3418
Langmuir 1999, 15, 3418-3423
Influence of Thermal Overlap Effects on the Ultrasonic Attenuation Spectra of Polydisperse Oil-in-Water Emulsions R. Chanamai, N. Herrmann, and D. J. McClements* Biopolymers and Colloids Research Laboratory, Department of Food Science, University of Massachusetts, Amherst, Massachusetts 01003 Received September 8, 1998. In Final Form: February 23, 1999 This study examines the influence of thermal overlap effects on the ultrasonic attenuation of polydisperse emulsions. The ultrasonic attenuation coefficient of a series of corn oil-in-water emulsions was measured as a function of droplet concentration (1-50 wt %), volume-surface mean droplet radius (r32 ) 0.12 and 0.58 µm), and frequency (1-120 MHz). There was good agreement between these measurements and predictions of the traditional ultrasonic scattering theory (UST) for the emulsions containing 0.58 µm droplets, but the UST greatly overestimated the attenuation of the emulsions containing 0.12 µm droplets, especially at higher droplet concentrations, due to thermal overlap effects. We show that the UST can be extended to account for these effects using a core-shell model and highlight the range of experimental conditions over which the existing UST can be successfully used.
Introduction Ultrasonic spectrometry is becoming increasingly popular as a means of characterizing the size and concentration of droplets in concentrated emulsions.1-3 Instruments that utilize this principle have recently become commercially available, which are capable of rapidly and nondestructively measuring the particle size distribution of emulsions and suspensions.4 These instruments measure the frequency dependence of the ultrasonic attenuation coefficient of an emulsion and then use an ultrasonic scattering theory (which describes the propagation of ultrasonic waves through an ensemble of particles) to interpret the resulting spectra.1,3 The concentration and size distribution of the droplets which gives the best fit between theory and experimental measurements are determined by computer software. The accuracy of this approach depends on how well the attenuation spectra predicted by the theory agree with the experimental measurements. The ultrasonic scattering theory (UST) that is most commonly used to interpret emulsion attenuation spectra is based on calculating the scattering coefficients of the individual droplets using the method of Allegra and Hawley5 and then inserting these values into a suitable multiple scattering theory.6-8 This approach has been shown to give excellent agreement with experimental measurements for a number of emulsion systems, often up to high droplet concentrations.9,10 Nevertheless, a number of recent studies have shown that this approach does not give good agreement with experimental mea* To whom correspondence should be addressed. (1) McClements, D. J. Langmuir 1996, 12, 3454. (2) Dukhin, A. S.; Goetz, P. J.; Hamlet, C. W. Langmuir 1996, 12, 4998. (3) Povey, M. J. W. Ultrasonic Techniques for Fluid Characterization; Academic Press: San Diego, CA, 1997. (4) Hackely, V. A.; Texter, J. J. Res. Natl. Inst. Stand. Technol. 1998, 103, 217. (5) Allegra, J. R.; Hawley, S. A. J. Acoust. Soc. Am. 1972, 51, 1545. (6) Waterman, P. C.; Truell, R. J. Math. Phys. 1962, 2, 512. (7) Lloyd, P.; Berry, M. W. Proc. Phys. Soc. London 1967, 91, 678. (8) Ma, Y.; Varadan, V. K.; Varadan, V. V. J. Acoust. Soc. Am. 1990, 87, 2779. (9) McClements, D. J.; Povey, M. J. W. J. Phys. D: Appl. Phys. 1989, 22, 38. (10) McClements, D. J. J. Acoust. Soc. Am. 1992, 91, 849.
