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Light scattering by large ellipsoidal particles. I. Rayleigh—Debye approach
Light Scattering by Large Ellipsoidal Particles I. Rayleigh-Debye Approach JOSIP J. PETRES
GJURO DEZELIC
AND
Department of Physical Chemistry, Institute "Rug jet Bo~kovig" and Department of Biocolloidal Chemistry, Andrija Stampar School of Public Hea#h, Faculty of Medidne, University of Zagreb, 41000 Zagreb Croatia, Yugoslavia Received April 15, 1974; accepted May 24, 1974 The angular dependence of the light scattered by monodisperse barium sulfate particles was determined at wavelengths of light 546 and 436 nm. The particles had the shape of prolate ellipsoids of revolution with different axial ratios. The results are discussed on the basis of the Rayleigh-Debye theory, and experimental and theoretical values of P (0) factors for ellipsoidal particles are compared. INTRODUCTION
At the present time there is no exact general theory, like Mie's theory for isotropic spherical particles, for the calculation of the intensities of the light scattered by randomly oriented ellipsoidal particles of arbitrary size and refractive index. Formally, the problem of the scattering of an electromagnetic plane wave by a triaxial ellipsoid was solved long ago by M6glich (1), but the explicit solutions of pertinent differential equations have not been obtained. In most papers light scattering by ellipsoidal particles was treated theoretically on the basis of the Rayleigh theory for ellipsoids small in comparison with the wavelength (2) or on the basis of the RayleighDebye theory which is applicable to particles with a refractive index sufficiently close to that of the surrounding medium. A brief discussion on the problem, especially concerning the light scattering by ellipsoids comparable to the wavelength, can be found in Kerker's book (3), or in papers of Stevenson (4) and Shatilov (5). The application and the range of validity of the Rayleigh-Debye theory were tested
extensively in the case of spherical and cylindrical particles by comparison of results obtained using the approximate formulae with values calculated by the exact theory (see e.g., Ref. (3)). However, there are rather few papers reporting on absolute measurements of the angular light scattering on ellipsoidal particles having dimensions and refractive indices as required by the RayleighDebye theory. Several years ago Koch (6) examined the applicability of the Rayleigh-Debye approach using suspensions of mitochondria and bacteria as model systems for ellipsoidal particles. Later experiments (7, 8) have shown that the Rayleigh-Debye approximation for calculating the intensity of scattered light at angles up to 30 ° applies relatively well to particles as large as bacteria (7), but a significant discrepancy was found between the theory and experimental measurements at high angles. Recently, Cross and Latimer (48) have used the Rayleigh-Debye approximation for both the homogeneous and the coated prolate ellipsoid to interpret the angular light scattering (10-90 °) from randomly oriented systems 296
Journal of Colloid and InterfaceScience,Vol. 5;0, No. 2, February 1975
Copyright © 1975 by Academic Press. Inc. All rights of reproduction in any form reserved.
ELLIPSOIDAL PARTICLE SCATTERING of heterogeneous Escherichia coli cells dispersed in water. The authors obtained a good theory-experiment agreement, especially for the coated ellipsoidal model. It seems that the most frequent source of difficulties in checking the Rayleigh-Debye approach for large ellipsoidal particles lies in nonadequate model systems. Objects such as mammalian mitochondria or bacteria (Escherichia coli being the most studied genus among them) are polydisperse systems regarding both the size and shape. Besides, these biological particles possess a heterogeneous internal structure which can influence light scattering in an unpredictable manner. The present paper describes our attempts to apply the Rayleigh-Debye theory in the evaluation of experimenta ! light scattering data obtained on a well-de•ed model system consisting of ellipsoidal particles large in comparison to the wavelength. For that purpose we chose monodisperse particles of barium sulfate which have been studied in our laboratory in the past years (9, 10). THEORETICAL
PART
The basic assumption for the applicability of the Rayleigh-Debye approach is that the "phase shift" corresponding to any point in the particle should be negligible, i.e., that 2 k h (m-- 1)<<1.
[1]
Here k is the propagation constant in the medium defined as 27r/X, where X is the wavelength of light in the medium, h is the longest dimension through the particle and m is the relative refractive index, i.e., the ratio re~n1, where m and nl are the refractive indices of particles and the dispersing medium, respectively. From the above criterion it is obvious that neither the particle size nor the relative refractive index can be too large. More detailed information about this can be found elsewhere (3, 11). In the Rayleigh-Debye approach the current formulae for the angular distributions of the intensities of the light scattered by an assembly
297
of optical isotropic and randomly oriented particles may be obtained in the way similar to that used for Rayleigh scatterers (see, e.g. Ref. (3)). Thus, if the incident light is unpolarized (subscript u), the angular dependence of the so-called total Rayleigh ratio (12) is given by R~(O) = K v P ( O ) c ( m -- 1)2
× (1 + cos2O). [2] Here, R~(v~) is the total Rayleigh ratio at the scattering angle v~ measured as the excess scattering intensity of the particles over that of pure dispersing medium, c denotes the concentration of particles in grams of scattering material per cubic centimeter of the suspension, and P(O) is the form factor. The quantity Kv is a parameter characteristic of the particular scattering system defined as K~ = 2~2V/X%,
[3]
where V is the volume of the scattering particle, X is the wavelength of light in the medium, and p2 is the particle density. In addition, the parameter K¢ should be multiplied by the density of the suspension, p12, if the particle concentration is to be expressed in grams of particles per gram of the suspension. If the particles have the shape of a prolate ellipsoid of revolution, the volume is given by V = (Tr/6)Db~D~, where Do and Db are the lengths of the long and short particle axes, respectively. If the incident light is plane polarized with respect to the scattering plane, the corresponding partial Rayleigh ratios will be V~(O) = 2 K c P ( O ) c ( m -
1)2
[4]
and Hh(~) = 2 K v P ( O ) c ( m -
1)2 cos20. [5-]
Here the subscripts h (horizontal) and v (vertical) refer to the state of polarization of the incident beam and the capital letters indicate the direction of polarization of the scattered component. Factor 2 in Eqs. [4"]
Journal of Colloid and Interface Science, VoL 50, No. 2, February 1975
298
PETRES A N D DEZELIC
EXPERIMENTAL
and [-5] comes from the definition Ro(o) = ½Ev~(e) + Hh(O)].
Preparation of Monodisperse Suspensions of Barium Sulfate
The form of the P(v~) function depends on the particle shape and on the scattering angle, The derivation of the P(v~) function for a system containing randomly oriented ellipsoids of revolution is a fairly difficult problem and generally no simple explicit expressions have been found. Guinier (13), followed by other authors (14-19), evaluated complex expressions for P(t~) in analytically different forms in connection with small-angle X-ray scattering. According to Debye (17), who derived the first three terms in a power series expansion of P(t~) for both the triaxial ellipsoids and ellipsoids of revolution, the P(t~) function for an ellipsoid of revolution with the semiaxes a, a, and c may be written as
Samples of barium sulfate suspensions were prepared by Takiyama's method of homogeneous precipitation (24). By varying this method it is possible to produce suspensions with monodisperse particles of different sizes and the approximate shape of a prolate ellipsoid of revolution. Details of the experimental procedure, especially of the preparation of stable suspensions using nonionic detergents as dispersing agents, are given in our previous paper (9). All suspensions were prepared under dustfree conditions, because it was not possible to clarify the suspensions by filtration after preparation without removal of a large quantity of the dispersed solid. The reaction solutions were carefully filtered through P(O) = 1 - (1/5)(ksa)~[1 + (~/3)] sintered glass filters G-5F manufactured by + (3/175)(ksa)4[1 + (2e/3) Schott and Gen., Mainz, Germany. Dust-free + (d/5)] - (4/4725)(ksa) 6 water used in the experiments was distilled from an all-glass still without glass joints. X E1 + (3e/3) + (3d/5) All glassware was cleaned with freshly dis-[- (d/7)] ~- . . . . E6] tilled dust-free acetone. The preparation of concentrated suspensions Here e is defined by the relation e = @2 _ a~)/ used as stock suspensions for light scattering a 2 = p2 _ 1, where p is the axial ratio, taken measurements was carried out in the following as the ratio of the semiaxis of revolution (c) manner. After completion of reaction the to the maximum radius of revolution (a), and obtained suspension was centrifuged (see s = 2 sin(O/2). It should be noted that Ref. (9)), the mother liquor was then removed, Eq. E6] is mathematically equivalent to the solution for P(t~) for ellipsoids of revolution and sedimented particles were redispersed by obtained by Roess and Shull (15), but their addition of the 0.05% aqueous solution of method includes the use of hypergeometric nonionic detergent Triton X-100 previously series, so that Debye's formula appears more saturated with barium sulfate. This solution suitable for numerical evaluation. Thus, was prepared from the original liquid Triton using Eq. [-6] Koch (6) computed and tabu- X-100 sample containing nearly 100% of lated the form factors for prolate ellipsoids the detergent material (Triton X-100--a of revolution with different axial ratios. trademark for octylphenoxyethanol, obtained Other tables and diagrams of P(O) functions by the courtesy of Rohm and Haas Co., for ellipsoids of revolution and triaxial el- Philadelphia, PA, U.S.A.) by dilution with lipsoids with different axial ratios, which are dust-free water. After dilution the detergent particularly important for interpreting the solution was shaken over night with solid results of small-angle X-ray scattering, are barium sulfate. The undissolved residue of barium sulfate was removed by centrifuging given in several other papers (15, 18-23). Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
ELLIPSOIDAL PARTICLE SCATTERING at about 500 )< g for 30 rain. The saturated supernatant was filtered through a G-SM filter immediately before use. The concentrations of barium sulfate stock suspensions were about (6-8) X 10.3 g solid/ cm ~ suspension. This was determined gravimetrically by evaporation to dryness of a known volume of the suspension (about 1015 ml). The dried residue was kept in vacuum over night at room temperature and then weighed. The result was corrected for the amount of the added detergent material. The error of these determinations was less than
±0.5%. Particle Si~e Determination The shape and size of barium sulfate particles were determined from electron micrographs taken by a Trtib, T~tuber and Co. electron microscope (Model KM-4) at a direct magnification 1,750:1. The particle size was measured directly from electron micrographic plates using a calibrated low power microscope at a magnification of 10:1 as described before (25). The precision of these measurements was 4-2 nm.
