J. Phys. Chem. 1983, 8 7 , 1614-1618
1614
Wavelength Dependence of the Turbidity of Spheroidal Particles Calculated in the Stevenson-Helier Approximation Yoh Sano and Masayukl Nakagakl' Institute for Plant Virus Research, Tsukuba Science Ctty, Yatabe, Ibaraki 305, Japan, and Facutty of Pharmaceutical Sciences, Kyoto Universtty, Sakyoku, Kyoto 606, Japan (Received: February 18, 1982; I n Final Form: November 23, 1982)
The wavelength exponent e of the turbidity defined by the equation, (r/c),, = kXo-', for spheroidal particles has been studied theoretically in the Stevenson-Heller (SH) approximation, that is, the approximation up to a8. The exponent t is constant and is equal to 4 in the Rayleigh-Gans approximation of spheroids (approximation of a6)depending neither on a,m, nor p , but in the SH approximation it decreases monotonously with a when m and p are kept constant. Here, a is the relative particle radius, m is the relative refractive index, and p is the axial ratio of the spheroidal particle. The wavelength exponent to be obtained experimentally, to, is not equal to t, because refractive indices, and therefore k as well, depend on the wavelength. The exponent to can be calculated if the dispersion of the refractive index is given by an equation such as the Cauchy equation. The numerical values of to for X, = 436 nm are also calculated for comparison with experimental data obtained for nucleoproteins such as viruses in water at 20 "C.
Introduction Turbidity measurements have often been used in the studies of biological dispersions such as suspensions of cells or viruses, or solutions of proteins. The wavelength dependence of the turbidity has been used to obtain the size and mass of macromolecules,1-5 and to correct the extinction of solutions for scattering contributions to obtain the true For spherical particles, Heller and c o - w ~ r k e rhave s ~ ~ ~calculated theoretical values of turbidity and its wavelength exponent with the Mie theory. Although the expressions for the wavelength exponents for isotropic rods, spheres, and random coils have been also evaluated by using the intraparticle interference functions applicable only for m N 1 and the refractive properties with Cauchy relation^,^ no reliable theoretical values for spheroidal particles are known as yet. Disregarding approximating treatments or treatments quantitatively correct only for limiting condition such as (a) for infinitely thin rods or infinitely thin disks, or (b) for bodies whose refractive index differs very little from that of the surrounding medium, a theoretical expression for the scattered light intensity of spheroidal particles was first given by Rayleighloand Gans" in the approximation of cy6, which will be referred to as the Rayleigh-Gam theory of spheroids (RGS theory) in order to avoid confusion with the well-known Rayleigh-Gans (RG) theory of scattering which involves the restrictive assumption that the refractive index of the particles is close to that of the surrounding medium (m N l),where
can be applied up to a length of the largest spheroid dimension of 113 of the wavelength in the medium without committing an error in excess of about 5%. This therefore opened up, to precise evaluation by means of light scattering, a potentially wide range of colloidal dimensions and molecular weight heretofore not accessible. The turbidity in this approximation was calculated by one of us.16 However, the wavelength exponent of the turbidity in this approximation has not been calculated yet. First of all, the theoretical equation for the wavelength exponent of the turbidity of spheroidal particles at random orientation is derived on the basis of the SH approximation which is applicable to particles as large as 0.1 pm in diameter, which means especially to particles of colloidal dimensions, and the numerical values calculated by this equation are discussed as a function of the axial ratio p of the spheroid p = a/b (2) where b is the length of semitransverse axis, and therefore p > 1 for prolate and p < 1 for oblate spheroids. Basic Relationships. The turbidity, r , is defined as the optical density per unit length of the optical path for colorless systems: T = -In ( I t / I o ) / l (3) Here Io and It are the intensities of incident and transmitted light and 1 is the length of the optical path in the turbid medium. The specific turbidity extrapolated to infinite dilution
a = 27ra/X
(1) P. Doty and R. F. Steiner, J . Chem. Phys., 18, 1211 (1950). (2) W. Heller, H. L. Bhatnager, and M. Nakagaki, J. Chem. Phys., 36, 1163 (1962). (3) R. D. Camerini-Otero, R. M. Franklin, and L. A. Day, Biochemistry, 13, 3763 (1974). (4) A. Cancelleii, C. Frontali, and E. Gratton, Biopolymers, 13, 735 (1974). (5) R. D. Camerini-Oteroand L. A. Day, Biopolymers, 17, 2241 (1978). (6) A. F. Winter and W. L. G. Gent, Biopolymers, 10, 1243 (1971).
