Optical Polydispersity and Contrast Variation Effects in Colloidal Dispersions

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

208, 487– 499 (1998)

CS985835

Optical Polydispersity and Contrast Variation Effects in Colloidal Dispersions Adolfo J. Banchio, Gerhard Na¨gele,1 and Andrea Ferrante Department of Physics, University of Konstanz, P.O. Box 5560, D-78457 Konstanz, Germany Received May 27, 1998; accepted August 25, 1998

We present a theoretical study on the effect of refractive index variations on static and dynamic light scattering in size-polydisperse suspensions of sterically and charge-stabilized colloidal particles with an internal optical structure (core-shell model) and size-dependent refractive indices. The equilibrium microstructure and the short-time dynamics of these optically, size-, and interaction-polydisperse systems are calculated using hypernetted chain and Percus–Yevick integral equation schemes. Our calculations show that, close to an index matching point, the scattered intensity I(k), the measurable structure factor SM(k), and the measurable hydrodynamic function HM(k) become very sensitive to the refractive index contrast with respect to the solvent. For this purpose, various definitions of index matching points are analyzed, and the strong relative enhancement of the incoherent part of the scattered intensity close to the matching points is discussed. For chargestabilized systems we show that the anomalous behaviour of I(k) and HM(k) in the matching regime of the solvent refractive index can be well described by a simple approximative scheme, which can be easily implemented. Consequences of our study for scattering experiments aimed to determine particle sizes or structural properties of colloidal dispersions are discussed. © 1998 Academic Press Key Words: optical polydispersity; contrast variation; index matching; light scattering.

I. INTRODUCTION

In recent years, a large effort has been made in studying the structural and dynamical properties of model-systems of sterically and charge-stabilized spherical colloidal particles (1, 2). Major experimental tools for investigating the microstructure and diffusion of these systems are static and dynamic light scattering (SLS and DLS) (1–13) as well as small-angle X ray (14 –16) and neutron scattering experiments (17, 18). Whereas the structural and dynamical properties of monodisperse colloids have been investigated in great detail, much less is known with regard to polydisperse suspensions and colloidal mixtures. In fact, most naturally occurring colloids and many synthesized colloidal suspensions are significantly polydisperse. Many colloidal suspensions which have been investigated are polydisperse with respect to particle sizes and pair interactions. 1

To whom correspondence should be addressed.

As a consequence, various theoretical and experimental investigations have been carried out to assert the influence of size- and interaction polydispersity on the static and dynamic scattering functions probed in light scattering (1–13, 18 –23), and good agreement between theory and experiment has been achieved for static and short-time dynamical properties (2, 5– 8, 10 –14, 18, 21). Besides being polydisperse in size and interaction, colloidal particles often exhibit an internal optical structure reflecting the method of particle synthesis. In other words, the particle refractive index n(r) depends on the position r inside the particle. Even for optically homogeneous particles, the refractive index can depend on the diameter s, i.e., n 5 n(s). We will refer to this additional type of polydispersity arising from a nonconstant n(r, s) as optical polydispersity. Very frequently, the various types of polydispersity are correlated with each other. Well-known examples of optically polydisperse colloids with core-shell structure are g-methacryloxypropyltrimethoxysilane(TPM)-coated charged silica spheres (3, 4, 13) and polyisobutene(PIB)-coated uncharged silica spheres (4). In both cases the particles have a radially symmetric core-shell structure of two different dielectric materials. Further examples of particles with core-shell structure are micellar solutions and water-in-oil microemulsions (8, 24, 25). There exist also systems of optically anisotropic particles, like partially crystalline fluorinated polymer particles (26). For such optically anisotropic particles it is possible to measure rotational diffusion using depolarized DLS (26 –28). In the interpretation of light scattering experiments on colloidal suspensions, it is often assumed that all particles have the same refractive index n and hence show no internal optical structure. For this idealized situation, quantities like the measurable static structure factor SM(k) and the measurable hydrodynamic function HM(k) are independent of the refractive index contrast n 2 ns between particles and suspending solvent (2). For optically polydisperse particles, this simplifying assumption is justified only when the refractive index distribution inside a particle is significantly smaller than the average contrast with the solvent. Strong optical contrast, however, gives rise to unwarranted multiple scattering of light, which invalidates the assumption of dominant single scattering underlying the conventional interpretation of scattering data. To avoid

487

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¨ GELE, AND FERRANTE BANCHIO, NA

multiple scattering in more concentrated systems, the optical contrast is varied, e.g., by changing the composition and/or temperature of the solvent so that according to some criteria (cf. Section III), an optimal index matching between particles and solvent is achieved. Close to such an index matching point, the optical structure of the particles becomes important and, as we will show in detail, it has to be accounted for in a correct interpretation of scattering data. The purpose of the present work is to provide a well-founded theoretical analysis of contrast variation on static and shorttime dynamic light scattering experiments in strongly correlated suspensions of spherically shaped particles. The particles are assumed to be optically, size, and interaction polydisperse. For this purpose, we will calculate the scattered intensity I(k), the measurable static structure factor SM(k) and, for charge-stabilized suspensions, the measurable hydrodynamic function HM(k). The static pair distribution functions of the charged particles interacting by a repulsive screened Coulomb potential are determined using the many-component hypernetted chain (HNC) integral equation scheme (2, 10 –13, 23). For suspensions of uncharged polydisperse spheres, we employ the well-known Percus–Yevick (PY) integral equation scheme (20, 22, 29). As typical models of optical polydispersity, we discuss the core-shell model and the so-called linear model of homogeneous spheres with linear dependence of the refractive index on the particle size. The distribution of particle sizes is modeled in our analysis by a two-parameter unimodal Schulz distribution (2, 23). We show that the measurable quantities I(k), SM(k), and HM(k) are very sensitive to the refractive index contrast with respect to the solvent close to an index matching point, due to the interplay of incoherent and coherent scattering contributions. Various definitions of points of optimal index matching are discussed, and it is described how these matching points can be determined. A simple decoupling approximation is proposed for charge-stabilized suspensions, which provides a good description of the anomalous behavior of SM(k) and HM(k) close to index matching and which can be easily implemented. In Section II, we address the salient concepts of the general theory of static and dynamic light scattering relevant for optically polydisperse dispersions. A discussion of various definitions of optical matching points and their experimental determination is given in Section III. We also introduce in this section a simple decoupling scheme for localizing the critical regime of strong index-matching and for calculating SM(k) and HM(k) in that regime. In Section IV, we explain the models of optical polydispersity and size polydispersity used in this work. Furthermore, we briefly summarize the methods used to calculate the microstructure and short-time dynamics of polydisperse systems. Our results for the static and dynamic properties are presented in Section V. A short summary and our final conclusions with regard to the relevance of measuring I(k), SM(k), and HM(k) to

obtain information on intraparticle and interparticle properties are contained in Section VI. II. LIGHT SCATTERING IN OPTICALLY POLYDISPERSE SYSTEMS

