Assessment of the stability of the turbidity spectrum method to the effect of the dispersion of the particulate substance and the medium

Abstract

Determinations of the parameters of disperse systems (particle radius r, mass My, and number N concentrations) by the turbidity spectrum method [I] usually omit any allowance for the dispersion of the particulate substance and the medium. In many cases this assumption is quite justifiable when the turbidity spectrum is measured far from absorption bands [2,3]. llowever, no assessment has yet been made of the general effect of omitting the dependence of the refractive indices of the particles ~ and of the dispersing medium ~o on the wavelength of the light % on light scattering data inversion by the turbidity spectrum method. This is part of the general question of the stability of methods for the inversion of light-scattering data (see [4], for example) and demands special analysis, which we provide here. In the calibration dependence of the wavelength exponent n on the relative particle size a = 2~r~o/~ (~ is the vacuum wavelength) the dispersion correction is given by [2,3] n ( ~ = no (~) F, PFz, ( I ) where no(a) is the dependence of the wavelength exponent on a without allowance for dispersion [1,2]; and FI, F2, and P are the correction coefficients appropriate to the particular system. The factor Fx allows for the dispersion of the medium: F~ = 1 d l n ~ o / d t n ~ . (2) The correction for the dispersion of the particulate substance is given by the product of F z = d m / d ~ , (3) P = 2 1 ( m 1 ) 5 ( m -1), (4) where m = ~/~o is the relative refractive index of the particles. Obviously the approximation of the function n(a) by the dispersion-free calibration no(a) [the assumption that n(a) = no(a)] introduces a systematic error into the determination of r, My, and N. If n is considered fixed, for example, determined experimentally, the error in r is 6r = ~ / % 1). 100%, (5) where a is the exact value of the parameter a found from Eq. (I) and ao is the root of the equation no (So) = n. ( 6 ) The errors ~My and 8N can be expressed in terms of ~r [I]: 8M v = {(1 qO.O18r)/(1 + no.0.016r) 1}. t00, %, (7) 6N = {1~1 + 0.016r)Z.(1 + no'0.016r) 1}. 100, %. (8) We can show that ~r corresponds to the relative increase in the parameter u, determined from the dispersion-free calibration, due to the increment Ano = no(a) -n(a): An0 (FT 1 1) n + PFz/FI. (9) Thus o u r p r o b l e m i s e s s e n t i a l l y e q u i v a l e n t t o c a l c u l a t i n g t h e s y s t e m a t i c e r r o r s i n t h e d e t e r m i n a t i o n o f r , t Iy , and N f r o m t h e c a l i b r a t i o n n o ( a ) i n t h e p r e s e n c e o f t h e s y s t e m a t i c a b s o l u t e e r r o r i n n g i v e n by Eq. ( 9 ) . We c a r r i e d o u t n u m e r i c a l c a l c u l a t i o n s o f t h e e r r o r s 6 r , 6My, and ~N o v e r a w i d e r a n g e o f v a l u e s o f n and w i t h d i f f e r e n t m f o r t h e s p e c i f i c e x a m p l e o f t h e s u s p e n s i o n o f p o l y m e r Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 31, No. I, pp. 117-121, July, 1979. Original article submitted June 19, 1978.