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Calibration of light-scattering instruments
JOVnNAL OF eoLLon) ANn INTerFACE scn~Nc]~ 21, 498-512 (1966)
CALIBRATION OF LIGHT-SCATTERING I N S T R U M E N T S IV. CORRECTIONS FOR REFLECTION EFFECTS 1 Josip p. Kratohvil 2 Deparlment of Chemistry, Clar~son College of Technology, Potsdam, New York Received February $2, 1965 ABSTRACT A more general expression than that given by Tomimatsu and Palmer is derived which corrects the light scattering intensity in different cells for reflections at the walls. Effects at both the glass-air and glass-liquid interfaces are considered. This expression requires intensity measurements at two supplementary angles and knowledge of the Fresnel coefficients. The magnitude of the correction has been calculated for values of the dissymmetry up to 26.5. For high dissymmetries, it is not possible to neglect the reflections at the liquid-glass interface as is usually done. Also, it is pointed out that there are upper limits for the dissymmetry beyond which it is not possible to apply these corrections. The importance of taking these corrections into account has been demonstrated by considering a few examples from the literature where application of a proper reflection correction will change the results significantly. INTRODUCTION T o m i m a t s u and P a l m e r (1) have recently discussed the reflection corrections in light scattering m e a s u r e m e n t s for the Brice-Phoenix p h o t o m e t e r (2). Their expressions are applicable, u n d e r certain assumptions, to glass cells of various shapes. W e have been working on the same problem in connection with light scattering studies on large particles (polymer latexes, hydroso]s), and would like here to provide (a) a m o r e general presentation of the physical problem; (b) some q u a n t i t a t i v e results for several situations c o m m o n l y encountered in practice; and (c) some applications of the expressions derived to certain light scattering results from the literature. T h e literature survey on the problem of reflection effects in light scattering m e a s u r e m e n t s is given b y T o m i m a t s u and P a l m e r (1, 3). 1 Supported by the Atomic Energy Commission Contract No. AT(30-1)-1801. 2 The author has had the pleasure of discussing the problern of reflections in light scattering measurements with Professor V. K. La Mer during the Los Angeles A.C.S. meeting in April, 1963. The preparation of this communication was stimulated by this discussion. 498
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS (*4 5} 135
*
.
..k.i/'T "~
*45
499
°
0,~
Fro. 1. Cross section of a cylindrical glass cell in the plane of scattering. The cell is equipped with plane parallel entrance and exit windows. All faces are clear, In the middle, the scattering envelope for the incident beam (2) (full curve) and for the reverse reflected incident light (2' + 3') (dashed curve) are pictured schematically. Tile angles designated with "minus" are those on the side facing the observer (detector). The angles in parentheses refer to the reverse envelope. The heavy arrows and unprimed numbers indicate the light rays incident on reflecting surfaces as well as those transmitted through them. The dashed arrows and primed numbers refer to the reflected rays. 1: t j o 1': f¢~Io; 1": f J o 2: t~t~Io 2': f ¢~t,,Io 3: tat~21To 3~: fatat[2Io 4: t~2tz~Io 2' -P 3': tat~(fz "~- hfa)Io For 5-9 see text. THE 1DFIYSICAL PROBLEM AND FORMULATIONS
The physical problem is illustrated schematically in Fig. 1, which represents the cross section of the glass cell in the plane of scattering. This applies to a symmetrical cylindrical transparent glass cell with flat entrance and exit windows. This geometry has been selected for two reasons. Firstly, in this type of scattering cell, tile maximum number of reflections are involved and the influence of these on the measured scattering properties is most pronounced. The expressions for other types of cells can easily be derived from this case. Secondly, the situation presented in Fig. 1 eor-
500
KRATOHVIL
responds very closely to a widely used commercial cell, the only difference being that these cells are usually made with a frosted inner back surface (4). As will be shown later, this frosting requires a slight change in the geometry of Fig. 1. If a parallel beam of light is perpendicularly incident on the entrance window, all the reflections may be treated as occurring at perpendicular incidence. Even those extreme scattered rays which originate in the finite illuminated volume located at the center of the cell and which reach the detector will have reflectivities and transmissivities almost the same as the rays which impinge on the cell walls at perpendicular incidence. This follows from Fresnel's equations, for which the reflectance remains very nearly constant (within 0.6 %) from perpendicular incidence to angles of incidence up to about 20 ° (5, 6). Also, over the same angular range, there is very little difference between the two polarized components. It is convenient that corrections for reflection effects are most important in the case of scattering by large particles. For such particles, the angle of acceptance of the phototube, and hence, the scattering volume, must be reduced below a certain value in order to obtain adequate angular resolution (7). This condition also assures that all reflections of concern occur at small angles of incidence. Fresnel's equations for the fractions of light, fa, and fs, reflected at perpendicular incidence at the glass-air and glassliquid interfaces, respectively, are given simply by f~ -- [(n~ - 1)/(ng + 1)]2;
[1]
f, = [(n~ - n,)/(no + n,)]~;
[2]
and
where ng and n~ are the refractive indices of the glass and the liquid. The transmission coefficients at the air-glass and liquid-glass interfaces are defined by t~ =
1 -
fo
[la]
tz =
1 -
fz.
[2a]
and
In Table I, the Fresnel coefficients are given at the two wavelengths (436 and 546 mu) most often used in light scattering measurements and for two types of glass, Pyrex and crown optical glass (Phoenix Co. Melt No. 2), of which light scattering cells are commonly made. fz was calculat.ed for water as the liquid in the cell. For most organic liquids, f~ is sufficiently close to zero and need not be considered because of the small difference between the refractive indices of the liquid and the glass. It is also assumed that the systems are nonabsorbing and scatter so
CALIBRATION OF LIGHT-SCATTERINGINSTRUMENTS
~01
TABLE I Refractive Indices of Water (25°C.) and Two Kinds of Glass Commonly Used for Light Scattering Cells, and the Corresponding Fresnel Coefficients at Two Wavelengths Dispersion was estimated from the data given in reference 6 for fused quartz and crown glass, respectively.
nz (water, 25°C.) n~ (Pyrex) f~ f~ no (optical glass, crown) f~ fz
436 mg
546 m~
1.340 1.482 O.0377 O.00253 1. 528 0.0436 O.00429
1.334 1.476 O.0370 O.00255 1.519 0.0424 O.00420
weakly that the light rays are not significantly attenuated on passage through the system. Furthermore, multiple reflections may be neglected (5, 8, 9). Owing to the reflections at the air-glass and glass-liquid interfaces of the flat entrance window of the cell, the incident intensity at the point of scattering is t~tffo, where I0 is the intensity of the incident radiation. Similarly, the intensity of the reversed incident beam, reflected at the exit window, is (t~hh + tat?f~)IO = tah(fz -t- t&)IO. This light reflected from the exit window will also be scattered but will have precisely the reverse angular dependence from that corresponding to incident light, as indicated schematically in Fig. 1. Therefore, the observed angular envelope will be the result of superposition of the two envelopes: the main one, corresponding to the incident intensity t~tffo, and the reverse envelope, eolTesponding to the reverse incident radiation t,h(f, ÷ hf~)Io. Furthermore, a fraction (f, + hfa) of the light scattered in the direction 0 is reflected at the liquidglass and glass-air interfaces at the exit point of the cell. Accordingly, the contributions from the main envelope and the reverse envelope must each be multiplied by the corresponding transmission h& • In Briee's working-standard method, which is commonly employed (1-3, 10), the galvanometer reading G~ for the transmitted intensity (0 = 0°), t~2tz2Io, is corrected to that corresponding to I0 by applying the factor T,. This is the apparent transmittance of the filled scattering cell measured in the photometer with the illuminated standard diffusor as a source. This factor is contained in the so-called residual refraction correction R ~ / R ¢ . When a symmetrical cell, symmetrically aligned, as in Fig. 1, is utilized, then the attenuation of the incident and scattered rays as they pass through the scattering system is canceled out. Thus, the true scattered intensity at an angle 0, Io, corresponding to I0, must be evaluated from
502
K~TORVm
the observed intensity Io', corresponding to the measured galvanometer reading Go. In addition to the contribution of scattering in a particular direction - 0 , the observed intensity at that angle, Io', will consist of several other contributions, depending upon the cell used. For the situation of Fig. 1, there will be four contributions. (a) Scattering of the incident beam of intensity tatzIo at angle - 0 gives a scattered beam of intensity tatzIo. This is corrected for the reflections at the exit point giving finally t~2tHo. (b) The scattering of the main beam at a scattering angle of + (180 - 0) is in turn reflected toward - 0 from the side opposite the detector. This is also corrected for reflections at the exit point and is given by t,2t~2(f, +
tzf~)I18o-o . (c) The scattering at - ( 1 8 0 -- 0) of reverse beam, corresponding to
tat~(fz + tJ~)Io, also contributes to the -O reading. When corrected for reflections at the exit point, this gives the same resul¢ as in (b) above. (d) The scattering at + 0 for the reverse envelope that is reflected from the back face toward - 0 , also corrected for reflections at the point of observation, gives ta2tz2(fz + t~f~)~Io. The observed intensity, I0 ~, is the sum of all four contributions, i.e.,
Io' = t~:t~2{[1 + (f~ + t~fa)2]Io + 2(f~ + t~f~)Ilso_o}.
