σp, of polystyrene-carboxylate is much less than those of the media, σm, and therefore Re[Ke] should be almost equal to -0.5 from eq 6. With this value, R was predicted as -0.16 in the case of re ) 1.501 µm and Urms ) 3.54 V regardless of frequency. However, the observed R values were positive in the frequency of 1 kHz e f e 10 kHz, and negative in the frequency of 56 kHz e f e 1 MHz. The anomalous dependence of DEP migration on frequency was observed in previous studies,15-22 but the effects of only ac frequency were investigated. We have examined here the effects of ac voltage, size of microparticle, kind of microparticle, and kind of electrolyte, as well as ac frequency and medium conductivity. Εfficiency of Dielectrophoretic Force, β. First, we must consider electric field lines in the DEP cell. They are mostly confined in the aqueous medium at low frequencies, because the conductivity of the cover and bottom glass plates is much less than those of the aqueous media. The permittivity of the glass (4.5o)33 is smaller than those of aqueous media in the higher frequency region, but it is not negligible. Therefore, the electric field lines are mainly distributed to the aqueous medium, but the electric field strength may be weaker than that in the lower frequency region. The frequency of turning point from the conductivity-controlled region to the permittivity-controlled (33) CRC Handbook of Chemistry and Physics, 62nd ed.; Weast, R. C.; Astle, M. J., Eds.; CRC Press: Boca Raton, FL, 1981; p E-57.

Figure 2. Dependence of the DEP mobility coefficient (R) of polystyrene-carboxylate microparticle (3.001 µm in diameter) on squared ac voltage (Urms2) in the KCl system. The solid lines are provided to guide the eye. The broken lines represent theoretical values calculated from eq 3 by substituting 1 and -0.5 for Re[Ke]. Positive DEP, f ) 10 kHz; negative DEP, f ) 1 MHz. (O) σm ) 3.11 × 10-4 S‚m-1, (0, 9) σm ) 1.42 × 10-2 S‚m-1, (4) σm ) 7.01 × 10-2 S‚m-1.

region can be estimated with eq 5 by substituting 0 and 4.5o (for glasses) for σp and p, respectively. Again, log fMW was 5.7 in the case of [KCl] ) 2.3 × 10-4 M (σm ) 2.39 × 10-3 S‚m-1), but their values were larger than 6.3 (higher than the upper limit of our experiments) in the other cases. From these calculations, we concluded that the electric field lines were confined in the aqueous medium in the most cases. Also, we should consider the relation between the size of microparticles and the metal plate thickness of the microelectrode. If the metal plate thickness of the microelectrode is larger than the particle diameter, the efficiency β should be 1. In fact, the thickness of metal plate (0.15 µm)25 is less than the microparticle diameter, and thus the electric field operating on the upper part of the single microparticle is weaker than on the lower part. This would cause a decrease in the efficiency, β. A kind of carbon microparticle (diameter 4.4 µm) possessing much higher conductivity (about 2 × 104 S‚m-1) was used as a sample, and its R values were found to be almost constant (0.55 ( 0.17 s-1) at Urms ) 3.54 V in the frequency range of 1 kHz-1 MHz.32 If β was 1, R was calculated to be 1.40 s-1 with eq 3 independent of f. From these, β was estimated to be 0.39 ()0.55/1.40).32 In the present study, we consider that β is approximately constant not depending on frequency, but more accurate investigations on β should be done. Effect of Applied ac Voltage on DEP Migration of Polystyrene-carboxylate Microparticle. Equation 3 means that R should be proportional to Urms2. Figure 2 shows the dependence of the observed R on Urms2 at various σm. As for the positive DEP region (f ) 10 kHz), the plots showed saturation curves with a saturated R value of about 0.8 s-1. In the case of the negative DEP (f ) 1 MHz), R was proportional to Urms2. At other σm (3.1 × 10-4 and 2.9 × 10-2 S‚m-1) in negative DEP, similar proportional relationships were obtained. The relations of eq 3 were superimposed in Figure 2 by substituting 1.0 and -0.5 for Re[Ke], which were the upper and the lower limits acceptable for Re[Ke], respectively. Comparison of the plots for the negative DEP suggests the applicability of eq 3. Those for the positive DEP are almost beyond the predicted ones, but R tends to increase with an increase in Urms2.

