The Dynamic Mobility

In the previous section we showed how a sinusoidal voltage pulse generates sinusoidal pulses of sound that appear to come from the electrodes. These ultrasound signals are measured in practice by attaching a pressure transducer to the material behind one of the electrodes. Two parameters can be obtained from this measurement the amplitude of the sound wave and the phase of the wave relative to the applied sinusoidal voltage. 

To determine the link between these measurements and the particle properties it is necessary to solve the mathematical equations that represent Rules 1 and 2 in the previous section. For the planar geometry described in that section, O Brien etal. [17]  

3) This result has appeared in many earlier publications, but the derivation was not published until 2003. 

The argument, or phase, of the dynamic mobility is a measure of how much the particle velocity lags behind the applied fielcL As we shall see, there are several reasons why the particle velocity may lag, but most often this lag is caused by the particle inertia at the MHz frequencies that are used for ESA measurements. 

4) This statement applies to thin-film electrodes. If the electrode is thick it is necessary to include the extra mass in the acoustic impedance formula. 

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Contreras A, Hale TK, Stenoien DL, Rosen JM, Mancini MA, Herrera RE (2003) The dynamic mobility of histone HI is regulated by cyclin/CDK phosphorylation. Mol Cell Biol 23(23) 8626-8636 Crosio C, Fimia GM, Loury R, Kimura M, Okano Y, Zhou H, Sen S, Allis CD, Sassone-Corsi P (2002) Mitotic phosphorylation of histone H3 spatio-temporal regulation by mammalian Aurora kinases. Mol Cell Biol 22(3) 874-885... 

We thus obtain the following expression for the dynamic mobility of a spherical polyelectrolyte ... 

Equation (25.45) is the required expression for the dynamic mobility of a soft particle, applicable for most practical cases. When co 0 fi 2, y 0, and F 0), Eq. (25.45) tends to Eq. (21.51) for the static case. When the polyelectrolyte layer... 

The original Acoustosizer used a single frequency whereas a later development has a range of 13 frequencies between 0.3 and 13 MHz. This allows the measurement of the dynamic mobility spectrum and the determination of the zeta potential and particle size. In order to invert the mobility spectrum into a size distribution a log-normal distribution of particle size is assumed. A comparison with photon correlation spectroscopy for determining particle size and laser Doppler anemometry for particle charge eonfirmed the results using ACS [266]. These and additional sedimentation measurements confirmed that changes in particle size and zeta potential due to dilution effects are likely to occur in aqueous and non-stabilized systems. 

Figure 7.19 Shear yield stress versus square of the zeta potential for the dispersions described in Fig. 7-18 at particle volume fractions (p = 0.184 and 0.213, or mass percentages of 57.0% and 61.4%. The zeta potential was obtained at low (p from the dynamic mobility. (From Leong et al. 1993, reproduced by permission of The Royal Society of Chemistry.)...

The dynamic mobility which is determined by the electro-acoustic effects differs from the low-frequency or DC mobility that is determined by electrophoresis... 

A linear relationship exists between the ESA or CVP amplitude and the volume fraction of the suspended particles. At relatively high-volume fractions, hydrodynamic and electric double-layer interactions lead to a non-linear dependence of these two effects on volume fraction. Generally, non-linear behavior can be expected when the electric double-layer thickness is comparable to the interparticle spacing. In most aqueous systems, where the electric double layer is thin relative to the particle radius, the electro-acoustic signal will remain linear with respect to volume fraction up to 10% by volume. At volume-fractions that are even higher, particle-particle interactions lead to a reduction in the dynamic mobility. 

Figure 2. Imaginary part of the complex susceptibility X (co) versus normalized frequency rjco for various values of the reaction field parameter. SoUd lines correspond to the matrix continued fraction solution, Eqs. (37) and (51) circles correspond to the smaU oscillation solution, Eq. (26) dashed lines correspond to the approximate small oscillation solution Eq. (53) and dotted lines correspond to the solution based on the dynamic mobility, Eq. (58).

ESA is directly proportional to the dynamic mobility, whereas UVP requires knowledge of the high-frequency conductivity of the slurry. For this reason, ESA is frequently used for routine analysis of powder slurries under processing conditions [75]. An example of the application of ESA to powder characterization is given in Fig. 4, where the effect of surface cleaning on a SisN4 powder is evidenced by a shift in pHiep of the aqueous slurry. This shift is caused by a reduction of the surface oxide layer thickness. 

These in-situ generated emulsions aided in tertiary recovery by improving the areal sweep efficiency of the alkaline slug and by improving the dynamic mobility ratio within the core. 

The dynamic mobilities of the cationic and anionic side chain of amphoteric copolymers of poly(sodium-2-methacryloyloxyethanesulfonate-co-2-metha-cryloyloxyethyltrimethylammonium iodide) (NaMES-METMAI) were estimat-... 

The particle property which is extracted from the measured ESA response is the dynamic mobility, of the drops. This is a complex quantity, having a magnitude and a phase angle (just as the ESA signal is a complex quantity). The magnitude of p j is analogous to the electrophoretic mobility obtained in, say, an electrophoresis experiment, where a d.c. field is applied. It is essentially determined by... 

