Refractive index of colloidal dispersions of spheroidal particles

Refractive Index of Colloidal Dispersions of Spheroidal Particles G. H. MEETEN Research Department, Polymer Science Division, Imperial Chemical Industries Limited, Welwyn Garden City, England Received August 15, 1979; accepted N o v e m b e r 27, 1979 The effect o f particle shape on the refractive index o f a colloidal dispersion of spheroidal particles is investigated theoretically, using the Rayleigh, Rayleigh- G a n s - Debye, and the anomalous diffraction light-scattering approximations. It is shown that departures from particle sphericity modify the dispersion refractive index, both size and shape being generally of importance. INTRODUCTION

Although a relatively easy measurement, few reliable and recent measurements of the refractive index n' (relative to vacuum) of colloidal dispersions have been reported (1, 2). For spheres, it has been shown theoretically (3, 4) that n' is strongly size dependent for wavelength-sized particles, hence being a useful sizing parameter. In contrast to turbidity (extinction), n' is particle-size independent for particle sizes considerably less than a wavelength, n' being proportional only to the volume fraction of the dispersed phase for a given dispersed material (1, 4). Hence n' is a useful parameter to measure total dispersed-phase concentration, ah:eady exploited empirically by a commercial instrument (5) and used in colloid particle sizing by hydrodynamic chromatography (6). Rayleigh (R) and Rayleigh-Gans-Debye (RGD) approximations have been shown (2, 4) to yield identical results. Also for spheres, the anomalous diffraction approximation (ADA) has been shown to agree closely with the exact Mie theory (4). A general result of both the ADA and the Mie theories is that the refractive index of a dispersion of spheres tends to zero in an oscillatory fashion as the sphere diameter increases.

Journal of Colloidand Interface Science, Vol. 77, No. 1, September1980

Apart from an elementary theory for very thin lamellae based on the ADA (2), no theory exists which explicitly incorporates nonspherical particle shape. This paper explores the shape and size dependence of n' for oblate and prolate spheroids, in the R, RGD, and AD approximations. In all cases the colloidal dispersion is taken to be a noninteracting, randomly orientated assembly of homogenous isotropic dielectric spheroids, each with a zero coefficient of imaginary refractive index. Thus any optical extinction is entirely conservative, i.e., originating from light scattering but not absorption. THEORETICAL

For any scattering particle the refractive index of the assembly of scattering particles may be written (2, 7b), n' = ml +

2~rmlNk-3(Im[S(O)]). [1]

Here ml is the continuous phase refractive index relative to vacuum, k is the wave vector in the dispersion, and N is the number of particles per unit volume. S(O) is the forward-scattering function as defined by Van de Hulst (7a). The angle braces in Eq. [1] indicate that an orientation average has been taken, in this paper over all possible orientations of the scattering particle,

0021-9797/80/090001-05502.00/0 Copyright © 1980by AcademicPress, Inc. All rights of reproductionin any form reserved.

2

G.H. MEETEN

equally weighted. Equation [1] for the general ellipsoid of semiaxes a, b, c may be written 3 (Im[S(O)]) n' = m l + , 2 mlv2 XaXbXe

[2]

where xa = ka etc. is analogous to the conventionally defined particle-size parameter x for spheres and v2 is the volume fraction of the dispersed phase. For dilute dispersions 1

dn'

n' - m 1

ml

dr2

mlv2

2

,

[3]

Rayleigh and Rayleigh-Gans-Debye Approximations

dr2

-

(~2 _

6

3(/x + -

ml

(/x -

1).

dr2

1)

2(/.t2 + 2)

[51

1

dn'

ml

dr2 = (/z - 1)"

(/x + 1)(/z2 + 5) 6(/x2 + 1)

[6]

For a lamella (very thin flat oblate spheroid; La = 1, L b = 0 ) , 1 dn'

ml dr2

For calculating n', the R and RGD have been shown to be identical approximations (2, 4). For the general ellipsoid one has (2)

mi

dn'

XaX~Xc

and the right-hand side of Eq. [3] is evaluated below for the R, RGD, and AD approximations.

dn'

1

For a rod (very long thin prolate spheroid; La = 0 , Lb = ½ ) , 3 (Im[S(O)])

1

+ Lc = 1 (7c). The uniformity of field required for the above description of shape requires in optics that the dielectric ellipsoids are much smaller than a wavelength. For a spheroid (ellipsoid of revolution) b = c, Lb = Lc, and a is the axis of revolution. For a sphere (La = L0 = ½), Eq. [4] yields

= (~z - 1)"

(/z + 1)(2/x2 + 1) 6/z2

[7]

dn'/dv2 ranks in magnitude as lamella > rod > sphere. 1)

Y~ [1 + Li(/z 2 -

1)1-1.

