Optical extinction, refractive index, and multiple scattering for suspensions of interacting...

$: Optical extinction, refractive index, and multiple scattering for suspensions of interacting colloidal particles$
Optical extinction, refractive index, and multiple scattering for suspensions ofinteracting colloidal particlesAlberto Parola, Roberto Piazza, and Vittorio Degiorgio

Citation: The Journal of Chemical Physics 141, 124902 (2014); doi: 10.1063/1.4895961 View online: http://dx.doi.org/10.1063/1.4895961 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic-field-dependent optical transmission of nickel nanorod colloidal dispersions J. Appl. Phys. 106, 114301 (2009); 10.1063/1.3259365 Effective interaction between large colloidal particles immersed in a bidisperse suspension of short-rangedattractive colloids J. Chem. Phys. 131, 164111 (2009); 10.1063/1.3253694 Laser tweezer microrheology of a colloidal suspension J. Rheol. 50, 77 (2006); 10.1122/1.2139098 Refractive-index-driven separation of colloidal polymer particles using optical chromatography Appl. Phys. Lett. 83, 5316 (2003); 10.1063/1.1635984 Tracer particle diffusion in crystal-like ordered colloidal suspensions J. Chem. Phys. 110, 1283 (1999); 10.1063/1.478181

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THE JOURNAL OF CHEMICAL PHYSICS 141, 124902 (2014)

Optical extinction, refractive index, and multiple scattering for suspensionsof interacting colloidal particles

Alberto Parola,1 Roberto Piazza,2,a) and Vittorio Degiorgio3

1Department of Science and High Technology, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy2Department of Chemistry (CMIC), Politecnico di Milano, Via Ponzio 34/3, 20133 Milano, Italy3Department of Industrial and Information Engineering, Università di Pavia, Via Ferrata 5a,27100 Pavia, Italy

(Received 17 July 2014; accepted 6 September 2014; published online 22 September 2014)

We provide a general microscopic theory of the scattering cross-section and of the refractive indexfor a system of interacting colloidal particles, exact at second order in the molecular polarizabilities.In particular: (a) we show that the structural features of the suspension are encoded into the forwardscattered field by multiple scattering effects, whose contribution is essential for the so-called “opticaltheorem” to hold in the presence of interactions; (b) we investigate the role of radiation reaction onlight extinction; (c) we discuss our results in the framework of effective medium theories, present-ing a general result for the effective refractive index valid, whatever the structural properties of thesuspension, in the limit of particles much larger than the wavelength; (d) by discussing strongly-interacting suspensions, we unravel subtle anomalous dispersion effects for the suspension refractiveindex. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895961]

I. STATEMENT OF THE PROBLEM

Scattering methods have long been a basic tool for theinvestigation of colloidal systems. The recent development ofoptical correlation techniques1–4 that, by successfully com-bining scattering and real-space visualization, allow to probethe microscopic Brownian dynamics still retaining the spatialresolution proper of a microscope, calls however for a criticalreassessment of the relation between scattering and imaging.A detailed analysis of the effects of the propagation through ascattering medium on the amplitude and phase of the transmit-ted wavefront is also of primary importance for digital holo-graphic techniques.5

The effect on the transmitted wavefront of the transitthough a scattering medium can be expressed by stating thatthe forward scattering pattern consists in a faithful reproduc-tion of the incident field that spatially superimposes with thetransmitted radiation, but with a different phase. The interfer-ence between this “simulacrum” of the incident field and theportion of the field which passes through the medium withoutbeing scattered yields both its phase delay in traversing themedium (thus fixing its refractive index) and, adding to thenon-radiative power loss due to absorption, the power reduc-tion of the transmitted field. For what concerns power loss,this is explicitly treated by the so-called “Optical Theorem”(OT), a general and extremely useful result that holds true notonly for electromagnetic radiation but also for matter waves.6

Consider the simple case of a plane wave with wave-vectorki = (ω/c) ki , where ki is a unit vector specifying the inci-dent direction, and polarized with the electric field along ni ,Ei(r, t) = niEi exp[i(ki · r − ωt)], which encounters a scat-tering and absorbing medium confined in a finite region ofspace around the origin. In far field, the radiation scattered

a)Electronic mail: [email protected]

along r = r/|r| with wavevector ks = ks r can be written as

Es(r, t) = nsEi

iS(ks , ki)

krei(k

sr−ωt), (1)

where ns is a vector normal to ks , which depends on both niand ks, and S(ks , ki), called the scattering amplitude, takes ingeneral complex values. Then, the OT states that the extinc-tion cross section is given by7

σext = 4π

k2(ni · ns)Re[S(ks = ki)]. (2)

This rather surprising result, which basically shows that eval-uating the total extinction (due to scattering and, possibly,absorption) of the radiation traversing the medium requiresto know the scattering amplitude only in the forward direc-tion ki , can be rather easily obtained8 by considering that, atsteady state, the change in the total energy density within aspherical region containing the whole scattering medium issolely due to the dissipative absorption processes taking placewithin it, and by equating the latter to the incident plus scat-tered energy flow through the spherical surface.9, 41

Most presentations of the OT consider an incoming planewave and a single scattering particle. Here we assume thatthe scattering volume contains a large number of particles inBrownian motion. The field scattered in any direction, exceptfor zero-scattering angle, is the sum of many uncorrelatedfields, it is a random process with zero-average and a two-dimensional Gaussian probability density. On the contrary,the fields scattered by individual particles in the forward di-rection have all the same phase, that is, the forward scatteredwave exactly reproduces the wavefront of the incident wave.In this paper, we will only deal with an incident plane wave,but the OT can be generalized to an arbitrary incident field (forinstance, a gaussian beam) by considering the angular spec-trum of the latter and applying Eq. (2) to each plane wave

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124902-2 Parola, Piazza, and Degiorgio J. Chem. Phys. 141, 124902 (2014)

component. Of course, in such a case the relation between ex-tinction and forward scattering amplitude applies only to thewhole far-field diffraction pattern.

The forward scattering amplitude contains more informa-tion than what simply conveyed by the OT. To see this, it issufficient to recall that, in a macroscopic description of thepassage of radiation though a material, the effects of prop-agation can be fully embodied into a complex refractive in-dex n = n + in′, whose real and imaginary parts are, respec-tively, related to dispersion and power loss. Then, both n andn′ can be formally linked to the real and imaginary parts ofS(ks = ki). Yet, such a relation would be of little practical in-terest unless we are able to evaluate S(ks = ki), which is theoverall forward scattering amplitude, in terms of the specificmicroscopic scattering and absorption events taking place inthe medium. Strenuous efforts to derive the macroscopic op-tical properties of a molecular fluid from microscopic scatter-ing events have spangled the history of physical optics (for areview of the early attempts, see for instance, the books byRosenfeld10 and Fabelinskii11), culminating in a series of im-pressive contributions by Hynne and Bullough,12–14 in whicha rigorous many-body electrodynamic theory is used to ob-tain consistent expressions for the refractive index, the ex-tinction coefficient, and the scattering cross section. Unfor-tunately, this powerful analysis, which was performed with avery sophisticated formalism and basically no approximation,leads to rather cumbersome general results that, as a matter offact, yield manageable expressions only for rather dilute realgases, where interactions are accounted for only at the levelof the second virial coefficient in a density expansion.

The situation looks however much more promising if weconsider a particulate medium, namely, a collection of indi-vidual scatterers, such as a suspension of colloidal particlesdispersed in a weakly scattering, non-absorbing solvent. Ourproblem can then be rephrased as follows: is there any way torelate the real and imaginary part of the refractive index of thewhole dispersion to the scattering properties of the individualscatterers? In the simple case of a thin slab of a medium con-sisting of a dispersion of identical scatterers illuminated bya monochromatic plane wave, the OT provides a straightfor-ward affirmative answer to this question, at least provided thattwo basic assumptions are satisfied:15

1. The scatterers are randomly arranged, namely, they donot display any structural correlation. This implies thatthe physical particles acting as scatterers interact veryweakly, so that any correlations in density fluctuationscan be neglected.

2. The incident field “seen” by a particle coincides with theexternal radiation, namely, any additional contributiondue to the surrounding scatterers is neglected. Providedthat we carefully specify that these contributions may bedue not only to radiating, but also to quasi-static fields inthe near-zone, which would actually be the case for thosescatterers that lying a distance r � λ from the particle,this loosely means that “multiple scattering” effects arenegligible.

In this case, assuming for simplicity that the scatterers areoptically isotropic, and indicating with θ the polar angle with

respect to the direction of ki , Eq. (2) reduces to

σext = 4πN

k2Re[s(0)], (3)

where N = ρV is the particle number in the volume V , k= 2π /λ, and s(0) is the amplitude of the field scattered in theforward direction θ = 0 by each single particle. This simplerrelation can be obtained by considering that (i) for indepen-dent particles, σ ext is the sum of the single-particle cross sec-tions; (ii) in the forward direction (and only in this direction)the scattering amplitudes are additive too, because all scatter-ing contributions add in phase, regardless of the positions ofthe particles in the scattering volume. By evaluating the phaseshift in propagation through the slab, the real and imaginaryparts of the effective refractive index of the medium are theneasily found to be16⎧⎪⎨⎪⎩

n = 1 − 2πρ

k3Im[s(0)],

n′ = 2πρ

k3Re[s(0)].

