MILTON LAIIKIN taneous Equations, Complex CLSME2." Since 'a" is a function of wavelength, this matrix may be solved for various values of wavelength. This has been done on an IBM 704. The generalization to the case of arbitrary thickness relationships between films may at times be convenient for theoretical purposes, but has little practical ad- vantage. It is more convenient to evaporate some multiple of 4 wavelength since this may readily be done by visual observation. If most of the film layers are 4-wave thickness, and if a one-wave film is desired, then it is only necessary in this program to call for four identical 4-wave films. Separate computation of transmission and reflection provides a convenient check on the accuracy of results. As can be seen from the following example for a seven- film case and in Table I, transmission plus reflection equals one. 0 O. S. Heavens, Optical Properties of Tlin Solid Films (Aca- demic Press, Inc., New York, 1955). JOURNAL OF THE OPTICAL SOCIETY OF AMERICA A seven-film case: Nt 2, N, 1. N2 2.- N3 1., N 4 2. N 5 1.1 N 6 2. N 7 1. N 8 2., N 9 1. 750 A 517 glass 37 zinc-sulfide film 38 magnesium-fluoride film 37 38 37 38 37 517 glass. LIMITATIONS One of the limitations of the method shown in Table I lies in the fact that the index of refraction for thin films 3 is not accurately known over a wide wavelength range. The preceding material assumes normal incidence. At other angles of incidence the equations become quite complicated. Likewise, they are for nonabsorbing media. VOLUME 50, NUMBER 7 JULY, 1960 Light Scattering of Monodispersed Polystyrene Latexes* MILTON KERKER AND EGON MATIJEVI6 Clarkson College of Technology, Potsd am, New York (Received February 12, 1960) The intensity of light scattered by four monodispersed polystyrene latexes has been measured at 450, 90°, and 1350. The polarization ratios and dissymmetries have been compared with newly computed Mie-theory functions. The range of validity of the Rayleigh-Gans theory for mnz= 1.20 has been discussed. A discrepancy between the experimental and theoretical values for the dissymmetry has been observed and accounted for in terms of secondary scattering. I. INTRODUCTION POLYSTYRENE latexes, consisting of monodis- persed, spherical particles whose sizes have been carefully determined by electron microscopy,' have been used in a number of light-scattering studies. Dandliker' has determined the particle size of a latex from the position of the first minimum in an angular- dependence curve. Heller and his associates 3 - 6 have carried out extensive calculations of Mie scattering functions and have studied the turbidity and 90° scattering for several polystyrene latexes. Goring, Senez, Mlelanson, and Huque 7 have measured the dissymmetry of a single latex at two wavelengths. * Supported in part by contract with U. S. Army Chemical Corps. 1 E. B. Bradford and J. W. Vanderhoff, J. Appl. Phys. 26, 864 (1955). Earlier references are given in this paper. 2 W. B. Dandliker, J. Am. Chem. Soc. 72, 5110 (1950). 3 R. M. Tabibian, W. Heller, and J. N. Epel, J. Colloid Sci. 11, 195 (1956). 4W. Heller and R. M. Tabibian, J. Colloid. Sci. 12, 25 (1957). 6 W. Heller and T. L. Pugh, J. Colloid. Sci. 12, 294 (1957). G R. M. Tabibian and W. Heller, J. Colloid. Sci. 13, 6 (1958). 7 D. A. I. Goring, M. Senez, B. Melanson, and M. M. Huque, J. Colloid. Sci. 12, 412 (1957). LaMer and Plesner 8 have observed the higher-order Tyndall spectra of a number of latexes, and one of us 9 has correlated their observations with the requirements of the Mie theory. Alfrey, Bradford, Vanderhoff, and Oster" 0 have measured the dissymmetry of several latexes and have calculated the particle sizes using the Rayleigh-Gans theory. Bateman, Wenick, and Eshler' have determined the particle size and concentration of four latexes from spectrophotometric data, using the Mie theory. Some of this light-scattering work has been carried out in order to provide additional substantiation of light-scattering theory using the known values of the latex particle sizes. In other cases, the light-scattering theory, combined with the experimental data, has been used to provide independent confirmation of the latex particle size. We have undertaken our work with another view in 8 V. K. LaMer and J. W. Plesner, J. Polymer Sci. 24,147 (1957). 9 M. Kerker, J. Polymer Sci. 28, 429 (1958). 10 T. Alfrey, Jr., E. B. Bradford, J. W. Vanderhoff, and G. Oster, J. Opt. Soc. Am. 44, 603 (1954). 11 J. B. Bateman, E. J. Weneck, and D. C. Eshler, J. Colloid. Sci. 14, 308 (1959). 722 Vol. 50
Transcript
MILTON LAIIKIN
taneous Equations, Complex CLSME2." Since 'a" is afunction of wavelength, this matrix may be solved forvarious values of wavelength. This has been done onan IBM 704.
