Acoustic Characterization of Concentrated ... - ACS Publications

Langmuir 2002, 18, 405-412

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Acoustic Characterization of Concentrated Suspensions and Emulsions. 2. Experimental Validation Alexander K. Hipp,† Giuseppe Storti, and Massimo Morbidelli* ETH Zu¨ rich, Department of Chemical Engineering, HCI, CH-8093 Zu¨ rich, Switzerland Received August 18, 2001. In Final Form: September 22, 2001 In the first part of this work, a core-shell extension of the isolated-particle framework of Epstein, Carhart, Allegra, and Hawley (ECAH theory) has been developed for the simulation of sound attenuation in concentrated emulsions and suspensions. In the present paper, the model predictions are validated by comparison with a systematic set of experimental data. The systems under consideration include silica/ water, poly(tetrafluoroethylene) (PTFE)/water, and corn oil/water at a range of particle sizes, particle volume fractions, and frequencies. These data cover a wide variety of situations where energy dissipation is dominated by viscoinertial losses, thermal losses, or combinations thereof, thus providing a significant test for the reliability and generality of the model. Moreover, the potential of the model as a basis for the estimation of particle sizes from acoustic data is investigated.

1. Introduction The acoustic analysis of suspensions and emulsions is difficult at high particle concentrations. In this case, the particles or droplets can no longer be assumed to behave as individual isolated objects, and a simple linear superposition of isolated-particle behavior is no longer adequate. In the colloidal particle size range, the source of highconcentration effects is the interaction between neighboring particles, whereas multiple scattering plays a negligible role.1 This means that the particles interact by penetrating each other’s thermal or viscous boundary layers and thus reduce the rate of energy dissipation with respect to the case of isolated particles. Only very few attempts have been made in the literature to describe this effect, for example Strout’s cell model2,3 or Hemar et al.’s core-shell model.4 However, these models are limited to viscoinertial attenuation (thus usually to solid/liquid suspensions) or thermal attenuation (thus usually to liquid/liquid emulsions), respectively. Moreover, by neglecting the redirection of waves, both models are limited to particle sizes much smaller than the wavelength. To find a comprehensive high-concentration approach for both suspensions and emulsions, which is also not limited to a particular particle size range, a core-shell model was developed in the first part of this work.5 This model is based on the general ECAH framework for isolated particles and thus accounts for attenuation caused

by viscoinertial, thermal, and redirective mechanisms. This model assumes that a given particle is surrounded by a shell of original dispersant in its immediate vicinity, which in turn is immersed in an unlimited effective medium accounting in an effective way for the effect of neighboring particles. The properties of this medium are computed using the same simple volume-average mixing rules as suggested in the original work.5 The size of the shell, and thus the proximity of the effective medium to the particle, is a function of the particle volume fraction φ. The following relation is used in this work:

rshell ) rp/φ1/3

This relation states that the volume of the dispersant is divided equally among the particles (radius rp). That is, the shell becomes infinitely large in the dilute limit and approaches the size of the particle for φ f 1. By implementation of the three-phase concept of the core-shell model into the framework of Epstein and Carhart6 and Allegra and Hawley,7 the complete flow field in the particle, shell, and effective medium can be determined, along with the underlying compressional, thermal, and shear contributions. By use of the coefficient of the outgoing compressional wave in the shell (An′′), the overall attenuation in the dispersion is obtained by means of the following relation5-7

Reff ) * To whom correspondence should be addressed. † Current address: Dow Europe, CH-8810 Horgen (Zurich), Switzerland. (1) Hipp, A. K.; Storti, G.; Morbidelli, M. On Multiple-Particle Effects in the Acoustic Characterization of Colloidal Dispersions. J. Phys. D: Appl. Phys. 1999, 32, 568. (2) Strout, T. A. Attenuation of Sound in High-Concentration Suspensions: Development and Application of an Oscillatory Cell Model. Ph.D. Thesis, Department of Chemical Engineering, University of Maine, Orono, ME, 1991. (3) Dukhin, A. S.; Goetz, P. J. Acoustic Spectroscopy for Concentrated Polydisperse Colloids with High Density Contrast. Langmuir 1996, 12, 4987. (4) Hemar, Y.; Herrmann, N.; Lemare´chal, P.; Hocquart, R.; Lequeux, F. Effective Medium Model for Ultrasonic Attenuation due to the Thermo-Elastic Effect in Concentrated Emulsions. J. Phys. II France 1997, 7, 637. (5) Hipp, A. K.; Storti, G.; Morbidelli, M. Acoustic Characterization of Concentrated Suspensions and Emulsions. 1. Model Analysis. Langmuir 2001, 17, 391.