[3]
Combining Eq. [3] with the corresponding expression for I~8o_o and substituting (fz -t- tzf~) = A, one arrives at the following expression for the true scattered intensity 1o, corresponding to the incident intensity I0 :
Io
=
tJt~2[(1 + A2)~ _ 4A2] •
Io'
y
[4]
Thus, in order to obtain the correct scattered intensity at an angle 0, one has to measure the intensities at two supplementary angles and to know the Fresnel coefficients. Equation [4] can be simplified considerably if contribution (d) is neglected. By summing up the first three contributions and repeating the procedure elaborated upon above, one can write instead of Eq. [4]: Ie --
1 (I0' -- 2AI;s0-e). t~2t~2(1 -- 4A 2)
[5]
The change in the numerical results is extremely small. For example, for f~ = 0.0370 and fz = 0.00255, the factor in front of the brackets has a value of 1.089 in Eq. [4] and 1.090 in Eq. [5], whereas the factor within the brackets amounts to 0.0788 and 0.0789 in the two equations. Consequently, ~11 considerations that follow will be based on the first three contributions only.
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS
503
It was mentioned earlier that the situation depicted in Fig. 1 does not correspond precisely to the one when cylindrical cells of Witnauer and Scherr (4) (Phoenix Co. catalog Nos. C-101 and C-105) are utilized, because these cells usually have frosted inner back surfaces. Since the light is diffusely reflected from such a surface, eliminating the fraction of scattered light reflected on the liquid-glass interface toward - 0 , it is obvious that the magnitude of the contribution (b) will be changed. However, it is not possible to evaluate this change quantitatively, because of the difficulty of calculating how much of the scattered light in + (180 - 0) direction is attenuated by reflection on the liquid-frosted glass interface before reaching the glass-air interface. Two other cases for the cylindrical cell of Fig. 1 should be considered. Firstly, one technique frequently used to eliminate the reflections is to paint the outside back surface dull black. However, the exit window is left clear; otherwise, Brice's working-standard method cannot be applied. For this case, only reflections at the liquid-glass interface from the backside of the cell are obtained. Therefore, contribution (b) will be smaller and equal to ta~tl2f~Ixso_o, leading to Io =
1 [Io' tJt~211 -- (fz + A) 2]
(f, + A)I~8o-o].
[6]
Secondly, if the coat of paint is applied to the inside of the back face, or if a piece of black dull plastic film is placed inside against the back curved face (1), the whole contribution (b) will be eliminated. Thus, Io -
1
ta~t~2(1
-
-
A s)
(Io' -
aIiso-o).
[7]
Another important case is the semioctagonM cell (e.g., Phoenix Co. catMog No. D-101) which is employed to measure the scattered intensities at 0 = 45 °, 90 °, or 135 °. The back face and the entrance or exit face of this cell form an angle of 90 ° and, in addition, the cemented joints are usually blackened. By tracing the rays and taking the actual dimensions of the cell and various components and distances of the Brice-Phoeuix instrument into account (11), it can be shown that rays scattered towards + 4 5 ° and +135 ° corners of the cell cannot reach the detector positioned at - 4 5 ° or - 1 3 5 ° by reflection. This has also been demonstrated experimentally by Tomimatsu and Palmer (1). Consequently, contribution (b) is entirely eliminated, regardless of whether the cell is coated with black paint or not, and Eq. [7] applies. Equations [5], [6], and [7] reduce to the corresponding expressions derived by Tomimatsu and Palmer (1) for the case treated by them in which only reflections at the glass-air interfaces are considered. For a cylindrical cell
504
KRATOI-IVIL
of Fig. 1, with clear or frosted back inner surface, the first three contributions discussed above will, then, reduce to: (a):
t~Io ;
(b):
t~2faI18o_o ;
and
(c):
ta2fJ18o_o.
From this, it follows that Io -
1 (Io' -- 2faI;8o-o). 1 - 2fa
[8]
It should be mentioned that Tomimatsu and Palmer (1) offered the experimental evidence that the frosted inside surface does not affect reflections on the glass-air interface. For a cylindrical cell with black painted back face, either outside or inside, contribution (b) is eliminated, and instead of Eq. [6] or [7], one obtains Ie -
1--
1
2fa
(Ie' - - / ~ I~8o-0),
[9]
which is also applicable to a semioctagonal cell for the same reasons that Eq. [7] was applicable. Except for the factor 1/(1 - 2f~), Eq. [9] was originally suggested by Sheffer and Hyde (12) and by Oth, Oth, and Desreux (13). The latter authors have also attempted to incorporate the liquid-glass reflections, with essentially the same result as Eq. [7], but without the factor 1/t~h2(1 - A2). Equations [8] and [9] differ slightly from those given by Tomimatsu and PMmer (1) in the first factor on the right-hand side of these equations. This difference arises because secondorder effects (f~ and higher powers in f~) have been consistently neglected. When the solvent is water, these become comparable in magnitude to values of fz. The above discussion should make apparent the kind of corrections that may be required in other situations in which the experimental procedure is changed or in which other reflecting surfaces are introduced (as, e.g., windows for thermostating ]ackets, polarizers or analyzers, various filters). It is worth while considering what happens when Iot = I~8o-o, i.e., when the observed angular envelope is symmetrical. If glass-air reflections only are considered, Eq. [8] gives Io = Io' at all angles. However, when Eq. [9] is applicable (cylindrical cell with back face painted black for all angles, or semioctagonal cell for angles - 4 5 ° and - 1 3 5 ° with opposite corners unpainted or painted and, with back face painted, for angle -90°), one obtains Io/Io' = t . / ( 1
--
2f~).
[10]
Tomimatsu and Palmer (1, 3) have already emphasized that at 0 = --90 ° and for cells with clear faces, all glass-air reflections are self-compensatory (neglecting f 2 and higher powers of fo). They also confirmed experimentally
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS
505
t h e v a l i d i t y of E q . [10] for all angles, as w e d i d for 90 ° in a p r e v i o u s p a p e r (14). F o r some a d d i t i o n a l e x p e r i m e n t s of this t y p e , see reference 15. Values of t h e r a t i o I o / I o ' for t h e ease w h e n t h e liquid-glass reflections are i n c l u d e d a r e listed in T a b l e I I . F o r t h e ease of Eq. [5], t h e r e is h a r d l y a n y a d d i t i o n a l effect, t h e r a t i o b e i n g 1.004 i n s t e a d of 1.000. T h e e q u a t i o n s d e v e l o p e d here for c o r r e c t e d i n t e n s i t i e s (Eqs. [4] t h r o u g h [9]) can b e p u t in t h e g e n e r a l f o r m : Io -- X ( I o '
-
[11]
YI;8o-o),
w h e r e X a n d Y a r e c o n s t a n t s d e p e n d i n g o n l y on fa a n d f~ for a p a r t i c u l a r g e o m e t r i c a l - o p t i c a l s i t u a t i o n . T a b l e I I s u m m a r i z e s t h e e q u a t i o n s for v a r i o u s eases a n d gives n u m e r i c a l v a l u e s of X a n d Y for fa = 0.0370 a n d f~ = 0.00255 ( P y r e x a n d w a t e r a t X0 = 546 m#), a n d t h e v a l u e s of t h e r a t i o I o / I o ' = X ( 1 - Y) w h e n Io' = Ii8o-e. I t is o b v i o u s t h a t besides t h e a n g u l a r s c a t t e r i n g intensities, o t h e r l i g h t s c a t t e r i n g functions, such as d i s s y m m e t r i e s a n d p o l a r i z a t i o n ratios, will also be influenced b y reflection effects. C o r r e c t e d d i s s y m m e t r i e s will a l w a y s be h i g h e r t h a n t h e o b s e r v e d ones. T h e m a n n e r in w h i c h t h e p o l a r i z a t i o n TABLE I I A S u m m a r y of Equations Developed for Various Geometrical-Optical Arrangements
The numerical values are those forf~ = 0.0370 andfz = 0.00255. t~ = 1 - f~ ; h = 1 - fz ; A = f ~ + h f ~ . IO/I'O = X
1/tJh~(1 - 4.4 2) Eq. [51: Cylindrical cell, clear faces 1/ta2h2[1 -- (fl + A) 2] Eq. [6]: Cylindrical cell, painted outside 1/&2ta2(1 -- A 2) Eq. [7] : Cylindrical cell, painted inside; or semioct, cell (45 °, 135 °) 1/(1 -- 2f~) Eq. [8]: Cylindrical cell, clear faces or frosted inside; fz = 0 1/(1 -- 2fa) Eq. [9]: Cylindrical cell, painted outside or inside, or semioct. cell;f~ = 0
Y
= 1.090
x(1 -
Y)
when,Io' =
2A = 0.0789
1.004
= 1.085 fz + A = 0.0420
1.039
= 1.085
A = 0.0395
1.042
= 1.080
2f~ = 0.0740
1.000
= 1.080
f , = 0.0370
1.040
506
KRATOI%IVIL
ratios are affected will depend n o t only on the a s s y m m e t r y of the scattering envelope b u t also on the intensities of b o t h the horizontal and vertical components of scattered light at two s u p p l e m e n t a r y angles, i.e., on four different intensities (this point will be illustrated in the last section). F o r corrected dissymmetries Zo one can write quite generally
Zo-
Io _ I o ' - - YI[+o-o _ Z / Y Ilso-o Iiso--o- YIo' 1- YZo"
[12]
where Zo' is the observed uncorrected d i s s y m m e t r y ( = Io'/I'~so_o). Similarly, for corrected polarization ratios one obtains
Ho Oo -
Vo
Ho'--
Vo'
-
-
YH~so-o ' ' YVlso-o
[13]
where H / and V / designate the observed horizontally and vertically polarized components and H0 and V0 are the corrected quantities. T~E N u ~ m c A L
ILLUST~mIO~S
I n Tables I I I and IV, values of the angular intensities corrected for reflections according to Eqs. [5] to [9] are listed for various a p p a r e n t dissymmetries. Corrected dissymmetries are also given. T h e reflectivities used were f~ = 0.0370 and f~ = 0.00255. I t is obvious t h a t these corrections are v e r y important, particularly TABLE I I I The Magnitude of the Reflection Correction on Scattered Intensities and Dissymmetries as Calculated from E,qs. [5] and [8] ! f~ = 0.0370;fz = 0.00255. In all cases it is assumed that 118o-o = 1.
