Langmuir 1999, 15, 3045-3049
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Articles Ultrasonic Wave Propagation in Silica Sols and Gels A. K. Holmes* and R. E. Challis School of Electrical and Electronic Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, U.K. Received October 6, 1998. In Final Form: February 5, 1999 Acoustic attenuation and phase velocity in the frequency range 2-50 MHz have been measured in a series of silica sols and gels with different ionic strengths, to investigate the effect of interparticle forces on acoustic propagation. While sols and gels at the same concentration but at different ionic strengths have widely differing viscosities, no significant differences were found in the acoustic properties. By comparison with previously published work on neutron scattering in silica sols and gels, this suggests that ultrasonic propagation in these colloids is not sensitive to interparticle forces.
1. Introduction A wide range of industrial products either pass through a colloid state during manufacture or are in a colloid state as the end product. Some examples are lubricants, ceramics, pharmaceuticals, and foods, and ultrasonics offers a possible means of monitoring their dispersed phase parameters and colloidal states. The validity of a number of models of acoustic propagation has been investigated in colloidal systems with industrial applications; for example, Davis1 applied the theory of Allegra and Hawley2 to coal slurries, McClements and Povey3 compared the same theory to emulsions of relevance to the food industry, and Harker et al. applied a hydrodynamic model to measurements on waste slurries.4 However, neither of these models incorporates the effects of interparticle forces. Harker and Temple5 suggested that interparticle forces might be important for small particles but did not include them in their model. Strout6 attempted to incorporate the effects of interparticle interactions into the hydrodynamic model using an oscillatory cell approach, and Dukhin and Goetz7 further developed this idea to show how interparticle forces might be incorporated. It is not known whether interparticle forces have any bearing on the acoustic properties of colloidal systems, but it has been suggested as a possible explanation for discrepancies between experimental data and the Epstein and Carhart8/ Allegra and Hawley (ECAH) model.9 In this paper we have attempted to investigate the effect of interparticle forces on acoustic propagation by carrying out broad-bandwidth ultrasonic measurements of absorption and phase velocity in silica sols and gels under (1) Davis, M. C. J. Acoust. Soc. Am. 1978, 64 (2), 406. (2) Allegra, J. R.; Hawley, S. A. J. Acoust. Soc. Am. 1972, 51 (5), 1545. (3) McClements, D. J.; Povey, M. J. W. J. Phys. D 1989, 22, 38. (4) Harker, A. H.; Schofield, P.; Stimpson, B. P.; Taylor, R. G.; Temple, J. A. G. Ultrasonics 1991, 29, 427. (5) Harker, A. H.; Temple, J. A. G. J. Phys. D 1988, 21, 1576. (6) Strout, T. A. Thesis, The University of Maine, 1991. (7) Dukhin, A. S.; Goetz, P. J. Langmuir 1996, 12, 4987. (8) Epstein, P. S.; Carhart, R. R. J. Acoust. Soc. Am. 1953, 25, 553. (9) Holmes, A. K.; Challis, R. E.; Wedlock, D. J. J. Colloid Interface Sci. 1994, 156, 339.
