Errors in extinction measurements as due to scattering of light by solutions of relatively small particles

JOURNAL OF COLLOID AND INTERFACESCIENCE 21, 1-8 (1966)

E R R O R S IN EXTINCTION M E A S U R E M E N T S AS SCATTERING OF LIGHT BY S O L U T I O N S RELATIVELY SMALL PARTICLES 1

DUE OF

TO

Karel P. M. Heirwegh ~ Laboratorium voor Biochemie (II), University of Leuven, Belgium Received February 26, 1965 ABSTRACT Extinctions measured on macromolecular solutions will generally be too low owing to the addition of scattered light to the transmitted light beam. E v e n in solutions with predominantly Rayleigh scattering the values are affected by a relative systematic error of 2% or more. The relative error is independent of the molecular weight for limitingly small particles. At infinite dilution it is maximal and solely determined by geometrical characteristics of the spectrophotometer used. INTRODUCTION

The extinction A' 3 of a solution will generally be due to absorption (Aa) and to scattering of light (As'). For solutions of proteins and of other kinds of macromolecules A' has been shown to result from the mere addition of Aa and As', from the visible to the near ultraviolet (1-3). From Mie's equation (4) it also follows that the influence of complex refractive indices will generally be negligible for proteins, even in the protein band (about 280 m~). Thus

A' = Aa -~- As'.

[1]

Unless special precautions are taken part of the laterally scattered light leaving the cuvette will reach the photocell (Fig. 1), thereby falsely increasing the transmitted light. This will consist of reflected light and of light scattered from the beam (primary scattering) and from the solution (rescattered light). Such an effect was first observed by Pitt (5) in 1938 and was used for the determination of color in turbid solutions (6-9). Important deviations from Beer's law encountered in a study of powder suspensions were shown to be due to the same phenomenon (10). Making use of suspensions of latex particles with diameters ranging from 163 to i This work was supported by a grant from the Nationaal Fonds voor Wetenschappelijk Onderzoek van Belgie. Present address: Laboratorium voor Fysiopathologie, Dienst voor Inwendige Geneeskunde, Rega Instituut, Minderbroedersstraat, Leuven, Belgium. 3 Quantities which are affected by the nonideal character of the solution will be marked by the index ('). 1

2

HEIRWEGH P z

Wl

I

dx

01

02

x

y .Di.

FIG. 1. Path followed by a light ray scattered from a line element dx situated in the incident light beam. R, the radius and 61, the apparent radius of the photocell window; ~(x), the limiting angle.

824 mt~, Heller and Tabibian (11) examined the effects of reflection, of primary scattering, and of rescattering of light on the determination of turbidity with an apparatus specially designed for that purpose. The applicability of the Beckman D U spectrophotometer for turbidity measurements in this particle range was also discussed. I t seemed of interest to consider the case of particles with predominant Rayleigh scattering to make possible an assessment of the importance of erroneous transmission on the determination of extinction in such cases. For this purpose some natural globular products with a diameter smaller than 40 m# were used. A mathematical formula developed to treat the phenomena further makes possible the estimation of erroneous transmission for limitingly small particles. EXPERIMENTAL

Helix pomatia hemocyanin (molecular weight 9 X l0 G) was prepared as previously described (12). A commercial glycogen preparation (Poulenc) 4, 4 A sample of this preparation was kindly given to us by Professor P. P~tzeys. In contrast to the natural product the preparation showed a pronounced concentration dependency of the specific scattering power (13).

ERRORS IN EXTINCTION MEASUREMENTS

3

purified by precipitation with methanol in the presence of a small amount of calcium chloride (13), was subjected to electrodialysis. Dilutions were prepared after centrifuging the glycogen solution twice in a Spinco Model L ultracentrifuge (4°C., 60 min., 25,000 g.). On a sample of the solution a straight log A' - log X plot was obtained between 320 and 546 m~. Glycogen concentrations were determined by dry weight determination. The average molecular weight of this preparation was shown to be about 3.5 X 106 by light-scattering measurements (14-16). Extinctions were measured at room temperature with 0.5- and 1-cm. cuvettes in a Beckman DU spectrophotometer with the sensitivity in the middle of its range. By means of a set of Bakelite rings and long thumb screws the distance Di between the window of the phototube and the entrance window W1 of the cuvettes (Fig. 1) was made adjustable to 7.43 (D2) and 11.43 cm. (Da). The usual distance D1 is equal to 3.43 cm. Corresponding values of the extinction are denoted by AI', A2', and A3'. For hemocyanin and for the glycogen preparation the difference between A2' and A3' was small as compared to that between AI' and A2'. By graphical extrapolation values of A' at infinite distance (zero solid angle) were then obtained. All diluted solutions were centrifuged for 20 rain. at 15,000 r.p.m, in a water-cooled Pirouette (Phywe, GSttingen, Germany) and carefully siphoned off prior to the extinction measurements. RESULTS AND DISCUSSION