Io'/I[so.o -To 1 2 3 4 5 6 7 8 9 10 11 12 12.67 13 13.50
1. 004 2. 093 3.182 4.272 5. 363 6. 453 7.542 8. 634 9.722 10.81 11.90 13.00 13,72
Eq. [Sj 118o-o 1. 004 0. 918 0. 832 O. 746 0. 660 0. 574 0.488 O. 402 O. 316 O.230 0.144 0.058 0
Io/118o-o 1. 000 2. 280 3. 825 5.727 8.126 11.25 15.46 21.48 30.77 47. O0 82.65 224.2 ¢~
Io
Eq. [8] I18o-o
Io/11~o-o
1. 000 2. 080 3.160 4.240 5. 320 6. 400 7.480 8.560 9.640 10.72 11.80 12.88
1. 000 0. 920 0. 840 O. 760 0. 680 0. 600 0.520 O.440 O. 360 O.280 0.200 0.120
1. 000 2. 261 3. 762 5.578 7. 824 10.67 14.39 19.46 26.77 38.30 59.00 107.3
13.96 14,50
0.040 0
348.9 ¢~
CALIBRATION OF LIGHT-SCATTERING
INSTRUMENTS
507
TABLE IV The Magnitude of the Reflection Correction on Scattered Intensities and Dissymmetries as Calculated from Eqs. [6], [7], and [9] f~ = 0.0370;fz = 0.00255. In all cases it is assumed t h at Izs0_0 ' = I.
Eq. [6]
[e'/
l~so-e 1 2 4 6 8 10 12 14 16 18 20 22 23.81 24 25.32 26 26.50
Eq. [7] Ilse-e
Eq. [9]
le
I18o-o Ie/I*so-e
Ie
Ie/Ilso-e
1.040 2.123 4.293 6.462 8.633 10.80 12.98 15.15 17.31 19.49 21.67 23.82 25.79
1.040 1.000 0.994 2.137 0.903 4.755 0.812 7.960 0.721 11.97 0.629 17.17 0.538 24.12 0.447 33.88 0.356 48.63 0.265 73.52 0.174 124.5 0.0825 288.7 0 co
1.042 2.128 4.297 6.465 8.638 10.81 12.98 15.15 17.32 19.49 21.67 23.82
1.042 1.000 0 . 9 9 9 2.131 0.914 4.702 0 . 8 2 8 7.810 0.742 11.64 0 . 6 5 6 16.48 0.571 22.73 0 . 4 8 5 31.23 0.399 43.42 0 . 3 1 4 62.05 0.228 95.00 0.142 167.8
26.00
0.05641461.0
27.42
0
I0
Ilso-e Ie/Ilso-o
1.040 2.120 4.280 6.440 8.600 10.76 15.08 17.24 19.40 21.56 23.72
1.040 1.000 1.000 2.120 0.920 4.652 0.840 7.665 o. 760 11.32 o. 680 15.83 o. 6oo 21.53 o. 520 29. oo o. 440 39.18 o. 360 53.87 0.289 77.00 0.200 118.6
25.88
0.120 215.7
28.04 28.58
0.040 701.0 0 co
12.92
co
when the dissymmetry is large. High dissymmetries such as those tabulated are often found for colloidal dispersions (hydrosols, polymer latexes, aerosols, biological corpuscles like mitochondria (16)), and even for some macromolecular solutions, as, e.g., glycogen (17). For such cases, careful consideration must be given to which form of the reflection correction is applicable. Tomimatsu (18) has already emphasized the differences between corrected intensities obtained using the formula of Sheffer and H y d e (12) and those obtained using his formulation. B y inspecting Tables I I I and IV, several observations are apparent. Firstly, when the refractive index of liquid is as low as that of water, reflections at the liquid-glass interfaces become important, especially for higher dissymmetries. The inclusion of these liquid-glass reflections leads to an increased value of Y. This has the apparent effect of increasing fa in those expressions which do not incorporate the liquid-glass reflection and explains the observation of Tomimatsu (18, and private communication), who, considering only the glass-air interface, presumed to measure the values of fa for Phoenix Pyrex cylindrical cells filled with water. He found an average value of 0.045 instead of 0.0377 at k0 = 436 mr* (Table I). Table IV also demonstrates that for high dissymmetries the results
508
~TO~VIL
differ when a coat of black paint is applied on the outside (Eq. [6]) or inside (Eq. [7]) of the back face. We have also indicated in Tables I I I and IV the limiting values of Io'/I~so_o for which the corrected IlS0-e becomes zero and hence the corrected dissymmetries become infinite. To measure dissymmetries higher than these limiting ones, it is necessary to eliminate, partly or completely, the source of the reflections either by changing the shape of the cell or by applying a black coat onto the back face and/or on the exit window. An alternative procedure is to insert black glass or plastic film within the cell, or to mount a Rayleigh horn. A word of caution is advisable in regard to a coat of paint on the back face. The paint used must be a nonglossy, dull black that adheres strongly to the glass and, upon drying, does not crack and transmit light. I t is always desirable to check experimentally, with the solvent or solution in the cell, how effective the coat of paint is in removing the reflections from a particular face. Otherwise, it appears to be a more reliable procedure to leave all the faces clear and to correct for the reflection effects with the appropriate equation. Since the corrected intensities in the backward directions and, thus, the dissymmetries are very sensitive to small changes in the observed dissymmetry, these should be measured as accurately as possible. For Ion~ I~_o = 10, the corrected dissymmetries are 47.0 (Eq. [5]) and 38.3 (Eq. [8], i.e., taking only glass-air reflections into account). However, if the observed dissymmetry had been 10.5, a corrected value of 46.7 is obtained from Eq. [8]. In other words, a 5 % experimental error in observed dissymmetry might result in a 25 % error in the corrected dissymmetry, and this in turn is equivalent, in this ease, to the effect of including the glass-water reflections. Of course, an experimental error of 5 % is not uncommon. By the same token, the refractive index of the glass must be known rather accurately, and its wavelength dependence must be considered. If, instead of A = 0.0370 for Pyrex glass (Table I) (X0 = 546 m~), a value off~ = 0.0377, corresponding to X0 = 436 m~, is used in correcting the observed dissymmetry of 10 by means of Eq. [8], a corrected value of 40.35 is obtained, instead of 38.3. It should be emphasized that for cells made of glass of still higher refractive index, the effects of the water@ass reflections become even more pronounced. For crown optical glass (Table I) at X0 = 546 m~ (fa = 0.0424 and fl = 0.0042), one obtains by applying Eq. [8] (glass-air reflections only) a corrected dissymmetry of 38.8 (see above) from an uncorrected dissylmnetry of 9.04. However, if full expression [5] is used, the corrected value is 55.6. Even for smaller observed dissymmetries, the differences in corrected values due to the inclusion of glass-water reflections are significant.