different conditions of ionic strength. Ramsey and Booth10 have previously shown from small angle neutron scattering (SANS) that the magnitude of the forces and to some extent the structure of the sol or gel are dependent on the ionic strength. We have repeated the work on silica sols and gels, but using ultrasonic measurements rather than SANS, to investigate the correlation between the two techniques, and the effect of the interparticle forces on the acoustic propagation. The ultrasonic measurements have also been modeled using the theory of Harker and Temple and Urick,11 to investigate whether relatively simple models of acoustic propagation, which ignore interparticle forces, give an adequate representation of acoustic propagation in silica sols with particle sizes e30 nm. 2. Materials and Methods 2.1. Preparation of Silica Sols and Gels. Silica sols of concentration 30% or 40% w/w were obtained commercially from the Aldrich Chemical Co. These were Ludox SM, HS, and TM grades (DuPont). Following the method of Ramsay and Booth,10 the stock sols were dialyzed repeatedly against dilute NaNO3 solutions of ionic strength 10-4 or 5 × 10-3 mol‚dm-3. Dialysis was assumed to be complete when the pH of the dialyzing electrolyte had fallen to pH 8.0 after 48 h of equilibration with the sols. All the measurements on sols in this paper were carried out at this pH. Sols were prepared with concentrations of 5% and 10% w/w in the case of Ludox SM and HS and with concentrations of 5, 10, 20, 30, 35, and 40% in the case of Ludox TM. Gels of concentration 30% and 35% were prepared from Ludox TM by dialysis against deionized water. The particle size of the sols was taken from the electron microscopy data of Ramsay and Booth, that is, 12, 16, and 30 nm diameter, respectively, for Ludox SM, HS, and TM. The particle sizes of the HS and TM grades were also checked by photon correlation spectroscopy (ALV 5000), and the Stokes diameters were found to be 18.8 ( 2 nm and 33.2 ( 2 nm, respectively. The viscosity of the sol samples was measured by U-tube viscometry. 2.2. Ultrasonic Measurement Technique. A detailed description of the technique can be found in ref 12. A small sample (10) Ramsay, J. D. F.; Booth, B. O. J. Chem. Soc., Faraday Trans. 1 1973, 79, 173. (11) Urick, R. J. J. Appl. Phys. 1947, 18, 983. (12) Challis, R. E.; Harrison, J. A.; Holmes, A. K.; Cocker, R. P. J. Acoust. Soc. Am. 1991, 90, 730.
10.1021/la981391m CCC: $18.00 © 1999 American Chemical Society Published on Web 04/01/1999
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of the colloid under test (25 cm3) is held in a Perspex test cell with aluminum end plates into which are mounted two 5 mm thick piezoelectric transducer elements. The test cell is housed in a temperature-controlled cabinet with the temperature regulated to 25 ( 0.1 °C. Acoustic pulses of 10 ns duration are transmitted through the colloid under test, and signals at the receiver transducer are amplified and digitized at 400 MHz. Signal processing is carried out on a host computer to correct for radiation coupling and transducer effects, and the frequency response of propagation through the test liquid is obtained from the fast Fourier transform (FFT) of the corrected signal. The phase velocity is calculated from the time-of-flight of the pulse and the corrected unwrapped phase spectrum, and attenuation is calculated from the corrected amplitude spectrum, giving velocity and attenuation spectra for the 2-60 MHz frequency range with 200 kHz point spacing. Due to the thickness of the transducers, their reverberation period of 1 µs is much greater than the input pulse duration (10 ns). When used in this manner, the frequency response of the transmitting transducer, when driven from a low-impedance source and for times less than one reverberation period, is flat throughout the measurement frequency range, and the frequency response of the receiver transducer is similar to that of a simple low-pass CR filter network, where C is the transducer capacitance and R is the electrical load resistance. A low-capacitance transducer material is used together with a load resistance designed to give a -3 dB point in the receiver frequency response at >60 MHz. The signal-to-noise ratio for a single pulse response, with water as the test liquid, is 30 dB at 5 MHz, and this is improved by a factor xn by forming a coherent average of n pulse responses. The signal-to-noise ratio deteriorates with increasing frequency due to the increasing attenuation coefficient of the test liquid, and the highest frequency at which measurements can be made depends on the attenuation coefficient of the colloid under test and the number of coherent averages taken of the received signal. The sampling time for a single pulse response is 10 ms, and the total sampling time depends on the number of coherent averages required. In this work, 10 000 responses were used in forming the coherent average, giving a total sampling time of 100 s and a signal-to-noise ratio of 70 dB at 5 MHz with water as the test liquid. The signal-to-noise ratio also deteriorates at the lower end of the measurement frequency range, due to the dominance of radiation coupling effects at low frequencies. Because the signal-to-noise ratio varies with frequency, it is not a simple matter to formally establish the accuracy and precision of the attenuation measurements. By observation we have established that the accuracy and precision are of the same order as the signal-to-noise ratio. Using a short input pulse enables the propagation velocity to be measured with an accuracy of (0.3 m‚s-1 across the measurement frequency range. Small changes in velocity as a function of frequency (velocity dispersion) are significant in comparing measured results with ultrasonic propagation theory.
