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Thermophoresis of aerosol particles at small knudsen numbers: Theory and experiment
J. Aerosol Sci., Vo| 13, No 4, pp 327 330, 1982.
0021-8502/82/0,10327-04 $03.00/0 Pergamon Press Ltd.
Pnnted in Great Britain
THERMOPHORESIS OF AEROSOL PARTICLES AT SMALL KNUDSEN NUMBERS: THEORY AND EXPERIMENT N. A. FUCHS Karpov Institute of Physical Chemistry, Moscow, U.S.S.R. (Received 25 N o v e m b e r 1981)
Abstract--The most reliable and exact m e t h o d of measuring the thermophoretic force consists of equilibrating single aerosol particles by an electrostatic force in a Millikan cell with heated upper and cooled lower electrodes. The "jet m e t h o d " of measuring thermophoretic velocity can involve considerable errors. The n u m e r o u s formulae published so far for particle thermophoresis at small K n u d s e n n u m b e r s with theoretical calculations of kinetic coefficients deviate considerably from experimental data obtained with sodium chloride particles,
The theory of thermophoresis of aerosol particles at large Knudsen numbers (Kn > 5) is elementary and agrees well with experimental data. Quite different is the situation in the theory ofthermophoresis at small Kn (Kn < 0.3) and relatively high thermal conductivity (2 >__10-2 cal/cm sec °C) of the particles. Many theoretical formulae for the thermophoretic velocity have been derived by a number of authors and this alone casts doubts on their validity, the more so since most of the authors did not compare them with experimental data. Measurements of the thermophoretic velocity v, or the thermophoretic force F, are possible by two methods. One of these is by using the Millikan cell with heated upper and cooled lower electrodes. Two versions are possible here. 1. After determining by Millikan's method the electric charge of the particle it is equilibrated in the cell in the absence of the thermal gradient, and the particle weight and hence its dimensions are determined. Then the particle is equilibrated in the presence of the thermal gradient and thus F, is determined. 2. The rate of particle sedimentation is measured in the presence and absence of the gradient and v, is found from the difference. The relation between F, and v, is given by the formula F, = 6rtrvttT/K where *7is the viscosity of the gaseous medium, r the particle radius and K the Cunningham-Millikan factor. In the second version the mean temperature of the sedimenting particle is higher than that of the gas, the particle heats the adjacent gas layer and this gives rise to an ascending convection microcurrent and thus diminishes the sedimentation velocity. In fact, in experiments of Schadt and Cadle (1961) the thermophoresis velocity determined by the second version was systematically lower than that calculated from the thermophoretic force found by the first method which is undoubtedly more reliable. Two series of experiments were made by the equilibration method at small Knudsen numbers with sodium chloride particles of thermal conductivity 2 = 10-2 cal/cm sec °C. In the work of Schadt and Cadle (1961) cubic crystalline particles were used in air, in the work of Jacobsen and Brock (1965)--spherical amorphous particles in argon. The particle radii in both cases were from 0.2 to ~ 1/~m. Measurements were made at gas pressures varying from 200 to 800 torr for each particle. The results of these measurements are shown in Fig. 1. The experimental points obtained by Jacobsen and Brock were taken directly from the laboratory journal* and those by Schadt and Cadle--from their paper after multiplying by 1.2--the ratio of argon and air viscosities because the reduced thermophoretic force F,/r 2 is proportional to the gas viscosity, independent of gas pressure and is a single-valued function of Kn. This important factor should be taken into account in the evaluation of theoretical expressions for F t . * These data were kindly communicated to the a u t h o r of this review by Prof. J. R. Brock. 327
328
-, ~, I-:~.~ HS
? E
i
Oi
02
03
Kn
Fig. 1. Reduced thermophoretic force. Sodium chloride particles in argon. A Experimental data of Schadt and Cadle (1961). Experimental data of Jacobsen and Brock (1965), [] r--0.436#m, • 550/~m, ×656/an, +840/~m, Ol.lOl/~m and O1.120/~m. Curve 2 is the theoretical curve derived by Jacobsen and Brock. The temperature gradient is given by G.
