10930
J. Phys. Chem. 1994, 98, 10930-10934
Sound Generation in Liquids by Electrically Driven Ions in Dilute Electrolytes N. Tankovsky’ and J. Burov Department of Solid State Physics and Microelectronics, Sofia University, 5 J. Bourchier Boulevard, Sofia-1126, Bulgaria Received: February 14, 1994; In Final Form: June 20, 1994@
A method for sound generation in liquids by applying an ac electric field to ionic solutions is presented. A phenomenological model is developed in qualitative agreement with the experimental results. The theoretical approach applies macroscopic continuum equations to the dynamics of an ion, but some microscopic features of the medium in the vicinity of a solvated ion are also taken into account. The results are useful for evaluating the electroacoustic effect in electrolytes and for obtaining information on electrolyte structure.
Introduction In a historical background the attempt to employ acoustical methods to study electrolyte properties originates with an early paper by P. Debye.’ In 1933 he predicted that the propagation of acoustic waves into electrolytes should induce an electric potential, which should have a maximum value at a distance equal to one-half of the acoustic wavelength. The effect, experimentally verified 16 years later,2 was called the “ionic vibration potential” and found application for evaluation of the partial molar volumes of solvated ions. Much easier because of its better efficiency, the same effect could be detected within colloidal solutions and was called the “colloidal vibration p~tential”.~ The reverse effect for excitation of acoustic waves in colloidal solutions by applying an ac electric field has been found as late as 1985 and has been called “electrosound a m p l i t ~ d e ” .The ~ double electric layer, which occurs at the interface between the two phases (solid particle-solvent) is of major importance for this effect. Correspondingly, it has found its main application as a device developed by Matec Applied Sciences5 for experimental evaluation of the electrokinetic potential and of the dynamic mobility of colloidal solutions. Completely different are the driving electroacoustic forces into ionic solutions, where the large local gradients of the physical fields and of the medium in the vicinity of an ion are of major importance. It is, however, astonishing that this problem has been treated neither theoretically nor experimentally until the last few years, when experimental evidence for acoustic wave generation by applying an ac electric field to electrolytes was pre~ented.~.’
Experimental Section The electroacoustic effect in electrolytes has been first experimentally verified with the help of a frequency-sweep technique6 and in capillary tubes, where the applied extemal electric field is nonhomogeneous. In a later work7 the same effect has been demonstrated with the help of the much more reliable pulse-modulation technique, where parallel-electrode electroacoustic cell geometry has been employed. In the present theoretical analysis we shall consider the simpler case of a homogeneous external electric field. The experimental setup and the apparatus used are shown in Figure 1. The electroacoustic cell denoted by B in Figure 1 is immersed in a water tub, ensuring excellent acoustical matching between the ionic solution in the cell and the water buffer. The electroacoustic cell, which is cylindrically shaped, is shown schematically in @
Abstract published in Advance ACS Abstracts, August 1, 1994.
0022-365419412098-10930$04.50/0
Figure 1. Experimental setup: (A) power amplifier; (B) electroacoustic cell, (1) main body, (2) massive metal electrode, (3) elastic emitting electrode, (4) openings; (C) pressure sensor, (1) PVDF coaxial cable; (D) low-noise amplifier; (E) acoustic waves absorber. the lower left side of Figure 1. The electric field vector is directed axially, along the x-coordinate axis, perpendicular to the electrodes, while the y- and z-coordinated axes are directed radially. A massive metal electrode (2) can be moved like a piston to vary the gap between the electrodes. The emitting, elastic electrode (3) is made out of a stretched rubber membrane covered with a thin layer of conducting silver paste. The ionic solution in the cell is driven by the electric field, and pressure waves are excited, which propagate through the water buffer and are detected by a piezosensor denoted by C in Figure 1. The sensor is made from a Raychem PVDF coaxial cable coiled like a spiral to ensure a large sensing area. More details of the experimental setup can be found in ref 7. In Figure 2 a plot taken from the oscilloscope screen is presented. The upper trace shows the cross-talk electrical signal followed by the acoustical signal at a time interval equal to the travel time of the acoustic wave from the electroacoustic cell to the pressure sensor. The lower trace shows the acoustical signal in the way it is digitized and saved for further processing. It can be seen that at some frequencies the acoustical signal is essentially nonlinear. A Fourier processing has been carried out for different kinds of electrolytes, and it has been found that both first and second harmonics (denoted as SI and S2, correspondingly) can be excited in the different frequency ranges, which proves the existence of both linear and quadratic mechanisms of sound generation. In Figures 3 and 4 the electroacoustic spectra of two electrolytes are presented: 0.01 N concentration of LiI and 0.01 N concentration of KBr, correspondingly. The dashed lines represent the first harmonic signal, and the continuous lines
0 1994 American Chemical Society
Sound Generation in Liquids
Phys. Chem., Vol. 98, No. 42, 1994 10931
J.