In Table I results are given for hemocyanin and for glycogen. For the usual setup of the Beckman DU spectrophotometer the decrease of the extinction A ' - AI'(A~A') sometimes amounted to 4 % of the extrapolated values A'. The interpretation of the data is complicated by the measurable contribution of secondary scattered light to the transmitted light beam and by the variable proportion of A s ' / A ' for hemocyanin. The phenomena due to Rayleigh scattering can be discussed as follows. A quantity called the primary scattering fraction ai is defined in terms of the energy Es' radiated from the light beam and of the part of it E~,i intercepted by the photocell, a~ = E~,~/E~'. The relationship between the intensity I~' falling onto a line element dx and the intensity io' radiated from it under an angle 0 and measured at a distance r (Fig. 1) is given by (16-18). I' ~e" = B 7~ P(0) (1 + cos2 0),

[2]

where P(8) is the particle scattering factor. Already at very low salt concentrations B is independent of x and ~ for solutions of globular macromolecules (18). An expression for E~' is then easily found: , 16 (1 - e-~'l) E~ = 3 - 7rBIo Q K' '

[3]

4

HEIRWEGH

TABLE I Extinction as a Function of the Distance D ~ : for Helix pomatia hemocyanin the series I and I I (0.1 M Sodium Acetate, pH 5.50) and I I I (0.1 M Phosphate Buffer, pH 7.10); for Glycogen the series I V (0.08 M NaCl) and V (~ < 0.001) A s'/A'

X(m#)

278 0.56

315

0.30

346

II

0.47

405

III

0.47

405

IV

1.00

330

c (g./t.)

l(cm.)

O. 500 0.202 1.000 0.500 0.500 0.500 0.202 0.202 1.000 O. 500 0.500 10.500 O. 202 i 1.000 1.002 1.001 0.500 ] 1.002 1.001 O. 500 1.000 1.000

At'

0.405 0.326 /

0.417 0.334 O. 506 O. 329 0.501 0.647 0.259 0.473 0.602 0.387 0.598 0.763 0.307

0.949 0.750 0.435 0.579 0.452

1.000

370

I 1.000 1.000

i 1.ooo V

330

1.00

370

1.000

1.013

i .000

0.720 0.536 0.594 0.424

1.000 1.000 1.000

A~ ~,AI' X 100

A'

0.972 0.753 0.437 0.583 0.454 0.263 1.021 0.727 0.541 0.599 0.428

0.973 0.753 0.438 O. 583 0.454 (0. 263) 1.023 O. 728 0.542 0.600 0.429

2.9 2.4 3.0 3.5 2.0 4.2 3.1 2.1 1.2 2.1 1.2 1.7 1.2 1.2 1.3 1.9 1.0 1.5 1.3 1.8 2.8 3.2 3.0 2.4 3.3 3.3 3.0 3.7 3.5 3.3 3.5

w h e r e K ' is d e f i n e d b y L a m b e r t ' s l a w (Ix' = I o ' e - K " * ) . T h e f u n c t i o n Q represents the particle dissipation factor (19), B y a s i m i l a r t r e a t m e n t i t c a n b e s h o w n t h a t E ' ,,~ is g i v e n b y El

,,~

=

2 r B . ]* . o . e --K* • l "5~

[4]

with 5,=

~)~

-

l +

~21-*/2

-

1 1+

7/

_1

dy,

ERRORS

IN EXTINCTION

MEASUREMENTS

5

where y -- D~ - x. T o perform the integration from 0 to ~(x) required to derive Eq. [4], the particle scattering factor P(O) and a cosine factor resulting from application of Lambert's law (i~.d = iote-~'"-x)/c°.°) were both put equal to 1. This is permissible as the error introduced is smaller than 1% (D~ 3.43 cm.; X 278 mu; rodlike particles with a length of 80 rnt~). By application of the principle of Fermat (20) a nearly linear relationship is found between (R and y for each set of the parameters D<, l, R, and n. R is the equivalent radius 5 of the photocell and n the refractive index of the solution relative to the environment. Equation [4] is then easily integrated by a graphical or a numerical procedure. From Eqs. [3] and [4] an expression is found for the primary scattering fraction