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS
509
As the data of Tables III and IV demonstrate, the corrected forward intensities are higher, and corrected backward intensities lower, than the corresponding measured intensities (see also reference 18). This could have an important implication with regard to sonie of the angular scattering data in the literature. Often one finds Zimm diagrams with pronounced downward curvature for the reciprocal intensities at backward angles. Sometimes this is attributed to the polydispersity of the material under study, as well as to branching or chain stiffness in case of polymer coils. Since reflection corrections for the backward intensities are greater than for those in forward direction, it is tempting to wonder how many of such "distorted" Zimm plots might be straightened out upon applying the proper correction. In connection with the angular measurements, it should be added that in almost all the light scattering instruments which have been described in the literature, it is not possible to measure the intensities at backward angles higher than about 135 °. Since the reflection corrections require the knowledge of scattered intensities at two supplementary angles, these cannot be evaluated for angles lower than 45 °. Fortunately, the general equation [11] reduces with good approximation to I~ = X I / f o r ~ < 90 °, particularly when the scattering envelope is highly asymmetric. APPLICATIONS
In this section we would like to consider briefly a few examples from the literature where the application of a proper reflection correction will change the results significantly. Although our suspicion is more than strong that there are many more examples, the information necessary to estimate whether a correction has been applied at all or even which expression would be the most appropriate is usually lacking. Tomimatsu (18) has applied Eq. [8] to correcting the Zimm plots of Gellert and Englander (19) for rabbit myosin A. The molecular weights derived from corrected data were about 15 % higher, and in agreement with those obtained by means of the sedimentation equilibrium method. The radius of gyration and the second virial coefficient were also affected. Chiang (20) attempted to explain discrepancies in the light scattering molecular weights of some polyethylene fractions as reported in the literature by correcting the data for reflection effects in the manner suggested by Othet al. (13). However, it seems to us that the equation given by Oth et al. is inadequate for the case Chiang discusses. Instead, our Eq. [8] is applicable. A most dramatic demonstration of the utmost importance of a proper evaluation of reflection effects is provided by some results on polystyrene latexes. Although De~eli6 and Kratohvil (21) have properly corrected the dissymmetry values (compare Tables VI and VII of reference 21), the
510
KRATOItVIL
reflection correction was not applied to the intensity curves from which the angular positions of maxima and minima in intensity were determined. For Dow polystyrene latex LS-067-A(D~ = 1.171t~) at X0 = 436 m~, theoretical calculations indicated that in the backward direction a maximum should occur at 130° and a minimum at 135°. These were not observed (compare Fig. 6 and Tables IXa and IXb of reference 21). However, after applying the appropriate correction (Eq. [8]), we now find the following intensities (on a relative scale) in the angular range 95 ° to 140° (the first number refers to uncorrected, the second to corrected intensity): 95°--4.17, 3.93; 100°--3.06, 2.53; 105°--2.72, 2.24; 110°--3.80, 3.37; 115°--3.72, 3.04; 120°--3.10, 2.00; 125°--3.55, 2.53; 130°--4.17, 2.88,135°--4.57, 1.60; 140°--6.17, 2.94. Thus, a maximum at 130 ° and a minimum at 135 ° now appear in accordance with the theoretical prediction for this latex. In previously published work from this laboratory on four polystyrene latexes (22), reflection corrections were completely ignored and large discrepancies between the experimental and theoretical vMues for dissymmetry were observed. These had been attributed to multiple scattering effects. However, we have now corrected the experimental values for reflection effects and the discrepancies have been reduced, although not completely eliminated. Indeed, the present disagreement may now be due to the acceptance of the electron-microscopic sizes, as supplied by Dow Co., as correct. By adjusting these sizes by only a few per cent, it is now possible to bring the corrected experimental values and the theoretical ones into very close agreement. We shall not elaborate on this point since it will be the subject of a more thorough discussion in a forthcoming publication. The same comments apply also to the results of Napper and Ottewill (23) on polystyrene latexes. These authors have asserted that in their experiments the reflection correction was negligible. However, although in the instrument they used (24) the cell was immersed in a liquid bath, the incident beam is still ultimately reflected back into the cell from the exit window of the bath. Finally, an example of the influence of reflection effects on the measured polarization ratios po will be given. A method of size distribution analysis based on the variation of polarization ratio with the angle of observation has been recently applied to La Mer sulfur sols (25). It had been stated then that the reflection effects on the measured o0 values were negligible. This conclusion was based on a check performed on several sols. However, at that time we did not recognize that the reflection correction depends on four different intensities (horizontally and vertically polarized components at two supplementary angles), and, therefore, must be checked for every case. All previous measurements discussed in reference 25 have been corrected accordingly, and in some cases significant differences have been found between the uncorrected and corrected o0 values. An example is given
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS
511
TABLE V Uncorrected and Corrected Polarization Ratios Po for a La Met Sulfur Sol (PS) PO
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130
Uncorrected
Corrected
1.211 0.765 0. 665 0.832 2.918 2.251 0.831 0.831 3.268 3.945 1.000 1.148 3. 553 4.378 1.925 2.745 2. 303 1.047 2.255 3. 873 1. 592
1.211 0.765 0.665 0.832 2.967 2.219 0.816 0.827 3.317 4.011 0.979 1.072 3.553 5.440 2.235 2.599 2.177 1.162 4.121 4. 537 1. 343
in Table V. The corrected p0 values may be either larger or smaller than the uncorrected ones. Again, the differences are very pronounced at the backward angles. Fortunately, however, in none of the cases analyzed were the parameters of the size distribution function significantly affected by these changes. I:~EFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
TOMIMATSU, Y., AND PALMER, K. J., or, Phys. Chem. 67, 1720 (1963). BtUCE, B. A., ttALWER, M., AND SPEISER, R., J. Opt. Soc. Am. 40,768 (1950). TOMIMATBU,Y., AND PALMER, K. J., or. Polymer Sci. 54, $21 (1961). WITNAUER,L. P., AND SCHERR, H. J., Rev. Sci. Instr. 2S, 99 (1952). JENXINS, F. A., AND WI~ITE, H. E., "Fundamentals of Optics," 2nd ed. McGrawHill, New York, 1950. "H~ndbook of Chemistry and Physics," 44th ed., p. 3090. The Chemical Rubber Publishing Co., Cleveland, 1963. TA~IBIAN, R. M., AND HELLER, W., J. Colloid Sci. 13, 6 (1958). DEBVE, P. P., Ph.D. Thesis, Cornell University, Ithaca, New York, 1944. STEIN, R. S., AND DorY, P., J. Am. Chem. Soc. 68, 159 (1946). TOMIMATSU,Y., AND PALMER, I~. J., J. Polymer Sei. 35, 549 (1959). I~ERKER, M., KRA~OHV~L, J. P., AND MATIJEVI~, E., or. Polymer Sci. A2, 303 (1964).
512
K~TO~WL
12. S~EFFE~, H., AND HYDE, J. C., Can. J. Chem. 30, 817 (1952). 13. OTH, A., OTH, J., AND DES~EUX, V., J. Polymer Sci. 10,551 (1953). 14. KEaxEa, M., K~ATO~VIL, J. P., OTTEWI•L, 1%. H., ~ I ) MATIJEVI6, E., J. Phys. Chem. 67, 1097 (1963). 15. MCI~TY~E, D., g. Res. N. B. Std. A68, 87 (1964). 16. GOTTERER, G. S., THOMSON,T. E., AND LEHNINGER,A. L., J. Biophys. Biochem. Cytol. I0, 15 (1961). 17. L~SKOV, R., AND I_V[ARGOLIASH,E., Bull. Res. Council Israel Sect. A 11,351 (1963). 18. TOm~TSV, Y., BiopoIymers 2, 275 (1964). 19. GELLEaT, M. F., AN~)ENGLANDER,S. W., Biochemistry 2, 39 (1963). 20. C~I~NG, I%., J. Polymer Sci. B2,855 (1964). 21. DE~ELIS, Gz., AND KRATO~VIL,J. P., J. Colloid Sei. 16,561 (1961). 22. KERIZ.E~,M., AND M~TIZEWd, E., J. Opt. Soc. Am. 50,722 (1960). 23. NAPPEd, D. H., AND OTTEWILL,R. H., J. Colloid Sci. 19, 72 (1964). 24. OTT:~WILL,R. It., AN1) PA~REIaA, It. C., J. Phys. Chem. 62,912 (1958). 25. KERKEE, M., DABY, E., COHEN, G., KRATOHVIL, J. P., AND M_&TIJEVIC, E., J. Phys. Chem. 67, 2105 (1963).