Light Scattering Measurements The light scattering measurements were made with a Brice-Phoenix Model 2000-DMS Light Scattering Photometer, manufactured by Phoenix Precision Instrument Co., Philadelphia, PA, U.S.A. A Philips SP 500 W watercooled super-pressure mercury lamp was used as the light source and therefore the instrument was specially recalibrated (26). The photometer was adjusted as described previously (27). The measurements were performed at wavelengths 546 and 436 nm. For polarization measurements polaroid sheets were used. Scattered light was measured at room temperature in the angular range of 30 ° to 135 ° using a cylindrical cell with the flat entrance and exit windows (Phoenix Catalog No. C-105). The cell was cleaned with freshly
299
condensed acetone in a device similar to that of Thurmond (28). For light scattering measurements a series of dilutions ranged from 1 X 10-5 to 7 )< 10-~ g cm -3 were prepared volumetrically by adding both glycerol and 0.05o-/o solution of Triton X-100 to a known volume of the concentrated stock suspension. Glycerol (p.a. redistiled sample, about 870/0, manufactured by E. Merck, Darmstadt, Germany) was previously saturated with barimn sulfate in a similar way as the detergent solution and was used after filtration through Selas 03 filter candles. The added amount of the detergent was small in comparison to the amount of glycerol and varied in respect to the used quantity of the stock suspension so that the total amount of the added detergent always exceeded 2~o vol relative to the final volume of each sample. The refractive index of dispersing media was therefore practically the same for all dilutions as verified by measuring the refractive index in several concentrations of the diluted suspension. The total and partial Rayleigh ratios were obtained experimentally by utilizing Brice's working--standard method (29). The absolute values of these ratios were computed by expressions given by Kratohvil and Smart (30). Reflection corrections were also included as was done by Kratohvil (31). In order to obtain the values corrected for multiple scattering at the finite concentration, the reciprocal specific total and partial Rayleigh ratios were plotted as a function of concentration at each of the measured angles and extrapolated to zero concentration. Corresponding theoretical values for the form factors of prolate ellipsoids of revolution were obtained by graphical interpolation from data reported in Koch's paper (6).
Determination of Refractive Indices The refractive indices of dispersing media were determined by an Abbe type refractometer at the wavelengths of light 546 and
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
300
PETRES AND DE2ELI¢
FIG. 1. Electron micrograph of barium sulfate particles; sample S-100. 436 nm. A high-pressure mercury lamp AH-4 equipped with filters for the isolation of the above wavelengths was used as the light source. The refractometer was calibrated with pure organic liquids of the known refractive index as described elsewhere (26). The error TABLE I PARTICLE SIZE DISTRIBUTION PARAMETERS OF MONODISPERSE BARIUM SULFATE SUSPENSIONS
in the readings on the refractometer scale was -4- (1-2) in the fourth decimal.
Determination of Density The densities of suspensions and dispersing media were determined by pycnometers at 25°C as described earlier (26). The density of barium sulfate particles was obtained by a pycnometric technique as suggested by Pugh and Heller (32).
Long axis Sample
N
S-115 S-116
Sample
b~ s(D~) (De). (rim) ( r i m ) (nm)
(D~)~ (rim)
P
300 287
245 284
254 290
1.089 1.046
N
Db (nm)
S(Db) (nm)
20 21
247 286
Short axis (Db)~, (Db)w ( r i m ) (nm)
P
S-115
300
121
12
122
126
1.115
S-116
287
120
9
121
123
1.052
RESULTS The typical form of monodisperse particles of one of barium sulfate suspensions used in our light scattering experiments can be seen in Fig. 1. It is of a nearly perfect ellipsoidal shape. Data on parameters of the particle size distribution for two samples of barium sulfate suspensions are given in Table I. Each sample is characterized by arithmetic mean lengths of the long and short axes of
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
ELLIPSOIDAL PARTICLE SCATTERING TABLE RESULTS
OF DETERMINATION SAMPLES OF BARIU~I
Sample
II
S-107 b
S-116 S-115
TABLE
III
REFRACTIVE INDICES • AND RELATIVE REFRACTIVE INDICES ~ OF DISPERSING M E D I A A N D B A R I U M SULFATE AT 25°C
OF DENSITY P2 OF SEVERAL SULFATE PARTICLES p12 a
301
p2
Refractive index, n W a v e l e n g t h (nm) 546 436
( g e m -3)
1.0204 1.0204 1.0032 1.0298
4.39 4.44 4.46 4.41
R e l a t i v e refractive index,b m Wavelength
(nm) 546
D e n s i t y of suspension measured at 25 °C. b Particle size of this s a m p l e : /), = 225 nm, /)b = 91 a m ,
particles, /3~ and /)b, respectively, obtained by the statistical particle size analysis of N counted particles, and by respective standard deviations of the distribution, s(D)~ and s(Db). The quantities (D~)~ and (D~)~ are the number average and weight average long axes, respectively, (Db)~ and (Db)~ are the number average and weight average short axes, respectively, and P is the polydispersity ratios defined as DJ/D~ 3 .All these quantities are defined and calculated as described elsewhere (25). I t is obvious from Table I that both samples of barium sulfate suspensions had a narrow particle size distribution as indicated by the P approaching unity and be regarded as monodisperse. Table I I gives data on both densities of barium sulfate suspensions (,012) and particles (p2) obtained at various barium sulfate particle sizes. The absence of a systematic variation of p2 in relation to the size of the particles indicates that their density does not change practically with the particle size. According to data in Table II, the average value for the TABLE
Barium sulfate particles Dispersing media Triton X-100 0.05% aq. solution Glycerol circ. 87%
436
1.594~
1.067 a
--
1.3337
1.3397
1.194
1.199
1.4536
1.4608
1.097
1.100
-
-
Calculated f r o m d a t a in Table I V b y Eq. [2]. b Defined a s npartioles/ndlspersing medium.
density of barium sulfate amounts to (4.43 4-0.03) g cm -3. This value is slightly lower (about 1.5%) than that of the mineral barite (33). This corroborates our previous finding that ellipsoidal barium sulfate particles have a sponge-like internal structure (10). Table I I I presents data on refractive indices n and relative refractive indices m of the dispersing media used in this study and of barium sulfate particles. Since there was no direct method for the determination of refractive indices of particles dispersed in such highly turbid suspensions, the refractive indices of barium sulfate particles were derived from (Ru(90)/C)o value determined by light scattering on particles of only one sample dispersed in glycerol (sample S-116) by using Eq. r2]. The parameters used for the calculation of these indices are given in Table IV. The P(0) values for 0 = 90 ° were IV
EXPERIMENTAL AND THEORETICAL D A T A USED I N EQ.
[-2] TO CALCULATE THE REFRACTIVE S-116) DISPERSED I N GLYCEROL
I N D E X OF
THE BARIUM SULFATE PARTICLES (SAMPLE W a v e l e n g t h in v a c u u m X, (nm)
546
436
[R~(90)/c-]o (g-1 cm2) K, X 10.3 (g-1 cm2) nx~ _P-l(t~) for 0 = 90°b
17.3 4- 0.2 4.84 -4- 0.09 1.4536 4- 0.0002 2.62
28.9 4- 0.5 12.1 4- 0.2 1.4608 4- 0.0002 4.23
a Refractive index of glycerol m e d i u m . b E x t r a c t e d f r o m K o c h ' s d a t a (6).
Journal o7 Colloid and Interjace Science, Vol. 50, No. 2, F e b r u a r y 1975
302
PETRES AND DEZELIC
soso~s-',00 .