A'//

a~ = 0.863(1/Q) (10~ - 1) 3,

[5]

as e-~'~ = 10-a'. An estimate of the total amount of erroneously transmitted light can be obtained from As' and A~A' following a treatment developed by Jullander (8) and by Jullander and Brune (9). If we take into account contributions to the transmitted beam of once and twice scattered light the following relationships are valid: I'

=

l0 -- I~ -- I~';

I<' = I' + (a
[6] ff,,II I' ,,II) ;

[7]

where I0 and I ' are the intensities of the incident and the transmitted light beam. Here I <' is the intensity intercepted by the photocell and I~ and I, represent the losses of the incident light due to absorption and to scattering (primary scattering). The part of the total primary (I~') and secondary scattered light (I:,H) which reaches the photocell is given by a<-I,' and O ~ i , I I " I '~ , I I • The rescattered light being proportional to the light radiated from the t beam (I~,H = ~'i ~p ), Eq. [7] can be rewritten as I '< = I ' + (a~ + / 3 , ) - I . '

[8]

with /3< = ~"a,.~x. The quantity/3<, which corrects for rescattered light, is called the secondary scattering fraction. If we make use of an alternative expression of the additivity relationship (Eq. [1]) I ' = I0 X 10- a '

and

I 0 - - I~ = I0 × 10-a~,

AiA' is given by A,A' = log ( I ( / I ' ) = log [1 + (a< + /3,)(10 a'' -- 1)1.

[9] [10]

6 As the window of the phototubes was an ellipse with small eccentricity an equivalent radius R was calculated corresponding to the radius of a circle with the same surface.

6

HEIRWEGH

T h e a p p r o x i m a t e form A,A' = 0.868(a, -t- f~,)(10 * ' ' -- 1)[2 -t- ( a , q- /3~)(10 A ' ' -

1)] -1

[11]

is valid w h e n A ; is smaller t h a n 0.8 a n d ( a i q- /~i) smaller t h a n 0.05. T h e e x p e r i m e n t a l scattering fractions ( a l q- /3~) for the glycogen solutions are slightly higher t h a n the theoretical values, a~, i n d i c a t i n g t h a t a m e a s u r a b l e a m o u n t of secondary scattered light is intercepted b y the photocell ( T a b l e I I ) . T h e differences, however, are too small to allow a n i n t e r p r e t a t i o n of the secondary scattering fractions, as the accuracy on TABLE II Comparison of Experimental (al + ~ ) and Theoretical (a,) Scattering Fractions for Solutions of Hemocyanin a (Series I) and of Glycogen (Series II) D1 3.43 cm.; R 1.1 cm.; n (water, 20°C) equal to 1.3642, 1.3543, 1.3520, 1.3499, 1.3452, and 1.3422 at 278, 315, 330, 346, 370 and 405 m~, respectively.

I

l(cm.)

~(m~)

AI'

A'

a~ -b ~

a~

~

1.000

315 346 405

0.500

278 315

0.491 0.595 0.172 0. 339 0.647 0.405 0.317 0.490 0.620 0.378 0.591 0.749 0.589 0.326 0.251 0.463 0.303 0.945 0.729 0.425 0.991 0.701 0.523 0.568 0.439 0.256 0.580 0.414

0.506 0.602s 0.173s 0. 343 0.659 0.417 0.329 0.501 0.647 0.387 0.598 0.763 0.596 0.334 0.259 0.473 0. 307 0.973 0.753 0.438 1.023 0.728 0.542 0.583 0.454 0.263 0.600 0.429

0.036 0.033 0.026 0. 023 0.027 0.120 0.047 0.025 0.046 0.058 0.031 0.042 0.022 0.105 0.045 0.028 0.036 0.008 0.012 0.018 0.008 0.015 0.019 0.013 0.019 0.025 0.016 0.021

0.016 0.014 0.024 0. 020 0.013 0.015 0.017 0.013 0.011 0.015 0.012 0.009 0.011 0.015 0.017 0.013 0.015 0.007 0.011 0.016 0.007 0.011 0.014 0.013 0.016 0.020 0.013 0.020

0.020 0.019 0.002 0. 003 0.014 0.105 0.030 0.012 0.035 0.043 0.019 0.033 0.011 0.090 0.028 0.015 0.021

346

0.202

II

1.000

405 278 315 346 330

370

Values of the particle dissipation factor Q were taken from Dory and Steiner (19) for spherical particles with a diameter of 42.5 m~ (21, 22). Values of A,'I/A ' were taken from a previous paper (12).

