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Observation of Field-Dependent Electrophoretic Mobility with Phase Analysis Light Scattering (PALS) John C. Thomas,* Bryan J. Crosby, Roland I. Keir,† and Kathryn L. Hanton Laser Light Scattering & Materials Science Group, University of South Australia, Mawson Lakes, Adelaide, SA 5095, Australia Received December 5, 2001. In Final Form: February 20, 2002 The measurement of electrophoretic mobility is commonly done by laser Doppler electrophoresis (LDE). Recently, a more sensitive technique, phase analysis light scattering (PALS), has been developed with particular benefits for application to nonpolar systems. We have carried out PALS measurements of the electrophoretic mobility (µ) of modified silica particles as a function of applied electric field. Our results are in poor agreement with LDE measurements on the same system. We show that this difference might result from the way in which PALS measurements are performed and suggest that, in this case, the known field-dependent mobility will cause artifacts in the PALS data. When the mobility is field-dependent, this factor must be accounted for appropriately in PALS data analysis.
Introduction The charge on colloidal particles suspended in liquids is a key determinant of the characteristics and behavior of such dispersions, e.g., particle charge determines dispersion stability (whether particles aggregate and sediment out), viscosity, extent of fouling, and so on. Particle charge in colloidal dispersions is conventionally monitored by determining the electrophoretic mobility of the colloids. The charging behavior of particles in nonpolar liquids is important for a number of technologies including toners for nonaqueous inks in printing applications1 and for stabilization and charging of dispersions in display technologies.2 Whereas particle charging behavior is reasonably well understood for aqueous systems, it remains problematic for nonpolar media. One reason for this is the difficulty of measuring or monitoring particle charge in these systems. Historically, electrophoretic mobilities in nonpolar media have been difficult to measure using conventional techniques such as microelectrophoresis or laser Doppler electrophoresis (LDE). This is because the mobilities encountered are typically 2 orders of magnitude smaller than those prevailing in polar liquids. Phase analysis light scattering (PALS) is a technique capable of measuring very small mobilities (∼10-13 m2/ V‚s). The development of the PALS technique has enabled rapid and reproducible measurements of small electrophoretic mobilities, as seen in nonpolar liquids, to be made.3 We have developed a PALS system in our own laboratory and have recently reported results from this technique using Irgalite Blue pigments in Isopar G4 and silica particles suspended in AOT/decane.5 * Corresponding author:
[email protected] www.laser. unisa.edu.au. † Current address: Institute of Physics, University of Fribourg, Chemin du Muse´e 3, Pe´rolles, 1700 Fribourg, Switzerland. (1) Bartchser, G.; Breithaupt, J. J. Imaging Sci. Technol. 1996, 40, 441. (2) Comiskey, B.; Albert, J. D.; Yoshizawa, H.; Jacobson, J. Nature 1998, 394, 253. (3) Miller, J. F.; Schatzel, K.; Vincent, B. J. Colloid Interface Sci. 1991, 143, 532. (4) Keir, R. I.; Quinn, A.; Jenkins, P.; Thomas, J. C.; Ralston, J.; Ivanova, O. J. Imaging Sci. Technol. 2000, 44, 528. (5) Keir, R. I.; Suparno; Thomas, J. C. Charging Behaviour in the Silica/Aerosol OT/Decane System. Langmuir 2002, 18, 1463.