Consider a polydisperse fluid system of spherical colloidal particles or microemulsion droplets. Such a system can be regarded as an m-component mixture. The particles forming component a, with a 5 1, . . . , m, are assumed to be identical, whereas particles belonging to different components can differ in terms of diameter sa, interaction, and scattering properties. The average scattered intensity I(k), measured in SLS experiments, is given, up to an irrelevant multiplicative constant, by (1, 2) I~k! 5 Nf 2~k!S M~k!,

[1]

where it is assumed that single scattering events prevail. Here, N 5 ¥am51 Na is the total number of spheres in the volume V, and (2, 21)

S M~k! 5

O ~x x ! m

1 f 2~k!

a b

1/ 2

f a~k! f b~k!S ab~k!

[2]

a,b51

is the measurable static structure factor at wave number k. The function SM(k), normalized so that SM(k 3 `) 5 1, is a linear superposition of partial static structure factors Sab(k), weighted by the partial molar fractions xa 5 Na /N and by the scattering amplitudes fa(k) of a-type particles. The partial static structure factors Sab(k) are related to the radial distribution function gab(r) of an a 2 b pair of particles as (2) S ab~k! 5 d ab 1 n~ x a x b! 1/ 2

E

dre ikzr@ g ab~r! 2 1#,

[3]

with n 5 N/V. Here, gab(r) is the conditional probability of finding a particle b a distance r apart from a particle a. Notice that Sab(k 3 `) 5 dab. Moreover,

O x f ~k! m

f n~k! 5

a

n

[4]

a51

is the nth moment of the distribution of scattering amplitudes. The key quantity determined in dynamic light scattering (DLS) is the electric field autocorrelation function I(k, t). For polarized single scattering, this quantity reads, up to a multiplicative factor (2, 20), I~k, t! 5 Nf 2~k!S M~k, t!,

[5]

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OPTICAL POLYDISPERSITY AND CONTRAST VARIATION

and the measurable dynamic structure factor SM(k, t) is given by the right hand side of Eq. [2] with Sab(k) replaced by its dynamic generalization, i.e., by the dynamic partial structure factors Sab(k, t). The dynamic structure factor Sab(k, t) describes the time correlations of density fluctuations of a- and b-particles for a given wavelength 2p/k. In a short-time DLS experiment, one determines the initially exponential decay of I(k, t), which is described for a polydisperse suspension by (2, 11) I~k, t! 5 I~k!e

2k2DM~k!t

.

[6]

From the initial decay rate, the k-dependent measurable diffusion coefficient DM(k) is obtained. This coefficient can be expressed as (2, 11, 13) D M~k! 5

H M~k! , S M~k!

[7]

i.e., in terms of the static quantity SM(k), and of the so-called measurable hydrodynamic function HM(k), with (11)

H M~k! 5

1 f 2~k!

O ~x x !

K

1 ~N a N b! 1/ 2

S

a b

1/ 2

f a~k! f b~k! H ab~k!.

Na

Nb

j51 l51

ab jl

a

f a~k! 5

[8]

b

L

where kˆ 5 k/k. The effect of the solvent-mediated hydrodynamic interactions (HI) on the translational motion of the particles is embodied in the hydrodynamic diffusivity tensors N Dab jl (R ), with j { a and l { b. These tensors provide a linear relation between the hydrodynamic forces acting on the N particles at a given configuration RN and the resulting translational particle velocities, in the form of a generalized Stokes law (1, 2, 11). Without HI, Hab(k) 5 Do(sa)dab, where Do(sa) 5 kBT/(3phsa) denotes the Stokesean diffusion coefficient of an isolated sphere of diameter sa in a fluid with shear viscosity h and temperature T. Contrary to Sab(k) and Hab(k), which are purely statistical

1 2es

E

dre ikzr@ e a~r! 2 e s#.

[11]

Va

Here, Va is the particle volume, es 5 n 2s is the solvent dielectric constant, and the factor 1/(2es) has been included into the definition of fa(k) for convenience. By adding and subtracting (6) the volume-averaged refractive index

ea 5

1 Va

E

dr e a~r!

[12]

in the integrand of Eq. [11], fa(k) is then conveniently partitioned in two parts according to (4)

~R N! z kˆ e ikz@Rj ~0!2Rl ~0!# , [9]

[10]

to hold for a 5 1, . . . , m, where l is the wavelength of the incident light, ns is the solvent refractive index, and na is the volume averaged refractive index of an a-type particle. The RGD condition is met, e.g., for particles with sufficiently narrow internal refractive index distribution in a nearly refractive index-matching solvent. Within the Rayleigh–Gans–Debye regime, the scattering amplitude of an a-sphere with (possibly orientationally averaged) radially symmetric refractive index distribution ea(r) 5 n 2a(r), where ea(r) is the local dielectric constant of particle a at a distance r from its center, is given by

a,b51

O O kˆ ? D

D

na 2p 21 !1 s l a ns

m

The partial hydrodynamic functions Hab(k) contain the configuration-averaged effect of the solvent-mediated hydrodynamic interactions on the collective short-time diffusion of aand b-particles. A statistical mechanical expression of Hab(k) is derived from an application of the generalized Smoluckowski equation which describes the temporal evolution of the probability density for the N-particle configuration RN 5 (R1, . . . , RN). Explicitly, we obtain (2, 11)

H ab~k! 5

mechanical quantities, it is apparent from Eqs. [1], [2], and [8] that I(k), SM(k), and HM(k) depend also on the scattering properties through the dependence of the scattering amplitudes fa(k) on the size and on the internal optical structure of the particles. In our subsequent analysis, we assume the Rayleigh–Gans– Debye (RGD) condition

f a~k! 5 f ah ~k! 1 f ainh~k!,

[13]

with f ah ~k! 5

na 2 ns 1 @ e 2 e s#B~k, s a! < B~k, s a! 2es a ns

[14]

and f ainh~k! 5

1 2es

E

dre ikzr@ e a~r! 2 e a#

Va

<

1 ns

E

Va

d 3re ikzr@ n a~r! 2 n a#,

[15]

¨ GELE, AND FERRANTE BANCHIO, NA

490

only. In this so-called decoupling approximation, SM(k) is approximated by SD(k), where (1, 2, 5, 7, 12, 21)

where

B~k, s a! 5

E

Va

d 3re ikzr 5 ps a3

j 1~k s a! . ksa

S M~k! < S D~k! ; X~k! 1 @1 2 X~k!#S I~k; s# !.