c)
1
(3)
xm*F*
The velocity given by the Urick equation is independent of frequency, whereas colloids in general exhibit frequency dependent velocity. 3.2. Harker and Temple Hydrodynamic Model. The model of Harker and Temple5 considers the bulk hydrodynamic properties of a colloid. The wavenumber of a longitudinal wave passing through the medium is obtained by solving continuity, momentum transfer, and compressibility equations appropriate to the medium. The complex wavenumber β is given by
β2 ) ω2m*
(
)
F1[F2(1 - φ - φS) + F1S(1 - φ)] F2(1 - φ)2 + F1[S + φ(1 - φ)]
(4)
where ω is the angular frequency, β ) ω/c(ω) + jR(ω) and S is given by
S)
(
)
(
)
9 δ δ2 9δ 1 1 - 2φ +j + 2 + 2 1-φ 4R 4 R R
where R is the particle radius and δ is the boundary layer thickness given by
δ)
2η xωF
where η is the fluid viscosity. 4. Theory of Interparticle Forces The thickness of the electrical double layer surrounding the silica particles, 1/κ, can be calculated from the DebyeHuckel inverse screening length equation13
κ)
( ) 2ci Ne2 0kT
1/2
(5)
where ci is the ionic strength of the solution, N is Avogadro’s number, e is the electron charge, 0 is the permittivity of free space, is the dielectric constant of the solvent, and k is Boltzmann’s constant. Therefore, a reciprocal relationship exists between the double-layer thickness and the ionic strength of the continuous phase. The surface potential ψ0 of particles is related to their surface charge zp by the equation
3. Theoretical Predictions of Ultrasonic Propagation
ψ0 ) zp/4π0R(1 + κR)
3.1. Urick Model. The simplest model of acoustic propagation in colloids is that of Urick,11 which assumes that a suspension of particles whose diameter is infinitesimally small compared to the wavelength of sound can be treated as a simple mixture with the effective density and compressibility given by the volume-averaged values:
Using SANS data, Penfold and Ramsay13 were able to estimate the surface charge zp on silica sol particles at different ionic strengths, enabling the surface potential to be calculated, and hence the standard Coulombic potential U(r) due to electrical repulsion between particles using the equation
F* ) (1 - φ)F1 + φF2
(1)
m* ) (1 - φ)m1 + φm2
4π0R2ψ20e-κ(r-2R) U(r) ) r
(2)
where φ is the volume fraction of suspended material, m1 and F1 are the compressibility and density of the fluid, and m2 and F2 refer to the suspended particles. The velocity is calculated from the volume-averaged density and compressibility:
(6)
(7)
This equation applies for distances between particle centers r greater than 2R; that is, the particles are not touching. Hence, the magnitude of the electrical potential between particles is greater for smaller particle separa(13) Penfold, J.; Ramsay, J. D. F. J. Chem. Soc., Faraday Trans. 1 1985, 81, 117.
Ultrasonic Wave Propagation in Silica Sols and Gels
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Figure 1. Viscosity of 30 nm silica sols plotted against particle concentration for sols of ionic strength 10-4 mol‚dm-3 (upper line) and 5 × 10-3 mol‚dm-3 (lower line).
tions, when there is greater overlap between their double layers. The results of Penfold and Ramsay show that the interparticle potential is greater for sols with surrounding electrolyte concentrations of 10-4 mol‚dm-3 versus 5 × 10-3 mol‚dm-3, and that it increases with particle concentration. For a simple cubic array of particles, the particle separation will be given by the equation
r)
[
]
4/3πR3 φ
Figure 2. Acoustic attenuation against frequency for 30 nm silica sols and gels: solid lines, sols of 10-4 mol‚dm-3 ionic strength; broken lines, 5 × 10-3 mol‚dm-3;dotted line, 35% w/w gel prepared by dialysis against deionized water (coincident with solid line).