CALIBRATION OF LIGHT-SCATTERING I N S T R U M E N T S IV. CORRECTIONS FOR REFLECTION EFFECTS 1 Josip p. Kratohvil 2 Deparlment of Chemistry, Clar~son College of Technology, Potsdam, New York Received February $2, 1965 ABSTRACT A more general expression than that given by Tomimatsu and Palmer is derived which corrects the light scattering intensity in different cells for reflections at the walls. Effects at both the glass-air and glass-liquid interfaces are considered. This expression requires intensity measurements at two supplementary angles and knowledge of the Fresnel coefficients. The magnitude of the correction has been calculated for values of the dissymmetry up to 26.5. For high dissymmetries, it is not possible to neglect the reflections at the liquid-glass interface as is usually done. Also, it is pointed out that there are upper limits for the dissymmetry beyond which it is not possible to apply these corrections. The importance of taking these corrections into account has been demonstrated by considering a few examples from the literature where application of a proper reflection correction will change the results significantly. INTRODUCTION T o m i m a t s u and P a l m e r (1) have recently discussed the reflection corrections in light scattering m e a s u r e m e n t s for the Brice-Phoenix p h o t o m e t e r (2). Their expressions are applicable, u n d e r certain assumptions, to glass cells of various shapes. W e have been working on the same problem in connection with light scattering studies on large particles (polymer latexes, hydroso]s), and would like here to provide (a) a m o r e general presentation of the physical problem; (b) some q u a n t i t a t i v e results for several situations c o m m o n l y encountered in practice; and (c) some applications of the expressions derived to certain light scattering results from the literature. T h e literature survey on the problem of reflection effects in light scattering m e a s u r e m e n t s is given b y T o m i m a t s u and P a l m e r (1, 3). 1 Supported by the Atomic Energy Commission Contract No. AT(30-1)-1801. 2 The author has had the pleasure of discussing the problern of reflections in light scattering measurements with Professor V. K. La Mer during the Los Angeles A.C.S. meeting in April, 1963. The preparation of this communication was stimulated by this discussion. 498
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS (*4 5} 135
*
.
..k.i/'T "~
*45
499
°
0,~
Fro. 1. Cross section of a cylindrical glass cell in the plane of scattering. The cell is equipped with plane parallel entrance and exit windows. All faces are clear, In the middle, the scattering envelope for the incident beam (2) (full curve) and for the reverse reflected incident light (2' + 3') (dashed curve) are pictured schematically. Tile angles designated with "minus" are those on the side facing the observer (detector). The angles in parentheses refer to the reverse envelope. The heavy arrows and unprimed numbers indicate the light rays incident on reflecting surfaces as well as those transmitted through them. The dashed arrows and primed numbers refer to the reflected rays. 1: t j o 1': f¢~Io; 1": f J o 2: t~t~Io 2': f ¢~t,,Io 3: tat~21To 3~: fatat[2Io 4: t~2tz~Io 2' -P 3': tat~(fz "~- hfa)Io For 5-9 see text. THE 1DFIYSICAL PROBLEM AND FORMULATIONS
The physical problem is illustrated schematically in Fig. 1, which represents the cross section of the glass cell in the plane of scattering. This applies to a symmetrical cylindrical transparent glass cell with flat entrance and exit windows. This geometry has been selected for two reasons. Firstly, in this type of scattering cell, tile maximum number of reflections are involved and the influence of these on the measured scattering properties is most pronounced. The expressions for other types of cells can easily be derived from this case. Secondly, the situation presented in Fig. 1 eor-
500
KRATOHVIL
responds very closely to a widely used commercial cell, the only difference being that these cells are usually made with a frosted inner back surface (4). As will be shown later, this frosting requires a slight change in the geometry of Fig. 1. If a parallel beam of light is perpendicularly incident on the entrance window, all the reflections may be treated as occurring at perpendicular incidence. Even those extreme scattered rays which originate in the finite illuminated volume located at the center of the cell and which reach the detector will have reflectivities and transmissivities almost the same as the rays which impinge on the cell walls at perpendicular incidence. This follows from Fresnel's equations, for which the reflectance remains very nearly constant (within 0.6 %) from perpendicular incidence to angles of incidence up to about 20 ° (5, 6). Also, over the same angular range, there is very little difference between the two polarized components. It is convenient that corrections for reflection effects are most important in the case of scattering by large particles. For such particles, the angle of acceptance of the phototube, and hence, the scattering volume, must be reduced below a certain value in order to obtain adequate angular resolution (7). This condition also assures that all reflections of concern occur at small angles of incidence. Fresnel's equations for the fractions of light, fa, and fs, reflected at perpendicular incidence at the glass-air and glassliquid interfaces, respectively, are given simply by f~ -- [(n~ - 1)/(ng + 1)]2;
[1]
f, = [(n~ - n,)/(no + n,)]~;
[2]
and
where ng and n~ are the refractive indices of the glass and the liquid. The transmission coefficients at the air-glass and liquid-glass interfaces are defined by t~ =
1 -
fo
[la]
tz =
1 -
fz.
[2a]
and
In Table I, the Fresnel coefficients are given at the two wavelengths (436 and 546 mu) most often used in light scattering measurements and for two types of glass, Pyrex and crown optical glass (Phoenix Co. Melt No. 2), of which light scattering cells are commonly made. fz was calculat.ed for water as the liquid in the cell. For most organic liquids, f~ is sufficiently close to zero and need not be considered because of the small difference between the refractive indices of the liquid and the glass. It is also assumed that the systems are nonabsorbing and scatter so
CALIBRATION OF LIGHT-SCATTERINGINSTRUMENTS
~01
TABLE I Refractive Indices of Water (25°C.) and Two Kinds of Glass Commonly Used for Light Scattering Cells, and the Corresponding Fresnel Coefficients at Two Wavelengths Dispersion was estimated from the data given in reference 6 for fused quartz and crown glass, respectively.
nz (water, 25°C.) n~ (Pyrex) f~ f~ no (optical glass, crown) f~ fz
436 mg
546 m~
1.340 1.482 O.0377 O.00253 1. 528 0.0436 O.00429
1.334 1.476 O.0370 O.00255 1.519 0.0424 O.00420
weakly that the light rays are not significantly attenuated on passage through the system. Furthermore, multiple reflections may be neglected (5, 8, 9). Owing to the reflections at the air-glass and glass-liquid interfaces of the flat entrance window of the cell, the incident intensity at the point of scattering is t~tffo, where I0 is the intensity of the incident radiation. Similarly, the intensity of the reversed incident beam, reflected at the exit window, is (t~hh + tat?f~)IO = tah(fz -t- t&)IO. This light reflected from the exit window will also be scattered but will have precisely the reverse angular dependence from that corresponding to incident light, as indicated schematically in Fig. 1. Therefore, the observed angular envelope will be the result of superposition of the two envelopes: the main one, corresponding to the incident intensity t~tffo, and the reverse envelope, eolTesponding to the reverse incident radiation t,h(f, ÷ hf~)Io. Furthermore, a fraction (f, + hfa) of the light scattered in the direction 0 is reflected at the liquidglass and glass-air interfaces at the exit point of the cell. Accordingly, the contributions from the main envelope and the reverse envelope must each be multiplied by the corresponding transmission h& • In Briee's working-standard method, which is commonly employed (1-3, 10), the galvanometer reading G~ for the transmitted intensity (0 = 0°), t~2tz2Io, is corrected to that corresponding to I0 by applying the factor T,. This is the apparent transmittance of the filled scattering cell measured in the photometer with the illuminated standard diffusor as a source. This factor is contained in the so-called residual refraction correction R ~ / R ¢ . When a symmetrical cell, symmetrically aligned, as in Fig. 1, is utilized, then the attenuation of the incident and scattered rays as they pass through the scattering system is canceled out. Thus, the true scattered intensity at an angle 0, Io, corresponding to I0, must be evaluated from
502
K~TORVm
the observed intensity Io', corresponding to the measured galvanometer reading Go. In addition to the contribution of scattering in a particular direction - 0 , the observed intensity at that angle, Io', will consist of several other contributions, depending upon the cell used. For the situation of Fig. 1, there will be four contributions. (a) Scattering of the incident beam of intensity tatzIo at angle - 0 gives a scattered beam of intensity tatzIo. This is corrected for the reflections at the exit point giving finally t~2tHo. (b) The scattering of the main beam at a scattering angle of + (180 - 0) is in turn reflected toward - 0 from the side opposite the detector. This is also corrected for reflections at the exit point and is given by t,2t~2(f, +
tzf~)I18o-o . (c) The scattering at - ( 1 8 0 -- 0) of reverse beam, corresponding to
tat~(fz + tJ~)Io, also contributes to the -O reading. When corrected for reflections at the exit point, this gives the same resul¢ as in (b) above. (d) The scattering at + 0 for the reverse envelope that is reflected from the back face toward - 0 , also corrected for reflections at the point of observation, gives ta2tz2(fz + t~f~)~Io. The observed intensity, I0 ~, is the sum of all four contributions, i.e.,
Io' = t~:t~2{[1 + (f~ + t~fa)2]Io + 2(f~ + t~f~)Ilso_o}.
