BRUNOH. ZIMMAND WALTERB. DANDLIKER
644
the average value off and f 2 are
f = J 10. f d P
(Za =--(a
F=
dP
$f2
0
(w
=
Jlf
-
I)! (a
-
1)!2
(Za (a
-
-
=: . _ _ _
l)! 1)!2
( a - l)!(a
(2a
xf”-’(1- f ) a - 1 df
- l)!o! 2a!
Vol. 58
a
=
+ 1)l -
+ I)!
-
(a
+ l)(U)
+
( ~ a 1)(2a)
The average deviation
1
za = 2
X
THEORY OF LIGHT SCATTERING AND REFRACTIVE INDEX OF SOLUTIONS OF LARGE COLLOIDAL PARTICLES BY BRUNO H. ZIMM AND General Electric Research Laboratories, The Knolls, Schenectady, N . Y .
WALTERB. DANDLIKER The Department of Biochemistry, University of Washington, Seattle, Washington Received April 19, 19.54
A general equation for the intensity of light scattered by a suspension of independently scattering particles is derived
+
VO,,= (Mcnz/h~4No)[4n2(dn/dc): (XO*/~)(T/C)~]. V0,vis the Rayleigh ratio of the vertically polarized component of the
excess scattered light a t 0 = 0 with the electric vector vertical in the incident light. The other quantities involvedare the molecular weight M , concentration c, refractive index n of the solution, the wave length Xa in vacuo, Avogadro’s number N o , the refractive index increment (dnldc) and the excess turbidity, r, due t o the solute. The second term of the equat,ion, whic!i is important in the case of very large particles, has usually been omitted in the past. For spheres, a general relationship between dnjdc, particle size 01 and refractive index of particle and medium can be deduced: dn/dc = 3nR(i1*),iZdD where R ( ~ I *is) obtained from tabulations of the Mie theory. The density of the particle is D and 01 = Z?rr/h where r is sphere radius and X is the wave length in the medium. For small spheres the refractive index equation reduces to the well known result: dn/dc = 3n/2D (m* l / m 2 2), where m is the relative refractive index.
-
+
Introduction Interpretations of light scattering studies on macromolecular and colloidal systems have used either of two theories. The study of very large spherical colloidal has been made possible by the Mie theory of scattering from spheres“’ and computations based on i t . 8 On the other hand, Einstein’s fluctuation theory,g and its limiting simple form for dilute solutions, the Rayleigh formula, lo have been used for smaller colloidal particles and high polymer molecules.11 An adaptation of a treatment given by Schuster12 allows one to connect these two theories and t o obtain a general relation between the scattering and refractive index of a colloidal solution and the size of the particles. This relation is the extension of the Rayleigh relation to very large particles. The ordinary Rayleigh relation is found to be valid only if the amount of light scattered per particle is not (1) V. K.La Mer and D . Sinclair, N.D.R.C. Report 57 (1941)and 1668 (1943). (2) I. Johnson and V. K. La Mer, J . A m . Chem. Soc., 69, 1184 (1947). (3) D . Sinclair and V. K.La Mer. Chem. Reus., 44, 245 (1949). (4) M. Kerker and V. K. La Mer, J . A m . Chem. Soc., 72, 3516 (1950). (5) W.B. Dandliker, ibid., 72, 5110 (1950). (6) G. Mie, Ann. Phyaik, 26, 377 (1908). (7) H.C.van de Hulst, “Optics of Spherical Particles,” Duwaer and Sons, Amsterdam, 1946. (8) “Tables of Scattering Functions for Spherical Particles,” National Bureau of Standards. Applied Mathematics Series 4 (1949). (9) A. Einstein, Ann. Physik, 33, 1275 (1910). ( 1 0 ) Rayleiah, PhiL Mae., 41, 447 (1871). (11) P . Debye, J. A p p l . Phgs.. 16, 338 (1944); G.Oster, Chem. Revs.. 43,319 (1948). (12) A. Schuster, ”An Introduction to the Theory of Optics,” Second Ed., Edward Arnold and Co.,London, 1920,p. 325.
too large, a phase shift of the scattered light having been ignored in its derivation. An approach related in some respects to the present one is given by van de Hulst.13 Theory Let the incident light be plane-polarized with the electric vector defined by Eo = Ro COS
(ut
-k ~ )
where z is measured from a point in the scattering medium and k is 27r times the reciprocal of the wave length. The light scattering from a single particle can be represented by
+
Ro[A cos (wt - k r ) B sin ( w t - k r ) ] f ( G , d ) / r ( 1 ) where r is the distance from the particle, 9 and 4 E.
