THE UNIVERSITY OF MICHIGAN' INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING LIGHT-SCATTERING BYEVERY DENSE MONODISPERSIONS OF LATEX PARTICLES Stuart W Churchill Go C. Clark Co M. Sliepcevich August, 1960 IP-449

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TABLE OF CONTENTS Page LIST OF FIGURES.......................... oiii INTRODUCTION........................... 1 EXPERIMENTAL e........................................... 2 Apparatus.........................................o. 2 Materials *.. *.....................................o Procedure.,............................................ 5 Determination of Particle Concentration and Separation... 6 RESULTS................................................ 7 CONCLUSIONS.................... 15 ii

LIST OF FIGURES Figure Page 1 Schematic Diagram of Collinating System.................. 3 2 Experimental All............................... 4 3 Experimental Transmissions, 0.814 i Particles............. 10 4 Experimental Transmissions, 1.171 t Particles............. 11 5 Effect of Particle Separation on Transmission, 0.814 A Particles................................ 12 6 Effect of Particle Separation on Transmission, 1.171 t Particles.................1....................... 13 -7 Effect of Particle Separation on Back Scattering Parameter 14 iii

INTRODUCTION The intensity of radiation within or at the boundaries of a dispersion of uniformly sized, nonabsorbing 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 conditions. The mean free path for scattering s = l/Nas = 4/NKsgd2 (1) where N is the number qf spheres per unit volume, as is the scattering cross section, Ks 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 = td/X, where X 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 s me n and a are established as illustrated by Scott and co-workers.(1, (3) Sinclair stated without documentation that optical interference between particles 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 of 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 millimeteres, using a beam of millimeter waves. (iii) Measurement of the transmission of a beam of monochromatic light through dense dispersions with particle concentration as a variable. Method (iii) was chosen because of its comparative simplicity and the more direct applicability of the results. -1

EXPERIMENTAL APPARATUS The equipment consisted Of a source and collimating.sytem, a receiver anid amplifying unit, and a cell and traversing mechanism, all located, in a dehumidified dark room at 180C, The source was a 50-candlepotwer, auto-I~eadlight bulb operated with a regulated power supply. The light beam was monochromaticized with interference filters, yielding a transmission of 45% at 5460 + 1500A and a band width of 120-140~A at the 22o5% transmission points, The collimating system is shown.in Figure 1 Diaphragms D1 and D2 reduced the stray 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 mm in diameter and 191.5 mm in focal length. With a 1/16-mm pinhole P the final beam had a dimaeter of 32 mm and a divergence of only 14.2 mino Such a high degree of collimation was rDt 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 tenstage 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 -= 4400 + 2500A with 10% of the maximum response at X = 3250 + 250~A and 6175 + 275~A. Voltage from a variable 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 Qresistor. The amplified signal was fed to the Y channel of an X-Y recorder. The cell is shown in Figure.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 follower 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 -2

Ds COLLIMATED LAMP LI L2 L3 Figure 1. Schematic Diagram of Collinating System

FIXED PART OF CELL -/,,PHOTOMULTIPLIER MOUNTING SCREWS,/ ^ o38mm i. D X 114mm. 1 4 HIGH _,. MOUNTING RING G ^ SET SCREW MOUNTING GLASS WINDOW - /BRACKETS MOVABLE PART!IA GLASS WINDOW OFF CELL CAPILLARY "- DRAIN TUBE 102mm I.D. X 63mm HIGH ALL PARTS ARE PLEXIGLAS EXCEPT GLASS WINDOWS Figure 2. Experimental All

-5up 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 diameter and 914 mm in focal length, was lo ated at the outlet window of the cello A pinhole.in the bahk of the camera opened to an opal glass optically coupled to the window of the photomultipliero This receiver system was surrounded 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 0.8144t with a standard deviation of 0.011O; the other, a mean diameter of 1.171 with a standard deviation of 0,0131. At X = 5460~A (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 1o for calculation of the transmissiono 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 quantity 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.

-6DETERMINATION OF PARTICLE CONCENTRATION AND SEPARATION Samples of the dispersion were withdrawn at theend and at two intermediate stagesof'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: - dIl RNaSIldl. (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. Eqno (2) can be integrated and rearranged in the form in (Io/I) = RNas(lm + lo) (3) where 1o is the unknown reference thickness for which the X channel of the recorder was set to zero, and 1m - ( - lo) is the measured distance, RNas and lo were calculated from Eqn. (3) and the data by least squares. Nas.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.o It should be noted that in water, and hence in this value of a, X = Xir/nwater = 5460/1.33 = 41050Ao N was in turn calculated using the theoretical values of 2,48 and 3 57 for Ks for the 0.814 and 1l171p particles, respectively. The volume fraction of solids x NJd3/6 was next calculated from the known particle diameters, The centre-to-centre distance between particles was calculated from the following expression for a rhombohedral array: 5 - (4/N)113 - (E/3 ix)~3d 2) 6 = 7/N) l/ d (T/5 fx1/5 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. = d is N =\n/d3 and the corresponding maximum x is I2/6 0.7405. The computed properties for the initial, undiluted dispersions are given in Table l. Values for the other traverses were obtained by multiplying the concentration by the corresponding dilution factor.

-7RESULTS The data were correlated in ter]s sof the two-flux model which has been discussed by Chu and Churchill 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 integrodifferential equation describing the radiant intensity in a dispersion then reduced 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 othera A finite source and dispersion of the same diameter with a perfect specular reflector atthe 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 re.sulting equation describing the forward component Il and the backward component 1I of the intensity are - dI1 = (BI1 + SIl - BI2)Nosdl (3) and dI2 = (BL2 + SI2 - BIl)Nadl (6) where B is the backward scattering coefficient for single scattering and S is the net sidewise scattering coefficient0 The boundary conditions are I1 =1.o0 at I 1 0, and 12 = 0 at 1 = 1t where lt is the thickness of the dispersiono S.olving these equations yields the following expression for the transmission: T Il(lt) = (7) Il(0) ch[p(lm+lo)] + q sh[p(lm+lo)] where p = NasS(2B+S and q = (B+S)/ S(2B+S).

