electroacoustic theory...

ELECTROACOUSTIC THEORY (extraction from Chapter 5, [52])

Electroacoustic phenomena, first predicted by Debye in 1933 [1] for electrolytes, arise from coupling between acoustic and electric fields. Debye realized that, in the presence of a longitudinal sound wave, any differences in the effective mass or friction coefficient between anions and cations would result in different displacement amplitudes. In turn, this difference in displacement would create an alternating electric potential between points within the solution. Indeed, this phenomenon is measurable and can yield useful information about the properties of ions. It is usually referred to as an "Ion Vibration Potential" (IVP). The first experimental observations of this IVP were reported by Yeager [2] in 1949, and by Derouet and Denizot [12] in 1951. A thorough theoretical treatment of the IVP phenomenon was given in 1947 by Bugosh et al [11]. There was a lot of interest in this effect in the 1950’s and 1960’s, because it was considered to be a very promising tool for characterizing ion solvation [2-12]. Zana and Yeager [10] summarized the results of this two decade effort in a review, which we follow here, in our presentation of IVP in this book. Sadly, this phenomenon has been largely forgotten, and virtually all of the interest in electroacoustic phenomena has shifted from pure electrolyte solutions to colloids.

Hermans [14] and Rutgers [15] in 1938 were the first to report a Colloid Vibration Potential (CVP). Since that time there have been several hundred experimental and theoretical works published. For brevity, we mention here only a few key papers. Enderby and Booth [16, 17] developed the first theory for CVP in the early 1950’s. The first quantitative experiments were made in 1960 by Yeager’s group [10]. In the early 1980’s Cannon with co-authors [18] discovered an inverse electroacoustic effect, that they termed ElectroSonic Amplitude (ESA). The first commercially available electroacoustic instrument was developed by Pen Kem, Inc [19]. There are now several commercially available instruments based on both electroacoustic effects. These are manufactured by Colloidal Dynamics, Dispersion Technology, and Matec.

After the basic works by Enderby and Booth [16, 17], the electroacoustic theory for colloids has been developing in two quite different directions. The original Enderby/Booth theory was very complex; a result of considering both surface conductivity and low frequency effects. In addition, it did not take particle-particle interactions into account, and consequently was valid only for dilute colloids. Hence, this Enderby/Booth theory required modifications and simplifications. The first extension of the Enderby-Booth approach was performed by Marlow, Fairhurst and Pendse [20]. They attempted to generalize it for concentrated systems using a Levine cell model [21]. This approach leads to somewhat complicated mathematical formulas, and perhaps this was the reason that it was abandoned. An alternative approach to electroacoustic theory was later suggested by O’Brien [22]. He introduced the concept of a dynamic electrophoretic mobility, m _d , and suggested a simple relationship between this parameter and measured electroacoustic parameters, such as Colloid Vibration Current (CVI) or Electric Sonic Amplitude (ESA). Later, O’Brien stated that his relationship is valid for concentrated system as well as dilute systems [34].

According to the O’Brien relationship, the average dynamic electrophoretic mobility, m _d , is defined as:

where the ESA is normalized by the applied external electric field, A (ω) is an instrument constant found by calibration, and F (Z_T, Z_s) is a function of the acoustic impedances of the transducer and the dispersion under investigation.

A similar expression can be used for the CVI mode:

Here the CVI is assumed to be normalized by the gradient of pressure (grad P), which, in the case of CVI, is the external driving force.

According to O’Brien, a complete functional dependence of ESA (CVI) on key parameters, such as zeta potential, particle size and frequency, is incorporated into the dynamic electrophoretic mobility. O’Brien stated that for all considered cases the coefficient of proportionality between ESA (CVI) and m _d, is frequency independent, and, in addition, is independent of particle size and zeta potential. This feature made the dynamic electrophoretic mobility an important and central parameter of the electroacoustic theory.

The first theory that relates this dynamic electrophoretic mobility to the properties of the dispersion medium and dispersed particles was initially created by O’Brien, but it neglected particle-particle interactions and, was therefore limited to the dilute case. We shall call this version the "dilute O’Brien theory". Later, Rider and O’Brien [23], Ohshima [24], and Ennis, Shugai and Carnie [25, 26] suggested modifications to extend this approach to concentrated systems.