=
are angles relating the directions and states of polarization of the incident and scattered rays, and f(tJ,+) is the function that describes the angular dependence of the scattering. We may set f ( 0 , 0) equal to unity. The quantity B allows for a change of phase. Now let a parallel beam of light traverse a scattering medium containing N independent particles per unit volume and consider the scattering occurring within a thin layer of thickness Ax, the x-axis being the direction of propagation of the beam. The method described in the appendix enables us to calculate the total forward scattering in the direction of propagation of the incident beam a t a distance z from the layer as
- k z ) - B cos(ot - k z ) ] (2) If the material is a solution, we take n as the refracEt, = RoXNAx[A sin
(wt
(13) H.C. van de Hulst, Physica, 16, 740 (1949).
LIQHTSCATTERING OF SOLUTIONS OF LARGECOLLOIDAL PARTICLES
Aug., 1954
tive index of the solvent alone while n' is that of the solution as a whole. The wave length in vacuo is Xo while the wave length in solution is X = Xo/n' a n d k = 27r/X. In the following discussion we shall neglect the scattering from the solvent itself so that the conclusions apply to the excess scattering due to the solute. I n the forward direction the scattered wave adds to the incident wave to give the transmitted wave. The constant part, (-ks), of the phase angleis the same in the inc,ident and scattered waves and may be dropped. The electric vector in the transmitted wave is then found to be E = Eo + Et. = Ro[(l - B N X A X )COS wt
+
(ANXAX)sin ut] ( 3 )
This may be reduced to
+
E = R~ ~ [ A N X A X P ( 1 - B N X A X YC O S
1
) -ANhAx BNXAX
(4)
or, since Ax may be made as small as desired E = Ro(1 - BNXAX) C O S (ut - A N X A X ) (5) From purely phenomenological theory we know that the transmitted wave is retarded in phase by an amount 6 compared to the incident wave, where 6 is related to Ax and the refractive indices by On the other hand from eq. 5 we see that S =
(7)
ANXAx
so that
645
These formulas have been derived for the case where only one kind of scattering particle is present. If there are several kinds of particles average values of A and B or A 2 and B 2must be used in eq. 5 and 13. The same thing must be done if the scattering is depolarized and depends on the orientation of the scattering particles. When the phase-shift terms containing B are negligible these averages are simple and lead to the usual formulas with the weight-average particle weights and Cabannes depolarization factors.11 However, if the terms containing B are important special assumptions must be made about the variation of B to obtain the averages. We do not care to investigate this subject further here, but restrict ourselves to the case of one type of particle and no depolarization of the forward-scattered light. Substituting the values of A and B from eq. 8 and 11, we obtain VO," =
4 ~ ~ n ' ~- (n)q n ~ Xo"
rtn'2
+4Nho2
(15)
The weight concentration, c, Avogadro's number,
NO,and the molecular weight, M , are now introduced, giving N = Noc/M. Also we assume n' n = c(dn/dc).