The generalization to the case of arbitrary thicknessrelationships between films may at times be convenientfor theoretical purposes, but has little practical ad-vantage. It is more convenient to evaporate somemultiple of 4 wavelength since this may readily be doneby visual observation. If most of the film layers are4-wave thickness, and if a one-wave film is desired, thenit is only necessary in this program to call for fouridentical 4-wave films.
Separate computation of transmission and reflectionprovides a convenient check on the accuracy of results.As can be seen from the following example for a seven-film case and in Table I, transmission plus reflectionequals one.
0 O. S. Heavens, Optical Properties of Tlin Solid Films (Aca-demic Press, Inc., New York, 1955).
One of the limitations of the method shown in Table Ilies in the fact that the index of refraction for thin films3
is not accurately known over a wide wavelength range.The preceding material assumes normal incidence. At
other angles of incidence the equations become quitecomplicated. Likewise, they are for nonabsorbing media.
VOLUME 50, NUMBER 7 JULY, 1960
Light Scattering of Monodispersed Polystyrene Latexes*MILTON KERKER AND EGON MATIJEVI6
Clarkson College of Technology, Potsd am, New York(Received February 12, 1960)
The intensity of light scattered by four monodispersed polystyrene latexes has been measured at 450, 90°,and 1350. The polarization ratios and dissymmetries have been compared with newly computed Mie-theoryfunctions. The range of validity of the Rayleigh-Gans theory for mnz= 1.20 has been discussed. A discrepancybetween the experimental and theoretical values for the dissymmetry has been observed and accounted forin terms of secondary scattering.
I. INTRODUCTION
POLYSTYRENE latexes, consisting of monodis-persed, spherical particles whose sizes have been
carefully determined by electron microscopy,' havebeen used in a number of light-scattering studies.Dandliker' has determined the particle size of a latexfrom the position of the first minimum in an angular-dependence curve. Heller and his associates3 -6 havecarried out extensive calculations of Mie scatteringfunctions and have studied the turbidity and 90°scattering for several polystyrene latexes. Goring,Senez, Mlelanson, and Huque7 have measured thedissymmetry of a single latex at two wavelengths.
* Supported in part by contract with U. S. Army ChemicalCorps.
1 E. B. Bradford and J. W. Vanderhoff, J. Appl. Phys. 26, 864(1955). Earlier references are given in this paper.
2 W. B. Dandliker, J. Am. Chem. Soc. 72, 5110 (1950).3 R. M. Tabibian, W. Heller, and J. N. Epel, J. Colloid Sci. 11,
195 (1956).4W. Heller and R. M. Tabibian, J. Colloid. Sci. 12, 25 (1957).6 W. Heller and T. L. Pugh, J. Colloid. Sci. 12, 294 (1957).G R. M. Tabibian and W. Heller, J. Colloid. Sci. 13, 6 (1958).7 D. A. I. Goring, M. Senez, B. Melanson, and M. M. Huque,
J. Colloid. Sci. 12, 412 (1957).
LaMer and Plesner8 have observed the higher-orderTyndall spectra of a number of latexes, and one of us9
has correlated their observations with the requirementsof the Mie theory. Alfrey, Bradford, Vanderhoff, andOster"0 have measured the dissymmetry of severallatexes and have calculated the particle sizes using theRayleigh-Gans theory. Bateman, Wenick, and Eshler'have determined the particle size and concentration offour latexes from spectrophotometric data, using theMie theory.
