Langmuir 2003, 19, 3953-3957
3953
Ultrasound Attenuation in Polystyrene Latexes Laurent P. Adjadj, Giuseppe Storti, and Massimo Morbidelli* Institute for Chemical and Bioengineering, ETH Zu¨ rich, CH-8093 Zu¨ rich, Switzerland Received November 22, 2002. In Final Form: February 4, 2003 The ECAH theory, named after the successive works of Epstein and Carhart and Allegra and Hawley, is a very powerful tool that allows the estimation of the particle size distribution in diluted dispersions from ultrasonic attenuation measurements. However, in the case of low-density contrast systems such as polymer latexes, discrepancies have been reported by various authors between the predicted and experimental data at the lower frequencies (below 20 MHz). To elucidate the reasons for such a mismatch, the role of the intrinsic attenuation of the polymer is investigated in this work. A new method for measuring the intrinsic attenuations using a commercial unit is developed and applied to polystyrene in the frequency range 1-120 MHz. A linear relationship between the sound attenuation and the frequency is obtained, which differs significantly from the previous data in the literature. With the use of these values for the intrinsic attenuation, it has been found that the predictions of the attenuation spectra in polystyrene latexes given by the ECAH model are in good agreement with the experimental data.
1. Introduction As a result of their penetration power, ultrasonic waves are widely used for numerous fast in situ and nondestructive analyses. Their applications range from process engineering to the material and food sciences. Some recent relevant examples for each of those fields include fluidizedbed control,1 investigations of shear and compressional behavior of different compounds, such as oils,2,3 and monitoring the cooling of fish.4 Further applications are described by Dukhin and Goetz.5 One important application of ultrasounds is the measurement of the particle size distribution, or PSD, and concentration in colloids and disperse systems, which may involve lubricants, polymers, ceramics, pharmaceuticals, cosmetics, and foods.5-7 When using ultrasound spectroscopy, the PSD is obtained by fitting the experimental attenuation data to complex models describing in detail the process of sound-wave propagation in disperse media.5 The basic piece of theory is the ECAH model, named after the successive works of Epstein and Carhart8 and Allegra and Hawley.9 It applies to the case of a single particle immersed in a continuous medium. This model is based on the simultaneous solution of the local mass-, momentum-, and energy-conservation equations in each phase. Several evolutions and adaptations of this model are described in refs 5, 7, 10, and 11. For diluted systems dispersed in water, two limiting behaviors can be identified * To whom correspondence should be addressed. (1) Cowan, M. L.; Page, J. H.; Weitz, D. A. Phys. Rev. Lett. 2000, 85, 453. (2) Verdier, C.; Longin, P. Y.; Piau, M. J. Non-Newtonian Fluid Mech. 1998, 76, 213. (3) Saggin, R.; Coupland, J. N. J. Am. Oil Chem. Soc. 2001, 78, 509. (4) Sigfusson, H.; Ziegler, G. R.; Coupland, J. N. Trans. ASAE. 2001, 44, 1235. (5) Dukhin, A. S.; Goetz, P. J. Ultrasound for characterizing colloids. Particle sizing, zeta potential, rheology; Elsevier: Amsterdam, The Netherlands, 2002. (6) Chanamai, R.; Alba, F.; McClements, D. J. J. Food Sci. 2000, 65, 507. (7) Povey, M. J. W. Ultrasonic techniques for fluids characterization; Academic Press: San Diego, CA, 1997; Vol. XIII. (8) Epstein, P. S.; Carhart, R. R. J. Acoust. Soc. Am. 1953, 25, 553. (9) Allegra, J. R.; Hawley, S. A. J. Acoust. Soc. Am. 1972, 51, 1545. (10) McClements, D. J. Adv. Colloid Interface Sci. 1991, 37, 33. (11) Hipp, A. K. Acoustic characterization of particulate systems. Principles, applications, and a new model for concentrated dispersions; Wissenschaftlicher Verlag Berlin WVB: Berlin, Germany, 2002; p 216.
