Particle Sizing in Colloidal Dispersions by ... - ACS Publications

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Langmuir 1999, 15, 2338-2345

Particle Sizing in Colloidal Dispersions by Ultrasound. Model Calibration and Sensitivity Analysis Alexander K. Hipp, Giuseppe Storti, and Massimo Morbidelli* ETH Zu¨ rich, Department of Chemical Engineering, Universita¨ tsstr. 6, CH-8092 Zu¨ rich, Switzerland Received August 18, 1998. In Final Form: December 30, 1998 In this paper, the use of ultrasonics for the determination of particle size distributions (PSDs) in suspensions and emulsions is discussed. Focusing on systems with a large density difference between the dispersed and continuous phases, a mathematical model is used to correlate the primary measurements the attenuation of acoustic waves as a function of frequencysto the PSD, and it becomes clear that the quality of the PSD thus obtained is directly related to the accuracy of the model parameters. To overcome possible problems in the case of incomplete physical-property information, we have developed a generalized procedure for particle sizing, based on a parametric sensitivity analysis and parameter calibration. The latter allows for computation of physical properties from acoustic data provided that the PSD is known. Moreover, particle sizing results are presented for dispersions of different sizes and materials, and compared to the findings of electron microscopy and dynamic light scattering.

1. Introduction The characterization of suspensions and emulsions is a promising application of ultrasonic techniques. Acoustic sensors are widely used for ultrasonic imaging in oceanography, medicine, and material science and have become an important tool in scientific and industrial areas (cf. Greguss1). Ultrasonics also entered the field of multiphase and colloidal systems, where a number of properties may be determined with this technique.2,3 Practical examples include quality control in the food industry,4-6 analysis of coal slurries,7 and monitoring of polymerization processes.8,9 The primary quantities of interest in disperse systems are the fraction of the dispersed phase, usually specified by its volume or weight, and the size distribution of the droplets or particles. In principle, both can be obtained from acoustic measurements, and the intrinsic advantages of ultrasonic techniques often prove highly beneficial, as they are fast, nondestructive, and reliable andsmaybe most importantlyscan be applied to opaque media. Especially in the field of particle sizing, where one is used to time-consuming sample treatment prior to the actual * Author to whom correspondence should be addressed. (1) Greguss, P. Ultrasonic Imaging: Seeing by Sound; Focal Press: New York, 1980. (2) McClements, D. J. Ultrasonic Characterization of Emulsions and Suspensions. Adv. Colloid Interface Sci. 1991, 37, 33. (3) Morbidelli, M.; Storti, G.; Siani, A. Monitoring Polymerization Reactors by Ultrasound Sensors. In Polymeric Dispersions: Principles and Applications; Asua, J. M., Ed.; Kluwer: Dordrecht, the Netherlands, 1997. (4) Javanaud, C. Applications of Ultrasound to Food Systems. Ultrasonics 1988, 26, 117. (5) McClements, D. J. Ultrasonic Characterization of Foods and Drinks: Principles, Methods, and Applications. Crit. Rev. Food Sci. Nutr. 1997, 37 (1), 1. (6) Povey, M. J. W., Mason, T. J., Eds. Ultrasonics in Food Processing; Blackie: Glasgow, U.K., 1998. (7) Davis, M. C. Coal Slurry Diagnostics by Ultrasound Transmission. J. Acoust. Soc. Am. 1978, 64 (2), 406. (8) Hauptmann, P.; Dinger, F.; Sa¨uberlich, R. A Sensitive Method for Polymerization Control based on Ultrasonic Measurements. Polymer 1985, 26 (11), 1741. (9) Apostolo, M.; Canegallo, S.; Siani, A.; Morbidelli, M. Characterization of Emulsion Polymerizations through Ultrasound Propagation Velocity Measurements. Macromol. Symp. 1995, 92, 205.

characterization, this enables one to characterize dispersions “as they are”. Two inherent properties can be attributed to the propagation of an acoustic wave: its speed and the energy that is absorbed (attenuation). The speed of sound is relatively easy to measure, and simple semiempirical theories are available for its prediction. Measurements at a single frequency are usually sufficient to obtain information about the volume (or weight) fraction of the dispersed phases, for example.10 However, the amount of information contained in an acoustic fingerprint may be maximized by varying the frequency. As it turns out, changes in the frequency result in large changes in the acoustic attenuation, whereas the response in the velocity is usually less pronounced.11 This makes acoustic attenuation particularly convenient for particle sizing. The possibility of using acoustic attenuation spectra for particle sizing has been considered earlier in the literature.2,7,12-22 Recent contributions report also on specific off-line and on-line applications of this technique.23-25 The present work reconsiders the underlying procedure, which allows one to obtain both the size (10) Urick, R. J. A Sound Velocity Method for Determining the Compressibility of Finely Divided Substances. J. Appl. Phys. 1947, 18, 983. (11) McClements, D. J.; Povey, M. J. W. Scattering of Ultrasound by Emulsions. J. Phys. D: Appl. Phys. 1989, 22, 38. (12) Ma, Y.; Varadan, V. K.; Varadan, V. V.; Bedford, K. W. Multifrequency Remote Acoustic Sensing of Suspended Materials in Water. J. Acoust. Soc. Am. 1983, 74 (2), 581. (13) Harker, A. H.; Temple, J. A. G. Velocity and Attenuation of Ultrasound in Suspensions of Particles in Fluids. J. Phys. D: Appl. Phys. 1988, 21, 1576. (14) Riebel, U.; Lo¨ffler, F. The Fundamentals of Particle Size Analysis by Means of Ultrasonic Spectroscopy. Part. Part. Syst. Charact. 1989, 6, 135. (15) Alba, F. Method and Apparatus for Determining Particle Size Distribution and Concentration in a Suspension using Ultrasonics. U.S. Patent No. 5 121 629, 1992. (16) Holmes, A. K.; Challis, R. E.; Wedlock, D. J. A Wide Bandwidth Study of Ultrasound Velocity and Attenuation in Suspensions: Comparison of Theory with Experimental Measurements. J. Colloid Interface Sci. 1993, 156, 261. (17) Holmes, A. K.; Challis, R. E.; Wedlock, D. J. A Wide-Bandwidth Ultrasonic Study of Suspensions: The Variation of Velocity and Attenuation with Particle Size. J. Colloid Interface Sci. 1994, 168, 339. (18) McClements, D. J. Principles of Ultrasonic Droplet Size Determination in Emulsions. Langmuir 1996, 12, 3454.

