Langmuir 1985,1, 361-365
36 1
Photophoresis of Irradiated Spheres: Evaluation of the Complex Index of Refraction+ Wayne M. Greene, R. Erik Spjut,* Ezra Bar-Ziv,f John P. Longwell, and Adel F. Sarofim Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received December 28, 1984. In Final Form: February 26,1985 The photophoreticforce on micron-sized particles within the continuum regime has been evaluated for a range of size parameters and complex indices of refraction. A dimensionless quantity, the asymmetry factor divided by the absorption efficiency, J/Qabs,which can be obtained both experimentallyand theoretically, is shown to provide a useful measure for determining the complex refractive index. For large size parameters, J / Q a bprovides a measure of the extinction coefficient, k ; it is insensitive to the real part of the refractive index, n, and depends weakly on the size parameter a. For small size parameters, J/Qabs depends strongly on a, n, and k . n and k can be obtained by measuring J/Qabaas a function of cy.
Introduction A particle heated by light absorption moves. This motion is known as photophoresis. The nature of photophoresis has been studied both the~reticallyl-~ and ex~erimentally."~In the continuum regime (particle size much bigger than the mean free path of the surrounding gas) the particle created a temperature gradient in the gas which causes the gas to flow along the particle surface. The hydrodynamics of this flow, as established by the temperature gradient in the particle, determine the magnitude of the photophoretic force.2 A knowledge of the radiant energy dissipation, or source function, as described in a companion papels would permit calculation of the photophoretic force for a given particle size and complex index of refraction, and, conversely, measurement of the photophoretic force for a range of particle sizes should permit calculation of the complex index of refraction for a given substance. Previous measurements of the refractive index from the photophoretic force were based on determining the reversal point whereas the present paper is based on the use of the recently acquired capability of measuring the temperatures of particles in an electrodynamic balance.
Theory The photophoretic force is a function of the size, shape, temperature, thermal conductivity,,and the complex index of refraction of a particle. It also depends on the pressure, viscosity, and composition of the surrounding gas. The wavelength, uniformity, and polarization of the incident radiation are important as well. This study is concerned only with the photophoretic force on a sphere within the continuum regime, i.e., for a Knudsen number, Kn, much less than unity. A particle that is heated nonuniformly creates a temperature gradient in the surrounding gas. The gradient results in a gas flow or creep from the cold side to the hot side. In 1878, Maxwell described the origii of the gas creep at the particle s ~ r f a c e .He ~ developed an expression for the creep velocity, KP aT, v = -(1) RT8P, a0 'Presented a t the symposium on "The Chemical Physics of Aerocolloidal Particles", 188th National Meeting of the American Chemical Society, Philadelphia, PA, Aug 26-31, 1984. On leave from the Nuclear Research Center-Negev, Beer-Sheva 84190,Israel.
*
0743-7463/85/2401-0361$01.50/0
where p is the gas viscosity, p g the gas density, and T, the particle surface temperature. Yalamov et al. have solved the mass, momentum, and energy conservation equations to yield an expression for the photophoretic force on a heated, nonvolatile particle irradiated by electromagnetic radiation.2
K is the coefficient of thermal slip. Maxwell estimated the value of this constant to be 3/4.9J0 I is the intensity of the incident radiation. Kp is the particle thermal conductivity. J is the asymmetry factor which describes the asymmetry of the temperature distribution on the particle surface. The asymmetry factor, J,ranges from -0.5 when all of the radiation is totally absorbed on the front surface to +0.5 when the radiation is totally absorbed on the back surface of the particle. Negative values of J correspond to particle motion away from the irradiating source, or positive photophoresis. Conversely, positive values of J imply that the particle motion is toward the irradiating source, or negative photophoresis. The asymmetry factor, J , therefore describes the qualitative nature of the photophoretic force.6J1 The asymmetry factor is a function of the dimensionless particle size, the complex index of refraction, and the normalized source function:2
J = (3/2)nka&'&=r
sin 20 B(r,O) dB dr
(3)
(1) Hettner, G. 2.Phys. 1962,37, 179-192. (2)Yalamov, Yu. I.; Kutukov, V. B.; Shchukin, E. R. J. Colloid Interface Sci. 1976, 57, 564-571. (3) Reed, L. D. J. Aerosol Sci. 1977,8, 123-131. (4) Arnold, $3.; Amani, Y. Opt. Lett. 1980, 5 , 242-244. (5) Arnold, S.;Amani, Y.; Orenstein, A. Reu. Sci. Instrum. 1980,51, 1202-1204. (6) Arnold, S.; Lewittes, M. J. Appl. Phys. 1982, 53, 5314-5319. (7) Pluchino, A. B.; Arnold, S. Abstr. Pap.-Am. Chem. SOC.1984, 188th, Coll 110. (8) Greene, W. M.; Spjut R. E.; Bar-Ziv, E.; Longwell J. P.; Sarofim A. F. J. Opt. SOC.Am., in press. (9) Maxwell, C. Philos. Tram. R. SOC.London 1879, 170, 231-256. (10) Work by (Derjaguin and Yalamov Derjaguin, V. B.; Yalamov, Yu. I. In "International Reviews in Aerosol Physics and Chemistry"; Hidy, E. G., Brock, J., Eds.; Permagon Press: New York, 1972)indicates that K can be higher than the value Maxwell obtained. (11)Pluchino, A. B. Appl. Opt. 1983,22, 103-106.
