BY FRANK T. GUCKER AND ROBERT L. ROWELL? Dept. of Chemistry, Indiana University, Bloomington, Indiana, U.S.A. Received 27th June, 1960 The light-scattering diagrams of single aerosol droplets of dioctyl phthalate $ have been determined by charging them, suspending then1 in an electrostatic field, illuminating them with monochromatic light of wavelength )I and measuring the light scattered into a photo- meter moving over the range of 40 to 140" from the direction of illumination. Particle radius r was calculated from rate-of-fall measurements and the Stokes-Cunningham- Millikan equation. Good agreement was obtained with scattering diagrams calcuIated from the Mie theory, which indeed provide a more sensitive measurement of particle size than do rate-of-fall measurements. Detailed calculations of scattered intensity according to the Mie theory have been made at angular intervals of 1" over the range of size parameter a = (27~rlA) = 01(0*1)30-0 for a refractive index of 1.486 and at several values of a for a refractive index of 1.50.The number of maxima in the scattering diagrams increases with a. Graphs of the angular position of the maxima in the two polarized components show that new maxima are formed by splitting where the curves for the two polarized components intersect and the size parameter is approximately divisible by ~14. Gustav Mie, in 1908,l obtained the first complete theoretical solution to the problem of the scattering of a plane electromagnetic wave by a spherical particle. A brief statement of some of the results of Me's work introduces the present investigation.A dielectric sphere of radius r is illuminated by a plane wave of natural (un- polarized) light of unit intensity and wavelength A. The intensity of the scattered light I observed at a distance R which is large compared to 2, at an angle 8 measured from the forward direction of propagation of the incident beam is given by I = (A2/8~2R2)(i1 + iz), where il is proportional to the intensity of light plane polarized with the electric vector perpendicular to the plane of observation (including the incident beam and the point of observation) and i2 is proportional to the intensity of light plane polar- ized with the electric vector perpendicular to il. The intensity functions, il and 2'2, are expressed in terms of infinite series dis- cussed by Gucker and Cohn 2 and given in the notation they suggested as The augmented angular functions, IIn and Tn, are formed from the first and second derivatives of Legendre polynomials of argument cos 0 and the amplitude functions, a, and b,, are derived from Wiccati-Bessel functions of arguments a = 2nr/A and * contribution no.958. 7 present address : Department of Chemistry, University of Massachusetts, Amherst, Massachusetts, USA. i.e., bis-2-ethylhexyl phthalate. 1852 86 LIGHT SCATTERING BY SINGLE AEROSOL DROPLETS ma, where rn is the re€ractive index relative to the medium. Penndorf and Gold- berg 3 found that the series converge rapidly after about (73- 1.2~) terms, and the summation is stopped at about that point, when further terms become inappreciable. The present work combines an experimental and theoretical study of light scattering from aerosol particles.We have made precise measurements of the angular variation of monochromatic light scattered by single droplets of dioctyl phthalate @QP) (bis-2-ethylhexyl phthalate), and computations OF the Me-theory intensity €unctions over the angular range 0" (1 ") 180", and size parameter a = 0.1 (0.1) 300. These detailed calculations allow a careful comparison of experi- ment with theory, and have revealed some unsuspected new relationships between size parameter and scattering diagrams. EXPERIMENTAL METHODS The object of the experimental work was to produce an apparatus capable of precise measurement of the angular intensity-distribution curve for a particlc of radius in the micron range.Such an instrument enables a quantitative test of various scattering theories and eventually the possibility of studying the scattering from substances with a complex index of refraction or of intricate shape, which are dimcult or impossible of theoretical evaluation. A detailed description of the first model of the apparatus, developed by Dr. James J. Egan, is given elsewhere.4 5 The principle of the measurement is to suspend a charged droplet of DOP in the electrostatic field of a Millikan-type apparatus, illuminate it with an intense beam of light, nearly parallel (0.8 O half-angle of divergence), and monochromatic (34-0 A half-band width) and measure the scattered light over the angular range (d = 40' to 140") with a photometar having an entrance aperture 4.1" wide and 6.0' high.The improved