Dynamic Electrophoretic Mobility of Colloidal Particles Measured by

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Dynamic Electrophoretic Mobility of Colloidal Particles Measured by the Newly Developed Method of Quasi-elastic Light Scattering in a Sinusoidal Electric Field D. Mizuno,* Y. Kimura, and R. Hayakawa Department of Applied Physics, Graduate School of Engineering, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan Received June 14, 2000. In Final Form: August 30, 2000 We developed a new method for measuring the dynamic electrophoretic mobility µ* and apparent diffusion coefficient D of colloidal particles in a sinusoidal electric field by employing the heterodyne method of light scattering. The frequency of the scattered light is sinusoidally modulated due to the Doppler effect if there is no Brownian motion of colloidal particles. The undesirable influence of Brownian motion is removed by squaring the time-domain signal, and µ* is obtained from the second and fourth harmonic components of signal detected with a lock-in amplifier. On the other hand, the influence of electrophoresis is excluded to obtain D by calculating the autocorrelation function of the component at the same frequency of the applied electric field. This method is applied to a suspension of latex particles, and the complex spectrum of µ* is successfully measured for the first time in a wide frequency range below 50 kHz. The obtained spectrum of µ* shows relaxation, and the apparent diffusion coefficient D has the applied field dependence. These findings are probably due to an anomalous double-layer polarization process which has been theoretically predicted. The applicability of our method to the measurement for concentrated colloidal suspensions through which laser light cannot pass is also discussed.

Introduction Colloidal particles in suspension are usually charged due to the desorption or adsorption of low molecular ions. They move toward the direction of the external electric field, and this phenomenon is called electrophoresis. The electrophoretic mobility µ is one of the experimentally obtainable values that offer information on the surface electrical properties of colloidal particles. Thus, the value of µ at low frequency has been intensively measured by the quasi-elastic light scattering in a dc electric field (QELS-DEF).1 Electrophoresis under the dc electric field has other important practical applications. For example, the difference of µ in swollen cross-linked gels or in a capillary tube is used for the separation of DNA and proteins. Such a technique increases its importance in recently prospering biotechnology such as human gene analysis.2 However, there are few pieces of experimental research3-6 which address the dynamical aspects of electrophoresis, although the frequency dependence of spherical colloidal particles is intensively studied theoretically.7,8 This is partly due to the difficulty in extracting information on electrophoresis alone under the strong Brownian motion of colloidal particles. Recently, Ito et al. proposed a new procedure9,10 to obtain the frequency(1) Schmitz, K. S. An Introduction to Dynamic Light Scattering by Macro-molecules; Academic Press: San Diego, 1990; p 319. (2) Viovy, J. L. Rev. Mod. Phys. 2000, 72, 813-872. (3) Hunter, R. J. Colloids Surf., A 1998, 141, 37-65. (4) Schmitz, K. S. Chem. Phys. 1983, 79, 297. (5) Schmitz, K. S. J. Chem. Phys. 1983, 79, 4029. (6) Schmitz, K. S.; Pontalion, T. J. J. Chem. Phys. 1995, 103, 794. (7) Mangersdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1998, 94, 2441-2452. (8) Mangersdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1998, 94, 2583-2593. (9) Ito, K.; Ooi, S.; Nishi, N.; Kimura, Y.; Hayakawa, R. J. Chem. Phys. 1994, 100, 6098. (10) Ito, K.; Ooi, S.; Kimura, Y.; Hayakawa, R. J. Chem. Phys. 1994, 101, 4463-4465.

dependent complex electrophoretic mobility µ* ()µ exp(iδ), where δ is the phase delay) and the diffusion coefficient D simultaneously by quasi-elastic light scattering in a sinusoidal electric field (QELS-SEF). Although this method has not yet been completely established experimentally, it has some advantages over the other conventional methods of relaxation spectroscopy in looking into the dynamics of colloidal suspensions. For example, dielectric relaxation spectroscopy has been generally applied to those systems, but it cannot separate the contribution from colloidal particles and that from low molecular weight ions. Moreover, it is not available for the measurement at low frequencies due to the electrode polarization or the large fluctuation of dc conductance. In this study, we developed a new method which can measure the dynamic electrophoretic mobility µ* and the apparent diffusion coefficient D, separately. The dynamic mobility is directly obtained from the signal of scattered light in the time-domain by using a lock-in amplifier after squaring the intensity of the scattered light to remove the random fluctuation of the signal which is originated from the Brownian motion of particles. On the other hand, temporal correlation of the analogue output of the lock-in amplifier is calculated to obtain the diffusion coefficient. The influence of electrophoresis is removed by shortening the time constant of the lock-in amplifier. By this new method, the dynamic electrophoretic mobility µ* in an aqueous suspension of latex particles is successfully measured in a much wider frequency range (5 Hz to 50 kHz) than those obtained by the method developed earlier.10 The measurement of µ* at much higher frequencies is also possible in principle, if faster detection devices are available. The observed mobility shows signs of relaxation at high frequencies, and the apparent diffusion coefficient depends on the applied electric field. These experimental results are probably ascribable to the anomalous conduction of a colloidal surface.7,11-13

