5892
J. Phys. Chem. 1996, 100, 5892-5897
Frequency Dependence of Ultrasonically Induced Birefringence of Rodlike Particles K. Yasuda,* T. Matsuoka, S. Koda, and H. Nomura Department of Chemical Engineering, School of Engineering, Nagoya UniVersity, Nagoya 464-01, Japan ReceiVed: September 18, 1995; In Final Form: December 11, 1995X
Ultrasonically induced birefringence of two rodlike particles of the hematite sols and the poly(tetrafluoroethylene) latex was investigated as a function of volume fraction, ultrasonic intensity, and frequency. The negative birefringence was observed for both samples, and the birefringence was proportional to the volume fraction and the ultrasonic intensity. The frequency dependence of the birefringence was observed in the wide ultrasonic frequency range from 5 to 225 MHz and did not agree with the Oka theory. The modified Oka theory for disclike particles was extended in view of application for rodlike particles. The birefringence measured was several tens of times larger than that calculated from the modified Oka theory for rodlike particles. The orientational relaxation time of the two particles estimated from the trace of the transient birefringence can be expressed in the form of the Debye-Einstein equation.
Introduction The double refraction is induced in liquids and solutions containing a certain amount of nonspherical particles as a result of the particle orientation due to ultrasonic waves. This was called the phenomenon of the ultrasonically induced double refraction.1 The early theoretical studies of the birefringence have been reviewed by Hilyard and Jerrard.2 The theory developed by Oka3 indicated that large disclike rigid particles align by hydrodynamic torque which is produced by the radiation pressure due to the passage of the ultrasonic wave. The normal of disclike particles is parallel to the ultrasonic field, and the sign of the birefringence of disclike particles is negative. The measurements of the birefringence of large rigid disclike particles were carried out on bentonite4 and gold sols.5,6 These experimental results show that the birefringence is proportional to the ultrasonic intensity as predicted by the Oka theory. However, the frequency dependence of the birefringence has been scarcely investigated because the measurements of the ultrasonic intensity at different frequencies are difficult. Recently, Ou-Yang et al.7 made use of the Raman-Nath diffraction effect to measure the ultrasonic intensity at the frequency range from 1 to 19 MHz and investigated the ultrasonically induced birefringence of gold sols as a function of solvent viscosity, particle size, and ultrasonic intensity and frequency. The birefringence increased with increasing ultrasonic frequency and decreasing viscosity, and the sign of the birefringence was positive. They modified the Oka theory to explain the sign of the birefringence and the dependence of the birefringence on the frequency and viscosity. For large rigid rodlike particles, no theoretical study of the ultrasonically induced birefringence has been reported. Experimental investigations on the birefringence of V2O5 sols1,8,9 have been carried out. Measurements8 by Petralia on V2O5 sols indicated that the birefringence depended on the viscosity and the sign of the birefringence was positive. In our previous paper,9 the ultrasonically induced birefringence of V2O5 sols was proportional to the ultrasonic intensity and the volume fraction, as predicted by the Oka theory. However, no experimental investigation of the frequency dependence of the birefringence of rodlike rigid particles has been carried out. The purpose of this work is to elucidate the frequency dependence of the ultrasonically induced birefringence of rodlike X
Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-5892$12.00/0
particles. The degree of birefringence for unit ultrasonic intensity has been estimated. For this purpose, we have measured the ultrasonic intensity from the Bragg and RamanNath diffraction intensity. The experimental results will be discussed on the basis of the modified Oka theory for rodlike particles. In addition, orientational relaxation times of rodlike particles have also been evaluated from the trace of the transient birefringence. Experimental Section Samples. The hematite (R-Fe2O3) sols of rodlike particles were prepared according to the method proposed by Ozaki et al.10 A mixture of 0.02 M FeCl3 and 4.5 × 10-4 M NaH2PO4 was forcedly hydrolyzed at 100 °C for 2 days. The particles were collected by