1/3
(8)
The average particle separations for silica sols determined by SANS by Penfold and Ramsay are very close to those predicted by eq 8, suggesting that the particle distribution has short range order and tends to form a simple cubic array-like structure. 5. Results 5.1. Viscosity. The viscosity of the 30 nm silica sols measured by U-tube viscometry is shown in Figure 1. While the viscosity remains low in the 5 × 10-3 mol‚dm-3 ionic strength sols up to 35% w/w, the viscosity of the 10-4 mol‚dm-3 sols increases sharply with concentration. The U-tube viscometer is strictly valid only for Newtonian fluids, as the shear rate varies across the capillary. It is possible that the higher concentration sols with greater viscosity exhibit non-Newtonian rheology. The viscosity of the samples prepared by dialysis against deionized water could not be measured by U-tube viscometry, but the samples could be seen to have gelled and could be inverted without any visible flowing of the gel. 5.2. Acoustic Absorption. The acoustic absorption in samples of continuous phase alone, that is, 10-4 or 5 × 10-3 mol‚dm-3 NaNO3, was found to be the same as that of pure water to within 5%. The acoustic absorption in the range 2-50 MHz of the 30 nm sols and gels is shown in Figure 2. It can be seen that for the 10% and 20% w/w sols, the attenuation is identical at the two different ionic strengths. In the case of the 35% w/w samples, the attenuation is identical in the 10-4 and 5 × 10-3 mol‚dm-3 sols and in the 35% w/w gel prepared by dialysis against deionized water, despite the large differences in viscosity between the three samples. In Figure 3, the attenuation in the 16 nm sols of 5% and 10% concentration is plotted. The solid lines represent the 5 × 10-3 mol‚dm-3 ionic strength sols, while the broken lines are for the 10-4 mol‚dm-3 sols. Again, the attenuation is virtually identical at the two different ionic strengths. Also shown is the attenuation predicted by the Harker and Temple model
Figure 3. Acoustic attenuation versus frequency for 16 nm silica sols: solid lines, sols of 10-4 mol‚dm-3 ionic strength; broken lines, 5 × 10-3 mol‚dm-3; dotted lines, predicted attenuation using the Harker and Temple model (eq 4); upper curves, 10% w/w concentration; lower curves, 5% w/w concentration. Table 1. Physical Constants for Silica Sols and Gels (25 °C) Required for the Harker and Temple Model parameter
symbol
propagation velocity density shear viscosity
c F ηs
units
water
silica
m‚s-1 1496.7a 5968.0b kg‚m-3 997b 2230c Pa‚s 8.91 × 10-4 b
a DelGrosso, V. A.; Mader, C. W. J. Acoust. Soc. Am. 1972, 52, 1442. b Kaye, G. W. C.; Laby, T. H. Tables of Physical and Chemical Constants, 16th ed.; Longman: England, 1995. c Experimental measurement.
(dotted lines), which in this case is in good agreement with the experimental data, although, for other particle sizes and concentrations, systematic differences were found between the Harker and Temple and ECAH models of acoustic propagation and the experimental data. This will be considered in detail in a further publication. The physical constants required for the Harker and Temple model are given in Table 1. 5.3. Ultrasonic Velocity. The propagation velocity of samples of continuous phase alone was found to be very close to that of pure water, with no significant velocity dispersion. The propagation velocities at 4 MHz for the two different ionic strengths and for the 30% and 35%
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Table 2. Propagation Velocity at 4 MHz for 30 nm Silica Sols and Gels
agreement despite the viscosity differences between the sols of different ionic strength and the gel.
propagation velocity
6. Discussion
silica conc w/w (%)
10-4 mol‚dm-3 ionic strength sol
5 × 10-3 mol‚dm-3 ionic strength sol
gel
0 5 10 20 25 30 35
1496.85 1492.90 1489.47 1485.96 1485.87 1488.02 1491.16
1497.03 1492.82 1489.77 1486.04 1486.74 1488.97 1491.32
1488.71 1491.42
Figure 4. Propagation velocity at 4 MHz for 5 × 10-3 mol‚dm-3 sols plotted against concentration: dots, experimental data; solid line, prediction using the Urick equation (eq 3).