10.1021/la000821h CCC: $19.00 © 2000 American Chemical Society Published on Web 11/03/2000

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The previous methods of QELS-DEF and QELS-SEF are applicable only to extremely dilute solutions, since the method needs to avoid the effect of multiple scattering. But in a practical sense, it is more convenient if the methods are also applicable to concentrated colloidal suspensions or in the system where the solvent or matrix scatters light strongly. The advantage of our newly developed method is not only its high sensitivity and high efficiency to remove noises but also its applicability to such practical uses. In the next section of this paper, the principle of the newly developed system will be explained. After that, the practical setup of the experimental system will be described and the results of the measured dynamic mobility and the apparent diffusion coefficient of the dilute colloidal suspension will be discussed. In the last section, we will report on the dynamic mobility in a concentrated suspension measured with a slightly modified experimental system.

|ES + ER|2 ) I0 + I + 2

ES(t) )

x∑ I

N

Here, it is assumed that every particle has the same mobility µ. In the case of the steady electric field E0 applied to a suspension, δrEn is given by

δrE ) µE0t

exp{i(q‚rn - ω0t)}

δrE ) µE0 sin(ω0t + δ)/ω0

|ES + ER|2 ) I0 + I + 2

I0 + I + 2

x

I0I N

∑n cos(q‚rn)

x

I0I N

∑n [cos(cn) cos(µqE0t) sin(cn) sin(µqE0t)] (6)

for QELS-DEF and

∞

2

|ES + ER|2 ) I0 + I + ESER* + ES*ER )

(5)

where δ is the phase delay to the applied field. By using eq 4 or 5, the intensity of the detected light is obtained as

(1)

where I ) ESES*, N is the number of particles in the scattering volume, and q is the scattering wave vector, which is determined by the direction of the scattered light. The magnitude q is written by means of the refractive index of the solvent n, the angle between the direction of incident and scattered light θ, and the wavelength of incident light λ as q ) 4πn sin(θ/2)/λ. The scattered light is mixed with the reference light and detected by the optical heterodyne technique to obtain the electrophoretic mobility. The temporal change of the intensity of the detected light is

(4)

On the other hand, when the sinusoidal field E ) E0 cos(ω0t) is applied to a suspension, δrEn is given by

|ES + ER|2 ) I0 + I + 2

n

∑

[cos(q‚r0n) × N n cos(q‚δrEn) - sin(q‚r0n) sin(q‚δrEn)] (3)

Principle of the Newly Developed QELS-SEF In this study, laser light scattering is used to measure the electrophoretic mobility and diffusion coefficient of colloidal particles in suspensions and its principle is explained below. Assuming that the system is composed of identical spherical particles without any internal degree of freedom, the scattered electric field ES(t) from the spheres whose position of the center of gravity is represented by {rn} is given by14

x

I0I

∑

x

I0I N

∑n [cos(cn){J0(z) + ∞

J2k(z) cos(2k(ω0t + δ))} + 2 sin(cn)

k)1

∑ J2k-1(z) ×

k)1

sin((2k - 1)(ω0t + δ))] (7) for QELS-SEF. In eqs 6 and 7, cn ) q‚r0n, z ) µq‚E0/ω0, and Jk is the Bessel function of k-th order. The signal detected by QELS-SEF is made up of two components, as shown in eq 7. One is the randomly fluctuating part due to the Brownian motion of colloidal particles, ∑n cos(cn) and ∑n sin(cn). The diffusion coefficient D can be obtained from the autocorrelation function of those parts. In the limit of E0 f 0, the usual Brownian motion is revived and the autocorrelation function of the intensity of the detected light is reduced to

〈cos(cn(t)) cos(cn(0))〉 ) 〈sin(cn(t)) sin(cn(0))〉 ∝ exp(-Dq2t) (8)

(2)

where ER is the electric field of the reference light and I0 ) ERER*. In equilibrium, rn(t) varies with time due to the Brownian motion of colloidal particles. When the particles are subjected to the external electric field E, another contribution δrE gives rise to displacement due to the electrophoresis. The total displacement of a colloidal particle is given as rn ) r0n + δrEn, where r0n is the position of a particle without an electric field. Equation 2 can be rewritten as (11) Shubin, V. E.; Hunter, R. J.; O′Brien, R. W. J. Colloid Interface Sci. 1993, 159, 174-183. (12) Arroyo, F. J.; Carrique, F.; Bellini, T.; Delgado, A. V. J. Colloid Interface Sci. 1999, 210, 194-199. (13) Midmore, B. R.; Pratt, G. V.; Herrington, T. M. J. Colloid Interface Sci. 1996, 184, 170-174. (14) Berne, B. J.; Pecora, R. Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics; John Wiely & Sons. Inc.: New York, 1976; Chapter 3.