and defined as “viscous” and “thermal”,9,12,13 depending on whether the density contrast (i.e., the density difference between the particles and the dispersing medium) is very large or very small, respectively. In the first case, the viscous-energy-dissipation mechanism prevails because the particles move with a velocity significantly different from that of the surrounding medium. For low-density contrast systems, such a velocity difference vanishes, so that the overall sound attenuation decreases and the contribution of the thermal-energy-dissipation mechanism becomes dominant. For high concentrations, particleparticle interactions must be taken into account,14 although the specific concentration value above which the system behavior cannot be described by the single-particle ECAH model varies, depending on the density contrast. These effects can be described using various theories and models, as described, for example, by Hipp et al.15 This issue is not further discussed in this work, where only diluted systems are considered. It is known that the ECAH model gives good agreement with the experimental data in the case of high-density contrast systems, for example, solid/liquid such as silica/ water13 or high-density polymers/water such as PTFE latexes.15 However, Hipp et al.11,16 reported discrepancies between the model and the experimental data for lowdensity contrast systems, such as poly(vinyl choride) (PVC) or polystyrene, particularly in the lower-frequency portion of the spectrum, that is, from 1 to 20 MHz. The case of the PVC latexes, as described by Storti et al.,16 is a good example of such a behavior. Allegra and Hawley, who already identified this behavior for polystyrene, suggested in their original work that some higher-attenuation modes (i.e., quadrupole oscillations related to A2 coefficients) not included in the current application of their theory could account for such discrepancies. This was, however, later discarded because the contributions of these second-order modes are very small at low frequencies.12,17,18 Kaatze et (12) Kaatze, U.; Trachimow, C.; Pottel, R.; Brai, M. Ann. Phys. (Leipzig) 1996, 5, 13. (13) Hipp, A. K.; Storti, G.; Morbidelli, M. Langmuir 1999, 15, 2338. (14) Harker, A. H.; Temple, J. A. G. J. Phys. D-Appl. Phys. 1988, 21, 1576. (15) Hipp, A. K.; Storti, G.; Morbidelli, M. J. Phys. D: Appl. Phys. 1999, 32, 568. (16) Storti, G.; Hipp, A. K.; Morbidelli, M. Polym. React. Eng. 2000, 8, 77.
10.1021/la026893l CCC: $25.00 © 2003 American Chemical Society Published on Web 03/25/2003
3954
Langmuir, Vol. 19, No. 9, 2003
al.12 proposed that this mismatch is explained by the multiple scattering of the thermal and viscous waves (some kind of overlap of the shear and thermal boundary layers of neighboring particles) or some clustering of the polystyrene particles. However, both these arguments should apply to relatively high-concentration systems, yet this behavior has been observed for very diluted systems. Holmes et al.18 suggested that the presence of some relaxation phenomena, intrinsic of the material considered, could cause this additional attenuation mode at low frequencies. Along the same line, Stieler et al.17 proposed, for the case of polystyrene latexes, the presence of “hindered conformational modes of motion of small segments of the polystyrene latex chains”. However, in the presence of low amounts of residual monomer, polystyrene is expected to behave as an ideal elastic solid, thus without any viscous contribution under the current experimental conditions (25 °C at 1-150 MHz). The aim of this work is to elucidate the reasons for such deviations between the experimental and the model results. Polystyrene latexes, which are low-density contrast systems, will be used for the experiments. Note that in this case because the effective attenuation is lower than that for high-density contrast systems, one should expect that minor dissipation mechanisms become of larger relative importance. 2. Experimental Section 2.1. Equipment. All ultrasonic spectroscopy measurements have been carried out using an ultrasizer unit (Malvern Ltd., U.K.). The attenuation spectra of various samples (pure liquids or suspensions) have been measured in the frequency range 1-150 MHz, depending on the operating conditions and nature of the sample. This same device, with a different operating mode, has been used to measure the intrinsic attenuation of bulk polymers at various frequencies. In any case, the temperature was set to 25 °C. The various latexes used in this work have been produced in an automated reactor (LabMax, Mettler-Toledo, Switzerland). It is a well-stirred, temperature-controlled glass reactor, with a 0.6-L vessel volume. Because all the reaction steps are computer controlled, an excellent batch-to-batch reproducibility is assured. The weight fraction of the polymer in each latex was measured thermogravimetrically with a halogen moisture analyzer HG53 (Mettler-Toledo, Switzerland). The final average particle size was measured by dynamic light scattering (Zetasizer 500, Malvern Ltd., U.K.). 2.2. Latex Synthesis. Two identical latexes (indicated as B1 and B2) have been produced to check the reproducibility of the measurements. The following recipe was used: 180 g of water, 120 g of styrene, 3.6 g of emulsifier (sodium dodecyl sulfate), and 0.6 g of initiator (sodium persulfate). In both cases, all the components were initially mixed in a beaker and then poured into the reactor vessel. The reaction temperature was set to 70 °C and kept at this level for 150 min at the constant stirring speed of 500 rpm. Note that most of the reaction was taking place in the first 40 min, as indicated by the jacket-temperature curve (not shown). After that, the temperature was raised to 80 °C to fully deplete the monomer. After dilution to approximately 3 L of volume, particle concentrations lower than 5% by weight were obtained (3.4 and 3.5% by weight for B1 and B2, respectively), thus fulfilling the requirement for diluted systems.14,19 The particle size is 70.0 nm by light scattering and 58.1 nm by ultrasound for B1 and 70.9 and 53.2 nm, respectively, for B2. A discussion about the causes for the discrepancies between the different particle-sizing techniques, such as light scattering and ultrasound, can be found elsewhere.12 (17) Stieler, T.; Scholle, F. D.; Weiss, A.; Ballauff, M.; Kaatze, U. Langmuir 2001, 17, 1743. (18) Holmes, A. K.; Challis, R. E.; Wedlock, D. J. J. Colloid Interface Sci. 1993, 156, 261. (19) Farrow, C. A.; Anson, L. W.; Chivers, R. C. Acustica 1995, 81, 402.
