BY S. W. CHURCHILL, G. C. CLARK,* AND C. M. SLIEPCEVICH 7 Dept. of Chemical and Metallurgical Engineering, The University of Michigan, Ann Arbor, Michigan, U.S.A. Received 20th June, 1960 The effect of particle separation distance on the light-scattering properties of dispersions of closely sized spheres was investigated by measuring the transmission as the concentra- tion was decreased by dilution. The data were corrclated in terms of a two-flux model. The coefficient in this model was observed to be essentially constant down to a centre- to-centre distance of about 1.7 diameters between particles and to vary less than 10 % down to a centre-to-centre distance of about 1.4 diameters, corresponding to 28 % solids by volume. Direct simulation of dilute aerosols having dimensions of the order of kilo- metres is therefore feasible with very dense hydrosols having dimensions of the order of millimetres.The intensity of radiation within or at the boundaries of a dispersion of uniformly-sized, non-absorbing spheres can be described in terms of the angular distribution for single scattering, the dimensions of the dispersion expressed in mean free paths for scattering, the source distribution, and the boundary con- ditions. The mean free path for scattering where Nis the number of spheres per unit volume, us is the scattering cross-section, K, is the scattering coefficient (the ratio of the scattering to the geometrical cross- section), and d is the diameter of the spheres. If the spheres are sufficiently far apart, the angular distribution for single scattering and the scattering coefficient are functions only of n, the refractive index of the sphere relative to the surrounding medium, and a = nd/A, where ;I is the wavelength of the radiation in the continuous medium.It is then possible to scale a dispersion in terms of mean free paths if the same n and CI are established as illustrated by Scott and co-workers;ls 2 Sinclair 3 stated without documentation that optical interference between par- ticles would be expected if the particles were less than 5 diameters apart. No measurements of interference or theoretical expressions for the effect have been found in the literature. The objective of this investigation was to determine the separation distance at which interference becomes appreciable and to measure the magnitude of the effect.Several possible methods of investigation were considered : (i) development of theoretical expressions for the two-body and multiple- body problems ; (ii) measurement of the radiant field around a set of two or more spheres with dimensions of the order of millimetres, using a beam of millimetre waves ; (iii) measurement of the transmission of a beam of monochromatic light through dense dispersions with particle concentration as a variable. * present address : Continental Oil Company, Ponca City, Oklahoma. -f present address : University of Oklahoma, Norman, Oklahoma. 192s. w. CHURCHILL, G. c . CLARK AND c . M. SLIEPCEVICH 193 Method (iii) was chosen because of its comparative simplicity and the more direct applicability of the results. EXPERIMENTAL APPARATUS The equipment consisted of a source and collimating system, a receiver and amplifying unit, and a cell and traversing mechanism, all located in a dehumidified dark room at 18°C.The source was a SO-candlepower, auto-headlight bulb operated with a regulated power supply. The light beam was monochromaticized with interference filters, yielding a transmission of 45 % at 5460flSOA and a band width of 120-140A at the 225 % transmission points. The collimating system is shown in fig. 1. Diaphragms D1 and D2 reduced the stray D, Ds FIG. 1 .-Schematic diagram of collimating system. light reaching the collimating lens, and diaphragm D3 limited the size of the collimated beam. The condensing lens L1 and L2 had focal lengths of 150 and 100 mm, respectively. The shutter was closed except during measurements.The collimating lens L3 was an achromatic, coated, telescope objective, 51 mrn in diameter and 191.5 rnm in focal length. FIXED PART OF CELL OTOMULTIPLIER MOUNTIN 3emm 1.0 x 114mm HIGH MOUNTING RING G G SCREWS MOVABLE PART / GLASS WINDOW OFF CELL f!'Ei::tA&E azmm I o x 63mm HIGH ALL PARTS ARE PLEXIGLAS EXCEPT GLASS WINDOWS FIG. 2.