(1)

3φ

∞

∑ (2n + 1) Re(An′′) + background attenuation

2kc′′2r3 n)0

(2)

where kc′′ is the compressional wavenumber of the dispersant (that is, the shell), given by ω/c′′ + iR′′, with the angular frequency ω ) 2πf, the intrinsic sound speed c′′, and the intrinsic attenuation R′′. The background attenuation, which is important for low-attenuating systems (typically emulsions), is φR′ + (1 - φ)R′′, that is, a volume-average of the intrinsic attenuations of the dispersed phase (R′) and of the dispersant (R′′). (6) Epstein, P. S.; Carhart, R. R. The Absorption of Sound in Suspensions and Emulsions. I. Water Fog in Air. J. Acoust. Soc. Am. 1953, 25 (3), 553. (7) Allegra, J. R.; Hawley, S. A. Attenuation of Sound in Suspensions and Emulsions: Theory and Experiments. J. Acoust. Soc. Am. 1972, 51 (5), 1545.

10.1021/la015541w CCC: $22.00 © 2002 American Chemical Society Published on Web 12/18/2001

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In the following, the core-shell model is validated experimentally for six different systems. These belong to the two categories of acoustic behavior and represent either “thermal” or “viscoinertial” systems depending on the dominating dissipation mechanism. First, the attenuation values predicted by the core-shell model as a function of particle volume fraction are compared with those observed experimentally. Second, the core-shell model is used for estimating particle size from sound attenuation spectra. In this context, specific attenution is given to the influence of the sample concentration on the estimated particle size. 2. Experimental Data In systems with dominant viscoinertial dissipation, which are usually characterized by a large difference in density, it has been shown that multiple-particle effects play an important role even at relatively low concentrations.1 In addition to the PTFE latex already studied in ref 1 (F′ ) 2.2 g/cm3, d ) 0.2 µm), this case is examined here for three monodisperse silica-in-water suspensions (F′ ) 2.2 g/cm3) of different particle size (d ) 0.1, 0.3, 0.4 µm, supplied by the Nissan Chemical Co.). The samples were analyzed at their original concentration (∼25 vol %) and diluted further to obtain measurements at 15 different concentrations for each sample. To investigate systems with dominant thermal dissipation, which are usually characterized by a small difference in density, data of Chanamai et al. are used.8 In this case, attenuation spectra of two corn-oil-in-water emulsions with different droplet size are considered (F′ ) 0.9177 g/cm3, d ) 0.2 and 1.2 µm), both analyzed at six different droplet concentrations, that is, 5, 10, 20, and 30 vol %, as well as 37 and 50 vol % for the first one, or 40 and 51 vol % for the second. The attenuation spectra of the silica suspensions are shown in Figures 1-3. Those of the PTFE and corn oil dispersions can be found in the above-mentioned original publications. In all cases, the measurements were carried out using a Malvern Instruments Ultrasizer.9-11 The solid contents indicated in Figures 1-3 were measured independently by gravimetry. It is interesting to note that at low concentrations (bottom portion of the figures), an increase in concentration always leads to an increase in attenuation. Instead, as a result of multiple-particle effects, the opposite may occur at high concentrations (top portion of the figures). Particle Sizes at Low Concentrations. It is wellknown that the isolated-particle model is quite reliable at sufficiently low particle concentrations and can thus be used as a basis for a particle size estimation procedure. As discussed in detail in an earlier contribution,12 such a procedure consists of estimating some characteristics of the particle size distribution, typically the average particle size d50, the standard deviation σ, and the particle volume (or weight) fraction, by fitting the acoustic attenuation (8) Chanamai, R.; Herrmann, N.; McClements, D. J. Influence of Thermal Overlap Effects on the Ultrasonic Attenuation Spectra of Polydisperse Oil-in-Water Emulsions. Langmuir 1999, 15 (10), 3418. (9) Alba, F. Method and Apparatus for Determining Particle Size Distribution and Concentration in a Suspension using Ultrasonics. US Patent 5,121,629, 1992. (10) Roberts, D. Ultrasound Analysis of Particle Size Distribution. Mater. World 1996, 4 (1), 12. (11) Alba, F.; Higgs, D.; Jack, R.; Kippax, P. Ultrasound Spectroscopy: A Sound Approach to Sizing of Concentrated Particulates. In Handbook on Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates; Hackley, V. A., Texter, J., Eds.; American Ceramic Society: Westerville, OH, 1998. (12) Hipp, A. K.; Storti, G.; Morbidelli, M. Particle Sizing in Colloidal Dispersions by Ultrasound. Model Calibration and Sensitivity Analysis. Langmuir 1999, 15 (7), 2338.