[3]
Combining Eq. [3] with the corresponding expression for I~8o_o and substituting (fz -t- tzf~) = A, one arrives at the following expression for the true scattered intensity 1o, corresponding to the incident intensity I0 :
Io
=
tJt~2[(1 + A2)~ _ 4A2] •
Io'
y
[4]
Thus, in order to obtain the correct scattered intensity at an angle 0, one has to measure the intensities at two supplementary angles and to know the Fresnel coefficients. Equation [4] can be simplified considerably if contribution (d) is neglected. By summing up the first three contributions and repeating the procedure elaborated upon above, one can write instead of Eq. [4]: Ie --
1 (I0' -- 2AI;s0-e). t~2t~2(1 -- 4A 2)
[5]
The change in the numerical results is extremely small. For example, for f~ = 0.0370 and fz = 0.00255, the factor in front of the brackets has a value of 1.089 in Eq. [4] and 1.090 in Eq. [5], whereas the factor within the brackets amounts to 0.0788 and 0.0789 in the two equations. Consequently, ~11 considerations that follow will be based on the first three contributions only.
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS
503
It was mentioned earlier that the situation depicted in Fig. 1 does not correspond precisely to the one when cylindrical cells of Witnauer and Scherr (4) (Phoenix Co. catalog Nos. C-101 and C-105) are utilized, because these cells usually have frosted inner back surfaces. Since the light is diffusely reflected from such a surface, eliminating the fraction of scattered light reflected on the liquid-glass interface toward - 0 , it is obvious that the magnitude of the contribution (b) will be changed. However, it is not possible to evaluate this change quantitatively, because of the difficulty of calculating how much of the scattered light in + (180 - 0) direction is attenuated by reflection on the liquid-frosted glass interface before reaching the glass-air interface. Two other cases for the cylindrical cell of Fig. 1 should be considered. Firstly, one technique frequently used to eliminate the reflections is to paint the outside back surface dull black. However, the exit window is left clear; otherwise, Brice's working-standard method cannot be applied. For this case, only reflections at the liquid-glass interface from the backside of the cell are obtained. Therefore, contribution (b) will be smaller and equal to ta~tl2f~Ixso_o, leading to Io =
1 [Io' tJt~211 -- (fz + A) 2]
(f, + A)I~8o-o].
[6]
Secondly, if the coat of paint is applied to the inside of the back face, or if a piece of black dull plastic film is placed inside against the back curved face (1), the whole contribution (b) will be eliminated. Thus, Io -
1
ta~t~2(1
-
-
A s)
(Io' -
aIiso-o).
[7]
Another important case is the semioctagonM cell (e.g., Phoenix Co. catMog No. D-101) which is employed to measure the scattered intensities at 0 = 45 °, 90 °, or 135 °. The back face and the entrance or exit face of this cell form an angle of 90 ° and, in addition, the cemented joints are usually blackened. By tracing the rays and taking the actual dimensions of the cell and various components and distances of the Brice-Phoeuix instrument into account (11), it can be shown that rays scattered towards + 4 5 ° and +135 ° corners of the cell cannot reach the detector positioned at - 4 5 ° or - 1 3 5 ° by reflection. This has also been demonstrated experimentally by Tomimatsu and Palmer (1). Consequently, contribution (b) is entirely eliminated, regardless of whether the cell is coated with black paint or not, and Eq. [7] applies. Equations [5], [6], and [7] reduce to the corresponding expressions derived by Tomimatsu and Palmer (1) for the case treated by them in which only reflections at the glass-air interfaces are considered. For a cylindrical cell
504
KRATOI-IVIL
of Fig. 1, with clear or frosted back inner surface, the first three contributions discussed above will, then, reduce to: (a):
t~Io ;
(b):
t~2faI18o_o ;
and
(c):
ta2fJ18o_o.
From this, it follows that Io -
1 (Io' -- 2faI;8o-o). 1 - 2fa
[8]
It should be mentioned that Tomimatsu and Palmer (1) offered the experimental evidence that the frosted inside surface does not affect reflections on the glass-air interface. For a cylindrical cell with black painted back face, either outside or inside, contribution (b) is eliminated, and instead of Eq. [6] or [7], one obtains Ie -
1--
1
2fa
(Ie' - - / ~ I~8o-0),
[9]
which is also applicable to a semioctagonal cell for the same reasons that Eq. [7] was applicable. Except for the factor 1/(1 - 2f~), Eq. [9] was originally suggested by Sheffer and Hyde (12) and by Oth, Oth, and Desreux (13). The latter authors have also attempted to incorporate the liquid-glass reflections, with essentially the same result as Eq. [7], but without the factor 1/t~h2(1 - A2). Equations [8] and [9] differ slightly from those given by Tomimatsu and PMmer (1) in the first factor on the right-hand side of these equations. This difference arises because secondorder effects (f~ and higher powers in f~) have been consistently neglected. When the solvent is water, these become comparable in magnitude to values of fz. The above discussion should make apparent the kind of corrections that may be required in other situations in which the experimental procedure is changed or in which other reflecting surfaces are introduced (as, e.g., windows for thermostating ]ackets, polarizers or analyzers, various filters). It is worth while considering what happens when Iot = I~8o-o, i.e., when the observed angular envelope is symmetrical. If glass-air reflections only are considered, Eq. [8] gives Io = Io' at all angles. However, when Eq. [9] is applicable (cylindrical cell with back face painted black for all angles, or semioctagonal cell for angles - 4 5 ° and - 1 3 5 ° with opposite corners unpainted or painted and, with back face painted, for angle -90°), one obtains Io/Io' = t . / ( 1
--
2f~).
[10]
Tomimatsu and Palmer (1, 3) have already emphasized that at 0 = --90 ° and for cells with clear faces, all glass-air reflections are self-compensatory (neglecting f 2 and higher powers of fo). They also confirmed experimentally
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS
505
t h e v a l i d i t y of E q . [10] for all angles, as w e d i d for 90 ° in a p r e v i o u s p a p e r (14). F o r some a d d i t i o n a l e x p e r i m e n t s of this t y p e , see reference 15. Values of t h e r a t i o I o / I o ' for t h e ease w h e n t h e liquid-glass reflections are i n c l u d e d a r e listed in T a b l e I I . F o r t h e ease of Eq. [5], t h e r e is h a r d l y a n y a d d i t i o n a l effect, t h e r a t i o b e i n g 1.004 i n s t e a d of 1.000. T h e e q u a t i o n s d e v e l o p e d here for c o r r e c t e d i n t e n s i t i e s (Eqs. [4] t h r o u g h [9]) can b e p u t in t h e g e n e r a l f o r m : Io -- X ( I o '
-
[11]
YI;8o-o),
w h e r e X a n d Y a r e c o n s t a n t s d e p e n d i n g o n l y on fa a n d f~ for a p a r t i c u l a r g e o m e t r i c a l - o p t i c a l s i t u a t i o n . T a b l e I I s u m m a r i z e s t h e e q u a t i o n s for v a r i o u s eases a n d gives n u m e r i c a l v a l u e s of X a n d Y for fa = 0.0370 a n d f~ = 0.00255 ( P y r e x a n d w a t e r a t X0 = 546 m#), a n d t h e v a l u e s of t h e r a t i o I o / I o ' = X ( 1 - Y) w h e n Io' = Ii8o-e. I t is o b v i o u s t h a t besides t h e a n g u l a r s c a t t e r i n g intensities, o t h e r l i g h t s c a t t e r i n g functions, such as d i s s y m m e t r i e s a n d p o l a r i z a t i o n ratios, will also be influenced b y reflection effects. C o r r e c t e d d i s s y m m e t r i e s will a l w a y s be h i g h e r t h a n t h e o b s e r v e d ones. T h e m a n n e r in w h i c h t h e p o l a r i z a t i o n TABLE I I A S u m m a r y of Equations Developed for Various Geometrical-Optical Arrangements
The numerical values are those forf~ = 0.0370 andfz = 0.00255. t~ = 1 - f~ ; h = 1 - fz ; A = f ~ + h f ~ . IO/I'O = X
1/tJh~(1 - 4.4 2) Eq. [51: Cylindrical cell, clear faces 1/ta2h2[1 -- (fl + A) 2] Eq. [6]: Cylindrical cell, painted outside 1/&2ta2(1 -- A 2) Eq. [7] : Cylindrical cell, painted inside; or semioct, cell (45 °, 135 °) 1/(1 -- 2f~) Eq. [8]: Cylindrical cell, clear faces or frosted inside; fz = 0 1/(1 -- 2fa) Eq. [9]: Cylindrical cell, painted outside or inside, or semioct. cell;f~ = 0
Y
= 1.090
x(1 -
Y)
when,Io' =
2A = 0.0789
1.004
= 1.085 fz + A = 0.0420
1.039
= 1.085
A = 0.0395
1.042
= 1.080
2f~ = 0.0740
1.000
= 1.080
f , = 0.0370
1.040
506
KRATOI%IVIL
ratios are affected will depend n o t only on the a s s y m m e t r y of the scattering envelope b u t also on the intensities of b o t h the horizontal and vertical components of scattered light at two s u p p l e m e n t a r y angles, i.e., on four different intensities (this point will be illustrated in the last section). F o r corrected dissymmetries Zo one can write quite generally
Zo-
Io _ I o ' - - YI[+o-o _ Z / Y Ilso-o Iiso--o- YIo' 1- YZo"
[12]
where Zo' is the observed uncorrected d i s s y m m e t r y ( = Io'/I'~so_o). Similarly, for corrected polarization ratios one obtains
Ho Oo -
Vo
Ho'--
Vo'
-
-
YH~so-o ' ' YVlso-o
[13]
where H / and V / designate the observed horizontally and vertically polarized components and H0 and V0 are the corrected quantities. T~E N u ~ m c A L
ILLUST~mIO~S
I n Tables I I I and IV, values of the angular intensities corrected for reflections according to Eqs. [5] to [9] are listed for various a p p a r e n t dissymmetries. Corrected dissymmetries are also given. T h e reflectivities used were f~ = 0.0370 and f~ = 0.00255. I t is obvious t h a t these corrections are v e r y important, particularly TABLE I I I The Magnitude of the Reflection Correction on Scattered Intensities and Dissymmetries as Calculated from E,qs. [5] and [8] ! f~ = 0.0370;fz = 0.00255. In all cases it is assumed that 118o-o = 1.