The other component of the detected signal is the response ∞ J2k(z) cos(2k(ω0t + to the sinusoidal external field, ∑k)1 ∞ δ)) and ∑k)1 J2k-1(z) sin((2k - 1)(ω0t + δ)), from which the dynamic electrophoretic mobility is obtained. However, one of them prevents us from extracting the information on the other one, especially in case of the two time scales, 1/Dq2 and 1/ω0, being almost equal. For example, the timedomain signal in this study loses the information on the phase delay δ of the dynamic mobility µ* within a time of 1/Dq2 due to the Brownian motion. In the previous method of QELS-SEF,4-6,15,16 one analyzes the power spectrum of the scattered light. From eqs 6-8, the power spectrum P(ω) of QELS-DEF is given as

P(ω) ) 2Dq2 (9) (I02 + 2I0〈I〉)δ(ω) + 2I0〈I〉 (ω - µqE)2 + D2q4

Electrophoretic Mobility of Colloidal Particles

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and that of QELS-SEF as

A12 )

P(ω) ) (I02 + 2I0〈I〉)δ(ω) +

[

∞

Dq2

∑

2Dq2

]

2I0〈I〉 J0 (z) + Jk (z) ω2 + D2q4 k)1 (ω - kω0)2 + D2q4 (10) 2

2

Hereafter, we call the term k ) l the l-th order term. Equations 9 and 10 are the same ones as those obtained by the different procedures in the previous papers.15 We also reported that the obtained power spectrum of QELSSEF has Lorentzian peaks at the harmonic frequencies of the applied sinusoidal electric field. Every peak is broadened due to the Brownian motion of particles, and its half-width equals Dq2. Therefore, the peaks are observable while the condition of Dq2 < ω0 is satisfied. Although the magnitude of the dynamic mobility µ* is obtained from the ratio of the height of peaks, this method does not give the information on the phase delay δ. Recently, Ito et al. proposed a new procedure9,10 to calculate the two-dimensional power spectrum, which is related to the two-dimensional correlation function via the WienerKhinchin theorem extended to the periodically stationary systems. By means of this method, they obtained the dynamic electrophoretic mobility µ* and the diffusion coefficient D simultaneously in the frequency range of up to 100 Hz. This method can be regarded as the extension of the fluctuation analysis, and there has been no direct procedure for measuring electrophoretic mobility µ* until now. Therefore, the frequency range and substances to which the previous method is applicable are limited because of certain serious problems, which will be discussed as follows. First, the spectrum density of noise is more or less dependent on the frequency, and the small peak under the noise level cannot be detected. Second, the intensity of the signal in the frequency range of ω1 < ω < ω1 + δω is measured to obtain P(ω1), where δω is the resolution of bandwidth determined by the observation time. Since δω must be chosen to remain much smaller than the peak width Dq2, the signal intensity becomes extremely low. The drift of colloidal particles, which is frequently observed in a high electric field, also makes another problem. If there is the drifting term δrd(t) ) vdt with velocity υd in the system studied, it makes every peak in eq 10 split into two. In fact, unexpected splitting or broadening of peak width has been frequently observed. So, we have developed a new measurement method to overcome all these problems by measuring the timedomain response to the sinusoidal electric field directly. First, the first- and second-order harmonic terms A1 and A2 in eq 7 can be extracted as

x ∑ x ∑

A1 ) 4

A2 ) 4

I0I [ N

I0I [ N

sin(cn)]J1(z) sin(ω0t + δ)

(11)

n

cos(cn)]J2(z) cos(2(ω0t + δ)) (12)

n

These components of output will average to zero by the lock-in detection because cos(cn) and sin(cn) are random variables which reflect the Brownian motion of colloidal particles. So, we squared these signals as (15) Imaeda, T.; Kimura, Y.; Ito, K.; Hayakawa, R. J. Chem. Phys. 1994, 100, 950-954.

A12 )

{∑(

16I0I J12(z) N

n

{∑(

16I0I J22(z) N

)}

1 - cos(2cn) 2

[

]

(13)

]

(14)

1 - cos (2ω0t + 2δ)

)}

1 + cos (2cn)

n

×

2

[

2 ×

1 + cos (4ω0t + 4δ) 2

Here, it is plausibly assumed that there are many particles having no positional correlation to each other in scattering volume. That is to say,

∑ cos(cn) cos(cm) ) 0, n*m ∑ sin(cn) sin(cm) ) 0

(15)

n*m

It is possible to detect invariable components appearing in the coefficients with a lock-in amplifier. The terms containing cos(cn) and sin(cn) are averaged out, and the detected signals are written as

A1′ ) -4I0〈I〉J12(z) cos(2ω0t + 2δ)

(16)

A2′ ) 4I0〈I〉J22(z) cos(4ω0t + 4δ)

(17)

The phase delay of 2δ and 4δ can be directly obtained with the two-phase lock-in amplifier. However, the amplitude of the output of the lock-in amplifier depends on the uncontrollable parameters such as the intensity of the laser light and the efficiency of the optical system. Therefore, the ratio of the amplitude of the first-order term to the second-order one -A2′/A1′ ) J22(z)/J12(z) is calculated to obtain the magnitude of the dynamic mobility |µ*|. In the above procedure, the information on the Brownian motion is excluded by squaring the detected signal to make a sensitive measurement of the dynamic mobility. On the contrary, the diffusion coefficient of colloidal particles can be obtained separately by eliminating the information on electrophoresis with a much simpler experimental setup. In this case, the signal in eq 11 is directly detected with a lock-in amplifier without the square-law operation. If the time constant tc of the lock-in amplifier is set shorter than the characteristic time of diffusion 1/Dq2, it is possible to observe the temporal change of the coefficient ∑n sin(cn)J1(z) which has been averaged out before. By calculating the autocorrelation function of the signal, the diffusion coefficient is obtained as shown in eq 8. It must be noted that this method is only available under the following conditions