Adjadj et al.
Figure 1. Procedure for the measurement of the intrinsic attenuation. For polystyrene (a), two polymer slabs of different thicknesses, x1 and x2, are placed between the two transducers set at a fixed distance l. For water (b), the transducers are set at two different distances. 2.3. Latex Treatments. Sonication of the latexes was performed by plunging the whole batch into an ultrasonic water bath (Sonorex Super RK510, Bandelin Electronics GmbH; delivering 320 W at 35 kHz). The typical duration of a sonication step was 45 min. The vacuuming of the latexes was performed in a custom-built setup. The dispersion is poured into a 6-L, three-necked flask connected to a gas/vacuum ramp and equipped with a cold trap to condense the evaporated materials. This setup allows for the reduction of the pressure down to 10 mbar and for feeds of different gases, such as air or nitrogen. The stripping of residual monomer can be enhanced by the introduction of a needle through a septum into the sample vessel: when the system is under vacuum, air is sucked through the needle and bubbles into the liquid. Temperature control and agitation are provided by a magnetic heating plate. 2.4. Measurement of the Intrinsic Attenuation. One of the physical properties needed in the ECAH model is the intrinsic attenuation in both the dispersing and the dispersed phases. Although reliable values are available for water, very little consistent data are found in the literature for polystyrene in the frequency range 1-100 MHz. The relationship reported by Allegra and Hawley,9 assuming that the intrinsic attenuation is proportional to the square of the frequency, differs from other literature data by a factor of between 3 and 10.17,18,20 Some other single-frequency data were published by Wada et al.21 and Kono.22 On the other hand, Zellouf et al.23 measured the intrinsic attenuation of several polymers other than polystyrene [poly(methyl methacrylate), polyamide, low-density polyethylene, and polypropylene]. All these show a close-to-linear behavior of the intrinsic attenuation as a function of the frequency in the investigated frequency range. Hipp11 postulated a power-law expression for the intrinsic attenuation as a function of the frequency and estimated the frequency exponent by fitting latex attenuation data. The obtained value is equal to 1 at low frequencies (50 MHz). Because of the lack of consistent data for the intrinsic attenuation, a specific procedure has been developed to take these measurements directly with the ultrasizer. A polystyrene plate is placed in the measuring chamber, perpendicular to the acoustic pathway, as shown in Figure 1a. The chamber is then filled with double-distilled water to ensure continuity between the emitter and the receiver. Then, the attenuation spectra are measured at a given distance l between the transducers, also referred to as the “acoustic pathway”. At (20) Tables of physical and chemical constants and some mathematical functions, 14th ed.; Kaye, G. W. C., Laby, T. H., Eds.; Longman: London, U.K., 1973; Vol. XI; Chapter 1. (21) Wada, Y.; Hirose, H.; Asano, T.; Fukutomi, S. J. Phys. Soc. Jpn. 1959, 14, 1064. (22) Kono, R. J. Phys. Soc. Jpn. 1960, 15, 718. (23) Zellouf, D.; Jayet, Y.; Saint-Pierre, N.; Tatiboue¨t, J.; Baboux, J. C. J. Appl. Phys. 1996, 80, 2728.