[16]

[18]

Here, Here, j1 is the first order spherical Bessel function, and na 5 (*Va d3rna(r))/Va is the volume averaged refractive index of an a-type particle. In Eq. [13], fa(k) has been partitioned into a homogeneous part, f ha(k), representing scattering from an optically homogeneous particle of dielectric constant ea and into an inhomogeneous part, f inh a (k), arising from the internal optical structure of an a-particle. The latter part, f ainh(k), does obviously not depend on the optical contrast of the particle relative to the solvent. Notice that f ainh(k 3 0) 5 0, expressing the fact that the internal structure is not resolved in the long wavelength limit. The second approximate equalities in Eqs. [14] and [15] apply only for small refractive index differences between na(r) and ns, i.e., when

e a~r! 2 e s 5 ~ n a~r! 2 n s!~ n a~r! 1 n s! < 2 n s~ n a~r! 2 n s! [17] holds. This further simplification is applicable, e.g., to TPM-coated silica particles in a nearly index-matching solvent as investigated in (3, 4, 13). For PIB-coated silica particles (4), microemulsions, or micellar solutions the internal optical structure is usually more pronounced, and the second equality in Eq. [17] is not valid. Then, the expressions in Eqs. [14] and [15] involving the dielectric constants instead of the refractive indices need to be used. When the internal refractive index distribution na(r) is quite narrow as compared to the average refractive index difference (contrast) to ns (the contrast being compatible with the RGD condition), then an a-particle scatters light like a homogeneous sphere with a size-dependent refractive index close to na. In the opposite regime of nearly index matched particles, where una 2 nsu/ns is small, the internal optical structure of the particles strongly influences, by means of the inhomogeneous part f inh a (k), the shape of I(k), SM(k), and HM(k). Depending on the proximity of the particle refractive indices to ns, the shapes of the measurable scattering functions can be strongly distorted away from the case of optically homogeneous spheres of equal refractive indices. We will discuss this interesting point in Section V, which contains our numerical results. III. INDEX MATCHING POINTS AND THE DECOUPLING APPROXIMATION

We obtain a particularly simple approximation for SM(k) by disregarding the size and interaction polydispersity in the particle correlations and by accounting for optical polydispersity

X~k! 5 1 2

#f ~k! 2

[19]

f 2~k!

is the incoherent scattering contribution with 0 # X(k) # 1, and SI(k, s# ) is the structure factor of the reference system of N ideally monodisperse particles of size s# 5 ¥m a51 xasa. Usually, SM(k) agrees well with its decoupling approximation SD(k) for charge-stabilized particles and hard spheres only for small polydispersities s # 0.05 (2, 12, 30). Here, s is the relative standard deviation of the particle size distribution. For nearly index matched particles, however, the decoupling approximation is applicable within good accuracy for significantly larger size-polydispersities, as our results will demonstrate. This finding arises from the fact that close to index matching, optical polydispersity entering the incoherent (intraparticle) scattering contribution becomes most important. Correspondingly, we can introduce a simple decoupling approximation for the partial hydrodynamic functions Hab(k), by accounting for size polydispersity only in the free-particle Stokesean diffusion coefficients D0(sa). With regard to potential and hydrodynamic interactions, the N spheres are treated as ideally monodisperse with common diameter s# . The decoupling approximation amounts to replace Hab(k) in Eq. [9] by H ab~k! < d abD 0~ s a! 1 d ab@D Is 2 D 0~ s# !# 1 ~ x a x b! 1/ 2H Id~k! [20] and consequently HM(k) by HD(k), where H M~k! < H D~k! 5 H 0M~k! 1 @D Is 2 D 0~ s# !# 1 @1 2 X~k!#H Id~k!.

[21]

Here,

H ~k! 5 0 M

1 f 2~k!

O x f ~k! D ~s ! m

a

2 a

0

a

[22]

a51

is the measurable hydrodynamic function without HI. It depends on quantities which can be determined by light scattering experiments. Furthermore, HI(k) 5 DIs 1 HId(k) denotes the hydrodynamic function of the ideally monodisperse reference system. This function is conveniently partitioned in a k-independent selfpart DIs 5 ^kˆ z D11(RN) z kˆ& and into a distinct part HId(k) 5

491

OPTICAL POLYDISPERSITY AND CONTRAST VARIATION

(N 2 1)^kˆ z D12(RN) z kˆ exp[ik z (R1 2 R2)]&, where Dij(RN) is the diffusivity tensor of the reference system. The self-part DIs is identified as the short-time self-diffusion coefficient of the reference system, since it determines the initial slope of the particle mean-square displacement ^[R1(t) 2 R1(0)]2&/6 of a reference particle (1, 2). The decoupling approximation for Hab(k) is justified for systems with weak hydrodynamic interactions dominated by the Stokesean friction, where the particles are nearly identical with respect to potential and hydrodynamic interactions. The decoupling approximation for HM(k) reduces for vanishing hydrodynamic interactions to H0M(k), since DIs 5 D0(s# ) and HId(k) 5 0 without HI. In the remainder of this section, we will discuss for polydisperse systems various possibilities of defining the point of optimal partial index matching with respect to the suspending solvent. As we shall see, the definition which should be employed is a matter of convenience and of the kind of experiment which is performed. Quite often, the index matching point is defined by that value ns 5 n(1) s of the solvent refractive index which is equal to the volume- and composition-averaged particle refractive index n# , viz.

O Vx m

n ~1! #; s 5 n

a51

a a

E

d 3r n a~r!.

[23]

Va

We refer to this definition of a solvent index matching point as IMP1. For particles which are optically polydisperse but otherwise identical, the forward scattering is minimized when ns 5 n(1) s . The scattered intensity is then given by I(k 5 0) 5 Nf 2(0). Furthermore, X(k) and SD(k) attain then at k 5 0 their common maximal value 1. We point out that the definition in Eq. [23] applies only to systems with small refractive index differences between na(r) and ns for all values of r. Otherwise, like in the case of PIB-coated silica particles discussed in Sections IV and V, the refractive indices in Eq. [23] should be replaced by the corresponding dielectric constants, i.e., na(r) by ea(r) and n# by e# . The index matching condition IMP1 is then defined as e(1) 5 e# . For given size distribution and na(r) (or ea(r)), the optimal solvent value ns (or es) according to IMP1 is easily computed. The index matching point can be defined alternatively as the solvent value ns 5 n(2) s , where (IMP2) X~k 5 0; n ~2! s ! 5 1

[24]