Mobilities of Single Microparticles

Figure 3. Dependence of the DEP mobility coefficient (R) of polystyrene-carboxylate microparticle on its radius (re) in the KCl system. The solid lines are provided to guide the eye. The broken lines represent theoretical values calculated from eq 3 by substituting 1 and -0.5 for Re[Ke]. Urms, 3.54 V; σm, (1.663.85) × 10-4 S‚m-1. (b) f ) 10 kHz, (0) f ) 100 kHz, (4) f ) 1 MHz.

Schnelle et al. reported a similar saturation in the plots of velocity against Urms2 for the negative DEP, and ascribed it to elevation of particles from electrodes.29 In our case of the positive DEP, the particles were slightly pushed to the bottom glass plate of the DEP cell and their velocity was retarded at the higher Urms, since the electric field at the surface of the bottom glass plate in the working area was stronger than that in medium above the surface. The R dependence on Urms2 implies that electric field was formed by the quadrupole microelectrode according to the theory. The following experiments were carried out at Urms ) 3.54 V (Urms2 ) 12.5 V2) where the linearity was satisfied for both positive and negative DEPs. Effect of Size of Polystyrene-carboxylate Microparticle on Its DEP Migration. The plot of R against re2 is shown in Figure 3 at f ) 10 kHz, 100 kHz, and 1 MHz. The relations between R and re2 predicted by eq 3 were also drawn in Figure 3 for the extreme cases of Re[Ke] ) 1.0 and -0.5. The positive R shows little dependence on the particle size, and this is inconsistent with eq 3. All the observed positive R values were larger than the theoretical ones. The result suggests that the DEP force does not operate on the particle volume directly (see eq 1). This may happen if the DEP force operates on the surroundings of the particle, whose size is larger than the particle and is constant independent of re. A proportional relationship between R and re2 was observed for the negative DEP of the particles of 0.96 and 1.501 µm. The migration of particles of 0.35 and 0.37 µm in radius was too slow to be measured under the same condition. The observed R values were within the range predicted by the theory. Therefore, it could be thought that the DEP force normally operated on the particle volume directly for the negative DEP. Some recent studies attempted to explain the anomalous positive DEP. Paul et al. postulated an increase in σp by the surface conductivity of particles.15 Paddison et al. introduced the nonlinear displacement vector to explain the positive DEP.29 However, they did not take the effect of the particle size into account. Since the effect of the particle size on the positive DEP velocity was not investigated quantitatively, the anomalous independence shown in Figure 3 is found in the present study for the first time. The reason is discussed below.

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Figure 4. Effect of frequency (f) of ac electric field and conductivity of medium on the DEP mobility coefficient (R) of polystyrene-carboxylate microparticle (3.001 µm in diameter) in the KCl system. (O) σm ) 2.39 × 10-3 S‚m-1; (b) σm ) 2.88 × 10-3 S‚m-1; (0) σm ) 4.17 × 10-2 S‚m-1; (9) σm ) 7.01 × 10-2 S‚m-1. pH was in the region of 6.1-7.0. The lines are fitting curves according to eq 9.

Figure 5. High positive DEP mobility coefficient (R) of polystyrene-carboxylate microparticle (3.001 µm in diameter) in hydrochloric acid. O, HCl; 0, KCl; b, KOH; 9, (C4H9)4NCl; the concentration of each electrolyte was 1.0 × 10-3 M. The solid lines for HCl, KCl, and KOH, and the dotted line for (C4H9)4NCl, are fitting curves according to eq 9.