-Experimental cell. With a 1/16-mm pinhole P the final beam had a diameter of 32 mm and a divergence of only 14.2 min. Such a high degree of collimation was not necessary for the transmission measurements but was desirable for the determination of particle concentration. The Du Mont 6291 photomultiplier used as a receiver is a ten-stage multiplier, 38 mm in diameter, with a flat, end-window type photocathode.The photocathode has a S-11 response characteristic; the maximum response is at A = 4400f250A with 10 % of the maximum response at h = 3250f250A and 6175f275 A. Voltage from a variable G194 LIGHT-SCATTERING BY DENSE DISPERSIONS power supply was fed to the photomultiplier through a step attenuator with resistances chosen to give an amplification of about 3 : 1 per step. The anode current was determined by measuring the potential drop across a 1000 52 resistor. The amplified signal was fed to the Y channel of an X-Y recorder. The cell is shown in fig. 2. The fixed part of the cell served as the receiver housing and as the upper boundary of the dispersion. The photomultiplier was optically coupled to the upper glass window of the cell with immersion oil.The movable part of the cell was attached to a platform which travelled on a screw turned by a hand crank. The screw was geared to a Helipot which served as a potentiometer with the output fed through a cathode followed to the X channel of the recorder. The recorder thus produced a continuous record of transmission as a function of cell thickness. As the cell thickness decreased, the excess dispersion flowed up around the lower plate of the fixed part of the cell and was thus optically decoupled from the dispersion remaining in the cell. All cell surfaces except the receiver window and the portion of the source window illuminated by the incident beam were painted with flat, black, acrylic resin. For determination of the particle concentration a camera with an achromatic, coated, telescope objective lens, 83 mm in diam.and 914 mm in focal length, was located at the outlet window of the cell. A pinhole in the back of the camera opened to an opal glass optically coupled to the window of the photomultiplier. This receiver system was sur- rounded by a black, light-tight housing. MATERIALS The dispersions were prepared from very uniformly sized polystyrene-latex spheres supplied by the Dow Chemical Company, Midland, Michigan. One batch had a mean diameter of 0.814 p with a standard deviation of 0.011 p ; the other, a mean diameter of 1.171 p with a standard deviation of 0.013 p. At X = 5460A (in air), the latex has a refractive index of 1.205 with respect to water and a negligible absorptivity.The spheres are stable in water and, since they are charged, do not agglomerate. PROCEDURE After optical alignment, the cell was closed to a thickness of about 0-5 mm and the Helipot shaft was adjusted to indicate a zero signal on the X channel of the recorder. About 50 ml of distilled water were added to the cell and the cell was opened until full deflection occurred on the X channel, corresponding to a cell thickness of about 4 mm. The amplified photomultiplier signal was then recorded as the cell was slowly closed to a zero signal on the X channel. This experiment provided the reference signal I0 for calculation of the transmission. After cleaning and drying the cell, 50 ml of a concentrated dispersion were added and the photomultiplier signal was again recorded as the cell was closed.The concentrated dispersion was next withdrawn from the cell to a reservoir, diluted with a measured quan- tity of water, mixed and returned to the cell, and a new traverse was carried out. Tests were made at twelve stages of dilution over a 10 : 1 range of concentration. All traverses were repeated as necessary to assure reproducibility and complete mixing. DETERMINATION OF PARTICLE CONCENTRATION AND SEPARATION Samples of the dispersion were withdrawn at the end and at two intermediate stages of dilution. After great dilution, traverses were made on these samples with the camera between the cell and the photomultiplier. The particle concentration was determined from these data and the modified form of the Bouguer-Beer law : -dI = R N ~ J d l , (2) where I is the collimated radiant flux density, R is a correction factor for the finite angle subtended by the receiver, and 1 is distance.Eqn. (2) can be integrated and rearranged in the form In (1011) = RNa,(l,+ 10) (3) where 20 is the unknown reference thickness