scattering properties of core-shell particles in plastic...

Scattering Properties of Core–Shell Particlesin Plastic Matrices

ALEX SMALL,1 SHENG HONG,2 DAVID PINE3

1Laboratory of Integrative and Medical Biophysics, National Institute of Child Health andHuman Development, National Institutes of Health, Bethesda, Maryland 20892

2Arkema Inc., 900 First Ave., King of Prussia, Pennsylvania 19406

3Department of Physics, New York University, New York, New York 10003

Received 15 July 2005; accepted 5 September 2005DOI: 10.1002/polb.20624Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: Blending submicron rubber particles with plastics can enhance themechanical strength of the composite material. However, the difference in refractiveindex between the particle and matrix scatters light, making the material more opa-que. We consider the possibility of reducing a particle’s scattering cross section byadding coatings. We find that adding coatings can reduce the amount of scattering bychanging the effective dielectric contrast between the particle and the matrix. Wealso found that, when the refractive index of the particle is very close to that of thematrix the order of the layers can have significant effects on the transmitted light.Such effects may be useful for engineering the optical properties of particle-dopedplastics. Resonant effects akin to those found in antireflection coatings on planar sur-faces are difficult to obtain and rarely provide a significant reduction in scattering.We discuss theoretical models that can qualitatively explain some of our results.VVC 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3534–3548, 2005

Keywords: blends; core–shell polymers; light scattering; optics; transparency

INTRODUCTION

Polymer scientists frequently find it useful toincorporate heterogeneous particles into a plas-tic matrix. Indeed, most commercial plasticstoday are inhomogeneous blends or alloys. Atypical example of such heterogeneous systemsis rubber-toughened plastics. The rubbery do-main enhances the mechanical strength of thecomposite material.1–6 However, incorporatingparticles into an otherwise homogeneous mate-rial causes light to be scattered rather than

transmitted, altering the appearance of the ma-terial. In many cases scattering is undesirable,as it tends to make the material more opaque,but in some cases scattering might be a tool thatone uses to tune the appearance of a material.

Most work on the optical properties of plasticcomposites has been guided by two observations:that smaller particles scatter more light thanlarger particles, and that decreasing the refrac-tive index contrast between the particle and thesurrounding matrix tends to reduce the amountof light scattered by a particle.7,8 These observa-tions, while valid under a wide range of condi-tions, are based on the assumption that the onlyadjustable variables are the size and refractiveindex of homogeneous particles. One could ask,however, if particles with more complex (i.e. lay-

Correspondence to: A. Small (E-mail: [email protected])

Journal of Polymer Science: Part B: Polymer Physics, Vol. 43, 3534–3548 (2005)VVC 2005 Wiley Periodicals, Inc.

3534

ered) morphologies might give us greater flexi-bility. It is known that a properly designed coat-ing on a planar surface can significantly reducethe amount of light reflected over a range ofvisible wavelengths and angles.9 Synthetically,particles with defined layer morphology are pos-sible by emulsion polymerization,10 suspensionpolymerization methods,11 or self-assembly ap-proaches.12,13 We will therefore explore the pos-sibility of producing antireflection coatings forsmall particles.

Most of the work done on scattering by coateddielectric spheres has been undertaken with thegoal of identifying and exploiting resonancesthat increase the scattering strength. Much ofthe current interest comes from colloidal chem-ists who are interested in self-assembly of co-lloidal crystals from particles that scatterstrongly.14 Other groups have studied fluores-cent particles made with multiple layers to takeadvantage of the different properties of the core,the fluorescent inner shell, and the surface.15

Astronomers have also taken an interest in core–shell particles, as the various layers of a particleof interstellar dust may yield information on theenvironments that the particle has encoun-tered.16

We are aware of only one study undertakenwith the goal of understanding how coatingsmight reduce scattering.17 That study focused onparticles with two layers (core and shell), a vol-ume-averaged dielectric constant close to that ofthe surrounding medium, and a diameter com-parable to or smaller than the wavelength oflight. We describe here a computational study orparticles with a wider range of sizes and refrac-tive indices, as well as the possibility of reduc-ing scattering by adding multiple shells to a coreparticle. We restrict our attention to materialsand particle sizes that are commonly encoun-tered when making composite plastics, and weuse our computational findings to inform gen-eral design rules.