0 1985 American Chemical Society
Greene et al.
362 Langmuir, Vol. I , No. 3, 1985 where M = n - ik and a = aD/X. h is the wavelength of the radiation. A spherical coordinate system is used with (r,O,$) as the designated coordinates. r is the reduced radial coordinate. B(r,O) is the &averaged normalized source function that describes the distribution of the radiant energy absorption within the particle. Calculation of B has been performed in a companion paper.8 For a given particle-gas system at a constant pressure the photophoretic force, eq 2 , depends on the surface temperature, light intensity, particle size, and complex index of refraction. Absolute measurement of the light intensity at the particle surface is difficult to perform. Therefore, we have developed a way to eliminate the necessity of measuring the intensity. For the usual case in which gas conductivity, Kg,is much smaller than the solid (or liquid) conductivity, K,, the maximum temperature difference within the particle is much smaller than the temperature differential between the particle and the gas. In this case the mean particle temperature, T,, can be calculated by a simple energy balance. For temperatures less than 2000 K, when the radiative heat loss by the particle can be neglected, the rate of energy absorption, qs, is equal to the convective loss, q,. With this assumption, qa =
nR2Qabsl
(4A)
4 , = 4nR2h(T,- T,)
(4B)
Qabsis the absorption efficiency, equal to the absorption cross section, Cab, divided by the geometrical cross section. Qab can be obtained from the Mie theory and is a function of the dimensionless particle size and the complex index of refraction.12 T, is the gas temperature far away from the particle. h is the convective heat transfer coefficient which is a function of the particle size and the thermal conductivity of the surrounding gas medium evaluated at the film temperature. For most situations involving small stationary particles, h = k fR. Combining eq 4A and B yields an expression for the intensity as a function of a , M , Tp,K,, and T,:
I=
4Kg(Tp- T,) __
RQabs
(5)
Substituting eq 5 into eq 2 yields
F
16np2K,(T, - T,)JK
= __-___ ph
--
PyTZJ(,Qabs
(6)
The light intensity has been replaced by a relation involving the particle temperature T,. Rearranging eq 6 yields
One of our goals was to evaluate the suitability of using photophoresis to calculate the complex index of refraction, M. The right-hand side of eq 7 contains only known (e.g., K g ,p,) or experimentally determined (e.g., T,, F)quantities. Uncertainty in the coefficient of thermal slip probably limits the accuracy of the determination to a factor of 2 at present. If J/Qabs,a dimensionless quantity, proves to be theoretically a strong function of M , then M could be (12) Kerker, M. “The Scattering of Light and other Electromagnetic Radiation“; Academic: New York, 1969.
determined by data fitting the experimental results with M as the fitting parameter.