10.1021/la981046x CCC: $18.00 © 1999 American Chemical Society Published on Web 03/13/1999

Particle Sizing in Colloidal Dispersions

distribution and the volume/weight fraction of the dispersed phase by fitting absorption spectra as predicted by a mathematical model to experimental measurements. Moreover, this paper is concerned with the physical properties that govern the propagation of ultrasonic waves. These properties also affect the theoretical relation between the attenuation spectrum and the particle size distribution and are therefore critical for particle sizing. To judge whether a specific parameter needs to be determined with accuracy, possibly by an independent measurement, or a rough order-of-magnitude estimate will do, a sensitivity analysis of the computed particle size distribution (PSD) with respect to the value of each parameter is presented. When needed, one can also take advantage of the physical-property dependence of the PSD result and compute unknown or uncertain physical properties in a “calibration procedure” from an expected PSD. On the basis of these considerations, a generalized procedure for particle sizing has been developed. Within the scope of this paper, examples are shown for a particular class of dispersions: those with high density contrast between the involved phases. In addition, by restricting ourselves to low particle concentrations, nonlinear effects due to multiple scattering or particleparticle interactions are excluded. Before entering the details of acoustic particle sizing, the underlying mathematical modeling is discussed, which includes the extension of the original single-size formulation to polydisperse systems. On the basis of this model, a generalized scheme is introduced to relate attenuation measurements, physical properties, and particle size. The following sections deal with the actual particle sizing, parametric sensitivity, parameter calibration, comparison to other techniques, and reproducibility of acoustic spectroscopy. 2. Modeling The propagation of an acoustic wave depends on the properties of the host materials. In general, both the phase and the amplitude of the wave change when it passes through either homogeneous or heterogeneous media, due to scattering and irreversible dissipation of energy. Scattering is similar to the scattering of light, in the sense that part of the wave is redirected at the particle boundary. Irreversible dissipation is composed of several mechanisms, primarily intrinsic, viscoinertial, and thermal losses. Intrinsic attenuation can be attributed to interactions at a molecular level and occurs in any kind of system, even in homogeneous ones. In addition to this, in heterogeneous systems the pressure fluctuations of the acoustic wave induce mechanical particle vibrations, which in turn cause viscoinertial losses. Finally, temperaturepressure coupling gives rise to irreversible heat flow (thermal losses). The behavior of a single spherical particle immersed in an acoustic field has been analyzed in a rigorous way by (19) Dukhin, A. S.; Goetz, P. J. Acoustic and Electroacoustic Spectroscopy. Langmuir 1996, 12, 4336. (20) Dukhin, A. S.; Goetz, P. J. Acoustic Spectroscopy for Concentrated Polydisperse Colloids with High Density Contrast. Langmuir 1996, 12, 4987. (21) Dukhin, A. S.; Goetz, P. J.; Hamlet, C. W., Jr. Acoustic Spectroscopy for Concentrated Polydisperse Colloids with Low Density Contrast. Langmuir 1996, 12, 4998. (22) Povey, M. J. W. Ultrasonic Techniques for Fluids Characterization; Academic Press: New York, 1997. (23) Pendse, H. P.; Sharma, A. Particle Size Distribution Analysis of Industrial Colloidal Slurries using Ultrasonic Spectroscopy. Part. Part. Syst. Charact. 1993, 10, 229. (24) Scott, D. M.; Boxman, A.; Jochen, C. E. Ultrasonic Measurement of Submicron Particles. Part. Part. Syst. Charact. 1995, 12, 269. (25) Scott, D. M.; Boxman, A.; Jochen, C. E. In-Line Particle Characterization. Part. Part. Syst. Charact. 1998, 15, 47.