We find that, in most cases, the generalguideline of minimizing the refractive index con-trast applies to coated particles as well as homo-geneous particles. Coatings that minimize thedifference between the matrix and the volume-averaged dielectric constant of the particle tendto produce the least scattering. Resonant effectsakin to those found in antireflection coatings onplanar surfaces are rare and usually weak whenpresent. We also find that a single coating isusually more effective at reducing scattering

than adding multiple coatings. The principalexception is when the volume-averaged dielectricconstant of a particle approximately matches thatof the matrix, in which case the scattering prop-erties can be sensitive to morphology (e.g., theorder of the layers). We will explain some ofthese findings in terms of analytical theories thathelp inform design rules.

Theory

Useful Parameters

For the purpose of this work, the three mostimportant characteristics of a particle are thescattering cross section (denoted r), asymmetryparameter (denoted g), and the volume-averageof the dielectric constant (denoted �e� eb).

18

The cross section r is proportional to the totalamount of light scattered by a particle whenilluminated by a plane wave, and it has units ofarea. Suppose that we illuminate a particle witha plane wave of intensity I0 (which has units ofenergy/area/time), and with an array of detec-tors we measure the power scattered in eachdirection. By integrating the measured over alldirections, we can obtain the total scatteredpower W (which has units of energy/time). Wecan then define the scattering cross section r as:

r � W=I0 ð1Þ

It is clear that r has units of area and is propor-tional to the total amount of light scattered by aparticle.

We will frequently find it convenient to dis-cuss the scattering efficiency Qscatt, which is justr normalized to the particle’s geometrical crosssection:

Qscatt � rpr2

ð2Þ

where r is the outer radius of the particle. Qscatt

has the advantage of being a dimensionlessnumber that we can use to compare particles ofdifferent sizes. It is important to note that Q fre-quently exceeds 1, and for very large particles,it tends toward 2. The reason for this is that, inaddition to scattering light incident on their geo-metrical cross section, particles also diffractlight at their edges, so a particle can effectively‘‘look’’ larger than its geometrical cross section.

The second parameter of interest, known asthe asymmetry parameter and denoted g, is a

SCATTERING PROPERTIES OF CORE–SHELL PARTICLES 3535

measure of the angular distribution of scatteredradiation (when the particle is illuminated by aplane wave), and is defined as:

g � hcos hi ð3Þ

where h is the scattering angle and the averageis taken over all directions, weighting accordingto the scattered intensity in each direction.

The last parameter, the average dielectriccontrast, is a measure of the difference betweenthe particle and the surrounding medium. Thedielectric constant of a medium is the square ofthe refractive index n and is denoted e. When itis spatially inhomogeneous it is often called thedielectric function rather than dielectric con-stant and denoted as e(r). The volume averageof e(r) is frequently denoted �e, and the averagedielectric contrast is just �e� eb, where eb is thedielectric constant of the surrounding back-ground medium. Finally, we will sometimesmake reference to the volume-averaged refrac-tive index of a particle, which we define as

ffiffi�e

p.

Light in a System of Particles

Once we know the scattering parameters g andr for a given particle morphology, we can askhow light will behave in a system of such par-ticles, and what the resulting appearance willbe. Qualitatively, if most photons entering thesystem are either unscattered or only scatteredby a single particle before exiting then the sys-tem will look transparent, with a slight sheendetermined by the wavelength dependence of r.

Quantitatively, most light will be eitherunscattered or scattered only once if the sampledimensions are comparable to or larger than themean free path ‘ : (rq)�1, where q is the num-ber density of particles per unit volume. Themean free path is the average distance that aphoton will travel before being scattered by aparticle. Once a sample’s thickness is largerthan ‘, photons will in general be scattered mul-tiple times before exiting the sample. Multiplyscattered photons can, under a very wide rangeof circumstances, be described as undergoing arandom walk.19 Photons executing a randomwalk exhibit many of the same properties asatoms and molecules executing random walks,and many of the same results from diffusiontheory apply.