(2) or, equivalently, where #f(k 5 0; n(2) is s ) 5 0. The value ns # easily obtained from plotting f(k 5 0) versus ns and locating the value where it vanishes. In general, n(2) s is different from n(1) s only for suspensions of particles which are different from each other not only in their optical properties. The definitions

IMP1 and IMP2 have the advantage that no structural infor(2) mation is needed for calculating n(1) s and ns . As a third possibility, we can define the index matching point by a value ns 5 n(3) s of the solvent refractive index, where the solid-angle integrated total intensity,

I tot 5

E

dV kI~k; n s! 5

4p

SDE

1 l 2p ns

2

4pns /l

dkkI~k; n s!,

[25]

0

is minimal. This definition of index matching is referred to as IMP3. Contrary to the matching point definitions discussed before, the index n(3) s depends also on the wavelength l of the incident laser beam. Experimentally, n(3) can be determined s directly from turbidimetric measurements. A theoretical computation of n(3) is more demanding than calculating n(1) and s s (2) ns , since information is needed on the pair correlation functions. The three definitions (IMP1–3) of index matching agree with each other only in the exceptional case of homogeneous spheres with narrow refractive index distribution, which are otherwise identical. If one is interested only in an estimate of the range of values ns (or es), where the optical contrast and therefore multiple scattering is small, any of the three definitions is equally good. However, if one is interested in single particle properties and aims at reaching an index matching point as accurately as possible, then the definition adopted for this point becomes crucial. For the example of a core-shell model and linear model, we will demonstrate that SM(k) and HM(k) show large variations in shape when ns is varied in a critical range of values located around the n(i) s . IV. MODEL SYSTEMS OF POLYDISPERSITY

In this work, we analyze two models of optical polydispersity, in combination with size (and related interaction) polydispersity described by an unimodal distribution function p(s). The distribution function p(s) is determined by the mean particle diameter s# and by the relative standard deviation s. As a first model of optical polydispersity we consider a core-shell model of colloidal spheres characterized by a radially symmetric dielectric constant distribution e(r) of the form (4, 24)

H

e 1, 0 # r # s / 2 2 d e ~r! 5 e , s / 2 2 d , r # s / 2 . 2

[26]

According to Eq. [26], an internal homogeneous spherical core of radius s/2 2 d and dielectric constant e1 is surrounded by a spherical shell of thickness d and dielectric constant e2. This model is appropriate for TPM- and PIB-coated silica spheres, like those analyzed in (4, 13), and for certain microemulsions or micellar suspensions (8, 24). The inhomogeneous part of the scattering amplitude f(k) in

¨ GELE, AND FERRANTE BANCHIO, NA

492

the core-shell model is given for an a-type particle, according to Eqs. [15, 16], by f inh~k! 5

e 2 2 e# e1 2 e2 B~k, s ! 1 B~k, s 2 2d!. es es

[27]

In our calculations we consider two different sets of parameters in the core-shell model. The first parameter set is chosen to mimic TPM-coated charged silica particles suspended in DMF (13) and reads: n1 5 1.451, n2 5 1.458, s# 5 124 nm, s 5 0.15, and d 5 6 nm for the constant thickness d of the TPM coating. For this set of parameters, the refractive index difference between core and shell is small enough that the linearization approximation in Eq. [17] can be safely used. As a consequence of keeping the shell thickness constant, the average dielectric constant and refractive index of a particle depend on the particle size. For the second parameter set, we choose values which are typical of uncharged PIB-coated silica spheres suspended in a toluene-tetrahydrofurane mixture as studied in Ref. (4): n1 5 1.45, n2 5 1.51, s# 5 312 nm, d 5 26 nm, s 5 0.11, and s 5 0.3. The second model of polydispersity we investigate represents a size-polydisperse suspension of optically homogeneous spheres with a size-related refractive index n(s). We assume n(s) to increase linearly in s according to

n ~ s ! 5 n ~ s# ! 1 p

F G

s 2 1 ; s min , s , s max s#

[28]

within an interval [smin, smax] of diameters, including s# as inner point, where the particle size distribution function p(s) is significantly different from zero. We refer to this model as the linear model. This rather schematic model is useful for illustrating the characteristic behavior of SM(k) and HM(k) close to optical matching. If we suppose that un(s) 2 nsu/ns ! 1 in the range of diameters where p(s) is significantly different from zero, then in the linear model the scattering amplitude is given as f~k! 5

S

D

p s 2 ss B~k, s !, ns s#

[29]

with

F

s s~ n s, p, n ~ s# !! 5 s# 1 1

G

n s 2 n ~ s# ! . p

on the optical parameters ns, p, and n(s# ) through the normalized matching diameter ss /s# . Two systems with equal size distributions but different values of n(s# ) and p have identical scattering functions SM(k) and HM(k), provided that the solvent refractive index is adjusted for each of the two systems such that the same diameter ss results. In the following section, we present results for the linear model corresponding to the system parameters s# 5 124 nm, s 5 0.15, n(s# ) 5 1.4516, and p 5 0.0138. The particle refractive index increases thus linearly from a minimal value of 1.445 at smin 5 0.516 3 s# to the maximal value 1.460 at smax 5 1.613 3 s# . These parameter values have been chosen to describe roughly the (average) refractive index dependence of suspensions of charged silica particles. The distribution of particle diameters is described throughout this work by an optimized m-component representation of an unimodal Schulz distribution (2, 23)

p~ s ; s# , s! 5

S D

F

G

st t11 t11 s ,t.0 exp 2 s# G~t 1 1! s# [31]

where ps(s; s# , s) is the probability density for particles with diameter s and G is the gamma function. The size distribution function is completely characterized by the mean diameter s# and by the relative standard deviation s. The latter is related to the width parameter t by @^ s 2& 2 s# 2# 1/ 2 5 @t 1 1# 21/ 2. s#

s5

[32]

The moments of the Schulz distribution are given by

^ s n& 5

E

`

d ss np~ s ; s# , s! 5

0

~n 1 t!! n s# , t!~t 1 1! n

[33]

with n 5 0, 1, 2, . . . . We approximate the continuous Schulz distribution by a histogram distribution

O x d~s 2 s ! m

[30]

According to Eq. [28], ss is determined as that particle diameter where the corresponding particle refractive index is completely matched to the solvent. The factor p/ns in Eq. [29] is cancelled, when calculating SM(k) and HM(k), due to the normalizing factor 1/f 2(k). Thus, SM(k) and HM(k) depend only

p~ s ! 5

a

a

[34]

a51

consisting of a small number m of components. The molar fraction xa and diameter sa of each component are determined from equating the first 2m moments of the discretized and continuous Schulz distribution, i.e., from