surements for concentrated monodisperse emulsions containing small droplets, i.e., r < 0.2 µm.11,12 It has been postulated that the disagreement between the predictions of the traditional UST and experimental measurements is due to thermal overlap effects, which are not accounted for in the theory.11,13 In the presence of an ultrasonic wave the temperature of an emulsion droplet fluctuates periodically relative to that of the surrounding liquid due to coupling of thermal and compressional waves.5 As a consequence, heat flows out of the droplet because of the temperature gradient at the droplet-liquid interface, which results in dissipation of some of the energy originally stored in the ultrasonic wave. The distance that the thermal wave penetrates into the surrounding liquid is characterized by the thermal skin depth
δT )
x
2τ1 216 ≈ nm ωF1CP,1 xf/MHz
(1)
where τ1 is the thermal conductivity, F1 is the density, and CP1 is the specific heat capacity of the continuous phase, ω is the angular frequency ()2πf), and f is the frequency.5,14 The approximate expression for δT is appropriate for pure water at room temperature. Thermal overlap effects will become important when δT g δ, where δ is half the average surface-to-surface separation between the droplets. Half the mean distance between droplets can be approximated by the following relationship15
δ)r
(
1 -1 φ1/3
)
(2)
where r is the droplet radius and φ is the disperse phase volume fraction. Thermal overlap effects should therefore (11) Herrmann, N. Application de Techniques Ultrasoneres a l′Etude de Dispersions; Ph.D. Thesis, Universite Louis Pasteur, Strasbourg, France, 1996. (12) Herrmann, N.; Boltenhagen, P.; Lemarechal, P. J. Phys. II 1996, 6, 1389. (13) McClements, D. J. Colloid Surf. 1994, 90, 25. (14) Isakovich, M. A. Zh. Eksp. Teor. Fiz. 1948, 18, 90. (15) Hemar, Y.; Herrmann, N.; Lemarechal, P.; Hocquart, R.; Lequeux, F. J. Phys. II 1997, 7, 637.
10.1021/la981195f CCC: $18.00 © 1999 American Chemical Society Published on Web 04/07/1999
Thermal Overlap Effects on Ultrasonic Attenuation Spectra
become important when
re
(
x
2τ1 1 -1 ωF1CP1 φ1/3
)
-1
≈
(
1 -1 xf/MHz φ1/3 216
)
A01 ) -
-1
nm (3)
that is, for small droplet sizes, low ultrasonic frequencies, or high disperse phase volume fractions. The approximation expression is for water at room temperature. The aim of this study is to systematically investigate the role of thermal overlap effects on the attenuation coefficient of oil-in-water emulsions and to determine whether the existing UST can be modified to take these effects into account by using a recently developed coreshell model15,16 Ultrasonic Scattering Theory Multiple Scattering Theory. A variety of multiple scattering theories are available to relate the ultrasonic properties of emulsions to their microstructure, composition, and thermodynamic properties.6-8 One of the most widely used is that of Waterman and Truell6
() K k1
2
)1+
4πNf (0)
+
2
k1
4π 2N2 2 [f (0) - f 2(π)] (4) k14
where f(0) and f(π) are the far-field scattering amplitudes of the waves scattered from the individual droplets
f(0) )
f(π) )
1
1
∞
∑ (2n + 1)An
(5)
∑ (-1)n(2n + 1)An
(6)
ik1 n)0 ∞
ik1 n)0
Langmuir, Vol. 15, No. 10, 1999 3419
where K is the complex propagation constant of the emulsion ()ω/c + iR), k is the complex propagation constant of the continuous phase ()ω/c1 + iR1), c is the ultrasonic velocity, R is the attenuation coefficient, i is -11/2, N is the number of droplets per unit volume of emulsion ()3φ/4πr3), and An values are the scattering coefficients of the individual droplets. The scattering coefficients of emulsion droplets are usually calculated using a theory that was originally developed by Epstein and Carhart17 and then modified and extended by Allegra and Hawley.5 The full solution of this theory involves solving a six-by-six complex simultaneous equation for each value of An (except for A0, where only a four-by-four equation needs to be solved). The ultrasonic velocity and attenuation coefficient of the emulsion are calculated from the complex propagation constant, K, using the following expressions: c ) Re(ω/K) and R ) Im(K). Monopole Scattering Coefficient in the Absence of Thermal Overlap Effects. In the long-wavelength limit, when the wavelength is much longer than the droplet size, the ultrasonic attenuation of an emulsion is usually dominated by the monopole scattering coefficient (A0). An explicit analytical expression for this term has been derived5
A02 ) -
(
)
ik1r F1 (k1r)2 - (k2r)2 3 F2
(
i(k1r)3(γ1 - 1) b12
1-
(7a)
)
β2CP1F1 2 H β1CP2F2
(7b)
where b1 ) (1 + i)r/δT, γ1 ()1 + Tβ21c21/CP1) is the ratio of specific heats, T is the absolute temperature, β is the cubical expansion coefficient, and H is a function of the thermodynamic properties of the oil and water (TD), the ultrasonic frequency, and the droplet size