Some of this light-scattering work has been carriedout in order to provide additional substantiation oflight-scattering theory using the known values of thelatex particle sizes. In other cases, the light-scatteringtheory, combined with the experimental data, has beenused to provide independent confirmation of the latexparticle size.
We have undertaken our work with another view in
8 V. K. LaMer and J. W. Plesner, J. Polymer Sci. 24,147 (1957).9 M. Kerker, J. Polymer Sci. 28, 429 (1958).10 T. Alfrey, Jr., E. B. Bradford, J. W. Vanderhoff, and G.
Oster, J. Opt. Soc. Am. 44, 603 (1954).11 J. B. Bateman, E. J. Weneck, and D. C. Eshler, J. Colloid.
Sci. 14, 308 (1959).
722 Vol. 50
LIGHT SCATTERING
mind, namely, to check the accuracy of our light-scattering equipment. For our theoretical functions, wehave used the Mie theory, which is as firmly establishedas the electromagnetic theory of light itself. Theparticle size of polystyrene latexes, determined byelectron microscopy, is also well established. Thus,they are ideal systems for calibrating light-scatteringequipment, provided the theoretical functions areavailable and certain pitfalls which will become ap-parent in the course of this paper are avoided.
We have measured the intensity of both polarizedcomponents of the light scattered by four polystyrenelatexes at 450, 1350, and 90°. The polarization ratiosand dissymmetries calculated from these data werethen compared with newly computed functions.
II. EXPERIMENTAL
Material
Four different polystyrene latex samples given to usby the Dow Chemical Company, having particlediameters 1380 A (standard deviation 62 A), 2640 A(60 A), 5110 A (74 A), and 5570 A (108 A), respectively,were utilized. Before use, the original samples werediluted about 100 times and centrifuged in a ServallUltraspeed centrifuge. The supernatant liquid wasfiltered through an ultrafine glass filter (Pyrex UF)under pressure of helium. The concentration of thesolid phase in the filtered solution was determined byweighing the residue on an Ainsworth microbalanceafter evaporation of an aliquot of the suspension invacuo. For light-scattering measurements, the filteredstock solution was further diluted with ultrafilteredwater. Diluted solutions were always carefully checkedfor the presence of dust particles.
The diameter of the latex particles was checked byelectron microscopy. The samples were prepared byevaporation of a drop of dilute suspension in vacuofrom a collodion-coated steel grid. The electron micro-scope was an RCA-EMT instrument. The calibrationof the instrument was performed by photographicexposure of a silica replica of a 30 000-line/in. grating.Examination of the electron micrographs showed thesame relative diameters for all four latexes as thosegiven by the Dow Laboratory. The absolute diameters,however, were somewhat smaller in all cases. Sinceindependent exposures of grating and samples cancause errors in absolute particle size determination,'we have used for calculation the values found byelectron microscopists of the Dow Physical ResearchLaboratory.
Light-Scattering Measurements
Light-scattering measurements were performed withthe Brice-Phoenix Photometer, 1000 series. Before use,the alignment of the instrument was checked and foundsatisfactory. However, the polaroids proved to bedeficient in two ways: (a) when crossed they did not
extinguish the light completely, and (b) they were notproperly oriented with respect to the horizontal plane.(It was noted that the orientation of one polaroid wasoff by more than 60). We replaced the polaroids by twoGlan-Thompson prisms. They were glued into theinner part of a holder consisting of two concentricbrass cylinders, which could be rotated with respect toeach other and then fixed at a desired orientation. Thisarrangement enabled us to orient the prisms veryaccurately, using the reflected beam at Brewster'sangle from a precision glass prism.