kHz
frequency
Figure 4. Electroacoustic spectrum of a 0.01 N water solution of KBr: first harmonic signal, dashed line; second harmonic signal,
continuous line. S, = B,
+ B,E + B#
(1)
where the values of the coefficients depend on the kind of electrolyte, on frequency, on concentration, on the kind of solvent, etc.
Phenomenological Theoretical Model I
I
*a,*
,
I
Figure 2. Oscilloscope plot of the acoustical signal.
Lil 0.01N
In the present paper a simple theoretical analysis is developed, which must he considered just as a first step to the theoretical explanation of the very complicated phenomena that occur in ionic solutions in the presence of an ac electric field. Although we have treated the ion as a macroscopic particle, whose dynamics is described by macroscopic continuum equations, which is a rather rough approximation, the obtained expressions are in qualitative agreement with the experimental relations (eq 1) and may he helpful for a better understanding of the electroacoustic mechanisms in electrolytes. We shall analyze all direct or indirect mechanisms of pressure generation in the vicinity of an ion when an ac electric field, E exp(iwt), is applied. The equation of motion for an ion having a mass mj, an effective solvation radius aj, and an electrical charge e can be written in a good approximation as follows: mj(dv/df) = eE - 6zqajvj
frequency
kHz
Figure 3. Electroacoustic spectrum of a 0.01 N water solution of L i I first harmonic signal, dashed line; second harmonic signal, continuous line
correspond to the second harmonic. It can he seen that different electrolytes have different spectra, but we have yet no explanation for the specific shapes of the obtained spectra. However, the eventual possibility to specify the presence and concentration of different ions in the solution from the spectra or to obtain information for the ion-ion or ion-solvent interactions in ionic solutions should be kept in mind. The experimentally obtained electric field dependences for low-concentration electrolytes could be described by p0lynon1als of the second degree such as the following:7
s,=A,+A,E+A,~
(2)
where q is the viscosity coefficient and vj is the velocity of the ion in the solvent. The index j = 1, 2 specifies the two kinds of ions: positive and negative. The ion velocity can he easily defined from (eq 2), remembering exponential time dependence exp(iwt): v. = J
eE eE 6mlaj iwmj = 6nqaj
+
I~
(3)
The upper approximation is justified since wmj > at/A. In particular, we can consider the geometry of the electroacoustic cell,7given in Figure 1, which is cylindrically shaped with diameter W and length L. First we shall sum all elementary sources arranged in a plane (y,z),with linear source intensity 1 - I . We integrate over all elementary waves generated by the ions. As far as we are interested, for the result at a great distance R, in the far field, only the phase shifts between the sources are taken into account. The amplitude of an elementary source is denoted by PO. The observation point is at distance
(22)
where A = [sin (kW sin q/2)]/[kWsin q / 2 ] is the antenna directivity characteristics. Next we sum all parallel plane sources (y,z), which are N = LIZ in number and are arranged along x at a distance 1 from one another. Finally the contribution from all elementary sources in the cell of volume W L can be given by
Po( ~ 2 u l 3 ) ~ i a ~ 2 ~ i ( o t - k R ) Although the thennoelastic force is directed radially to the axis x, the radiating ion is small enough to allow application of eq 18 in the far field, and after the corresponding substitutions we obtain
10933
(23)
Now we can express the pressure generated by all the ions in the electroacoustic cell due to all four effects analyzed above. The first harmonic signal is given by the relation
Correspondingly, for the second harmonic signal we obtain