-8" Values of the parameters p and q and. the unknown reference distance 1o were determined by least squares on an IBM 650 computer using the method proposed by Scarborough(7) for non-linear equations. Values of BK5 and Sks were then computed from the previously determined values of N and the dilution factors, Although the computed values of BKs were in all cases about 1000 times the values of SKs, the inclusion of S in the model resulted in a distinctly better representation for the datao The experimental transmissions and curves representing Eqno (7) are plotted against 1 for the two particle diameters in Figures 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 1o2%o The experimental transmissions are replotted against NKstd21.t/4 in Figures 5 and 6 using values of Ks for isolated spheres, For a dilute dispersion, this abscissa corresponds to the cell thickness in mean free paths for scattering; for concentrated dispersions Ks, 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 transmissions are included0 It should be noted that the data for different concentrations cover different ranges of the abscissa; for example, the data for the most dilute dispersion extend only over the lowest tenth of the abscissa0 If there were no.optical interference be tween particles, all data for a given particle size should lie along a single curve0 Thus the spread 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 insofar as sidewise losses and other non.-idealities in the experiment are negligible or the same from traverse to traverseo 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 + 20% for both particle sizeso A more critical test of interference is provided by Figure 7 in which the product of the coefficients B and Ks is plotted against 6/d for both particle sizeso This plot should be independent of sidewise losses from the cell.o Insofar as the modified two-flux moded represents the physical situation, BKs is the fraction of the geometrically obstructed light which is scattered into the backward hemisphere by a single particle, Since B, S and Ks occur in Eqno (7) only as the products a BKs and SKS, the separate effects of particle separation on B and Ks cannot be deciphered from the data of this experiment0 For both particle sizes BKs appears to be essentially constant down to a 6/d of about 1,7, then to decrease to a minimum, to increase to a maximum and finally to decrease againo The magnitude of this variation is.only about + 10% and undoubtedly is due in part to

-9experimental error, The uncertainty ih the computed values of BKs: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 derived4 Additional details concerning the equipment, procedures and data are given by Clark (8) TABLE I Properties of Undiluted Dispersions dj~,u N, particles/cm3 x 5d_ 0,814 9.81 x 1011 0.278 1.385 14171 3.23 x 1011 0.272:1.395

-10I03 S~ 0.125 0 0"1. 16 lo3L 0.200.. ". 250 0.$00 ""`o0.357 ( ^- ^ ^0.417 0.500 -10 ~'~'~~~~~~~~0.588 0.714 0,.833 1.000 I I I I 0 I 2 3 45 Figure 3. Experimental Transmissions, 0.814 p Particles

~~~~~~~-1~~-~~i~~~~~ ~DILUTION 10 FACTOR. 100 0124 0.125,~ -~"-e 0.167 H" 0.200 ~I0_, "-0. 250 ~ 0.357 0.588 -I - I I I 0.714 0.833 1.000 0 I 2 3 4 5 CELL THICKNESS. mm. Figure 4. Experimental Transmissions, 1.171 p Particles

-122.99 o 1.65 0 1.55 A 1.47 + 1.39 * I0" z 0 0) C~, (n "t \A 0 1000 2000 3000 4000 5000 6000 7000 NK 77 d2/4 Figure 5. Effect of Partiole Separation on Transmission, 0.814 Particles

-13S/d 3.01 0 1.67 a 1.56 A 1.48 + 1.39 * I0 z I00oir. 10 0 1000 2000 3000 4000 5000 6000 7000 N Ksr'd2l/4 Figure 6. Effect of Particle Separation on Transmission, 1.171 t Particles

10 9 / d= 1.171/.L 88~/ /')- "' - "/ ^ I 0 6 x 7d x 7" d= 0.8141 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.0 E/d Figure 7. Effect of Particle Separation on Back Scattering Parameter

CONCLUSIONS The modified two-flux model was found to provide an excellent representation for the datao The observed variations in BKs and T with concentration are surprisingly small, considering the very small distances separating the particles. The limiting 5/d above which optical interference between particles can be neglected is apparently about 107 rather than 5 as postulated by Sinclairo(1) Therefore dispersions of spheres as concentrated as 15% solids can be used to simulate dilute dispersions without correction for interference between particleso This research was supported in part by National Science Foundation Research Grant G10006. Computer time was donated by the Continental Oil Company. The suggestions of Profo Co M. Chu, and the assistance.of Drso Ro H. Boll, J. Ho Chin, Bo Ko Larkin and Jo Ao Leacock, and Messrso P, Ho Scott and Jo Chen are gratefully acknowledged, 1 Scott, Clark and Sliepcevich, J. Physic. Chem., 1955, 59, 849. 2 Scott and Churchill, Jo Physic. Chem., 1958, 62, 13000 3 Sinclair, Handbook on Aerosols (UoSo Atomic Energy Comm., Washington, Do C., 1950), Chaps. 5-8. 4 Gumprecht and Sliepcevich, J. Physic. Chemo, 1953, 57, 90. 5 Chu and Churchill, I, R. E. Trans. 1956, AP-4, 142. 6 Larkin and Churchill, J. Amero Inst. Chem. Engrs., 1959, 5, 467. 7 Scarborough, Numerical Mathematical Analysis, (Oxford Univ. Press. 2nd edo, 1950). 8 Clark, Ph. Do Thesis (Univ. of Michigan, Ann Arbor, Michigan, 1960). -15

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