(1)
Here a is the length of semiaxis of symmetry for the spheroidal particle and X is the wavelength of light in the medium. The difficulty with the RGS theory is that it considers only dipolar scattering, i.e., it applies only to spheroids which are very small compared to the wavelength. Recently, the theory has been extended to a8 by Stevenson12and Hellerl3-I5to extend the range of this theory by taking into account magnetic dipole radiation and electric quadrupole radiation. Tests of this extension of the RGS theory, which shall be referred to as the Stevenson-Heller (SH) approximation, made it likely that it ~-
*Address correspondence t o the author a t the Faculty of Pharmaceutical Sciences, Kyoto University, Sakyoku, Kyoto 606, Japan.
(7) G. Holzwarth, D. G. Gordon, J. E. Mcginnes, B. P. Dorman, and M. Maestre, Biochemistry, 13, 126 (1974). (8) H. Eisenberg, R. Joseph, and E. Reisler, Biopolymers, 16, 2773 (1977). (9) W. Heller, J . Chem. Phys., 40, 2700 (1964). (IO) L. Rayleigh, Phil. Mug., 44, 28 (1897). (11) R. Gans, Ann. Phys. (Leipzig),37, 881 (1912). (12) A. F. Stevenson, J . Appl. Phys., 24, 1134, 1143 (1953). (13) W. Heller and M. Nakagaki, J . Chem. Phys., 60, 3889 (1974). (14) M. Nakagaki and W. Heller, J . Chem. Phys., 61, 3297 (1974). (15) W. Heller and M. Nakagaki, J . Chem. Phys., 61, 3619 (1974). (16) M. Nakagaki, J . Phys. Chem., 84, 1587 (1980).
0022-3654/83/2087-1614$01.50/00 1983 American Chemical Society
The Journal of Physlcal Chemistry, Vol. 87, No. 9, 1983
Wavelength Exponent for Spheroids
(T/c)~,where c is the concentration in g/mL, is related to Z by the following equation: (T/
= (X 2 N ~ 2?rM2) /
C)o
1815
4.0
(4)
where M2 and N A are the molecular weight (particle weight) and Avogadro number. A dimensionless quantity Z is expanded in a power series of a: =
+ 28,'
(5)
The first term is treated by the Rayleigh-Gans theory of spheroidslOJ1which considers scattering by an oscillating electric dipole. The second term represents the contributions of a magnetic dipole and an electric quadrupole, which was first derived by Stevenson12and then discussed by Heller and co-workers.13-15 The treatment including up to the second term is called the Stevenson-Heller approximation. The values of and Z8have been calculated by using the dimensionless scattering intensity (i) for the spheroidal particle16 26 (4/3)[A2 (2/3)AB + (1/3)B2] (6a) Z8 = -(8/9)(1 - l/p2)[(11/75)A2
+ (31/375)AB +
(1/25)B2] + (8/9)[A(3C + D)+ B(C
+ D)]
(6b)
Here A, B , and C are the functions of m and p as defined by eq 7-10 in ref 13, where m = n/no (7) and n and no are the refractive indices of the spheroidal particle and of the surrounding medium, respectively. Wavelength Exponent for Spheroids. The usual experimental parameter for the wavelength dependence is the negative slope, to,of the logarithm of the turbidity (or optical density) against the logarithm of the experimental wavelength (wavelength in vacuo), A,,, where A, = noh (8) This dimensionless slope to is called the wavelength exponenta2 The method of particle size determinations by means of the wavelength exponent was introduced by Heller and co-workers for spherical particles by means of the Mie theory.2 However, no reliable theoretical values of the wavelength exponent for spheroidal particle are known as yet. For accurate particle size determinations of spheroidal particles, it is necessary to derive the theoretical %(a,m,p)values for various a,m, and p values. The theoretical equations for the wavelength exponent of spheroidal particles at random orientation can be derived on the basis of the Stevenson-Heller approximation (i.e., a8theory). The wavelength exponent to defined by to = -d In (T/c)O/d In A,, (94 is rewritten by using eq 4 and 8 to = -2 2 d In no/d In Xo - d In Z(m,a,p)/d In Xo (9b) Remembering that m, no, and a are functions of Xo one obtains €0 = tF1 - tzF2 (10) where e = a In X / a In (Y - 2 (114
+
tz
= a In X/am
F1= 1 - d In no/d In X, Fz = dm/d In X,
(Ilb) (IlC)
Old)
W
3.5
3.0 0.5
0
a
1.0
1.
Flguro 1. Variatbn of e with a of spherical particles @ = 1) for various m values. The broken line is in the RGS approximation and the soli lines are in the SH approximation. The Mie values at various m values are as follows: (X) 1.05; (A)1.10; (0) 1.20; (0) 1.30.