Results and Discussion The variation of J and Qabas well as J / Q a bwith M and a are described below. J is calculated from the Mie theory
by integrating the normalized source function with a 9sin (20) weight as defined by Yalamov et a1.2 The methods for the integration and the calculation of the normalized source function are presented elsewhere.8 Qab is calculated through the use of the f i s t two boundary coefficients from the Mie theory. The sensitivity of J/Qabsto M is detailed below. Figure 1presents an example of the calculation of J and Qabsas a function of a for n = 1.3 and k as a parameter varying from 0.01 to 10. In Figure l A , for k = 0.01, it is seen that a positive J and hence negative photophoretic force exist up to large values of a. The curve reaches a maximum at a = 15 and then tends to decrease. It is expected that a larger value of a,J and the photophoretic force will undergo reversal. As k is increased to a value of 0.1, J and the photophoretic force have a maximum and a reversal a t a lower value of a than for k = 0.01. In addition, this maximum is of smaller magnitude. J becomes more negative as a is increased. The changes in J/Qabswith both increases in a and in k can be explained by the associated preponderance of absorption on the irradiated side of the particle. In general, the photophoresis becomes more positive ( J decreases) as a and k (for a > 5 ) increase. For k’s greater than about 5 the trend is reversed. A t high values of k the particle absorbs energy on both the front and back surfaces due to surface wave refraction, and, therefore, J and the photophoretic force are reduced in magnitude.8 The absorption efficiency, Qab, describes the normalized cross-sectional area of absorption. In Figure lB, all curves are seen to pass through a maximum at some value of a and decrease to an asymptotic value. For low k’s, Qabs increases with increases in k due to the increased absorption. For high k’s, Qabsdecreases as k is increased. A greater resistance to light penetration causes a smaller absorption penetration depth, and thus more light scattering occurs. The trends of J and Qabs with a , for various values of k , are similar. Figure 2 presents J/Qabaas a function of a for different values of k as calculated from the J and Qabs presented in Figure 1. The general trend of JIBab with a is monotonic in k as expected. Figures 2-4 present J/Qabas a function of a for varying n and k. Two distinct regions of J/Qabsdependence on a exist. For a greater than 7 , J/Qabsappears to behave monotonically in a for given values of n and k . Oscillatory behavior is observed a t small a. Values of J/Qab,at high a’s are presented in Figure 5 , as a function of a , for selected values of n and k . In this regime, J/Qabsis relatively insensitive to n and strongly dependent on k . The dependence on a is quite weak. A similar plot of J/Qabeas a function of a, for different values of n and k, is shown in Figure 6 for the small range. As k increases for constant n, the oscillatory pattern is damped out. As n increases at a given k , the oscillatory amplitude and frequency increase. J/Qabsvs. a at k = 10 shows that the oscillatory behavior is entirely damped out and that J/Qab,shows little dependence on n. J/Qabsis a sensitive measure of the complex index of refraction. Discussion of a possible experimental mea-
Langmuir, Vol. 1, No. 3, 1985 363
Photophoresis of Irradiated Spheres I
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surements of J/Qabsfollows. J / Q a b is determined experimentally by the measurement of the photophoretic force, the particle size, and the gas and particle temperatures in conjunction with eq 7. Arnold et al. have measured the photophoretic force by electrically levitating a particle and measuring the force parallel to the gravitation field.@ They used a Millikan-type, parallel-plate configuration. Phillip has used an electrodynamic balance for measuring the weight of a ~article.'~Also, Spjut et al. have developed a similar configuration but with temperature measurement capabilities.14 It is suggested that one of the configurations (13) Phillip, M. A. M. S. Thesis, Massachusetts Institute of Technology, Cambridge,MA, 1981. (14) Spjut, R. E.; Sarofim, A. F.; Longwell, J. P. Langmuir, preceding
paper in this issue.
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Figure 5. J/Q* as a function of a,for a > 7 where J/Qabshows a weak dependence on n and a strong dependence on k.
be used to measure the photophoretic force, particle size, and temperature simultaneously. In Figure 7, the lumped-parameter model calculation of the intensity necessary to achieve 1000 K for various indices of refraction and a is shown. The experimentalist should use Figure 7 to choose an intensity region to yield a large enough difference in temperature to be accurately measured (greater than 500 K)14 for use in eq 7. It was noted that for high a J / Q a b is insensitive to the real part of the complex index of refraction, n. In Figure 8, J/QabSvs. k , for varying a,is presented. J/Qabsis relatively insensitive to a but is a strong function of k. k can be determined by measurement of J/Qab in the large range and one a is needed.
364 Langmuir, Vol. 1, No. 3, 1985
Greene et al.
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