centrifugation and washed five times with water. The above experimental condition gives R-Fe2O3 particles of fairly narrow size distribution with the length of the major axis 0.55 µm and the axial ratio 5.5, as reported by Ozaki et al.10 The R-Fe2O3 sols at pH ) 3.5 were prepared by using HCl aqueous solution, and the volume fraction of the sols was varied from 9.4 × 10-8 to 2.4 × 10-5. The poly(tetrafluoroethylene) (PTFE) latex of prolate particles was kindly supplied by Asahi-ICI Fluoropolymers Co., Ltd. The electron micrograph of PTFE particles indicated the length of the major axis ranged from 0.3 to 1.2 µm with a mean of 0.55 µm (the standard deviation 0.19 µm) and the average axial ratio of 1.9. The volume fraction of PTFE latex was varied from 1.3 × 10-7 to 8.2 × 10-4. Birefringence Measurement. Experimental details of the setup are described in our previous papers.9,11 The direction of the incident light is perpendicular to the direction of the sound wave propagation. The light from a He-Ne laser (632.8 nm, 5 mW) passes through the polarizer with an angle of polarization at 45°, sample cell, λ/4 plate, and analyzer with a small offset angle β from its extinction position. If the acoustic field is applied to solutions, the optical phase retardation δ is produced and the intensity of light passed through the analyzer increases. For |β| , 1 and |δ| , 1, the intensities of light passed through the analyzer are given by
I ) I0((δ/2)2 + βδ + β2) + Ib
(1)
where I0 and Ib are the light intensities in the absence of the © 1996 American Chemical Society
Ultrasonically Induced Birefringence of Rodlike Particles
J. Phys. Chem., Vol. 100, No. 14, 1996 5893
sound wave with the polarizer and with the analyzer being parallel and perpendicular, respectively. From eq 1, the phase retardation is given as
δ)
I+ - I2I0β
(2)
where the light intensities with offset angles (β are described by I+ and I-, respectively. The birefringence ∆n is related to the phase retardation δ as follows.
∆n ) λδ/2πd
(3)
where d is the optical path length and λ the wavelength of the light. If the orientational motion of monodisperse particles is expressed in terms of a single relaxation process, the extinction curve of the birefringence can be given as
∆n(t) ) ∆nmax exp(-t/τ)
(4)
where τ is the orientational relaxation time and ∆nmax the maximum value of ∆n(t). In our experiments, the ultrasonic frequency was varied from 5 to 225 MHz. The pulse length of the ultrasonic wave for measurements on R-Fe2O3 sols was 40 ms at a low repetition rate (4 Hz) to avoid the ultrasonic heating and streaming. The pulse length on PTFE latex was 180 ms, and the pulse repetition rate was 1 Hz. The measurements were carried out in the temperature range from 10 to 50 °C. The experimental data were stored in the digital storagescope, and the values averaged for 100 runs were taken. Ultrasonic Intensity Measurement. The ultrasonic intensity in the sample cell must be measured precisely to investigate the frequency dependence of the birefringence for unit ultrasonic intensity. In this work, the birefringence in the wide frequency range from 5 to 225 MHz was investigated. The measurements of the ultrasonic amplitude were carried out on the basis of the phenomena of the light diffraction due to sound waves. The diffraction in the frequency range from 5 to 225 MHz is named the Raman-Nath12 or the Bragg diffraction,13 depending on the magnitude of the parameter Q,14 given as
Q ≡ 2πdλ/nsΛ2
(5)
where Λ is the wavelength of the ultrasound and ns the refractive index of solution. The conditions of Q e 0.5 (f e 5 MHz) and Q g 4π (f g 25 MHz) correspond to the Raman-Nath and the Bragg regimes, respectively. For Q e 0.5, the diffracted light intensity Im of the mth order is related to the ultrasonic amplitude A as
Im/II ) Jm2(BA)
(6)
where Jm is the Bessel function of the mth order, II is the intensity of the incident light, and B is given by
B)
( )
2 2 2πd (ns - 1)(ns + 2) 2 λ 6ns FsVs3
Figure 1. (a) Trace of transient ultrasonically induced birefringence of R-Fe2O3 sols of volume fraction 4.7 × 10-6 at 25 °C (25 MHz, 0.005 W‚cm-2). (b) Applied ultrasonic pulse.
1/2
(7)
where Fs is the density of solution and VS the sound velocity of solution. For Q g 4π, the diffracted light intensities of the 0th order and +1th order are given as
I0d/II ) cos2(BA/2), I+1d/II ) sin2(BA/2)
(8)
Figure 2. Birefringence against ultrasonic intensity at 25 °C (25 MHz): (b) R-Fe2O3 sols of volume fraction 4.7 × 10-6; (0) PTFE latex of volume fraction 7.6 × 10-5.