About 40 experimental points pertaining to different particle sizes and gas pressures were obtained by Jacobsen and Brock. As can be seen from Fig. 1 these points fall well on one curve: the standard deviation of the points from this curve does not exceed 1%, i.e. these data are very exact. When taking into account the difference in the shape and density of the particles in experiments of Schadt and Cadle, and Jacobsen and Brock the agreement between the results of both series of experiments can be regarded as satisfactory. It follows from these results that for NaC1 particles at Kn > 0.06 F r and, as we shall see later, vr increases monotonically with Kn. As pointed out by Deryaguin and Yalamov (1972) a serious error can occur in the equilibration method, due to the ascending thermal gas slip along the nonuniformly heated side walls of the cell. Therefore in the middle of the cell there must be a descending gas current, which leads to too high values of F,. However, as seen from Fig. 1, the experimental points obtained with particles of different size lie on the same curve whereas if such a current occurred in those experiments the downward drag caused by it should increase with particle size. Evidently, this effect did not exceed the accidental errors of the measurements. Another possible source of error in the work of Jacobsen and Brock is that the size of the particles were not measured directly but calculated from their weight. The density p of the amorphous NaCI spheres was obtained by extrapolation to 25 °C of the plot for melted NaC1 vs temperature, which gave 0.93 of the density of crystaUine NaC1. However the error in such a determination of the density and size of particles result in changing both F,/r z and K n in the same direction and thus partially compensate for each other. In the jet method of measuring v, aerosol is introduced isokinetically through a narrow horizontal slit into a laminar current of a filtered gas flowing through a plane-parallel channel with heated upper and cooled lower walls. The aerosol is preheated to the temperature of the ga s at the slit. The distance between the slit and the entrance to the channel is such that atthe place of introduction the profile of the flow velocity in the channel becomes parabolic and the temperature gradient across the channel is constant. The sedimentation velocity of the thin aerosol jet is measured. Many errors may occur in this method. 1. The error due to microconvection, already mentioned. 2. The error caused by thermal slip along the side walls (also mentioned).
Thermophoresis of aerosol particles at small Knudsen numbers
329
3. The aerosol jet, especially a monodisperse one, carries gas down with it so that the observed sedimentation velocity is the sum of the particle sedimentation relative to the gas and the downward flow of the medium itself. This effect increases with the aerosol concentration in the jet and its cross section. 4. Due to the gradient of the flow velocity in the channel the particles rotate with an angular velocity equal to one half of the gradient (Fuchs, 1964). 5. A polydisperse aerosol jet becomes diffuse. Experiments by Deryaguin et al. (1966) were performed with polydisperse aerosols (r = 0.2-0.6 #m) obtained by spraying a solution of NaCI and drying the droplets in air at atmospheric pressure. At such small r the effect of gravitational sedimentation could be ignored and in the absence of a thermal gradient the aerosol jet was strictly horizontal. Its rate of fall due to thermophoresis was measured with a horizontal microscope from its vertical shift relative to the distance from the point of introduction into the channel. The absence of the effect (3) was proved by the observation that the change of aerosol concentration from 103 to 106/cm 3 did not change the sedimentation velocity. The overall error in these measurements is by estimation of the above authors 8-13 %. Measurements by Prodi et al. (1979) were made with more or less monodisperse condensation NaC1 aerosols with r from 0.17 to 0.85/~m in the air at atmospheric pressure. The rate of fall of the jet was calculated from the distance from the point of introduction to the sediment on the lower wall. Effect (3) was not taken into account and as neither the jet thickness nor the aerosol concentration are indicated in the paper it is impossible to estimate the magnitude of this effect. The results of the two latter publications are shown in Fig. 2. We see that the values of vt found by means of the jet method are much higher than those calculated from F d r 2. From all that has been said above it follows that former values are erroneous although the cause