1 1 > tc > 2 ω Dq 0

(18)

QELS-SEF in a Dilute Suspension of Latex Particles 1. Dynamic Mobility µ*. The experimental setup of QELS-SEF developed in this study is shown in Figure 1. Incident laser light (He-Ne, λ ) 6328 Å) is crossed with reference light in the sample cell, in which parallel plate platinum electrodes are set. The scattered light mixed with the transmitted reference light is detected with the optical heterodyne technique and amplified with

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Mizuno et al.

Figure 1. Block diagram of the newly developed QELS-SEF in this study.

a preamplifier (AD624). The harmonic frequency component of the applied electric field in the detected signal is clipped out by using a band-pass filter (NF3628). The filtered signal is squared with a square-law detector made up from an analogue multiplier (AD633) and is detected with a lock-in amplifier (SR830). The sinusoidal electric field generated by a synthesizer (HP33120A) is amplified with a high-voltage amplifier (HEOPS) and applied to the sample. The applied field is monitored on a digitizing oscilloscope (HP54501A) and is also used as a reference signal for the lock-in detection after passing through the filter. Since the frequency of signal is 2ω0 for the firstorder term and 4ω0 for the second-order one, the secondand fourth-order harmonic detection of the lock-in amplifier is used. There might be an extra phase shift or change of amplitude due to the frequency dependence of devices such as the filter and the multiplier. Therefore, they are corrected by measuring the reference signal instead of the detected one. We used extremely dilute aqueous suspensions (about 0.001% in volume fraction) of polystyrene latex particles (Dow. Co. Ltd) with a diameter of 0.6 µm and a surface charge density of 4.3 × 10-2 C/m2. The latex particles are purified by mixed-bed ion-exchange resin before they are suspended in distilled water. The scattering angle is set at θ ) 10°, which corresponds to the scattering wavenumber q ) 2.3 × 106 m-1. The applied electric field dependences of the amplitude of the first- (A1′) and the second-order response (A2′) at 240 Hz are shown in Figure 2. According to eqs 16 and 17, the amplitudes of A1′ and A2′ in the region of linear electrophoresis (z , 1) are proportional to E02 and E04, respectively. The magnitude of the electrophoretic mobility can be obtained from the ratio of |A2′|/|A1′| as

µ)4

x

|A2′| ω0 |A1′| qE0

(19)

µ ) 3.8 × 10-8 C m/(N s) is obtained from the data in Figure 2a. Instead of repeating this operation at every frequency, |A1′| at the respective frequency and that at 240 Hz are compared to evaluate µ on the plausible assumption that there is little change in the scattering intensity and the efficiency of the optics. The measuring time is shortened in this way. Equation 19 is satisfied as far as the value of z is small enough, but it does not hold at low frequency and under a strong electric field. The

Figure 2. Applied electric field dependence of the amplitude of the first A1′ and second-order terms A2′ in the dilute latex suspension. (a) The amplitudes A1′ and A2′ at 240 Hz in the region of linear electrophoresis (z , 1). (b) The amplitudes of A1′ at 15 Hz. The dotted curve in part b is that of eq 16 with µ ) 3.8 × 10-8 C m/(N s). The solid line is the best fitted curve of eq 20 with µ0 ) 3.7 × 10-8 C m/(N s) and σ ) 7.3 × 10-9 C m/(N s).

applied field dependence of the first-order response |A1′| in a wider range of electric field at 15 Hz is shown in Figure 2b as an example. The dotted line in Figure 2b is the theoretically predicted one by eq 16 for µ ) 3.8 × 10-8 C m/(N s). In practice, every particle has a different mobility. Provided that the distribution of mobility is Gaussian, the signal intensity for the first-order A1′ is proportional to

1

(

∫0 exp -

x2πσ

∞

)

(µ - µ0)2 2σ2

J1(z)2 dµ

(20)

where µ0 is the average of mobility and σ is its standard deviation. The solid line in Figure 2b is the best fitted curve of eq 20 with µ0 ) 3.7 × 10-8 and σ ) 7.3 × 10-9 C m/(N s). The average value of the mobility µ0 is in good agreement with that obtained from the data in Figure 2a. Hereafter, the magnitude of the mobility obtained from the electric field dependence in the linear region is safely used without scanning the wide range of the electric field. The distribution of mobility has not been derived from the spectrum of the conventional QELS-DEF because of the broadening of the peak width caused by the diffusion of particles or the so-called snapshot time for each particle to leave away from the scattering volume. The standard deviation σ obtained in this study is free from these influences, but it is larger than expected. In the region of high electric field, the apparent electrophoretic mobility depends on the applied electric field and has a larger value than that in the linear region. That explains the rapid decrease of |A1′| in the high electric field. Therefore, the distribution of mobility looks slightly broad and asymmetric. By changing the angular frequency ω0, the frequency spectrum of the dynamic mobility µ* is measured as shown in Figure 3. The dc electrophoretic mobility is measured by QELS-DEF (Otsuka elec. ELS-800) as 3.7 × 10-8 C