I
~-
'
;ko " 546 nrn
w/~./~" Triton X- 100 ~=2 ~ ( ' / " TritonX-IO0 sat. B~SO4 o A CX 105 gcm-3 FIG. 2. Effect of dissolution of b a r i u m sulfate p a r -
(sample S-100) on the intensities of scattered light; dilution of the stock suspension with 0.05% aq. solution of Triton X-100 saturated (circles) and unsaturated (triangles) with barium sulfate. ticles
extracted from Koch's theoretical data (6)• The refractive indices obtained in this single sample of barium sulfate particles are listed in Table I I I and served as optical constants for subsequent exeriments. Since the relative refractive index m has been derived from the same relation used in other light scattering evaluations, one must not mistake it as an absolute optical constant. The values of m in this paper have to be regarded merely as •
"
.
[,¢1
BaSOa S - I 1 6 ;~o= 546 nm m = 1,097
5
,o
~"-~-----~,~
5
120
o. 3 2 ~
6
0
"------~'---"~-~----'-~-----~-c----o~ 0
2
50
<'-" 40 3
4
5
6
C x 105 g cm -3
FIG. 3. Reciprocal specific total Rayleigh ratios dependence of the scattering angIe and the concentration of barium sulfate suspension; sample S-116 dispersed in glycerol and measured at 546 nm.
c/R~,(O) in
some "Rayleigh-Debye" relative refractive indices. The preliminary experiments have shown that the most suitable concentration range for light scattering measurements was from 1 X l0 -5 to 6 X 10.5 g cm -4. The solubility of the barium sulfate in water and 87% glycerol at 25°C is 2.23 X l0 -6 g m1-1 (34) and 1.3 X 10.6 g m1-1, respectively. The later value was estimated on the basis of data reported by Collins and Leineweber (35). Thus, in order so prevent the dissolution of particles during the dilution of a concentrated stock suspention with 0.05 aq. solution of Triton X-100 or 87% glycerol, it was necessary to use the dispersing media previously saturated with barium sulfate. The effect of the dissolution of particles on the intensities of scattered light can be seen in Fig. 2. The concentrated stock suspension of barium sulfate (sample S-100, particle size /5~ = 300 nm, /5~ = 140 nm) was diluted in one case with 0.05% aq. solution of Triton X-100 saturated with barium sulfate and in the other case with the same Triton X-100 solution without addition of barium sulfate. I n the latter case a strong decrease of the specific total Rayleigh ratio, R,,(90)/c ascribed to the dissolution of particles, especially in the concentration range below 2 X 10-~ g cm-% was found, so that the extrapolation to zero concentration becomes impossible. The experimental values of form factors P(O) for both systems S-116 and S-115 were obtained from specific total and partial Rayleigh ratios extrapolated to the zero concentration, (R,, (O)/C)oand (V~ (O)/C)o,using Eqs. [2-] and [4-]. For that purpose graphs c/R~(O) or c/V~(O) vs. c for various angles were drawn. The typical graph of this type is shown in Fig. 3. All (c/Ru(O))o and (c/V~(O))o values were calculated by the first order least squares method. The average standard deviation on the values obtained was less than ± 2 % . Experimental values of form factors for barium sulfate particles dispersed in the
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
ELLIPSOIDAL PARTICLE SCATTERING
II
t
BoSO/+
'
S-II6
iOI
axial ratio
9~-
;ko(rim)[Ru(~')/c]c=o
|
5/+6
L
/+36 5/+6
7 ~-
p= 2.36
/
/'
/,
o
/
•
1
J'"
/"
[Vv~#)/c]~ °
•
/+36
/
/
/+ 3 2 I
0
I
2
3
X
FIG. 4. Theoretical curves of P-I(O) functions for spheres, rods and prolate ellipsoids of revolution with the axial ratio p = 2.36; points are experimental P-I(O) values obtained on barium sulfate particles (sample S-116) dispersed in glycerol from (R~(O)/c)o (circles) and (V~(O)/c)o (triangles) data; measured at
546 nm (open points) and 436 nm (filledpoints).
Open and filled points denote data measured at 546 and 436 nm. From Figs. 4 and 5 it is obvious that there is good agreement between experimentally obtained form factors and theoretical P-I(O) curves for ellipsoids of revolution with the given axial ratio (2.36 for S-116 and 2.03 for S-115). The statistical analysis has shown that the average percent deviations of the experimental from theoretical values (calculated for both wavelengths 546 and 436 nm) amount to -0.73~o for S-116 and - 2 . 8 % for S-115. We thought it interesting, from the point of view of the application of the RayleighDebye theory, to compare deviations of the experimental form factors from the theoretical values if for the same particle size the relative refractive index is increased by a change in the dispersing medium. To this end we calculated the P(O) values from the specific total Rayleigh ratios (R~(O)/c)o determined in the i
II
BaSO/+ axial
I0
glycerol medium (S-116 and S-115) were compared with theoretical P(O) functions for spheres, rods and prolate ellipsoids of revolution with axial ratios 2.36 and 2.03. A graphical presentation of the results obtained is shown in Figs. 4 and 5. Theoretical values of P-l(0) functions for spheres and rods are taken from data given by Doty and Steiner (36) while those for prolate ellipsoids of revolution are derived from Koch's data (6). Parameter x is given as ksR~, where R~ is the radius of gyration. This parameter is given uniformly for all particle shapes allowing a graphical presentation in one diagram. In the case of a prolate ellipsoid of revolution with the axial ratio p and the semiaxis a the radius of gyration is defined as a[(2 + f2)/S]L Points in Figs. 4 and 5 are experimental P-I(O) values obtained from (R~(O)/C)o (circles) and (V~(O)/c)o (triangles) data.
303
S-II5
ratio
;%(nm) 546 /+36
9 8
p=2.03 [Ru(~)/c]c= ° o
[Vv(#)/c] c_o v
5/+6 /+36
7
$
A6
3_ 5 4 3
rods
2 1
0
I
X
2
3
FIG. 5. Theoretical curves of 2-1(0) functions for spheres, rods and prolate ellipsoids of revolution with the axial ratio p = 2.03; points are experimental P-I(O) values obtained on barium sulfate particles (sample S-f15) dispersed in glycerol from (R,(O)/c)o (circles) and (V~(O)/c)o (triangles) data; measured at 546 nm (open points) and 436 nm (filled points).
Journal of Colloid and InterJace Science, Vol. 50, No. 2, February 1975
304
PETRES AND DEZELIC i
i
i
i
/
BaS04 S-II6
7
/ ~olnm)
rn
546
[Ru(,Y)/C]c: 0
I. 097 1.194
5
o •
axial ratio
•
.
•
•
e •
p:236
/o °
/°
o
o
I
0
/ • •
I
2
r
1
3
/.
Xe2
FIG. 6. Comparison of experimental p - l ( O ) values obtained on barium sulfate particles (sample S-116) at 546 n m from (R~(O)/c)o for m = 1.194 (fiIled circles; 0.05% aq. solution of Triton X-100) and m = 1.097 (open circles; glycerol) with theoretical P-I(O) values calculated for prolate ellipsoids of revolution, p = 2.36.