(4)

It is useful to recall that, when n′ �= 0, the intensity of aplane wave propagating in the medium along z decreases asI = I0exp (−γ z), where the extinction coefficient

γ = 2kn′ = 4πρ

k2Re[s(0)] = σext

V(5)

is simply the extinction cross section per unit volume.Equation (4) has been used to investigate the effects of

particle size on refractive index and extinction by using theexpression for scattering amplitude obtained from the generalMie theory for light scattering from non-interacting spheri-cal particles.17 However, repulsive and attractive interparti-cle forces are well known to strongly affect (the former byincreasing, the latter by reducing) the transmittance of lightthrough a colloid. Moreover, Eq. (4) suggests that also the realpart n of the refractive index should not be immune from in-teraction effects. It is then very tempting to scrutinize whetheran effective refractive index could be defined in the interactingcase too, provided that the expression for the single-particleforward scattering amplitude is suitably revisited to accountfor the structure of the medium.

The goal of this work is to extend the OT approach tothe case of interacting colloidal particles, and to apply ourresults to investigate the contribution of correlations to therefractive index of a suspension. We shall confine our inves-tigation to suspensions of spherical particles in the colloidalsize range that, though possibly concentrated in terms of par-ticle volume fraction φ, are still dilute in terms of numberdensity ρ = φ/v, where v is the volume of a single particle.This restriction allows to describe interparticle forces usingsimple model pair potentials and correlation functions, an ap-proach which is generally unsuited to properly describe thestructure of dense molecular fluids. In addition, we shall sys-tematically adopt the lowest order approximation for the op-tical properties of the scatterers in which interactions effectsdo nevertheless show up which, as we shall see, amounts toa second order approximation in the optical polarizability α.Even within these approximations, however, an explicit evalu-ation of S(0) casts new light on the optical mechanism leading


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to the formation of the transmitted wavefront, and highlightsa rather unexpected role played by multiple scattering, usu-ally just regarded as a nuisance in light scattering studies. Akey result of our investigation is indeed that, when consider-ing the radiation strictly scattered in the forward direction, thecontribution from multiple scattering events, even when neg-ligible at finite q, is conversely found to be crucial to figureout why the forward scattering amplitude, and therefore lightextinction, depends on interparticle interactions.

This paper is then organized as follows. The microscopicapproach we use and the approximations we make are intro-duced in Sec. II, where we first evaluate the electric field inthe forward direction due to the superposition of the incidentfield with the waves generated by point-like oscillating parti-cles. By considering a slab geometry, we show that a micro-scopic expression for the refractive index of a suspension ofuncorrelated point-like particles is fully consistent at all or-ders with the high frequency limit of the Clausius-Mossotti(CM) formula, namely, the Lorentz-Lorenz expression18

n2 − 1

n2 + 2= 4π

3ρα, (6)

provided that the expression for the dipole polarizabilityincludes the contribution from the reaction radiation field,namely, the self-action of the dipole on itself.

The correlation contribution to the scattering amplitudeand to the refractive index for the general case of a homoge-neous but correlated distribution of point-like dipoles is de-rived in Sec. III, and used to check that the forward scatteringamplitude is rigorously linked to σ ext by the Optical Theorem.Such an explicit comparison yields an interesting conceptualconsequence: particle spatial correlations are “encoded” intothe forward scattering amplitude only via the additional con-tribution to the incident field brought in by the secondaryfields scattered by those dipole lying within a close-by regionwith a size comparable to the correlation range of the medium.In particular, we discuss the limits of validity of the CM ap-proximation in terms of the ratio of the correlation length ξ ofthe system to the wavelength λ of the incident light.

In Sec. IV, we first extend the former results to a systemof particles of finite size in vacuum, comparing in particularthe limits for small and large particle size both in the absence(Sec. IV A) and in the presence (Sec. IV B) of interparticleinteractions. Extension of these results to the case of particlesdispersed in a solvent is made in Sec. IV C, where we showthat this is straightforward provided that the latter is assumedto be an uncorrelated dielectric medium. In Sec. IV D, weframe our results within the context of effective medium the-ories, which is made, showing in particular that a very generalresult for the effective static dielectric constant, which is ex-act at 2nd order in polarizability whatever the structural cor-relations of the suspension, fails in the optical regime whenλ � ξ , and has to be substituted by a novel, equally generalexpression for the effective refractive index. Illustrative exam-ples of the contribution of interparticle interactions to the con-centration dependence of the refractive index are presented inSec. IV E for the specific case of hard-sphere interactions.In particular, by investigating the strongly correlated case of acolloidal fluid in equilibrium with a colloidal crystal, we show

that, whenever the peak of the structure factor S(q) falls withinthe detectable q-range, the refractive index displays a peculiar“anomalous dispersion” region where it behaves similarly tothe refractive index of a Lorentz oscillator close to resonance.Experimental conditions in which these effects could be ob-served are finally discussed in Sec. IV F.

II. SYSTEM OF POINT-LIKE PARTICLES

The purpose of this section is to describe the totalfield scattered by a system of point-like polarizable particles(namely, simple dipoles) by explicitly taking into account thecontribution to the incident field on each single dipole dueboth the other surrounding dipoles and to the self-action ofthe dipole on itself. Consider then a collection of N oscillat-ing dipoles made of a mobile charge e and a fixed charge −elocated at fixed positions Ri, with a spatial distribution to bespecified later. Defining the instantaneous dipole moment ofa given particle as

p(t) = p0e−iωt , (7)

the electric field in r generated by the oscillating dipole placedat the origin has the form8

Eμ

d (r, t) = ei(kr−ωt)k3�μν(r) pν0 , (8)

where k = ω/c, the radial unit vector is nμ = rμ/r, and thedimensionless tensor �(r) is defined by

�μν(r)= (3nμnν −δμν)

[1

(kr)3− i

1

(kr)2

]− 1

kr(nμnν −δμν) .

(9)Here and in the following, Greek superscripts refer to the spa-tial components (x, y, z) and the summation over repeatedGreek indices is understood. It is useful to observe right fromthe start that the 2nd and 3rd term in �μν(r), which respec-tively account for the field in the so-called “intermediate” and“radiation” zones,8 are of order r/λ and (r/λ)2 with respectto the electrostatic part decaying as r−3. When we take intoaccount spatial correlations, the relative contribution of thesetwo terms will be found to increase with the correlation lengthξ of the system.

If we include the presence of an external linearly polar-ized plane wave of the form

Eμ

0 (r, t) = E μei(k·r−ωt) (10)

with kμE μ = 0, the total electric field, due to the externalsource and the collection of dipoles, is then

Eμ(r, t) = Eμ

0 (r, t) +∑

j

Eμ

d (r − Rj , t) (11)

= Eμ

0 (r, t)+k3∑

j

eik|r−Rj|�μν(r−Rj ) pν

0j e−iωt ,

(12)

where Ed(r − Rj, t) is the contribution to the electric field in rdue to the dipole in Rj. Now we introduce the polarizability α

by assuming that the moment pj of the dipole in Rj is propor-tional to the local electric field due to the other charges (i.e.,the dipoles represent polarizable point-like objects),

pj (t) = p0j e−iωt = α Ej (Rj , t), (13)


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where the subscript j in Ej(Rj, t) means that the contributiondue to the dipole in Rj has to be subtracted, and α depends ingeneral on the frequency ω.

Yet, as we already mentioned, a consistent treatmentrequires to take also into account the action on the oscillat-ing charge of the field emitted by itself, namely, of the so-called “radiation reaction field” which, in the absence of non-radiative dissipation, provides the only mechanism for powerloss. The question of the back-reaction of the radiated fieldonto the motion of a charge is one of the most challengingproblems in electrodynamics, since it leads to an equation,originally derived by Lorentz19 and then generalized to therelativistic case by Dirac,20 which, containing the derivativeof r, causes serious difficulties due to the appearance of “run-away” solutions showing an exponential increase of r evenin the absence of external fields. Nevertheless, for a chargeoscillating at non-relativistic speed, the ingenious approachdevised by Lorentz safely allows to include radiation reactioneffects at lowest order by introducing an imaginary additivecontribution to the particle polarizability15

α → α = α

[1 + i

2

3k3α

]. (14)

In the following, the physical counterpart of the elementarydipoles we introduced will be atoms or molecules excited at afrequency ω far from any electronic or vibrational transition,so that α will be taken as a real quantity.

Since we are considering point-like dipoles, the only in-trinsic length scale in the problem is the wavelength λ of theincident radiation. It is then suitable to define a dimension-less polarizability αd = αk3 (where the subscript d stands for“dipoles”) that, substituted into Eq. (12) using (14), yields

Eμ(r, t) = Eμ

0 (r, t) + αd

∑j

eik|r−Rj|�μν(r − Rj ) Eν

j (Rj , t).

(15)This is an equation for the electric field E(r, t), which can besolved by iteration. To second order in the scaled polarizabil-ity αd the explicit solution is

Eμ(r, t)

= Eμ

0 (r, t) + αd

∑j

e−ik|r−Rj|�μν(r − Rj ) Eν

0 (Rj , t)

+α2d

∑j �=l

eik|r−Rj|eik|R

j−R

l|�μν(r−Rj )�νσ (Rj −Rl)E

σ0 (Rl , t).

(16)

The second order approximation in αd given by Eq. (16)will be particularly useful in what follows both to describea system of interacting dipoles, and to extend our results tothe case of finite-size particles in Sec. IV, where, in the caseof non-interacting colloids, we shall also check for consis-tency with the exact Mie results obtained within a contin-uum approach. However, it is interesting to point out thatEq. (15) is also the starting point of a non-perturbative in-vestigation of the dispersion relation which characterizes themedium, which fully justifies the CM relation for a systemof uncorrelated dipoles, and actually generalized it to accountfor the extinction contribution brought in by radiation reaction

effects. For a monochromatic perturbation, the time depen-dence is factorized as

Eμ

j (r, t) = Eμ

j (r)e−iωt . (17)

Moreover, by evaluating the electric field at the position ofthe ith particle and subtracting the singular contribution dueto the ith dipole, we get

Eμ

i (Ri) = Eμ

0 (Ri) + α∑j �=i

eik|Ri−R

j|�μν(Ri − Rj ) Eν

j (Rj ).