apparatus used in this work employs a precision goniometer (0.2" accuracy) and dual photometers, a measuring photometer which responds to the scattered light and a reference photometer which monitors the attenuated incident radiation. Each photometer employs an RCA type 1P21 multiplier phototube, the output of which is fcd into a bucking circuit (based on Egan's design) which allows cancellation of the back- ground current. The resultant signals are amplified by Leeds and Northrup type 9836 micromicroampere amplifiers and fed into a factory-modified Leeds and Northrup Speedo- max 6 recorder capable of recording the output from either channel or their ratio. An advantage of this arrangement is that the ratio of the scattered light to the attenuated incident light is independent of fluctuations in the intensity of the incident beam.The angular position of the measuring photometer at each end of the range is indicated on the recording paper by means of a solenoid marginal pen actuated by an electromechanical trigger pulse from the precision goniometer and associated electronic circuit. COMPUTATIONS The augmented angular functions, IIn and Tn, have been calculated at intervals of 1" to fifteen decimal places for orders n = 1 to 43. Amplitude functions, a, and b,, given to ten digits, for refractive indices 1.33, 1.40, 1.44, 1.486 and 1.50 and for size parameters a = 0.1 (0.1) 30.0 have recently been published in five parts by Penndorf and Goldberg.3 We used the amplitude functions (part 4, rn = 1.486; part 5, m = 1.50 corresponding to the refractive index of DOP for two wavelengths of visible light) along with our augmented angular functions, to calculate the complex amplitudes of the intensity functions, il and i2, the functions themselves (equal to the square of the modulus), and their sum, il+ i2 = ie.The calculations for m = 1.486 cover the range of a = 0.1 (0- 1) 30 and 6 = 0" (1") 180" and are recorded on 54,300 ( 3 0 0 ~ 181) IBM cards, each containing the complex amplitudes, Re(il), Im(il), Re(i2), and Im(i2), and the intensity functions, il, i2, and is, for a particular a and 6. The work was done in double precision and the results are in floating-point notation with eight sig- nificant figures and the floating point index punched for each number.Similar calculations for rn = 1.50 have been made over the complete angular range at intervals of 2 9 for ct = 10.0, 12.0, 15.8, 19.2, 20.5 and 209, in connectionF. T . GUCKER AND R. I,. ROWELL 18; with Egan's measurements,s and at intervals of 1" for a = 15.4 (0.1) 18.8 anc 20.2 (0.1) 20.6 in connection with OUT experimental work (see below). The results of the calculations show that the shape of the angular intensity distribution curves and the angular positions of the maxima are very sensitive tc small changes in the size parameter a and in the refractive index rn. l ~ i ~ [ 0 angle, 0 FIG. 1 .-Mie-theory scattering diagram showing total intensity function, ie = i~ +i2, over the range in scattering angle 0 = 50" to 120" for a size parameter a = 18.4 and refractive indices m = 1.486, dashed curve, and 1-50, solid curve.angle, 6' FIG. 2.-Angular position of the maxima in the total intensity function, it, = i l f i 2 , for a refractive index of 1-486 over the range in size parameter, a = 18.0 to 22.0, and scattering angle, 0 = 50" to 100". Fig. 1 shows a comparison on a semi-logarithmic scale of a portion of the angular intensity distribution for size parameter a = 18.4 and refractive indices of 1.486, dashed curve, and 1-50, solid curve. A logarithmic scale seemed ap- propriate for the intensity functions because of the large range in the values of the resonance peaks and valleys. Although the curves are roughly the same, the small difference in refractive index appreciably affects both the amplitude of the resonance peaks and the angular position of thezmaxima.188 LIGHT SCATTERING BY STNGLE AEROSOL DROPLETS The junior author 6 has compared the scattering diagram for io over the corn- plcte angular range for a = 20.9, m = 1-58 with five scattering diagrams for m = 1-486: a = 20.9, 21.0, 21.1, 21.2 and 21.5, and found best agreement for a = 21.1, which suggests that diagrams having the same valuc of ma (31.35 in these cases) would have about the same structure.The behaviour of the maxima for different size parameters can be coiivenieiitly summarized by plotting the angular position of the maxima on a size-angle diagram. A limited region is shown in fig. 2, which covers the angular range, 8 = 50' to lQQ", and the size range a = 18.0 to 22.0.The pattern is somewhat irregular but shows that the increase in number of maxima with particle size is due to their migration away from the