Figure 1. Attenuation spectra of silica in water (0.1 µm) at different particle weight fractions w. The attenuation was scaled by the frequency (in MHz) for convenience; the lines are simple interpolations. The attenuation of pure water (- - -) is shown as reference.

spectrum. In the present cases, the required physical property parameters are taken from ref 12 for water, silica, and PTFE, and from Chanamai et al.8 for corn oil. This leads to the results of Table 1, which were obtained at the lowest available particle concentrations using the isolated particle model of Epstein/Carhart6 and Allegra/Hawley.7 The values are in good agreement with the reference data. The particle/droplet concentrations of these samples are sufficiently low to neglect multiple-particle effects, which, as will be shown later, become significant only well above the concentrations considered here, that is, well above 0.5 vol % for the viscoinertial systems and 5 vol % for the thermal ones. 3. A Priori Prediction of Attenuation vs Concentration An effective way to determine the extent of the effect of multiple-particle interactions is to switch from a frequency-based to a concentration-based representation. For the samples of the preceding section, the attenuation is shown as a function of the particle volume concentration in Figures 4-9 for various fixed frequencies. The experimental data (interpolated between the closest frequencies in the attenuation spectra) are shown as symbols, while the predictions of the theoretical models are plotted as lines. The particle size distributions (d50, σ) assumed in the simulations and the model parameters are the same ones as reported and used in Table 1. Note that both isolated-particle and core-shell model were used in a fully predictive mode, using the same parameter values without any adjustment. By inspection of Figures 4-9 it can be concluded that a very good agreement between the core-shell model

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Table 1. Characteristics of the Particle Size Distributions reference values silica/water (0.1 µm) PTFE/water (0.2 µm) silica/water (0.3 µm) silica/water (0.4 µm) corn oil/water (0.2 µm) corn oil/water (1.2 µm) a

acoustic spectroscopy

dPCS (nm)

φ/w

d50 (nm)

σ

φ/w

122 249 326 426 240a 1160a

-/1.00% -/1.05% -/1.06% -/1.12% 5/-% 5/-%

112 239 329 433 226 1150

0.05 0.05 0.05 0.08 0.63 0.47

0.44/0.97% 0.49/1.11% 0.50/1.10% 0.51/1.12% 4.84/4.48% 4.99/4.61%

Reference 8.

Figure 2. Attenuation spectra of silica in water (0.3 µm) at different particle weight fractions w. Notation as in Figure 1.

Figure 3. Attenuation spectra of silica in water (0.4 µm) at different particle weight fractions w. Notation as in Figure 1.

predictions and the experimental data is given for all systems under consideration. Whereas the ECAH model yields the expected linear behavior (which is useful only for very dilute systems), the core-shell model follows the experimental points closely, both at high frequencies, where the attenuation is linear for most of the concentration range, as well as at low frequencies, where both coreshell model and experiments deviate significantly from linearity. Also the fact that for viscoinertial systems, the deviations from linearity become significant at lower frequencies than for thermal systems is correctly predicted by the model.

The quality of the particle sizing results obviously depends on the accuracy of the underlying theoretical model. In particular, the analysis of concentrated samples is strongly affected by multiple-particle effects. To illustrate this point, results from both the isolated-particle and the core-shell model are discussed in the following. As expected, by ignoring multiple-particle effects, the isolated-particle model fails to give reliable particle sizing results at higher concentrations. Particle Sizing with the Isolated-Particle Model. When using the isolated-particle model for the six systems under investigation at different particle volume fractions, one obtains the PSD characteristics shown in Figure 10. In particular, this includes the particle weight fraction w on the right-hand side and the PSD on the left-hand side of the figure. The PSD is represented by bars corresponding to d45, d50, and d55. It is seen that for all systems, the procedure works well at very low particle concentrations: the correct particle size d50 is estimated (compared to the reference values in Table 1), and also the particle weight fraction is obtained correctly (the correct values are indicated by dashed lines in the right-hand side of the figure). For the PSD broadness, the obtained results are satisfactory, although the comparison is really possible

4. Particle Sizing at High Concentrations As mentioned above, to obtain particle sizes from the attenuation spectra, a theoretical model (isolated-particle or core-shell) is fitted to the attenuation spectrum of the system under investigation.12 In this work, monomodal log-normal particle size distributions are assumed for simplicity, and the models were fed with the same parameters as in the previous sections. This size estimation procedure needs to be carried out individually for each system and typically yields the size d50, the broadness σ, and the particle weight fraction w simultaneously.