(18)For given positions of the N dipoles of the medium (R1···RN),this is a set of linear equations for the 3N unknowns E

μ

i (Ri).Here we want to analyze the possible solutions in the bulk,i.e., the monochromatic waves which can propagate in themedium. Consider then a planar slab of thickness h, placedorthogonally to the direction of propagation z of an incidentlinearly polarized plane wave E

μ

0 (r) = εμ

0 eikz. The field insidethe slab at the position Zi can always be written as the super-position of two counter-propagating components (the trans-mitted and the reflected wave) with configuration dependentamplitude

Eμ

i (Ri) = ε+μ

i eiqZi + ε

−μ

i e−iqZi , (19)

where q is a complex quantity to be specified later represent-ing the average wave vector of the propagating field inside themedium. By substituting this parametrization into Eq. (18),we obtain

ε+μ

i eiqZi + ε

−μ

i e−iqZi

= εμ

0 eikZi + α

∑j �=i

eik|Ri−R

j|�μν(Ri − Rj )

× [ε+νj eiqZ

j + ε−νj e−iqZ

j

]. (20)

Now we fix the position Ri = r of the ith molecule and aver-age over the equilibrium distribution of the other particles ofthe medium. By defining ε±μ(r) = ⟨

ε+μ

i

⟩, Eq. (20) becomes

ε+μ(r)eiqz + ε−μ(r)e−iqz

= εμ

0 eikz + αρ

∫dr′g(r − r′)eik|r−r′|�μν(r − r′)

×[ε+ν(r′)eiqz′ + ε−ν(r′)e−iqz′]

, (21)

where ρ is the number density of particles in the system, andg(r) is the pair distribution function. The domain of integra-tion in Eq. (21) coincides with the whole volume of the slab.In going from Eqs. (20) to (21) we assumed that the configu-rational average of the field amplitude ε±ν

j at fixed position ofboth the ith and jth particles, coincides with the previously de-fined local field amplitude ε±ν(r′), for every choice of r. Weanticipate that this assumption is in fact correct to second or-der in an expansion in powers of the molecular polarization.Finally, we look for a homogeneous solution of the integralEq. (21): ε±μ(r) = ε±μ. We will show that the existence ofsuch a solution requires a suitable choice of the, up to nowarbitrary, propagation wave-vector q.

Note that g(0) = 0 due to the presence of a hard core:this prevents the occurrence of spurious singularities in theintegration domain. Next we write g(r) = 1 + h(r), where


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h(r) is non-zero only at short range (i.e., only for r com-parable to the molecular diameter). The first contribution(g(r) = 1) accounts for the average particle distribution, whilethe residual term, containing h(r), provides the correlationcontribution to the propagating wave. Let us examine the un-correlated term with the supplementary caveat to exclude aninfinitesimal neighborhood of r = 0. By explicitly performingthe integrals we obtain the following set of four consistencyconditions:

ε+

k − q+ ε−

k + q= − k2ε0

2παρ, (22)

ε+e−i(k+q)h

k + q+ ε−ei(k−q)h

k − q= 0, (23)

ε±

k + q+ ε±

k − q− 2

3

ε±

k= − k2ε±

2παρ, (24)

which are necessary and sufficient for the validity of Eq. (20)for all z. The condition expressed by Eq. (22) is equivalentto the extinction theorem: the incident wave of wave-vector kdoes not propagate in the medium because it is exactly can-celed by the contribution of the oscillating dipoles. Moreover,together with Eq. (23), it provides the amplitudes of the wavespropagating in the direction of the incident signal (ε+) and inthe opposite direction (ε−). Finally, the last two equations (24)allow to fix the wave-vector q of the wave at frequencyω = kc propagating inside the medium. By defining the com-plex refractive index n via q = nk, the final result is

n2 − 1

n2 + 2= 4π

3αdρd = 4π

3αρ, (25)

where ρd = k−3ρ is a dimensionless dipole density. No-tably, Eq. (25) is a generalized Clausius–Mossotti (or, better,Lorentz-Lorenz) relation for the complex refractive index n,which includes the effects of radiation reaction through theimaginary part of α. Expanding n at second order in αρ, us-ing (14) with α real, and equating the real and imaginary parts,we obtain ⎧⎪⎨⎪⎩

n = 1 + 2παρ + 2

3(παρ)2,

n′ = 4π

3ρk3α2.

(26)

The real part coincides with the expansion at 2nd order of theusual CM formula, whereas the dissipative radiation-reactionterm contribute only to attenuation. When this result is in-serted into Eq. (4), it yields an explicit expression for the scat-tering amplitude of a single non-interacting dipole

s0(0) = 2

3k6α2 − ik3α, (27)

to leading order in α (linear for the imaginary, and quadraticfor the real part). From the OT we then get the correct extinc-tion cross section σ ext = (8π /3)Nα2k4 for Rayleigh scatteringfrom independent particles with a size much smaller than λ.

III. CORRELATED FLUID OF POINT-LIKE DIPOLES

We now consider a correlated dielectric medium start-ing from the general expression (20). The weight function

h(r − r′) is appreciably different from zero only in a smallneighborhood of r, therefore if the observation point r isplaced in the bulk, we can extend the integral to the wholespace, neglecting the effects of the boundary surfaces. Theresulting consistency condition, which corrects the Lorentz-Lorentz formula (25) for a correlated fluid, is obtained by in-cluding into Eqs. (22)–(24) a correlation integral C(q, k),

1 = 4παρ

[1

n2 − 1+ 1

3+ C(q, k)

], (28)

C(q, k) = k3

4π

∫dr e−iqzh(r)�xx(r)eikr , (29)

where, as usual, an infinitesimal neighborhood of r = 0 isexcluded from the integration domain. The correlation inte-gral C(q, k) can be conveniently expressed in terms of Fouriertransforms. However, due to the slow decay of the tensor�xx(r), it is customary to introduce a converging factor e−ηr,where the limit η → 0+ at the end of the computation is un-derstood. The resulting expression reads

C(q, k) = 1

2q2

∫dp

(2π )3h(p)

[k2q2 + (q2 − p · q)2

|q − p|2 − k2 + iη− q2

3

],

(30)

where the complex wavevector q = nk is directed along z.Equation (28), with the definition (30), implicitly relates thecomplex refraction index n to the microscopic (complex) po-larizability α in a correlated fluid of number density ρ. Tosecond order in the scaled polarizability, the correlation in-tegral (30) can be evaluated at q = k. In this case, Eq. (28)explicitly provides n as a function of α with the result

n2 = 1 + 4παρ

1 − 4παρ[

13 + C(k, k)

] , (31)

which reduces to (25) for C = 0 and represents an approx-imate, non-perturbative expression of the complex refractiveindex of a correlated medium. Expanding again to second or-der, we obtain the exact lowest order correction to the refrac-tive index in a correlated fluid

n = 1 + 2παρ + 2

3(παρ)2 + 8(παρ)2C. (32)

Recalling that αρ = αdρd , we point out that this expansioncan be regarded as valid at second order in αd with no restric-tion on the value of the scaled density ρd (namely, it is nota low-density expansion). The formal expressions of the realand imaginary part of C read

Im C = 1

16πk3

∫ 2k

0dp ph(p)

[2k4 − k2p2 + p4

4

],

(33)

Re C = 1

16π2k3

∫ ∞

0dp ph(p)

[8

3k3p − kp3

+(

2k4 − k2p2 + p4

4

)ln

p + 2k

|p − 2k|]

.

Here and in the following we drop the momentum dependenceof the correlation integral, setting C = C(k, k). To this orderof approximation, both the refractive index n and the forward


193.204.33.52 On: Wed, 24 Sep 2014 10:15:25


scattering amplitude S(0) acquire an additive contribution dueto correlations {

δS(0) = −iα2k3V C,

δn = 2πα2C,(34)

where V is the volume of the sample. Notably, if we define anexcess scattering amplitude per particle, δs(0) = δS(0)/N, sothat s(0) = s0(0) + δs(0), Eq. (4) remains then formally valid,although of course the effective forward scattering amplitudes(0) actually depends on ρ and on the specific structure of themedium via the correlation integral C.

It is useful to point out that the correlation contributionto the imaginary part n′ of the refractive index in Eq. (33)depends only on those values of p that are smaller than themaximum wave-vector 2k (corresponding to a scattering an-gle θ = π ) falling within the experimentally detectable range.Although this is seemingly not the case for Re C, we shall seein Sec. IV E that the actual occurrence or not of a peak of thestructure factor S(q) within the accessible range q ≤ 2k doesappreciably influence the value of the refractive index n. No-tice also that Eq. (32) provides a quantitative explanation ofthe reason why the Lorentz-Lorenz expression for the refrac-tive index of a molecular fluid is often a very good approxi-mation, even in the presence of consistent correlations. For kξ� 1, where ξ is the correlation length defined as the distancewhere h(r) becomes negligible,21 the correlation coefficient Cis indeed easily found to behave as (kξ )2 ∼ (ξ /λ)2. Then, pro-vided that ξ is of the order of the molecular size (which isusually the case, unless the system is close to a critical point),correlation corrections are small.