To determine the precise relationship between the ESA signal and the dynamic mobility one must solve the set of differential equations given by O Brien in his 1990 paper (10). For the AcoustoSizer that problem is simplified by the geometry because the electrode dimensions and separation are both large compared to the wavelength of the sound (of millimeter order at the frequencies used). In that case the relation is given by O Brien et al. (12) as ... 

The analysis of the relationship between the dynamic mobility and the particle properties has been made possible by the development of special procedures for dealing with systems in which the double layer around the particle or droplet is thin compared to the radius of curvature. The double-layer thickness is measured by the Debye-Hiickel parameter k which is related to the ionic strength of the electrolyte (13). For a 1 mM solution of a 1 1 electrolyte, the double-layer thickness, k, is about 10 nm and it decreases as the square root of the concentration, so for a 0.1 M solution it would be about 1 nm. The double layer is regarded as thin if the ratio of radius to thickness (ka) exceeds about 20 and that will be the case for most normal emulsions at most electrolyte concentrations. 

O Brien has shown (10) that for a dilute suspension of spherical particles (less than about 4% by volume, say) wifli thin double layers, the dynamic mobility is related to the particle properties as follows ... 

The factor (1 in Eq. (2) measures the tangential electric field at the particle siuface. It is this component which generates the electrophoretic or electroacoustic motion. For a fixed frequency, it can be seen from Eq. (4) that (1 +J) depends on the permittivity of the particles and on die function X - Kg/K a, where Ks is the surface conductance of the double layer X measures the enhanced conductivity due to the charge at the particle surface. It is usually small unless the zeta potential is very high, so for most emulsions with large ka, X has a negligible effect. The ratio fp/f is also small for oil-in-water emulsions. Equation (4) can then be reduced to/= 0.5 and hence the dynamic mobility becomes ... 

Figure 3 Comparison of theoretical (10) and experimental values of the magnirnde and phase of the dynamic mobility for a silica sol of radius 300 nm.

One immediate effect of increasing the particle concentration in the emulsion is that the acoustic impedance, Zg, can no longer be approximated as equal to that of the dispersion medium. Since Eq. (1) remains valid at all concentrations commonly encountered, it is important that the correct value of Zg is used, so that the correct value of the dynamic mobility is obtained from the measured ESA signal. In principle, the value of Zg for the emulsion could be a complex function of the frequency and the properties of the suspension, but the exact behavior is of little consequence for measurements with the AcoustoSizer, since it measures the value at each frequency before calculating from ESA signal. 

Figure 8 (a) Comparison of the magnitude of the dynamic mobility of the emulsion with the calculated values for low (103 mV) and high (175 mV) zeta potential (b) the same for the phase angles. 

Fignre 9 (a) Magnitude of the dynamic mobility as a function of frequency for various volume fractions for a particle of radius 1 um (b) phase angles for the same conditions as in (a). 

Assuming (as it is reasonable) that for conditions in which the approximation ko 5> 1 is valid, the dynamic mobility also contains the (1 — Cq) dependence displayed by the static mobility (Equation (3.37)), one can expect a qualitative dependence of the dynamic mobility on the frequency of the field as shown in Figure 3.14. The first relaxation (the one at lowest frequency) in the modulus of u can be expected at the a-relaxation frequency (Equation (3.55)) as the dipole coefficient increases at such frequency, the mobility should decrease. If the frequency is increased, one finds the Maxwell-Wagner relaxation (Equation (3.54)), where the situation is reversed Re(Cg) decreases and the mobility increases. In addition, it can be shown [19,82] that at frequencies of the order of (rj/o Pp) the inertia of the particle hinders its motion, and the mobility decreases in a monotonic fashion. Depending on the particle size and the conductivity of the medium, the two latter relaxations might superimpose on each other and be impossible to distinguish. 

FIGURE 3.14 Modulus of the dynamic mobility and real part of the induced dipole coefiBcient as a function of frequency for particles 500 nm in radius in 0.1 mmol/1 KCl solution. Zeta potential —150 mV. 

Dukhin et al. [83-85] have performed the direct calculation of the CVI in the situation of concentrated systems. In fact, it must be mentioned here that one of the most promising potential applicabilities of these methods is their usefiilness with concentrated systems (high volume fractions of solids, 4>) because the effect to be measured is also in this case a collective one. The first generalizations of the dynamic mobility theory to concentrated suspensions made use of the Levine and Neale cell model [86,87] to account for particle-particle interactions. An alternative method estimated the first-order volume fraction corrections to the mobility by detailed consideration of pair interactions between particles at all possible different orientations [88-90]. A comparison between these approaches and calculations based on the cell model of Zharkikh and Shilov [91] has been carried out in Refs. [92,93],... 

R. J. Hunter, Recent developments in the electroacoustic charactiaisation of colloidal suspensions and emulsions. Colloids Surf. A 141(1), 37-65 (1998). doi 10.1016/S0927-7757(98)00202-7 C. Knosche, Mdglichkeiten und Grenzen der elektroakustischen Spektroskopie zur Gewinnung von Partikelgrofieninformationen. PhD thesis, Technische Universitiit Dresden, 2001 M. Loewenberg, R.W. O Brien, The dynamic mobility of nonspherical particles. J. Colloid Interface Sci. 150(1), 158-168 (1992). doi 10.1016/0021-9797(92)90276-R R.W. O Brien, Electro-acoustic effects in a dilute suspension of spherical particles. J. Fluid Mech. 

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