of the discrepancy cannot be established without special measurements. The lower limit of measurements of Ft/r 2 by the equilibration method reached so far at atmospheric pressure corresponds to r ~ 1/~m. For still larger particles it is possible to estimate qualitatively the thermophoresis velocity from the experiments with thermal precipitators with a heated wire. It is well known that in precipitates obtained in this instrument a segregation of particles with different size occurs. For instance in the experiments of Fuchs and Yankovskii (Fuchs, 1964) with polydisperse NaCI aerosols it was found that the windward (directed against the flow) side of the precipitate contains chiefly particles with r < 0.1/am, the middle part--those with r - 0.1--0.5/~m and the lee side 0.3-1/~m in accordance with the findings dealt with above. A few observations made with still larger particles show that at r = 5-10/am the efficiency of
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02
03
Kn
Fig. 2. Thermophoretic velocity. Sodium chloride particles in air. Experimental data x Prodi et aL (1979);O Deryaguin et al. (1966). Curve I is obtained by recalculation from curve 1 in Fig. 1. Curve lI--theoretical curve derived by Deryaguin and Storozhilova (1972), curve III--by Deryaguin et aL (1966), curve IV--by Deryaguin and Yalamov (1965), curve V--by Deryaguin and Yalamov (1972), curve VI--by Bakanov et al. (1979), and curve VII--by Poddoskin et al. (1980) and Yalamov et aL (1980).
330
;
~ l
F:t (HS
thermal precipitators is considerably lower than 100 Oo but at usual flow rates through them (about 10 cm3/min) by no means equal to zero. It follows from all the foregone that the thermophoresis velocity fas opposite to photophoresis) of NaC1 particles at K n >- 0.01 is positive, i.e. directed to a lower temperature and increases monotonically with rising K n . In all probability the same can be said about aerosols of other materials with heat conductivity .~ 10-2 cal/cm sec C . We emphasize that the negative thermophoresis was never observed in the experiment. As regards the theory of thermophoresis, we wish to point out that the analysis of formulae derived for the thermophoretic velocity and of the assumptions on which they are based is not the aim of this paper. We shall confine ourselves to comparing some of these formulae with the experimental data obtained with NaC1 particles. For the ratio of heat conductivities of these particles and o f the air we shall take a value 200. It makes no sense to take an exact value for this ratio as it will not affect the results. We do not wish to encumber this paper with these mostly very complex formulae and will only discuss the results of numerical calculations. We begin with the thermophoretic force. In Fig. 1 beside the experimental curve is the curve plotted according to the theoretical formula [-7] in the paper of Jacobsen and Brock. We see that at K n = 0.1 the experimental value is half the theoretical. In the same paper there is a more complex formula [8] which by choosing the values of four kinetic coefficients by the best fit method agrees well with experimental data at K n <_ 0.25 but this formula is obviously an empirical one. The same can be said about a very complex formula by Vestern and W a l d m a n n (1977). The agreement with experiment was achieved only by choosing the best fit values o f coefficients C t and Ch. According to the expression for F t / r 2 derived by Sone and Aoki (1977), at 0.7 x 10 -:~ < K n < 0.5 thermophoresis is negative, i.e. the particles are moving towards a higher temperature in obvious contradiction to the experiment. According to Dwyer (1967)at the value of the coefficient of thermal accommodation of gas molecules on the particles greater than 0,5, thermophoresis is negative in the range 0.8 x 1 0 : ~, K n < 0.5. In Fig. 2 the curve I, as stated above, was obtained from the experimental curve for F , r-'. It can be called an "ideal" curve of the thermophoretic velocity, as it can apparently be obtained only in the absence of all interfering factors. The remaining curves are plotted in accordance with the formulae derived in the papers mentioned in the subscript to Fig. 2. As can be seen from this figure, there is no agreement whatever between these formulae as well as between them and the experimental data. The question a b o u t the thermophoresis of very large particles ( K n < 0.06) remains open. Its reliable experimental solution is possible only by the equilibration method at high gas pressures where one can work with relatively small particles easily equilibrated in a Millikan cell. Unfortunately, such experiments are still lacking.