Two different light-scattering cells were used in theseexperiments. One was the Brice-Phoenix C-101 cylin-drical cell with flat entrance and exit windows. Whenthe back wall of this cell was painted black it showed nospurious reflections, and when tested with Ludoxsuspensions showed no 135°-45° dissymmetry.
The second cell was a square brass cell, built in ourLaboratory, which we designed primarily for light-scattering measurements of aerosols. The cell had twomounted windows (at 0 and 900) made of opticalsilica glass and a Rayleigh horn in the 1800 position,and therefore could be used only at 900. The data usingboth cells at 900 for the same suspension were in goodagreement.
The latex suspensions were always prepared in anumber of dilutions. Upon continued dilution, a regionof concentration was reached where the ratio of thescattered intensity to concentration remained constant,and these values were used in further calculations.
The light-scattering intensities were measured at45, 900, and 1350 for both the 436 mg and the 546 mglines of the mercury spectrum. Only the latter dataare utilized in this work. Altogether, nine readings weretaken for each latex at each angle and wavelength, viz.,all possible combinations of the incident and scatteredbeams unpolarized, polarized horizontally, and polarizedvertically. Thus, a quantity such as the 135°-45°dissymmetry of the vertically polarized componentcould be calculated in a number of ways, viz., with thepolarizer in the vertical position and the analyzer out,with the polarizer out and the analyzer in the verticalposition, and with both polarizer and analyzer in thevertical position. The dissymmetries and polarizationratios calculated from these various data were in goodagreement, and the results presented below are averagesof measurements of this type.
For the dilute latex suspensions and with crossedGlan-Thompson prisms, no significant signal wasobtained. This indicated the absence of depolarization,as is required for Mie scatterers.
III. COMPUTATIONS
The computation of light-scatteringfunctions from theM\ie theory,2 which is the general theory of scatteringby spherical particles, is quite tedious, and the number
12 G. Mie, Ann. Physik 25, 377 (1908).
July 1960 723
M. KERKER AND E. MATIJEVIC
TABLE I. Functions of partial derivatives of Legendre polynomialsfor the arguments cos45' and cos135'.
of published tables is still small.'3 Gumprecht andSliepcevich14 and Pangonis, Heller, and Jacobson'- havecalculated light-scattering functions for the index ofrefraction corresponding to that for polystyrenesuspended in water and irradiated with light of wave-length 5461 A, viz., fi= 1.206. The tables of Gumprechtand Sliepcevich cover large values of the size parametera, while those of Pangonis, Heller, and Jacobson coverthe range of small values in which we are here interested,since they correspond to the sizes of the polystyrenelatex particles; a= 27rr/X, where r is the particle radiusand X the wavelength of the light in the medium.
We had studied the scattered light at angles ofobservation 450, 900, and 135°, and therefore neededthe angular distribution functions or intensity functions,i, and i2 , for these angles. These are defined by
cc
i| jj*|2.. E {An7rn+Bn[x7rn -(1-x2)7r,]} 2 (1)n=1
i2 Ii 2* 12= E {A [x1r 0- (1-x 2)7>r.1]+Bn) 12. (2)n=1
An and Be are the light-scattering functions tabulatedby Pangonis, Heller, and Jacobson. The other functions
13H. C. Van de Hulst, Light Scattering by Small Particles (JohnWiley & Sons, Inc., New York, 1957).
14 R. 0. Gmprccht and C. M. Sliepccvich, ight ScatteringFunctions for Spherical Particles (University of Michigan Press,Ann Arbor, Michigan, 1951).
1' W. J. Pangonis, W. Heller, and A. W. Jacobson, Table of LightScattering Functions for Spherical Particles (Wayne State Univer-sity Press, Detroit, Michigan, 1957).
in the preceding equations are defined as follows:
7n =0Pn(x)/dx7T,_ = d2Pn (X)/d 2
x= cosy,
(3)
(4)
(5)
where P(x) is the Legendre polynomial of degree nand y is the angle between the direction of propagationof the scattered light and the reversed direction ofpropagation of the incident beam.