Effect of Medium Conductivity on DEP Migration of Polystyrene-carboxylate microparticle. By adding KCl, the conductivity of medium, σm, was changed from 2.39 × 10-3 to 7.01 × 10-2 S‚m-1. Figure 4 shows the variation of R with σm as a function of ac frequency. As noticed in this figure, the positive and negative DEPs were observed in the lower and higher frequency regions, respectively. The inflection points were about 104.5 Hz, showing little dependence on σm. The positive R values decreased with an increase in σm, whereas the negative R was slightly influenced by σm. The quantitative analysis of the plots in Figure 4 is shown below. Effect of Kind of Electrolyte on DEP Migration of Polystyrene-carboxylate and Polystyrene Microparticles. HCl, KOH, or (C4H9)4NCl was added as an electrolyte instead of KCl. Interestingly, the R value of polystyrene-carboxylate microparticle depended on the kind of electrolyte as well as on the σm value. Figure 5 shows the dependence of R on the ac frequency in 1.0 × 10-3 M HCl, KCl, KOH, or (C4H9)4NCl suspension. The upper limit of R was predicted from eq 3 as 0.32 s-1 by substituting Re[Ke] to 1. The highest R value in the HCl system and the higher R values than the upper limit would

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Figure 6. Dependence of R values on the electric mobility of cation (u+) of the electrolyte. The dotted line is drawn with the least-squares method by assuming a linear relationship. Urms, 3.54 V; f, 1 kHz; O, polystyrene-carboxylate; b, polystyrene.

deeply relate to the anomalous independence of R on the particle radius (see Figure 3). As for the polystyrene microparticle, a quite similar enhancement of R by HCl was also observed (not shown). We suspected unfavorable bulk flow or convection at first. We already showed that the temperature rise of the sample suspension in the DEP cell was negligibly small.25 Joule heating sometimes leads to the temperature heterogeneity of solution, and to bulk flow.5 The σm value for this HCl system is close to that for one of the KCl systems in Figure 4 (open square), but the R value for the HCl system is about 2 times larger than that for the KCl system. Again, it is confirmed that the bulk flow caused by Joule heating is negligible. A decrease in solvent viscosity (η) can raise R values, but a selective decrease in η in the HCl system would not be realized. As a consequence, the anomalous increment of R is not ascribable to the variation in σm. Since sulfonate groups as well as carboxylate groups exist on the surface of the polystyrene-carboxylate microparticle,34,35 cations will exist around the anionic groups even at pH 3. The surface of the polystyrene microparticle was not treated artificially, but sulfonate groups exist on its surface, which comes from polymerization initiators, such as potassium persulfate.34,35 Therefore, we paid attention to the movement of cations around the negatively charged microparticle as well as the surface conductivity in the lower frequency region. Figure 6 shows the plot of the R value at 1.0 kHz against the electric mobility (velocity per unit field) of the cations, u+.36 The R value monotonically increases with an increase in u+, and the dotted line is drawn by assuming a linear relationship between them. The intercept was calculated to be 0.42 ( 0.07 s-1, which corresponded to the R value with a cation of no electric mobility. Interestingly, this intercept is close to the upper limit of R, 0.32 s-1, as mentioned above. These facts strongly suggest that the movement of cations around a particle enhanced the positive DEP. In the following section, the positive DEP behaviors of the microparticles are discussed with the movement of cations. (34) Zhao, J.; Brown W. J. Colloid Interface Sci. 1996, 179, 255. (35) Furusawa, K. University of Tsukuba, private communication. (36) u+ values were calculated as 3.6 × 10-7, 7.6 × 10-8, and 2.0 × 10-8 m2/V s for H+, K+ and (C4H9)4N+, respectively, from the molar conductivity of the cations, λ+, at infinite dilution at 25 °C with the equation, λ+ ) zFu+, where z is charge and F is Faraday constant. λ+ values were obtained from CRC Handbook of Chemistry and Physics, 72nd ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1991; p 5-96.

Tsukahara et al.

