secondary scattering by kraut_dandliker

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JOURNAL OF POLYMER SCIENCE TOT,. XVIII, PAGES 563-572 (1953)

A Study of Secondary ScatteringJ. KRAUT and W. B. DANDLIKER, Department of Biochemistry,University of Wa shingto n, Seattle, W ashin gton

INTRODUCTIONLight-scattering studies usually are concerned with turbidities of theorder of wi th ordinary cell dimensions, two related,convenient approximations may then be made: first, tha t the turbidity,

7, times the path length is equal to the ratio of the scattered energy to theincident energy, and, second, that rnultiple scattering may be neglected.For high turbidities, however, allowances must be made for the attenua-tion of the incident and scattered rays as they pass through the solution aswell as for the increase in scattered intensity due to multiple scattering.The use of Ludox colloidal silica sols of high turbidity for the calibration oflight-scattering is an instance in which the above effectsmust be taken into account.A recent treatment of the scattering from very turbid solutions has beenmade by Maron and LOU,^ but they have considered only the first of theabove mentioned effects, which we shall call here the attenuation effect.In the present work this has been compensated for by the procedure em-ployed in the measurement of scattered intensities, thus enabling us tostudy the secondary scattering itself.We show both theoretically and experimentally that for high turbiditiesthe secondary scattering makes an important contribution proportionalto 9, and gives rise to large, concentration-dependent depolarizationsnoted previously for Ludox sols.

cm.-l or less.

THEORETICALWe imagine a Cartesian coordinate system oriented so that the incidentbeam is propagated in the negative x-direction, while the observer issituated far out on the positive y-axis. The unpolarized incident beam isconsidered to consist of two polarized beams of equal intensity with the

electric vector of the first in the direction of the z-axis and that of the secondin the direction of the y-axis.The intensity of primary scattering in an arbitrary direction by a volumeelement of a scattering medium containing small isotropic particles is givenby :(1)37. I cos2 1L.& = - - dv87r r z

563


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.i64 .T. KRAUT AND W . B . DANDLIKER

where d i is the scattered intensity a t a distance r from the volume element,T is the turbidity of the medium, I is the intensity of the incident polarizedbeam, 1c. is the angle between the electric vectors of the incident and scat-tered radiation, and dz! is the volume of the scattering element.' Thus theintensity of primary scattering from a volume element dv l arriving at asecond such element dv2 when dz!l is irradiated by an unpolarized incidentbeam is given by the sum of:

and

where p is the distance between the two volume elements and their Cartes-ian coordinates are denoted by xl,yl,zl and xl,y2,~2,espectively. The sub-scripts 21 and h identify the coiitributions made by the vertically andhorizontally polarized components of the uripolarized incident beam.A second application of equation (1) gives for the intensity di ' of thedoubly scattered ray arriving a t tlie observer's position a t y = R on the y-axis the sum of:

and

The total intensity of secondary scattering may now be found by integra-tion of di' = di: + dii, over the regions appropriate to the geometry of thelight-scattering instrument. The elements dv, extend throughout thevolume of scattering material illuminated by the incident beam, while theelements du2 extend throughout tlie volume of scattering material viewedby the photomultiplier. For the easily integrated case of a small il-luminated volume surrounded b, a large spherical volume of scatteringmaterial, the whole of which is vieby ed by the photomultiplier, one obtains :

33160a' (90) = ~ UT'

where S'(90) is the Rayleigh ratio for secondary scattering a t ninety de-grees to the incident beam and u is the radius of the surrounding sphere ofscattering matter. While this simplified model for the actual geometryof the light-scattering instrument appears to be excessively crude, it is feltthat any improvement in the theory which might be obtained by perform-ing the above integration for the actual experimental conditions would notwarrant the additional labor involved. This expectation is apparentlyborne out by the da ta reported in t,he next section.


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A STU D Y O F SEC ON D A R Y SC A TTER I N G 56 5Com bining equa tion (6) with th e usual expressioti for th e Hayleigh ra tiofor pr imary scatter ing, the Rayleigh ratio for to tal scatter ing a t ninetydegrees is given b y :

11R(90) = 37 (1 + m )6 r ( 7 )We have neglected the factor e - r p in equations (2) an d (3) and similarfactors in equations (4) an d ( 5 ) , arising from a tte nu ati on in traversing t hedistance p, as well as tertiary and higher scat tering effects. Th ese wouldmerely introduce terms containing higher powers of T into equations (6)and ( 7 ) . Equat ion (7) s therefore an approximation for th e rang e where

ur


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LA.