Equation 10 is equivalent to eq 6 in ref 2, if we assume that the e at p = 1is equal to n in ref 2, and 2: is a function of a,m, and p for the spheroidal particles. From eq l l a and 5, t in the SH approximation is given as follows: t
=4
+ 2LY2Z8/(Z6+ (U2Z8)
(12)
In the Rayleigh-Gans approximation (RGS approxima= 0, t is, therefore, always 4 at any a, m, tion) where and p value. Since t is a hypothetical value of the wavelength exponent obtained by neglecting the dispersion of refractive index, the actual value to should be calculated. For this purpose, it is necessary, according to eq 10, to know the variation of the turbidity with m and the dispersion of m and of no with the wavelength in addition to t. The first of these three correction factors, t2,will, in general, be by far the most important one. It can be evaluated, again on the basis of the SH approximation, for various ranges of a, m, and p values within which the exponent method seems to be of practical interest. The dispersion corrections F , and F2 depend on the individual characteristics of the spheroids and of the solvent. These dispersion corrections, therefore, must be determined for each particular system unless this information is available from the literature.
Numerical Results and Discussion Variation of the Exponent t. Figure 1 shows the variation of the exponent t with a of spherical particles (p = 1)for various m values. Each curve is calculated from eq 12 together with the following equations at p = 1: (2&,=,= (4/3)[(m2 - l)/(m2 + 2)12
(13a)
= - ( 8 / 5 ) [ ( m 2- l)/(m2 + 2)]'(2 - m2)/(m2+ 2)
(13b) The broken line shows that t in the RGS approximation is always constant at 4, while solid curves in the SH approximation decrease with a because 2 8 C 0 in eq 12. In Figure 1, the Mie values are also plotted from ref 2, showing that the Mie values also decrease with a. It is seen
1616
Sano and Nakagaki
The Journal of Physical Chemistry, Vol. 87, No. 9, 1983 I
I
I
I
0 7".0.8'.L
0.9
'\\-
m =1.2 0 Flgure 2. Variation of the coefficient t with a values of the prolate spheroids for various axial ratios and m = 1.20 in the SH approximatlon. Open circles are Mle values. Curve A indicates the limiting value at p m. The horizontal broken line represents the RGS approximation.
-
that the SH approximation agrees well with the Mie values at various m up to at least a < 0.5, and that the values in the SH approximation are a little larger for m > 1.20 and a little smaller for m < 1.20, in the region where 0.7 < a < 1.50. The agreement between them is best when m is about 1.20. This result agrees well with that of the specific turbidity.16 In the region where 0.5 < a < 0.7, the Mie values show extreme oscillations, but the exponent in the SH approximation decreases monotonously with a. Figure 2 shows the variation of the exponent t with a for prolate spheroids of various axial ratio p and of a given m value, the case of m = 1.20 being shown here as an example. At a certain p value, t decreases monotonously with increase in a,and the decrease of E from the RGS value (shown by the horizontal broken line, which is constant at 4 irrespective of a) is larger as a becomes larger. The difference between the RGS and the SH approximations becomes smaller when the particles are deformed from the spherical shape; the change is rapid up to p = 2 , and for p > 2 the change is slow, passing through a maximum point of e at about p = 3 for m = 1.20. For the thin rodlike molecules having the limiting value of p m, E is calculated with the equations
-
p4Z6"= (4/81)[(m2 - l)/(m2
+ 1)I2(m4+ 2m2 + 9) (14a)
p4Z," =
-(8/30375)[(m2 - l)/(m2
+ 1)I2(15m4+ 32m2 + 273)
-
(14b) The t value thus calculated for p m and shown with curve A is much greater than that of the spherical particle ( p = 1) as is seen in Figure 2. Figure 3 shows the dependence of the exponent 6 on the axial ratio p of the prolate spheroids for various a values and m = 1.20. The values in the SH approximation show a stronger dependence on the axial ratio up to at least 4. When a particle is deformed from a sphere to a prolate spheroid (p > l),the E value increases steeply at first and the increase is the larger as a becomes larger. Then, t passes through a maximum at about p = 3 and then decreases monotonously with increase of p , and the
0
5
p
10
a3
Figure 3. Variation of the coefficient t with axial ratio p of the prolate spheroids for various a values and m = 1.20. The broken line is in the RGS approximation and the solid lines are in the SH approximation. The horizontal lines on the right side indicate the limiting values at p
-
m.