Polystyrene Latexes
Langmuir, Vol. 19, No. 9, 2003 3955
Table 1. Physicochemical Parameters Used in Model Simulations Continuous Phase: Water speed of sound32 [m/s] 32,33 density [kg/m3] viscosity32 [kg/(m s)] thermal conductivity32,33 [J/(s m K)] heat capacity (cp)33 [J/(kg K)] intrinsic-attenuation exponent24 intrinsic attenuation at 1 Hz24 [Np/m] thermal volumetric expansion32 [1/K]
1497 997 8.90 × 10-4 0.5952 4178 2 2.2 × 10-14 2.56 × 10-4
Dispersed Phase: Polystyrene speed of sound34 [m/s] density34 [kg/m3] shear rigidity9 [kg/(m s2)] thermal conductivity9 [J/(s m K)] heat capacity (cp)34 [J/(kg K)] thermal volumetric expansion34 [1/K]
2365 1053 1.27 × 109 0.115 1195 2.63 × 10-4
PSD Data weight fraction particle diameter [nm] standard deviation
Figure 2. Comparison between the experimental (0, B1; ], B2) and calculated (s, ECAH model using the Allegra and Hawley9 intrinsic-attenuation data) attenuation spectra.
3.5 × 10-2 70 0
each specific frequency value, the measured overall attenuation σi (in dB) can be expressed as the sum of three different contributions:
σi ) σsys + xiRPS + (l - xi)RW
(1)
where σsys indicates the contribution due to the equipment nonidealities (e.g., nonlinearity of the transducer-set response, reflection in the measuring cell, different types of scattering at the plate-water interface, etc.), while RPS and RW represent the intrinsic attenuation of polystyrene and water, respectively (in dB per unit length) and xi is the thickness of the polymer plate. By measuring the overall attenuation for two different plate thicknesses, the unknown system contribution σsys can be eliminated, and the following expression for RPS is obtained:
RPS )
σ1 - σ2 + RW x1 - x2
(2)
where the subscript i identifies a polymer plate of a given thickness. Commercial polymer plates were used for the measurements (“crystal” grade, amorphous polystyrene plates, Good Fellow, U.S.A.). This approach was first validated by measuring the attenuation of pure water. In this case, as shown in Figure 1b, the overall attenuation σi for a given acoustic pathway li is given by
σi ) σsys + liRW
(3)
By subtraction of two spectra measured at two different pathways, σsys can be eliminated, and we obtain
RW )
σ1 - σ2 l1 - l2
(4)
3. Results and Discussion First, the two latexes were analyzed with the ultrasizer just after production. The acquired attenuation spectra were compared with the predictions of the ECAH model,11 using for polystyrene the intrinsic-attenuation data reported by Allegra and Hawley,9 for water those by Pinkerton,24 and for all the other physicochemical parameters the values reported in Table 1. In Figure 2, the agreement between the attenuation spectra of the two latexes shows the good reproducibility of the experimental data, in particular at high frequencies. On the other hand, the comparison with the model results shows a significant discrepancy at frequencies below 20 MHz. This is the (24) Pinkerton, J. M. M. Proc. Phys. Soc., London 1949, 62B, 129.
Figure 3. Overall attenuation spectra in the low- (top) and high- (bottom) frequency ranges for polystyrene slabs. Plate thicknesses: [, 1.2 mm; 0, 2 mm; 2, 6 mm; ×, 10 mm.
disagreement discussed above that has been reported by several authors and is typical of low-density contrast systems, such as many polymer latexes.11 3.1. Role of the Intrinsic Attenuation. To understand the reason for this disagreement, we investigate the role of the intrinsic attenuation of the polymer, for which, as discussed above, no reliable data are available in the literature. Using the approach described in the Experimental Section, the attenuation spectra were measured for four different polystyrene plates. Figure 3 shows the raw spectra in the low (part a) and high (part b) frequency ranges. For both frequency ranges, each independent pair of attenuation data has been used with eq 2 to produce six polystyrene intrinsic-attenuation spectra for each frequency range, the averages of which are indicated in the logarithmic plot shown in Figure 4a. The sets of data at low and high frequencies have been linearly interpolated, leading to calculated slopes quite close to unity, that is, 0.987 for low and 1.096 for high frequencies, with R2 values of 0.985 and 0.993, respectively, where R2 is defined as follows: 2
R
(Yi - Y ˆ i)2 ∑ )1∑Yi2 - (∑Yi)2/n
(5)
Therefore, a simple linear dependence of the intrinsic attenuation upon the frequency was assumed, similar to the results of Zellouf et al.23 and some other relationships described by Stieler et al.17
3956
Langmuir, Vol. 19, No. 9, 2003
Adjadj et al.