4 5

1

Luc

%02 06 1

2 18 18 24 30

36 02

06 1

218

18

24 30 36

2

T(m. x 2

967 1

414

28

25

22

24 1

127 5

0 70 93 86 10

15

3

Gx1

72 14

36

57

70

69

80

10

19

46

90

19

19

12

29

20.n

TABI

LGHTSCT

NGDATAONLUDOXSLCSOLS

4

Gx02

72 12

34

52

76

65

80

95

20

46 90 11

10

13

13

29

5Gx1

01 04 16

33 65 53

77 00 00 01 03 07 05 07 0915

6 G/7

24

25

28

30 34 32

35 37 1

617 18 1

819

18

19

20

7

GJ

24

25

27 29 32

31

32

34 17

17

18

18

19

19 19 19

8

GhT

00 00 01

01 02 02 03 03 00 00 00 00 00 00 00 00

9

G/-

(WTO

01

02

05

07 1

109 1

214

00

00

01

02

02

02

03

03

10

GT-

(G,TO

01 02

04

06 09

08 09 11

00

01

01

02

03

02

02

03

1(

(10)

02

03

05

08 1

210

12

15

00 01

02

02

03

03 03 04

12

(11)9)

11 11 10

11

10

11

10

10

22

15

12

13

13

13

10

12

11

(o) (8 48

35

35

36

32

33

31

32

62

13

41

44

40

41

36

41

14

Px02

1$

27

47

63

89 79 9.

45

07 13 2

11

27 38

33

40 3216


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A STUDY O F SECONDARI- SCATTERING 569source provided th e required sp ectra l lines. T he turbidi ties are listed incolumn 2 of Tab le I.I t should be mentioned th at the exact %slues of the turbidities in column2 are sorne\$hat in do ubt. I t was found th at , although th e use of a dia-phragm of fixed dimensions in the spectrophotometer caused a small butmeasurable change in the results, the sign and magnitude of the changevaried erratically w tli th e dimensions of th e diaphragm . Similarly, th eturbidities measured in a 2 cm. cell were consistently a few per ce nt largerthan those measured in the 10 cm . cell. Blackening th e ou ter surface ofthe cells also caused a slight increase in th e measured turbid ity. Sincethere is no clear criterioii for using oiie set of cond itions rath er th a n a no the r,the results of measurements made in the blackened 10 cm . cell witho ut adiaphragm were chosen arbitrarily for listing in Table I. In any event , i tis unlikely th a t ail error greate r tha n 570 is involved.Columns 6, 7 , an d 8 are obtained by dividing the G-values of columns3, 4, and 5 , respectively, by T . T he results are plotted wmus T in Figures1,2, an d 3. T h e points represciiting the solution containing sa lt, however,have been omitted since they seein to deviate considerably from their ex-pected positions, a nd furthermore, as pointed ou t above, this solution ex-hibited some dissymm etry of scattering.Columns 9 and 10 of Ta ble I give the differences obtai ned by subtra ctingthe intercepts at T = 0 from the experimental values of G / T and G J T .These interce pts are evidently identical within experimental error an d ha vethe values ( G / T ) ~ (Gr/7)" = 2.30 fo r X = 4358 A. and 1.64 fo r X =

I I5 10 15T ~ ~ ~ ,cm-')x102

Fig. 2. Similar to Figure 1 exceptthat G , was obtained with a verticallyoriented Polaroid over the photom ul-tiplier. The result was then multipliedby one-half.

10 20 3

3 -

L\c9

5 10 Ir546,ern-') x 10'

Fig. 3 . Similar to Figure 1 exceptthat Cn was obtained with a hori-zontally or iented Polaroid over th ephotomultiplier. Th e result was thenmultiplied by one-half.