In conventional microelectrophoresis or LDE, an effectively constant electric potential (and hence field) is applied to a sample via electrodes, and the resulting particle velocity/mobility is measured.6 The potential is typically a 1- or 2-Hz square wave, and the data are collected during one or both of the half-cycles when the field is constant. For PALS measurements, the electrode drive is typically a 30-Hz sinsusoid, and the data collection is synchronized to this waveform. Clearly, in the latter case, data are collected in the presence of a (sinusoidally) varying electric field. In the (usually assumed) case of materials for which the electrophoretic mobility does not depend on the applied field and/or small enough applied fields, this is not important, and PALS determines the correct mobility (as determined by other techniques). However, there are materials for which the mobility might not be field-independent. An obvious important example is toner systems, which are typically used in combination with high electric fields and which might display a pronounced dependence of mobility on applied field. Jin et al.7 performed LDE measurements to determine the electrophoretic mobility of CN-surface-modified silica particles dispersed in Isopar M as a function of the applied electric field strength over a range of aerosol OT concentrations. They observed a pronounced variation of the electrophoretic mobility, µ, with electric field, E. The mobility was negative. Broadly speaking, the data indicated that µ was constant at lower field strengths (∼10 kV/m), but then rose monotonically to a higher (more negative) plateau value at ∼200 kV/m. A model of field strength “stripping” the charge from the colloidal particles, thus increasing their zeta potential and hence µ, was invoked to describe the data. We have made measurements on a similar silica system using the PALS technique and found that, although, in our case, a field dependence was observed, the rate of increase of µ with E was less rapid and a plateau was not reached. Rather, the measured mobility reached a maximum value at ∼600 kV/m and then began to decrease. Thus, we needed to look more closely at the PALS (6) Uzgiris, E. E. Prog. Surf. Sci. 1981, 10, 53. (7) Jin, F.; Davis, H. E.; Evans, D. F. In IS&T’s Fourteenth International Congress on Digital Printing Technologies; Society for Imaging Science and Technology: Springfield, VA, 1998; pp 206.
10.1021/la011758e CCC: $22.00 © 2002 American Chemical Society Published on Web 04/30/2002
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measurement and data analysis technique to reconcile the results. The current theory of PALS3 explicitly assumes that the electrophoretic mobility does not depend on the applied electric field. In this case, the measured signal (the amplitude-weighted phase difference or AWPD) is a simple cosinusoid, at the same frequency as the electrode drive, that can be fitted to determine the (constant) mobility. When the mobility is field-dependent, we anticipate that the AWPD function is not a simple cosinusoid. In fact, we have observed that the AWPD function contains what appears to be harmonics for samples measured at high electric field, where we believe there is a field-dependent mobility. This harmonic distortion was not removed by increased averaging, which would normally remove noise from the signal. Thus, we inferred that the field-dependent mobility was producing the distorted AWPD function. This is analogous to the effect in Fourier transform rheology that indicates a strain-dependent viscosity, as addressed extensively by Wilhelm et al.8 Clearly, in such cases, the existing PALS analysis model should not be used, and an alternative must be developed for the technique to be applied legitimately.
For
E(t) ) E0 sin(ωEt + φ)
we have
〈Q(t) - Q(0)〉 N
〈Q(t) - Q(0)〉 )
∫0tqA(t′) µ(E) E(t′) dt′
(1)
Here, is the amplitude-weighted phase difference, q is the magnitude of the scattering vector, A(t) is the amplitude of the complex Doppler signal, µ(E) is the (field-dependent) electrophoretic mobility, and E(t) is the applied electric field. Assuming that A(t) is not fielddependent but maintaining a field-dependent mobility, we can proceed similarly to Miller et al.3 and write
∫
t
µ(E) E(t′) dt′ 0
) 〈A〉qµ*
Ik )
I1 )
k)1
N
I2 ) )
[ ∫[
∫0t e t
0
Rk[
(8) Wilhelm, M.; Maring, D.; Spiess, H.-W. Rheol. Acta 1998, 37, 399.