Figure 1 is a block diagram intended to classify the complicated relationships between the various electroacoustic theories.

Because it appeared to yield a desirable electroacoustic theory for the concentrated case, the O’Brien approach appeared superior to the Enderby-Booth approach. However, one important question remained unanswered. In principle, these two approaches must give the same result, but it had not been clear if this was indeed the case. It is obvious that such a comparison needed to be done. It would provide strong support for both theories if the two approaches merged.

Figure 1 Various versions of the electroacoustic theory for colloids.

The situation is even more complicated because O’Brien’s approach has been developed mostly for ESA, whereas the Enderby-Booth approach was primarily used to explain the (inverse) effect of CVP or CVI. For dilute systems this difference does not present any problem. However, it does for concentrates, because the inertial frame of references can be different for the ESA and CVI effects.

All of these issues have been addressed recently by an international group of scientists: Shilov from Ukraine, Ohshima from Japan, and A.Dukhin and Goetz from the USA [27-29]. This group has generalized the Enderby-Booth approach to the point that it should now be equivalent to the latest version of O’Brien’s approach. What is of utmost importance, is that the group has also developed an electroacoustic analog of Smoluchowski’s equation. As described earlier (Chapter 2), Smoluchowski’s equation is known to be valid for any concentration and any particle shape, provided that the DL is thin (κa>>1) and the surface conductivity is negligible (Du<<1). By making these same two assumptions, this group derived a general theory for electroacoustics, that is valid for any concentration or particle shape. The theory requires no other assumptions. We shall call this theory the "Smoluchowski dynamic electroacoustic limit" (SDEL). The SDEL is a low frequency asymptotic solution and is a natural test for all possible electroacoustic theories. Hence, any other proposed theory must reduce to the SDEL under the conditions of κa>>1 and Du<<1, and at sufficiently low frequencies. The SDEL serves as a verification criterion for any proposed electroacoustic theory in the same manner that Smoluchowski’s equation serves for electrophoretic theories.

Later Shilov and others further generalized the CVI theory to include particle inertia, surface conductivity, Maxwell-Wagner DL relaxation, thermodiffusion, and barodiffusion [27-30]. This modern CVI theory satisfies the Smoluchowski dynamic electroacoustic limit.

We are not aware of any direct comparison between this SDEL and any theory based on the O’Brien approach. Experimental tests by Ennis, Shugai and Carnie [25, 26] of the latest theory for the dynamic mobility indicate a good correlation even at very high volume fractions for the thin DL. This strongly suggests that the theory may be in compliance with the SDEL; otherwise one would not expect such a good correlation with experiment.

In summary, the connection, between the modern version of the O’Brien approach to ESA theory and modern versions of the Enderby-Booth approach to CVI theory remain uncertain. Although both are able to independently fit corresponding experimental data, the path of theoretical affinity between the two approaches has yet to be resolved.

We emphasize here the CVI theory for the following reasons:

It is known that in the radio frequency domain electric fields might affect the surface properties and cause strange kinetic and thermodynamic effects [31, 32], including oscillation of the zeta potential. The use of ultrasound as the driving force is therefore a better choice than an electric field as required in the ESA method.

When using ultrasound there is an opportunity to calibrate an absolute power using the internal reflection inside of the transducer.

The frame of references is well defined for CVI, but problematic for ESA.

We suggest those readers who are interested in the details of the electroacoustic theory of ESA read the original papers [33-40] or a review written by Hunter [41].

The Low frequency electroacoustic limit - Smoluchowski limit, (SDEL)

There is a way to derive an expression for CVI using a well-known Onsager reciprocal relationship [42, 43, 44]. This relationship is certainly valid in the stationary case, but much less is known about its validity for alternating fields. This uncertainty cautions us to use the relationship only in the limiting case of very low frequencies, or at least frequencies much lower than both the characteristic hydrodynamic frequency, w _hd , and the electrodynamic Maxwell-Wagner frequency, w _MW. It thus follows that:

(4) where K_m is the electric conductivity of the medium, and e ₀ and e _m are dielectric permittivities of the vacuum and medium, respectively.