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570 -1. KRAUT AND W . R . DANDLIKER5461 A. I K ~he next section these intercepts will bc used to compute thecalibration constants of the light-sc*atteritig cell, arid the data of columns9 and 10 will then give the total and tlic vertically polarind secmdaryscattering ititensities, rcspectivel) .Figure 3 shows that the interccpt of Gh/r a t r = 0 is very nearly zero.It may be assumed, then, that the values in colurnri 8 are proportional tothe horizontally polarized secondary scattering as they stand.Column 11 is obtained by adding the figures in columns 8 and 10, henceshould be proportional to the total secondary scattering and should com-pare with column 9. The comparison is made in column 12 by dividing thefigures of column 11 by those in 9.In column 13 we give the ratio of the vertically polarized secondaryscattering to the horizontally polarized secondary scattering by dividingcolumn 10 by colunin 8.Finally, the apparent depolarizations, pu are given in column 14 by takingthe ratio of column 5 to column 4.

DISCUSSIONThe linearity of the plots of Figures 1 , 2 , and 3 provides confirmation ofthe dependence of the intensity of secondary scattering on the square ofthe turbidity. The slight upward curvature tha t becomes apparent above

turbidities of 0.15 cm.-' probably is due to the influence of tertiary andhigher order scattering. This may mean that a better expression forR(90)a t exceedingly high turbidities ould be of the form :

37.1 6 ~(90)= _ _ e a r

rather than that given by equation (7). Indeed, workers at FordhamUniversity have successfully used a somewhat similar expression to correctfor higher order scattering.'

If we define the calibration constant of the light-scattering cell to bek = R(90)/G, it may be obtained from the intercepts of Figure 1 as:k = 3 / [ 1 6 ~ ( G / ~ ) o ]

For h = 4358 A., k has a value of 2.59 X l o p 2 and for h = 5461 A ., thevalue is 3.64 x 1 0 F . Subtraction of the intercepts from the experimentalvalues of G /T and multiplication by kr gives the Rayleigh ratio for second-ary scattering. Thus the numbers in column 9 of Table I should be pro-portional to the total intensity of secondary scattering, while those incolumns 8 and 10 should similarly be proportional to the intensities of thehorizontally polarized and vertically polarized secondary scattering com-ponents. The sum of the figures iu columns 8 and 1 0 then should be equalto those in column 9. Column 12 sliows tha t this sum is greater by about9% for the blue line and about 26'% for the green. This deviation is notquite as serious as it may at first appear, since a 3 7 , change in the interceptof the plot of G JT for the blue arid a 5y; change for the green would suf-


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A S'I'UDY O F S J X O N D A R Y SCA17'E:HING .571fice to produce substantial agreement. The dificulty remains, however,to explain why ( G 0 / 7 ) 0 might difTer by a few per cent from (G /T) , while(Ch/.r)o s clearly very nearly zero. A possible explanation is that intro-duction of the Polaroid filter over the photomultiplier causes a slightchange in the effective calibration constant of the system. Support forthis possibility is afforded by the observation that the Polaroids are in-deed not perfectly flat and free from distortions, and thus very well mightbe responsible for a few per cent change in the calibration constant.We may compute the value of u in equations (6 ) , ( l l ) , nd (10) from thelimiting slopes of Figures 1, 2, and 3, and the above calibration constants.The results are summarized in Table 11. These results are of the magnitude

TABLE TIEXPERIMENTAL. T ~ ~ ~ ~ ~F uA, A . Figure 1 Figure 2 Figure 3

4358 1 . 74 cm. 1.88 cm. 2.26 cm.5461 1.'74cm. 2 03 CI11. 2 . 2 4 cm.

to be expected for a 3 x 3 cm. square cell. In view of the approximatenature of the model assumed in obtaining equations (6) and ( l o ) , the agree-ment among the experimental values of u is quite satisfactory. It appearstha t the treatment we have presented here is capable of providing a reason-ably accurate picture of the effects of secondary scattering.It is also evident that the intrinsic depolarization of Ludox solutionsmust be less than 0.4%, if we consider the probable maximum value ofthe intercept of Figure 3 to be about (Gh/7) , ,= 0.01 for the 4358 A. curve.Even this small remaining depolarization is probably to be explained bythe fact that the receiver views light scattered at angles slightly differentfrom ninety degrees, as has been suggested by Geiduschek.8 The hori-zontally polarized component of the scattered light a t ninety degrees,therefore, is essentially a direct measure of the intensity of secondaryscattering. Hence the use of the Cabannes factor in computing the tur-bidity from scattering measurements, although involving an erroneousinterpretation, amounts to a partial correction for the secondary scattering,as was recognized by Mommaerts.'According t o equations (10) and ( l l ) , the ratio of the intensity of thevertically polarized component of the secondary scattering to the intensityof the horizontally polarized component should be 4.5. Column 13 ofTable I shows that we have found this ratio to be about 3.5 for the blueline and 4.4 for the green. Again, the agreement is considered to be satis-factory. It may be pointed out that this partial vertical polarization ofthe secondary scattering can be readily verified in a qualitative way byvisual observation of the halo s~irroiiiidiiig l i t : primary beam through arotating Polaroid fi1tt.r.