Dynamic Diffusion Cloud of Cations around Microparticle. Ions around a charged particle form the Stern and Gouy layers or the double layer. The thickness of the double layer around a particle (that is, Debye length) is about 10 nm at 1 × 10-3 M 1:1 electrolyte. However, this is not a dynamic but a static value. We should discuss here the rapid response of the ions in the double layer to an applied ac electric field, that is, the dynamic behavior of the ions and its dynamic effect on the particle conductivity. Ions in the double layer produce a dynamic diffusion cloud under an ac electric field, since they can move faster than the particle. The cloud generated by the movement of ions leads to high permittivity of the suspension.20,23,37-40 Some researchers considered that the surface conductivity caused by ions around charged particles brought about the positive DEP of the particle,15,20,22 but the dynamic diffusion cloud as well as the motion of ions was not included in discussion of DEP behaviors. The dynamic surface conductivity of a particle including ions in the double layer can be expressed as15

σp )

2λ0 rDEP(1 + ω2τ2)

+ σp,h

(7)

λ0 is the limiting value of the surface conductivity of the particle at lower frequency (ω f 0), σp,h is the limiting value of the conductivity of the particle at higher frequency (ω f ∞), and τ is the relaxation time. Paul et al. used 9.1 × 10-8 S for the typical λ0 value of microparticle, and it was slightly dependent on the model.15 This value was also employed in the present study. To fit the experimental data to a reasonable function and to understand the anomalous positive DEP, we introduce an effective DEP radius, rDEP, of a Debye-type function for the first time:

rDEP )

rDEP,l - re 1 + ω2τ2

+ re

(8)

where rDEP,l is the limiting value of the DEP radius at lower frequency (ω f 0), τ is the relaxation time that is the same value as that in eq 7, and re is the real radius of the microparticle. This type of function of ω and τ often appears in discussion of the motion of ions around a particle and in the expression of permittivity and conductivity of a suspension with particles.5,28,30 When ω f 0, σp and rDEP approach (2λ0/rDEP,l + σp,h) and rDEP,l, respectively. When ω f ∞, they go toward σp,h and re, respectively, which correspond to the intrinsic values of the microparticle. In the Stokes equation (eq 2), re was used at any frequency; this means that the friction force from a medium operates on the particle directly.41 With these assumptions, R can be rewritten as

2rDEP3mUrms2Re[Ke] R)β 3reηd4

(9)

(37) Grosse, C.; Foster, K. R. J. Phys. Chem. 1987, 91, 3073. (38) Fixman, M. J. Chem. Phys. 1983, 78, 1483. (39) Hu, Y. Langmuir 1998, 14, 271. (40) Dukhin, S. S.; Shilov, V. N. Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes; Halsted: New York, 1974. (41) Alvarez, O.; Brodwick, M.; Latorre, R.; McLaughlin, A.; McLaughlin, S.; Szabo, G. Biophys. J. 1983, 44, 333.

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Table 1. The Limiting Value of σp at Lower Frequency, the Limiting Value of Conductivity (σp,h) at Higher Frequency, the Limiting Value of the DEP Radius (rDEP,l) at Lower Frequency, and the Relaxation Time (τ) of the Polystyrene-carboxylate Microparticle (3.001 µm in Diameter) in the KCl System [KCl]/M 2.3 × 10-4 /S‚m-1

σm σp(ωf0)a/S‚m-1 σp,h/S‚m-1 rDEP,l/µm τ/µs rb a

2.0 × 10-3

10-3

10-2

2.39 × (9.6 ( 0.4) × 10-2 0.001 ( 0.001 1.92 ( 0.08 9(2 0.966

2.9 × 10-3

2.88 × (8 ( 1) × 10-2 0.01 ( 0.01 2.5 ( 0.2 4(1 0.955

10-2

4.17 × (1.10 ( 0.05) × 10-1 0.022 ( 0.004 2.06 ( 0.06 8(1 0.998

5.0 × 10-3 7.01 × 10-2 (1.0 ( 0.4) × 10-1 0.03 ( 0.02 3(1 7(5 0.836

Calculated with eq 7. b Correlation coefficient.