These substitutions give
The Refrabtive Index of a Suspension of Spheres The results of Mie'sc exact theory of the scattering from spheres may be related to VO,,.in terms of the nomenclature of La Mer and Sinclair' by the equation Vo,, = cNoX2( IR(il*)
+ iI(ii*)[ ) ' / ~ T ' M
(17)
The intensity of the transmitted beam I , the time where the scattering functions R(il*)and I(&*)arc t o be evaluated for scattering in the direction of the average of E2,is incident beam. For small particles we know ( T / c ) E a = Ro'(1 - BN>Ax)'/2 = &'(l I is small so that the first term in eq. 16 must be 2BNXAx)/2, (9) predominant for small particles. The same is t r w while that of the incident beam I is equal to Ro2/2. for R ( i l * ) ,as we may see from the tabulations of the The fractional decrease in intensity per unit length Mie theory. Thus from a comparison of eq. 17 with eq. 16 one finds that is defined as 7
=
(AI/I)AX
(10)
so B = r/2NX
(11)
Thus we have expressed the two scattering constants A and B in terms of the refractive index increment of the solution and the turbidity, 7 . We may also relate to our formulas the observed scattering a t small angles. The scattered field from one particle with 9 and 4 equal to zero and observed a distance z from the particle is, from eq. 1 E.,o = R o [ Acos (ut
- kx) + R sin (wt - k x ) l / x
(12)
The scattered intensity IB,ois proportional to the time average of E,$ which is Ia.0 = E,.,'
=
+ B2]/2x2
Ro2[A2
(13)
The "Rayleigh ratio" is the quantity Vo,, = NIa,ox2/Io = N ( A a
+ B")
(14)
The symbol V0,"indicates that the vertically polarized component scattered a t zero angle with vert& arclly polarieed incident light is being used,
and The relation between particle size and the quantities R(il*) and I(&*)may be conveniently represented by a type of plot used by van de Hulst.' Figure 1 gives the plot for different values of m, the refractive index of the sphere divided by n. Values of p = 47rr(m - l ) / X , the phase shift, are shown along the curves, where r is the radius of the sphere. The points for m = 1.55 were taken from the tables8 while the curve m = 1 was calculated from the equations given by van de Hulst.' The reader should note that the quantity [ R ( i l * ) ] / a 2 corresponds to ImA of van de Hulst while our [ - I ( i l * ) ] / a 2corresponds to his ReA. (ais r / 2 aX.) From these curves we can show that (dnldc) depends not only upon the refractive index of the particle and medium but also upon the particle size, The general relationship for isotropic spheres
BRUNO H. ZIMMAND WALTERB. DANDLIKER
Vol. 58
I .o
0.5
2L 1: I
I
0.5 I.o - ~ ( i i ) / ~ z . Fig. 1.-Amplitude function for forward scattering for two values of the relative refractive index (see eq. 18 and 19). Values of p = 4ar(m - l ) / X are given along the
curves.
is given by eq. 18. If D is the particle density, the molecular weight can be written
which with eq. 18 gives dn =-3n’R(il*) dc 2a3~
(21)
R(il*)-m2-1 03 mz+ 2
which reduces the general eq. 21 to
dc
3n’ m2 - 1
=
20 2-)(
Equation 23 is equivalent to the refractive index equations given by Heller14and by Ewart, Roe, Debye and McCartney. l5 Discussion It is evident from eq. 16 that the calculation of molecular weights from measurements of the Rayleigh ratio involves a term in ( T / c ) as well as the usual one in (dn/dc). The ( T / C ) term, which arises from a phase shift in the scattering, is usually negligible, but must be taken into account when very large particles are encountered. It was important in the case of a polystyrene for example. The use of eq. 21 and 23 for the refractive index can be illustrated by application to the sulfur hydrosols studied by La Mer and co-workers. Figure 2 shows the quantity .
. .
plotted as a function of p = 4nr(m - 1)/X for various values of m. We can determine from Fig. 2 that for sulfur sols in water (dn/dc) is zero a t the green Hg line (A, = 5461 8.) when the particle radius is 2880 A. It may be possible to test these W.Heller, Phys. Rev.. 68, 5 (1945). (IS) R. H. Ewctrt, C. P. Roe, P. Debye and J. Rb MoCartney,lJ. Chrm. PA@&,14b 687 (1946). (14)
0
I
I
I
2
4
6
e.
Fig. 2.-Dependence of the refractive index derivative (dnldc) on size and relative refractive index, m.