Expression (34) can be readily shown to be fully consis-tent with the Optical Theorem. We first evaluate the real partof Eq. (34) through Eq. (33), retaining the correlation contri-bution and radiation-reaction effects,

Re[S(0)]=Nα2

{2

3k6+ ρ

4

∫ 2k

0dq qh(q)

[2k4−k2q2+ q4

4

]},

(35)

where the first term comes from radiation reaction in the firstorder contribution, while the second from Eq. (33). It is con-venient to change the integration variable to q = 2ksin (θ /2)with θ ∈ (0, π ). Equation (35) then becomes

Re[S(0)] = Nα2k6

{2

3+ ρ

4

∫ π

0dθ sin θ (1 + cos2 θ )h(q)

}= N

α2k6

4

∫ π

0dθ sin θ (1 + cos2 θ )[1 + ρ h(q)].

(36)

Calling θ the scattering angle and ϕ is the angle betweenthe scattering plane and the polarization vector, the differen-tial cross section for Rayleigh scattering from a collection ofdipoles is given by8

dσ

d�= Nα2

(ω

c

)4(1 − sin2 θ cos2 ϕ) [1 + ρ h(q)], (37)

where q = 2ksin (θ /2) and, for a harmonically bound oscil-lator of elementary charge e excited at a frequency ω much

lower than its natural frequency ω0, α = −e2/(mω20). Putting

again s(0) = S(0)/N, we immediately verify via an integrationof Eq. (37) on the solid angle d� = sin θ dθ dϕ, namely, byaveraging over all possible orientations of the incident fieldwith respect to the scattering plane, that Eq. (3) is satisfied byour final expression (36).

IV. COLLOIDAL SUSPENSIONS

Up to now we considered just point-like polarizable par-ticles, i.e., particles whose size is much smaller than the wave-length of the incident field. However, if we are interestedin colloidal suspensions, we have to deal with polarizablespheres whose size may be comparable to or even larger thanthe optical wavelength. To this aim, we model each particlep as a homogeneous dielectric sphere of radius a made of Mpolarizable molecules. On a microscopic scale, the system isagain described by a collection of point-like dipoles, whosespatial distribution clusters however into spherical units cen-tered around the position of the center of mass of each sin-gle colloidal particle. The derivation of Secs. II and III istherefore still valid, provided the polarizability α is the mi-croscopic polarizability of each molecule, the density ρ is thenumber density of molecules, related to the colloidal particledensity ρp by ρp = ρ/M and the distribution function h(r) hasa non-trivial structure, appropriate for the underlying “clusterfluid.”

Let us consider a collection of N spherical parti-cles, characterized by a normalized probability distributionPp(R1···RN). Each polarizable molecule is identified by its po-sition rl

m, where l = 1···N labels the colloid and m = 1···M thespecific molecule in the colloid. If the molecules are homoge-neously distributed inside each sphere in an uncorrelated way,their probability density in space is given by

P ({rlm})=

∫dR1 · · · dRN Pp(R1 · · · RN )

∏l,m

θ (a−|rlm−Rl |)v

,

(38)where θ (x) is the Heaviside step function and v = (4π/3)a3 isthe particle volume. The molecular distribution enters our ex-pressions through the correlation integral Eq. (29) where weused the standard definition of radial distribution function22

ρ2 g(R − R′) =⟨∑

i �=j

δ(R − Ri) δ(R′ − Rj )

⟩. (39)

Now, this expression must be generalized to

ρ2 g(r − r′) =⟨ ∑

(l,m)�=(l′,m′)

δ(r − rlm) δ(r′ − rl′

m′ )

⟩, (40)

where the average is taken according to the probability distri-bution (38). In performing the average, we must consider twopossibilities in the summation over particle pairs:

� l = l′ (and then m �= m′). These terms take intoaccount spatial correlations among molecules insidethe same sphere, induced by their confinement. The


193.204.33.52 On: Wed, 24 Sep 2014 10:15:25


resulting contribution to ρ2g(r − r′) is

ρ M1

v2

∫dR θ (a − |r − R|) θ

(a − |r′ − R|) . (41)

The convolution integral is easily performed in Fourierspace by introducing the form factor

F (q) = 1

v

∫dR θ (a − r) eiq·R = j1(qa)

qa, (42)

where

j1(x) = 3sin x − x cos x

x3

is the 1st order spherical Bessel function of the firstkind.

� l �= l′. This term takes into account the correlations be-tween molecules belonging to different colloids. Theresulting contribution is

ρ2 1

v2

∫dRdR′ gp(R − R′) θ

×(a − |r − R|)θ (a − |r′ − R′|), (43)

where gp(r) is the distribution function of the colloidalparticles.

In summary, our final expressions for the correlation contri-bution to the refractive index (33) are still valid with the sub-stitution

ρ2 h(q) → M2 ρp F (q)2 [1 + ρp hp(q)]. (44)

We note two main differences with respect to the previous ex-pressions: (i) the presence of the form factor F(q)2 and (ii) theadditive contribution (the unity in the square bracket). The lat-ter takes care of the scattering from pairs of molecules insidethe same colloid, which in turns provides the second ordercontribution in the Mie scattering of each colloidal particle.23

The factorization in Eq. (44) of the intermolecular correla-tion function into a form factor F(q) and a structure factorSp(q) = 1 + ρphp(q) follows from the assumption (38) ofhomogeneous and uncorrelated molecular distribution insideeach colloidal particle. In the case of correlated distributionof molecules, no simplification occurs but the general expres-sion of the correlation integral (33) still holds, leading to theOptical Theorem (36) and (37). It is also important to no-tice that we have in this case an intrinsic structural lengthscale, given by the particle size a. It is then suitable to per-form the expansion in terms of the particle polarization perunit volume αp = Mα/v, a dimensionless quantity that playsthe same role as αd for point-like dipoles. Substituting (44)into the correlation integral (33), we find at 2nd order in αp,

n = 1 + 2πφ αp + 2π(π

3φ2 + Cφ

)α2

p, (45)

where we have defined a dimensionless complex correlationfactor C = Cr + iCi , with

Cr = v

4πk3

∫ ∞

0dq qF 2(q)[1 + ρph(q)]

×[

8

3k3q − kq3+

(2k4−k2q2+ q4

4

)ln

q+2k

|q−2k|]

,

(46)

Ci = v

4k3

∫ 2k

0dq qF 2(q)[1+ρphp(q)]

[2k4−k2q2+ q4

4

].

We stress again that Eq. (45) is valid, at second order in αp/v,for any value of φ.

For the real part n of the refractive index and the ex-tinction coefficient γ = 2kn′, which are the experimentallyobserved quantities, Eq. (45) yields⎧⎨⎩n = 1 + 2πφ αp + 2π

(π

3φ2 + Crφ

)α2

p,

γ = 4πkφCi α2p.

(47)

For an easier comparison with the experimental data, andto check for consistency in the absence of interparticle inter-actions with the continuum Mie theory, it is useful to intro-duce the index of refraction np of the material constituting thecolloidal particle. Expanding the CM equation inside the par-ticle at second order in the refractive index contrast �np = np− 1, the particle polarizability per unit volume is easily foundto be given by

αp = 1

4π

[2�np − (�np)2

3

]. (48)

Retaining for consistency only terms to order (�np)2, Eq. (47)becomes⎧⎪⎪⎨⎪⎪⎩

n = 1 + φ �np +[φ − 1

3+ Cr

π

]φ

2(�np)2,

γ = kφ

πCi (�np)2 = 2φ

λCi (�np)2.

(49)

Notice that for Cr = 0 the refractive index is given by

n = 1 +(

1 − �np

6

)�npφ + O(φ2), (50)

which therefore differs at first order in φ, even in the absenceof both intra- and inter-particle correlations, from the simpleexpression n = 1 + �npφ, obtained by volume-averaging therefractive indices of particle and solvent (which is converselycorrect for polarizabilities).

For what follows, it is also useful to introduce, as cus-tomary in light scattering theory, an “efficiency factor” Qext,defined as the ratio of σ ext to the total geometric cross-sectionNπa2 of the particles. Taking into account the definition of γ

in (5), we have

Qext = σext

Nπa2= 4a

3φγ, (51)

so that, from the second of (49),

Qext = σext

Nπa2= 4ka

3πCi(�np)2. (52)


193.204.33.52 On: Wed, 24 Sep 2014 10:15:25


A. Non-interacting particle limit and comparisonwith Mie theory

We first examine the limit, denoted by the superscript“0”, in which inter-particle correlations can be neglected, thatis obtained by setting h(q) ≡ 0 in Eq. (46),⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

C 0r = 1

3x3

∫ ∞

0dy yF 2(y)

[8

3x3y − xy3

+(

2x4 − x2y2 + y4

4

)ln

y + 2x

|y − 2x|]

,

C 0i = π

3x3

∫ 2x

0dy yF 2(y)

[2x4 − x2y2 + y4

4

],

(53)

where x = ka and y = qa. This single-particle approximationwill be compared to the Mie solution, expanded at 2nd orderin �np. It is worth considering the cases of particles muchsmaller or much larger than the wavelength separately.

a. Small particles (x � 1). In the limit x → 0, the realand imaginary parts of the correlation factor in (53) are easilyfound to be ⎧⎪⎪⎨⎪⎪⎩

C 0r −−−→

x→0

88π

75x2,

C 0i −−−→

x→0

8π

9x3.