Table 2. The Limiting Value of σp at Lower Frequency, the Limiting Value of Conductivity (σp,h) at Higher Frequency, the Limiting Value of the DEP Radius (rDEP,l) at Lower Frequency, and the Relaxation Time (τ) of the Particles in Various Electrolyte Systems electrolyte (1 × 10-3 M) HCl

KCl

KOH

(C4H9)4NCl

σm/S‚m-1 σp(ωf0)/S‚m-1 σp,h/S‚m-1 rDEP,l/µm τ/µs r

Polystyrene-carboxylate (3.001 µm in Diameter) 4.03 × 10-2 1.42 × 10-2 1.99 × 10-2 (5.9 ( 0.2) × 10-2 (7.6 ( 0.7) × 10-2 (1.01 ( 0.06) × 10-1 0.03 ( 0.00 0.003 ( 0.007 0.013 ( 0.005 5.5 ( 0.2 2.5 ( 0.1 2.07 ( 0.07 11.9 ( 0.2 4(1 7(3 1.000 0.963 0.992

9.33 × 10-3 (9 ( 1) × 10-2 0.004 ( 0.008 2.2 ( 0.2 20 ( 10 0.934

σm/S‚m-1 σp(ωf0)/S‚m-1 σp,h/S‚m-1 rDEP,l/µm τ/µs r

4.02 × 10-2 (4.1 ( 4.6) × 10-2 0.03 ( 0.03 20 ( 60 6(6 0.996

Polystyrene (2.836 µm in Diameter) 1.52 × 10-2 (8 ( 3) × 10-2 0.01 ( 0.02 2.4 ( 0.3 11 ( 7 0.946

9.58 × 10-3 (8 ( 2) × 10-2 0.00 ( 0.02 2.5 ( 0.3 5 ( 10 0.907

Table 1 shows the parameters obtained by the nonlinearleast-squares fitting of eq 9 for the points in Figure 4. For all instances, the correlation coefficients are sufficiently high. The lines in Figure 4 reproduce the observed points well, which were drawn with eq 9 and the parameters obtained. The determined values, σp,h, rDEP,l, and τ, are close to each other independent of the KCl concentrations. It is reasonable that the σp,h values are close to 0. The rDEP,l values are larger than the re value (1.501 µm), and τ values are in the range of 4-9 µs. The limiting σp values at ω f 0 are larger than σm values for all cases, and this mainly causes the positive DEP in the lower frequency region. The decreasing tendency of R with the increase in σm for the positive DEP (Figure 4) is understandable from the fact that the σp value in the lower frequency region remains constant regardless of the increase in σm, and thus the difference, (σp - σm), in the Clausius-Mossotti factor (eq 6) becomes small. Table 2 includes the parameters obtained from the data of polystyrene-carboxylate and polystyrene microparticles. For all instances, high correlation coefficients were obtained. The lines in Figure 5 reproduce the observed points well, which were drawn with eq 9 and the parameters. Again, the σp,h values are close to 0 and the τ values are in the range of 4-20 µs. The rDEP,l values in the KCl and KOH systems are close to each other, but those for the HCl system are larger than those for the other systems. The electric mobility of H+ is much higher than those of K+ and (C4H9)4N+, and this may be the cause of the larger value of rDEP,l in the HCl system. Now little dependence of R on re2 for the positive DEP (Figure 3) can be explained with the generation of the dynamic diffusion cloud, whose size would have little dependence on re and strongly depends on the mobility of cations. According to the parameters, we propose a model of the dynamic diffusion cloud of cations as Figure 7. The thickness of the dynamic double layer is determined by

2.01 × 10-2 (9 ( 1) × 10-2 0.01 ( 0.01 2.3 ( 0.1 9(3 0.982

Figure 7. Schematic drawing of a dynamic diffusion cloud of cations around a negatively charged microparticle. Cations around the particle can move longer at lower frequency and create the dynamic diffusion cloud (its radius is rDEP, upper), but can slightly move at higher frequency (lower).