[4]

i=a,b,e

Here /x is the relative refraction index m2/ml, m2 being the disperse-phase refractive index relative to vacuum. The coefficients La, Lb, and L~ are defined as follows. An externally applied electric field Eo parallel to the a axis of the ellipsoid produces a uniform field

E = Eo - LaP~co inside the ellipsoid, P being the dielectric polarization of the ellipsoid's dielectric and ~0 the permittivity of free space. Similar equations exist for b and c axes. The coefficients depend on the axes ratio of the ellipsoid and they are related by L~ + L~ Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980

Anomalous Diffraction Approximation S(O) for spheroids at arbitrary orientation is given by Latimer (8), who expresses S(O) for spheroids in terms of an equivalent sphere. Using his Method I (Method II gives identical results in the ADA), the equivalent sphere radius is as = bp(cos 2 tO + p2 sin 2 tO)-l/2,

[8]

where p is the axes ratio a/b and tO is the angle between the a axis and the light propagation direction. For spheres of radius as AD theory gives (2) Im[S(O)] = k202( sinO pZ

cos P ) , p

[9]

where p is the phase lag relative to the

DISPERSION OF SPHEROIDALPARTICLES continuous medium of a diametral ray through the equivalent sphere, 47r as(/~ - 1). k

~.

0.1

1 dn' mi~ 2

[101

p = --

3

----

For a randomly oriented set of spheroids, Im[S(O)] is averaged over all solid angles by (Im[S(O)]) = ~

1

j

J

,

0-05

Ia I Im[S(O)]df~ 0

= Im

fo

0

~ S(O) sin 6dO.

[11]

Equations [31, [8], [9], [10], and [11] then may be used to obtain (1/mO'(dn'/dv2) in the ADA. Expansion and term by term integration gives the series,

dn' ml dr2 3

,~(-1)'+J[(j

2xa j~"l

7_ 1)!]2[2xb(/z -

1)]2J-1

[(2j - 1)!]2(2j + ])

J 22z-2.(2 j _ 2I)!p2j-21+l x ~

[12]

[(j - I)!] 2

1--1

Expansion of the sums in Eq. [12] gives

dn' dr2

1

ml

DS~2~2/2P 2 8 j ~ a ~ b / 1 7 5 + -525 -+

16 ) .525p . .2.

[13]

TO compare particles of the same volume but different shape the relevant size parameter is x (Xaa~) 1/3, and Eq. [131 becomes =

1 m i

xp

50

FIG. 1. Dependence of (l/m1).(dn'/dv2) on volumeaveraged particle-size parameter x for prolate and oblate spheroids. Dotted line is for the oblate spheroid p = ½. The relative refractive index is/z = 1.1. For a general study of how (1/ml)'(dn'/ numerical integration of Eq. [11] has been carried out. Some results are shown in Figs. 1, 2, and 3. It is relevant to note that for commonly encountered values of relative refractive index the ADA and exact Mie theories of refraction are in fair agreement. It follows that the Latimer ADA/Mie hybrids (8) as applied to refraction will not give very different results from the pure ADA. DISCUSSION

1) - ( t x - 1)axaxb 2p + - 15 15p

= (/x-

+(tX--

20 ~

dr2) varies with particle size and shape a

1

_

10

dn' ---(t*dr2

1)-(~-

1lax2 2p

4

( 1 5 + - -15p )

+(/z -

× ( 2p2 + 8 + \ 175

1)a

525

1)512213x4

16 ) . 52-5p2

[14]

Comparison of Rayleigh/RGD and AD Results For particles of any shape, Eqs. [5], [6], and [7] show that in the RA/RGDA, (l/m1) "(dn'/dv2) tends to (/L - 1) as/z tends to unity. The same result follows in the ADA for small particles (x ---> 0) from Eq. [13] or [14]. Thus for small, weakly refracting particles there is exact agreement between the RA/RGDA and the ADA. As /z departs from unity the agreement becomes approximate, e.g., for/~ = 1.2 the biggest difference is - 4 % (spheres). Thus as x ---> 0, the ADA overestimates dn'/dv2 by a few percent compared with the RA/ RGDA. The RA/RGDA and the ADA for Journal of Colloid and Interface Science, VoL 77, No. 1, September 1980

4

G.H. MEETEN

0-1

"•

0.05~

o

1 dn'

'

