PARTICLE SIZE AND OPTICAL PROPERTIES OF EMULSIONSL EMERSON D. BAILEY, J. BURTON NICHOLS, AND ELMER 0. KFZAEMER
The Experimental Station of E . I . duPont de Nemours and Co., Inc., Wilmington, Delaware RECeiVed June 11, 1938
Numerous efforts have been made to calculate theoretically the dependence of the light-scattering power of suspensions upon the size and refractive index of the particles, the refractive index of the medium, and the wave length of light used. For infinitely small, non-absorbing particles in a non-absorbing medium, Rayleigh’s treatment (9) is generally judged satisfactory, but if the particle size approaches the order of magnitude of the wave length of light, or if the particle is colored, the theoretical calculation of scattering becomes much more difficult and a matter of approximation (1, 4, 5, 6, 8, 9, 11). On the other hand, the empirical correlation of experimental data on light scattering with relevant physical factors has on the whole been unsatisfactory, owing to insufficiently accurate information on particle size. As a rule, investigators have had to be satisfied to assume their suspensions to contain particles of uniform size, although as a matter of fact, any artificial and practically all natural suspensions contain particles of definitely non-uniform size. Fortunately, the Svedberg ultracentrifuge can under favorable conditions provide the required information, but rather involved calculations are required to extract the desired relations between light scattering and particle size, owing to the fact that the ultracentrifuge does not give directly a particle-size distribution. Specifically, a particle-size distribution may be expressed as a relationship between dcldr and r, where dc/dr is proportional to the weight of particles having a radius between r and r dr. The Ultracentrifuge, however, gives the relation between d(kc)/dr and T , where k is the apparent absorption coefficient of a suspension of particles of radius r. For a suspension of colorless particles in a colorless medium, the “absorption” is of course entirely due to scattering, and the “absorption coefficient” in such cases provides a measure of the light-scattering
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Presented a t the Thirteenth Colloid Symposium, held a t St. Louis, Missouri, June 11-13,1936. Contribution No. 172 from the Experimental Station of E. I. duPont de Nemours and Co., Inc. 1149
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E. D. BAILEY, J. B. NICHOLS, AND E . 0. KRAEMER
power of the suspension. The relation between d(kc)/dr and T w e call a “weight-optical distribution,” and since k in general varies with r, the “weight-optical distribution” normally is not identical with the true “weight distribution” curve ( 7 ) . In this paper, a method will be described for calculating, with the aid of a simple mechanical product-integraph, the relationship between the “absorption Coefficient” k and the radius from a series of weight-optical distribution curves. In addition, the light-scattering powers of a series of emulsions of constant particle-size distribution but different refractive index relations are reported i n terms of apparent absorption coefficient k , and a general relation connecting the apparent absorption coefficient with particle size, the refractive indicea of particles and medium, and the wave length is dedured by means of the product-integraph. THE PRODUCT-INTEGRAPH
AND ITS USE
The most direct way of determining the relation between absorption coefficient (as a measure of light scattering) and radius for a certain kind of particle in a medium of specified refractive index is by the analysis of a series of w4ght-optical distribution curves for suspensions with different particle sizes. If a given weight-optical distribution be represented by the equation d(kc)/dr = fn(r) and t h r variation of the absorption coefficient k with r be represented by
Ilk
=
F(r)
then, in general, it follows that
Each weight-optical distribution is represented by a particular f ( r ), and our task is to calculate F(T)from the series of simultaneous integral equations equal in number to thc experimentally determined weight-optical distributions. Since the solution of such equations with the required accuracy is impractical by algebraic means, a mechanical product-integraph bascd upon the principles of a two-dimensional moment balance was devised for the purpose. The inonlent balance was simply a drawing board supported on a fulcrum and carrying a system of coordinates. The r-axis (or abscissa) coincides JTith the fulcrum. The function. are divided into a number of equal areas (e.g., 10) and arc represented by a series of equal weights (for instance, 10 g. each). Each weight is placed a t the mean r value for the area it represents. On the left-hand side of the board, 10 cm. from the
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fulcrum, a weight of 100 g., representing unit moment, is placed. On the right-hand side of the board, perpendicular to the fulcrum, F ( r ) is measured along the ordinates. When the weights representing any f ( r ) are placed a t their appropriate r-positions and their ordinates adjusted until the board balances, the corresponding equation of the set is satisfied, and F ( r ) is the curve drawn through the positions of the weights. Each of the experimental functions f ( r ) is put on the moment balance separately, and F ( r ) is adjusted until, by a process of trial and error, an F(r) curve is found which will satisfy all of the equations. The resulting curve represents the relation between l / k and radius, which of course can be readily converted, if desired, into a curve of k versus radius. With a moment balance sensitive to differences in moments of 0.3 per cent, an F(r) curve can be found that satisfies the experimental data to 3 to 4 per cent. A specific example of the use of this product-integraph for determining the relationship between “absorption coefficient” and particle radius has previously been presented (3), and a more detailed description of the theory of the product-integraph is published elsewhere (2). THE EFFECT OF REFRACTIVE INDEX AND PARTICLE SIZE UPON ABSORPTION COEFFICIENT (LIGHT SCATTERING)
Inasmuch as the light-scattering efficiency of a particle is affected by particle shape and is complicated by double refraction in solid particles, emulsions were chosen as representing ideal conditions. A stock emulsion of Nujol in 76 per cent glycerol containing 1 per cent of Castile soap was prepared and thoroughly homogenized. From this stock emulsion, a series of emulsions of the same particle-size distribution but with markedly different light-scattering properties was prepared by reduction of the refractive index of the medium by dilution with various proportions of glycerol and water. Six emulsions with the dispersion medium varying from 15 to 76 per cent of glycerol were so prepared, corresponding to a range in refractive index of the medium from 1.359 to 1.444 a t a wave length of 444 millimicrons, where the absorptions and weight-optical distributions were measured. At the same wave length, the refractive index of the Nujol is 1.488. The weight-optical distribution curves of the six emulsions were determined with the Svedberg low-speed ultracentrifuge in ,the usual manner and are presented in figure 1. The areas of these curves are equal to the light absorptions, measured also in the ultracentrifuge within an aperture of about 4”,with a parallel beam of incident light (see table 1). The “absorption coefficients” are defined in the conventional way and refer to a suspension containing 0.01 cc. of particles per 100 cc. of suspension. In contrast to the case described above, the F ( r ) function is different for
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E. D. BAILEY, J. B. NICHOLS, AND E. 0. ERAEMER
each of these emulsions, owing to the differences in the refractive indices, but the function relating concentration and radius (i.e., dc/dr versus T ) is the same. It was found that the differences in light-scattering efficiency
FIG.1. Weight-optical distribution curves of series of emulsions with the same particle-size distribution but different refractive index of the medium.