Figure 5. Velocity dispersion (i.e. velocity minus velocity at 4 MHz) for 30 nm silica sols and gel: dotted lines, sols of 10-4 mol‚dm-3 ionic strength; broken lines, 5 × 10-3 mol‚dm-3 sols; solid line, 30% w/w gel prepared by dialysis against deionized water.
gels are given in Table 2. It can be seen that, for each concentration, the velocities are generally in agreement regardless of ionic strength, bearing in mind the experimental error associated with the technique of (0.3 m‚s-1. Two of the samples have velocities differing by more than the expected error (the 30% 5 × 10-3 and 35% 10-4 mol‚dm-3 samples), and this is likely to be due to temperature fluctuations. The velocity at 4 MHz for the sols of ionic strength 5 × 10-3 mol‚dm-3 at different concentrations is shown in Figure 4 together with the velocity given by the Urick equation (eq 3). Figure 5 shows the velocity dispersion, plotted as velocity minus velocity at 4 MHz, for the 30 nm sols of 10, 20, and 30% concentration and for the 30% gel. It can be seen that, as in the case of the attenuation data, the curves are in close
The viscous properties of silica sols and gels are dependent on the repulsive forces between particles due to their electric double layers. From eq 5 the double-layer thickness is 30.5 nm at 10-4 mol‚dm-3 ionic strength, but only 4.3 nm at 5 × 10-3 mol‚dm-3. The SANS results of Penfold and Ramsay13 for 16 nm silica sols show that the equilibrium distance between the particles is close to that which would be predicted for a simple cubic array using eq 8. If this is applied to the 30 nm sols whose viscosity is plotted in Figure 1, then the distance between the surfaces of the particles, r - 2R, is estimated to be 55 nm at 5% w/w concentration, reducing to 12 nm at 35% w/w. Hence, there is a large increase in viscosity in the 10-4 mol‚dm-3 sols with concentration because the double layers of adjacent particles overlap, but there is no overlap in the 5 × 10-3 mol‚dm-3 sol even at 35% w/w. The Coulombic potential U(r) between particles can be estimated from the data of Penfold and Ramsay for 16 nm sols using eq 7 and is found to increase with particle concentration and at lower ionic strength. The potential for a 7.8% volume fraction sol at 10-4 mol‚dm-3 was approximately 20 times greater than that for a 6.1% sol at 5 × 10-3 mol‚dm-3. However, the ultrasonic absorption of 5% and 10% w/w sols of 16 nm particle diameter, as shown in Figure 3, is almost identical for the two different ionic strengths. In Figure 2 the acoustic absorption for the 30 nm sols and gels again appears to be independent of ionic strength. If the absorption is compared for 35% sols at 10-4 and 5 × 10-3 mol‚dm-3, where Figure 1 shows the viscosity to differ by an order of magnitude, there is no difference in the absorption, and the attenuation of a 35% gel is also identical. The same is true of the velocity dispersion, as shown for the 30% sols and gel in Figure 5, and the low-frequency propagation velocities in Table 2, which do not show any significant differences between samples of different ionic strengths. It is perhaps surprising that the acoustic properties of sols and gels should be so similar, despite the obvious differences in the viscosities of the samples. However, Ramsay and Booth10 concluded from their neutronscattering work that there is no abrupt change from short to long range order in the transition from a silica sol to a gel and that there is little change in the relative disposition of the particles. The equilibrium particle separations were found to be similar to those that would be predicted for a simple cubic array in both sols and gels. The differences in viscosity between sols and gels are apparently dependent only on the strength of the repulsive force between particles. We have previously shown that the acoustic propagation in fully and partially flocculated emulsions is dependent on the structure of the flocs.14 This leads to the important conclusion that the acoustic properties of colloids are sensitive to changes in structure but not to interparticle forces. This implies that no mechanism exists by which acoustic absorption or phase velocity is affected by the perturbation of the equilibrium spacing of particles in the interparticle force field by an acoustic wave. The particle sizes of the silica sols used here (