parallel way for particles dispersed in glycerol (m = 1.097 at the wavelength 546 nm) and 0.05% aq. solution of Triton X-100 (m = 1.194). A graphical presentation of the results obtained for the sample S-116 is seen in Fig. 6. The values on the abscissa are defined as x, 2 = (ksa)% where a = Db/2. As is well seen from Fig. 6, the experimental values obtained for m = 1.194, unlike those obtained for m = 1.097, are in poor agreement with the theoretical curve. The statistical analysis has shown that the average percent deviation of the experimental from theoretical values amounts to + 3 5 % in the former and about --1.8°-/o in the latter case, by which the individual deviations for m = 1.194 increase with the increase in the scattering angle from -15~o for 0 = 40 ° to + 4 6 % for v~ = 90 °. DISCUSSION For lack of an exact theory we have tried to explain light scattering on ellipsoidal particles by means of the Rayleigh-Debye theory. I t is clear that, with regard to the size of the particles studied, the relative refractive index must be fairly close to 1 in order to meet the condition determining the validity of the Rayleigh-Debye theory, i.e., that 2 k h (m -- 1) << 1. Our estimate was that m might be small enough if barium sulfate particles
were dispersed in the glycerol medium provided the refractive index of particles did not exceed the refractive index of barite, the natural crystal form of barium sulfate. The value of the refractive index for barite (37), for instance, for the light wavelength 546 nm amounts to 1.643 (calculated as the geometrical mean of the refractive indices n,, n¢, nr, because barite has two optic axes). For the same wavelength the refractive index for 87% glycerol is 1.4536, thus m being 1.13, while the particles dispersed in 0.050-/0 aq. solution of Triton X-100 would have m = 1.23. From the conclusions made in papers (38-40) concerning the range of the validity of the Rayleigh-Debye theory it appears clear that the possibility of the use of the RayleighDebye theory could be expected only for m _< 1.13. In addition, practically it may be assumed that the refractive index of the particles of barium sulfate obtained by homogeneous precipitation is lower than that of barite. In the earlier studies of the ultrathin sections of such particles it has already been shown that these particles possess a porous internal structure with the size of pores being about 30 A (10). In view of this and the results of some other authors (41-43) who have shown that the precipitates of barium sulfate
Journal oJ Colloid and Interface Science, Vol. 50, igo. 2, February 1975
ELLIPSOIDAL PARTICLE SCATTERING obtained by different types of homogeneous precipitation may be chemically heavily contaminated by foreign ions from the mother liquor and water molecules, it has been assumed that it is pores that are primarily the sites where the occluded material can be found (10). Thus, it is logical that the inside of the particles should be nonhomogeneous with regard to the refractive index. Besides, the composition of the reaction mixture in the process of precipitation barium sulfate particles by the Takiyama's method of homogeneous precipitation (24) is such that it may be assumed that the refractive index of nonhomogeneities within the particles, water included, is lower than that of barite. In view of this, it is very probable that the relative refractive index for particles dispersed in the glycerol medium is below 1.13. The refractive indices for barium sulfate particles determined from light scattering measurements are by about 3% lower than the barite refractive index. This is in agreement with our earlier claim about the nonhomogeneity of the particle inner structure. It should be noted that several years ago Lewis and Lothian (44) found that the refractive index for the barium sulfate particles examined amounted to 1.52 in the range of light wavelengths from about 0.41 to 0.78 ~m. The authors assumed that the particles (prepared from barium chloride and sulfuric acid in the presence of hydrochloric acid--following Andreasen (46)) had a spherical shape (the mean radius, ¢ = 4.6 Fzm) and calculated the refractive index by the expression deduced by van de Hulst (45) for the total scattering coefficient if r/X is much greater than unit and when m approaches unity. The obtained result for the refractive index was explained as an intermediate value between the refractive index for water and crystal barium sulfate, because on the basis of data on the specific gravity determined by the sedimentation velocity, Lewis and Lothian estimated that the particles contained about 60o-/ovoI water. Our values for the refractive index of barium
305
sulfate particles are relatively high in relation to those obtained by Lewis and Lothian but a complete comparison between our results and theirs is not possible because the methods used for the particle preparation as well as the form of the particles obtained were different. Moreover, in the course of some detailed investigations of the morphological properties of the barium sulfate particles (9, 10, 47) obtained under different precipitation conditions (24, 46) we have observed that regardless of their size, the particles appear as more or less three-axial forms. For this reason we are inclined to think that the particles studied by Lewis and Lothian could only approximately be considered spherical, so that the theory applicable for spherical particles could not be wholly applied to the particles in question. Generally, a good agreement between the experimental points for both samples of barium sulfate dispersed in glycerol (S-116 and S-115) and the theoretical curve P-l(v~) for prolate ellipsoids of revolution was obtained. Some greater deviations (but no more than 15%) of experimental from theoretical values were only observed at angles 0 ~< 50 ° and t~ >/ 130 ° (see, Figs. 4 and 5). The most probable and most important reasons for it may be an imperfect monodispersity of the samples and the particle deviation from the form of a perfect prolate ellipsoid of revolution. SUMMARY The angular dependence of the light scattered by monodisperse barium sulfate particles was determined at wavelengths of light 546 and 436 nm. The particles had the shape of prolate ellipsoids of revolution with the longer axis from 245 to 300 nm and the shorter axis from 120 to 140 nm. The axial ratio varied from 2.03 to 2.36. The particles were dispersed in 0.05% aq. solution of the nonionic detergent Triton X-100 or in 87% glycerol. The results are discussed on the basis of the RayleighDebye theory, and experimental and theoretical values of form factors P(O) for ellipsoidal particles are compared. It is found
Journal of Colloid and InterJace Science, Vol. 50, No. 2, February 197.5
306
PETRES AND DE2ELIC
t h a t it is possible to a p p l y the R a y l e i g h - D e b y e t h e o r y to i n t e r p r e t light s c a t t e r i n g on large b a r i u m sulfate p a r t i c l e s if those are dispersed in the glycerol m e d i u m . I n this case a good a g r e e m e n t between e x p e r i m e n t a l and theoretical values was o b t a i n e d . REFERENCES 1. MOEGLICn, F., Ann. Physik 83, 609 (1927). 2. LORD IRAYLEIGH,Phil. Mag. 44, 28 (1897). 3. KERKER, M., "The Scattering of Light and Other Electromagnetic Radiation." Academic Press, New York, 1969. 4. STEVENSON,A. F., J. Appl. Phys. 24, 1143 (1953). 5. SHATILOV,A. V., Opt. Spektrosk. 9, 86 (1960). 6. KocH, A. L., Biochlm. Biophys. Acta 51, 429 (196l). 7. Koch, A. L., AND EHRE~ELD, E., Biochim. Biophys. Acta 165, 262 (1968). 8. KocE, A. L., J. Theor. Biol. 18, 133 (1968). 9. PETRES, J., DEZELIC, GJ., AND TE~AK, B., Croat. Chem. Acta 38, 277 (1966). 10. PETRES, J., DE~ELId, GJ., AND TE~AK, B., Croat. Chem. Acta 41, 183 (1969). 11. VAN DE HULST, H. C., "Light Scattering by Small Particles." John Wiley and Sons, Inc., New York, 1957. 12. DE~ELId, GJ., Pure Appl. Chem. 23, 327 (1970). 13. GUINIER, A., Ann Phys. (Paris) 12, 161 (1939). 14. PATTERSON,A. L., Phys. Rev. 56, 972 (1939). 15. RoEss, L. C., AND SKULL, C. G., J. Appl. Phys. 18, 30 (1947). 16. POROD, G., Acta Phys. Austr. 2, 255 (1948). 17. EDSALL, J. T., AND DANDLIKER, W. B., Forlschr. Chem. Forsch. 2, 1 (1951). 18. SAITO, N., AND IKEDA, Y., J. Phys. Sac. Yap. 6, 305 (1951). 19. SCH~nDT, P. W., AND HmHT, R., JR., J. Appt. Phys. 30, 866 (1959). 20. MAL~ON, A. G., Acta Crystallogr. 10, 639 (1957). 21. BEIDEE, G., BISCHOF, M., GLATZ, G., POROD, G., VON SACKEN, J. CH., AND WAWRA,H., Z. Elektrochem. 61, 1311 (1957). 22. MITTELBACH, P., AND POROD, G., Acta Phys. Austr. 15, 122 (1962). 23. MITTELBACtI, P., Acta Phys. Austr. 19, 53 (1964). 24. TAKIYA~A, K., Bull. Chem. Soc. Yap. 31, 950 (1958).
25. D~EEId, GJ., WRtSCHER, M., D~WD~, Z., AND KRATOHVIL,J. P., Kolloid Z. 171, 42 (1960). 26. gEG~OVI6, N., AND D~ELId, GJ., Croat. Chem. Acta 45, 385 (1973). 27. PETRES, J. J., AND DEW,LId, GJ., J. Polym. Scl. Part C 42, 1181 (1973). 28. THURMOND, C. D., J. Polym. Sci. 8, 607 (1952). 29. BRINE, B. A., HALWER, M., AND SPEISER, R., Y. Opt. Soc. Amer. 40, 768 (1950). 30. KRATOHVIL,J. P., AND SMART, C., J. Colloid Sd. 20, 875 (1965). 31. KRATOHVIL, J. P., J. Colloid Interface Sci. 21, 498 (1966). 32. PuGI.I, T. L., AND HEELER, W., J. Colloid Sci. 12, 173 (1957). 33. WALTON, G., AND WALDEN, G. H., J. Amer. Chem. Soc. 68, 1742 (1946). 34. SEIDELL, A., ASrD LINK~, W. F., "Solubilities of Inorganic and Metal-Organic Compounds." D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. 35. COLLINS,F. C., AND LEINEWEBER~J. P., J. Phys. Chem. 60, 389 (1956). 36. DOTY, P., AND STErnER, R. F., J. Chem. Phys. 18, 1211 (1950). 37. OPTISC~E KONSTANTEN in Landolt-Boernstein, "Zahlenwerte und Funkfionen," II Band, 8. Tell, 6. Auflage, Berlin/GStfingen/Heidelberg, 1962. 38. DE'ELIte, GJ., AND KRATOttVIE, J. P., Kolloid Z. 173, 38 (1960). 39. HELLER, W., J. Chem. Phys. 42, 1609 (1965). 40. KERKER, M., FARONE, W. A., AND MATIJEVIC, E., J. Opt. Soc. Amer. 53, 758 (1963). 41. PAUEIK,F., BUZ~.GH,E., PCEOS, L., AND ERDE¥, L., Acta Chim. (Budapest) 38, 311 (1963). 42. Em)EY, L., PAUEIK,F., BUZAGH,E., ANDP6EOS, L., Acta Chim. (Budapest) 41, 109 (1964). 43. BUZ3.GtI-GERE, E., PAULIK, F., AND ERDEY, L., Talanta 13, 731 (1966). 44. LEwis, P. C., AND LOTmAN, G. F., Brit. J. Appl. Phys., suppl. 3, S 71 (1954). 45. VAN DE HUEST, H. C., see Ref. (11), p. 172. 46. ANDREASEN, A. H. M., SKEEL-CItRISTENSEN,K., AND KJAER, B., Kollold Z. 104, 181 (1943). 47. PETRES, J. J., DE~ELI(~, GJ., AND 'I'EZAK, B., Croat. Chem. Acta 40, 213 (1968). 48. CROSS, D. A., AND LATIMER, P., Appl. Opt. 11, 1225 (1972).