centre and the appearance of new maxima in the centre. COMPARISON OF EXPERIMENT WITH THEORY FOR io The problem of obtaining a precise comparison of experiment with theory was complicated by the facts that the particle diameters were so small that a cor- rection had to be applied to Stokes's law, the light was nearly but not precisely monochromatic, and the observing aperture had a small but finite width. From measurements of the rate of fall of the DOP particle we calculated its radius from the Stokes-Cunningham-Millikan (SCM) equation, Here u is the terminal velocity, p2 and p1 the densities of the particle and medium (air), g the acceleration of gravity, r the particulate radius, q the coefficient of viscosity, and ~ C M the Cunningham-Millikan correction lactor, v = 2 ( ~ , - p1)8r2/9v(1 + f C d (4) fcM = (l/r)(A+Be-Cr/'), ( 5 ) where I is the mean free path.For DOP particles in air we used Millikan's values 7 for the constants A , B and C which he determined for oil drops falling in air and which Rosenblatt and LaMer 8 have found applicable to tricresyl phosphate droplets falling in air. The value of rn = 1.486 was first chosen as most representative of the refractive index of DOk for visible light, and this value was used for our complete calculations. Later, however, we found the refractive index of a sample of DOP at 25°C was 14840 for sodium-D light, but 1.4980 for light of wavelength 4371.3A.6 Ac- cordingly, we used our detailed calculations for m := 1.486 as a first approximation and made selected new calculations for nz = 1.50 to improve the cornparison of experimental results with theory.One particle we studied had a radius of 1.44,~ as determined by rate-of-fall mcasurements and the SCM equation, and a value of a = 20.8 based upon the mid-band wavelength of 4371 8, for A. Comparison of the positions of the light- scattering maxima with fig. 2 gave as a first approximation good agreement for a = 20.4 (m = 1.486). The IBM 650 programme of Gucker and Engle and the published amplitude functions of Penndorf and Goldberg 3 were then used to cal- culate intensity functions for a refractive index of 1-50 in this region. The experimental and theoretical curves were compared on a semi-logarithmic scale similar to fig.1 in an intermediate graph (not shown). The principal ad- vantage of the logarithmic scale is that it allows comparison of the shape of the experimental and theoretical curves without the necessity of a normalizing factor ; its chief disadvantage is that it gives too much weight to the smaller experimental numbers. The comparison gives the best agreement for a light scattering a of 20.3 (rn = 1-58). The result is consistent with the previous approximation using data based on m = 1.486. The difference of 0-5 a between the SCM value of 20.8 and light- scattering value of 20.3 is within the expected range of error inasmuch as Millikan 7 believed his constants to be accurate to f 2 %, the error in the time-of-fall measure- ment was about 1 %, and accurate Cunningham-Millikan constants (whichF.T. GUCKER AND R. L. ROWELL 189 depend on the mean free path and the nature of the collision between the molecules of the medium and the particle) have never been determined for DQP. Fig. 3 shows a comparison of the experimentd (solid) and theoretical (dashed) curves on a linear scale. The experimental curve was obtained from the difference of two measurements: the angular response with the particle illuminated by intense blue light (4371.3 A with half-band width of 344 A) minus the background, which is the angular response with no particle in the cell. The background con- sisted of dark current arid photocurrent from light scattered by the solid parts of the cell and the air, Most of the d.c.component of the background was reduced with the bucking circuits during a run. Similar experimental scattering diagrams have been obtained 4 for three other particles of DOF with radii 0.4, 1.3 and 2.3 p. angle, 0 FIG. 3.-Comparison of experiment, solid curve, with theory, dashed curvc. The curves in fig. 3 have been normalized by equating the areas shaded by the first two maxima, which are the most reliable. The disagreement is due chiefly to low signal-to-noise ratio and the strong dependence of the shape of the curves on the scattering angle and the particle size. Work is now in progress on a more refined comparison made by integrating the theoretical curves first over the range of size parameter a (approximately 0.2 a) corresponding to the small band-width of the incident radiation and then for each angular position over the range of angle corresponding to the finite width of the observing aperture (4.1").The self-consistency of the present