the moving distance of cations that are swung by an applied ac electric field. Under the lower frequency conditions, the ions in the diffusion cloud move longer and thus a larger dynamic diffusion cloud was generated. This movement also causes the higher surface conductivity. At the higher frequency, the ions cannot move and remain close to the particles, leading to rDEP ) re. The moving distance of cations in the DEP cell was calculated with the electric mobility of cations, applied ac voltage, ac frequency f, and the electric field strength in the DEP

3872

Langmuir, Vol. 16, No. 8, 2000

cell.42 At R ) 30 µm, Urms ) 3.54 V, H+ and K+ can move by 1.5 and 0.3 µm, respectively, at f ) 1 kHz, but they can only move by 1.5 and 0.3 nm, respectively, at f ) 1 MHz. These calculations suggest the above assumption. Briefly, we discuss the dynamic diffusion cloud by comparison with the static double layer. rDEP,l should agree with the value of the static double layer (ca. 10 nm at 1.0 × 10-3 M ionic strength) without any electric field. At a zero ac frequency, that is, direct current, the ions in the double layer can move around the particles to form a larger ion cloud, and hence rDEP,l approaches not the static double layer value but a larger value. This would be suggested by eq 7, in which σp does not approach the intrinsic conductivity of the particle but a larger value. The rDEP,l is approximately constant independent of the KCl concentration by considering the standard deviation in Table 1, not similar to the static double layer, whose size shrinks with an increase in ionic strength. As mentioned above, the thickness of the dynamic double layer corresponds to the moving distance of cations, and therefore the rDEP,l may not depend on the ionic strength. Our empirical treatment will be applicable to the results of Santamarı´a et al.22 and Kaler et al.16 Santamarı´a et al. reported that the valences of cation of electrolyte influenced the positive DEP behavior of PVC particles that possessed negative charges on them, whereas valences of anion had no effects. Kaler et al. found that the H+ concentration in the aqueous solution had a large influence on the positive DEP behavior of Canola protoplasts in lower ac frequency region. Conclusions 1. The DEP mobility coefficient, R, of the polystyrenecarboxylate particle was approximately proportional to the squared ac voltage, Urms2, for the positive DEP, but showed a saturation. The saturation would be caused by (42) Huang, Y.; Pethig, R. Meas. Sci. Technol. 1991, 2, 1142.

Tsukahara et al.

the fact that the thickness of the electrode was less than the particle size. As for the negative DEP, R was proportional to Urms2. This dependence of R confirmed that the electric field formed by the quadrupole microelectrode was just as that expected by the theory. 2. We proposed a DEP radius (rDEP) of a Debye-type function for charged particles to fit the R dependence on the ac frequency (f). The observed dependence, for example, an anomalous positive DEP observed at f e 10 kHz, could be reproduced by the combination of rDEP and surface conductivity of particles. From these parameters, we postulated a dynamic diffusion cloud of cations around the negatively charged particle. The features of the dynamic diffusion cloud were as follows: (a) the conductivity of the diffusion cloud was higher than the medium conductivity; (b) the size of the diffusion cloud was larger than that of the particle and (c) it was determined by the moving distance of cations in the cloud, which was controlled by the electric mobility of the cations; (d) the size of the diffusion cloud was of a function of a unique relaxation time. With the model, we could also explain the increasing tendency of R in the order of electrolyte, HCl . KCl ≈ KOH g (C4H9)4NCl, as well as the little dependence of R on the particle radius, re. 3. A negative DEP observed for f g 56 kHz was caused by the lower conductivity of particles than the medium conductivity, and the R value was proportional to the squared re. These were expected by standard DEP theories. 4. The DEP technique in the lower frequency region has a high potential to determine the surface conductivity of particles and to characterize the particle surface. In the higher frequency region, the conductivity of particles is distinguishable by DEP, which was demonstrated recently by us.32 Acknowledgment. The authors are grateful to Shimadzu Co. for making the DEP cell by photolithography. LA980441K