493

OPTICAL POLYDISPERSITY AND CONTRAST VARIATION

O m

x a~ s a! l 5 ^ s l&,

[35]

a51

with ^s l& according to Eq. [33] and l 5 0, 1, . . . , 2m 2 1. Five components are usually sufficient for the calculation of the static structure factor SM(k) in suspensions of polydisperse hard spheres or polydisperse charged spheres with s # 0.3. Close to index matching, it was occasionally necessary to employ up to nine components. Our results for SM(k) and HM(k) remain unchanged when the number of components is further increased. After having explained the models of optical and size polydispersity used in this work, we describe now how SM(k) and HM(k) are calculated from the knowledge of the effective pair potentials. For charge-stabilized suspensions, the effective pair potential uab(r) between two particles of components a and b is given by the hard-core repulsion superimposed by a repulsive screened Coulomb potential of the DLVO form (2) u ab~r! 5 L BZ 2 k BT

1 21 2 ek

sa 2

11k

ek

sa 2

sb 2

11k

sb 2

e 2kr r

[36]

valid for particle separations r . [sa 1 sb]/2. Here, LB 5 e2/(eskBT) is the Bjerrum length, e the elementary charge, T the temperature, and Z is the effective particle charge in units of e. We only consider salt-free suspensions of strongly correlated particles with univalent counter-ions. The screening parameter k is then determined through k2 5 4pLBnZ. For dispersions of charge-stabilized silica spheres described within the core-shell and the linear models, we choose LB 5 1.93 nm corresponding to a dimethylformamide (DMF) fluid at room temperature. We further assume an effective particle charge Z 5 220 independent from the particle size, in accordance with a recent study on TPM-coated silica spheres dispersed in DMF, where experimental and theoretical work have been combined (13). The partial static correlation functions Sab(k) and gab(r) needed to determine SM(k) and HM(k) in case of charge stabilized suspensions are calculated using an m-component HNC integral equation scheme (2, 23), with a Schulz histogram distribution as described in Eq. [34]. In decoupling approximation, the static structure factor SI(k, s# ) and the related radial distribution function gI(r; s# ) of the one-component reference system are calculated using the one-component HNC. With regard to our calculations of HI(k) and the partial hydrodynamic functions Hab(k) entering into HM(k), we employ the assumption of pairwise additive hydrodynamic interactions plus a far-field expansion of the hydrodynamic two-body mobility tensors as described and justified for charge stabilized dispersions in detail by Na¨gele et al. (2, 11, 13).

Concerning the core-shell model of uncharged PIB-coated silica spheres, the particles are assumed to behave like an m-component suspensions of hard spheres, with diameters sa determined according to Eq. [35]. The partial static structure factors are calculated by employing the multi-component Percus–Yevick integral equation scheme (PY) (29). We finally point out that the solvent refractive index n(2) s of the matching point definition IMP2 can be analytically expressed in terms of the parameters s and s characterizing the (continuous) Schulz distribution. Using Eq. [33] for the moments of the Schulz distribution, n(2) is determined at weak s optical particle–solvent contrast in the linear model as

n ~2! # ! 1 3ps 2. s 5 n~s

[37]

For the core-shell model, we obtain after a lengthy but straightforward calculation the result

e ~2! s 5 e 1 2 ~ e 1 2 e 2!

d 1 2 1 1 2s s#

F

3 62

S DG

d 12 d 8 1 2 2 1 1 s s# 1 1 s s#

2

,

[38]

(2) 2 with e(2) s 5 (ns ) . Eq. [38] is valid for any particle–solvent contrast compliant with the RGD condition. In the calculation of n(3) according to the matching criterion IMP3, we have s employed a numerical minimization of Itot(ns), with I(k; ns) determined for charged and for hard spheres using the HNC and PY integral equation schemes, respectively.

V. RESULTS AND DISCUSSION

We present here our numerical results on the solvent refractive index dependence of the static and short-time correlation functions SM(k), I(k), and HM(k). For the core-shell model of charge-stabilized TPM-coated silica spheres, we obtain the following values for the indexmatched solvent refractive index ns according to the matching (2) (3) point criteria IMP1–3: n(1) s 5 1.4529, ns 5 1.4528, and ns 5 1.4523 for l 5 459.9 nm. These values of ns correspond to a Schulz-distributed dispersion of polydispersity s 5 0.15, total volume fraction f 5 0.063, and effective charge number Z 5 (2) (3) 220 (13). Notice that contrary to n(1) s and ns , ns depends also on f, l, and Z. The numerical differences in the n(i) s appear in the fourth decimal point, since n1 5 1.451 is different from n2 5 1.458 only at the third decimal point. The values for n(i) s in case of the core-shell model for PIB-coated uncharged silica particles with n1 5 1.45, n2 5 1.51, l 5 436 nm (4), and f 5 0.3 are determined as n(i) s 5 1.4755, 1.4750, and 1.4649 for i 5 1, 2, 3 and s 5 0.11. For larger polydispersity s 5 0.3, we obtain n(i) s 5 1.4768, 1.4720, and 1.4614, for i 5 1, 2, 3. Fig. 1 shows the Schulz size distribution function p(s) for

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494

FIG. 1. Size distribution function p(s) and refractive index distributions n(s) and n# (s) of the linear and core-shell models of optical polydispersity respectively versus normalized particle diameter s. Solid line, continuous Schulz distribution for polydispersity s 5 0.15. Dashed line, n(s) of the linear model for n(s# ) 5 1.4516 and p 5 0.0137. Notice that n(s) varies linearly only in the diameter range where p(s) is significantly different from zero; n(s) 5 1.445 for s/s# # 0.516 and n(s) 5 1.460 for s/s# $ 1.613. Dashed– dotted line, volume-averaged refractive index n# (s) of the core-shell model for system parameters describing TPM-coated charged silica particles with s# 5 124 nm and d 5 6 nm.

s 5 0.15 plotted versus the normalized particle diameter s/s# . The figure includes further the size dependence of the particle refractive index n(s) from the linear model given in Eq. [28], with the values of n(s# ) and p provided in the figure caption. For the core-shell model, the corresponding quantity to compare with is the volume averaged particle refractive index 1 n# ~ s ! 5 Vs