Values of the functions of the partial derivatives ofthe Legendre polynomials for y=90' were obtainedfrom the tables of Gumprecht and Sliepcevich.6 Thosefor 1350 and 450 were computed by us with the aid ofthe following recursion formulas:
and the values ro= , r = 1, ro'=0, rl'=0. The compu-tations were carried out independently by each author,and the results compared. These results, which can beused to calculate intensity functions at 1350 and 450for any available scattering functions, are presentedin Table I.
Intensity functions were then calculated from thefunctions in Table I and the light-scattering functionsof Pangonis, Heller, and Jacobson, and these are givenin Tables II-IV. The largest value of a was chosen asseven, since this corresponds to the largest size ofmonodisperse polystyrene latex particles which we hadavailable. Ashley and Cobb7 have recently presentedintensity functions for refractive index, m= 1.20 forvarious values of y and a. We duplicate their effortfor a= 1, 2, 3, 5 at y= 900. Our results differ signifi-cantly from theirs for i2 at a= 1 and for i at a= 3. Thephysical meaning of the intensity function, i is givenas follows. When a particle is illuminated by plane-polarized light of unit intensity whose electric vectoris perpendicular to the scattering plane, the intensityof scattered light per unit solid angle, polarized in thesame sense, is given by 2i,/8r 2. The correspondingquantity for horizontally polarized light is given byX
2 i2 /87r2 . R(i,*), I(ij*), R(i 2 *), I(i 2 *) are the real andimaginary parts of the complex amplitude functions,il* and i2 *, and are necessary in order to define thephase relations of the scattered light, e.g., in order tocalculate the Stokes parameters. 8 ,9
Some time after completion of the foregoing compu-tations, we had occasion to program for the computationof intensity functions by the IBM 704 computer. Wewere then able to obtain machine-computed functions
16 R. 0. Gumprecht and C. M. Sliepcevich, Functions of PartialDerivatives of Legendre Polynomials (University of MichiganPress, Ann Arbor, Michigan, 1951).
17 L. E. Ashley and C. M. Cobb, J. Opt. Soc. Am. 48, 261 (1958).18 F. Perrin, J. Chem. Phys. 10, 415 (1942).19 M. Kerker and V. K. LaMer, J. Am. Chem. Soc. 72, 3516
(1950).
724 Vol. 50
LIGHT SCATTERING
TABLE II. Angular-distribution functions for y=45', m= 1.20.
for the precise a values of our four latexes, viz., a= 1.06,2.02, 3.91, 4.26, which we had previously obtained byinterpolation from Tables II-IV. These results arepresented in Table V. In this case, the Pangonis, Heller,and Jacobson light-scattering functions were notutilized, since our program was for the completecomputation and the machine input consisted only ofthe index of refraction and the value of a. We found thatalthough most of the values of i and i 2 which had beenobtained by interpolation agreed well with those ofTable V, in some cases the interpolated values were inerror by as much as 20%. This emphasizes the erraticvariation of these functions even through an intervalof only 0.2 of an a unit.
The availability of these scattering functions nowmakes possible a precise evaluation of the range ofapplicability of the Rayleigh-Gans (R-G) theory toscattering by polystyrene latex. Van de Hulst2 0 hascautioned that the R-G theory is only valid for smallvalues of the parameter, a(m- 1). LaMer and Plessner,8and Kerker9 have pointed out the failure of the R-Gtheory in the interpretation of higher-order Tyndallspectra of polystyrene latex, and Nakagaki and Heller2 '
20 H. C. Van de Hulst, Recherches astronomiques de l'Observa-toire d'Utrecht, XI, Part 1, Optics of Spherical Particles, Amster-dam, 1956.
21 M. Nakagaki and W. Heller, J. Chem. Phys. 30, 783 (1959).
have demonstrated its failure in describing the back-ward scattering, i.e., in the direction toward the incidentbeam. However, because Mie functions were notavailable, Dandliker2 did use the R-G theory in some ofhis calculations; and Alfrey, Bradford, Vanderhoff,and Oster10 used it to determine the size of polystyrenelatex particles from dissymmetry measurements.