The most important parameter to describe arandom walk is not the average distance be-

tween scattering events ‘. Rather, it is the aver-age distance that a photon travels before itsdirection of propagation is randomized, which isgiven the name transport mean free path anddenoted with the symbol ‘*. If the particles scat-ter isotropically then that distance will be equalto ‘, but if scattering is biased in the forwarddirection, then a photon may have to undergoseveral scattering events (and, on average, travela distance ‘* > ‘) before its direction of propaga-tion is randomized. Clearly the value of ‘* willdepend on the angular distribution of scatteredradiation, and it turns out that the relationshipbetween ‘* and ‘ is

‘� ¼ ‘=ð1� gÞ ð4Þ

Qualitatively, the relationship between ‘* and ‘is similar to the relationship between the bondlength and persistence length in polymers.

Materials tend to look white when the samplethickness is larger than ‘*. The fraction of inci-dent light transmitted through a sample ofthickness L � ‘� is proportional to ‘*/L. Poly-mers that have a natural color due to absorptionof light also tend to assume a less saturated(less colorful) appearance when doped with scat-tering particles. For both of these reasons, thenatural question to ask will be how can we mod-ify the properties of polymer particles to maxi-mize ‘* and hence minimize r(1 � g).

Theory of Scattering from LayeredSpherical Particles

The parameters r and g can be calculated bysolving Maxwell’s equations for a plane wave in-cident on a layered spherical particle. The exactanalytical solution for scattering from a homoge-neous particle was originally worked out byMie,20 and is outlined in a number of goodbooks.18,21 The theory of scattering from a lay-ered particle was first worked out by Aden andKerker22 for a particle with a single coating, andvarious algorithms have been developed sincethen to describe scattering from particles with anarbitrary number of layers. We use the algorithmof Wu and Wang23 to perform the calculations.The algorithm is described in detail in theirpaper; what we present here is an outline of theMie theory in a form that will be useful for mak-ing analogies with waves in planar dielectric coat-ings.

3536 SMALL, HONG, AND PINE

We consider a spherical particle with N layers,with layer 1 being the core and layer N the out-ermost shell. (See Fig. 1) Each layer has radius rjand refractive index nj (and hence dielectric con-stant ej ¼ nj

2). The particle is embedded in abackground matrix of refractive index nb (anddielectric constant eb ¼ nb

2). Because of sphericalsymmetry, the incident, scattered, and internalfields can be expanded as a superposition of vec-tor spherical harmonics, special functions similarto the spherical harmonics used to solve theSchrodinger equation for the hydrogen atom.

We have to solve a boundary value problem todetermine the expansion coefficients of the vec-tor spherical harmonics. Using notation similarto that of Smith and Fuller,24 we use the symbolNmp

(3) (k0nr,h,/) for waves radiating outward fromthe origin and Nmp

(3)* (k0nr,h,/) for waves converg-ing inward toward the origin, where the super-script * denotes complex conjugation, k0 is 2pdivided by the wavelength of light in vacuum, nis the refractive index of the medium that thewave is traveling in, and r, h, and / are theusual spherical coordinates. The subscript prefers to the polarization of the wave and is 1for transverse magnetic waves and 2 for trans-verse electric waves. The subscript m refers tothe order of the Hankel function of the first kind(a type of spherical Bessel function) governing

the radial dependence of the vector sphericalharmonic.

At large distances from the origin the Hankelfunctions are proportional to expð6ik0nrÞ

k0nr, where

the sign of the exponent depends on whetherthe wave radiates outward from the origin (þ)or converges inward toward the origin (�).When the thickness of a layer is much smallerthan the distance r from the origin we can ap-proximate the radial dependence of the field inthat layer with a sinusoidal function, a fact thatwe will use later in our discussions of the proper-ties of particles large compared with the wave-length of light.

The plane wave incident on the particle canbe written in the following form:

Einc ¼ ex expðik0nbzÞ

¼ 1

2

X2p¼1

X1m¼1

qmp Nð3Þmpðk0nbr; h;/Þ

�

þNð3Þ�mp ðk0nbr; h;/Þ

�ð5Þ

where qmp ¼ �[inþp (2n þ 1)/n(n þ 1)].24 Wehave written this field in terms of incoming andoutgoing spherical waves for convenience whenmaking analogies with waves in planar dielectriccoatings. The scattered field can be written as

Escatt ¼X2p¼1

X1m¼1

qmpampNð3Þmpðk0nbr; h;/Þ ð6Þ

where amp is proportional to a scattering ampli-tude. (The factor of qmp has been inserted for con-venience when matching boundary conditions.)Calculating the set of coefficients {amp} is the prin-ciple task when studying scattering by a particle,as these coefficients completely determine thescattered field. Once the scattered field is knownit is straightforward to calculate the parameters rand g.