Langmuir, Vol. 15, No. 7, 1999 2339 Table 1: Physical Properties in Acoustic Spectroscopya continuous phase (water) density shear viscosity speed of sound intrinsic attenuation heat capacity thermal conductivity thermal expansion

dispersed phase (droplets/particles)

F η c R cp k β

density shear viscosity/modulusb speed of sound intrinsic attenuation heat capacity thermal conductivity thermal expansion

F′ η′/µ′ c′ R′ c′p k′ β′

a Primed quantities denote the properties of the dispersed phase. The shear viscosity is used for liquids, and the shear modulus for solids.

b

Epstein and Carhart for the liquid-liquid case.26 Allegra and Hawley generalized their treatment to solid particles.27 Even though other approaches exist, such as the coupled-phase theory,13,20 the model of Epstein/Carhart and Allegra/Hawley was chosen because of its more general formulation. The approach of these authors can be summarized as follows. Instead of considering each of the contributing effects individually, wave equations are derived from the equations of change, that is from the conservation laws of mass, momentum, and energy. The solution of these wave equations can be written as an infinite series of spherical Bessel functions, Legendre polynomials, and unknown expansion coefficients. The latter can be obtained from the continuity conditions at the particle surface, that is from the continuity of (1) normal and tangential velocity, (2) normal and tangential stress, and (3) temperature and heat flux. Repeated evaluation of these conditions yields the expansion coefficient of each term in the infinite series. Apart from some typographical errors, the full representation of this problem can be found in the original article27 (eqs 8a-f of their paper). Table 1 shows the physical properties involved. Proper evaluation of these parameters is crucial for the application of this technique, as will be discussed in detail in this paper. In principle, the full solution of the wave equations can be generated. However, for the purpose of particle sizing, it is sufficient to compute the expansion coefficients of the compressional wave equationsthe “scattering coefficients” An, with n ) 0 ... ∞swhich can directly be related to the bulk properties of the acoustic wave:

() K

2

)1-

k

24iφ

∞

∑ (2n + 1)An

k3d3 n)0

(1)

or

() K k

2

)1-

24i

[

k3d3

1+

]

F′ (1 - w) F

w

-1 ∞

∑ (2n + 1)An

(2)

n)0

where K is the bulk wavenumber, as defined by K ) ω/ceff + iReff, from which effective values for both attenuation and sound speed can be extracted. k ) ω/c + iR is the continuous-phase wavenumber, φ is the dispersed-phase volume fraction, w is its weight fraction, d is the particle diameter, ω is the angular frequency (2πf), and i ) (-1)1/2. The densities F and F′ are defined in Table 1. Note that Allegra and Hawley’s expression27 (eq 9 of their paper) is (26) 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. (27) Allegra, J. R.; Hawley, S. A. Attenuation of Sound in Suspensions and Emulsions: Theory and Experiments. J. Acoust. Soc. Am. 1972, 51 (5), 1545.

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Hipp et al.

a simplified version of eq 1, which can be reobtained through a first-order Taylor expansion at φ ) 0. The infinite series of eq 1 converges rapidly, so that only a few scattering coefficients need to be retained. In the cases studied in this paper, the largest contribution was due to A1 (which is typical for high-density-contrast dispersions), followed by A0 and A2. Higher-order terms did not affect the result. However, it is well-known that, even in similar dispersions, a larger number of coefficients might be required for very large particles (d . 1 µm).27 Equation 1 has its origin in the general wave theory and can also be found in electromagnetic (optical) problems.28 It contains the implicit assumption that each particle is immersed in the undisturbed field and thus ignores the effect of other particles. This “single-scattering” assumption is met only in the limit of a vanishing number of particles. Several authors have attempted to include multiple scattering in their analysis and have derived expressions that differ from eq 1 by higher-order terms in the volume fraction.28-33 However, since the present study is carried out at rather low volume concentrations (around 0.5%), the single-scattering relationswhich gives reasonable results even at higher concentrationssis used throughout this work. 3. Size Distributions The theoretical treatment of Epstein and Carhart26 and Allegra and Hawley27 is designed for a single particle size. However, the extension to a number of particles is straightforward.18 It relies on the fact that the acoustic field is linear and so are the wave equations and the boundary conditions used to compute the scattering coefficients An. Therefore, the solutions may be superimposed for different particle sizes, and the sum of scattering coefficients in eq 1 may be generalized as follows:

1

∞

∑

d3n)0

(2n + 1)An w

∫d

pv(d) d3

∞

∑ (2n + 1)An(d) dd

(3)

n)0

In this equation, pv(d) is the probability that the system contains particles of a certain range of diameters (d ... d + dd), in the units of inverse meters, normalized to unity: ∫pv(d) dd ) 1. The probability density is based on the particle volume, as indicated by the index v. In the monodisperse case, pv(d) becomes a Dirac delta, and the validity of relation 3 becomes apparent by the equality

∫0∞f(d) δ(d - dm) dd ) f(dm) where f(d) is the integrand of relation 3 without pv(d). The probability density pv(d) can take any form that represents the distribution of particle sizes. A common assumption is that the logarithm of the particle diameter (28) Waterman, P. C.; Truell, R. Multiple Scattering of Waves. J. Math. Phys. 1961, 2 (4), 512. (29) Lloyd, P.; Berry, M. V. Wave Propagation through an Assembly of Spheres. IV. Relations Between Different Multiple Scattering Theories. Proc. Phys. Soc. 1967, 91, 678. (30) Twersky, V. Acoustic Bulk Parameters in Distributions of PairCorrelated Scatterers. J. Acoust. Soc. Am. 1978, 64 (6), 1710. (31) Tsang, L.; Kong, J. A.; Habashy, T. Multiple Scattering of Acoustic Waves by Random Distribution of Discrete Spherical Scatterers with the Quasicrystalline and Percus-Yevick Approximation. J. Acoust. Soc. Am. 1982, 71 (3), 552. (32) Ma, Y.; Varadan, V. V.; Varadan, V. K. Multiple Scattering Theory for Wave Propagation in Discrete Random Media. Int. J. Eng. Sci. 1984, 22, 1139. (33) Ma, Y.; Varadan, V. K.; Varadan, V. V. Comments on Ultrasonic Propagation in Suspensions. J. Acoust. Soc. Am. 1990, 87 (6), 2779.