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
GJURO DEZELIC
AND
Department of Physical Chemistry, Institute "Rug jet Bo~kovig" and Department of Biocolloidal Chemistry, Andrija Stampar School of Public Hea#h, Faculty of Medidne, University of Zagreb, 41000 Zagreb Croatia, Yugoslavia Received April 15, 1974; accepted May 24, 1974 The angular dependence of the light scattered by monodisperse barium sulfate particles was determined at wavelengths of light 546 and 436 nm. The particles had the shape of prolate ellipsoids of revolution with different axial ratios. The results are discussed on the basis of the Rayleigh-Debye theory, and experimental and theoretical values of P (0) factors for ellipsoidal particles are compared. INTRODUCTION
At the present time there is no exact general theory, like Mie's theory for isotropic spherical particles, for the calculation of the intensities of the light scattered by randomly oriented ellipsoidal particles of arbitrary size and refractive index. Formally, the problem of the scattering of an electromagnetic plane wave by a triaxial ellipsoid was solved long ago by M6glich (1), but the explicit solutions of pertinent differential equations have not been obtained. In most papers light scattering by ellipsoidal particles was treated theoretically on the basis of the Rayleigh theory for ellipsoids small in comparison with the wavelength (2) or on the basis of the RayleighDebye theory which is applicable to particles with a refractive index sufficiently close to that of the surrounding medium. A brief discussion on the problem, especially concerning the light scattering by ellipsoids comparable to the wavelength, can be found in Kerker's book (3), or in papers of Stevenson (4) and Shatilov (5). The application and the range of validity of the Rayleigh-Debye theory were tested
extensively in the case of spherical and cylindrical particles by comparison of results obtained using the approximate formulae with values calculated by the exact theory (see e.g., Ref. (3)). However, there are rather few papers reporting on absolute measurements of the angular light scattering on ellipsoidal particles having dimensions and refractive indices as required by the RayleighDebye theory. Several years ago Koch (6) examined the applicability of the Rayleigh-Debye approach using suspensions of mitochondria and bacteria as model systems for ellipsoidal particles. Later experiments (7, 8) have shown that the Rayleigh-Debye approximation for calculating the intensity of scattered light at angles up to 30 ° applies relatively well to particles as large as bacteria (7), but a significant discrepancy was found between the theory and experimental measurements at high angles. Recently, Cross and Latimer (48) have used the Rayleigh-Debye approximation for both the homogeneous and the coated prolate ellipsoid to interpret the angular light scattering (10-90 °) from randomly oriented systems 296
Journal of Colloid and InterfaceScience,Vol. 5;0, No. 2, February 1975
Copyright © 1975 by Academic Press. Inc. All rights of reproduction in any form reserved.
ELLIPSOIDAL PARTICLE SCATTERING of heterogeneous Escherichia coli cells dispersed in water. The authors obtained a good theory-experiment agreement, especially for the coated ellipsoidal model. It seems that the most frequent source of difficulties in checking the Rayleigh-Debye approach for large ellipsoidal particles lies in nonadequate model systems. Objects such as mammalian mitochondria or bacteria (Escherichia coli being the most studied genus among them) are polydisperse systems regarding both the size and shape. Besides, these biological particles possess a heterogeneous internal structure which can influence light scattering in an unpredictable manner. The present paper describes our attempts to apply the Rayleigh-Debye theory in the evaluation of experimenta ! light scattering data obtained on a well-de•ed model system consisting of ellipsoidal particles large in comparison to the wavelength. For that purpose we chose monodisperse particles of barium sulfate which have been studied in our laboratory in the past years (9, 10). THEORETICAL
PART
The basic assumption for the applicability of the Rayleigh-Debye approach is that the "phase shift" corresponding to any point in the particle should be negligible, i.e., that 2 k h (m-- 1)<<1.
[1]
Here k is the propagation constant in the medium defined as 27r/X, where X is the wavelength of light in the medium, h is the longest dimension through the particle and m is the relative refractive index, i.e., the ratio re~n1, where m and nl are the refractive indices of particles and the dispersing medium, respectively. From the above criterion it is obvious that neither the particle size nor the relative refractive index can be too large. More detailed information about this can be found elsewhere (3, 11). In the Rayleigh-Debye approach the current formulae for the angular distributions of the intensities of the light scattered by an assembly
297
of optical isotropic and randomly oriented particles may be obtained in the way similar to that used for Rayleigh scatterers (see, e.g. Ref. (3)). Thus, if the incident light is unpolarized (subscript u), the angular dependence of the so-called total Rayleigh ratio (12) is given by R~(O) = K v P ( O ) c ( m -- 1)2
× (1 + cos2O). [2] Here, R~(v~) is the total Rayleigh ratio at the scattering angle v~ measured as the excess scattering intensity of the particles over that of pure dispersing medium, c denotes the concentration of particles in grams of scattering material per cubic centimeter of the suspension, and P(O) is the form factor. The quantity Kv is a parameter characteristic of the particular scattering system defined as K~ = 2~2V/X%,
[3]
where V is the volume of the scattering particle, X is the wavelength of light in the medium, and p2 is the particle density. In addition, the parameter K¢ should be multiplied by the density of the suspension, p12, if the particle concentration is to be expressed in grams of particles per gram of the suspension. If the particles have the shape of a prolate ellipsoid of revolution, the volume is given by V = (Tr/6)Db~D~, where Do and Db are the lengths of the long and short particle axes, respectively. If the incident light is plane polarized with respect to the scattering plane, the corresponding partial Rayleigh ratios will be V~(O) = 2 K c P ( O ) c ( m -
1)2
[4]
and Hh(~) = 2 K v P ( O ) c ( m -
1)2 cos20. [5-]
Here the subscripts h (horizontal) and v (vertical) refer to the state of polarization of the incident beam and the capital letters indicate the direction of polarization of the scattered component. Factor 2 in Eqs. [4"]
Journal of Colloid and Interface Science, VoL 50, No. 2, February 1975
298
PETRES A N D DEZELIC
EXPERIMENTAL
and [-5] comes from the definition Ro(o) = ½Ev~(e) + Hh(O)].
Preparation of Monodisperse Suspensions of Barium Sulfate
The form of the P(v~) function depends on the particle shape and on the scattering angle, The derivation of the P(v~) function for a system containing randomly oriented ellipsoids of revolution is a fairly difficult problem and generally no simple explicit expressions have been found. Guinier (13), followed by other authors (14-19), evaluated complex expressions for P(t~) in analytically different forms in connection with small-angle X-ray scattering. According to Debye (17), who derived the first three terms in a power series expansion of P(t~) for both the triaxial ellipsoids and ellipsoids of revolution, the P(t~) function for an ellipsoid of revolution with the semiaxes a, a, and c may be written as
Samples of barium sulfate suspensions were prepared by Takiyama's method of homogeneous precipitation (24). By varying this method it is possible to produce suspensions with monodisperse particles of different sizes and the approximate shape of a prolate ellipsoid of revolution. Details of the experimental procedure, especially of the preparation of stable suspensions using nonionic detergents as dispersing agents, are given in our previous paper (9). All suspensions were prepared under dustfree conditions, because it was not possible to clarify the suspensions by filtration after preparation without removal of a large quantity of the dispersed solid. The reaction solutions were carefully filtered through P(O) = 1 - (1/5)(ksa)~[1 + (~/3)] sintered glass filters G-5F manufactured by + (3/175)(ksa)4[1 + (2e/3) Schott and Gen., Mainz, Germany. Dust-free + (d/5)] - (4/4725)(ksa) 6 water used in the experiments was distilled from an all-glass still without glass joints. X E1 + (3e/3) + (3d/5) All glassware was cleaned with freshly dis-[- (d/7)] ~- . . . . E6] tilled dust-free acetone. The preparation of concentrated suspensions Here e is defined by the relation e = @2 _ a~)/ used as stock suspensions for light scattering a 2 = p2 _ 1, where p is the axial ratio, taken measurements was carried out in the following as the ratio of the semiaxis of revolution (c) manner. After completion of reaction the to the maximum radius of revolution (a), and obtained suspension was centrifuged (see s = 2 sin(O/2). It should be noted that Ref. (9)), the mother liquor was then removed, Eq. E6] is mathematically equivalent to the solution for P(t~) for ellipsoids of revolution and sedimented particles were redispersed by obtained by Roess and Shull (15), but their addition of the 0.05% aqueous solution of method includes the use of hypergeometric nonionic detergent Triton X-100 previously series, so that Debye's formula appears more saturated with barium sulfate. This solution suitable for numerical evaluation. Thus, was prepared from the original liquid Triton using Eq. [-6] Koch (6) computed and tabu- X-100 sample containing nearly 100% of lated the form factors for prolate ellipsoids the detergent material (Triton X-100--a of revolution with different axial ratios. trademark for octylphenoxyethanol, obtained Other tables and diagrams of P(O) functions by the courtesy of Rohm and Haas Co., for ellipsoids of revolution and triaxial el- Philadelphia, PA, U.S.A.) by dilution with lipsoids with different axial ratios, which are dust-free water. After dilution the detergent particularly important for interpreting the solution was shaken over night with solid results of small-angle X-ray scattering, are barium sulfate. The undissolved residue of barium sulfate was removed by centrifuging given in several other papers (15, 18-23). Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
ELLIPSOIDAL PARTICLE SCATTERING at about 500 )< g for 30 rain. The saturated supernatant was filtered through a G-SM filter immediately before use. The concentrations of barium sulfate stock suspensions were about (6-8) X 10.3 g solid/ cm ~ suspension. This was determined gravimetrically by evaporation to dryness of a known volume of the suspension (about 1015 ml). The dried residue was kept in vacuum over night at room temperature and then weighed. The result was corrected for the amount of the added detergent material. The error of these determinations was less than
±0.5%. Particle Si~e Determination The shape and size of barium sulfate particles were determined from electron micrographs taken by a Trtib, T~tuber and Co. electron microscope (Model KM-4) at a direct magnification 1,750:1. The particle size was measured directly from electron micrographic plates using a calibrated low power microscope at a magnification of 10:1 as described before (25). The precision of these measurements was 4-2 nm.