(54)

Substituting in Eqs. (49) and (52), we find the limitingbehaviour⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

n0 −−−→x→0

1 + φ �np − (�np)2

6φ(1 − φ) + 44

75(�np)2φ x2,

γ 0 −−−→x→0

8

9kx3φ (�np)2,

Q 0ext −−−→

x→0

32

27x4 (�np)2,

(55)

where in the first equation we have also retained the lowest-order dependence on x, for later convenience. Reassuringly,Q 0

ext coincides with the efficiency factor for Rayleigh scatter-ers, namely, for particles much smaller than the wavelength.15

It is also very interesting to notice that, using Eq. (4) the realpart of the scattering amplitude can be written

Re s0(0) = 8

27(�np)2x6 = (2/3) (vαp)2 k6, (56)

which, comparing with Eq. (27), is identical to the radiationreaction contribution from a single, point-like dipole of polar-izability vαp. This result is equivalent to the brilliant conclu-sion reached by Lorentz: the radiation reaction from a spher-ical radiator with fixed polarizability does not depend on itssize, provided that the latter is much smaller than the wave-length. It also clarifies, however, a subtle feature of the gen-eral results obtained for colloidal fluids. In deriving Eq. (45),we have actually disregarded the radiation reaction term ofeach polarizable molecule because, due to the presence of thedimensionless factor αk3 � 1 in Eq. (14), this gives a negligi-ble contribution to the scattering of the whole colloidal parti-cle. Surprisingly, therefore, while the extinction from a distri-bution of uncorrelated point-like dipoles is solely due to radi-ation reaction, when the same dipoles “cluster” into uniform

spherical particles this contribution becomes vanishinglysmall. The microscopic approach we followed shows that itis again multiple scattering (in the generalized sense stated inSec. I) that, due to internal correlations, generates a “collec-tive” radiation reaction effect, leading to finite extinction.

b. Large particles (x � 1). In the opposite case x → ∞,the real and imaginary parts of the correlation factor in Eq.(53) can be readily evaluated at leading order in x in terms ofsimple integrals of j1(y) with the result⎧⎪⎪⎨⎪⎪⎩

C 0r −−−→

x→∞7π

3,

C 0i −−−→

x→∞3π

2x

(57)

which, using Eq. (49), yields⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

n0 −−−→x→∞

1 + φ�np + φ

(1 + φ

6

)(�np)2,

γ 0 −−−→x→∞

3φ

2ax2(�np)2 = 2πρa4k2(�np)2,

Q 0ext −−−→

x→∞2x2(�np)2.

(58)

The expression for γ 0 and Q 0ext in (58) are however quite sus-

picious: in fact, they highlight a severe limit in the quadraticexpansion we use. Indeed, from (58) Q 0

ext grows without lim-its with x, whereas in Mie theory Q 0

ext −−−→x→∞

2, whatever the

value (even complex) of the particle refractive index.24 Ac-tually, the efficiency factor in (58) coincides with the valueobtained in the Rayleigh–Gans (RG) approximation of the ex-act Mie solution, which requires both �np and the maximumphase delay δ = 2x�np that the incident field undergoes intraversing the particle to be small.15 The second condition, inparticular, is equivalent to assume that the incident radiationon each volume element of the particle coincides with the ex-ternal field. In our description, this means that, for δ � 1, in-ternal multiple scattering contributions are negligible, so thatintra-particle correlations are only related to the geometri-cal arrangement of the elementary scatterers expressed by theform factor. Moreover, since for large particles Q 0

ext � δ2/2,the RGD condition is met only when Q 0

ext � 1, namely, whenthe extinction cross-section is substantially smaller than thegeometrical “shadow” of the particle.

As a matter of fact, in the double limit x → ∞, �np → 0,made by keeping δ finite, known in the light scattering jargonas the “anomalous diffraction” limit, it is possible to find anexact solution for s(0), given in our notation by (see Sec. 11.22in van de Hulst15),

s(0) = x2

(1

2+ i

eiδ

δ+ 1 − eiδ

δ2

), (59)

which yields, for the efficiency factor

Q 0ext (δ) = 4

x2Re s(0) = 2 − 4

δsin(δ) + 4

δ2(1 − cos δ),

(60)whereas Q 0

ext (δ) −−−→δ→0

δ2/2, for δ � 1 the scattering cross sec-

tion per particle πa2Q 0ext correctly converges to twice the ge-

ometrical shadow. This finite limiting value, which does not


193.204.33.52 On: Wed, 24 Sep 2014 10:15:25


FIG. 1. Comparison between of the efficiency factor Q0ext obtained from Eq.

(60) (full line), and the 2nd order approximation in �np from Eq. (52), for�np = 0.05 (open dots) and �np = 0.2 (full dots). The broken line shows

the Rayleigh–Gans approximation Q0ext = δ2/2. The region with δ ≤ 1 is

expanded in the inset.

depend on �np25 and corresponds to the limit of diffraction

optics, can be recovered only by resumming all orders in �np,however small they are, and is therefore missing in our anal-ysis. Technically, this is due to the fact that the Mie solution,expressed as a series depending on the two parameters np andx, is not absolutely convergent, therefore, exchanging the lim-its x → ∞ and np → 1 is therefore not permitted.

The efficiency factor obtained from Eq. (60), whose com-plex oscillating behavior can be regarded as the effect ofthe interference between the transmitted and the diffractedfields, is contrasted in Fig. 1 with the full numerical so-lution of Eq. (52) in the absence of inter-particle correla-tions, Q 0

ext = (2�np/3π ) C 0i δ. The plot shows that, for �np

= 0.05, the latter is very close to the RG limit given by Eq.(58). As shown in the inset, the range of validity of our 2ndorder approximation shows extends up to values of δ � 1 or,equivalently, for values of the efficiency factor Q 0

ext � 0.5.Notice, however, that for the larger value �np = 0.2, includedfor later convenience, differences are more marked.

Luckily, the evaluation of dispersion effects does not ar-guably suffer from this limitation. Indeed, the expression re-fractive index in (58) does not depend on the particle size,and should give the correct limiting behavior for x → ∞(at 2nd order in �np). This is also suggested from the limit-ing behavior of the refractive index obtained from (59) usingEq. (4),

n = 1 + φ�np − (2/5) x2(�np)3 + O(�n5p),

which does not contain terms in (�np)2, and depends on par-ticle size only at order (�np)3 and higher. Trusting this ansatz,in what follows we shall mainly focus on the effect of inter-particle interactions on the refractive index of the suspension,limiting the discussion of extinction properties to dispersionsof particles with a size a � λ/(4π�np).

B. Refractive index of interacting colloids:An exact limit

For small particles, including interparticle interactionsdoes not substantially modify the behavior of the refractiveindex given by Eq. (55), since the real part of the correlationfactor is still found to be proportional to x2. Yet, Cr specifi-cally depends on the nature of interparticle forces: the case ofhard-sphere suspensions will be discussed in Sec. IV E.

Remarkably, however, in the opposite limit of ka → ∞the real part of the correlation integral can be analyticallyevaluated at any particle volume fraction. In fact, in this limitEq. (46) becomes

Cr = 7v

6π

∫ ∞

0dq q2F (q)2[1 + ρph(q)] (61)

which, by use of the convolution theorem can be written as

Cr = 7π

3v

∫dr

∫dr′θ (r−a)θ (r ′−a)[δ(r−r′)+ρp h(r−r′)],

(62)where use has been made of the definition of the form factorF(q) in (42). The domain limitation induced by the presenceof the θ function implies that |r − r′| < 2a and thereforeh(r − r′) = −1 in the whole integration domain for colloidsprovided of a hard core contribution. This immediatelyyields

Cr = 7π

3(1 − φ) = (1 − φ)C 0

r . (63)

When this result is substituted in the general expression(49) for the real part of the refractive index, we obtain theexact limit of n for ka → ∞ to second order in the particlepolarizability,

n = 1 + φ�np + φ(1 − φ)(�np)2. (64)

Note that this asymptotic result is valid for any specific formof the interparticle interactions, provided the latter contain ahard core contribution.

C. Inclusion of the solvent

The former results have been obtained for particles sus-pended in a vacuum. Nevertheless, once the refractive indexhas been expressed in terms of continuum electrodynamicsquantities such as np, inclusion of the effects of a solvent, act-ing as a homogeneous, non-absorbing medium of refractiveindex ns, is straightforward. Equations (55) and (64) retainindeed their validity provided that we simply make the sub-stitutions n → n/ns, np → np/ns. Besides, in the presence ofthe solvent the incident wave-vector should be written as k =2πns/λ. Putting �nps = np − ns, the general expression forthe complex refractive index in Eq. (49) becomes

n

ns

= 1 + φ�nps

ns

+ φ

2

[φ − 1

3+ C

π

] (�nps

ns

)2

. (65)

Note that Eq. (65) is a 2nd order expansion in �nps/ns, whichdoes not require np − 1 � 1 and ns − 1 � 1 separately. In the


193.204.33.52 On: Wed, 24 Sep 2014 10:15:25


limits ka = 0 and ka → ∞, we have therefore

n = ns + φ�nps − φ(1 − φ)

6ns

(�nps)2(ka = 0), (66)

n = ns + φ�nps + φ(1 − φ)

ns

(�nps)2(ka → ∞). (67)

A note of caution is however appropriate, since the contin-uum electrodynamics approach fully neglects fluctuations. Itis then worth wondering whether this simple way to accountfor the presence of the solvent holds true also in the presenceof correlations, by considering again the problem in a micro-scopic perspective. This is done in the Appendix, where weexplicitly show that Eq. (66) is rigorously true only providedthat the solvent is regarded as a uniform, uncorrelated dielec-tric medium.

D. Effective medium approach

A colloidal suspension of particles at volume fractionφ in a solvent at volume fraction 1 − φ is actually a com-posite medium. It is then useful to try and frame our resultswithin the problem of “homogenization” of a heterogeneousmedium, which basically consists in mapping the latter intoa homogeneous structure by defining “effective,” global ma-terial properties.26 For what follows it is useful to point outthat most of the approaches has addressed the case wherethese material properties are response functions to an exter-nal field which is uniform, or slowly-varying over the lengthscales that characterize the microscopic structure of the het-erogeneous medium. This is the case of the static dielectricconstant, but also of several other physical quantities such asthe thermal and low-frequency electric conductivities, or evenof mechanical quantities such as the elastic stress tensor.