The field in the jth layer of the particle canbe written in the following form:

EðjÞint ¼

X2p¼1

X1m¼1

qmp uðjÞmpN

ð3Þmpðk0njr; h;/Þ

�

þmðjÞmpNð3Þ�mp ðk0njr; h;/Þ

�; 2 � j � N ð7Þ

Figure 1. Diagram of layered particle with varia-bles rj, nj, and nb illustrated.


Eð1Þint ¼

X2p¼1

X1m¼1

qmpuð1Þmp Nð3Þ

mpðk0n1r; h;/Þ�

þNð3Þ�mp ðk0n1r; h;/Þ

�ð8Þ

where ump(j) and vmp

(j) are coefficients of the out-going and incoming fields, and once again thefactor of qmp has been included for ease whenmatching boundary conditions. Finally, the mag-netic field H in any layer can be obtained usingFaraday’s law: H ¼ �i

k0r�E.

In eq 8 we impose our first boundary condi-tion. At the core of the particle ump

(j) ¼ vmp(j). An

incoming spherical wave front that convergestoward the center of the particle will, uponreaching the center, diverge outward, and so theamplitudes of the incoming and outgoing fieldsmust be equal at the center of the particle. Theother boundary conditions that we must applyare that the transverse components of the elec-tric and magnetic fields are continuous acrossthe boundary between layers j and j þ 1 (1 � j� N � 1) and between layer N and the sur-rounding matrix. These conditions yield a set of2N equations that we can solve to obtain the setof coefficients {amp}, which can then be used toobtain r and g for a given particle morphology.

The process that we have just described is ingeneral tedious, but excellent software is avail-able to implement these calculations.

EXPERIMENTAL

Computational Methods

We calculated r and g for a variety of particlemorphologies. We used code developed by Vosh-chinnikov16,22 to implement the algorithm of Wuand Wang.23 Our only significant modification tothe code, aside from customizing the input andoutput formats, was to allow for the possibilitythat the matrix might have a larger refractiveindex than any layer of the particles. In Vosh-chinnikov’s code it is assumed that the refrac-tive index of the matrix is smaller than therefractive index of any layer in the particle, andso the algorithm only checks the layers of theparticle when searching for the largest refrac-tive index in the problem. We modified the codeto also check the matrix refractive index. Re-sults for homogeneous particles and particleswith a single coating were checked against the

freeware program Scatlab, and agreement wasfound to at least one part in 10�3.

Parameters

All calculations assume light with a wavelengthof 550 nm, the center of the visible range.Searching for a resonance by varying the par-ticle size is equivalent to searching for reso-nance by varying k. Also, the typical designproblem facing a plastics engineer is one wherethe wavelength range of interest is fixed, andthe adjustable parameters are the thicknesses ofthe various particle layers. One can determinethe optical properties at some other wavelengthk2 = 550 nm by rescaling the diameters (andlayer thicknesses) of the particles described hereby a factor k2/550 nm.

For convenience we assume a particle volumefraction / ¼ 5% relative to the matrix. One cancalculate ‘ and ‘* at some other volume fraction/2 by rescaling ‘ or ‘* by a factor 0.05//2, aslong as /2 � 0.1. For volume fractions above 0.1,particle positions become correlated because ofexcluded volume effects, and correlations canmodify scattering in a manner that tends toincrease ‘*. Also for convenience, we do all cal-culations for particles in a matrix with a refrac-tive index of 1.500. This is comparable to typicalrefractive indices of commercially important poly-mers. To implement the systems described herein a matrix with some other refractive index n2,rescale the refractive index of each layer by afactor n2/1.500, and rescale the particle diam-eters and layer thicknesses by a factor (n2/1.500)�1.