H ) F(TD,f,r)
(8)
An expression for H is given in the Appendix of McClements.18 Monopole Scattering Coefficient in the Presence of Thermal Overlap Effects. The theory developed by Allegra and Hawley5 assumes that the droplets are surrounded by an infinite liquid, i.e., each droplet is assumed to act independently of its neighbors. This theory would therefore be expected to breakdown when the thermal waves generated by neighboring droplets overlap with one another, i.e., at low frequencies, small droplet sizes, and/or in highly concentrated emulsions.10-12 This effect has been accounted for theoretically using a “coreshell” model.15,16,19 In this theory, it is assumed that each emulsion droplet is surrounded by a shell of the continuous phase, which is itself surrounded by an effective medium, whose properties are determined by the composition of the whole emulsion. The thickness of the shell (δx) is approximately equal to half the average distance between the droplets in an emulsion (δ) and is given by eq 2.15 When thermal overlap effects are included, the value of H in the A02 term (eq 7b) becomes a function of the thermodynamic properties of the oil and water phases, the ultrasonic frequency, the droplet size, and the thickness of the shell:
H ) F(TD,f,r,δx)
(9)
(7)
An expression for the H term that includes thermal overlap effects has recently been derived (eq 7, ref 16). Polydisperse Emulsions. The core-shell model was originally developed for monodisperse emulsions.15,16,19 In practice, most emulsions are polydisperse and so it is necessary to extend the theory to take into account a particle size distribution. The calculation of the ultrasonic properties of an emulsion requires knowledge of the surface-to-surface separation between the droplets. For monodisperse emulsions, this value is related to the radius and volume fraction of droplets by the expression φ ) r3/(r + δx)3 (eq 2). In a polydisperse emulsion, it is necessary to make some assumption about the spatial distribution of the droplets within the emulsion in order to estimate the surface-to-surface separation. One approach is to assume that the surface-to-surface separation of the droplets is independent of their particle size, i.e., the thickness of the shells surrounding the droplets in the core-shell model is independent of droplet size, then
(16) McClements, D. J.; Hemar, Y.; Herrmann, N. J. Acoust. Soc. Am. 1999, 105, 915. (17) Epstein, P. S.; Carhart, R. R. J. Acoust. Soc. Am. 1953, 25, 553
(18) McClements, D. J. Adv. Colloid Interface Sci. 1991, 37, 33. (19) McClements, D. J.; Hemar, Y.; Herrmann, N. J. Phys. D: Appl. Phys. 1998, 31, 2950.
A0 ) A01 + A02
3420 Langmuir, Vol. 15, No. 10, 1999 N
φ)
njr3j ∑ j)1
( ( ) N
)
N
nj(rj + δx)3 ∑ j)1
Chanamai et al.
Φj ∑ j)1
(rj + δx) r3j
3
-1
(10)
where rj, nj, and Φj are the radius, number per unit volume, and fractional volume of droplets in the jth size class and N is the total number of size classes. When φ and the droplet size distribution (rj vs Φj) are known, it is possible to calculate δx by solving eq 10. The value of the monopole scattering coefficient is then described by the following equation: N
A0 )
∑ j)1
Φj A0(rj,δx) rj 3
(11)
Correction Factor for Thermal Overlap Effects. The most accurate description of the ultrasonic properties of an emulsion (in which thermal overlap effects are negligible) is to calculate all of the significant An terms by solving the full six-by-six complex simultaneous equations of Allegra and Hawley5 and then inserting them in the multiple scattering theory (eq 4). At present, the influence of thermal overlap effects has not been directly incorporated into the full ultrasonic scattering theory. Nevertheless, we can still correct the full theory by developing an empirical correction factor that is based on the long wavelength explicit expressions given above
A0,corrected ) A0,full
A′0,LWL A0,LWL
(Vacuum oven 3610, Lab-Line Instruments, Inc., Melrose Park, IL) which involved weighing the emulsions before and after drying to confirm the correct oil content in the emulsions. In all cases, the droplet concentrations determined by this technique agreed with the initial amount of oil added to the emulsion. Particle Size Determination by Light Scattering. The size of the droplets in the emulsions was measured using a commercial particle-sizing instrument based on static light scattering (Horiba LA-900, Irvine, CA). A refractive index ratio (refractive index of oil divided by that of water) of 1.08 was used by the instrument to calculate the particle size distribution. Measurements are reported as the surface-volume mean radius: r32 ) ∑niri3/∑niri2, where ni is the number of droplets of radius ri. To prevent multiple scattering effects the emulsions were diluted with distilled water prior to analysis so that the droplet concentration was