Discussion and Conclusions In the present analysis the ion-ion interactions have been ignored. Experimentally, these interactions can be minimized by applying a dc electric field, causing space separation of the ions of opposite charge. In Figure 5 3-D experimental graphics are shown, presenting the electroacoustic spectrum of a 0.01 N water solution of Na acetate as a function of the applied dc voltage. In Figure 5a the first harmonic signal, $1, is presented, and one can notice an essential amplitude amplification as a dc electric field is applied. This result is in qualitative agreement with formula 24, because after the space separation of the ions of opposite charge their contributions to the linear mechanisms of pressure excitation should be summed and not subtracted. In Figure 5b the second harmonic signal, S2, is shown, and no amplitude amplification due to a dc field can be observed. The obtained expressions 24 and 25 are polynomials of the second degree relative to E, which is in agreement with the experimentally obtained dependencies (eq 1). A simple, approximate check of the theoretical model can be performed by evaluating the ratio of the coefficients in the linear and quadratic terms in relations 24 and 25 and their comparison with the corresponding experimental coefficients. From eqs 1, 24, and 25 one can write
2eTdv: B,IB, = 7KUjVEj
Tankovsky and Burov
10934 J. Phys. Chem., Vol. 98, No. 42, 1994
gradient V q in eq 27 can be evaluated with help of an approximate formula giving the radial dielectric constant versus radius for a monovalent ion in water, as presented in”
4 $0,
~ ( r=) q,(1 - 16e-’,25r)
(29)
where r is in angstrom units. After substituting the numerical values of the parameters in eqs 26 and 27 one can evaluate the ratios A,/A2 = lo-’ and BlIBz lop3,and this result coincides within an order of magnitude accuracy with the experimental values. The obtained relations are useful for a better understanding and numerical evaluation of the electroacoustic effect in electrolytes. The authors believe that this effect, after more thorough development, should prove to be a new and useful method for studying both simple electrolytes and more complicated ionic structures in solutions.
Acknowledgment. An acknowledgment is due to M. Nikolov for his help in the experimental measurements. We are also grateful to the National Fund “Scientific Research” at the Bulgarian Ministry of Education and Science for financial support. References and Notes
Figure 5. Na acetate (0.01 N) electroacoustic spectrum dependence of the applied dc voltage.
To evaluate dddg in eq 26, we can differentiate the ClausiusMosotti equation and obtain
ddde =
+
NAa(€ 2)2 9644
where NA is Avogadro’s number, M is the molar weight of the solvent, and a is the polarizability of the solvent, which for polar molecules like water can be approximated as a = pz/ 3kT, p being the dipole moment of the solvent molecule. The
(1) Debye, P. J . Chem. Phys. 1993, I , 13. (2) YeGer, E.; Bugosh, J.iHovorka, F.; McCarthy, J. J . Chem. Phys. 1949, 17, 411. (3) Vidts, J. Meded. Kl. Vlaam. Acad. Wet.,Lett. Belg., K1. Wet. 1945, 3, 5. (4) Oja, T.; Peterson, G.; Cannon, D. United States Patent 4,497,208, 1985. (5) Cannon, D. Matec Applied Sciences, 75 South Street, Hopkins, MA 01748. ( 6 ) Tankovsky, N.; Pelzl, J. Physical Acoustics-Fundamentals and Applications; Plenum Press: New York, 1991. (7) Tankovsky, N. J. Appl. Phys. 1994, 75 (2), 1239. (8) Clay, C.; Medwin, H. Acoustical Oceanography: Principles and Applications; John Willey & Sons, 1977. (9) Landau, L.; Lifshitz, E. Electrodynamics of Continuous Media; Nauka: Moskva, 1982; in Russian. (10) Carslaw, H. S.; Jeager, J. C. Conduction of Heat in Solids; Clarendon: Oxford, 1959. (1 1) Krestov, G . Ion Solvation; Nauka: Moskva, 1987; in Russian. (12) Pau, P. C.; Berg, J. 0.;McMillian, W. G. J . Phys. Chem. 1990,94 (6), 2671.