In the case of a very dilute suspension of spherical parti-cles in a solvent, the problem is conceptually analogous to thediscussion of a system of uncorrelated point dipoles made inSec. II, provided that each particle is attributed a polarizabil-ity per unit volume αp = (εp − εs)/(εp + 2εs). It is thereforenot surprising that Maxwell, who first explicitly tackled thisproblem,27 obtained a result that can be written, for the caseof the effective dielectric constant ε∗ we are discussing

ε∗

εs

= 1 + 2βφ

1 − βφ, (68)

where β = (εp − εs)/(εp + 2εs), which is strictly related tothe CM equation.26 As Maxwell already pointed out, how-ever, Eq. (68) is valid only at first order in φ, hence it shouldconsistently be written

ε∗

εs

= 1 + 3εp − εs

εp + 2εs

φ + o(φ). (69)

A straightforward way to prove (69) consists in noticing that,from the definition of the effective dielectric constant and in-dicating with E0 an external uniform field, we must have28

(ε∗ − εs)E0 = 1

V

∫V

dr [D(r) − εsE(r)]

= ρp

∫v

dr (εp − εs) E(r),

where E(r) and D(r) are the local, fluctuating electric and dis-placement fields, V is sample volume, and the last equality isbecause the averaged quantity differs from zero only withinparticle volume v. Then, if we assume that the field incidenton particles coincides with the external field (namely, if weneglect the additional contributions due to the other particles),the field inside a dielectric sphere is also uniform, and givenby E(r) ≡ [3εs/(εp + 2εs)]E0, wherefrom Eq. (69) immedi-ately follows.

In the presence of correlations, expressions which arevalid to higher order in φ can be found only for specific ge-ometries, although rigorous upper and lower limits for ε∗,such as the Hashin-Shtrikman bonds, can be given.26 A veryinteresting situation is however that of a “weakly inhomoge-neous” medium, which for the present purposes we identifywith a suspension of colloidal particles made of a materialwith dielectric constant εp, which does not differ too muchfrom the dielectric constant εs of the suspending medium.Denoting by ε = φεp + (1 − φ)εs the volume average of thedielectric constants (which is the expression at lowest orderin �εps = εp − εs for the dielectric constant of the mixture),

and by (δε)2 = ε2 − (ε )2 = φ(1 − φ)(�εps)2 its mean square

fluctuation, one finds, at second order in �εps,26, 28, 29

ε∗ = ε − (δε)2

3ε= ε − φ(1 − φ)

3ε(�εps)

2. (70)

Notice that this expression, originally derived by Braun29 us-ing an approach closely resembling the one we used in Sec. II,is valid whatever the spatial correlations of the particles and,in particular, for any value of the particle volume fraction φ.Remarkably, Eq. (70) also coincides with the 2nd order ex-pansion in �εps of Eq. (68), a result which has however beenderived in the uncorrelated, single-particle limit φ → 0. Thismeans that, at this order of approximation in �εp, the staticdielectric constant is not affected by correlations.30

As we anticipated, however, Eq. (70) requires the appliedelectric field to be slowly-varying on the microscopic struc-tural length scales of the suspension (the particle size, or ingeneral the correlation length for interacting particles): it isthen very useful to investigate whether Eq. (70) still holds atoptical frequencies, namely, for the refractive index n = √

ε.This is readily found to be true in the limit ka → 0, where,according to Eq. (54), C 0

r vanishes as (ka)2: it is indeed easyto show that Eq. (70), written in terms of the refractive in-dices np = √

εp, ns = √εs , and expanded at second order in

�nps, coincides with Eq. (66). Hence, at 2nd order in the po-larizability difference, the refractive index of a suspensionof particles small compared to the wavelength satisfies theLorentz-Lorenz equation at any volume fraction. Accordingto our results, this does not hold true for finite values of ka,where system-specific effects of the intra- and inter-particlecorrelations should be expected. Remarkably, however, a dis-tinct limiting behavior, which is still independent from thenature and strength of particle interactions (provided that thelatter have a hard-core contribution) and valid for any vol-ume fraction, is reached at large ka. Notice in particular thatnot only the amplitude, but also the sign of the quadraticcorrection in Eq. (67) differs from the CM expression.


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FIG. 2. Inset A: Single-particle (Mie) limit of the real part C 0r of the corre-

lation factor of hard spheres for several values of x = ka. (Body) Full cor-relation contribution Cr, scaled to C 0

r and plotted as a function of φ for thesame values of x. The full and dotted lines, respectively, show the limitingbehavior for x → ∞ and x → 0. Inset B: Slope of Cr versus x2 in the limit x→ 0, plotted on a semi-log scale and fitted with a single exponential.

Equation (67) is then a very general result for the effectiverefractive index of a weakly inhomogeneous 2-componentsmedium that, at variance with Eq. (66), applies when the fieldvaries on much shorter spatial scales than the microscopiccorrelation length of the system. The fact, that is, does notdepend on the structural organization of the medium, but onlyon the volume fractions of the two components, suggests, thatis, should also be obtained from phenomenological but moregeneral arguments.

E. Correlation effects on the refractive indexfor hard spheres

For intermediate values of x, correlation effects on therefractive index become system-specific: it is particularlyinstructive to examine these effects for a fluid of monodis-perse hard spheres of radius a. Consider first the single-particle (Mie) limit discussed in Sec. IV A, where only intra-particle correlations are taken into account. The inset A inFig. 2 shows that, in agreement with Eqs. (54) and (57),the real part C 0

r of the correlation factor, which vanishes forx → 0 (the “Clausius–Mossotti” limit), progressively growswith x, asymptotically approaching the value C 0

r = 7π/3. Aswe already mentioned, even in the presence of inter-particleinteractions Cr retains, for small values of ka, a quadratic be-havior, Cr = cx2. For hard spheres, Inset B shows that, to agood degree of approximation, the slope c decreases expo-nentially up to φ � 0.4, starting from the value C 0

r = 88π/75given by Eq. (54). The fractional contribution Cr/C 0

r due tointer-particle correlation is conversely shown in the body ofFig. 2 as a function of the particle volume fraction, for severalvalues of x. Starting from the limiting behavior shown in insetB (dotted line), Cr/C 0

r is seen to rapidly approach, by in-creasing x, the asymptotic behavior Cr/C 0

r = 1 − φ given byEq. (63). For x � 5, as a matter of fact, Cr is a remarkably lin-ear function of particle volume fraction, showing that for largex only the Mie contribution and excluded volume effects are

FIG. 3. Panel (a) Excess correlation contribution n − n, where n = ns+ �n

psφ, to the real part of the refractive index for suspensions of

polystyrene particles (np = 1.59) in water (ns = 1.33), corresponding to thevalues of x in the legend. The full and dashed lines, respectively, correspondto the limits x → 0 in Eq. (66) and x → ∞ in Eq. (67). The dependence onvolume fraction of the dimensionless extinction coefficient, γ λ, and of thescattering efficiency scaled to the Mie value, Q

ext/Q0

ext , are shown in (b)and (c), respectively.

relevant. For 1.5 � x � 2, however, the trend of Cr/C 0r ver-

sus φ is rather peculiar: for instance, the curve for x = 1.6,which at low φ lies below the curve for x = 2.5 as expected,crosses the latter at φ � 0.3, reaching a consistently highervalue at the maximum packing fraction φ � 0.5 of the stablefluid phase.

These distinctive structural effects are better investigatedby considering a specific case, which also allows to inquirehow consistent and experimentally detectable are correlationcontributions to the refractive index. As a colloidal systemof practical relevance, we focus on suspensions of monodis-perse polystyrene (PS) latex particles (np � 1.59) in water(ns � 1.33). For this colloid, despite a substantial refrac-tive index mismatch between particle and solvent, the termswe neglect should not be larger than 20% of the 2nd orderterm in �nps/ns. Panel (a) in Fig. 3 shows that, for x = 0.5(a � 0.06λ), the difference n − n between the suspensionrefractive index and the value obtained by simply averagingover volume fractions, n = ns + �npsφ, is pretty close to thequadratic term in Eq. (66), whereas for x = 25 (a � 3λ) it al-ready approaches the limiting expression given by Eq. (67).Panel (a) also shows that for x = 2.5 (a � 0.3λ), a valuesufficiently large for Cr/C0

r to show a linear trend in φ (seeFig. 2), n − n is a quadratic function of φ, as expected fromEq. (65).

For what concerns extinction, the value x = 25 (δ � 10)is far too large to be discussed within our approximation.Fig. 1 shows however, that this is still reasonably feasiblefor x = 2.5 (δ � 1), which can then be compared to thebehavior for x = 0.5 (δ � 0.2), corresponding to particles


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which are much smaller than the wavelength. For particles ofthis size, the scattered intensity is basically independent fromthe scattering wave-vector q, and proportional to the productof the volume fraction times the osmotic compressibility ofthe suspension. Using the Carnahan-Starling equation of statefor hard spheres,31 both σ ext and γ should then be propor-tional to

φ

(∂�

∂φ

)−1

= φ + 2φ2(4 − φ)

(1 − φ)4. (71)

Panel (b) in Fig. 3, where this functional behavior is comparedto the dimensionless quantity γ λ, shows that this is indeed thecase, to a good degree of approximation. Notice in particularthat the extinction coefficient displays a strong maximum fora particle volume fraction which is very close to the knownvalue φ � 0.13 where the scattering from small hard-spherespeaks. A similar non-monotonic trend is observed for x = 2.5too, but the value where γ peaks is shifted to the consistentlyhigher value φ � 0.27. As we shall shortly investigate in moredetail, γ is eventually determined by the that part of the struc-ture factor that is detected within the experimentally accessi-ble q-range, and this strongly depends on particle size. Thestrong effect of interparticle interactions on extinction is bet-ter appreciated in Panel (c), where we plot the volume frac-tion dependence of the ratio of the scattering efficiency Qextobtained form Eq. (52) to its value Q 0

ext in the absence ofinter-particle correlations. For both values of x, Qext stronglydecreases with φ, reaching a value at φ = 0.5 that is aboutseven times smaller than Q 0

ext for x = 2.5, and as much as 50times smaller for x = 0.5.