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Figure 4. Attenuation as a function of particle volume fraction at various frequencies for silica in water (0.1 µm), using the core-shell (s) and the ECAH (- - -) model. The experimental data (symbols) are taken from Figure 1.


Figure 5. Attenuation as a function of particle volume fraction at various frequencies for PTFE in water (0.2 µm), using the core-shell (s) and the ECAH (- - -) model. The experimental data (symbols) are taken from ref 1.


only on qualitative terms since most of the considered systems are substantially monodisperse. However, as the particle concentration increases, the situation changes and the quality of the estimates deteriorates significantly. In particular, we see that the larger the particle concentration, the more the average particle size and the particle weight fraction are underestimated. This is a consequence of the fact that the isolated-particle model overestimates sound attenuation by neglecting particle-particle interactions. An exception

is the 1.2 µm corn oil sample, which because of its thermal nature and its relatively large droplets actually shows only a rather low influence of multiple-particle effects, as is also evident from the attenuation values shown in Figure 9. It is worth pointing out that all results in Figure 10 were obtained with a good fit between theoretical and experimental attenuation spectra, as shown for silica/ water in Figures 1-3. This confirms that, as demonstrated


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Figure 8. Attenuation as a function of particle volume fraction at various frequencies for corn oil in water (0.2 µm), using the core-shell (s) and the ECAH (- - -) model. The experimental data (symbols) are taken from Chanamai et al.8

Figure 10. Particle sizing using the isolated-particle model; three estimated parameters: d50, σ, and particle weight fraction w. The results are connected by lines for convenience. The dashed lines on the right label the case where the ordinate values are equal to those of the abscissa.

Figure 9. Attenuation as a function of particle volume fraction at various frequencies for corn oil in water (1.2 µm), using the core-shell (s) and the ECAH (- - -) model. The experimental data (symbols) are taken from Chanamai et al.8

Figure 11. Particle sizing using the isolated-particle model as in Figure 10, but two estimated parameters: d50 (left) and particle concentration (right). The broadness of the size distribution σ is set to the values reported in Table 1 for all concentrations.

earlier for a different case (parameter mismatch, section 6 of ref 12), a good model fit is not sufficient to guarantee accurate estimates but that one depends on quality of the underlying model of sound attenuation. As discussed in detail in ref 12, one of the limits of the estimation procedure is that it cannot estimate too many variables at the same time. It it therefore interesting to analyze the estimates obtained when decreasing the number of degrees of freedom. The case shown in Figure

10 corresponds to three degrees of freedom, because the PSD (d50, σ) as well as the particle concentration (w) are estimated simultaneously. These can be reduced to two, either by setting the broadness σ (Figure 11) or the particle concentration w (Figure 12) to the values listed in Table 1. In Figure 13, both simplifications are combined, leaving only one degree of freedom. It is seen that in all cases, the obtained estimates exhibit the same inaccuracies as those shown in Figure 10. This confirms that their origin is

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Figure 12. Particle sizing using the isolated-particle model as in Figure 10, but two estimated parameters: d50 and σ. The concentrations w are set to the values of Table 1. Broken lines indicate poor fit of the attenuation spectra.

Figure 14. Particle sizing using the core-shell model; three estimated parameters: d50, σ, and particle weight fraction w. The results are connected by lines for convenience. The dashed lines on the right label the case where the ordinate values are equal to those of the abscissa.

Figure 13. Particle sizing using the isolated-particle model as in Figure 10, but one estimated parameter: d50. Both particle concentration w and broadness σ are set to the values of Table 1. Broken lines indicate poor fit of the attenuation spectra.

Figure 15. Particle sizing using the core-shell model as in Figure 14, but two estimated parameters: d50 (left) and particle concentration (right). The broadness of the size distribution σ is set to the values reported in Table 1 for all concentrations.

indeed the underlying isolated-particle model, and not a limitation of the adopted parameter estimation procedure. It is also worth mentioning that when fixing the particle weight fraction, the remaining degrees of freedom were often not sufficient to provide a good fit of the experimental attenuation spectra. In particular, when an error (defined as the sum of squared errors divided by the number of observations) greater than 5% was obtained by the fitting procedure, the corresponding estimates were indicated by broken lines in Figure 12 and 13.