is normally distributed, which is described by a lognormal distribution,

pv(d) )

[ (

)]

1 1 ln d - ln d50 exp 2 σ dx2πσ

2

(4)

with the median d50 and the standard deviation σ. The corresponding cumulative distribution is given by

Pv(d) )

[

∫0dpv(d′) dd′ ) 21 + 21 erf x22

]

ln d - ln d50 σ

(5) The median d50 is also called the 50% fractile, since half of the particles fall below this threshold. 4. Numerical Techniques The model as discussed in the previous sections allows for the computation of attenuation spectra given the particle size distribution (d50, σ) and the dispersed phase weight fraction w. However, in practical particle sizing, the problem is inverted: the attenuation spectrum is available through experimental measurements, and one wants to calculate PSD and weight fraction. This is the “primary problem” of particle sizing. The process of moving from the solution of a model to its actual input is called inversion. For complex problems, one often needs to rely on a nonlinear regression algorithm that determines the unknown quantitiessin the present case PSD/weight fractionsby fitting theoretical model predictions to the measured attenuation data. Even though finding the PSD/weight fraction is the primary concern of particle sizing, one should bear in mind the underlying physical properties. These parameters are involved whenever the model is evaluated, either to predict attenuation spectra or to estimate PSD/weight fraction. Since complete literature data may not always be available, it is important to note thatsin mathematical termss there is no intrinsic difference between calculating PSD/ weight fractions using physical properties (as in the primary problem) and computing physical properties from the PSD/weight fraction. Therefore, there are three ways of thinking of this problem (Figure 1): (1) From a given PSD/weight fraction and given physical properties, one can compute attenuation spectra (“modeling problem”, Figure 1a). (2) From measured attenuation spectra and given physical properties, one can estimate PSD/weight fraction (“primary problem”, Figure 1b). (3) From measured attenuation spectra and given PSD/weight fraction, one can estimate physical properties (“secondary problem”, Figure 1c). To examine this strategy, a computer code was developed that incorporates the physical properties on the same level as the PSD/weight fraction, that is, as possibly unknown quantities. The parameter vector is thus defined as

p ) [d50, σ, w,

r PSD/weight fraction

F, η, c, R, cp, k, β

r physical properties continuous phase

F′, η′ or µ′, c′, R′, r physical properties dispersed phase c′p, k′, β′] The notation corresponds to the one introduced in Table 1. Note that some of the above quantities, for example R


Langmuir, Vol. 15, No. 7, 1999 2341 Table 2. Physical Properties of Water and Silica at 25 °C density37-40 shear viscosity39,40 speed of sound27,37 intrinsic attenuation27,40,41 silica density35-37

Figure 1. Three ways of seeing the problem.

and R′, are in principle frequency-dependent and should be evaluated accordingly. A nonlinear regression algorithm (GREG34) allows for an estimation of any combination of these parameters. The portion of the parameter vector to be estimated this way can be defined as punknown. Well-established properties are excluded from the optimization, labeling the corresponding part of the parameter vector as ppreset. In this framework, two operation modes are of immediate practical interest: I. Estimation Mode

punknown ) [d, σ, w] ppreset ) [physical properties] II. Calibration Mode

punknown ) [some physical properties] ppreset ) [d, σ, w, other physical properties] The use of both modes is illustrated in the following sections. First, the estimation mode is used to characterize a colloidal dispersion and to study the related parametric sensitivity. After that, the computation of physical properties in the calibration mode is illustrated, which leads to a generalized procedure for particle sizing by acoustic spectroscopy. 5. Particle Sizing in High-Density-Contrast Dispersions For the materials under investigation, the theoretical attenuation as computed from eq 1 is dominated by the A1 term. In the long-wavelength regime, that is, for particle sizes much smaller than the wavelength, this term can be attributed to viscoinertial losses.11 Other attenuation (34) Stewart, W. E.; Caracotsios, M.; Sørensen, J. P. Parameter Estimation from Multiresponse Data. AIChE J. 1992, 38 (5), 641.