In the regime 0.5 e Q e 4π, numerical computation results reported by Klein and Cook15 were used in the estimation of the ultrasonic amplitude. The ultrasound intensity can be obtained from the ultrasonic amplitude as follows: W ) A2/2. Throughout this work, the ultrasonic intensity was less than 0.05 W/cm2. Results Figure 1 displays the representative trace of the transient ultrasonically induced birefringence of R-Fe2O3 sols and the waveform of the pulse used. The decay of the birefringence is due to the particle orientational relaxation, and a saturation of the birefringence is observed after about 15 ms. The waveform of the birefringence of PTFE latex is similar to that of R-Fe2O3 sols. In both samples, the signs of the birefringence are negative. The sign of the birefringence depends on the orientational direction of the particles and whether the particle is a conductor or not.7 As R-Fe2O3 and PTFE particles are nonconductors, the negative sign of the birefringence means that the major axis of the particles is aligned perpendicularly to the direction of the ultrasonic wave propagation. Figure 2 shows the birefringence as a function of ultrasonic intensity. The birefringence is proportional to the ultrasonic intensity for both cases. The birefringence normalized for unit ultrasonic intensity is plotted against the volume fraction in Figure 3. As the slope of Figure 3 is unity, the birefringence of both particles is proportional to the volume fraction. The
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Yasuda et al.
Figure 3. Birefringence per ultrasonic intensity plotted against volume fraction at 25 °C (25 MHz): (b) R-Fe2O3 sols; (0) PTFE latex.
Figure 6. Plots of ∆nT/W against the ultrasonic frequency. (A) R-Fe2O3 sols of (volume fraction ) 4.7 × 10-6). Temperature: (O) 50 °C; (0) 25 °C; (4) 10 °C. (B) PTFE latex (volume fraction ) 8.2 × 10-4). Temperature: (b) 50 °C; (2) 10 °C.
Figure 4. Frequency dependence of the birefringence for unit ultrasonic intensity for R-Fe2O3 sols of volume fraction 4.7 × 10-6. Temperature: (b) 50 °C; (0) 25 °C; (4) 10 °C.
Figure 5. Frequency dependence of the birefringence for unit ultrasonic intensity for PTFE latex of volume fraction 8.2 × 10-4. Temperature: (b) 50 °C; (4) 10 °C.
birefringence of R-Fe2O3 sols is larger than that of PTFE latex, since the refractive index and axis ratio of the former are larger than those of the latter. Figures 4 and 5 illustrate the frequency dependence of the birefringence of R-Fe2O3 sols and PTFE latex normalized for unit ultrasonic intensity ∆n/W. For both samples, the ∆n/W increases with the frequency and the value of [∂(∆n/W)∂f] decreases with increasing frequency. Figure 6 shows the plot of ∆nT/W against the ultrasonic frequency in order to compen-
Figure 7. Orientational relaxation time for R-Fe2O3 sols and PTFE latex against η/kBT. (A) R-Fe2O3 sols (volume fraction: (0) 2.4 × 10-5; (O) 4.7 × 10-6). (B) PTFE latex (volume fraction: (9) 4.1 × 10-3; (b) 7.6 × 10-5; ([) 2.6 × 10-6). The orientational relaxation times for R-Fe2O3 sols obtained by transient magnetic birefringence17 measurements are also included. Field strength: (4) high; (3) low.