Particle Sizing with the Core-Shell Model. An improved performance is obviously expected when using the core-shell model, since a very good prediction of the attenuation-concentration behavior was found for this model as shown in Figures 4-9. Instead, the results in Figure 14 for the simultaneous estimation of PSD (d50, σ) and concentration (w) seem no better than those of the isolated-particle model: there still is a strong concentration effect in the obtained estimates, and one observes a significant underestimation at larger particle concentra-


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15) results in a slight improvement, but only for the viscoinertial systems (silica, PTFE); the corn oil estimates actually become worse. Instead, presetting the weight fraction w (Figures 16 and 17) leads to very good results for all samples: The estimated particle sizes are relatively constant throughout the entire range of particle concentrations. These results indicate that particularly at high concentrations, the amount of information contained in an attenuation spectrum is not sufficient to simultaneously estimate particle size and particle weight fraction. This is not actually a severe limitation of ultrasonic particle sizing, since obtaining the particle concentration from other techniques is relatively straightforward. When concentrated suspensions and emulsions are analyzed, the particle concentration should therefore be considered as an input parameter to the acoustic particle sizing procedure. 5. Conclusions

Figure 16. Particle sizing using the core-shell model as in Figure 14, but two estimated parameters: d50 and σ. The concentrations w are set to the values of Table 1.

Figure 17. Particle sizing using the core-shell model as in Figure 14, but one estimated parameter: d50. Both particle concentration w and broadness σ are set to the values of Table 1.

tions of both average particle size (except as before corn oil 1.2 µm) and particle weight fraction. In the smallest silica sample, this effect is even more pronounced than in the analogous isolated-particle case Figure 10. However, in this case the problem occurs because of the excessive number of degrees of freedom. That is, the estimation procedure based solely on the attenuation spectra does not possess sufficient information to estimate three parameters simultaneously. This is evidenced in Figures 15-17, where either the broadness, σ, or the particle concentration, w, or both are set to the values reported in Table 1. Presetting the broadness σ (Figure

In this work, the performance of the core-shell model for the prediction of attenuation and particle size distributions was evaluated for several concentrated suspensions and emulsions. Systems were investigated where the attenuation is dominated both by viscoinertial dissipation (silica/water, PTFE/water) and by thermal dissipation (corn oil/water). In all cases, a very good prediction of the acoustic attenuation was found, both at low and at high particle concentration. In particular, the nonlinear attenuationconcentration behavior is predicted correctly in the entire frequency range. It can therefore be concluded that the core-shell concept is indeed able to properly account for multiple-particle effects in concentrated suspensions and emulsions. It is particularly remarkable that this model operates entirely on a predictive basis, since it requires only information related to the isolated-particle behavior as input parameters, that is, the typical parameters of the ECAH model. The core-shell model was also used for particle sizing. It was found that in the presence of multiple-particle effects, the amount of information contained in an attenuation spectrum is limited, that is, the simultaneous extraction of both particle size and particle weight fraction is not possible. The reason for this is an intrinsic ambiguity in the attenuation spectra, where at high particle/droplet volume fractions, size, and concentration are no longer independent parameters. Nevertheless, when the particle concentration is treated as a known parameter and evaluated independently through some other technique, the core-shell model gives very good results in the entire range of concentrations. Compared to the results of the isolated-particle model, the influence of the particle concentration is substantially reduced. Glossary An′′ c′′ d d45, d50, d55 i kc′′ r rp rshell w

nth-order scattering coefficient in shell sound speed in shell (original dispersant) diameter percentiles of size distribution imaginary unit compressional wavenumber in shell radius particle radius shell radius particle weight fraction

412 R′ R′′ Reff F′ σ φ ω

Langmuir, Vol. 18, No. 2, 2002 attenuation in particle attenuation in dispersant attenuation in overall dispersion particle density standard deviation of log-normal distribution particle volume fraction angular frequency, ω ) 2πf

Hipp et al.

Acknowledgment. Financial support from Malvern Instruments Ltd. and stimulating discussions with Robert Jack and Fraser McNeil-Watson from the same company are gratefully acknowledged. This work was supported by the Swiss National Science Foundation (Grant 20-61883.00). LA015541W

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