Light Scattering Measurements The light scattering measurements were made with a Brice-Phoenix Model 2000-DMS Light Scattering Photometer, manufactured by Phoenix Precision Instrument Co., Philadelphia, PA, U.S.A. A Philips SP 500 W watercooled super-pressure mercury lamp was used as the light source and therefore the instrument was specially recalibrated (26). The photometer was adjusted as described previously (27). The measurements were performed at wavelengths 546 and 436 nm. For polarization measurements polaroid sheets were used. Scattered light was measured at room temperature in the angular range of 30 ° to 135 ° using a cylindrical cell with the flat entrance and exit windows (Phoenix Catalog No. C-105). The cell was cleaned with freshly
299
condensed acetone in a device similar to that of Thurmond (28). For light scattering measurements a series of dilutions ranged from 1 X 10-5 to 7 )< 10-~ g cm -3 were prepared volumetrically by adding both glycerol and 0.05o-/o solution of Triton X-100 to a known volume of the concentrated stock suspension. Glycerol (p.a. redistiled sample, about 870/0, manufactured by E. Merck, Darmstadt, Germany) was previously saturated with barimn sulfate in a similar way as the detergent solution and was used after filtration through Selas 03 filter candles. The added amount of the detergent was small in comparison to the amount of glycerol and varied in respect to the used quantity of the stock suspension so that the total amount of the added detergent always exceeded 2~o vol relative to the final volume of each sample. The refractive index of dispersing media was therefore practically the same for all dilutions as verified by measuring the refractive index in several concentrations of the diluted suspension. The total and partial Rayleigh ratios were obtained experimentally by utilizing Brice's working--standard method (29). The absolute values of these ratios were computed by expressions given by Kratohvil and Smart (30). Reflection corrections were also included as was done by Kratohvil (31). In order to obtain the values corrected for multiple scattering at the finite concentration, the reciprocal specific total and partial Rayleigh ratios were plotted as a function of concentration at each of the measured angles and extrapolated to zero concentration. Corresponding theoretical values for the form factors of prolate ellipsoids of revolution were obtained by graphical interpolation from data reported in Koch's paper (6).
Determination of Refractive Indices The refractive indices of dispersing media were determined by an Abbe type refractometer at the wavelengths of light 546 and
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
300
PETRES AND DE2ELI¢
FIG. 1. Electron micrograph of barium sulfate particles; sample S-100. 436 nm. A high-pressure mercury lamp AH-4 equipped with filters for the isolation of the above wavelengths was used as the light source. The refractometer was calibrated with pure organic liquids of the known refractive index as described elsewhere (26). The error TABLE I PARTICLE SIZE DISTRIBUTION PARAMETERS OF MONODISPERSE BARIUM SULFATE SUSPENSIONS
in the readings on the refractometer scale was -4- (1-2) in the fourth decimal.
Determination of Density The densities of suspensions and dispersing media were determined by pycnometers at 25°C as described earlier (26). The density of barium sulfate particles was obtained by a pycnometric technique as suggested by Pugh and Heller (32).
Long axis Sample
N
S-115 S-116
Sample
b~ s(D~) (De). (rim) ( r i m ) (nm)
(D~)~ (rim)
P
300 287
245 284
254 290
1.089 1.046
N
Db (nm)
S(Db) (nm)
20 21
247 286
Short axis (Db)~, (Db)w ( r i m ) (nm)
P
S-115
300
121
12
122
126
1.115
S-116
287
120
9
121
123
1.052
RESULTS The typical form of monodisperse particles of one of barium sulfate suspensions used in our light scattering experiments can be seen in Fig. 1. It is of a nearly perfect ellipsoidal shape. Data on parameters of the particle size distribution for two samples of barium sulfate suspensions are given in Table I. Each sample is characterized by arithmetic mean lengths of the long and short axes of
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
ELLIPSOIDAL PARTICLE SCATTERING TABLE RESULTS
OF DETERMINATION SAMPLES OF BARIU~I
Sample
II
S-107 b
S-116 S-115
TABLE
III
REFRACTIVE INDICES • AND RELATIVE REFRACTIVE INDICES ~ OF DISPERSING M E D I A A N D B A R I U M SULFATE AT 25°C
OF DENSITY P2 OF SEVERAL SULFATE PARTICLES p12 a
301
p2
Refractive index, n W a v e l e n g t h (nm) 546 436
( g e m -3)
1.0204 1.0204 1.0032 1.0298
4.39 4.44 4.46 4.41
R e l a t i v e refractive index,b m Wavelength
(nm) 546
D e n s i t y of suspension measured at 25 °C. b Particle size of this s a m p l e : /), = 225 nm, /)b = 91 a m ,
particles, /3~ and /)b, respectively, obtained by the statistical particle size analysis of N counted particles, and by respective standard deviations of the distribution, s(D)~ and s(Db). The quantities (D~)~ and (D~)~ are the number average and weight average long axes, respectively, (Db)~ and (Db)~ are the number average and weight average short axes, respectively, and P is the polydispersity ratios defined as DJ/D~ 3 .All these quantities are defined and calculated as described elsewhere (25). I t is obvious from Table I that both samples of barium sulfate suspensions had a narrow particle size distribution as indicated by the P approaching unity and be regarded as monodisperse. Table I I gives data on both densities of barium sulfate suspensions (,012) and particles (p2) obtained at various barium sulfate particle sizes. The absence of a systematic variation of p2 in relation to the size of the particles indicates that their density does not change practically with the particle size. According to data in Table II, the average value for the TABLE
Barium sulfate particles Dispersing media Triton X-100 0.05% aq. solution Glycerol circ. 87%
436
1.594~
1.067 a
--
1.3337
1.3397
1.194
1.199
1.4536
1.4608
1.097
1.100
-
-
Calculated f r o m d a t a in Table I V b y Eq. [2]. b Defined a s npartioles/ndlspersing medium.
density of barium sulfate amounts to (4.43 4-0.03) g cm -3. This value is slightly lower (about 1.5%) than that of the mineral barite (33). This corroborates our previous finding that ellipsoidal barium sulfate particles have a sponge-like internal structure (10). Table I I I presents data on refractive indices n and relative refractive indices m of the dispersing media used in this study and of barium sulfate particles. Since there was no direct method for the determination of refractive indices of particles dispersed in such highly turbid suspensions, the refractive indices of barium sulfate particles were derived from (Ru(90)/C)o value determined by light scattering on particles of only one sample dispersed in glycerol (sample S-116) by using Eq. r2]. The parameters used for the calculation of these indices are given in Table IV. The P(0) values for 0 = 90 ° were IV
EXPERIMENTAL AND THEORETICAL D A T A USED I N EQ.
[-2] TO CALCULATE THE REFRACTIVE S-116) DISPERSED I N GLYCEROL
I N D E X OF
THE BARIUM SULFATE PARTICLES (SAMPLE W a v e l e n g t h in v a c u u m X, (nm)
546
436
[R~(90)/c-]o (g-1 cm2) K, X 10.3 (g-1 cm2) nx~ _P-l(t~) for 0 = 90°b
17.3 4- 0.2 4.84 -4- 0.09 1.4536 4- 0.0002 2.62
28.9 4- 0.5 12.1 4- 0.2 1.4608 4- 0.0002 4.23
a Refractive index of glycerol m e d i u m . b E x t r a c t e d f r o m K o c h ' s d a t a (6).
Journal o7 Colloid and Interjace Science, Vol. 50, No. 2, F e b r u a r y 1975
302
PETRES AND DEZELIC
soso~s-',00 .
I
~-
'
;ko " 546 nrn
w/~./~" Triton X- 100 ~=2 ~ ( ' / " TritonX-IO0 sat. B~SO4 o A CX 105 gcm-3 FIG. 2. Effect of dissolution of b a r i u m sulfate p a r -
(sample S-100) on the intensities of scattered light; dilution of the stock suspension with 0.05% aq. solution of Triton X-100 saturated (circles) and unsaturated (triangles) with barium sulfate. ticles
extracted from Koch's theoretical data (6)• The refractive indices obtained in this single sample of barium sulfate particles are listed in Table I I I and served as optical constants for subsequent exeriments. Since the relative refractive index m has been derived from the same relation used in other light scattering evaluations, one must not mistake it as an absolute optical constant. The values of m in this paper have to be regarded merely as •
"
.