F η c R/f 2 F′

997.05 kg/m3 890.3 × 10-6 Pa s 1497 m/s 2.20 × 10-14 Np s2/m 2200 kg/m3

mechanisms and their underlying parameters therefore play a minor role. As a result, parameters such as those purely related to the thermal behavior (cp, k, β, cp′, k′, β′) do not affect the particle sizing procedure. Moreover, when turning to the properties of the solid particles, it is demonstrated in the following section that also its acoustic properties (c′, R′) and the shear modulus µ′ are of negligible influence. Therefore, only 5 model parameters (F, η, c, R, F′) out of the total number of 14 remain to be specified. This makes ultrasonic particle sizing of high density contrast dispersions particularly convenient, especially since four parameters belong to the usually aqueous and thus wellknown dispersant. The dispersions analyzed in this paper fall into this category and can thus be characterized by a minimum number of physical properties. Note that the behavior might become more complex when particles are not much smaller than the wavelength.18,27 However, this occurs only for rather large particles, since, even for frequencies as high as 150 MHz, the wavelength in water is not below 10 µm. In the usual frequency range, the wavelength and thus the allowable maximum particle size are even larger. The sample studied in this section is a colloidal silica sol (silicon dioxide) from Nissan Chemical Industries, Ltd. Sols of this kind are stable aqueous dispersions of amorphous particles, in this case of approximately 0.3 µm in diameter; other sizes and materials follow in the remainder of this paper. Because of its amorphous structure, the density of silica is usually tabulated as 2.2 g/cm3, slightly lower than those of crystalline forms of silica.35-37 The properties of water were taken from standard references and deviate only slightly from the values used in Allegra and Hawley’s original publication.27 As seen in Table 2, all parameters except R are independent of frequency. For the latter, a proportionality to f 2 has been assumed. Figure 2 shows the measured and the theoretical attenuation spectra after optimization. The experimental data were recorded at 25 °C ((0.1 °C) using a Malvern Instruments Ultrasizer.15,42 The optimization resulted in a PSD/weight fraction of d50 ) 312 nm, σ ) 0.05, w ) 1.16%; the complete PSD is reconstructed from these numbers in Figure 3 (eq 4). A reasonable agreement between the attenuation curves in Figure 2 indicates the validity of the solution and is a necessary condition for reliable particle sizing. However, due to the ill-posed (35) Ullmann’s Encyclopedia of Industrial Chemistry, 5th ed.; VCH: New York, 1993; Vol. A23. (36) Kirk-Othmer: Encyclopedia of Chemical Technology, 4th ed.; Wiley: New York, 1997; Vol. 21. (37) CRC Handbook of Chemistry and Physics, 78th ed.; CRC Press: Boca Raton, FL, 1997. (38) Perry’s Chemical Engineers’ Handbook, 7th ed.; McGraw-Hill: New York, 1997. (39) Ullmann’s Encyclopedia of Industrial Chemistry, 5th ed.; VCH: New York, 1996; Vol. A28. (40) Franks, F., Ed. Water: A Comprehensive Treatise; Plenum Press: New York, 1972; Vol. 1: The Physics and Physical Chemistry of Water. (41) Pinkerton, J. M. M. The Absorption of Ultrasonic Waves in Liquids and its Relation to Molecular Constitution. Proc. Phys. Soc. London 1949, 62B (2), 129. (42) Roberts, D. Ultrasound Analysis of Particle Size Distribution. Mater. World 1996, 4 (1), 12.

2342 Langmuir, Vol. 15, No. 7, 1999

Hipp et al. Table 3. Error in PSD/Weight Fraction When Using Manipulated Physical Propertiesa parameter (off by +10%) density (particles) density (water) speed of sound (water) intrinsic attenuation (water) shear viscosity (water) a

Figure 2. Experimental attenuation spectrum (symbols) and theoretical prediction after optimization (line) for silica 0.3 µm. The corresponding PSD/weight fraction is d50 ) 312 nm, σ ) 0.05, w ) 1.16%.

Figure 3. Particle size distribution pv(d) for silica 0.3 µm as estimated by acoustic spectroscopy. The graph corresponds to the optimization in Figure 2, that is, to log-normal parameters of d50 ) 312 nm and σ ) 0.05.

nature of the problem, it should be kept in mind that the corresponding combination of PSD/weight fraction might not be unique. In the case of multiple solutions, the optimization is expected to be rather sensitive to the initial conditions, running into the various suboptimal solutions in an arbitrary way. Fortunately, such behavior was never observed in any of the cases studied; the results reported in this paper were found to be independent of the choice of initial conditions. 6. Sensitivity Analysis In this section, we study the effect of the model parameters on the particle sizing result and investigate the robustness of the model with respect to parameter uncertainties. This is to give guidelines for the specification of physical properties, pointing out where research for accurate numbers is worthwhile and where it is not. To simulate parameter uncertainties, each parameter is manipulated by 10%. By repeating the analysis step, 9 out of the total number of 14 parameters (Table 1) were found to have no significant effect on the estimated PSD/ weight fraction, as already stated in the preceding section. An error in these parameters shows up in or behind the fourth digit of d50, σ, and w, which is below the overall accuracy of the technique and can thus safely be neglected. The reason for this behavior is that the viscoinertial loss mechanism can be attributed (in the long-wavelength regime) to the first-order scattering coefficient A1,11 whichsin the system under examinationsrepresents the dominating factor in the series of eq 1. When following

residual ξ w (%) (eq 6) (%)

d50 (%)

σ

F′ F c R/f 2

-5 0 0 +5

0.05* 0.05* 0.05* 0.05*

-16 +22 +10 -5

,5 ,5 ,5 ,5

η

+5

0.05*

0

,5

An asterisk denotes a lower limit.

the solid-particle approximation for this case, as suggested by Allegra and Hawley27 (eq 15 of their paper), one ends up with a rather simple relationship between only a few physical properties and the overall attenuation. As a matter of fact, in our cases, particle sizing results obtained with this approximation for A1 and the neglect of other coefficients (A0, A2, ...) were within a few percent of those obtained from the general theory as described and used throughout this paper. Standard Degrees of Freedom. The effect of the remaining parameters is shown in Table 3, where the value of each physical property is increased by 10%, one after another. The resulting PSD/weight fractions were compared to those obtained previously and expressed in terms of a relative change. One can observe an approximately linear propagation of the parameter error into the result: for example, a 10% change in silica density results in a 5% change in the estimated diameter and a 16% change in w. The standard deviation σ remains untouched, even though this is probably due to the narrowness of the sample; in the case of a broad PSD, one would expect changes in σ as well. An important observation is related to the residual model error as reported in the last column of Table 3. This quantity is a per-point average of the error squares and is a measure for the discrepancy between the experimental and theoretical attenuation spectra, defined as

ξ)

x ∑( 1

N

Ni)1

)