ERRORS IN EXTINCTION MEASUREMENTS

7

(a~ q- fl~) is severely limited by the error in &A'. For hemocyanin rescattering of light was important, especially at 278 mt~. A qualitative analysis of the concentration and the wavelength dependency of the fl~-values also showed that they are predominantly due to secondary scattering. A study of the phenomenon of rescattering of light could possibly be made along these lines. Consider the limiting case, for which the selective absorption is zero (A' = A,') and the rescattered light negligible (fl~ << a~). It is further supposed that the greatest dimension of the particles does not exceed one twentieth of the incident wavelength (Q ~ 1 ). These conditions are reasonably well approximated in the nonabsorbing region of the visible and the near- ultraviolet for low concentrations of most globular proteins. An expression for the relative error A,A'/A,' is obtained by combination of Eqs. [5] and [11].

A~A'/A/

= 0.75(5~//)[2 +

0.864A/(5~/1)] -1.

[12]

The relative error is independent of the molecular weight of the lightscattering particles. At infinite dilution it is maximal and solely determined by the geometry of the spectrophotometer, except for a minor change in the apparent radius ~ of the window of the phototube. With a setup such as used in the Beckman DU spectrophotometer 5~/l is about 0.073, 0.060, and 0.054 for 1-cln., 0.5-cm., and 0.2 cm.-cuvettes, respectively. The relative error is about 2.5% F at infinite dilution and slowly decreases with increasing values of A/. or practical work with synthetic or biological polymers the error could easily be made negligible by reducing the size of the window of the phototube, e.g., to R about 0.3 cm. (introduction of optical stops). For a given set of experimental conditions and at a given value of As' the error will decrease with increasing values of Aa. It will increase if the scatterers do not behave like ideal point oscillators (Q < 1 ) and if rescattering of light becomes important. The situation at high particle weights (diameter from 163 to 824 mt~) is well illustrated by the observations of Heller and Tabibian (11). ACKNOWLEDGMENTS We wish to express our t h a n k s to Professor R. Lontie for his interest in this work. Our t h a n k s are also due to Drs. A. R a h m a n and E. De Wolf for helpful discussions of problems involved in the m a t h e m a t i c a l t r e a t m e n t . REFERENCES 1. 2. 3. 4.

SCHRAMM,G., AND DANNENBERG,H., Ber. Deut. Chem. Ges. 77, 53 (1944). TREIBER, E., AND SCHAUENSTEIN, E., Z. Naturforsch. 4b, 252 (1949). SCI:[AUENSTEIN, E., AND BAYZER, H., J. Polymer Sci. 16, 45 (1955). MIE, G., Ann. Physik 9.5, 377 (1908).

8 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

HEIRWEGH PITT, Phot. J. 78, 486 (1938) (after Dognon, 1943). DOGNON, A., Rev. opt. 19,205 (1940). DOGNON, A., Rev. opt. 22, 9 (1943). JULLANDER, I., Acta Chem. Scand. 3, 1309 (1949). JULLANDER, I., AND BRUNE, K., Acta Chem. Scand. 4,870 (1950). RosE, H. E., Nature 169,287 (1952). HELLER, W., AND T.~BIBIAN, R. M., J. Colloid Sei. 12, 25 (1957). HEmW'~.GH, K., BORGINON, H., AND LONTIE, R., Biochim. Biophys. Aeta 48, 517 (1961). PUTZEYS, P., and VERHOEVEN, L., Rec. Tray. Chim. 68,817 (1949). P•TZEYS, P., AND BROSTEAUX,J., Trans. Faraday Soc. 31, 1315 (1935). PUTZEYS, P., AND DORY, E., Ann. Soc. Sci. Bruxelles 60, 37 (1940). PUTZEYS, P., Exposes Ann. Biochem. Med. 11, 29 (1950). DOTY, P., ANDEDSALL,J. T., Advan. Protein Chem. 6, 35 (1951). DowY, P., ANDSTEINER, R. F., J. Chem. Phys. 17,743 (1949). DOTY, P., AND STEINER, R. F., J. Chem. Phys. 18, 1211 (1950). MORGAN,J., "Introduction to Geometrical and Physical Optics." McGraw-Hill, New York, 1963. SCHRAMM,G., AND BERGER, G., Z. Naturforsch. 7b, 284 (1952). CLAESSON, I. M., Arkiv Kemi 10, Nr. 1 (1956).