Io'/I[so.o -To 1 2 3 4 5 6 7 8 9 10 11 12 12.67 13 13.50
1. 004 2. 093 3.182 4.272 5. 363 6. 453 7.542 8. 634 9.722 10.81 11.90 13.00 13,72
Eq. [Sj 118o-o 1. 004 0. 918 0. 832 O. 746 0. 660 0. 574 0.488 O. 402 O. 316 O.230 0.144 0.058 0
Io/118o-o 1. 000 2. 280 3. 825 5.727 8.126 11.25 15.46 21.48 30.77 47. O0 82.65 224.2 ¢~
Io
Eq. [8] I18o-o
Io/11~o-o
1. 000 2. 080 3.160 4.240 5. 320 6. 400 7.480 8.560 9.640 10.72 11.80 12.88
1. 000 0. 920 0. 840 O. 760 0. 680 0. 600 0.520 O.440 O. 360 O.280 0.200 0.120
1. 000 2. 261 3. 762 5.578 7. 824 10.67 14.39 19.46 26.77 38.30 59.00 107.3
13.96 14,50
0.040 0
348.9 ¢~
CALIBRATION OF LIGHT-SCATTERING
INSTRUMENTS
507
TABLE IV The Magnitude of the Reflection Correction on Scattered Intensities and Dissymmetries as Calculated from Eqs. [6], [7], and [9] f~ = 0.0370;fz = 0.00255. In all cases it is assumed t h at Izs0_0 ' = I.
Eq. [6]
[e'/
l~so-e 1 2 4 6 8 10 12 14 16 18 20 22 23.81 24 25.32 26 26.50
Eq. [7] Ilse-e
Eq. [9]
le
I18o-o Ie/I*so-e
Ie
Ie/Ilso-e
1.040 2.123 4.293 6.462 8.633 10.80 12.98 15.15 17.31 19.49 21.67 23.82 25.79
1.040 1.000 0.994 2.137 0.903 4.755 0.812 7.960 0.721 11.97 0.629 17.17 0.538 24.12 0.447 33.88 0.356 48.63 0.265 73.52 0.174 124.5 0.0825 288.7 0 co
1.042 2.128 4.297 6.465 8.638 10.81 12.98 15.15 17.32 19.49 21.67 23.82
1.042 1.000 0 . 9 9 9 2.131 0.914 4.702 0 . 8 2 8 7.810 0.742 11.64 0 . 6 5 6 16.48 0.571 22.73 0 . 4 8 5 31.23 0.399 43.42 0 . 3 1 4 62.05 0.228 95.00 0.142 167.8
26.00
0.05641461.0
27.42
0
I0
Ilso-e Ie/Ilso-o
1.040 2.120 4.280 6.440 8.600 10.76 15.08 17.24 19.40 21.56 23.72
1.040 1.000 1.000 2.120 0.920 4.652 0.840 7.665 o. 760 11.32 o. 680 15.83 o. 6oo 21.53 o. 520 29. oo o. 440 39.18 o. 360 53.87 0.289 77.00 0.200 118.6
25.88
0.120 215.7
28.04 28.58
0.040 701.0 0 co
12.92
co
when the dissymmetry is large. High dissymmetries such as those tabulated are often found for colloidal dispersions (hydrosols, polymer latexes, aerosols, biological corpuscles like mitochondria (16)), and even for some macromolecular solutions, as, e.g., glycogen (17). For such cases, careful consideration must be given to which form of the reflection correction is applicable. Tomimatsu (18) has already emphasized the differences between corrected intensities obtained using the formula of Sheffer and H y d e (12) and those obtained using his formulation. B y inspecting Tables I I I and IV, several observations are apparent. Firstly, when the refractive index of liquid is as low as that of water, reflections at the liquid-glass interfaces become important, especially for higher dissymmetries. The inclusion of these liquid-glass reflections leads to an increased value of Y. This has the apparent effect of increasing fa in those expressions which do not incorporate the liquid-glass reflection and explains the observation of Tomimatsu (18, and private communication), who, considering only the glass-air interface, presumed to measure the values of fa for Phoenix Pyrex cylindrical cells filled with water. He found an average value of 0.045 instead of 0.0377 at k0 = 436 mr* (Table I). Table IV also demonstrates that for high dissymmetries the results
508
~TO~VIL
differ when a coat of black paint is applied on the outside (Eq. [6]) or inside (Eq. [7]) of the back face. We have also indicated in Tables I I I and IV the limiting values of Io'/I~so_o for which the corrected IlS0-e becomes zero and hence the corrected dissymmetries become infinite. To measure dissymmetries higher than these limiting ones, it is necessary to eliminate, partly or completely, the source of the reflections either by changing the shape of the cell or by applying a black coat onto the back face and/or on the exit window. An alternative procedure is to insert black glass or plastic film within the cell, or to mount a Rayleigh horn. A word of caution is advisable in regard to a coat of paint on the back face. The paint used must be a nonglossy, dull black that adheres strongly to the glass and, upon drying, does not crack and transmit light. I t is always desirable to check experimentally, with the solvent or solution in the cell, how effective the coat of paint is in removing the reflections from a particular face. Otherwise, it appears to be a more reliable procedure to leave all the faces clear and to correct for the reflection effects with the appropriate equation. Since the corrected intensities in the backward directions and, thus, the dissymmetries are very sensitive to small changes in the observed dissymmetry, these should be measured as accurately as possible. For Ion~ I~_o = 10, the corrected dissymmetries are 47.0 (Eq. [5]) and 38.3 (Eq. [8], i.e., taking only glass-air reflections into account). However, if the observed dissymmetry had been 10.5, a corrected value of 46.7 is obtained from Eq. [8]. In other words, a 5 % experimental error in observed dissymmetry might result in a 25 % error in the corrected dissymmetry, and this in turn is equivalent, in this ease, to the effect of including the glass-water reflections. Of course, an experimental error of 5 % is not uncommon. By the same token, the refractive index of the glass must be known rather accurately, and its wavelength dependence must be considered. If, instead of A = 0.0370 for Pyrex glass (Table I) (X0 = 546 m~), a value off~ = 0.0377, corresponding to X0 = 436 m~, is used in correcting the observed dissymmetry of 10 by means of Eq. [8], a corrected value of 40.35 is obtained, instead of 38.3. It should be emphasized that for cells made of glass of still higher refractive index, the effects of the water@ass reflections become even more pronounced. For crown optical glass (Table I) at X0 = 546 m~ (fa = 0.0424 and fl = 0.0042), one obtains by applying Eq. [8] (glass-air reflections only) a corrected dissymmetry of 38.8 (see above) from an uncorrected dissylmnetry of 9.04. However, if full expression [5] is used, the corrected value is 55.6. Even for smaller observed dissymmetries, the differences in corrected values due to the inclusion of glass-water reflections are significant.