E

Vs

F

G

2d 3 d r n ~r! 5 n 2 2 ~ n 2 2 n 1! 1 2 , s 3

average refractive index n# (s) ' n# (s# ). The iso-scattering wavenumber ki is defined by the first zero of f h(k). For this wavenumber, f(ki) 5 f inh(ki) so that the scattering amplitude becomes contrast independent. As a consequence, SM(ki) and HM(ki) are nearly independent of ns provided that the size polydispersity is small enough for n# (s) ' n# (s# ) to be valid in the diameter range where p(s) . 0. The second moment, f 2(k), the measurable structure factor, SM(k), and the average intensity, I(k), of the core-shell model of TPM-coated silica spheres with s 5 0.15, Z 5 220, and f 5 0.063 are displayed in Fig. 3 for various values of ns. The functions I(k) and SM(k) are calculated using the HNC integral equation scheme. Results are shown for these functions at ns 5 1.4500 and at larger values of ns up to n(2) 5 1.4528 where the index matching point IMP2 occurs. This point is characterized by X(k 5 0; n(2) s )5 1 and consequently by SD(k 5 0) 5 1. Here SD(k) is the decoupling approximation of SM(k) defined in Eq. [18]. The fact that SM(k 5 0; n(2) s ) ' 1 in Fig. 3 indicates that the decoupling approximation is a good approximation for SM(k) at least at small values of k, and for values of ns close to n(2) s . We will discuss below this interesting finding in more detail. At the lowest considered value ns 5 1.4500, the particle– solvent contrast is so pronounced that fa(k) ' f ha(k), i.e., the a-type particles in the m-component mixture can be considered as homogeneous spheres of refractive index na. Under this condition, f 2(k) is monotonously decaying in the k-interval displayed in the figure. Notice that kmax 5 4pns /l ' 0.004 for ns 5 1.45 and l 5 496 nm, providing an upper bound for the range of wavenumbers scanned in a light scattering experiment (13). Consequently, the iso-scattering wavenumber is here outside the experimentally accessible k-window.

[39]

with particle volume Vs 5 (ps3)/6, and n(r) given by Eq. [26] with e replaced by n. From Eq. [39], it follows that n# (s) ' n1 for s @ d and n# (s) 5 n2 when s 5 2d, i.e., when the whole particle consists only of the coating material. As noticed from Fig. 1, there is only a small variation of n# (s) in the diameter range where p(s) is significantly larger than zero. The homogeneous and inhomogeneous parts, f h(k) and inh f (k), of the scattering amplitude f(k) of a core-shell particle with diameter s 5 s# and n# (s) . ns are depicted in Fig. 2. The inhomogeneous part, f inh(k), does not depend on the optical contrast with the solvent and vanishes for k 3 0, since the internal shell structure is not resolved when k s# ! 1. Since n2 . n# . n1, f inh(k) is negative in the experimentally accessible k-range (cf. Eq. [27]). Contrary to f inh(k), the homogeneous part f h(k) and hence f(k) depend on the solvent contrast, i.e., on n# (s) 2 ns. For large contrast (still within the validity of the RGD regime) it follows that f(k) ' f h(k), and the particles act in this case like homogeneous spheres with nearly constant

FIG. 2. Scattering amplitude f(k) 5 f h(k) 1 f inh(k) of the core-shell model with n1 5 1.451, n2 5 1.458, s 5 s# 5 124 nm, and d 5 6 nm (TPM-coated silica spheres). Dashed line, homogeneous part f h(k); dashed– dotted line; inhomogeneous part f inh(k); solid line, total scattering amplitude f(k).

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495

FIG. 4. Measurable hydrodynamic function HM(k) normalized by the corresponding function H0M(k) without hydrodynamic interactions. System parameters as in Fig. 3. FIG. 3. Second moment f 2(k) of the scattering amplitude distribution, measurable static structure factor SM(k), and average scattered intensity I(k) versus wavenumber k, for solvent refractive indices ns as indicated in the figure. The results for f 2(k) and I(k) are presented in arbitrary units on a logarithmic scale. The dashed-dotted curves correspond to the index matching point IMP2, where X(k 5 0) 5 1 and SD(k 5 0) 5 1. System parameters correspond to core-shell model of TPM-coated silica spheres with s 5 0.15, f 5 0.063, and Z 5 220.

The incoherent part, f inh a (k), contributes more strongly to fa(k) inh h when ns approaches n(2) s . Recall from Fig. 2 that f a (k) ( f a(k)) is negative (positive) in the k-interval shown in Fig. 3. Due to partial cancellation of the two contributions to fa(k), the quantities f 2(k), SM(k), and HM(k) become very sensitive to the value of ns when ns is near to the matching points n(i) s . The strong dependence of these scattering functions on ns is particularly pronounced at k 5 0 and around the principal peak position km of SM(k). Note the overall flattening in the undulations of the scattering intensity I(k) and, in particular, the decrease of the principal peak height of SM(k) as compared to ns 5 1.4500. In Fig. 4, we present the corresponding results for the measurable hydrodynamic function HM(k), normalized by the hydrodynamic function H0M(k) without HI. Apparent from this figure is the strong similarity in the dependence of HM(k)/H0M(k) and of SM(k) on k and ns. Similarly to SM(k), HM(k)/H0M(k) can be strongly distorted in the index matching regime of ns-values. We proceed now to asses the accuracy of the decoupling approximation for SM(k) and HM(k) in case of charge-stabilized dispersions of coated silica-spheres close to index matching. The advantage of the decoupling approximation is that only the single static correlation function of the one-component reference system is required instead of the m(m 1 1)/2 pair correlation functions needed for the m-component histogram representation of the Schulz distribution. Moreover, size- and

optical polydispersity enter into the description only through the incoherent scattering contribution X(k). At the matching points IMP1–3, the comparison within the core-shell model between SM(k) and its decoupling approximation SD(k) is depicted in Fig. 5 for s 5 0.15. The decoupling approximation reproduces the shape of SM(k) quite well in the index matching regime, in particular for values of k smaller than the position of the principal peak and for wavenumbers where X(k) ' 1. The peak position km of SM(k) is slightly overestimated in the decoupling approximation, and there are also slight differences in the peak heights of SD(k) and SM(k). We notice further that SD(k) is dominated by the incoherent scattering contribution, i.e., SD(k) ' X(k). The reason for this

FIG. 5. Measurable static structure factor SM(k) at index matching points IMP1–3, in comparison with the corresponding results, SD(k), in decoupling approximation. System parameters as in Fig. 3.