decrease is the larger as a becomes larger. When a particle is deformed from a sphere to an oblate spheroid (p < l), the t value decreases rapidly, and the decrease is the steeper as a becomes larger. According to eq 12 the t value diverges to --m when CY approaches a. = (&/ (-&))'/', where the denominator in eq 12 is equal to zero. In the region a > a. the t value cannot be assigned with significance because the turbidity calculated by eq 4 and 5 becomes negative in this region. This unreasonable result for the turbidity to be negative at small p and large a values has already been presumed from Figure 5 of ref 16. We must use a higher approximation than the SH one in this region. General theories without any size or refractive index limitations have also been published: for example, for the full boundary value the solution by Asano and Yamamoto17 and/or for the extended boundary condition method (EBCM) the solution by Barber and Yeh.', Therefore, theories such as the full boundary value solution or the EBCM solution will be adapted to some regions of oblate spheroids or to the regions of prolate spheroids of a >> 1. A comparison of these theories with the SH approximation will be discussed elsewhere. However, the SH approximation is interesting because of its applicability to particles of colloidal dimensions as large as 0.1 pm in diameter. In Figure 4 solid curves show the condition a = ol, as a function of axial ratio p for spheroids of various m values. The value of a0 is the larger as m becomes larger. If we assume from the previous paper16 that the SH approximation is applicable to the region a I1 and p (= a 3 / p 2 )I1, which is the region below both the horizontal broken line and the broken curve in Figure 4, the negative turbidity in question occurs only above the solid curves in the range of p < 0.1. The SH approximation can, therefore, be applied to the region below the solid curves shown as a hatched area in Figure 4. The effect of m on the variation of the exponent t with axial ratio for the prolate spheroids at a given LY value is (17) S. Asano and G. Yamamoto, Appl. Opt., 14, 29 (1975). (18)P. Barber and C. Yeh, A p p l . O p t . , 14, 2864 (1975).
The Journal of Physical Chemistry, Vol. 87, No. 9, 1983 1617
Wavelength Exponent for Spheroids
4c i
I
l
I
I
I
I
I
I I
CJ W
2c
Figure 4. Variation of a. (where the denomlnator In eq 12 is equal to zero) with axial ratio p for spheroids of various m values. The dotted curve shows the relation of p (= a 3 / p 2 )= 1. The SH approximation is applicable in the hatched region. I
I
\
’.5O
C
1.1
I
1.3
I
m
1.
Flgure 6. Variation of coefficient c p with m values of the spheroids. The solid line is calculated from eq 16 and broken line is calculated from eq 17. The horizontal bars define the range between the limiting oblate and prolate spheroids.
The value of e2 at constant a decreases with increasing m, as shown in Figure 6. At any given m, the entire range of t2 values found between both limiting cases of the prolate ( p m) and the oblate (p 0) spheroids and 0 4 a 4 1 in the SH approximation is represented by a vertical bar cut off by narrow horizontal bars. The solid curve connecting the bars is satisfied by the equation +
+
t2
= [2/(m - l)] - (m - 1)
(16)
The Mie value shown with dotted curve in Figure 6, on the other hand, is satisfied by the equation2 E2(Mie) = [2/(m - l)] - 5(m - 1)
‘
3.6tl.05
0
a~0.7 I
5
I
p
10
Figure 5. Variation of coefficient e with axlal ratio p of the prolate spheroids for varlous m values and a = 0.7.
shown in Figure 5, the case of a = 0.7 being shown here as an example. The exponent t shows the intense dependence on the m values, i.e., with increasing axial ratio it increases monotonously for m < 1.15, but it decreases monotonously for m > 1.35. For 1.20 4 m 4 1.30, t takes a maximum in the region of about p = 2-3, and the maximum point shifts to a smaller p value as the m value increases. After the maximum, the e value then decreases monotonously with increase of p. Needless to say, the exponent t at any p value is larger as the m value becomes larger, for particles of the same size. According to eq 13b, the term Zaat p = 1approaches zero when m approaches 4 2 and changes its sign when m exceeds 4 2 . The E value at p = 1,therefore, reaches and then exceeds 4 when m exceeds 1.414. Variation of the Correction Factor e2. The correction factor e2, defined by eq l l b , is easily obtained from the available Z(m,p,a) data according to
(17)
It is seen that the difference between the SH approximation and the Mie value becomes larger as m increases. The deviation of the SH value from Mie’s value is 8.9% at m = 1.20, and is less than 5 % if m is smaller than 1.153. variation of the Exponent to. The usual experimental parameter for the wavelength exponent is not t but to. It is, therefore, important to obtain theoretical values for the wavelength exponent to for the spheroidal particles. According to the definition of the wavelength exponent to in eq 10, it is necessary to know, in addition to E and t2, the dispersion corrections Fl and F2 Dispersion data are either given or may be expressed in terms of the Cauchy equation
no(Xo)= no(Xl)(al+ bl/Xo2)
(18a)
n(X1)(a2+ b2/Xo2)
(18b)
n(X0) =
no&) and n(Xl)are the solvent and the particle refractive index at a reference wavelength, X1. In order to evaluate
F1and F2,one may use eq 18 provided al, bl, a2, and b2 are known or determined. If the medium is water, the values for al and bl for water in the range 3950-6700 A are available from the “International Critical Tables”.19 Likewise, for many biopolymers it has been found that eq 18 is well fit. A Cauchy-like relationship has been also used (19) “International Critical Tables of Numerical Data, Physical Chemistry and Technology”,Vol. VII, McGraw-Hill,New York, 1933, p 13.