Finally, for the most part we will considerparticles with outer diameters between 50 nmand 5 lm, and cores or layers with refractiveindices between 1.35 and 1.65. These parame-ters are representative of the range of materialsfrequently used in commercial plastics.

RESULTS

Our key findings are

1. Coatings on small to midsize (d � 50 nmto 1 lm) particles tend to decrease r andbring g closer to 0, provided that the coat-ing decreases the dielectric contrast be-tween the medium and the particle. A sin-gle coating is generally more effective than


multiple coatings, and the scattering prop-erties tend not to depend sensitively on thethickness of the coating if the volume-aver-aged dielectric constant of the particle dif-fers from that of the surrounding matrixby more than �0.02.

2. When the volume-averaged dielectric con-stant of the particle matches or nearlymatches that of the surrounding matrix, rand g can depend sensitively on particlemorphology.

3. For larger particles, the effect of a coatingcan be more complicated, but typically r isunaffected while g can be modified slightly.

Homogeneous Particles

We begin by showing in Figure 2 a plot of Qscatt

versus particle diameter for homogeneous sp-heres with refractive indices 1.400 and 1.594.These numbers were chosen because the dielec-tric contrast is the same in both cases, and inmany cases Qscatt and r are proportional to thesquare of the dielectric contrast. We see that aslong as the particle diameter is less than 2 lm,the high-index and low-index particles scatterwith the same efficiency. This fact is useful forstudying coated particles. If we want to askwhether coatings are more useful on core par-ticles with refractive indices higher or lowerthan that of the surrounding medium, it helpsto compare core particles with similar scatteringproperties. For particle diameters larger than2 lm, the high-index and low-index particles dif-fer in their scattering efficiencies. The differenceis due to interference effects: waves that scatter

from different portions of the particle interferewith each other, and the phase shift betweendifferent waves depends on the refractive indexof the particle.

Single Coatings on Particles Smaller thanor Comparable to k

Arbitrary Choices of Refractive Indices

We now consider the effect of adding a singlecoating to a spherical particle. We begin bystudying core particles with refractive indices of1.400 and 1.594 and outer diameters of 100,320, and 1000 nm. We chose these parametersbecause homogeneous high-index and low-indexparticles of these sizes and refractive indiceshave nearly identical scattering efficiencies.

On the low-index particles (n1 ¼ 1.400) weconsider coatings with refractive indices 1.350,1.449, and 1.546. On the high-index particles (n1

¼ 1.594) we consider coatings with refractiveindices 1.449, 1.546, and 1.650. The values1.449 and 1.546 were chosen to facilitate com-parisons with antireflection coatings on planarsurfaces: these are the optimum refractive indi-ces for antireflection coatings on planar samplesof the low-index and high-index materials (res-pectively) surrounded by a medium of back-ground refractive index nb ¼ 1.500. (The opti-mum refractive index ncoat for an antireflectioncoating on a slab of refractive index ncore in abackground medium of refractive index nb isH(ncorenb).) The values 1.350 and 1.650 werechosen arbitrarily, to study the effect of a coat-ing that provides a larger contrast comparedwith the surrounding matrix.

Figures 3–5 show our results. We calculatedthe scattering efficiency versus the volume frac-tion / of the core material for the particle sizes,refractive indices, and coatings discussed in theprevious paragraph. In every case, we find thatadding a coating with a refractive index closerto that of the surrounding matrix reduces thescattering efficiency, while adding a coating witha refractive index further from that of the sur-rounding matrix increases the scattering effi-ciency of the particle. The scattering efficiencyis minimized when the volume average of thedielectric constant is approximately equal to thedielectric constant of the surrounding medium.(We did not sample enough data points to deter-mine if the dielectric constant must be exactlymatched to minimize Qscatt, but for our practical

Figure 2. Scattering efficiency Qscatt versus particlediameter for homogeneous spheres of different refrac-tive indices in a medium of refractive index nb ¼ 1.5.


concerns any small discrepancy is irrelevant.)Varying the volume fraction of the core onlyminimizes Qscatt when it is possible to make thevolume-averaged dielectric contrast zero. Other-wise varying the volume fraction of the core (or,equivalently, the thickness of the particle) onlychanges the scattering efficiency monotonically.