k
dt′
(7)
i(ωEt′+φ)
(8)
]
- e-i(ωEt′+φ) 2i
2
dt′
]
e2i(ωEt′+φ) + e-2i(ωEt′+φ) - 2 -4
[
2
dt′
]
-1 e2i(ωEt′+φ) e-2i(ωEt′+φ) + - 2t′ 4 2ωEi -2ωEi
[
]
-1 e2i(ωEt+φ) - e-2i(ωEt+φ) t 2 4ωE 2i
)
t sin 2(ωEt + φ) 2 4ωE
[
∫0t e
)3
i(ωEt′+φ)
[
[
]
- e-i(ωEt′+φ) 2i
3i(ωEt′+φ)
∫0t e
1 4
)-
)-
∑ ∫0 E (t′) dt′] k)1
) 〈A〉qµ*
]
)
I3 )
k
[
- e-i(ωEt+φ) 2i
i(ωEt+φ)
∫0t e
∫0tsin(ωEt′ + φ) dt′
(4) t
∑ RkE0 Ik
cos φ - cos(ωEt + φ) ) ωE
(3)
∫0t[µ* ∑ RkEk-1(t′)]E(t′) dt′
(6)
k
We arbitrarily limit k to a maximum value of 5 and consider only terms up to fifth order in E0. The Ik can be evaluated for these terms as follows
N
〈Q(t) - Q(0)〉 ) 〈A〉q
dt′
where
(2)
where µ* is the electrophoretic mobility at low (zero) applied electric field. Note that R1 ) 1 so that, in the limit of low (zero) applied electric field, the mobility will be µ*. Now, the AWPD is given by
]
2i
k
k)1
N
∑ RkEk-1(t) k)1
[
ei(ωEt+φ) - e-i(ωEt+φ)
N
For the point of illustration, we assume a polynomial field dependence of the mobility so that
µ(E) ) µ*
∫0
t
RkE0k
k)1
) 〈Q(t) - Q(0)〉 ) 〈A〉q
∑
) 〈A〉qµ*
Phase Analysis Light Scattering Theory In a PALS experiment, the quantity ultimately measured is the mean value of the AWPD signal. This is conveniently determined by temporal averaging over many equivalent cycles of the electric field applied to the electrodes, E(t), to obtain
(5)
ei(ωEt+φ) - e-i(ωEt+φ) ) E0 2i
t
(9) 0
3
dt′
]
- e-3i(ωEt′+φ) 2i ei(ωEt′+φ) - e-i(ωEt′+φ) dt′ (10) 3 2i
[
]
∫0tsin 3(ωEt′ + φ) - 3 sin(ωEt′ + φ) dt′
1 4
]
cos φ - cos(ωEt + φ) 4ωE cos 3φ - cos 3(ωEt + φ) 12ωE
Observation of Electrophoretic Mobility with PALS
I4 )
[
i(ωEt′+φ)
∫0t e
[
]
- e-i(ωEt′+φ) 2i
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4
dt′
]
+ e-4i(ωEt′+φ) 2 e2i(ωEt′+φ) + e-2i(ωEt′+φ) + 3 dt′ 4 2 (11) 1 t ) 0 cos 4(ωEt′ + φ) - 4 cos 2(ωEt′ + φ) + 8 3 dt′ )
4i(ωEt′+φ)
∫0t e
1 8
[
]
∫
)
I5 )
sin 2φ - sin 2(ωEt + φ) 4ωE sin 4φ - sin 4(ωEt + φ) 3t + 32ωE 8
[
i(ωEt′+φ)
∫0t e
[
]
- e-i(ωEt′+φ) 2i
5
dt′
]
- e-5i(ωEt′+φ) 2i e3i(ωEt′+φ) - e-3i(ωEt′+φ) + 5 2i i(ωEt′+φ) e - e-i(ωEt′+φ) dt′ 10 2i (12) 1 t sin 5(ωEt′ + φ) - 5 sin 3(ωEt′ + φ) + ) 0 16 10 sin(ωEt′ + φ) dt′ )
5i(ωEt′+φ)
∫0t e
1 16
[
[
]
]
∫
[
]
cos φ - cos(ωEt + φ) )5 8ωE cos 3φ - cos 3(ωEt + φ) + 5 48ωE cos 5φ - cos 5(ωEt + φ) 80ωE
[
]
The terms for k ) 1-5 can be collected, and we can write the AWPD signal as
〈Q(t) - Q(0)〉 ) 〈A〉qµ* (8R1E0 + 6R3E03 + 5R5E05) [cos φ - cos(ωEt + φ)] + 8 ωE
{
(R2E02 + R4E04) [sin 2φ - sin 2(ωEt + φ)] 4 ωE (4R3E03 + 5R5E05) [cos 3φ - cos 3(ωEt + φ)] 48 ωE R4E04[sin 4φ - sin 4(ωEt + φ)] + 32 ωE
(13)
}
R5E05[cos 5φ - cos 5(ωEt + φ)] (4R2E02 + 3R4E04)t + 80 ωE 8
Equation 13 reveals several things. First, we observe that, for an assumed polynomial field dependence of the mobility, we anticipate that the AWPD function will contain a fundamental term at the electrode drive frequency and harmonic components. Second, we see that
there is a term linear in t that causes a slope to appear in the AWPD signal. This slope could be mistakenly ascribed to a sedimentation component and nulled out, as suggested by Miller et al.3 We note that most smooth functions can be approximated by a polynomial of sufficiently high order, so that the behavior predicted above should be quite general for a system that has a fielddependent mobility. It can be seen in eq 13 that the terms associated with increasing powers of the applied field are manifested in the AWPD in two ways. As mentioned above, they give rise to higher-frequency components directly in the function, and perhaps more importantly, they also couple back and make a contribution to the fundamental and other lower-frequency terms. This effect might be more important than the introduction of the high-frequency components, as they are weighted much more strongly in the fundamental. For example, the relative weighting of the 4th harmonic (5ωE term) directly in the AWPD function compared with its contribution back into the fundamental is (5/8)/(1/80) ) 1/50. This means that a high-frequency component might be affecting the magnitude of the fundamental long before it is observable as a distinct spectral component. In other words, even though these components might not be directly detectable in the AWPD, they will already be biassing the measured value of the mobility as it is determined from the magnitude of the fundamental in the AWPD function. In summary, for even a simple field dependence of the mobility, we expect harmonics of the electrode drive to appear in the AWPD signal at high applied electric fields. Experimental Section PALS Measurements. Figure 1 shows a schematic of our PALS apparatus, which is based on the initial design of Miller et al.3 The system