REFERENCES Bakanov, S. P., Deryaguin, B. V. and Roldugin. V. 1. t1979) Usp.Bz. Nauk 129, 255. Deryaguin, B. V., and Yalamov Yu. t1965) J. Colloid Sci. 20, 555. Deryaguin, B. V., Storozhilova, A. i. and Rabinovich, Ya. 1. (1966) J. Colloid lnterJace Sci. 21, 25. Deryaguin, B. V. and Storozhilova,A. 1. (1972) Assessment of Airborne Particles, p. 116.Thomas, Springfield.Illinois. Deryaguin, B. V. and Yalamov, Yu. !1972) International Reviews in Aerosol Physics and Chemistry, Vol. 3 Dwyer, H. A. (1967) Physics Fluids 10, 976. Fuchs, N. A. (1964) The Mechanics of Aerosols, Vols 11 and 16. Pergamon Press, Oxford. Jacobsen, S. and Brock, J. R. (1965) J. Colloid Sci. 20, 544. Poddoskin, A. B., Yushkanov, A. A. and Yalamov, Yu. I. I1980) Zh. tekh. Fiz, 50. 157. Prodi, F., Santachiara, C. and Prodi, V. 11979) ,/. Aerosol Sci, 10, 421. Schadt, S. C. H. and Cadle, R. D. (1961) J. phys. Chem. 65, 1683. Sone, Y. and Aoki, K. (1977) Rarefied Gas Dynamics, Part 1, p. 417. New York. Vestern, H. and Waldmann, L. (1977) Physica 86A, 303. Yalamov, Yu., Poddoskin, A. B. and Yushkanov, A. A. (1980) Dokl. Akad. Nauk SSSR 254, 343.
0021-8502/82/0,10327-04 $03.00/0 Pergamon Press Ltd.
Pnnted in Great Britain
THERMOPHORESIS OF AEROSOL PARTICLES AT SMALL KNUDSEN NUMBERS: THEORY AND EXPERIMENT N. A. FUCHS Karpov Institute of Physical Chemistry, Moscow, U.S.S.R. (Received 25 N o v e m b e r 1981)
Abstract--The most reliable and exact m e t h o d of measuring the thermophoretic force consists of equilibrating single aerosol particles by an electrostatic force in a Millikan cell with heated upper and cooled lower electrodes. The "jet m e t h o d " of measuring thermophoretic velocity can involve considerable errors. The n u m e r o u s formulae published so far for particle thermophoresis at small K n u d s e n n u m b e r s with theoretical calculations of kinetic coefficients deviate considerably from experimental data obtained with sodium chloride particles,
The theory of thermophoresis of aerosol particles at large Knudsen numbers (Kn > 5) is elementary and agrees well with experimental data. Quite different is the situation in the theory ofthermophoresis at small Kn (Kn < 0.3) and relatively high thermal conductivity (2 >__10-2 cal/cm sec °C) of the particles. Many theoretical formulae for the thermophoretic velocity have been derived by a number of authors and this alone casts doubts on their validity, the more so since most of the authors did not compare them with experimental data. Measurements of the thermophoretic velocity v, or the thermophoretic force F, are possible by two methods. One of these is by using the Millikan cell with heated upper and cooled lower electrodes. Two versions are possible here. 1. After determining by Millikan's method the electric charge of the particle it is equilibrated in the cell in the absence of the thermal gradient, and the particle weight and hence its dimensions are determined. Then the particle is equilibrated in the presence of the thermal gradient and thus F, is determined. 2. The rate of particle sedimentation is measured in the presence and absence of the gradient and v, is found from the difference. The relation between F, and v, is given by the formula F, = 6rtrvttT/K where *7is the viscosity of the gaseous medium, r the particle radius and K the Cunningham-Millikan factor. In the second version the mean temperature of the sedimenting particle is higher than that of the gas, the particle heats the adjacent gas layer and this gives rise to an ascending convection microcurrent and thus diminishes the sedimentation velocity. In fact, in experiments of Schadt and Cadle (1961) the thermophoresis velocity determined by the second version was systematically lower than that calculated from the thermophoretic force found by the first method which is undoubtedly more reliable. Two series of experiments were made by the equilibration method at small Knudsen numbers with sodium