sate for the 1/kBT factor related to thermal randomization.7 The increment of temperature leads to the slight increase of birefringence. It can be seen in Figure 1 that the birefringence decays after the ultrasonic field disappears. This decay reflects the orientational relaxation process of the particles. The orientational relaxation time of the particles is obtained by fitting the extinction process of the birefringence to eq 4. The orientational relaxation times of R-Fe2O3 sols and PTFE latex are plotted against η/kBT in Figure 7, where η is the solvent viscosity, T the temperature, and kB the Boltzmann constant. Figure 7
Ultrasonically Induced Birefringence of Rodlike Particles
J. Phys. Chem., Vol. 100, No. 14, 1996 5895
demonstrates that the orientational relaxation time is proportional to η/kBT. The orientational relaxation times of R-Fe2O3 and PTFE particles in dilute solution follow the Debye-Einstein equation,
τ ) ηV*/kBT
(9)
where V* is the effective volume of a particle. The effective volumes of R-Fe2O3 and PTFE particles estimated from the slopes in Figure 7 are 1.5 × 10-20 and 7.6 × 10-20 m3, respectively. According to Perrin,16 V* of a prolate particle is given as
V* )
G(p) )
(
)
1 - p4 8πa3 9 (2 - p2)G(p) - 1
[
1 + (1 - p ) 1 ln 2 1/2 p (1 - p )
(10)
]
2 1/2
(11)
where p is the axial ratio b/a and a and b are the radii of the major and minor axes of the particle, respectively. For PTFE latex, 2a calculated from eqs 10 and 11 is 0.80 µm with p ) 1/1.9. The size of PTFE particles obtained from electron microscopy is within 0.3-1.2 µm, with an average value of 0.55 µm, and it is smaller than the size estimated from the birefringence measurements. This disagreement may be responsible for estimation of an average particle size. The average size of PTFE particles obtained from electron microscopy is indicated as the number average particle size. On the other hand, the larger particles have a great influence on the decay of the birefringence, and the average size of PTFE particles estimated from the birefringence measurement is larger than the number average particle size. For R-Fe2O3 particles, the value of 2a is 0.55 µm with p ) 1/5.5 and is consistent with the value reported by Ozaki.10 Discussion Birefringence of Rodlike Particles. The Oka theory3 indicates that the birefringence is proportional to the ultrasonic intensity and volume fraction. In his theory, the solvent is assumed to be an ideal fluid and the birefringence does not depend on the ultrasonic frequency nor the solvent viscosity. In this work, the birefringence of R-Fe2O3 and PTFE particles is proportional to the ultrasonic intensity and volume fraction and depends on the frequency and viscosity. This means that the birefringence of rodlike particles is not fully interpreted by the Oka theory. Recently, Ou-Yang et al.7 modified the Oka theory to explain the frequency and viscosity dependence of the birefringence of conducting disclike gold sols. We have extended the modified Oka theory for disclike particles in view of application for rodlike particles. The detailed derivation is described in the section of the Appendix. When the particle length is much smaller than the ultrasonic wavelength, the birefringence of rodlike particles is given by (see the Appendix)
∆n )
4Xπab2∆GφA2F2 15kBTVS0ns
(12)
where ∆G is the optical anisotropy, φ the volume fraction, and VS0 the sound velocity of the solvent. F is given as
Figure 8. Plots of normalized birefringence ∆nTVskB/A2φ against RL: (b) R-Fe2O3 sols; (0) PTFE latex. Dotted (Fe2O3 sols) and solid (PTFE latex) curves show the theoretical values.
F)
F - F0 V-u ) V F + i(ζ/Vω)[(1 + R 1/2) - i(R L
1/2
L
+ γRL)] (13)
with RL ) a2ωF0/2η, where F and F0 are the densities of particles and solvent, respectively, V is the fluid velocity, u is the translational velocity of the particle, ζ is the Stokes’ drag coefficient, ω is the ultrasonic angular frequency, V is the particle volume, and γ is a constant. The 2πa values of both particles investigated here are smaller than the ultrasonic wavelength by more than a factor of 10 in the frequency below 100 MHz. The reduced birefringence ∆nTVSkB/A2φ can be estimated from the results in Figures 4 and 5. The results are shown in Figure 8, where the theoretical line is included. For R-Fe2O3 sols, the experimental values are about 20 times larger than the theoretical values, and for PTFE latex, the experimental values are about 100 times as large as the theoretical values. The large deviations from the theoretical values cannot be explained in terms of the modified Oka theory for rodlike particles, even if one takes into account the experimental errors and estimation errors in the refractive index and the size of the particle. In the Oka theory, the turning torque which aligns the disclike particles was derived from the Eular equation for an ideal fluid and the ultrasonically induced birefringence was predicted to be proportional to F2, namely, [(V - u)/V]2. The value of F in an ideal fluid does not depend on the frequency and viscosity. Ou-Yang et al. modified the Oka theory to interpret the frequency dependence of the birefringence of disclike particles in viscous fluids. In the modified Oka theory, the parameter F in the Oka theory was replaced by that derived from the NavierStokes equation for viscous fluids. However, the turning torque also should be modified on the basis of the Navier-Stokes equation. In addition, under oscillatory motion, the flow around the rodlike particles is complicated and the flow may be in eddy or stagnation. A new theory of the ultrasonically induced birefringence should be required for the case of rodlike particles after taking into account of the turning torque in viscous fluids and the boundary condition of fluids around the rodlike particles. Orientational Relaxation Times of Rodlike Particles. As pointed out in a previous section, the extinction of the transient ultrasonically induced birefringence reflects the orientational relaxation process. Birefringence induced by magnetic and electric fields also involves the same information as the ultrasonically induced birefringence. James17 estimated the orientational relaxation time of R-Fe2O3 sols from the measure-
5896 J. Phys. Chem., Vol. 100, No. 14, 1996
Yasuda et al.