produces a moving fringe pattern, and the phase shift between the light scattered from the particles and the moving fringe is detected. Light from a 5-mW HeNe laser is passed through a long-focal-length lens (f ) 850 mm) and into a 50% beam splitter. The two resulting beams are passed through acousto-optic modulators, or Bragg cells (Isomet model 1205C2). One Bragg cell is driven by one channel of an acousto-optic driver (Brimrose model FFA-80) at 80 MHz. The other Bragg cell is driven by the other, single-sideband channel of the acoustooptic driver at a frequency of 80 MHz plus a modulation frequency provided by the lock-in amplifier (Stanford Research Systems model SR830). For the current experiments the modulation frequency was set at 33 kHz. Thus, one laser beam is frequencyshifted by 80 MHz and the other by 80 MHz + 33 kHz, and the 33-kHz difference frequency gives rise to the moving fringe pattern. Mirrors are then used to direct the two beams so that they intersect in the center of the sample cell. The beams cross at an angle in the cell of 10.6°, giving rise to a set of moving fringes spaced 3.4 µm apart. The scattering cell contains a pair of parallel-plate palladium electrodes with a separation of 2.18 mm and a width of 6 mm. The electrodes are driven by a high-voltage amplifier (Trek model 610D), which, in turn, is driven by a programmable function generator (Stanford Research Systems model DS335). The electrode drive in these experiments was a 30-Hz sinusoid. The forward-scattered light from the particles is collected by a fiber optic probe consisting of a SELFOC lens and a single-mode fiber9 and detected by a photomultiplier tube (Hammamatsu model R649). The photomultiplier output is fed to the lock-in amplifier, which performs lock-in detection against the 33-kHz modulation frequency. The averaging time constant for the output of the lock-in amplifier was set at 100 µs. The x and y outputs of the lock-in amplifier are acquired by the control computer via a highspeed data acquisition card (National Instruments model PCI(9) Suparno; Duerloo, K.; Stamatelopolous, P.; Srivastva, R.; Thomas, J. C. Appl. Opt. 1994, 33, 7200.
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Figure 1. Phase analysis light scattering (PALS) system. AOM, acousto-optic modulator; AOD, acousto-optic driver; SELFOC, fiber optic probe detector; PMT, photomultiplier tube; FG, function generator; TC, temperature control unit; GPIB, general-purpose interface bus controller; NIDAQ, data acquisition cards; PC, laboratory computer. 6035E). A GPIB controller card (National Instruments model AT-GPIB/TNT) is used for computer control of both the lock-in amplifier and the function generator. To facilitate measurement of harmonic distortion of the AWPD function for PALS, some modifications to the data collection and analysis are incorporated. First, data are only sampled for periods that are multiples of the electric field period. Second, oversampling of the lock-in amplifier output was included into the data capture algorithm to increase the signal-to-noise ratio, as inspired by the high-sensitivity Fourier transform rheology of Wilhelm et al.10 Also, the underlying slope on the AWPD function was removed. This means that the slope of the AWPD function will not introduce artifactual frequency components into the analysis results. Miller et al.3 suggested that these slopes in the AWPD function were due to constant velocities imposed on particles by thermal vortices. As we showed above, a slope on the AWPD function is also attributable to field-dependent µ. Note, however, that this term depends only on the even Rk and will be small. Samples. Silica spheres with nominal diameter 7 µm with -CN surface terminations (Mac-Mod catalog no. 820962102) were used in this work. Aerosol OT (AOT) (Fluka, ∼98% pure) was purified by dissolution in methanol (Univar, analytical grade) and tumbled with activated charcoal. This dispersion was filtered and centrifuged to obtain a clear supernatant. The methanol was extracted by rotary evaporation, and any remaining methanol was removed by heating in an oven at 120 °C for 24 h. The purified AOT was kept in a desiccator. AOT solutions for the electrophoretic mobility measurements were prepared from a stock solution of 5 mM AOT in Isopar M, a light paraffin oil. The stock was then tumbled overnight. Dispersions were made by adding 10 mg of silica particles to 10 mL of the AOT solution in a clean, dry glass vial so that the final silica concentration was 1 mg/mL. The dispersions were then tumbled overnight. A sample of the dispersion was then put into a clean quartz cuvette for measurement in the PALS apparatus.