chloride particles of thermal conductivity 2 = 10-2 cal/cm sec °C. In the work of Schadt and Cadle (1961) cubic crystalline particles were used in air, in the work of Jacobsen and Brock (1965)--spherical amorphous particles in argon. The particle radii in both cases were from 0.2 to ~ 1/~m. Measurements were made at gas pressures varying from 200 to 800 torr for each particle. The results of these measurements are shown in Fig. 1. The experimental points obtained by Jacobsen and Brock were taken directly from the laboratory journal* and those by Schadt and Cadle--from their paper after multiplying by 1.2--the ratio of argon and air viscosities because the reduced thermophoretic force F,/r 2 is proportional to the gas viscosity, independent of gas pressure and is a single-valued function of Kn. This important factor should be taken into account in the evaluation of theoretical expressions for F t . * These data were kindly communicated to the a u t h o r of this review by Prof. J. R. Brock. 327
328
-, ~, I-:~.~ HS
? E
i
Oi
02
03
Kn
Fig. 1. Reduced thermophoretic force. Sodium chloride particles in argon. A Experimental data of Schadt and Cadle (1961). Experimental data of Jacobsen and Brock (1965), [] r--0.436#m, • 550/~m, ×656/an, +840/~m, Ol.lOl/~m and O1.120/~m. Curve 2 is the theoretical curve derived by Jacobsen and Brock. The temperature gradient is given by G.
About 40 experimental points pertaining to different particle sizes and gas pressures were obtained by Jacobsen and Brock. As can be seen from Fig. 1 these points fall well on one curve: the standard deviation of the points from this curve does not exceed 1%, i.e. these data are very exact. When taking into account the difference in the shape and density of the particles in experiments of Schadt and Cadle, and Jacobsen and Brock the agreement between the results of both series of experiments can be regarded as satisfactory. It follows from these results that for NaC1 particles at Kn > 0.06 F r and, as we shall see later, vr increases monotonically with Kn. As pointed out by Deryaguin and Yalamov (1972) a serious error can occur in the equilibration method, due to the ascending thermal gas slip along the nonuniformly heated side walls of the cell. Therefore in the middle of the cell there must be a descending gas current, which leads to too high values of F,. However, as seen from Fig. 1, the experimental points obtained with particles of different size lie on the same curve whereas if such a current occurred in those experiments the downward drag caused by it should increase with particle size. Evidently, this effect did not exceed the accidental errors of the measurements. Another possible source of error in the work of Jacobsen and Brock is that the size of the particles were not measured directly but calculated from their weight. The density p of the amorphous NaCI spheres was obtained by extrapolation to 25 °C of the plot for melted NaC1 vs temperature, which gave 0.93 of the density of crystaUine NaC1. However the error in such a determination of the density and size of particles result in changing both F,/r z and K n in the same direction and thus partially compensate for each other. In the jet method of measuring v, aerosol is introduced isokinetically through a narrow horizontal slit into a laminar current of a filtered gas flowing through a plane-parallel channel with heated upper and cooled lower walls. The aerosol is preheated to the temperature of the ga s at the slit. The distance between the slit and the entrance to the channel is such that atthe place of introduction the profile of the flow velocity in the channel becomes parabolic and the temperature gradient across the channel is constant. The sedimentation velocity of the thin aerosol jet is measured. Many errors may occur in this method. 1. The error due to microconvection, already mentioned. 2. The error caused by thermal slip along the side walls (also mentioned).