results strongly substantiates the angular intensity distribution predicted by the Mie theory and independently confirms the Stokes-Cunningham-Millilcan law. Indeed, precise analysis of more of the light- scattering data should enabIe us to test the correctness of the relative angular intensity distributions predicted by the Mie theory for different sized particles, and calculate accurate values for the Cunningham-Millikan constants using a particle radius determined from the angular position of the maxima in the intensity dis- tribution. The method of using light-scattering data to determine particle size is promising since an uncertainty of 0-1 a corresponds to only 0.807 p in r.190 LIGHT SCATTERING BY SINGLE AEROSOL DROPLETS BEHAVIOUR OF THE MAXIMA IN il AND i2 The migration of' the maxima on the size-angle diagram for the total intensity (fig.2) coupled with the several abrupt changes and " splittings " in the pattern, led us to conclude that there might be two strong and opposite trends corresponding to the two plane-polarized components. Investigation of the data for the plane-polarized components revealed un- dulating curves, similar to fig. 1, and a plot of the angular position of the maxima on a size-angle diagram in fig. 4 showed the expected trends : general migration of the maxima in il, solid curve, toward the forward direction and general migration of the maxima in i2, dashed curve, toward the backward direction.In addition, some regions of the i2 data and more regions of the il data showed splitting with increasing size parameter. angle, 6 FIG. 4.-Angular position of the maxima in il, solid curve and i2, dashed curve, €or a refractive index of 1.486 over the range in size parameter, a = 18.0 to 22-0 and scattering angle, 0 = 50" to 100". Fig. 5 shows a photograph of a three-dimensional model of the intensity function il over the range of 18.5 to 19.1 in a and 50" to 100" in 8. Here the maxima form the ridges, and the model gives some insight into the nature of the splitting process. The first two ridges on the left are fairly uniform in height over this small range in size parameter, while the others slope down through cols which shift toward larger values of 8 and a.As the size parameter increases from 18.5 to 18.7, the last two ridges on the right slope down and a deep valley develops between them. With further increase in a, a shoulder appears on the right slope of the col in the second ridge from the right, and grows into a new ridge. The two ridges formed by the splitting then migrate in opposite directions as do those on either side. An inspection of fig. 4 shows that splitting occurs approximately where the curves for il and i2 intersect at a value of a divisible by n/4, which corresponds to " standing-wave positions " for both polarized components of the scattered light, i.e., where 1 sphere diameter equals $, +, etc., the wavelength of the incident light in air.For the region studied, the splittings were found to occur within the angular range 8 = 78-93' for il, 50-54" for i2, and 66-76' for i,g. We are now engaged inFIG. 5.-Mie-theory intensity iunctlon il over the range in size parameter, cc = 18.5 to 19.1 and scattering angle, 0 = 50" to 100". [To face p. 190F. T. GUCKER AND R. L. ROWELL 191 determining the pattern of the maxima for the complete angular range, 8 = 0-180", and for the size parameter range, a = 0.1-30.0. It is a pleasure to acknowledge the support of the National Science Foundation for a grant that made this research possible, and the courtesy of Dr. Rudolph Penndorf of the Geophysics Research Division of the Air Force Cambridge Research Center who sent us the theoretical values of the amplitude functions before their publication. We also should like to thank Ora May Engle of the Research Corn- puting Center of Indiana University for the computations of the scattered in- tensities on the IBM 650, and Maurice Williams for his help in the design and construction of the apparatus. 1 Mie, Ann. Physik, 1908, 25, 377. 2 Gucker and Cohn, J. Colloid Sci., 1953, 8, 555. 3 Penndorf and Goldberg, Nevi Tables of Mie Scattering Functions for Spherical Particles, Geophysical Research Papers No. 45, Parts 1 through 5 (as ASTIA docu- ments from U.S. Department of Commerce, Office of Technical Services, Wash- ington 25, D.C.). Parts 4 and 5, for refractive indices 1-486 and 1.50 respectively, are ASTM documents no. AD-98778 and AD-98771. 4 Egan, P k D . Thesis (Indiana University, 1954). 5 Gucker and Egan, J. ColZoid Sci., accepted for publication. 6 Rowell, P h B . Thesis (Indiana University, 1960). 7 Millikan, Physic. Rey., 1911, 32, 349; 1923, 22, 1. 8 Rosenblatt and LaMer, Physic. Rev., 1946, 70, 385.