[,¢1
BaSOa S - I 1 6 ;~o= 546 nm m = 1,097
5
,o
~"-~-----~,~
5
120
o. 3 2 ~
6
0
"------~'---"~-~----'-~-----~-c----o~ 0
2
50
<'-" 40 3
4
5
6
C x 105 g cm -3
FIG. 3. Reciprocal specific total Rayleigh ratios dependence of the scattering angIe and the concentration of barium sulfate suspension; sample S-116 dispersed in glycerol and measured at 546 nm.
c/R~,(O) in
some "Rayleigh-Debye" relative refractive indices. The preliminary experiments have shown that the most suitable concentration range for light scattering measurements was from 1 X l0 -5 to 6 X 10.5 g cm -4. The solubility of the barium sulfate in water and 87% glycerol at 25°C is 2.23 X l0 -6 g m1-1 (34) and 1.3 X 10.6 g m1-1, respectively. The later value was estimated on the basis of data reported by Collins and Leineweber (35). Thus, in order so prevent the dissolution of particles during the dilution of a concentrated stock suspention with 0.05 aq. solution of Triton X-100 or 87% glycerol, it was necessary to use the dispersing media previously saturated with barium sulfate. The effect of the dissolution of particles on the intensities of scattered light can be seen in Fig. 2. The concentrated stock suspension of barium sulfate (sample S-100, particle size /5~ = 300 nm, /5~ = 140 nm) was diluted in one case with 0.05% aq. solution of Triton X-100 saturated with barium sulfate and in the other case with the same Triton X-100 solution without addition of barium sulfate. I n the latter case a strong decrease of the specific total Rayleigh ratio, R,,(90)/c ascribed to the dissolution of particles, especially in the concentration range below 2 X 10-~ g cm-% was found, so that the extrapolation to zero concentration becomes impossible. The experimental values of form factors P(O) for both systems S-116 and S-115 were obtained from specific total and partial Rayleigh ratios extrapolated to the zero concentration, (R,, (O)/C)oand (V~ (O)/C)o,using Eqs. [2-] and [4-]. For that purpose graphs c/R~(O) or c/V~(O) vs. c for various angles were drawn. The typical graph of this type is shown in Fig. 3. All (c/Ru(O))o and (c/V~(O))o values were calculated by the first order least squares method. The average standard deviation on the values obtained was less than ± 2 % . Experimental values of form factors for barium sulfate particles dispersed in the
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
ELLIPSOIDAL PARTICLE SCATTERING
II
t
BoSO/+
'
S-II6
iOI
axial ratio
9~-
;ko(rim)[Ru(~')/c]c=o
|
5/+6
L
/+36 5/+6
7 ~-
p= 2.36
/
/'
/,
o
/
•
1
J'"
/"
[Vv~#)/c]~ °
•
/+36
/
/
/+ 3 2 I
0
I
2
3
X
FIG. 4. Theoretical curves of P-I(O) functions for spheres, rods and prolate ellipsoids of revolution with the axial ratio p = 2.36; points are experimental P-I(O) values obtained on barium sulfate particles (sample S-116) dispersed in glycerol from (R~(O)/c)o (circles) and (V~(O)/c)o (triangles) data; measured at
546 nm (open points) and 436 nm (filledpoints).
Open and filled points denote data measured at 546 and 436 nm. From Figs. 4 and 5 it is obvious that there is good agreement between experimentally obtained form factors and theoretical P-I(O) curves for ellipsoids of revolution with the given axial ratio (2.36 for S-116 and 2.03 for S-115). The statistical analysis has shown that the average percent deviations of the experimental from theoretical values (calculated for both wavelengths 546 and 436 nm) amount to -0.73~o for S-116 and - 2 . 8 % for S-115. We thought it interesting, from the point of view of the application of the RayleighDebye theory, to compare deviations of the experimental form factors from the theoretical values if for the same particle size the relative refractive index is increased by a change in the dispersing medium. To this end we calculated the P(O) values from the specific total Rayleigh ratios (R~(O)/c)o determined in the i
II
BaSO/+ axial
I0
glycerol medium (S-116 and S-115) were compared with theoretical P(O) functions for spheres, rods and prolate ellipsoids of revolution with axial ratios 2.36 and 2.03. A graphical presentation of the results obtained is shown in Figs. 4 and 5. Theoretical values of P-l(0) functions for spheres and rods are taken from data given by Doty and Steiner (36) while those for prolate ellipsoids of revolution are derived from Koch's data (6). Parameter x is given as ksR~, where R~ is the radius of gyration. This parameter is given uniformly for all particle shapes allowing a graphical presentation in one diagram. In the case of a prolate ellipsoid of revolution with the axial ratio p and the semiaxis a the radius of gyration is defined as a[(2 + f2)/S]L Points in Figs. 4 and 5 are experimental P-I(O) values obtained from (R~(O)/C)o (circles) and (V~(O)/c)o (triangles) data.
303
S-II5
ratio
;%(nm) 546 /+36
9 8
p=2.03 [Ru(~)/c]c= ° o
[Vv(#)/c] c_o v
5/+6 /+36
7
$
A6
3_ 5 4 3
rods
2 1
0
I
X
2
3
FIG. 5. Theoretical curves of 2-1(0) functions for spheres, rods and prolate ellipsoids of revolution with the axial ratio p = 2.03; points are experimental P-I(O) values obtained on barium sulfate particles (sample S-f15) dispersed in glycerol from (R,(O)/c)o (circles) and (V~(O)/c)o (triangles) data; measured at 546 nm (open points) and 436 nm (filled points).
Journal of Colloid and InterJace Science, Vol. 50, No. 2, February 1975
304
PETRES AND DEZELIC i
i
i
i
/
BaS04 S-II6
7
/ ~olnm)
rn
546
[Ru(,Y)/C]c: 0
I. 097 1.194
5
o •
axial ratio
•
.
•
•
e •
p:236
/o °
/°
o
o
I
0
/ • •
I
2
r
1
3
/.
Xe2
FIG. 6. Comparison of experimental p - l ( O ) values obtained on barium sulfate particles (sample S-116) at 546 n m from (R~(O)/c)o for m = 1.194 (fiIled circles; 0.05% aq. solution of Triton X-100) and m = 1.097 (open circles; glycerol) with theoretical P-I(O) values calculated for prolate ellipsoids of revolution, p = 2.36.