The Onsager relationship provides the following link between the quasi-stationary streaming potential CVP, the effective pressure gradient, Ñ P_rel , which moves liquid relative to the particles, the electroosmotic current, <I> ,and the electroosmotic flow, <V>:

Let us use the Onsager relationship in consideration of the CVP effect for a macroscopically small element of the suspension’s volume (i.e. the element, which is small with respect to the length of ultrasonic wave, but which contains a majority of particles). To use Equation 5, we need to know the effective gradient of pressure, which provides the velocity of liquid passing around the particles in a vibrated element of the suspension’s volume. This value can be easily obtained following the "coupled phase model" that characterizes the particles motion in a sound field in concentrated colloids. In the quasi-stationary (low-frequency) case, the effective gradient of pressure is equal to the specific (per unit volume of suspension) friction force exerted on the particles, which is g (u_p - u_m). This force is a part of the pressure gradient that moves the particles relative to the liquid. In the extreme case of low frequency, coupled phase model leads to the following expression for this effective pressure gradient:

(6) In addition, we can use the fact that the expression on the left hand side of Equation 5 is the electrophoretic mobility, m , divided by the conductivity of the system, K_s. As a result, we obtain the following expression for CVI:

(7) This expression specifies the Colloid Vibration Current at the low frequency limit. This means that m is the usual stationary case electrophoretic mobility. We can thus use the Smoluchowski law for electrophoresis [45] in the form that is valid for concentrated systems:

Here we used two conditions, that restrict the applicability of Smoluchowski law. As a result the asymptotic value of the CVI at low frequency is:

This yields the following asymptotic value for the dynamic electrophoretic mobility that is defined according to the O’Brien relationship (2):

(10) The density dependent multiplier appears because, according to O’Brien’s relationship, the dynamic electrophoretic mobility is defined in relationship to the density contrast between the particle and the media. This multiplier disappears in dilute systems, but is very important in concentrates, since it conveys additional volume fraction dependence.

Equation 10 is very important because it provides a test for the electroacoustic theory. The theory is supposed to be valid for small particles with a thin DL (κa>>1), and negligible surface conductivity (Du<<1). If these three conditions are met, Equation 10 is then valid for any volume fraction, and any particle shape and particle size. This makes it analogous to the Smoluchowski theory of microelectrophoresis.

There should be a similar equation for ESA, perhaps with a somewhat different density and volume fraction dependence, reflecting the difference in the inertial frame of references between CVI and ESA, but as yet there is none.

The Colloid Vibration Current in concentrated systems

A new theory for Colloid Vibration Current has been created in close collaboration with Prof. Vladimir Shilov, and would have been impossible without his major contribution.

We retain the O’Brien expression for introducing dynamic mobility as follows:

We also retain the same structure for the dynamic electrophoretic mobility expression, presenting as separate multipliers both the inertial effects (function G) and the electrodynamic effects (function 1+F). However, in contrast to the dilute case, functions G and F for concentrated systems depend on the particle concentration. There is also an additional density dependent multiplier,, which is equal to the ratio of the particle velocity relative to the liquid, and to the particle velocity relative to the center of mass of the dispersion. The convenience of the introduction of such a multiplier, which differs from unity only for concentrated suspensions, follows from the exact structure of Smoluchowski’s asymptotic solution for , given by Equation 10. The corresponding equation, which in a convenient way reflects simultaneously limiting transformations both to Smoluchowski’s asymptotic solution (Eq.10), and to O’Brien’s asymptotic solution is given as follows:

The generalization for the case of polydisperse systems is given by:

The new values of the functions G and F are given with the following equations:

where ; , , φ_i and Du_i = κ^σ /K_m a_i is the volume fraction and Dukhin number for the ith fraction of the polydisperse colloid, correspondingly, and φ is the total volume fraction of disperse particles. Special functions H and I are given in the book [52].

These expressions are restricted to the case of a thin DL and are valid for a broad frequency range, including the Maxwell-Wagner relaxation range. They take into account both hydrodynamic and electrodynamic particle interaction, and are valid for polydisperse systems without making any superposition assumption.

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