We have calculated intensity functions at yy= 450,90°, and 1350 for several values of a between 0.2 and4.2 using the following R-G theory relations:
(8)
(9)
Z is the quantity 2 a sin (0), where 0 is the supplementof -y. J(Z) is the Bessel function of order. The factor[2(m- 1)]2 is usually approximated by (M 2 - 1)2. How-ever, for m as large as 1.20 this introduces a 20% error,and we have therefore used the exact factor.
The agreement between the R-G and Mie theory isbest at -y=13 5 ' and poorest at Y=45°. At y= 4 5 0, iicalculated from the R-G theory for a= 1.2 is 96%of the value obtained using the Mie theory. The corre-sponding values at a= 1.5, 1.8, and 4.2 are 58%, 190%,and 0.08%, respectively. Obviously, the R-G theoryhas little validity at -y= 450 for values of a> 1.2.
i,=i 1o2 6(-1)][(X/2Z) (Z)]2/Z2
i2 =il COS'0.
July 1960 725
"V2M. KERI(ER AND E. XIATIJEVIV
TABLE III. Angular-distribution functions for 'y= 90°, mo 1.20.
At = 1350 the agreement is much better. Up toa= 1.5 the difference is below 10%, and it is not untila> 4 that the values differ by as much as 50%. It islikely that at greater scattering angles the agreementwould be even better. However, since the functionsthemselves do not vary markedly with a at theseforward angles, much greater precision is necessary inorder to make use of data at these angles for particle-size determination.
IV. RESULTS AND DISCUSSION
In Tables VI and VII, we have compared our experi-mental values of the polarization ratio and dissymmetrywith these quantities calculated from the Mlie theory.The particle sizes determined by electron microscopyhave been used in the calculations. The polarizationratio, p=il/i 2 , is the ratio of the intensities of thehorizontally polarized to the vertically polarizedscattered light. The dissymmetry, Di, is the ratio of theintensity of vertically polarized light scattered at 1350to that scattered at 45°. D2 is the correspondingquantity for horizontally polarized scattered light.
The agreement between the experimental and theo-retical values of p and p35 is quite satisfactory. Theagreement of the values of po for a= 1.06 is especiallystriking when one considers that two beams beingcompared in the polarization ratio differ by a factor
of 2000. However, the values of p45 for a= 2.02 and 4.26are in poor agreement with the calculated values.Also, all of the dissymmetry values differ quite ap-preciably from those calculated from theory.
It may be that the discrepancy found for both p45 andDi and D2 is due to some error in the scattered in-tensities at 45'. If this is the case, then the scatteredintensity at 450 would be in error precisely by a factorgiven by the ratio of the experimental to the theoreticalvalue of D. Accordingly, we have calculated correctedvalues of P45 from the following relation:
These corrected values of P45 have been tabulated inthe last column of Table VI, and their agreement withthe theoretical values is remarkably good.
We have also analyzed our results using a secondapproach, namely, by calculating the values of a fromthe experimental data and comparing them with theelectron microscope values. This was done by comparingthe experimental values with the appropriate p vs aand D vs a functions. The results are presented inTable VIII. Although a is not single valued, an un-ambiguous selection can be made, since the chosenvalue must fit the several independent experimental
726 Vol. 50
LIGHT SCATTERING
TABLE IV. Angular-distribution functions for -y= 1350, m= 1.20.
quantities. For example, although p45(corr)=0.107 and this is often very erratic. For example, becausecorresponds to a= 3 .21 as well as a= 2 .02 , the former P135 is nearly constant in the neighborhood of a=1,possibility must be eliminated, since it is inconsistent data of highest accuracy still lead to a very high degreewith the other experimentally observed values for a at of uncertainty in the value of a. On the other hand, a135° and 90°. 23% discrepancy between the experimental and theo-
The required accuracy in the experimental data for retical values for p at a= 4.2 6 gives rise to only aan accurate determination of a depends upon the 1.5% error in the determination of a. In view of this,functional dependence of the scattering functions on a, it is apparent that a correct determination of a does not
TABLE V. Additional angular-distribution functions for in= 1.20.