Ri,exp - Ri,th Ri,exp

2

(6)

where N is the number of observations in a spectrum. In the present case, none of the parameter errors result in a poor fitsthe model is flexible enough to compensate for any error in the input parameters. Such agreement is misleading, since one is tempted to believe in the estimated result without reflecting on its quality. For this reason, it is essential either to double-check part of the result by independent measurements or to reduce the amount of information to be extracted. Reduced Degrees of Freedom. Thus far, ultrasonic particle sizing was presented as a technique that yields both PSD (d50, σ) and weight fraction w. This allows three degrees of freedom to the optimization routine, which is enough to converge to small residual model errors regardless of incorrect physical properties. Therefore, when dealing with uncertain model parameters, the degrees of freedom should be decreased to reduce the risk of erroneous results. For example, in the case where one is interested primarily in the PSD, the degrees of freedom can easily be reduced by measuring the particle concentration by some other means. To demonstrate this strategy, we set the weight fraction to 1.16% and repeated the sensitivity analysis. The value of this parameter was obtained previously from acoustic spectroscopy (Figures 2 and 3). The resulting parametric


Langmuir, Vol. 15, No. 7, 1999 2343

Table 4. Error in PSD When Using Manipulated Physical Properties (w Preset to 1.16%)a d50 (%)

σ

residual ξ (eq 6) (%)

F′ F c R/f 2

-36 +17 +8 -3

0.5** 0.05* 0.05* 0.21

6.2 11.4 5.9 4.5

η w

+5 -22

0.05* 0.42

,5 4.2

parameter (off by +10%) density (particles) density (water) speed of sound (water) intrinsic attenuation (water) shear viscosity (water) weight fraction

a An asterisk denotes a lower limit, and two asterisks an upper limit.

Table 5. Parameter Calibration for Silica 0.3 µm estimated parameters none, all parameters set to literature values (Table 2) F′ ) 2200.2 kg/m3 F′ ) 2199.9 kg/m3, F ) 996.9 kg/m3 F′ ) 2211.2 kg/m3, c ) 1514 m/s F′ ) 2172.4 kg/m3, F ) 825.6 kg/m3, c ) 2020 m/s F′ ) 2118.5 kg/m3, c ) 1359 m/s, R/f 2 ) 2.105 × 10-14 (Np s2)/m F′ ) 2127.2 kg/m3, F ) 955.5 kg/m3, R/f 2 ) 2.124 × 10-14 (Np s2)/m F′ ) 2128.1 kg/m3, F ) 951.9 kg/m3, c ) 1509 m/s, R/f 2 ) 2.126 × 10-14 (Np s2)/m

residual ξ (eq 6) (%) 2.255 2.255 2.255 2.251 2.157 2.133 2.132 2.132

sensitivities are shown in Table 4. The behavior is as expected as the residuals ξ have become larger than those in the previous case (Table 3). However, the overall situation has improved: the parameter values are of poor quality, so is obviously the PSD information, but in turn, the model cannot fit the experimental data. Therefore, the model now shows the poor conditions, which is preferable to the case where one obtains somewhat better results, but without indications of error. 7. Parameter Calibration and General Procedure for Particle Sizing Parameter calibration is a way to estimate physical properties. This option is useful for dispersions of unknown or uncertain properties which are well-characterized in terms of particle size distribution and concentration. However, it is important to focus on the essential model parameters, since those of negligible influence do not affect the results of acoustic spectroscopy (as discussed previously) and can therefore not be estimated using the modelbased technique presented here. The sensitivity analysis of the previous section proves to be a useful tool in this context, indicating the following ranking of parameters for the present case: (1) silica density F′; (2) water density F; (3) water speed of sound c; (4) water attenuation R; (5) water shear viscosity η. The idea behind parameter calibration is to let the optimization routine find values for the unknown quantities that lead to the best fit between model and experiments. During this search, both PSD and weight fraction are set to the expected values. In the present case, the numbers of the preceding sections were used for this purpose, that is, d50 ) 312 nm, σ ) 0.05, and w ) 1.16% (Figures 2 and 3). The resulting parameter values are shown in Table 5, where the initial situation (no unknown parameters) is displayed in the first row. In the following rows, more and more physical properties are estimated. Note that the residual ξ is constantly decreased as more degrees of freedom are given. It becomes clear from this example that two physical properties can easily be obtained. However, when asking