for which the X channel of the recorder was set to zero, and I, = (I-lo) is the measured distance. RNys and lo were calculated from eqn. (3) and the data by least squares. Nus was then calculated taking R = 0.998, corresponding to the angle of 47.8 min subtended by the receiver, N = 1.205 and the appropriate value of a. It should be noted that in water,s. w. CHURCHILL, G. c. CLARK AND c. M. SLIEPCEVICH 195 and hence in this value of a, h = Xair/nwater = 5460/1.33 = 4105A.N was in turn calculated using the theoretical values of 2.48 and 3.57 for K, for the 0.814 and 1.171 p particles, respectively. The volume fraction of solids x = N7rd3/6 was next calculated from the known particle diameters. The centre-to-centre distance between particles was calculated from the following expression for a rhombohedra1 array : 6 = (J21N)) = (~/3J2x)*d. (4) Since the particles are charged, this arrangement, which gives the maximum possible distance between particles for a given concentration, may be approached as the particle concentration increases to the limit. This limit for 6 = d is N = 2/%/d3 and the cor- responding maximum x is q / 2 / 6 = 0.7405. The computed properties for the initial, undiluted dispersions are given in table 1. TABLE 1 .-PROPERTIES OF UNDILUTED DISPERSIONS 4 P N, particIes/cm3 X 0.814 9.81 x 1011 0.278 1.385 1.171 3.23 X 1011 0.272 1.395 Values for the other traverses were obtained by multiplying the concentration by the corresponding dilution factor. RESULTS The data were correlated in terms of the two-flux model which has been’discussed by Chu and Churchill 5 and others, and successfully used by Larkin and Churchill 6 and others for multiple scattering.In this model the angular distribution of radiation scattered by a single sphere is represented by forward and backward components. The integro- differential equation describing the radiant intensity in a dispersion then reduces to two ordinary differential equations for the forward and backward components of the intensity.The idealized experiment would have consisted of an infinite layer of dispersion with an infinite, collimated source at one face and a totally absorbing surface at the other. A finite source and dispersion of the same diameter with a perfect specular reflector at the circumference would produce the same transmission as the infinite system. The experimental transmission obtained in this investigation would be expected to be somewhat less than in the idealized case because of the finite dimensions of the source and dispersion, and the failure of the dispersion beyond the circumference of the source to act as a perfect reflector. A correction for the net sidewise loss of radiation was therefore incorporated in the two-flux model. The resulting equation describing the forward component 11 and the backward component 12 of the intensity are and where B is the backward scattering coefficient for single scattering and S is the net sidewise scattering coefficient.The boundary conditions are I1 = 1.0 at I = 0, and 12 = 0 at I = Zt where It is the thickness of the dispersion. Solving these equations yields the following expression for the transmission : I,(&> 1 11(0) - cosh [p(Z, + lo)] + q sinh [-(I, + lo)]’ T=-- (7) where p = Nos JS(2B + S ) and q = (B + S)/d S(2 B + S). Values of the parameters p and q and the unknown reference distance lo were deter- mined by least squares on an IBM 650 computer using the method proposed by Scar- borough 7 for non-linear equations. Values of BK, and SK, were then computed from the previously determined values of N and the dilution factors.Although the computed values of BK, were in all cases about 1000 times the values of SK,, the inclusion of S in the model resulted in a distinctly better representation for the data.,196 LIGHT-SCATTERING BY DENSE DISPERSIONS t- i 1 I I 0 I 2 3 4 cell thickness, mm FIG. 3.-Experimental transmissions. 0.814 p particles. I , 0 .,4 !3 WJ *i 3 5 -2 10 -3 10 0 I 2 3 cell thickness, nun FIG. 4.-Experimental transmissions. 1.171 p particles.S . W. CHURCHILL, G . C. CLARK AND C. M. SLIEPCEVICH 197 The experimental transmissions and curves representing eqn. (7) are plotted against 2, for the two particle diameters in fig. 3 and 4. The precision of the data and the excellent representation obtained with eqn.