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ever, the relaxation of mobility observed in this study occurs at higher frequency than that expected by this process and the relaxation is probably due to the faster process. Taking the double-layer polarization process into account, the dynamic mobilities for various values of κa were calculated numerically7,8,26 and analytically.17 O’Brien showed17 that the electrophoretic mobility for κa . 1 is written as

µ) Figure 3. Frequency dependence of the magnitude µ and the phase shift δ of the dynamic mobility µ* in the dilute latex suspension. The solid line is the best fitted curve of eq 21. The dotted line is the low-frequency limit of µ measured by QELSDEF.

m/(N s), which is shown as a dotted line in Figure 3 and makes good agreement with that measured by newly developed QELS-SEF. It seems that there is no frequency dependence of the amplitude µ, and the phase delay δ is almost zero under 1 kHz. Since the half width of a bandpass filter becomes narrower than the peak width Dq2 (∼5 s-1) at 8 Hz, all of the signal cannot pass through the filter and the apparent decrease in µ is caused at this frequency. Measurement of µ at much lower frequency is also possible, since this apparent phenomenon disappears if the band-pass filter is replaced with a low pass filter. Relaxation is found in Figure 3 at higher frequencies than 1 kHz. Two possible relaxation processes are proposed17-19 for an isolated colloidal particle corresponding to the two characteristic lengths, the particle radius a, and Debye length κ-1. The faster process concerned with the relatively small length κ-1 is called double-layer polarization. The counterions in the double layer are swept across the surface of colloidal particles under a external electric field, and the double layer is polarized. It is theoretically shown that the characteristic time of this process is about /K∞ ≈ 1/κ2Di, where and K∞ are the dielectric constant and the conductivity of bulk solution and Di is the diffusion coefficient of the low molecular weight ions. This process has been frequently observed12,20 as the high-frequency dispersion of dielectric permittivity (so-called Maxwell-Wagner relaxation), and it is believed that the electrophoretic mobility will also show the dispersion by the same mechanism.21 During this faster process, electrical current is carried only by the counterions inside the double layer, whereas the same amount of current is carried by both the co-ions and the counterions outside the layer. Therefore, the total density of ions increases at one side of the particle and it decreases at the other side. This unbalance of total ion density makes another diffusive flux and is relaxed by spreading out into the surrounding solution. This process, called concentration polarization, is related to the much longer length a + κ-1, and its characteristic time τd ≈ a2/Di of polarization is about 10-4 s. The dielectric dispersion by this process has been rigorously investigated.22-25 How(16) Schurr, J. M.; Schmitz, K. S. Ann. Rev. Phys. Chem. 1986, 37, 271. (17) O’Brien, R. W. J. Fluid Mech. 1988, 190, 71-86. (18) Hinch, E. J.; Sherwood, J. D.; Chew, W. C.; Sen, P. N. J. Chem. Soc., Faraday Trans. 2 1984, 80, 535-551. (19) Minor, M.; van der Linde, A. J.; van Leeuwen, H. P.; Lyklema, J. J. Colloid Interface Sci. 1997, 189, 370-375. (20) O’Brien, R. W. J. Colloid Interface Sci. 1986, 113, 81-93. (21) O’Brien, R. W.; Cannon, D. W.; Rowlands, W. N. J. Colloid Interface Sci. 1995, 173, 406-418. (22) Delgado, A. V.; Arroyo, F. J.; Gonzalez-Caballero, F.; Shilov, V. N.; Borkovskaya, Y. B. Colloids Surf. A 1998, 140, 139-149.

2ζ [1 + f(λ,ω′)] 3η

(21)

where η is the shear viscosity of bulk solution and ζ is the zeta potential of a colloidal particle and

( (

f(λ,ω′) )

)

p

1 + iω′ - 2λ + iω′

2(1 + iω′) + 2λ + iω′ +

)

p

(22)