Figure 4. Intrinsic-attenuation data for polystyrene and their interpolations. Experimental data (top) at the (O) low and (]) high frequencies, with the solid line corresponding to the trend lines (slopes: 0.987 at low and 1.096 at high frequencies); experimental data (bottom) gathered over the entire frequency range from (2) this work and (+) Kaye and Laby20 and calculated results from (s) eq 6 and (- - -) Allegra and Hawley.9
Figure 5. Attenuation spectra for polystyrene latexes. Experimental data: [, B1; 4, B2. Calculated ECAH model using the polystyrene intrinsic-attenuation data: (- - -) Allegra and Hawley9 and (s) eq 6.
Figure 4b shows a comparison between the experimental intrinsic-attenuation data and the fitted linear relation valid in the entire frequency range at 25 °C:
RPS ) 0.285f
(6)
where the intrinsic attenuation is in dB/cm and the frequency is in MHz. In the same figure, the relationship reported by Allegra and Hawley9 and single data point given by Kaye and Laby20 are also shown. It is apparent that the disagreement of both of them with the data measured in this work, that is, eq 6, is substantial. We can now investigate the impact of the different intrinsic-attenuation-versus-frequency relations on the calculated attenuation spectra for the polystyrene latex. In Figure 5, the experimental data are compared with the spectra computed using for the intrinsic attenuation of polystyrene either eq 6 (continuous curve) or the relation of Allegra and Hawley (broken curve) and for all the other parameters the values summarized in Table 1. It is found that the use of eq 6 significantly reduces the disagreement between the experimental and the model results, particularly in the low-frequency range but also at high frequencies.
Figure 6. Effects of the various treatments on the attenuation spectra for polystyrene latexes. Batch 1 (top), experimental data: 0, first measurement (same as that for B1 in Figure 2); O, second measurement; [, after sonication (same as that for B1 in Figure 5). Batch 2 (bottom), experimental data: ], first measurement (same as that for B2 in Figure 2); /, after styrene addition; 4, after styrene removal and sonication (same as that for B2 in Figure 5). For both the top and the bottom plots, the solid line is the calculated spectrum (same as that in Figure 5).
3.2. Role of Gas Bubbles. One important aspect in measuring the attenuation spectra in latexes is to accurately check the reproducibility of the data. For example, Figure 6a shows two spectra (open circles and solid diamonds) taken in latex B1 a few minutes apart from each other. It is seen that, while the data at a frequency above about 20 MHz are superimposed, some deviations occur at low frequencies. A similar behavior was also reported by Hipp.11 We show in the following discussion that this behavior can be due to the presence of gas bubbles in the system. Dukhin and Goetz5 discuss the effect of bubbles on ultrasound attenuation. To affect the measurements in the 0.6-60-MHz range, the bubble size should be in the 0.1-10-µm range. Thermodynamic considerations show that such small bubbles are not stable in pure water. Epstein and Plesset25 and Ljunggren and Eriksson26 used several models to evaluate their lifetimes. They showed that a 10-µm bubble is expected to vanish in less than 10 s and a 10-nm one within a few microseconds. However, the presence of hydrophobic interfaces, such as those at the polymer-particle surfaces, may have a stabilizing effect. Tyrrell and Attard27 showed some atomic force microscopy pictures of nanobubbles on a flat hydrophobic surface, which, on the other hand, seemed to be “ephemeral and easily destroyed”. Some vapor/gas cavities can be observed between two hydrophobic surfaces (bead/plate or bead/bead).28,29 The formation of such bridging cavities is suspected to be involved in the long-range attraction between approaching hydrophobic surfaces.30 Although these cavities are not spherical, their dimensions are in the micrometer range. The structure of the hydrophobic surface can influence the formation and stabilization of bubbles, and in fact, curved interfaces cause the contact (25) Epstein, P. S.; Plesset, M. S. J. Chem. Phys. 1950, 18, 1505. (26) Ljunggren, S.; Eriksson, J. C. Colloid Surf., A 1997, 130, 151. (27) Tyrrell, J. W. G.; Attard, P. Phys. Rev. Lett. 2001, 8717, 176104. (28) Yaminsky, V. V.; Yushchenko, V. S.; Amelina, E. A.; Shchukin, E. D. J. Colloid Interface Sci. 1983, 96, 301. (29) Yushchenko, V. S.; Yaminsky, V. V.; Shchukin, E. D. J. Colloid Interface Sci. 1983, 96, 307. (30) Parker, J. L.; Claesson, P. M.; Attard, P. J. Phys. Chem. 1994, 98, 8468.