Results PALS measurements were made on samples of 1 mg/ mL CN-Si suspended in decane with 5 mM AOT for field amplitudes from 50 kV/m to 1 MV/m. In all cases, a negative mobility was observed. Figure 2 shows the resulting mobility measured as a function of electric field amplitude. These values are determined by simply analyzing the data using our standard curve-fitting model for a field independent mobility. The mobility is seen to (10) Wilhelm, M.; Reinheimer, P.; Ortseifer, M. Rheol. Acta 1999, 38, 349.
Figure 2. Electrophoretic mobility determined by phase analysis light scattering as a function of the applied electric field.
increase (become more negative) with applied field, reaching a maximum at 600 kV/m, and then decrease. During the course of these measurements, it was observed that, as the field amplitude was increased above ∼400 kV/m, the AWPD functions began to deviate visibly from the sinusoidal shapes observed at low fields. Apparently, higher-frequency components were increasingly present in the data. This can be seen in Figure 3, which shows representative measured AWPD functions for increasing values of applied field. We performed a fast Fourier transform (FFT) analysis on these data to determine the spectral components present. The spectra corresponding to each of the AWPD functions are shown on the right-hand side of Figure 3. Note that the vertical axes have a logarithmic scale. At 100 kV/m, the AWPD data really only contain one component, the fundamental at the electrode drive frequency of 30 Hz. As the applied field is increased, harmonics of the fundamental begin to appear and become significant above ∼400 kV/m. This can be readily observed in Figure 4, which shows the amplitudes of the harmonics as a percentage of the 30-Hz fundamental. As can be seen, the harmonics are all at the few-percent level below a field of about 400 kV/m. However, beyond this value, the harmonic content increases rapidly, and at 1 MV/m, the second harmonic (3ωE component) is 36% of the amplitude
Observation of Electrophoretic Mobility with PALS
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Figure 3. (a) AWPD functions for the silica/aerosol OT/decane system as a function of applied electric field. (b) Amplitude spectra of AWPD functions.
Conclusion
Figure 4. Relative amplitudes of harmonics contained in the AWPD function as a function of the applied electric field.
of the fundamental. Interestingly, the odd harmonics (3ωE, 5ωE, 7ωE) dominate the even harmonics, so that, at 1 MV/ m, the odd harmonics are 57% of the fundamental amplitude whereas the even harmonics are only ∼3%. It is also apparent from Figures 3b and 4 that to analyze the AWPD data and extract the mobility correctly, it will be essential to consider harmonic components up to high order (∼7ωE).
We have observed that, when PALS measurements are made on a sample that has a field-dependent mobility, the resulting AWPD function contains harmonics of the electrode drive frequency. These harmonics become increasingly apparent as the applied field amplitude is increased and the AWPD function becomes distinctly nonsinusoidal. This is in agreement with our analysis of the AWPD function for a field-dependent mobility. This analysis predicts that the AWPD function will contain harmonics of the electrode drive, as observed. This observation and prediction have not been reported previously. It is clear from the present work that the PALS technique must be used with caution for a sample with a field-dependent mobility. Furthermore, if the PALS technique is used, an appropriate analysis that accounts for the different form of the AWPD function must be used to recover meaningful results. Acknowledgment. This research was supported in part by the Australian Technology Network (ATN) Small Research Grants Scheme. LA011758E