where ω′ ) ω/K∞ ≈ ω/κ2Di and KS is the surface conductance.27 The attainable value of κa in our experiment is no smaller than 1 even if ionic strength is lowered as much as possible.28 Therefore, eq 22 is applicable to our experimental results. The presence of relaxation shows that λ is no smaller than unity. Moreover, eqs 21 and 22 show that the Smoluchowski equation µ ) ζ/η is applicable only in the case of λ , 1 or ω′ . 1. Therefore, the conventional method of dc electrophoresis may underestimate ζ for a sample with large λ like our sample. In our case, ζ is obtained by using the high-frequency value of dynamic electrophoretic mobility in the Smoluchowski equation. If the current is carried only by ions above the shear plane (ions below the shear plane are immobile), λ is theoretically estimated to be extremely smaller than 1 except in the case of large ζ and small κa.29,30 Therefore, the faster relaxation process is believed to be difficult to observe. However, it has been reported11,12,21 recently that mobile ions below the shear plane cause anomalously large surface conduction in some polymer latex systems. This will enhance the relaxation of the faster process, and this is why the relaxation is observed in our experiment. The solid line in Figure 3 is the best fitted curve of eq 21 with λ ) 5.4, /K∞ ) 5.1 × 10-6 s, and ζ ) 300 mV. The values of and K∞ of the same sample are directly measured and we obtained the value of /K∞ ≈ 5.3 × 10-6 s at 2.4 kHz. This is in good agreement with that estimated from the relaxation behavior of mobility. In fact, very high but credible values for λ and ζ are obtained from the wide band spectrum of dynamic electrophoretic mobility µ*, which may be due to the low ionic strength in our sample.11,12 Moreover, these large values for λ and ζ make electrophoretic mobility less sensitive to the slower process,18 which is also consistent with our experiment. On the other hand, the conventional method of dc electrophoresis underestimates ζ, as we mentioned before, and yields the usual value of ζ ) 44 mV from the Smoluchowski equation. In any case, more precise measurement with controlling the ionic strength of latex in (23) Fixman, M. J. Chem. Phys. 1983, 78, 1483. (24) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204. (25) Grosse, C.; Foster, K. R. J. Phys. Chem. 1987, 91, 3073-3076. (26) Mangersdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1992, 88, 3567. (27) Dukhin, S. S. Adv. Colloid Interface Sci. 1993, 44, 1-134. (28) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1996, 77, 18971900. (29) Bickerman, J. J.Kolloid Z. 1935, 72, 100. (30) Bickerman, J. J. Trans. Faraday Soc. 1940, 36, 154.

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electric field increases, the apparent diffusion coefficient D of the respective sample monotonically increases beyond the value predicted by the Einstein-Stokes relation. The deviation is almost proportional to the square of the electric field E02. The apparent diffusion coefficient of colloidal particles in a sinusoidal electric field has been scarcely discussed, but we suggest some possible explanations to this experimental result as below. If the ions under the shear plane are mobile, as mentioned before, the effective charge of the colloidal particle fluctuates and it couples to the external electric field, which may give another contribution to the motion of particles. The mathematical description is given by the Langevin equation in one dimension including the fluctuation of ion concentration in the sinusoidal electric field as

γυ ) -(Q - δq(t))E0 cos(ω0t) + δR(t)

(25)

where γ is the drag coefficient and Q is the average charge of the colloidal particle. δR(t) and δq(t) are the fluctuating force and the fluctuating part of the charge of particles, respectively. δR(t) and δq(t) are the random variables which satisfy the following relations as Figure 4. (a) Correlation function 〈Y(t)‚Y(0)〉 at 1 kHz and E0 ) 1.2 kV/m. The solid line is the best fitted curve of eq 8. (b) Applied field dependence of the apparent diffusion coefficient D of the latex particles with a diameter of 0.83 µm (open circles) and 0.6 µm (filled circles).

a wider frequency range is necessary to discuss the relaxation behavior of µ*. 2. Diffusion Coefficient. It is also possible to measure the apparent diffusion coefficient accurately with almost the same system as that in Figure 1 without a square-law detector. The time constant of a lock-in amplifier tc is extremely shortened to satisfy the condition of eq 18, and the fluctuating part of eq 11 is extracted. The autocorrelation function g(1)(t) of the out-of-phase component Y for the output of the lock-in amplifier is calculated as

x

Y ) -4

I0I [ N

∑n sin(cn)]J1(z) cos δ

(23)

The quadrature part is relevantly employed, since the component in phase with the reference has a small signal intensity due to the small value of δ and the magnitude of the output has the same value for the signals 180° apart from each other. The measured correlation function g(1)(t) at 1 kHz and E0 ) 1.2 kV/m is shown in Figure 4a. The sample is the same as that used in the measurement of mobility. The time constant of the lock-in amplifier tc is set to 3 ms, which satisfies the condition of eq 18. The solid line in Figure 4a is the best fitted curve of eq 8 with Dq2 ) 5.77 s-1. The diffusion coefficient is estimated as D ) 8.23 × 10-13 m2/s by the Einstein-Stokes relation

kBT D) 6πηa

(24)

where η is the solvent viscosity, kB is the Boltzmann constant, and it is slightly smaller than the measured one. The apparent diffusion coefficients measured at various applied fields E0 are shown as the open circles in Figure 4b. The filled circles are the diffusion coefficients measured for the carboxylate-modified polystyrene latex with a diameter 0.83 µm (Rhone-Poulenc Co.). This sample is neutralized by sodium hydroxide solution and diluted with a solution of 1 mM sodium chloride. As the applied

〈δR(t)〉 ) 0, 〈δR(t1)‚δR(t2)〉 ) 2kBTγδ(t1 - t2)

(26)

〈δq(t)〉 ) 0, 〈δq(t1)‚δq(t2)〉 ) kBTξ/τ exp{-|t1 - t2|/τ} (27) where ξ and τ denote the intensity and the characteristic time of charge fluctuation. Using these relations, the mean square displacement is calculated and the apparent diffusion coefficient Dapp is obtained as