Thermophoresis of aerosol particles at small Knudsen numbers
329
3. The aerosol jet, especially a monodisperse one, carries gas down with it so that the observed sedimentation velocity is the sum of the particle sedimentation relative to the gas and the downward flow of the medium itself. This effect increases with the aerosol concentration in the jet and its cross section. 4. Due to the gradient of the flow velocity in the channel the particles rotate with an angular velocity equal to one half of the gradient (Fuchs, 1964). 5. A polydisperse aerosol jet becomes diffuse. Experiments by Deryaguin et al. (1966) were performed with polydisperse aerosols (r = 0.2-0.6 #m) obtained by spraying a solution of NaCI and drying the droplets in air at atmospheric pressure. At such small r the effect of gravitational sedimentation could be ignored and in the absence of a thermal gradient the aerosol jet was strictly horizontal. Its rate of fall due to thermophoresis was measured with a horizontal microscope from its vertical shift relative to the distance from the point of introduction into the channel. The absence of the effect (3) was proved by the observation that the change of aerosol concentration from 103 to 106/cm 3 did not change the sedimentation velocity. The overall error in these measurements is by estimation of the above authors 8-13 %. Measurements by Prodi et al. (1979) were made with more or less monodisperse condensation NaC1 aerosols with r from 0.17 to 0.85/~m in the air at atmospheric pressure. The rate of fall of the jet was calculated from the distance from the point of introduction to the sediment on the lower wall. Effect (3) was not taken into account and as neither the jet thickness nor the aerosol concentration are indicated in the paper it is impossible to estimate the magnitude of this effect. The results of the two latter publications are shown in Fig. 2. We see that the values of vt found by means of the jet method are much higher than those calculated from F d r 2. From all that has been said above it follows that former values are erroneous although the cause of the discrepancy cannot be established without special measurements. The lower limit of measurements of Ft/r 2 by the equilibration method reached so far at atmospheric pressure corresponds to r ~ 1/~m. For still larger particles it is possible to estimate qualitatively the thermophoresis velocity from the experiments with thermal precipitators with a heated wire. It is well known that in precipitates obtained in this instrument a segregation of particles with different size occurs. For instance in the experiments of Fuchs and Yankovskii (Fuchs, 1964) with polydisperse NaCI aerosols it was found that the windward (directed against the flow) side of the precipitate contains chiefly particles with r < 0.1/am, the middle part--those with r - 0.1--0.5/~m and the lee side 0.3-1/~m in accordance with the findings dealt with above. A few observations made with still larger particles show that at r = 5-10/am the efficiency of
? E
O}
02
03
Kn
Fig. 2. Thermophoretic velocity. Sodium chloride particles in air. Experimental data x Prodi et aL (1979);O Deryaguin et al. (1966). Curve I is obtained by recalculation from curve 1 in Fig. 1. Curve lI--theoretical curve derived by Deryaguin and Storozhilova (1972), curve III--by Deryaguin et aL (1966), curve IV--by Deryaguin and Yalamov (1965), curve V--by Deryaguin and Yalamov (1972), curve VI--by Bakanov et al. (1979), and curve VII--by Poddoskin et al. (1980) and Yalamov et aL (1980).