TABLE 1 Absorption coe.ficients of Nujol emulsions ~
~
EMULSION
ABSORPTION COEFFICIENT
NELA -2B -2c
0 0293 0 0656 0 110
,
-
I
EMULSION
ABBORPTION COEFFICIENT
NE-ID -2E -2F
0 162 0 236 0 300
as determined by the refractive indices could be reduced to a common denominator, as it were, by use of the dimensionless factor
m2 - 1 no where m is the ratio of the refractive index of the particle to that of the medium, no is the refractive index of the medium, h is the wave length of the light used in vacuum, no/h is therefore the wave length of the light used in the medium, and r is the particle radius. This factor we call the
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“relative optical radius.” The abscissa values of the weight-optical curves were converted to a ‘lrelative optical radius” basis. The ordinates of the weight-optical distributions were in turn multiplied by the corresponding radii. The resulting curves (figure 2) could then be treated in the same fashion as the f(r) functions discussed in the section on the product-integraph and its use, and a curve obtained with the prod-
(&).? FIG.2. Projected area concentration in per cent versus relative optical radius
FIG.3. Generalized curve relating the absorption constant of a suspension with the concentration, particle size, refractive indices of particle and medium, and the wave length.
uct-integraph relating l / ( k r ) and the “relative optical radius.” Figure 3 presents kr versus “relative optical radius.” k in this case refers to the absolute L‘absorption coefficient” of a suspension containing 0.01 cc. of particles per 100 cc. of suspension measured under the specified conditions of illumination and angle of collection. kr we refer to as the “absorption’ coefficient per unit projected area.” THQ JOURNAL OB PEYBICAL C m m I s T a Y , VOL. 40, NO.
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E. D. BAILEY, J. B . NICHOLS, Ah-D E. 0. XRAEMER
Presumably, figure 3 should describe the light absorption for the indicated range of relative optical radius, regardless of the values of the individual quantities entering into the relative optical radius. Experimentally, however, the actual range in values covered is as follows:
r = 20 to 270 mp m = 1.03 t o 1.94 no = 1.359 t o 1.444 h = 444 mp Further work wjll be required before we can be sure that the method here employed for correlating the various factors involved is really as generally valid as it now appears.
FIG.4. The veight-distribution curve for the series of emulsions, as calculated f i o n i the neight-optical curves of figure 1 and the generalized curve of figure 3.
Figure 4 presents the weight-distribution curves for the emulsions used, as calculated from the experimental weight-optical distribution curves by means of the generalized curve of figure 3. The relation of these experimental results to theoretical calculations of light scattering will be discussed in a later publication. REFERENCES (1) ANDREEV: J. Gen. Chem. U. S. S. R. 6, 529 (1935)
(2) BAILEY:To appear in Rdv. Sci. Instruments.
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(3) BAILEY,NICHOLS,AND KRAEMER: Paper presented a t the Eighty-eighth Meeting of the American Chemical Society, held in Cleveland, Ohio, September, 1934. (4) BLUMER:Z. Physik32,119 (1925);38,304,920(1926). (5) CASP~RSSON: Kolloid-Z.60,151 (1932);66, 162 (1933). (6) MIE: Ann. Pbysik [4126, 377 (1908). J. Phys. Chem. 36,328 (1932). (7) NICHOLS,BAILEY,AND KRAEMER: (8) PUTZEYS AND BROSTEAUX: Trans. Faraday SOC.31,1314 (1935). (9) RAY:Proc. Indian Assoc. Cultivation Sci. 7 , 1 (1921); 8,23 (1923). Phil. Mag. [5] 47,375 (1899). (10) RAYLEIGH: (11) SHOULEJKIN: Phil. Mag. 48,2,307 (1924).