ments of the transient magnetic birefringence. He used the R-Fe2O3 sols with the average length of the major axis 0.54 µm and the axial ratio ca. 5, which are nearly equal to those used in this work. Although, as is shown in Figure 7, the relaxation time obtained from the magnetic birefringence depends on the magnetic intensity, the values are close to those from the ultrasonically induced birefringence within experimental error. James explained the intensity dependence of the relaxation time as follows. The larger particles are preferentially oriented in the low magnetic fields, while under the high magnetic fields all the particles are oriented. In the ultrasonically induced birefringence, the relaxation time is independent of the ultrasonic intensity investigated here. Irrespective of their size, the particles can be oriented by the radiation pressure. As is shown in Figure 7, the orientational relaxation times of both R-Fe2O3 and PTFE particles can be expressed by the Debye-Einstein equation. This fact means that the orientational motion of these particles is not affected by the interparticle interaction. In other words, information of the orientational motion of isolated particles with large anisotropy is obtained by the ultrasonically induced birefringence, since the measurements in very dilute solution are possible. Acknowledgment. We would like to express our sincere thanks to Prof. Ozaki, Yokohama City University, for his helpful comments concerning preparation of R-Fe2O3 sols and Mr. Matsuoka, Asahi-ICI Fluoropolymers Co. Ltd., for providing PTFE latex samples and their electron micrograph. We wish to thank Mr. Takahashi and Mr. Tachibana, Work Shop for Experimentation and Practice, School of Engineering, Nagoya University, for making the acoustic cell. This work is partly supported by a Grant-in-Aid from the Ministry of Education, Science and Culture (No. 06640743). Appendix Modified Oka Theory for Rodlike Particles. Application of ultrasonic fields will induce the birefringence ∆n, which is given by
∆n ) 2π∆GΘφ/ns
X)
[
(A2)
[ ] ] [ ] ]
23 1 1+ - + (1 - 2) ln 3 2 1- 1 1+ 1+ 1 - 23 + (1 - 2) ln - (1 - 2) ln 2 1- 2 1- (A3) 2π3
[
[ ] ][
∆n )
with ) (a2 - b2)1/2/a and where θ is an angle between the
4Xπab2∆GφA2F2 15kBTVs0ns
(A4)
For spheroid particles, ∆G is obtained by an extension of the theory of Peterlin and Stuart23 as
∆G )
n02(n2 - n02)2(N2 - N1) [4πn02 + (n2 - n02)N1][4πn02 + (n2 - n02)N2]
(A5)
where n and n0 are the refractive index of particles and solvent, respectively, and for prolate particles, N1 and N2 are shape factors along a and b, respectively. N1 and N2 are obtained by
(
N1 ) 2π(µ2 - 1) µ ln
4π - N1 µ+1 - 2 , N2 ) µ-1 2
)
(A6)
with µ ) a/(a2 - b2)1/2. Recently Ou-Yang et al.7 indicated that the suspended particles which execute translational oscillations in fluid are subject to the drag force that is shown in terms of the librational Reynolds number.24 They supposed that the translational motion is much faster than the orientational motion, and the equation of translational motion for particles in dilute region is expressed as
F)
F - F0 F + i(ζ/Vω)[(1 + RL1/2) - i(RL1/2 + γRL)]
(A7)
At this point we extend the modified Oka theory for disclike particles in view of application to prolate particles. When the prolate particles are translating in the direction normal to its major axis, ζ is given25 as
ζ)
(A1)
where Θ is the mean orientation of the particles. The mean orientation can be calculated from the distribution function of particles, which is obtained by considering the potential energy due to the torque caused by the applied field.18,19 Oka3 assumed that the passage of acoustic waves sets up the radiation pressure so that large rigid disclike particles are subject to the turning torque that will align them. When the particle length is much smaller than the ultrasonic wavelength, the ultrasonic torque is estimated by considering the Rayleigh disc20,21 problem. We assumed that prolate particles such as R-Fe2O3 sols and PTFE latex are also subject to the torque due to radiation pressure. Ko¨nig22 derived the time-averaged torque |M| for prolate particles as
|M| ) XF0ab2(V - u)2 sin 2θ,
major axis of prolate spheroids and the direction of ultrasonic propagation. If the ultrasonic energy is small compared with the thermal disturbance, the birefringence is expressed as
16πηab2 (χ + b2R)
(A8)
where
dx R ) ab2∫∞0 2 (a + x)1/2(b2 + x)2
(A9)
dx χ ) ab2∫∞0 2 (a + x)1/2(b2 + x)
(A10)
and
If RL ) ∞, the parameter F in viscous fluids is reduced to that in an ideal fluid. For an ideal fluid, F is obtained from the inertia coefficient of a prolate particle.