parallel way for particles dispersed in glycerol (m = 1.097 at the wavelength 546 nm) and 0.05% aq. solution of Triton X-100 (m = 1.194). A graphical presentation of the results obtained for the sample S-116 is seen in Fig. 6. The values on the abscissa are defined as x, 2 = (ksa)% where a = Db/2. As is well seen from Fig. 6, the experimental values obtained for m = 1.194, unlike those obtained for m = 1.097, are in poor agreement with the theoretical curve. The statistical analysis has shown that the average percent deviation of the experimental from theoretical values amounts to + 3 5 % in the former and about --1.8°-/o in the latter case, by which the individual deviations for m = 1.194 increase with the increase in the scattering angle from -15~o for 0 = 40 ° to + 4 6 % for v~ = 90 °. DISCUSSION For lack of an exact theory we have tried to explain light scattering on ellipsoidal particles by means of the Rayleigh-Debye theory. I t is clear that, with regard to the size of the particles studied, the relative refractive index must be fairly close to 1 in order to meet the condition determining the validity of the Rayleigh-Debye theory, i.e., that 2 k h (m -- 1) << 1. Our estimate was that m might be small enough if barium sulfate particles
were dispersed in the glycerol medium provided the refractive index of particles did not exceed the refractive index of barite, the natural crystal form of barium sulfate. The value of the refractive index for barite (37), for instance, for the light wavelength 546 nm amounts to 1.643 (calculated as the geometrical mean of the refractive indices n,, n¢, nr, because barite has two optic axes). For the same wavelength the refractive index for 87% glycerol is 1.4536, thus m being 1.13, while the particles dispersed in 0.050-/0 aq. solution of Triton X-100 would have m = 1.23. From the conclusions made in papers (38-40) concerning the range of the validity of the Rayleigh-Debye theory it appears clear that the possibility of the use of the RayleighDebye theory could be expected only for m _< 1.13. In addition, practically it may be assumed that the refractive index of the particles of barium sulfate obtained by homogeneous precipitation is lower than that of barite. In the earlier studies of the ultrathin sections of such particles it has already been shown that these particles possess a porous internal structure with the size of pores being about 30 A (10). In view of this and the results of some other authors (41-43) who have shown that the precipitates of barium sulfate
Journal oJ Colloid and Interface Science, Vol. 50, igo. 2, February 1975
ELLIPSOIDAL PARTICLE SCATTERING obtained by different types of homogeneous precipitation may be chemically heavily contaminated by foreign ions from the mother liquor and water molecules, it has been assumed that it is pores that are primarily the sites where the occluded material can be found (10). Thus, it is logical that the inside of the particles should be nonhomogeneous with regard to the refractive index. Besides, the composition of the reaction mixture in the process of precipitation barium sulfate particles by the Takiyama's method of homogeneous precipitation (24) is such that it may be assumed that the refractive index of nonhomogeneities within the particles, water included, is lower than that of barite. In view of this, it is very probable that the relative refractive index for particles dispersed in the glycerol medium is below 1.13. The refractive indices for barium sulfate particles determined from light scattering measurements are by about 3% lower than the barite refractive index. This is in agreement with our earlier claim about the nonhomogeneity of the particle inner structure. It should be noted that several years ago Lewis and Lothian (44) found that the refractive index for the barium sulfate particles examined amounted to 1.52 in the range of light wavelengths from about 0.41 to 0.78 ~m. The authors assumed that the particles (prepared from barium chloride and sulfuric acid in the presence of hydrochloric acid--following Andreasen (46)) had a spherical shape (the mean radius, ¢ = 4.6 Fzm) and calculated the refractive index by the expression deduced by van de Hulst (45) for the total scattering coefficient if r/X is much greater than unit and when m approaches unity. The obtained result for the refractive index was explained as an intermediate value between the refractive index for water and crystal barium sulfate, because on the basis of data on the specific gravity determined by the sedimentation velocity, Lewis and Lothian estimated that the particles contained about 60o-/ovoI water. Our values for the refractive index of barium
305
sulfate particles are relatively high in relation to those obtained by Lewis and Lothian but a complete comparison between our results and theirs is not possible because the methods used for the particle preparation as well as the form of the particles obtained were different. Moreover, in the course of some detailed investigations of the morphological properties of the barium sulfate particles (9, 10, 47) obtained under different precipitation conditions (24, 46) we have observed that regardless of their size, the particles appear as more or less three-axial forms. For this reason we are inclined to think that the particles studied by Lewis and Lothian could only approximately be considered spherical, so that the theory applicable for spherical particles could not be wholly applied to the particles in question. Generally, a good agreement between the experimental points for both samples of barium sulfate dispersed in glycerol (S-116 and S-115) and the theoretical curve P-l(v~) for prolate ellipsoids of revolution was obtained. Some greater deviations (but no more than 15%) of experimental from theoretical values were only observed at angles 0 ~< 50 ° and t~ >/ 130 ° (see, Figs. 4 and 5). The most probable and most important reasons for it may be an imperfect monodispersity of the samples and the particle deviation from the form of a perfect prolate ellipsoid of revolution. SUMMARY The angular dependence of the light scattered by monodisperse barium sulfate particles was determined at wavelengths of light 546 and 436 nm. The particles had the shape of prolate ellipsoids of revolution with the longer axis from 245 to 300 nm and the shorter axis from 120 to 140 nm. The axial ratio varied from 2.03 to 2.36. The particles were dispersed in 0.05% aq. solution of the nonionic detergent Triton X-100 or in 87% glycerol. The results are discussed on the basis of the RayleighDebye theory, and experimental and theoretical values of form factors P(O) for ellipsoidal particles are compared. It is found
Journal of Colloid and InterJace Science, Vol. 50, No. 2, February 197.5
306
PETRES AND DE2ELIC
t h a t it is possible to a p p l y the R a y l e i g h - D e b y e t h e o r y to i n t e r p r e t light s c a t t e r i n g on large b a r i u m sulfate p a r t i c l e s if those are dispersed in the glycerol m e d i u m . I n this case a good a g r e e m e n t between e x p e r i m e n t a l and theoretical values was o b t a i n e d . REFERENCES 1. MOEGLICn, F., Ann. Physik 83, 609 (1927). 2. LORD IRAYLEIGH,Phil. Mag. 44, 28 (1897). 3. KERKER, M., "The Scattering of Light and Other Electromagnetic Radiation." Academic Press, New York, 1969. 4. STEVENSON,A. F., J. Appl. Phys. 24, 1143 (1953). 5. SHATILOV,A. V., Opt. Spektrosk. 9, 86 (1960). 6. KocH, A. L., Biochlm. Biophys. Acta 51, 429 (196l). 7. Koch, A. L., AND EHRE~ELD, E., Biochim. Biophys. Acta 165, 262 (1968). 8. KocE, A. L., J. Theor. Biol. 18, 133 (1968). 9. PETRES, J., DEZELIC, GJ., AND TE~AK, B., Croat. Chem. Acta 38, 277 (1966). 10. PETRES, J., DE~ELId, GJ., AND TE~AK, B., Croat. Chem. Acta 41, 183 (1969). 11. VAN DE HULST, H. C., "Light Scattering by Small Particles." John Wiley and Sons, Inc., New York, 1957. 12. DE~ELId, GJ., Pure Appl. Chem. 23, 327 (1970). 13. GUINIER, A., Ann Phys. (Paris) 12, 161 (1939). 14. PATTERSON,A. L., Phys. Rev. 56, 972 (1939). 15. RoEss, L. C., AND SKULL, C. G., J. Appl. Phys. 18, 30 (1947). 16. POROD, G., Acta Phys. Austr. 2, 255 (1948). 17. EDSALL, J. T., AND DANDLIKER, W. B., Forlschr. Chem. Forsch. 2, 1 (1951). 18. SAITO, N., AND IKEDA, Y., J. Phys. Sac. Yap. 6, 305 (1951). 19. SCH~nDT, P. W., AND HmHT, R., JR., J. Appt. Phys. 30, 866 (1959). 20. MAL~ON, A. G., Acta Crystallogr. 10, 639 (1957). 21. BEIDEE, G., BISCHOF, M., GLATZ, G., POROD, G., VON SACKEN, J. CH., AND WAWRA,H., Z. Elektrochem. 61, 1311 (1957). 22. MITTELBACH, P., AND POROD, G., Acta Phys. Austr. 15, 122 (1962). 23. MITTELBACtI, P., Acta Phys. Austr. 19, 53 (1964). 24. TAKIYA~A, K., Bull. Chem. Soc. Yap. 31, 950 (1958).
25. D~EEId, GJ., WRtSCHER, M., D~WD~, Z., AND KRATOHVIL,J. P., Kolloid Z. 171, 42 (1960). 26. gEG~OVI6, N., AND D~ELId, GJ., Croat. Chem. Acta 45, 385 (1973). 27. PETRES, J. J., AND DEW,LId, GJ., J. Polym. Scl. Part C 42, 1181 (1973). 28. THURMOND, C. D., J. Polym. Sci. 8, 607 (1952). 29. BRINE, B. A., HALWER, M., AND SPEISER, R., Y. Opt. Soc. Amer. 40, 768 (1950). 30. KRATOHVIL,J. P., AND SMART, C., J. Colloid Sd. 20, 875 (1965). 31. KRATOHVIL, J. P., J. Colloid Interface Sci. 21, 498 (1966). 32. PuGI.I, T. L., AND HEELER, W., J. Colloid Sci. 12, 173 (1957). 33. WALTON, G., AND WALDEN, G. H., J. Amer. Chem. Soc. 68, 1742 (1946). 34. SEIDELL, A., ASrD LINK~, W. F., "Solubilities of Inorganic and Metal-Organic Compounds." D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. 35. COLLINS,F. C., AND LEINEWEBER~J. P., J. Phys. Chem. 60, 389 (1956). 36. DOTY, P., AND STErnER, R. F., J. Chem. Phys. 18, 1211 (1950). 37. OPTISC~E KONSTANTEN in Landolt-Boernstein, "Zahlenwerte und Funkfionen," II Band, 8. Tell, 6. Auflage, Berlin/GStfingen/Heidelberg, 1962. 38. DE'ELIte, GJ., AND KRATOttVIE, J. P., Kolloid Z. 173, 38 (1960). 39. HELLER, W., J. Chem. Phys. 42, 1609 (1965). 40. KERKER, M., FARONE, W. A., AND MATIJEVIC, E., J. Opt. Soc. Amer. 53, 758 (1963). 41. PAUEIK,F., BUZ~.GH,E., PCEOS, L., AND ERDE¥, L., Acta Chim. (Budapest) 38, 311 (1963). 42. Em)EY, L., PAUEIK,F., BUZAGH,E., ANDP6EOS, L., Acta Chim. (Budapest) 41, 109 (1964). 43. BUZ3.GtI-GERE, E., PAULIK, F., AND ERDEY, L., Talanta 13, 731 (1966). 44. LEwis, P. C., AND LOTmAN, G. F., Brit. J. Appl. Phys., suppl. 3, S 71 (1954). 45. VAN DE HUEST, H. C., see Ref. (11), p. 172. 46. ANDREASEN, A. H. M., SKEEL-CItRISTENSEN,K., AND KJAER, B., Kollold Z. 104, 181 (1943). 47. PETRES, J. J., DE~ELI(~, GJ., AND 'I'EZAK, B., Croat. Chem. Acta 40, 213 (1968). 48. CROSS, D. A., AND LATIMER, P., Appl. Opt. 11, 1225 (1972).
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975