Polystyrene Latexes
angle to be more favorable.31 The main factors triggering the in situ formation of bubbles are stirring and irregularities at the hydrophobic surface.30,31 To check for the presence of bubbles, latex B1 was treated in an ultrasonic bath for 45 min and then the attenuation spectrum shown in Figure 6a (open squares) was measured. These measurements have been repeated for weeks, and no significant change has been found. Furthermore, measurements using only water with an emulsifier produced spectra stable in time, even without the sonication treatment, that could be superimposed on the ones for pure water. This seems to confirm that the irreproducibilities of the experimental data, particularly at low frequencies, shown in Figure 6a are related to the presence of microbubbles, which are stabilized by the polystyrene particles. In conclusion, sonication for about 1 h is advisable before taking measurements, at least when operating with polymer latexes. Other usual precautions to minimize air incorporation, such as stirring-rate minimization, use of doubly distilled water that is still hot (more than 50 °C) from the distillation device, nitrogen flushing of the reactor, and stripping of the emulsifier, have been found to be insufficient to avoid these irreproducibilities. 3.3. Role of Residual Monomer. In polymer latexes, it is likely to find some residual monomer, which, as in the case of bubbles, can affect their acoustic properties. In the latexes considered in this work, the polymerization reaction has been pushed to 97% conversion (measured by thermogravimetry), and residual monomer has been removed by stripping so that no styrene could be detected in the end products. Therefore, to analyze the role of residual monomer, 400 ppm of styrene, that is, 1.2 g in the total 3-L diluted sample, have been added to the latex B2. This was then stirred for 1 h and left at room temperature for 3 days to reach equilibrium. Figure 6b (31) Eriksson, J. C.; Ljunggren, S. Langmuir 1995, 11, 2325. (32) CRC Handbook of Chemistry and Physics, 81st ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2000; Sec. 6, 10. (33) Kell, G. S. In Water: a comprehensive treatise; Franks, F., Ed.; Plenum Press: New York, 1972; Vol. 1. (34) Ultramod ’98, version 1.10; Malvern Instruments Ltd: Worcestershire, U.K., 1998 (values taken form the material database).
Langmuir, Vol. 19, No. 9, 2003 3957
shows the effect of the styrene addition: open diamonds refer to the original latex and stars to the latex after monomer addition. As a double-check, 10 successive steps of vacuuming were applied to remove the added styrene. A significant oscillatory behavior was observed during this treatment, most probably indicating the presence of bubbles induced by the applied vacuuming. As usual, stable measurements were obtained only after 1 h of sonication. The corresponding spectrum is shown by the open triangles in Figure 6b, which is in good agreement with the data of the original latex as well as with the ECAH-model results (solid line). On the other hand, it has not been possible to compare the model results with the spectra obtained in the presence of monomer because no information is available on the effect of monomers on the relevant physicochemical parameters of polymers, such as, for example, shear rigidity and intrinsic attenuation. 4. Conclusion It has been confirmed that, in the case of dilute systems, where particle-particle interactions are negligible, the ECAH model gives a reliable description of the soundattenuation spectra, provided that accurate values of all the involved physicochemical parameters are used. An experimental method to measure the intrinsic-soundattenuation spectra of bulk polymers has been proposed. The obtained results, in contrast with previous data reported in the literature, show a linear dependence of the intrinsic attenuation upon the frequency in the typical frequency range of ultrasound spectroscopy. Using these intrinsic-attenuation data, it has been possible to accurately reproduce the experimental attenuation spectra in polystyrene latexes. This result is an important prerequisite to using sound-attenuation spectra to estimate PSDs in polymer latexes. Acknowledgment. The financial and technical support of Malvern Instruments Ltd. is gratefully acknowledged. This work was also supported by Grant NF 2-7772900 of the Swiss National Science Foundation. Alexander Hipp is especially thanked for his support. LA026893L