(

Dapp ) 1 +

)

E02ξ 1 ‚ D 2γ 1 + (ω τ)2 0 0

(28)

where D0 is the diffusion coefficient without the external field. The correction term to D0 increases in proportion to E02. Therefore, the coupling of the external field and the fluctuation of the charge below the shear plane may contribute to the electric field dependence of Dapp, in the case of τ , 1/ω0. The other possibility is so-called dielectrophoresis.31,32 Particles in inhomogeneous ac field move toward or away from the regions of high field depending on the difference between the polarizability of the particles and that of the medium. The dielectrophoretic force Fd acting on a spherical particle (permittivity p) suspended in water (permittivity ) can be written31 as

[

Fd ) 2πa3 Re

]

p - ‚∇E02 p + 2

(29)

This effect may be caused by a leakage field in our experiment and also causes a correction which is proportional to E02 in the apparent diffusion coefficient. Joule heating is also another explanation for the increase of Dapp. However, its contribution may not be so large, since the electric field dependence of Dapp is less pronounced in the highly conducting sample (0.833 µm latex in 1 mM sodium chloride solution), as shown in Figure 4. In any case, the previous QELS-SEF cannot give an exact (31) Sauer, F. A.; Schlogl, R. N. Interactions between Electromagnetic Fields and Cells; Plenum: New York, 1985; p 203. (32) Eppmann, P.; Pruger, B.; Gimsa, J. Colloids Surf. A 1999, 149, 443-449.


Langmuir, Vol. 16, No. 24, 2000 9553

Figure 6. Power spectrum of the multiply scattered light in the concentrated latex suspension under the applied electric field of 4.8 kV/m at 600 Hz.

Figure 5. Block diagram of the improved QELS-SEF for the concentrated latex suspension. The illustration of the multiple scattering is schematically drawn at the upper right side of the diagram.

mobility if any of these phenomena arise, but our new measurement of mobility is not influenced at all by these effects. QELS-SEF in Concentrated Suspension of Latex Particles As the concentration of colloidal particles increases, the suspension becomes turbid and incident laser light is scattered out after multiple scattering processes by particles, as schematically drawn at upper right in Figure 5. Recently, QELS has been extended to a very high multiple scattering medium, and it is called diffusing wave spectroscopy (DWS).33 The electric field of the scattered light in DWS is generally given by the extension of eq 1 as

ES(t) )

∑p Ep exp (iq0φp)

(30)

where φp denotes the scattering path length, Ep is the amplitude of the field from path p at the detector, and q0 is the wavenumber of incident light. Equation 30 shows that the scattered light loses its memory due to the change of φp. The correlation time of DWS is much shorter than that in a dilute suspension because all the Brownian motion of colloidal particles along the scattering path during the multiple reflection contributes to the variation of the path length. Therefore, the power spectrum of the concentrated solution is extremely broadened and lowered, which has been recently investigated theoretically34-36 by introducing the characteristic length called the transport mean free path l*. The length l* is defined as the (33) Weitz, D. A.; Pine, D. J. In Dynamic Light Scattering; Brown, W. Ed.; Oxford University Press Inc.: New York, 1993; Chapter 16. (34) Vera, M. U.; Durian, D. J. Phys. Rev. E 1996, 53, 3215. (35) Pine, D. J.; Weitz, D. A.; Chaikin, P. M.; Herbolzheimer, E. Phys. Rev. Lett. 1988, 60, 1134. (36) Zhu, J. X.; Pine, D. J.; Weitz, D. A. Phys. Rev. A 1991, 44, 3948.

distance at which incident light loses its memory of the initial direction by multiple scattering, and it is closely related to the properties of the turbid system such as the size, concentration, or scattering cross section of colloidal particles. Therefore, it is expected to obtain these properties from DWS, but it has not been generally verified experimentally because of the troublesome problems such as the absorption of light, many-body interaction of particles, and interference effects in strong scattering media. Regardless of such uncertainty of DWS, the newly developed system can yield dynamic electrophoretic mobility in such a concentrated suspension with a slight modification. To explain its principle, φp in eq 30 is divided into two parts as

φp ) φp0 + δφpE

(31)

where δφpE is the change of the scattering path length due to the electrophoresis and φp0 is the path length without electrophoresis. If there is no Brownian motion, each colloidal particle is translated parallel to each other, as shown in the upper right of Figure 5. Therefore, the scattering path length during the multiple scattering is preserved and δφpE can be written as

