NearMegahertz Sonochemistry
In the past, it was argued that 20-50 kHz was the optimal frequency range for sonochemistry [1,33]. More recently, however, a large number of experiments have shown that this is often not the case. Generally, sonochemistry progresses as well, if not better, at the near-megahertz frequencies (i.e., 1001000 kHz; see Sec. II.C.1). It is expected that near-megahertz ultrasonic frequencies will soon become as important to sonochemistry as 20-50 kHz has been in the past. Generally, higher-frequency ultrasound behaves differently because the structures involved are large in terms of the ultrasonic wavelength. For this reason, we will briefly introduce a few equations, which should give the sonochemist a passing knowledge of the behavior of beams of sound at the near-megahertz frequencies [33,34]. These dimensional relationships have been summarized in Fig. 3.
Sonochemistry is conducted at such high sound amplitudes that water can no longer be considered a linear material [35]. This means, among other things, that the local velocity of the ultrasound in the solution is dependent on the local sound amplitude. It is thus difficult to mathematically predict the resulting sound pattern at points removed from the transducer, even if the sound pattern is well known near the transducer. On the other hand, sound propagation through a linear material is well known and can be computed very accurately. Thus, most attempts at mathematically predicting nonlinear sound fields will only go so far as to predict the field in a linear material, and then to assert that the nonlinear field is similar. This is the approach followed here, and nonlinear effects will be discussed where possible. The acoustical wave velocity c in liquids is approximately defined by the equation c2 = B/p, where B is the bulk modulus, which has the units of pressure required to compress (or shrink in volume) the bulk fluid by a known amount, and p is the mass density (mass per unit volume) of the liquid [34]. Note that the presence of any bubbles in the solution will
- Figure 3 Beam geometry of linear ultrasound.
decrease both B and q, generally by unequal amounts, with a net effect of reducing c. For instance, B for pure water is 316,000 psi, and because of this large bulk modulus, water is often considered incompressible. It is obvious that the addition of even a few bubbles to the water would dramatically increase this compressibility and dramatically decrease B, while only affecting q by a small amount. Such bubbles are easily generated by the passing of large waves through the solution, as was described in Sec. II.A.
The wavelength of linear or low-amplitude sound in water [m] is: k = c/v. Here, the wave velocity c is approximately 1500 m/sec, and v is the frequency in Hertz. Sound from a flat source (called a piston source if round, or a plate source if rectangular) does not retain the shape of the source as it propagates away, but spreads out fairly predictably. The rate of this spreading is often stated as the beamwidth, which means the width of a cone [°], that contains one-half the power of the beam. This width of this cone follows the relationship sin(h) = k/d, where d is the diameter or width of the source [m]. However, this equation loses its utility for d smaller than one wavelength because there are no angles h with sin(h) greater than one. When this condition is encountered, it simply means that the flat source is tending to become a spherical radiator and to project its sound equally in all directions. Let us consider a numerical example: the wavelength of a 20kHz beam in water is 7.5 cm, or 2.95 in. The wavelength of 500 kHz sound is 3 mm, or 0.118 in. Comparing a 1-in.-diameter plate source at 20 and 500 kHz, we find that the 20-kHz beam is approaching a spherical radiator, whereas the 500-kHz beam is concentrated into a fairly narrow beam only 7° wide.
Sound does not begin to spread exactly at the transducer's surface. The closest distance from the transducer where one will find a well-formed beam is called the farfield distance, and is approximately d2/k m from the transducer. The farfield distance is sometimes called the ''d2 over k distance." From this point to infinity (or until the beam encounters a reflector or refractor), the beam retains the same general shape, spreading as predicted by its beamwidth. The farfield distance of our 20-kHz example is about 9 mm, and about 21.5 cm for the 500-kHz source. By contrast, the nearfield of a source is the region near the transducer. In this region, the sound field basically looks like the source, and the nearfield is sometimes called the shadow zone. As one moves away from the transducer, there is a point, at about one-third of the way to the farfield distance, where the sound field's width has shrunk to just over half of the plate's width. There is a commensurate increase in intensity to about three times the level at the transducer. This point in the beam is sometimes called the main axial maximum of the beam. Many high-frequency sonochemistry sources will ca-vitate the water at this point. However, this point is always moved closer to the transducer by the nonlinear, self-focusing action of the intense sound field [36]. At higher frequencies, it is sometimes possible to conduct sono-chemistry at this point in the beam, without further focusing.
Between the main axial maximum and the farfield distance, the beam widens and takes on the familiar beam pattern. This pattern consists of a main lobe surrounded by many sidelobes of lower intensity. It is helpful to know that, at the farfield distance, the main lobe is again about the width of the source, and the intensity is about 80% of the intensity at the transducer face (not including losses due to absorption). This information has been condensed into Figs. 3 and 4. Fig. 4 illustrates a computer simulation from a three-dimensional solution of the wave equation considering a 1-in. square plate transducer operating at 500 kHz and utilizing only the propagating terms. Ignoring the evanescent wave contribution results in a small error, as can be seen in Fig. 4, where the intensity at the transducer face is incorrectly predicted to be slightly less than one. The x-axis has been normalized by dividing distance by the farfield distance; the maximum on the x-axis (one)
Range divided by farfield distance Figure 4 Normalized intensity for a plate transducer, 1x1 in., 500 kHz.
corresponds to the farfield distance (21.5 cm). The main axial maxima can be seen at a distance of about 0.33 d2/l. From this point outward, the intensity is seen to decrease in a way that approaches the familiar 1/r2 pattern. In summary, the ultrasound originating from the transducer is ''choked down'' to three times the intensity at one-third the farfield distance, then reaches about 80% of the original size and intensity at the farfield. By conducting sonochemistry anywhere between the transducer and the far-field, one is assured that the intensity is somewhere between 0.8 and three times the intensity at the source.
Cavitation begins at much smaller intensities when low sound frequencies are applied. Fig. 5 describes how the threshold intensity increases with increasing frequency. Drawing a vertical line at approximately 20 kHz, as one moves up this vertical line, wave intensity increases [W/cm2]. The first thing one encounters as the intensity is increased is the curve for ''aerated water,'' or water saturated with air. The intensity at this point is sufficient to produce cavitation as desorbed air contributes to bubble nucleation. As one continues to increase intensity, one will encounter the curve for degassed cavitation. This intensity is the absolute maximum intensity allowed (at standard conditions) for sound traveling in water at this frequency. Most of sonochemistry are performed at intensity levels between these two values.
The discussion above of the main axial maximum is a good example of a nonlinear effect. As the sound reaches the main axial maximum, the intensity can become sufficient to approach the lower curve on Fig. 5, and
Figure 5 Variation in threshold intensity with frequency. (From Ref. 1.)
Frequency (Hz)
Figure 5 Variation in threshold intensity with frequency. (From Ref. 1.)
gaseous cavitation can begin. This creates local bubbles in the water, which reduce the bulk modulus of the water, thus reducing the local sound velocity. Because there are more bubbles in the center of the beam where the intensity is highest, the sound waves travel slowest at the beam's center and ''self-focusing'' occurs, which steers more sound towards the center of the beam, and so on. The result of these effects is that the apparent main axial maxima are moved closer to the transducer source.
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