F)
F - F0 R F+ F 2-R 0
(A11)
Comparing eq A7 with eq A12 leads to γ given as follows.
γ)
R(χ + b2R) 6a2(2 - R)
(A12)
The frequency dependence of the birefringence of prolate,
Ultrasonically Induced Birefringence of Rodlike Particles namely, rodlike particles, can be expressed in terms of F, which is obtained by substituting eqs A8-A10 and A12 into eq A7. References and Notes (1) Kawamura, H. Kagaku (Tokyo) 1938, 7, 6, 54, 139 (in Japanese). (2) Hilyard, N. C.; Jerrard, H. G. J. Appl. Phys. 1962, 33, 3470. (3) Oka, S. Kolloid Z. 1939, 87, 37; Z. Phys. 1940, 116, 632 (in German). (4) Jerrard, H. G. Ultrasonics 1964, 2, 74. (5) Yasunaga, T.; Tatsumoto, N.; Inoue, H. J. Colloid Interface Sci. 1969, 29, 178. (6) Lipeles, R.; Kivelson, D. J. Phys. Chem. 1980, 72, 6199. (7) Ou-Yang, H. D.; MacPhail, R. A.; Kivelson, D. Phys. ReV. 1986, A33, 611. (8) Petralia, S. NuoVo Cimento 1940, 17, 378 (in Italian). (9) Yasuda, K.; Matsuoka, T.; Koda, S.; Nomura, H. Jpn. J. Appl. Phys. 1994, 33, 2901. (10) Ozaki, M.; Kratohvil, S.; Matijevic, E. J. Colloid Interface Sci. 1984, 102, 146. (11) Koda, S.; Koyama, T.; Enomoto, Y.; Nomura, H. Proc. 12th Symp. Ultrasonic Electronics, Tokyo, 1991. Jpn. J. Appl. Phys., Part 1 1992, 31, Suppl. 31-1, 51.
J. Phys. Chem., Vol. 100, No. 14, 1996 5897 (12) Raman, C. V.; Nath, N. S. N. Proc. Ind. Acad. Sci. 1935, A2, 406. (13) Bragg, W. H.; Bragg, W. L. X-rays and Crystal Structure; Bell: London, 1915. (14) Scruby, C. B.; Drain, L. E. Laser Ultrasonics; Adam Hilger: Bristol, 1990. (15) Klein, W. R.; Cook, B. D. IEEE Trans. Sonics Ultrason. 1967, SU-14, 123. (16) Perrin, F. J. Phys. Radium 1934, 5, 497; 1936, 7, 1 (in French). (17) James, R. O. Colloids Surf. 1987, 27, 133. (18) Debye, P. Polar Molecules; Dover: New York, 1929. (19) Polhman, R. Z. Phys. 1937, 107, 497 (in German). (20) King, L. V. Proc. R. Soc. London Ser. A 1935, 153, 1, 17. (21) Rayleigh, J. W. S. Theory of Sound, 2nd ed.; Dover: New York, 1945. (22) Ko¨nig, W. Ann. Phys. Chem. 1891, 43, 43 (in German). (23) Peterlin, A.; Stuart, H. A. Z. Phys. 1939, 112, 129. (24) Landau, L. D.; Lifshiftz, P. M. Fluid Mechanics; Pergamon Press: Oxford, 1959. (25) Lamb, S. H. Hydrodynamics, 6th ed.; Cambridge University Press: Cambridge, 1932.
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