© Elsevier ScientificPublishingCompany,Amsterdam- Printed in The Netherlands BBA 37107 MEASUREMENT OF PARTICLE SIZES BY ELASTIC AND QUASI-ELASTIC LIGHT SCATTERING* CARL HOLT, THOMAS G. PARKER and DOUGLAS G. DALGLEISH Hannah Research Institute, Ayr, KA6 5HL (U.K.) (Received December 5th, 1974) (Revised manuscriptreceivedApril 10th, 1975)

SUMMARY A method of measuring an average particle radius in a highly polydisperse dispersion using the wavelength dependence of turbidity is described. For particles which are no larger than 0.3 of the wavelength of light used, a polynomial representation of the scattering cross-section can be used. For larger particles, more extensive numerical calculations are required. The use of the method is illustrated by determining the average particle radius of casein micelles by both elastic and quasi-elastic light scattering techniques. A polydisperse homogeneous sphere model is found to be a reasonably accurate representation of casein micelles. Several modifications of the model which would improve the agreement between the two techniques are mentioned briefly.

INTRODUCTION In earlier works [1-3] attempts were made to measure the average particle size of casein micelles using quasi-elastic light scattering. The results of these studies gave average particle radii appreciably in excess of those found by electron microscopy [4, 5], with the exception of the results of Carrol et al. [12]. It was considered advisable to develop alternative and, if possible, independent methods of measuring particle sizes, suitable for use with casein micelles. Conventional light scattering techniques at finite angles are widely used for measuring particle mass and radius, but usually special apparatus is essential. Doty and Steiner [6] and others [13-15] have described a method of obtaining the same information by using a conventional spectrophotometer to measure the variation of turbidity with wavelength. This paper describes the application of their technique to casein micelles in conjunction with measurements of particle size by quasi-elastic light scattering. THEORY (1) Elastic light scattering

For monodisperse particles of molecular weight M, concentration c and with * H.R.I. Reprint No. 870

284 a refractive index n' close to that of the solvent, the turbidity (z) is given by:

Hc r

--

1 MQ

- -

(1)

where

32~3nZ(dn/dc)2 H --

3NA;t~

(2)

n is the refractive index of the solvent, NA is Avogadro's number, 2o is the wavelength of light in vacuo and dn/dc is the specific refractive index increment. The function denoted by Q results from internal interference of light scattered by the particle at all angles 0.

Q=~3 f]P(O)sinO(l+ cos20)dO

(3)

P(O) is the particle scattering factor. The wavelength dependence of Q can be used to determine particle sizes in a similar way to the use of P(0) at finite scattering angles. Doty and Steiner [6] showed that: [4_

dlogQ

dlogr

d log ;t

d log~ + 4

(4)

but this assumes that 2d log[n(dn/dc)]/d log 2 is negligible compared to fl over the wavelength range of interest. This is true with, for example, colloidal silica at visible wavelengths, but in general Eqn 4 must be modified to d l o g z ._4__[~ d log 2

y

(5)

where

d log[n(dn/dc)] 7

2

(6)

d log2

Examination of tabulated values of dn/dc for many proteins [7] shows that both dn/dc and its wavelength dependence are almost constant, giving a value of --0.3 for 7 over the wavelength range 436-546 nm. To measure the radius of a particle it is necessary in general to have a model of the overall shape of the particle and the distribution of mass within it. For homogeneous spheres,

p(R,O)=(3~_(sinx

_ xcos x)) 2

(7)

where x-R

4~rnR s i n ( O ) 20

particle radius

(8)

285 Doty and Steiner [6] integrated Eqn 3 by a numerical method to obtain a relationship between fl and R. In fact the integration of Eqn 3 was carried out by Lord Rayleigh in 1914 [8], in connection with the equivalent problem of the extinction cross-section of homogeneous spheres. His result may be re-expressed as: Q(a)

8ct -4

27

(2.5 -}-

sin° 8

a

a2

(4

))

-k 2G(a) ~ - -- 1

(9)

where G(a) ~- In a q - / ' -- cl(tt) F ---- Euler's constant = 0.5772...

(lO)

c,(a) ~ - - f a cost t dt

(11)

a

(12)

---- 2x/sin (0/2)

Differentiation of Eqn 9 gives the relationship between fl and R which is: 1 + 4a-z[12 + 3cos a -- 8G(a)] + 8a-asin a + 8a-4124G(a) -- 25 + 17cos a] 0.5 + 2a-215 -- 4G(a)] -- 4a-asin a + 4a-417cos a -- 7 + 8G(a)]

fl=

This relationship is shown in Fig. I (z ---- c~). (13) The treatment so far has neglected the effects of polydispersity. Eqn 1 still applies to each species and since turbidities are additive: "r = S ' c t = H S c~M~Q~ i

(14)

i

Multiplying both sides by Sc~/.SciM~ gives:

Hc "r

Xci l SciM~ i

Xc~M.~ l 1 -- - Sc~M~QI M~,Q~

(15)

l

where the subscripts w and z refer to weight average and z average quantities, respectively. Eqn 9 may be expressed as a power series in a 2 which takes the form: Q ( a ) = , S C i a 2t

(16)

i

and the corresponding expression for a polydisperse system, expressed in terms of the z average of a z is: Q ( a ) = L'C, Fi(~2)~

(17)

i

where the factors Fi which modify the coefficients Ci in the expansion may be ex-

286 p Z=O0

1.6

Z=O 1.4 Z=2

1.2

1.0

0.8

0.6

O.z:

0.2'

I

o.~

o.~

o'.~

oA

Fig. 1. The effect of particle radius and polydispersity o n the wavelength dependence of turbidity, y axis,/3; x axis, (RZ)z*/2. Labels on the three lines are (from top to b o t t o m ) z =- c~, z = 9, z = 2.

pressed in terms of the moments of the distribution in particle sizes. For example, using moments defined by: oo

Ak

fl w(R)M(R)(R -- R) k dR

08)

where w(R) is the weight fraction of particles of radius R to R + dR and the average radius k is given by: 3O

(19)

= fo w(R)M(R)R dR the first three modifying factors are:

t:,

l

(20)

[~4

F2

~_ 6A2R2 _ _ 4A3R _]_ A4 (A2 q_ ~z)2

/ ~ + 15A 2/~4

F3

___

20.d 3R3 + 15A 4/~2

(dZ _~ /~2)3

(21) - -

6A 5/~ + A 6

(22)

For a given distribution, I = F1