330
;
~ l
F:t (HS
thermal precipitators is considerably lower than 100 Oo but at usual flow rates through them (about 10 cm3/min) by no means equal to zero. It follows from all the foregone that the thermophoresis velocity fas opposite to photophoresis) of NaC1 particles at K n >- 0.01 is positive, i.e. directed to a lower temperature and increases monotonically with rising K n . In all probability the same can be said about aerosols of other materials with heat conductivity .~ 10-2 cal/cm sec C . We emphasize that the negative thermophoresis was never observed in the experiment. As regards the theory of thermophoresis, we wish to point out that the analysis of formulae derived for the thermophoretic velocity and of the assumptions on which they are based is not the aim of this paper. We shall confine ourselves to comparing some of these formulae with the experimental data obtained with NaC1 particles. For the ratio of heat conductivities of these particles and o f the air we shall take a value 200. It makes no sense to take an exact value for this ratio as it will not affect the results. We do not wish to encumber this paper with these mostly very complex formulae and will only discuss the results of numerical calculations. We begin with the thermophoretic force. In Fig. 1 beside the experimental curve is the curve plotted according to the theoretical formula [-7] in the paper of Jacobsen and Brock. We see that at K n = 0.1 the experimental value is half the theoretical. In the same paper there is a more complex formula [8] which by choosing the values of four kinetic coefficients by the best fit method agrees well with experimental data at K n <_ 0.25 but this formula is obviously an empirical one. The same can be said about a very complex formula by Vestern and W a l d m a n n (1977). The agreement with experiment was achieved only by choosing the best fit values o f coefficients C t and Ch. According to the expression for F t / r 2 derived by Sone and Aoki (1977), at 0.7 x 10 -:~ < K n < 0.5 thermophoresis is negative, i.e. the particles are moving towards a higher temperature in obvious contradiction to the experiment. According to Dwyer (1967)at the value of the coefficient of thermal accommodation of gas molecules on the particles greater than 0,5, thermophoresis is negative in the range 0.8 x 1 0 : ~, K n < 0.5. In Fig. 2 the curve I, as stated above, was obtained from the experimental curve for F , r-'. It can be called an "ideal" curve of the thermophoretic velocity, as it can apparently be obtained only in the absence of all interfering factors. The remaining curves are plotted in accordance with the formulae derived in the papers mentioned in the subscript to Fig. 2. As can be seen from this figure, there is no agreement whatever between these formulae as well as between them and the experimental data. The question a b o u t the thermophoresis of very large particles ( K n < 0.06) remains open. Its reliable experimental solution is possible only by the equilibration method at high gas pressures where one can work with relatively small particles easily equilibrated in a Millikan cell. Unfortunately, such experiments are still lacking.
REFERENCES Bakanov, S. P., Deryaguin, B. V. and Roldugin. V. 1. t1979) Usp.Bz. Nauk 129, 255. Deryaguin, B. V., and Yalamov Yu. t1965) J. Colloid Sci. 20, 555. Deryaguin, B. V., Storozhilova, A. i. and Rabinovich, Ya. 1. (1966) J. Colloid lnterJace Sci. 21, 25. Deryaguin, B. V. and Storozhilova,A. 1. (1972) Assessment of Airborne Particles, p. 116.Thomas, Springfield.Illinois. Deryaguin, B. V. and Yalamov, Yu. !1972) International Reviews in Aerosol Physics and Chemistry, Vol. 3 Dwyer, H. A. (1967) Physics Fluids 10, 976. Fuchs, N. A. (1964) The Mechanics of Aerosols, Vols 11 and 16. Pergamon Press, Oxford. Jacobsen, S. and Brock, J. R. (1965) J. Colloid Sci. 20, 544. Poddoskin, A. B., Yushkanov, A. A. and Yalamov, Yu. I. I1980) Zh. tekh. Fiz, 50. 157. Prodi, F., Santachiara, C. and Prodi, V. 11979) ,/. Aerosol Sci, 10, 421. Schadt, S. C. H. and Cadle, R. D. (1961) J. phys. Chem. 65, 1683. Sone, Y. and Aoki, K. (1977) Rarefied Gas Dynamics, Part 1, p. 417. New York. Vestern, H. and Waldmann, L. (1977) Physica 86A, 303. Yalamov, Yu., Poddoskin, A. B. and Yushkanov, A. A. (1980) Dokl. Akad. Nauk SSSR 254, 343.