\Ye con(-ludeby pointitig out tlie hcaring of this s tudy on tlie usual light-scattering procedures. First, when the iiifluence of tlic attenuation effecthas been eliminated by the experimental technique, calibration constants


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572 J. KRAUT AND W. B. DANDLIKEHare properly obtained by a linear extrapolation of ( G / T ) o zero turbidity.Second, th e slope of th e extrapolation curve m ay be m inimized by reducingthe volume of scattering medium viewed by th e photomultiplier. Th ird,Geiduscheks metho d8 for correcting depolarizations for secondary s ca tte r-ing and other spurious effects may be conveniently implemented by theuse of a plot such as Figure 3. T he value of Gh obtained from such acalibration cu rve a t the turbidity of th e solution un der invest.igation shouldbe subtracted from th e observed Gh t o o btain th e tru e horizontally polar-ized com ponent of the primary scattering.

The authors gratefully acknowledge financial support frorn th e U . S. Public HealthService, National Institutes of Health, and the Initiative 171 Fund of the State ofWashington.

References1 . W. I?. 1% .M. Mommaerts, J . Colloid Sci., , 71 (19.52).2. A. Oth, J. Oth, and V. Desreux, J . Polymer Sci., 0, 551 (1953).3. S. H. Maron and R. L. 1.1.Lou, J . Polymer Sci., 4 , 29 (1954) .4. B. I . Zimm, R. S. Stein, and P. Doty, Polymer Bull., 1, 90 (1945) .5. B. A. Brice, M. IIaIwer, and R. Speiser, J . O p t . Soc. A m . , 40, 768 (1950) .6 . B. A. Brice, G.C. Nutting, and M. IIalwer, J . Am . Chem.SOC.,5 , 824 (1953) .7 . S. N. Timasheff, personal communication.8 . E. P. Geiduscheh, J . Polymer Sci., 3, 408 (1954).

SynopsisA quantitative treatment of the influence of secondary scattering on light scattering is

presented and experimentally verified. It is shown th at secondary scattering accountsfor the apparent shortening of the cell dimensions and for the large apparent depolariza-tions found previously by other workers when using highly turbid solutions as calibra-tion standards. Methods for obtaining the correct calibration constant and depolariza-tion a re suggested.

RCsumCUn traitement quantitatif de Iinfluence de la diffusion secondaire sur la diffusion

lumineuse est pr6sentb e t v6rifi6 expCrimentalement. On montre que la diffusionsecondaire est responsirble du raccourcissement apparent des dimensions de la cellule etpour les dhpolarisatioris importantes mais apparentes, decrites pr6cCdemment pardautres chercheurs lorsquils utilisent des solutions de turbidit6 ClevCe pour calibrerleur appareil . [Jne mcithode dextrapolation est sugg6rCe; elle permettrait dobtenir laconstante d e calibrage corrrcte et de mesurer la dhpolarisation r6elle.

ZusammenfassungEs wird eine quantitative Berechnung des Einflusses von sekundiirer Streuung auf

Lichtstreuung gegeben und experimentell gepruft. Es wird gezeigt, dass sekundareStreuung die scheinbare Verkurzung der Zellen-Dimensionen und die grossen schein-baren Depolarisationeri erkllrt, die fruher von anderen Autoren gefunden wurden,wenn st ark triibe LGsungen als Kalibrierungsstandard verwendet wurden. Methodenzur Erhaltung der korrekt.en Kalihrierungskonstanten und Depolarisationen werdenvorgeschlagen.Received January 19, 1955Revised April 7, 1955

secondary scattering by kraut_dandliker

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