TABLE VIII. Comparison of values of a calculatedfrom various sources.
Fromelectron
microscope
1.062.023.914.26
From From FromP45 P45 (corr) p135
Below 0.40 1.00 to 1.10 1.00 to 1.252.09 or 1.49 2.02 2.03
3.93 3.91 3.743.99 4.28 4.04
necessarily provide an unambiguous test of the accuracyof the light-scattering data or theory. A more reliabletest would be a comparison of the type presented inTable VI, where any errors in experimental techniqueor theory would show up as a discrepancy between theexperimental and theoretical values of p9o, p135,, andP45(corrected). The computations of functions pre-sented in this paper and the availability of mono-dispersed polystyrene latex now make such a comparisonfeasible for scattering at y= 4 5 ', 900, and 1350, usingeither the vertical or horizontal components.
V. SECONDARY SCATTERING
A possible explanation of the discrepancy found forP45, Di, and D2 may be multiple scattering. This isusually minimized by working at low latex concen-tration and narrow light-beam width. However, withthe large dissymmetries encountered here, the scatteringin the forward directions is quite intense even whenconcentration and beam width have been reduced tothe point where that in the backward directions becomescomparable to the solvent scattering. In such a case,if the ratio of the multiple forward-to-backward scatter-ing is less than the dissymmetry of the primary scatter-ing, the measured dissymmetry will be reduced, andP45 may be altered.
We have worked out a theory of multiple scatteringfor a restricted case and have applied it to this problem.We only consider secondary scattering for light beamsof infinitesimal thickness. The model for the secondaryscattering at y=450 is shown in Fig. 1. A particle atposition 1 in the primary beam scatters through angle012 and is secondarily scattered by a particle in position2 through angle . It can be shown that in general4=225-6, so that the intensity scattered at 7=450 bythis encounter is proportional to iOi22r-, where io is theintensity function for y=6 for either horizontally orvertically polarized light, as the case may be. The sum
TABLE VII. Comparison of theoretical and experimental valuesof the dissymmetry (1350/450).
DI (vertically polarized) D2 (horizontally polarized)7 Exptl Theor D, Exptl Theor D
where 0' and " are determined by the location ofposition 1 in the incident beam. It can be shown thatfor primary scattering at 7y=450
045'= tan-1[0.70711/(R-0.29289)] (11)
045" = tan-li0.70711/(1.70711-R)], (12)
where R is the distance of position 1 from the entranceof the incident beam into the scattering cell, expressedas the fraction of the radius of this cell, i.e., R=a/r,where a is the actual distance. Since we must considerthe secondary scattering emanating from the fulldistance of the incident beam in the cell, R will varyfrom 0 to 2. The total intensity of secondary scatteringin the direction -y=45° is obtained by integrating overall positions of R and is given by
2 045t�
i 5(45)= JfIoi 2 i225_oddR. (13)
At 7-=1350, the corresponding quantities are
2 0135"
i 3(135)= j ioi31'5.od~dR (14)O °35'
0135'=tanl[O. 70711/(R- 1.70711)] (15)
0135" = tan-'[0.70711/ (0.29289-R)]. (16)
The dissymmetry of the secondary scattered light isdefined as
D,=i,(135 0)/i 5(45 0), (17)
where the appropriate functions must be used for eachof the polarized components.
In order to compute D8, we have obtained intensityfunctions for the values of a corresponding to thelatexes we have investigated. These functions werecomputed for 100 intervals, as well as at 450 and 1350,using the IBM 704 computer. Each of the integrationsindicated in Eqs. (13) and (14) was carried outgraphically.