δφpE ) δφpEf + δφpEl ) q‚δrE/q

(32)

where δφpEf and δφpEl are the change of the path length before and after the multiple scattering. Equation 32 is the same as that in the single scattering. Therefore, eq 7 is also available in the concentrated suspension only if cn is replaced by qφp0, and multiple peaks appear at the harmonic frequencies of the applied electric field in the power spectrum of the QELS-SEF. The profile of each peak is not Lorentzian but has the same shape as that observed in the DWS spectrum theoretically. However, the peaks other than the 0-th order one cannot be observed by the same setup34-36 of the usual DWS because the peaks of higher order are completely hidden behind the extremely broadened tail of the 0-th order peak. This undesirable effect can be removed to some extent by focusing the incident laser light and reducing the scattering volume about 50 µm3. That is because the width of the DWS spectrum is closely related to the average number of scattering events, which is almost determined by l* in the usual DWS settings. However, in our case, the signal light is accidentally emitted out and detected before scattered at sufficient times, which is expected from l* inside a cell. Thus, the power spectrum obtained in the concentrated suspension of latex particles with a diameter of 0.6 µm (Dow. Co. Ltd., volume fraction φ ) 1%) is shown in Figure 6. We applied an electric field of 4.8 kV/m at 600 Hz where z ≈ 0.1 is close to the upper limit of the linear electro-

9554


Figure 7. Dynamic mobility µ* in the concentrated latex suspension. The filled circles are the magnitude µ measured with the system of Figure 1. The open circles are those measured with the improved system of Figure 5. The filled squares are the phase delay δ measured with the system of Figure 1.

phoretic region. Focusing the laser light makes the peaks in the power spectrum rather sharp due to the reduced size of the scattering volume, and the small peak of the first order can be observed as shown by the arrow in Figure 6. However, the previous QELS-SEF is not useful in this situation because the exact profile of the power spectrum is not known any longer. Our newly developed system does not need the precise form of the power spectrum and any information of D and l* or the size of the scattering volume in order to extract the dynamic mobility of colloidal particles. The filled circles in Figure 7 are the measured dynamic mobility in the concentrated suspension of latex particles. The incident and reference laser beam is focused on the same spot at the inner surface of the parallel plate glass cell whose inside surface is coated with ITO (indium tin oxide). The reference light reflected by the inner surface of the cell and the backward scattered light are automatically mixed. The mixed signal is detected by the heterodyne technique at a scattering angle θ ) 100°. The magnitude of the projection component of the scattering wave vector q in the direction of the applied electric field is 2.3 × 107 m-1. We measured the same sample as that used in Figure 6. The mobility µ decreases under 1 kHz, but the phase shift remains almost zero. The half width of the bandpass filter used in our system is fixed at about one tenth of the center frequency nω0. Therefore, at low frequencies it becomes narrower than the half width of the peaks in the power spectrum in concentrated suspension and the whole signal cannot pass through the filter. This causes the apparent decrease of the magnitude under 1 kHz, and it is in good agreement with the half width (about 100 Hz) of the 0-th and first-order peak in Figure 6. To improve this unfavorable situation, the amplitude of the incident light is modulated at the angular frequency Ω with a chopper. As the amplitude of the detected signal is also modulated at Ω, the n-th order signal appears at the frequencies Ω ( nω0. Either the frequency of (Ω + nω0)/2π or (Ω - nω0)/2π is always set around at 500 Hz by

Mizuno et al.

controlling Ω. The referenced signal of the lock-in amplifier was made by multiplying the monitored signal of the chopper with the monitored signal of the output of the high-voltage amplifier. The block diagram of the improved system is shown in Figure 5. The open circles in Figure 7 represent the mobility measured with the improved system. It is obvious that the apparent decrease of µ at low frequency disappears. The dynamic mobilities in the dilute (Figure 3) and the concentrated (Figure 7) suspensions are not so much different from each other. Reed and Morrison37,38 have shown for the colloidal suspension with a thin double layer that the hydrodynamic and electrostatic interactions cancel one another even in highly concentrated systems, and the effect which must be taken into account is only the reverse flow of fluid displaced by moving particles. Therefore, the correction for the mobility in the concentrated suspension of volume fraction φ is on the order of φ,37,38 which is within the experimental error in our system. The phase delay δ shows the relaxation behavior at higher frequencies than 10 kHz, and its characteristic frequency seems to be higher than that in the dilute sample. This may be due to the difference in conductivity of a bulk solution. The ions dissolved from latex particles may also give contributions to the conductivity. Therefore, the relaxation by the anomalous surface conduction mentioned before probably goes to the higher frequency according to eq 22. Conclusion We developed a new method to measure the dynamic mobilities and diffusion constants of colloidal particles in a much wider frequency range than that of the previously developed methods. The frequency spectrum of mobility in dilute and concentrated suspensions shows signs of relaxation at frequencies higher than 50 kHz, and the apparent diffusion coefficient depends on the applied electric field. These results are ascribable to the polarization of the double layer amplified by the anomalous conduction below the shear plane. Unfortunately, the present study that was originally planned to observe a much slower process (concentration polarization) is insufficient for further quantitative analysis. At present, the highest frequency which our system attains (50 kHz) is only determined by the performance of the high-voltage amplifier and the lock-in amplifier. It is possible to extend the available frequency range because the signal-to-noise ratio is sufficiently large even at 50 kHz. More detailed discussion will be given elsewhere with further experiments at a controlled ion concentration and in a much wider frequency range. LA000821H (37) Reed, L. D.; Morrison, F. A. J. Colloid Interface Sci. 1976, 54, 117. (38) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1987, 115, 422-436.

Dynamic Electrophoretic Mobility of Colloidal Particles Measured by

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