In the limit for very small spheres
dn
81%
predictions by direct refractive index measurements. Heller14found that eq. 23 gave unreasonable values for the refractive index of large particles when applied to measured values of (dn/dc). He proposed an empirical correction equation which gives values approximating those from eq. 21 if the corrections are small. This may be regarded as support for eq. 21. Some confusion may arise over two related points: the difference between the ((observed” scattering a t small angles and E+,of eq. 2, and the requirement that the solution be composed of independent, i.e., non-interacting, particles. The “observed” scattered intensity, from one particle was multiplied by N , the number of particles, to give the total scattering. This procedure cannot be used when the angle is so small that there is a correlation of phase between the waves scattered by the different particles. I n the latter case the amplitudes must be added, as is done for Et, in the Appendix. However, angles small from an observational standpoint are usually still large enough for the random positions of the particles to cause random phases in the scattered light so that the intensities, not the amplitudes, are to be added. The latter can only be done if the positions of the particles are strictly uncorrelated, hence the condition of non-interaction of the particles. I n principle, this condition may always be satisfied by diluting the solution sufficiently. Appendix Derivation of the Amplitude and Phase of the Light Scattered Forward by a Thin Slab of Material.--Schusterl2 obtained Rayleigh’s formula for the relation between the intensity of the scattered light and the refractive index of a medium containing independent scattering centers by summation
of the amplitudes of the waves scattered in the direction of theincident beam. The summation was assumed to be analogous to the familiar Fresnel zone procedure so that the results of the latter could be used without further analysis. It is not clear that the assumption is correct, since the Fresnel procedure assumes that the scattering decreases uniformly to zero as the angle increases so that the contribution of the outermost zone is negligible. The scattering does not generally go to zero a t large angles, however. We attempt to improve the derivation in the following. The problem is made as simple as possible by considering a parallel plane-polarized beam of light traversing a non-absorbing medium of constant refractive index n containing N scattering particles per unit volume. The apparent refractive index of the medium and scatterers as a whole is n'. The beam is moving in the direction of increasing x. We consider the effect on the beam of a thin slab of thickness Ax lying on the y-z plane and bounded at the edges by the curve C (Fig. 3). We take Ax small enough so that the retardation of the beam by the scattering particles in the slab, (n' - n)Ax, is much smaller than the wave length, A. From the macroscopic point of view such a slab affects the beam in three ways: it scatters a small amount of light t o the side, reducing the intensity of the beam by the amount TAX; it retards the phase of the beam by the amount 27r(n' - n)Ax/Xo, and finally, it produces a diffraction pattern dependent on the shape of its boundary C. If we had introduced surfaces across which the refractive index changed we would also have had to consider reflection. OseenlG has already treated the reflection problem, which is not of primary interest for this paper. Our problem here is t o find the relation between the scattering and the intensity and phase changes of the main beam and to separate these from diffraction effects. We take polar coordinates for points in the y-z plane, with u the radius vector and (b the angle between the radius and the z-axis. It is convenient t o represent the incident light apart from the oscillating time factor by the real part of (A-1)
&e-ika
and the scattered light from a single particle a t a distance r by the real part of RoAf(fi,+)e-i(kr
f
e)/r
(A-2)
with E
= tan-l ( B / A )
(A-3)
which corresponds with eq. 1 of the main part of the paper. If there are N particles per unit volume, the average number whose centers lie in a unit area of slab is NAx. We want the total amplitude of scattering at the point II: on the x-axis. This is found by summing the individual amplitudes from all the particles in the slab, ie., by the integral E.
647
LIGHTSCATTERING OF SOLUTIONS OF LARGE COLLOIDAL PARTICLES
Aug., 1954
= RoANAxe-it
JL?(
rY,d)(e-ikr/r)ududd
2
;'1
Fig. 3.-Diagram for the derivation of Schuster's formula for the scattering by a thin layer.
distance to the farthest point. The integration can be divided into two parts: one over the region with u < a and the other over the remainder. I n this form the scattered amplitude is E.
- 2rRoANAxe-ie
{La
g(u)(e-ikr/r)u du
+
Jb.h(u)(e-ikr/T)u d u l
(A-5)
where
g
= 1
J-