To highlight distinctive correlation effects, it is particu-larly useful to investigate the behavior of the refractive in-dex for φ = 0.5, which is the limiting volume fraction ofa hard spheres fluid, as a function of the scaled particlesize x. In Panel (a) of Fig. 4, the difference �n betweenthe refractive index n and its volume-average approximationn = ns + �nps is contrasted to the single-particle Mie limit

FIG. 4. Panel (a) Refractive index increment �n = n − n for a suspensionof polystyrene particles at φ = 0.50 (fluid phase, dots) and at φ = 0.55 (FCCcolloidal crystal, triangles), compared to the values �n = n0 − n obtainedby neglecting inter-particle structural correlation (Mie limit, squares). Thescattering efficiency Qext is shown in the inset. Panel (b) Detailed behaviorof �n in the region 1 ≤ x ≤ 2.5, compared to the behavior of the structurefactor of the HS fluid, calculated using the Verlet–Weis approximation22 andevaluated at the maximum experimentally detectable wave-vector q = 2x/a= 4πns/λ (see text).

�n = n0 − n. For comparison, we also include the corre-sponding data for the face-centered-cubic crystal at φ = 0.55in equilibrium with the fluid phase. It is important to pointout that the latter assumes orientation symmetry, and there-fore corresponds to the data that would be obtained for arandomly-oriented polycrystalline sample. Several features ofthe plot are worth be pointed out. First, n − n is substantiallysmaller for both the fluid and colloidal crystal phase than inthe non-interacting approximation. In particular, the quadraticincrease of �np at small x, expected from the discussion of Crin Fig. 2, is more than one order of magnitude weaker than inthe Mie case. The effect of repulsive HS interactions is evenmore dramatic on extinction: the inset in Panel (b) shows in-deed that, in the fluid phase (extinction in the crystal vanished,since it is assumed as ideal), the scattering efficiency is ex-tremely small for x � 1, rising for x � 2 to values which arestill several times smaller than in the Mie theory. A very pe-culiar feature is finally the presence of a strong peak for thecrystal phase, and of a less pronounced “bump” for the fluidphase, occurring in the region 1.5 � x � 2 where we alreadypointed out a peculiar behavior of Cr (see Fig. 2). The originof this rather surprising effect can be grasped by consider-ing Panel (b). There, together with an expanded view of the“bump” region, we plot the structure factor S(q) for a HS fluidat φ = 0.50, calculated at the wave-vector q = 2k = 4πns/λcorresponding to the backscattering condition θ = π . In otherwords, for a given experimental value x∗, fixed by both theparticle size and the incident wavelength, only those wave-vectors q of the structure factor with q ≤ 2x∗/a fall within thedetectable range and contribute to the scattering cross section.From the plot, we clearly see that the refractive index increaseis associated with (and actually slightly anticipates) the pro-gressive “entrance” of the first peak of S(q) in the detectablerange. A further interesting observation comes from noticingthat, for a fixed particle size, the curve is basically a plot of therefractive index versus the inverse of the incident wavelength.Then, the trailing part of the peak which follows the maxi-mum, shown with open dots in Panel (b), corresponds to aregion where the refractive index increases with increasing λ,which is the hallmark of an anomalous dispersion region.32

As a matter of fact, the overall trend strongly resembles thebehavior of the refractive index n(ω) of a Lorentz oscillatorclose to its natural resonance frequency ω0: n already showsan increase for ω < ω0, followed by an anomalous disper-sion region where n(ω) is a decreasing function of ω, and bya final recovery. No true absorption is however present in theproblem we are considering. This finding seems to suggestthat, besides in resonant absorption, anomalous dispersionmay take place in the presence of any process in the medium,such as scattering, that lead to extinction of the incident field.

F. Feasibility of the experimental determinationof correlation effects

It is useful to inquire whether and how correlation effectson the refractive index can be experimentally investigated. InPanel (a) of Fig. 3 the correlation contribution to n reaches,for x = 2.5 and φ � 0.3, a value of about 4 × 10−3, which


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is well within the accuracy of a good refractometer. However,Panel (b) shows that in these conditions extinction is quitelarge, giving an extinction length γ −1 � 10λ � 30a: investi-gations should then be performed using method exploiting alow penetration depth (see below). Given the low extinction,correlation effects are arguably much easier to be detected inthe very large φ limit discussed in Fig. 4. For instance, for x= 1, where in fluid phase Qext � 10−3, corresponding to anextinction length still as large as about 300λ, the refractiveindexes of both the fluid and the crystal phase already dif-fer from the Mie prediction by about 10−2, and of 6 × 10−4

between themselves.Unfortunately, accurate data on the refractive index of

dense colloids are scarce, and not very recent.33, 34 More-over, as common in light scattering practice, experimentsmostly focused on measuring the refractive index incrementdn/dφ (or, more usually dn/dc, where c is the concentration inmass/volume), which amounts to implicitly assume that n islinear in φ. However, at first order in φ, the correlation fac-tor Cr reduces to the Mie limit C 0

r , and inter-particle correla-tions show up only at order φ2. It is however worth mention-ing a rather surprising result obtained long ago by Okubo,35

which may be related to the results we discussed in Fig. 4. Bymeasuring the refractive index of strongly deionized chargedPS suspensions, Okubo observed indeed a substantial peak inthe refractive index, located very close the transition betweenthe fluid and the colloidal crystal phase. Unfortunately, thisearly investigation has not been further pursued, at least toour knowledge.

Modern approaches based on fiber optic sensing,36 ormade in a total internal reflection configuration,37 should pro-vide a sufficient accuracy to detect correlation effects evenfor strongly turbid samples. Yet, when using methods relyingon so small penetration depths, care should be taken to avoidprobing correlation effects on the particle distribution at theinterface between the solution and the sensor wall, rather thanbulk structural properties. An interesting alternative wouldbe using a novel optical correlation method recently intro-duced by Potenza et al.,38 which consists in measuring thetwo-dimensional power spectrum P(qx, qy) of the transmittedbeam intensity distribution on a plane placed at close distancez from the sample. By means of the optical theorem, one findsfor N identical scatterers

P (qx, qy) = 4π2

k2|Ns(0)|2 sin2

(q2z

2k− ϕ

), (72)

where q2 = q2x + q2

y and (in our notation) ϕ = arg[s(0)].Equation (72) describes a fringe-like pattern characterized bya phase shift ϕ which is directly related to the ratio betweenthe imaginary and the real part of the scattering amplitude.One of the major advantages of the method is that it can beapplied to very turbid samples too, since multiple scatteringyields only a constant background that can be easily sub-tracted out. The technique has successfully been applied toinvestigate dilute colloidal suspension.38 At sufficiently highφ, however, inter-particle correlation effects should yield no-ticeable deviations with respect to the Mie expression used toevaluate the fringe pattern.

V. CONCLUSIONS

In this article, we have presented a general microscopictheory for the attenuation and the phase delay suffered by anoptical plane wave that crosses a system of interacting col-loidal particles, deriving an expression for the forward scat-tered wave, exact at second order in the molecular polarizabil-ity, which explicitly takes into account the interactions amongall induced dipoles. Whereas previously available treatmentshave separately discussed either attenuation (neglecting cor-rections due to radiation reaction) or refractive index (usingsome variant of the Lorentz-Lorenz formula and ignoring in-terparticle correlations), our approach treats on an equal ba-sis the real and the imaginary part of the refractive index. Indetail:

� We have investigated the role of radiation reaction onlight extinction, showing that the structural features ofthe suspension are encoded into the forward scatteredfield by multiple scattering effects, whose contributionis essential for the so-called “optical theorem” to holdin the presence of interparticle interactions. The localfield acting on a specific dipole is the sum of the exter-nal field plus all the fields due to the presence of all theother oscillating dipoles within the scattering volume.Our treatment considers the average local field, whichis polarized as the external field, while the fluctuationsof the local field, not discussed here, give rise to whatis usually called multiple scattering.

� In the case of negligible interparticle interactions, ourresults are found to be consistent, at second order in thepolarizability, with the exact Mie theory for sphericalparticles.

� We have discussed our results in the framework of ef-fective medium theories, presenting a general result forthe effective refractive index valid, whatever the struc-tural properties of the suspension, in the limit of a par-ticle size much larger than the wavelength.

� In the case of correlated particles we found that sig-nificant corrections to the value of the refractive indexexist when the x parameter is of the order of one, thatis, when the particle size is comparable to the wave-length of light.

� By treating concentrated hard-sphere suspensions, wehave unraveled subtle anomalous dispersion effects forthe suspension refractive index and we have discussedthe feasibility of an experimental test of our calcula-tions.