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS
509
As the data of Tables III and IV demonstrate, the corrected forward intensities are higher, and corrected backward intensities lower, than the corresponding measured intensities (see also reference 18). This could have an important implication with regard to sonie of the angular scattering data in the literature. Often one finds Zimm diagrams with pronounced downward curvature for the reciprocal intensities at backward angles. Sometimes this is attributed to the polydispersity of the material under study, as well as to branching or chain stiffness in case of polymer coils. Since reflection corrections for the backward intensities are greater than for those in forward direction, it is tempting to wonder how many of such "distorted" Zimm plots might be straightened out upon applying the proper correction. In connection with the angular measurements, it should be added that in almost all the light scattering instruments which have been described in the literature, it is not possible to measure the intensities at backward angles higher than about 135 °. Since the reflection corrections require the knowledge of scattered intensities at two supplementary angles, these cannot be evaluated for angles lower than 45 °. Fortunately, the general equation [11] reduces with good approximation to I~ = X I / f o r ~ < 90 °, particularly when the scattering envelope is highly asymmetric. APPLICATIONS
In this section we would like to consider briefly a few examples from the literature where the application of a proper reflection correction will change the results significantly. Although our suspicion is more than strong that there are many more examples, the information necessary to estimate whether a correction has been applied at all or even which expression would be the most appropriate is usually lacking. Tomimatsu (18) has applied Eq. [8] to correcting the Zimm plots of Gellert and Englander (19) for rabbit myosin A. The molecular weights derived from corrected data were about 15 % higher, and in agreement with those obtained by means of the sedimentation equilibrium method. The radius of gyration and the second virial coefficient were also affected. Chiang (20) attempted to explain discrepancies in the light scattering molecular weights of some polyethylene fractions as reported in the literature by correcting the data for reflection effects in the manner suggested by Othet al. (13). However, it seems to us that the equation given by Oth et al. is inadequate for the case Chiang discusses. Instead, our Eq. [8] is applicable. A most dramatic demonstration of the utmost importance of a proper evaluation of reflection effects is provided by some results on polystyrene latexes. Although De~eli6 and Kratohvil (21) have properly corrected the dissymmetry values (compare Tables VI and VII of reference 21), the
510
KRATOItVIL
reflection correction was not applied to the intensity curves from which the angular positions of maxima and minima in intensity were determined. For Dow polystyrene latex LS-067-A(D~ = 1.171t~) at X0 = 436 m~, theoretical calculations indicated that in the backward direction a maximum should occur at 130° and a minimum at 135°. These were not observed (compare Fig. 6 and Tables IXa and IXb of reference 21). However, after applying the appropriate correction (Eq. [8]), we now find the following intensities (on a relative scale) in the angular range 95 ° to 140° (the first number refers to uncorrected, the second to corrected intensity): 95°--4.17, 3.93; 100°--3.06, 2.53; 105°--2.72, 2.24; 110°--3.80, 3.37; 115°--3.72, 3.04; 120°--3.10, 2.00; 125°--3.55, 2.53; 130°--4.17, 2.88,135°--4.57, 1.60; 140°--6.17, 2.94. Thus, a maximum at 130 ° and a minimum at 135 ° now appear in accordance with the theoretical prediction for this latex. In previously published work from this laboratory on four polystyrene latexes (22), reflection corrections were completely ignored and large discrepancies between the experimental and theoretical vMues for dissymmetry were observed. These had been attributed to multiple scattering effects. However, we have now corrected the experimental values for reflection effects and the discrepancies have been reduced, although not completely eliminated. Indeed, the present disagreement may now be due to the acceptance of the electron-microscopic sizes, as supplied by Dow Co., as correct. By adjusting these sizes by only a few per cent, it is now possible to bring the corrected experimental values and the theoretical ones into very close agreement. We shall not elaborate on this point since it will be the subject of a more thorough discussion in a forthcoming publication. The same comments apply also to the results of Napper and Ottewill (23) on polystyrene latexes. These authors have asserted that in their experiments the reflection correction was negligible. However, although in the instrument they used (24) the cell was immersed in a liquid bath, the incident beam is still ultimately reflected back into the cell from the exit window of the bath. Finally, an example of the influence of reflection effects on the measured polarization ratios po will be given. A method of size distribution analysis based on the variation of polarization ratio with the angle of observation has been recently applied to La Mer sulfur sols (25). It had been stated then that the reflection effects on the measured o0 values were negligible. This conclusion was based on a check performed on several sols. However, at that time we did not recognize that the reflection correction depends on four different intensities (horizontally and vertically polarized components at two supplementary angles), and, therefore, must be checked for every case. All previous measurements discussed in reference 25 have been corrected accordingly, and in some cases significant differences have been found between the uncorrected and corrected o0 values. An example is given
CALIBRATION OF LIGHT-SCATTERING INSTRUMENTS
511
TABLE V Uncorrected and Corrected Polarization Ratios Po for a La Met Sulfur Sol (PS) PO
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130
Uncorrected
Corrected
1.211 0.765 0. 665 0.832 2.918 2.251 0.831 0.831 3.268 3.945 1.000 1.148 3. 553 4.378 1.925 2.745 2. 303 1.047 2.255 3. 873 1. 592
1.211 0.765 0.665 0.832 2.967 2.219 0.816 0.827 3.317 4.011 0.979 1.072 3.553 5.440 2.235 2.599 2.177 1.162 4.121 4. 537 1. 343
in Table V. The corrected p0 values may be either larger or smaller than the uncorrected ones. Again, the differences are very pronounced at the backward angles. Fortunately, however, in none of the cases analyzed were the parameters of the size distribution function significantly affected by these changes. I:~EFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
TOMIMATSU, Y., AND PALMER, K. J., or, Phys. Chem. 67, 1720 (1963). BtUCE, B. A., ttALWER, M., AND SPEISER, R., J. Opt. Soc. Am. 40,768 (1950). TOMIMATBU,Y., AND PALMER, K. J., or. Polymer Sci. 54, $21 (1961). WITNAUER,L. P., AND SCHERR, H. J., Rev. Sci. Instr. 2S, 99 (1952). JENXINS, F. A., AND WI~ITE, H. E., "Fundamentals of Optics," 2nd ed. McGrawHill, New York, 1950. "H~ndbook of Chemistry and Physics," 44th ed., p. 3090. The Chemical Rubber Publishing Co., Cleveland, 1963. TA~IBIAN, R. M., AND HELLER, W., J. Colloid Sci. 13, 6 (1958). DEBVE, P. P., Ph.D. Thesis, Cornell University, Ithaca, New York, 1944. STEIN, R. S., AND DorY, P., J. Am. Chem. Soc. 68, 159 (1946). TOMIMATSU,Y., AND PALMER, I~. J., J. Polymer Sei. 35, 549 (1959). I~ERKER, M., KRA~OHV~L, J. P., AND MATIJEVI~, E., or. Polymer Sci. A2, 303 (1964).
512
K~TO~WL
12. S~EFFE~, H., AND HYDE, J. C., Can. J. Chem. 30, 817 (1952). 13. OTH, A., OTH, J., AND DES~EUX, V., J. Polymer Sci. 10,551 (1953). 14. KEaxEa, M., K~ATO~VIL, J. P., OTTEWI•L, 1%. H., ~ I ) MATIJEVI6, E., J. Phys. Chem. 67, 1097 (1963). 15. MCI~TY~E, D., g. Res. N. B. Std. A68, 87 (1964). 16. GOTTERER, G. S., THOMSON,T. E., AND LEHNINGER,A. L., J. Biophys. Biochem. Cytol. I0, 15 (1961). 17. L~SKOV, R., AND I_V[ARGOLIASH,E., Bull. Res. Council Israel Sect. A 11,351 (1963). 18. TOm~TSV, Y., BiopoIymers 2, 275 (1964). 19. GELLEaT, M. F., AN~)ENGLANDER,S. W., Biochemistry 2, 39 (1963). 20. C~I~NG, I%., J. Polymer Sci. B2,855 (1964). 21. DE~ELIS, Gz., AND KRATO~VIL,J. P., J. Colloid Sei. 16,561 (1961). 22. KERIZ.E~,M., AND M~TIZEWd, E., J. Opt. Soc. Am. 50,722 (1960). 23. NAPPEd, D. H., AND OTTEWILL,R. H., J. Colloid Sci. 19, 72 (1964). 24. OTT:~WILL,R. It., AN1) PA~REIaA, It. C., J. Phys. Chem. 62,912 (1958). 25. KERKEE, M., DABY, E., COHEN, G., KRATOHVIL, J. P., AND M_&TIJEVIC, E., J. Phys. Chem. 67, 2105 (1963).