1

0 10 2 0 ~ 0

50

Fro. 2. Dependence of (1/ml). (dn'/dv2) on xb for the sphere (p = 1) and various oblate spheroids. Circles are f o r p = 1/100 and squares f o r p = ½. The relative refractive index is/x = 1.1.

small x also differ slightly in their prediction of the shape dependence of dn'/dv2. Whereas the ADA predicts no shape dependence for very small particles, Eq. [4] predicts a shape dependence through the axes ratio functions La and L~. However, for commonly encountered values of /x, Eqs. [5], [6], and [7] for spheres, thin rods, and flat lamellae show a shape dependence of only a few percent in dn'/dv2. Within this few percent variation, ( - 1 % for /x = 1.1), Eq. [14] for very small values of x shows that dn'/dv2 in the ADA ranks in magnitude as spheres > rods > lamellae. This ranking is reversed in the R/RGD approximations. As x increases, Fig. 1 shows that this ranking depends on particle size.

dependence of dn'/dv2 may be derived from the x dependence since p oc x(/x - 1). Thus the data for /x = 1.2 may be found from Fig. 1 by dividing the numbers on the abscissa by (1.2 - 1)/(1.1 - 1), i.e., 2. For particles of equal volume, Fig. 1 shows that prolate and oblate spheroids of axes ratio 2 and V2 have a closely similar dn'/dv2 up to x ~ 20. Even for p = 8 and p = l~, the refraction up to x ~ 10 is not greatly different. Figure 2 for the sphere and a range of ellipsoids shows that for oblate ellipsoids it is the major dimension and hence xo which controls the refraction, the axes ratio variation having a negligible effect for p < ½, i.e., for increasingly oblate spheroids. An analogous behavior is not shown by prolate spheroids in Fig. 3, indicating that in this case the major dimension (or xa) alone cannot be regarded as the controlling parameter. CONCLUSION

The ADA, in common with the RA/ RGDA, shows that refraction of randomly oriented spheroids is fairly shape dependent even when particles of equal volumes are compared. Unlike the RA/RGDA and in common with Mie theory for spheres, a strong size dependence of refraction is shown in the ADA for particles of all shapes. Increasing departure from sphericity

0-1

1 dn' mldV2

Shape and Size Dependence in the ADA Figure 1 shows (1/ml)'(dn'/dv2) calculated for various shapes of particle. When particles of equal volume but different shape are compared, the oscillatory dependence of dn'Mv2 shown by the sphere disappears as the particle shape increasingly deviates from spherical. In common with the sphere, dn'Mv2 for all shapes tends to zero as the particle size increases. When dn '~dr2 is zero the particle may be regarded as a pure diffractor of light. In the ADA, the /x Journal o f Colloid and Interface Science, Vol. 77, No. 1, September 1980

0.0.5

0 0 10 2 0 ~ 4 0 I

I

,d

I

50

FIG. 3. Dependence of ( l / m 0 .(dn'Mv2) o n x a for the sphere (p = 1) and various prolate spheroids. The relative refractive index is tz = 1.1.

DISPERSION OF SPHEROIDAL PARTICLES damps the oscillations in the decay of refraction to zero with increasing particle size. Like turbidity, the particle size and shape dependence of refraction could be used to measure these parameters. Compared with turbidity, refraction is independent of particle size and shape within a few percent for small (x ~ 1) particles. Hence for small particles refraction measures the total dispersed-phase concentration. In contrast to turbidity, refraction is mainly determined by the smaller particles of any particle-size distribution. Combined with turbidity, refraction for spheres has been used to simultaneously determine particle size and refractive index (9). The theory and data presented herein enable refractive index measurements to be furthered as a useful technique.

5

REFERENCES 1. Kerker, M., and Chou, A., J. Phys. Chem. 60, 562 (1956). 2. Champion, J. V., Meeten, G. H. and Senior, M., J. Chem. Soc. Faraday Trans. H 74, 1319 (1978). 3. Zimm, B. H., and Dandliker, W. B., J. Phys. Chem. 58, 644 (1954). 4. Champion, J. V., Meeten, G. H., and Senior, M., J. Colloid Interface Sci. 72, 471 (1979). 5. Editorial, J. Phys. E 12,445 (1979). 6. Silebi, C. A., and McHugh, A. J.,J. Appl. Polymer Sci. 23, 1699(1979). 7. Van de Hulst, H. C., "Light Scattering by Small Particles," (a) p. 28; (b) p. 33; (c) p. 70. Wiley, New York, 1957. 8. Latimer, B., J. Colloid Interface Sci. 53, 102 (1975). 9. Champion, J. V., Meeten, G. H., and Senior, M., J. Chem. Soc. Faraday Trans. I1 75, 184 (1979).

Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980