It is finally useful to point out that the general approachwe have followed can in principle be extended to investigateother interesting physical problems. Strong analogies exist forinstance between the scattering of (vector) electromagneticand (scalar) ultrasonic waves from a particle dispersion. Al-though in the case of ultrasonic scattering no analogous ofpoint-like dipoles exists, an approach formally identical tothe Mie scattering theory can be developed, once the wholecolloidal particle is assumed to be an elementary scatterer,responding to the incident acoustic field via its density andcompressibility difference with the solvent.39 An extension of


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the approach we developed for the refractive index might thenprovide an explicit expression for the dispersion of the soundspeed in a correlated suspension. It is however worth pointingout that, in acoustic scattering, absorption effects are usuallyfar from being negligible.

Similarly, finding the thermal conductivity of a suspen-sion within an effective medium approach is formally analo-gous to evaluate its dielectric constant in the long-wavelengthlimit. In Sec. IV D, we have shown that, for a weakly inhomo-geneous medium, the static dielectric constant is not affectedby correlation, and is always given by Eq. (70): the same re-sult should then hold for the thermal conductivity. However,the case of a dispersion of correlated particles with a thermalconductivity much higher than the base fluid (metal nanopar-ticles, for instance) could still be investigated by a suitableextension of the general equations (28) and (30), at least nu-merically. This may shed light on the highly debated prob-lem of the so-called “anomalous” enhancement of thermalconductivity in nanofluids.40

APPENDIX: REFRACTIVE INDEX OF BINARYMIXTURES AND SUSPENSIONS

For a homogeneous, uncorrelated mixture of point-likeparticles with polarizabilities α1, α2 and number densities ρ1,ρ2, a straightforward generalization of Eq. (16), applied to aslab geometry for an incident field of the form (10), yields, at2nd order in the polarizability,

nd = 1 + 2π (α1ρ1 + α2ρ2) + 2

3π2(α1ρ1 + α2ρ2)2, (A1)

where the superscript “d” is to remind that Eq. (A1), whichjust states that in the absence of correlation the polarizabil-ity per unit volume of the mixture is additive, applies onlyto point-like dipoles. Let us now identify species 2 withmolecules constituting the colloidal particle. When correla-tions are included, by generalizing the procedure discussed inSec. IV, we obtain a general expression of the complex re-fractive index of colloidal particles embedded in a correlatedfluid

n = 1 + 2π (α1 ρ1 + αp φ) + 2

3π2(α1 ρ1 + αp φ)2

+2π{α2

1 ρ21 C[h11(q)] + α2

p φvC[F 2(q)Spp(q)]

+2α1αp ρ1φ C[F (q)h1p(q)]}, (A2)

where we have defined a correlation functional C[f(q)] thatgeneralizes expression (30) to

C[f (q)] = 1

4π2k2

∫dq

[k4 + (k · q)2

q2 − k2 + iη− k2

3

]f (|k − q|).

(A3)In the special limit where we identify type-1 particles withthe molecules of the solvent, considered as an incompressiblecontinuum, the number density of the solvent ρs in the freevolume V (1 − φ) is related to ρ1 by

ρs = ρ1

1 − φ(A4)

and the correlation functions h11(q), and h1p(q) can be relatedto that of the colloidal particle hpp(q) ≡ h(q),

h11(q) = ρpv2

(1 − φ)2F 2(q)[1 + ρp h(q)], (A5)

h1p(q) = − v

1 − φF (q)[1 + ρp h(q)]. (A6)

When these expressions are inserted into Eq. (A2) we find

n = 1 + 2π [αsρs(1 − φ) + αp φ]

+2

3π2[αsρs(1 − φ) + αp φ]2. (A7)

It is easy to show that this form is in fact fully equivalent toEq. (65), when the latter is expanded to second order in ns− 1 and np − 1.

1A. Duri, D. A. Sessoms, V. Trappe, and L. Cipelletti, Phys. Rev. Lett. 102,085702 (2009).

2R. Cerbino and V. Trappe, Phys. Rev. Lett. 100, 188102 (2008).3S. Buzzaccaro, E. Secchi, and R. Piazza, Phys. Rev. Lett. 111, 048101(2013).

4R. Piazza, in Colloidal Foundations of Nanoscience, edited by D. Berti andG. Palazzo (Elsevier, Amsterdam, 2014), Chap. 10.

5T. Zhang and I. I. Yamaguchi, Opt. Lett. 23, 1221 (1998).6R. G. Newton, Am. J. Phys. 44, 639 (1976).7The factor k = 2π /λ introduced the denominator of Eq. (1) makes S(k

i, k

s)

dimensionless. By factoring out an imaginary unit, which is related tothe Gouy phase shift accumulated in far-field between a spherical anda plane wave, the amplitude function coincides in the short-wavelengthlimit (aside from polarization effect) with a standard normalized diffrac-tion pattern.15 Note that in terms of the vector scattering amplitudef = i(S/k)n

scommonly used in particle scattering, the optical theorem

reads σext

= (4π/k2)Im[ni· f(k

s= k

i)].

8J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1998).

9For optically anisotropic (birefringent) media, the directions of polarizationof the incident and scattered field in the forward direction do not coincide,namely, the scattered field contains a “depolarized” component that, be-ing perpendicular to n

icannot of course interfere with the incident beam.

Nevertheless, Eq. (2) states that the total scattering cross section (includ-ing that due to depolarized scattering) is still accounted for by the solepolarized component. The reason for this correct, but apparently paradoxi-cal result is discussed in a recent publication41 concerning light scatteringfrom anisotropic colloidal particles.

10L. Rosenfeld, Theory of Electrons (North-Holland, Boston, 1951).11I. L. Fabelinskii, Molecular Scattering of Light (Plenum Press, New York,

1968).12F. Hynne and R. K. Bullough, Philos. Trans. R. Soc. A 312, 251 (1984).13F. Hynne and R. K. Bullough, Philos. Trans. R. Soc. A 321, 305

(1987).14F. Hynne and R. K. Bullough, Philos. Trans. R. Soc. A 330, 253 (1990).15H. C. van de Hulst, Light Scattering by Small Particles (Plenum Press, New

York, 1968).16Because of the choice we made for the phase of the incident and scattered

field, the scattering amplitude we define is the complex conjugate of theone defined by van de Hulst.15

17J. V. Champion, G. H. Meeten, and M. Senior, J. Colloid Interface Sci. 72,471 (1979).

18M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, Ox-ford, 1980), pp. 100–104.

19H. A. Lorentz, The Theory of Electrons and Its Applications to the Phe-nomena, of Light and Radiant Heat (Forgotten Books, reprinted in Leipzigby Amazon Distribution, 2012).

20P. A. M. Dirac, Proc. R. Soc. Lond. Ser. A 167, 148 (1938).21Formally, ξ can be defined as ξ = ∫

drr2h(r)/∫

drrh(r), which foran exponentially-decaying correlation function coincides with the decaylength.


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22J. P. Hansen and I. R. McDonald, Theory of Simple Liquids: With Applica-tions to Soft Matter, 4th ed. (Academic Press, New York, 2013).

23Notice that, as a matter of fact, F(q) is the purely “geometrical” form factorobtained for the scattering from a uniform sphere in the Rayleigh–Gansapproximation by assuming that the incident field on each volume elementis the unperturbed external field.

24This is known as the “extinction paradox,” since a very large particle appar-ently “casts a shadow” which is the double of its geometrical cross section.Consistency with the “macroscopic” experience is recovered by noticingthat half of the scattered light is scattered in an extremely narrow diffrac-tion cone around the forward direction, which could be excluded only usinga detector with an acceptance angle ϑ � λ/2a.

25Of course, if �np is identically equal to 0, Q0ext must vanish. What (60)

actually means is that Q0ext (δ) is discontinuous: lim

δ→0Q0ext(δ) �= Q0

ext(0).26K. Z. Markov, Heterogeneous Media: Modelling and Simulation

(Birkauser, Boston, 1999), Chap. 1.27Maxwell actually discussed the problem of the electrical conductivity of

a matrix containing spherical inclusions. His results were later extendedto the optical properties of metal films by James Clerk Maxwell Garnett,who owes his rather curious name to the admiration of his father, WilliamGarnett, for his own friend and mentor J. C. Maxwell.

28L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media(Pergamon Press, New York, 1960).

29W. F. Brown Jr., J. Chem. Phys. 23, 1514 (1955).

30It is useful to notice that, at 2nd order in �εp, also the rigorous upper andlower limits for ε∗ given by the Hashin-Shtrikman bonds coincide.

31N. F. Carnahan and K. E. Starling, J. Chem. Phys. 51, 635 (1969).32From Panel A, anomalous dispersion is of course present, and even much

more pronounced, for the crystal phase. However, since we simply modelthe system as an FCC crystal at T = 0, neglecting therefore the “rattling”motion of the particles at finite T, we cannot make a realistic correlationwith the behavior of S(q). Notice however that, in spite of the fact thatin this idealized model the structure factor peaks are discontinuous deltafunctions, the peak in the refractive index is finite.

33G. H. Meeten and A. N. North, Meas. Sci. Technol. 2, 441 (1991).34M. Mohammadi, Adv. Colloid Interface Sci. 62, 17 (1995).35T. Okubo, J. Colloid Interface Sci. 135, 294 (1990).36A. Banerjee et al., Sens. Actuat. B 123, 594 (2007).37Y. E. Sarov et al., Nano Lett. 8, 375 (2008).38M. A. C. Potenza, K. P. V. Sabareesh, M. Carpineti, M. D. Alaimo, and M.

Giglio, Phys. Rev. Lett. 105, 193901 (2010).39P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New

York, 1968), Chap. 8.40J. Eapen, R. Rusconi, R. Piazza, and S. Yip, Trans. ASME J. Heat Transf.

132, 102402 (2010).41V. Degiorgio, M. A. C. Potenza, and M. Giglio, “Scattering from

anisotropic particles: A challenge for the optical theorem?” Eur. Phys. J.E 29, 379–382 (2009).


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