(7) are apparent. The standard deviations for the 26 traverses averaged about 1.2 %. The experimental transmissions are replotted against NK,7rd2Zt/4 in fig. 5 and 6 using values of K, for isolated spheres. For a dilute dispersion, this abscissa corresponds to the cell thickness in mean free paths for scattering; for concentrated dispersions K,, and hence the mean free path, may be somewhat different. Due to compression of the data in this form, only data for selected concentrations and curves for the extreme trans- missions are included. It should be noted that the data for different concentrations N&?rd21/4 FIG. 5.-Effect of particle separation on transmission. 0.814 p particles. S/d: 299 0 1.65 0 1.55 A 1.47 + >O 1.39 cover different ranges of the abscissa ; for example, the data for the most dilute dispersion extend only over the lowest tenth of the abscissa. If there were no optical interference between particles, all data for a given particle size should lie along a single curve.Thus the spread of the data and curves indicates the magnitude of the interference in so far as sidewise losses and other non-idealities in the experiment are negligible or the same from traverse to traverse. The transmission appears to increase and' then to decrease as the particle separation distance is decreased, but the magnitude of the variation is less than f20 % for both particle sizes. A more critical test of interference is provided by fig. 7 in which the product of the coefficients B and Ks is plotted against 6/d for both particle sizes.This plot should be independent of sidewise losses from the cell. In so far as the modified two-flux model represents the physical situation, BK, is the fraction of the geometrically obstructed light which is scattered into the backward hemisphere by a single particle. Since B, S and K, occur in eqn. (7) only as the products BK, and SK,, the separate effects of particle separatio198 LIGHT-SCATTERING BY DENSE DISPERSIONS 10 ,,.f I I I I I I 0 1000 2000 3000 4000 5 0 0 0 6 0 0 0 7( NKsrrd21J4 ? - FIG. 6.-Effect of particle separation on transmission. 1.171 p particles. 8 / d : 3.01 0 1.67 1-56 A 1.48 + 1.39 @ 1.0 1.5 2 0 2 5 30 W FIG. 7.-Effect of particle separation on back scattering parameter. 0, d = 0.814 p ; 0, d = 1.171 p.S .W. CHURCHILL, G . C. CLARK AND C. M . SLIEPCEVICH 199 on B and Ks cannot be deciphered from the data of this experiment. For both particle sizes, BK, appears to be essentially constant down to a 8/d of about 1-7, then to decrease to a minimum, to increase to a maximum and finally to decrease again. The magnitude of this variation is only about j l 0 % and undoubtedly is due in part to experimental error. The uncertainty in the computed values of BK, is greater than the uncertainty in the measurements of transmission and distance, but is difficult to estimate because of the non-linearity of the equations from which BK, is derived. Additional details concerning the equipment, procedures and data are given by Clark.* CONCLUSIONS The modified two-flux model was found to provide an excellent representation for the data. The observed variations in BK, and T with concentration are sur- prisingly small, considering the very small distances separating the particles. The limiting 6/d above which optical interference between particles can be neglected is apparently about 1-7 rather than 5 as postulated by Sinclair.1 Therefore dis- persions of spheres as concentrated as 15 % solids can be used to simulate dilute dispersions without correction for interference between particles. This research was supported in part by National Science Foundation Research Grant G1006. Computer time was donated by the Continental Oil Company. The suggestions of Prof. C. M. Chu, and the assistance of Dr. R. H. Boll, Dr. J. H. Chin, Dr. 13. K. Larkin, Dr. J. A. Leacock, and- Messrs. P. H.! Scott and J. Chen are gratefully acknowledged. 1 Scott, Clark and Sliepcevich, J. Physic. Chem., 1955, 59, 849. 2 Scott and Churchill, J. Physic. Chem., 1958, 62, 1300. 3 Sinclair, Handbook on Aerusuls (U.S. Atomic Energy Corn., Washington, D.C. 4 Gumprecht and Sliepcevich, J. Physic. Chem., 1953, 57, 90. 5 Chu and Churchill, I.R.E. Trans., 1956, AP-4, 142. 6 Larkin and Churchill, J. Amev.!Inst. Chem. Engrs., 1959, 5, 467. 7 Scarborough, Numerical Mathematical Analysis (Oxford Univ. Press, 2nd ed., 1950). 8 Clark, Ph.D. Thesis (Univ. of Michigan, Ann Arbor, Michigan, 1960). 1950), chap. 5-8.