496

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behavior of SD(k) is that SI(k; s# ) ' 0 provided k is sufficiently smaller than km. Likewise, the approximate equality SI(k ! km; s# ) ' 0 is due physically to the small relative osmotic compressibility of the strongly repelling particles in case of chargestabilized suspensions. The incoherent scattering function X(k) and the second moment f 2(k) of the scattering amplitude distribution for a Schulz-distributed mixture of optically homogeneous TPM-coated silica spheres are in general monotonously decreasing functions in k up to the wavenumber kmax ' 4 3 1023 Å21. In contrast to dispersions of size-distributed homogeneous particles represented by the linear model where X(k) is usually monotonously decreasing, the X(k) of the core-shell model can be strongly nonmonotonic within [0, kmax]. In particular, X(k) can exhibit a pronounced peak of value one at some nonvanishing k . 0 within the detectable k-range. This wavenumber is determined according to Eq. [19] by setting #f(k) 5 0. For sufficiently small s, this wavenumber is approximately equal to the k-value where the first zero of f(k; s# ) appears (cf. Fig. 2). However, the measurable structure factor is approximately equal to one in the range of wavenumbers where X(k) ' 1, since SM(k) ' SD(k) 5 1 for wavenumbers with X(k) 5 1. In our discussion of Fig. 2, we noted that contrary to f inh(k), f h(k) depends on the contrast (n(s) 2 ns). Consequently, the k-value where f(k; s# ) crosses the zero line and hence the k-value where X(k) becomes maximal shifts with changing contrast. Therefore, the wavenumber regime where the oscillations of SM(k) in the core-shell model are flattened, i.e., where X(k) ' 1, depends on the contrast. Similarly to the findings for the SM(k) of coated silica dispersions, there is also good overall agreement in the index matching regime between HM(k) and its decoupling approximation HD(k), the latter calculated according to Eq. [21]. This finding is illustrated in Fig. 6, where ns 5 n(2) s has been selected. For comparison, Fig. 6 includes also the hydrodynamic function HI(k) of the ideally monodisperse reference system. In Figs. 7a and 7b, we compare SM(k) with SD(k) for the core-shell model of charged silica particles, both at k 5 0 and at the peak position km of SM(k), illustrating the dependence on the solvent refractive index ns. Figure 7a nicely demonstrates the enormous variation of SM(k 5 0) and the nearly perfect agreement with the decoupling approximation in the regime of critical index matching located around the n(i) s . Figure 7b shows the corresponding finding at k 5 km. Sufficiently outside the ns-regime of strong variations of SM(k), which is indicated by the double arrow, SM(k) is only weakly dependent on ns, and it deviates here more strongly from SD(k). Since 0 # X(k) # 1 and, usually, SI(k 5 0) # 1, it follows that SD(k 5 0) # 1. The maximal value one for SD(k 5 0) and hence for SM(0) ' SD(0) is thus reached at the refractive index ns ' n(2) s where X(0) 5 1 (cf. Fig. 7a). The minimum of SM(0) and of SM(km) occurs at ns ' n(3) s . According to Eq. [25], this finding is consistent with a minimized total scattering intensity Itot. Without showing the corresponding results for HM(k)/H0M(k) as a function of ns, we only mention here that at k 5 0 and km,

FIG. 6. Comparison between decoupling approximation result HD(k) and HM(k), the latter calculated using a 5-component histogram representation of the Schulz size distribution. Also shown in the figure is the normalized hydrodynamic function HI(k) of the monodisperse reference system employed in the decoupling approximation. System parameters as in Fig. 3 with ns ' n(2) s .

nearly perfect agreement is reached in the region of critical index matching between HM(k)/H0M(k) and the normalized decoupling approximation result HD(k)/H0M(k). In summary, we can state for the core-shell model of charge-stabilized dispersions that the simple decoupling approximation provides a semiquantitative description of SM(k) and HM(k) in the index matching regime. We discuss now our results for contrast variation effects in the core-shell model of uncharged PIB-coated silica spheres (4). In our calculations of I(k) and SM(k), we employ the analytic PY solution for mixtures of hard spheres in the way described in Section IV. We use here up to nine components in the histogram size distribution. We do not present theoretical results for HM(k) since a theory for the partial hydrodynamic functions of nondilute hard sphere mixtures incorporating the full many-body (i.e., nonpairwise additive) hydrodynamic interactions has not been developed to date. This is contrary to monodisperse hard sphere dispersions, where lattice-Boltzmann computer simulations (31, 32) and calculations of HI(k) based on the so called dg-method of Beenakker and Mazur (33, 34) have been reported. The dg-method accounts in an approximative way for many-body HI, and it is in good agreement with computer simulation results for f , 0.4. In nondilute mixtures of hard-spheres one needs to account for the complicated short-ranged many-body part of the hydrodynamic interactions since the relative probability of finding two particles is largest at contact. The second moment, f 2(k), depends on the model adopted for the optical polydispersity but it does not depend on the microstructure. Hence, the behavior of f 2(k) for PIB-coated hard-spheres as a function of k and ns is in principle the same as shown in Fig. 3 for charged particles. However, the mean

OPTICAL POLYDISPERSITY AND CONTRAST VARIATION

497

FIG. 7. Dependence of SM(k) and SD(k) on ns for (a) k 5 0 and (b) k 5 km. Here, km is the wavenumber where SM(k) attains its maximum. The double arrow encloses the range of critical index matching. Note the strong variations of the measurable static structure factor in the critical ns-range. System parameters, aside from ns, as in Fig. 3.

diameter s# of the PIB-coated particles is more than twice as large as that of the TPM-coated particles, so that the isoscattering wavenumber is located now within the accessible wavenumber range. Therefore, we observe now in Fig. 8 a nonmonotonic behavior of f 2(k) in case of homogeneous spheres with the same refractive index n 5 n# (s# ) (solid line).

FIG. 8. Second moment f 2(k), SM(k), and I(k) versus k for values of ns as indicated in the figure. The intensity I(k) and f 2(k) are plotted in arbitrary units. System parameters correspond to core-shell model for uncharged PIB-coated silica spheres with n1 5 1.45, n2 5 1.51, s# 5 312 nm, d 5 26 nm, total volume fraction f 5 0.3, and s 5 0.11. The solid lines refer to the case of homogeneous spheres with same refractive index n 5 n# (s# ).

Results for f 2(k), SM(k), and I(k) for the core-shell model of PIB-coated silica spheres are shown in Fig. 8, for various values of ns and polydispersity s 5 0.11. The solid lines represent f 2(k), SM(k), and I(k) for homogeneous spheres with the same refractive index. The dotted lines are the corresponding findings at the matching point IMP3. Here, we observe at ns 5 1.4745 that SM(k 5 0) is somewhat larger than one, in disagreement with the decoupling approximation where, as noted before, SD(k) , 1 when SI(k) , 1. In the Percus–Yevick approximation for hard-spheres, SI(0) 5 (1 2 f)4/(1 1 2f)2 ' 0.09 at f 5 0.3. The long-wavelength limit of SM(k) as a function of ns is included in Figs. 9a and 9b for s 5 0.11 and s 5 0.3, respectively. Included in these figures are the three s-dependent values of the solvent refractive index ns compliant with the matching criteria IMP1–3. Even for a comparatively modest polydispersity s 5 0.11, there are significant deviations between SM(0) and its decoupling approximation not only outside but also inside the matching regime. The maximum of SM(0) as a function of ns is underestimated in the decoupling approximation, and the position and half-width of the peak associated with the maximum is overestimated. The differences between SM(k) and SD(k) at k 5 0 are more pronounced at larger polydispersity s 5 0.3 (cf. Fig. 9b). We only show here for conciseness the case k 5 0. From our numerical analysis we conclude that the decoupling approximation is a rather poor approximation for the correlation functions of uncharged polydisperse dispersions. Contrary to charge-stabilized dispersions, this is the case even in the index matching regime where SM(0) varies strongly with ns. The most probable distance between two charged particles does not coincide with the distance of minimal approach allowed by the hard cores. This is the reason why the decoupling approximation performs far better in the matching regime in