1618
The Journal of Physical Chemistry, Vol. 87, No. 9, 1983
4,!
I
I
E
W
m U
c
0
0
W
4.:
4.
5
p
10
C
Flgue 7. Variation of coefficienteo in the SH approximation wlth axial ratio p of the prolate spheroids for various LY values and m = 1.20. The horizontal lines on the right side indicate the limiting values at p
-
m.
to express the wavelength dependence of the refractive index increment
dn,(xo)/dc = [dn,(hl)/dcl(ai+ b 2 / / b 2 )
(19)
where n, is the refractive index of the suspension. The values of ai and b i were obtained for several proteins by Perlman and Longthworthm and for many nucleoproteins such as viruses by Camerini-Otero et al.,3-5in which both cases were close to each other over the range 350-700 nm. Thus, with the approximation that the wavelength dependence of the refractive index increment is equal to that of ( n - no),one obtains F1
= 1 + a1/2
F2 = ( m- 1)(a1 - a2)/2
(20a) @Ob)
where a1 and cy2 are equal to 4bl/(al&2 + b,) and 4 b i l (a2'X,2 b2/), and these values have been calculated by Camerini-Otero et al. over the range 300-700 nm (Table I11 of ref 5). Equation 20 means that the dispersions of no and n make positive contributions to the wavelength exponent.
+
(20) G. E. Perlman and L. G. Longthworth, J. A m . Chem. Soc., 70, 2719 (1948).
Sano and Nakagaki
If we assume that the wavelength of light in vacuo Xo is 436 nm and the medium is water at 20 "C whose refractive index is 1.3345 and m = 1.20, both dispersion coefficients F1and F2 are calculated by using the a1and a2 values. The eo value can, therefore, be obtained for spheroidal particles with these parameters. The results are shown in Figure 7 . For the calculations the values used are a, = 0.9922 and bl = 2.3 X lo3 nm2 for the aqueous solvent, a2/ = 0.925 and b i = 2.2 x lo4 nm2 for spheroidal particles. Figure 7 shows the variation of the wavelength exponent eo at 436 nm in the SH approximation against p with various a values. The to at 436 nm increases remarkably when the spheroid deforms from a sphere to a prolate one, but no appreciable change of eo is seen when p has exceeded 3 for small a values. For large a and large p values, eo decreases with p and therefore the p , associated with to at a constant a, is double valued. This would add a difficulty in the determination of the axial ratio for large prolate spheroids, even if the molecular size a is known independently. When a particle is deformed from a sphere to an oblate spheroid (p < l),the to value decreases rapidly, and the decrease is the steeper as the a becomes larger. This is due to the fact that the denominator in eq 12 tends to rapidly approach zero for larger a and smaller p values. Because the wavelength dependences of the refractive index of the polymer and of the solvent are small, the dispersion coefficient F1 is almost equal to 1, and F2 is constant for a given wavelength such as 436 nm. The dependence of e2 on a and p is small, and the second term on the right-hand side in eq 10 is almost constant. Therefore, Figure 3 is almost superimposable with Figure 7 with a shift of the ordinate. If the light scattering spheroids are homogeneous compact particles, the value of a immediatelygives information on their dimension along their symmetry axis. In conjunction with a knowledge of p which may be obtained by way of electron microscopy, their volume is also determined. In general, the determination of the partial specific volume of compact materials is easy, so that the theoretical variable m is obtained with the well-known quantities (dnldc) and no,21which are closely related to m.22 For macromolecules, a knowledge of a and p immediatelygives information on the molecular weight M , by using the partial specific volume.I6
Acknowledgment. The authors express their hearty thanks to Professor Wilfried Heller who has kindly encouraged us to proceed with this work. (21) W. Heller and E. S. E. Schwartz, J.Polym. Sci., 10, 1903 (1972). (22) W. Heller, J. Phys. Chem., 69, 1123 (1965).