for a larger number, the optimization tends to find values that are far from the expected ones (which are known in this case but not in general). Parameter values computed this way cannot be considered reliable, because even though they lead to an improvement of the residual model error, they are incorrect and unlikely to be useful in other situations. Therefore, we suggest the following general strategy be applied when a new combination of materials is considered for particle sizing by acoustic spectroscopy: (1) Analyze the experimental attenuation spectrum of the new sample by computing d50, σ, and w using the best physical-property information available. If the agreement between model and experiment is satisfactory, continue by checking the result (step 2); otherwise, proceed with a parameter calibration (step 3). (2) Check the results by determining part of them from an independent source (d50, σ, or w). These parameters can be excluded from the optimization. If the residual model error stays reasonably small when estimating the remaining parameters and the obtained numbers are close to the initial values estimated in step 1, the results can be accepted. Otherwise, a parameter calibration (step 3) is necessary after all. (3) Calibrate physical properties. Rank the parameters according to their sensitivity and optimize one or more of them using d50, σ, and w as obtained from independent sources. It is important to not blindly trust a result simply because it was obtained with a good model fit. Instead, one needs to be aware of the large number of degrees of freedom (d50, σ, w) which may lead to artificial solutions where parameter uncertainties are compensated for by incorrect answers. Therefore, at least part of the solution should be verified as suggested above. 8. Comparative Analysis In this section, the results of acoustic spectroscopy are compared to those of other techniques. Among the large number of particle sizing techniques, dynamic light scattering (DLS or photon correlation spectroscopy, PCS, Malvern Instruments Zetasizer 5000) and electron microscopy (transmission/scanning) were chosen as references. The particle concentration of each sample was determined by thermogravimetry. The following samples were considered: (1) colloidal silica, approximately 0.1 µm in diameter; (2) colloidal silica, approximately 0.3 µm in diameter, monodisperse; (3) colloidal silica, bimodal; (4) PTFE, approximately 0.25 µm in diameter, monodisperse. The silica dispersions are sols of spherical particles manufactured by Nissan Chemical Industries. Sample 1 contains particles of a range of sizes, from about 50 to 110 nm, as detected in electron micrographs. Sample 2 is the dispersion already discussed earlier, a virtually monosize standard. Sample 3 is a bimodal sample with a dominating narrow mode at approximately 0.5 µm. A secondary mode is located at around 0.2 µm, however making up only less than 5% of the overall particle volume. Sample 4 contains nearly spherical poly(tetrafluoroethylene) particles of narrow size distribution, as supplied by Ausimont. The density of both silica35-37 (Table 2) and PTFE43 is indicated as approximately 2.2 g/cm3. Acoustic spectroscopy has been applied in two ways, in the standard mode and the reduced mode. The former refers to particle sizing with unknown PSD and particle (43) Mark, J. E., Ed. Physical Properties of Polymers Handbook; American Institute of Physics: New York, 1996.

2344 Langmuir, Vol. 15, No. 7, 1999

Hipp et al.

Table 6. Comparison of Results Obtained through Acoustic Spectroscopy to Those from Electron Microscopy, Dynamic Light Scattering (Photon Correlation Spectroscopy), and Thermogravimetrya Silica 0.1 µm acoustic spectroscopy median diameter d50 (nm) (PCS:dPCS) standard deviation σ (PCS:PI) weight fraction w (%)

standard mode

reduced mode

TEM

DLS/PCS

83 0.05 1.14

82 0.09

81 0.19

112 0.10

gravimetry

1.17 Silica 0.3 µm

acoustic spectroscopy median diameter d50 (nm) (PCS:dPCS) standard deviation σ (PCS:PI) weight fraction w (%)

standard mode

reduced mode

SEM/TEM

DLS/PCS

312 0.05 1.16

314 0.05

346/354 0.05

355 0.06

gravimetry

1.15 Silica 0.2/0.5 µm


standard mode

reduced mode

TEM small/large mode

DLS/PCS

428 0.26 1.14

432 0.05

215/483 0.18/0.02

419 0.16

gravimetry

1.10 PTFE 0.25 µm


standard mode

reduced mode

TEM small/large mode

DLS/PCS

239 0.05 1.11

251 0.05

266 0.09

249 0.04

gravimetry

1.05

Modes of acoustic spectroscopy: standard mode, three degrees of freedom (d50, σ, w), standard physical properties of Table 2; reduced mode, as for standard mode, but w preset to gravimetric results. a

concentration (d50, σ, w), whereas, in the latter, the degrees of freedom are reduced by setting the weight fraction w to a given value. A reduced mode of this kind may be useful if part of the result is already known from an independent measurement. As discussed previously, such a practice may improve the robustness of acoustic particle sizing with respect to incorrect physical-property information. The common starting point of both operation modes is the measurement of the acoustic spectrum, which in the present cases was performed at 25 °C ((0.1 °C) using a Malvern Instruments Ultrasizer.15,42 In Table 6, the results are reported in the first two columns and compared to those of microscopy (column 3) and dynamic light scattering/photon correlation spectroscopy (column 4). Standard Operation Mode. Acoustic spectroscopy is generally in agreement with other methods. The particle size d50 and the weight fraction w are within about 10% and 5% of the reference techniques, respectively. Furthermore, in the case of narrow samples (silica 0.3 µm; PTFE), acoustic spectroscopy detects the width of the distribution in the correct way. However, the values of the standard deviation σ for the smallest silica (0.1 µm) and the bimodal silica (0.2/0.5 µm) are debatable. In the first case, acoustic spectroscopy does not detect the broadness properly, possibly due to a reduced sensitivity for the smaller particles (e.g. < 80 nm). In this case, this technique would have cut off the small-size part of the PSD. For the bimodal silica, the presence of small particles seems to be reflected by a relatively large value of the standard deviation. Note that this result can in principle be improved by using a bimodal PSD in the estimation procedure. Although possible, this approach has not been examined in this work. Reduced Degrees of Freedom. This mode results in only minor changes as compared to the previous operation