The calculated values of D. are presented in TableVII. The results confirm the mechanism of secondaryscattering as a possible cause of the discrepancy between
FIG. 1. Model for secondary scattering at -y=45O. Radiationscattered from 1 at angle 012 to 2 where it is secondarily scatteredat angle 0.
the experimental and theoretical values of the dis-symmetry. For example, for a= 1.06 and 2.02, theexperimentally observed dissymmetry is intermediatebetween the D and D8, the corresponding values forthe primarily and secondarily scattered radiation, inagreement with the requirement of the relation.
D (exptl) = [i (135°) + is (135°) X f]/[i(45-)+i,(45 0)Xf] (18)
where f is a factor which is dependent on the latexconcentration and the beam geometry. The values ofD, for a = 3.91 are less than the theoretical dissymmetrybut are still greater than the experimental dissymmetry.In this case Eq. (18) could not be applied quantitatively.For a= 4.26 the experimental dissymmetry of thevertically polarized component is greater than thetheoretical value and, in agreement with our model, thevalue of D, is still greater than both of these values.For the horizontal component at a= 4 .2 6 , the modelbreaks down since D, is larger than the theoreticalvalue, whereas the experimental value is smaller.However, the breakdown in this single case should benot too surprising in view of the simplicity of the modeland the complexity of the scattering pattern at a valueof a as high as 4.26. In addition to secondary scattering,a more refined model would also have to take the finitebeam width, the finite solid angle of reception at thephotomultiplier tube, and multiple scattering intoconsideration, but this problem would be formidableindeed. It is interesting to note that our experimentallydetermined dissymmetries were independent of concen-tration at the low concentration range where we worked.This would indicate that both is and i which appearin Eq. (18) are linear functions of concentration in
agreement with our idealized model for secondaryscattering.
This dissymmetry effect has been encountered byother workers, but because they did not have the Mietheory intensity functions available they were notin'a position to recognize the effect. For polystyrenelatexes with a= 1.26, 1.30, and 2.11, Alfrey, Bradford,Vanderhoff, and Oster' obtained dissymmetries of2.67, 3.25, and 15.9, using unpolarized light. Thedissymmetries calculated from the Mie theory havethe correspondingly higher values of 3.19, 3.52, and34.6, respectively, in agreement with the trends wehave observed in the foregoing. Similarly, for a latex ofa=0. 6 7 , Goring, Senez, Melanson, and Huque7 reporta dissymmetry of 1.23, again with unpolarized light,compared to the Mie theory value of 1.33.
VI. CONCLUSION
In view of the error induced in the 45°-135° dis-symmetry and possible error in the polarization ratio at450, which we believe to be due to multiple scattering,these quantities must be treated with due deference forMie scatterers.
One way of avoiding this difficulty might be torestrict all measurements to the forward directions.Thus one might work with the 120°-150° dissymmetry.
Another alternative would be to determine a fromthe polarization ratio at 1350 and 900. This could beused to calculate the theoretical dissymmetries D1 andD2, and then a corrected polarization ratio at 450 couldbe calculated, as we have done above. If this polarizationratio is consistent with the a already determined, theanalysis could then be considered as probably correct.
ACKNOWLEDGMENT
We are grateful to Dr. J. W. Vanderhoff of the DowChemical Company for a supply of monodispersedpolystyrene latexes, and to Dr. K F. Schulz for assist-ance in some of the experimental work.
Note added in proof.-It has only recently come to ourattention that Heller and Nakagaki22 have publishedtables of i and i2 for -ry=4 5 ' and 1350 and m=1.20.However, their tables do not include R(il*), (ii*),R(i 2 *), and I(i2*) as ours do. We also note major dis-crepancies between their results and ours, especially forii at y=45° and a=1. 2 , 2.2, 4.2, 6.0, 6.2; for i2 at,y=45' and a = 2 .0, 2.2, 6.0, 6.2; and for i at y=135°and a= 6.0. We have rechecked our results and are con-vinced that they are correct. The agreement with ourresults in the other cases is within 0.3%.
22 W. Heller and M. Nakagaki, J. Chem. Phys. 31, 1188 (1959).