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498

FIG. 9. SM(0) and SD(0) for uncharged PIB-coated silica spheres as functions of ns, for polydispersities (a) s 5 0.11 and (b) s 5 0.3. Indicated in the figure are further the index matching points corresponding to IMP1–3. Aside from s, all system parameters as in Fig. 8.

the case of the charge-stabilized systems considered earlier; the assumption of uncorrelated particle sizes and positions inherent in the decoupling approximation is then less severe. While the decoupling approximation cannot be used for a quantitative prediction of the strong variations of SM(k) in the index matching regime, it is still useful for approximately locating that part of the matching regime where SM(0) ' 1. Finally, we briefly address the linear model of charged silica particles introduced in Section V. Results for f 2(k), SM(k), and I(k) for various values of ns located inside and outside the matching regime are summarized in Fig. 10. The system parameters are listed in the figure caption together with the values of the three matching refractive indices n(i) s . The solid lines are the result for optically homogeneous spheres with the same refractive index n 5 n(s# ). Contrary to f 2(k) and I(k), SM(k) is independent of ns in this case. The dashed lines correspond to ns ' n(3) s . Contrary to the core-shell model, in the linear model both f 2(k) and X(k) are usually monotonously decaying functions for wavenumbers k , 4 3 1023 Å21 accessible by light scattering. This finding is due to the absence of an incoherent scattering amplitude contribution in the linear model. At low optical contrast the linear model shows the same qualitative trends as the core-shell model, i.e., the oscillations in SM(k) and I(k) are significantly diminished as compared to the strongcontrast case. The overall flattening of SM(k) at low contrast is more pronounced in the linear model (cf. Fig. 1). As for the core-shell model, there is also very good agreement between SM(k), HM(k), and the corresponding decoupling approximations in the matching point regime of low optical contrast. VI. CONCLUSIONS

We have theoretically analyzed the influence of contrast variation on the static and short-time dynamical properties of

polydisperse colloidal dispersions. As models of optical polydispersity, the core-shell model and the linear model were considered together with a discretized Schulz distribution of particle sizes. Realistic system parameters have been selected in our calculations corresponding to dispersions of coated silica spheres (3, 4, 13). The static pair distribution functions needed

FIG. 10. Second moment f 2(k), SM(k), and I(k) of the linear model of charged silica spheres. Values of ns as indicated in the figure. The system parameters are n(s# ) 5 1.4516, p 5 0.0137, s# 5 124 nm, s 5 0.15, Z 5 220, and f 5 0.063. The index matching points n(i) s according to IMP1–3 are given (2) (3) for this model by n(1) s 5 1.4516, ns 5 1.4525, and ns 5 1.4533 for l 5 495.9 nm. The solid lines refer to the case of homogeneous spheres with same refractive index n 5 n(s# ).

OPTICAL POLYDISPERSITY AND CONTRAST VARIATION

for calculating SM(k), I(k), and HM(k) were computed for charge-stabilized dispersions by employing the multicomponent HNC integral equation scheme, with Yukawa pair potentials of DLVO type. For hard sphere-like particles we have used the many-component analytic PY solution. The hydrodynamic function HM(k) was obtained for charge-stabilized dispersions on the basis of a far-field approximation for the hydrodynamic diffusivity tensors. Since the definition of an optical index matching point is not unique in the presence of correlated size- and optical polydispersity, three distinct definitions (IMP1–3) have been examined. It was demonstrated that, within a critical regime of ns-values including the matching points n(i) s (for i 5 1, 2, 3), the shapes of the measurable quantities SM(k), I(k), and HM(k) can be strongly distorted from the case of strong optical contrast or optically homogeneous colloidal spheres of equal refractive index. By generalizing the decoupling approximation of SM(k) to HM(k), we have shown for charge-stabilized dispersions with modest sizepolydispersity (s , 0.2) that this simple scheme provides a semiquantitative description in the critical range of solvent refractive indices. The range of overall low optical contrast with respect to the solvent can be conveniently located using the criterion IMP2, where only the knowledge of #f(k 5 0) 5 0 is needed without any information about the microstructure of the particles. The decoupling approximation is found to be substantially less accurate in case of polydisperse hard spheres due to strong correlations between the particle sizes and positions. However, the decoupling approximation remains useful even for hard spheres to locate roughly the regime of critical index matching. In the critical ns-regime, one cannot determine the size-polydispersity, mean hydrodynamic radius, and the effective charge of correlated optically inhomogeneous particles or particles with size-related refractive indices using the simple assumption of a homogeneous sphere model. This assumption leads to incorrect results. If one wants to avoid experimentally the critical ns-regime with its strong sensitivity on optical polydispersity, calculations of SD(k) and X(k) with the assessment of the ns-range where SD(k) varies strongly can prove to be very helpful. However, if one is particularly interested in the critical regime, then optical polydispersity has to be accounted for in full detail. From our analysis it also follows that it is very difficult to infer interparticle (i.e., microstructural) properties from measuring SM(k) and HM(k) without further knowledge of the optical properties and of the size distribution of the particles. For identical microstructures (i.e., identical gab(r) and Sab(k)) HM(k) and SM(k) can vary strongly in shape when the optical properties of the particles or the solvent contrast are changed. The amount of microstructural information obtained from SM(k) and HM(k) is particularly low at small wavenumbers at low optical contrast. In fact, for charge-stabilized systems and low optical contrast we have shown that SM(k) ' SD(k) ' X(k) at k , km, but the incoherent scattering function X(k) contains no microstructural information at all. On the other hand, the

499

finding SM(k) ' X(k) at k , km is useful for extracting some information on intraparticle properties in (nondilute) dispersions of strongly correlated particles. ACKNOWLEDGMENTS A.J.B. acknowledges financial support by the DAAD. We further acknowledge useful discussions with J. Bergenholtz and R. Klein (University of Konstanz), A. Vrij (Utrecht University), and B. Weyerich (now Nortel-Dasa).

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