mode, which supports the reliability of the obtained estimates. Changes can be observed in the standard deviation of the smallest and the bimodal silica (0.1 µm, 0.2/0.5 µm) and in a slight modification of the median diameter of PTFE. Other Sizing Techniques. Comparative analyses of particle size methods are difficult to carry out, since, in general, different methods give different results (cf. Collins44). Even if the size range of each technique is appropriate, differences occur because different dimensions of the particle are measured, different assumptions are made, or the sample is handled in different ways. Moreover, comparison between particle size methods suffers from the different scales on which the sizes of the particles are weighted. Acoustic spectroscopy yields the particle size distribution on a volume basis, whereas transmission microscopy (TEM) measures the projected perimeter and therefore the particle cross section. The numbers in Table 6 reflect the log-normal parameters of the corresponding distributions (d50 and σ in eqs 4 and 5). As opposed to that, dynamic light scattering (DLS/PCS) gives an harmonic intensity average (dPCS). This quantity is characteristic of this technique, and a direct relation between dPCS and d50 exists only in limiting cases.45,46 The same is true for the “polydispersity index” (PI) obtained by this technique, which is analogous but not identical to the standard deviation σ. These differences become important for broad distributions, as in the case of the (44) Collins, E. A. Measurement of Particle Size and Particle Size Distribution. In Emulsion Polymerization and Emulsion Polymers; Lovell, P. A., El-Aasser, M. S., Eds.; Wiley: New York, 1997. (45) Finsy, R.; De Jaeger, N. Particle Sizing by Photon Correlation Spectroscopy. Part II: Average Values. Part. Part. Syst. Charact. 1991, 8, 187. (46) ISO 13321. Particle Size AnalysissPhoton Correlation Spectroscopy; International Organization for Standardization: Geneva, Switzerland, 1996.


Langmuir, Vol. 15, No. 7, 1999 2345

Table 7. Repeated Measurements by Acoustic Spectroscopya 1

a

2

3

4

5

6

median diameter d50 (nm) standard deviation σ weight fraction w (%)

83 0.05 1.15

Silica 0.1 µm 84 84 0.05 0.05 1.14 1.14

82 0.05 1.16

83 0.05 1.15

84 0.05 1.12


315 0.05 1.15

Silica 0.3 µm 308 312 0.15 0.05 1.18 1.16

312 0.05 1.16

312 0.05 1.16

312 0.05 1.16


426 0.27 1.15

Silica 0.2/0.5 µm 428 428 0.24 0.28 1.14 1.15

430 0.27 1.14

429 0.20 1.13

430 0.28 1.15


240 0.05 1.11

PTFE 0.25 µm 239 241 0.05 0.06 1.11 1.11

237 0.11 1.12

241 0.05 1.10

238 0.08 1.11

Each column corresponds to an attenuation spectrum that was measured and analyzed independently.

smallest (0.1 µm) and the bimodal silica (0.2/0.5 µm). This is shown by the fact that the small-size mode of the latter is visible on micrographs but practically vanishes on a volume or intensity basis. Therefore, both acoustic spectroscopy and light scattering do not resolve the bimodality and suggest some overall average. It is therefore not surprising that the results of acoustic spectroscopy, electron microscopy, and light scattering do not directly coincide. For example, the median size d50 is slightly lower than the averages determined by electron microscopy and light scattering. However, this needs to be interpreted as due to inherent properties of the methods and does not indicate inaccuracy. Note that forcing acoustic spectroscopy to estimate larger sizes (by changing the underlying physical properties) is not useful. To increase the estimated size by the appropriate amount (while leaving the weight fraction intact), unrealistic changes in the physical properties are required, for example η ) 1100 × 10-6 Pa s or, alternatively, F′ ) 1.83 g/cm3 combined with c ) 980 m/s. This corresponds to relative changes with respect to the literature values (Table 2) of +24%, -17%, and -35%, respectively. 9. Reproducibility of the Results An important feature of any characterization technique is its reproducibility, that is, how well consecutive analyses of the same sample agree with each other. Widespread results are of little help, even though they might occasionally match those of other techniques. Procedures that continuously lead to similar numbers are far more useful, in principle even at the price of a limited agreement with reference techniques. This fact is particularly true in the field of particle sizing, where it is generally agreed that the results are a function of the technique by which they are obtained.44 To examine this aspect for acoustic spectroscopy, a series of successive measurements is listed in Table 7. Six independent attenuation spectra were recorded for each sample and processed in the standard operating mode using the physical properties of Table 2. The closeness of the numbers documents the excellent reproducibility of acoustic spectroscopy.

10. Conclusions In this work, the acoustic attenuation of several aqueous dispersions was measured, and a suitable mathematical model used to convert these data into PSDs and volume/ weight fractions. The model, extended to the polydisperse case, was discussed with respect to its model parameters. A generalized scheme was presented that enables one to analyze the problem from an arbitrary point of view, that is, by focusing either on attenuation, on PSD/weight fraction, or on the model parameters. The behavior of the model was demonstrated for the special case of dispersions having large density contrast between the particles and the continuous phase. The physical properties governing the numerical conversion between measured attenuation and PSD/weight fraction were examined, and it was found that in the present case only a small subset needs to be taken into account. Their influence on the calculated result was demonstrated in a sensitivity analysis. In addition, a calibration procedure was presented that allows for computation and/or tuning of the key parameters. As shown for several sizes and materials, the results of acoustic spectroscopy are highly reproducible and compare favorably to those of other methods. The differences between acoustic spectroscopy, electron microscopy, and dynamic light scattering were typically found to be less than 10%, which can be attributed to the nature of the measurement techniques. Acknowledgments. Financial support and stimulating discussions with the members of the Ultrasizer Group at Malvern Instruments Ltd. are gratefully acknowledged. The authors would also like to thank the Laboratory for Electron Microscopy (EM 1) of the ETH Zu¨rich for its help with the micrographs and Nissan Chemical Industries Ltd. and Ausimont for providing the sample materials. This work was supported by Grant NF 2100-050504 of the Swiss National Science Foundation. LA981046X

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