We can deduce that the mechanism by whichcoatings reduce the scattering efficiency of acore–shell particle is very different from themechanism by which antireflection coatings re-duce scattering from planar surfaces. Antireflec-tion coatings on planar surfaces operate bydestructive interference of scattered waves. In-terference effects are very sensitive to phaseshifts and coating thicknesses. In Figures 3–5,however, the minimum scattering efficiency doesnot occur at the same coating thickness irrespec-tive of particle diameter. Rather, it always oc-curs at approximately the same volume fraction

of core material. In all of the cases that we haveexamined involving particle diameters less than2 lm (including cases not shown here), the scat-tering efficiency is minimized when the volume-averaged dielectric constant of the particlematches or nearly matches that of the surround-ing medium. Later we shall discuss a simpleanalytical model that predicts this behavior.

We now consider how coatings on particlesaffect transport of light through a system of par-ticles. We calculate ‘* for light propagatingthrough a system of core shell particles, varyingthe particle morphology. We will concentrate onparticle morphologies that minimize the scatter-ing efficiency. We do not assume that minimiz-ing Qscatt maximizes ‘*, since ‘* depends on theangular distribution of scattered radiation aswell as the total amount of light scattered.

In Figure 6(a) we calculate ‘* for light propa-gating through a system of core–shell particles

Figure 3. Scattering efficiency Qscatt of a core–shellparticle versus volume fraction of core material. Par-ticles have an outer diameter of 100 nm and coatingswith various refractive indices, and are embedded ina background medium of refractive index nb ¼ 1.50.



with a core that has a refractive index of 1.4and a coating that has a refractive index of1.546, and we vary the particle diameter andthe volume fraction of the core material. InFigure 6(b) we do the same, except that the corematerial has a refractive index of 1.594 and thecoating has a refractive index of 1.449. For bothparticle types, we find that the morphology thatminimizes Qscatt sharply maximizes ‘* only inthe case of the smallest particles (100-nm diam-eter). In this case, ‘* is enhanced by nearly 3orders of magnitude. For larger particles (320-and 1000-nm diameters) ‘* is only weakly af-fected by the presence of a coating.

Volume-Averaged Dielectric Constant NearlyMatches Matrix

In many industrial applications, such as impact-modified transparent poly(methylmethacrylate)and poly(vinyl chloride), multi-layered core–shell

particles are used, and the refractive indices andvolume fractions of the layers frequently result ina volume-averaged dielectric constant that nearlymatches that of the surrounding matrix. In suchcases, the volume-averaged dielectric constant ofthe particle is not the only important parameterfor coated particles smaller than or comparable tok. We have found that when the volume-averageddielectric constant of a small particle nearlymatches the dielectric constant of the surroundingmedium, the order of the layers can significantlyaffect Qscatt and g and hence ‘*.

As an example, in Figure 7 we show calcula-tions of l* for core–shell particles in which thecore and shell have equal volume fractions, andone of the layers has a refractive index of 1.45

Figure 6. Transport mean free path l* for light dif-fusing through a system of core–shell particles. Weshow calculations of l* for a variety of particle sizesand morphologies (background medium nb ¼ 1.5). (a)Particles with a core material having a refractiveindex 1.4 and coating having a refractive index 1.546.(b) Particles with a core material having a refractiveindex 1.594 and coating having a refractive index1.449.



while the other layer has a variable refractiveindex. The outer diameter is 140 nm. We seethat interchanging the core and shell materialscan, in some cases, alter ‘* by a factor of 10.Depending on the situation the weakest scatter-ing can be obtained with either a high-index coreor a high-index shell. In all cases, ‘* is quitelarge, on the order of a millimeter or more.

We attribute this morphology-dependent phe-nomenon to the effective size of the particle.When the refractive index of the shell is close tothat of the matrix, incoming light waves ‘‘see’’

an effectively smaller particle, because the outershell scatters only weakly. If we then inter-change the core and shell materials, the newparticle will ‘‘appear’’ larger to incoming lightwaves because the edges will scatter morestrongly. Interestingly, this effect can cause ‘* tobe shorter for particles on which the shell’srefractive index nearly matches that of the sur-rounding medium: a smaller particle scatterslight less efficiently (which tends to increase ‘*),but it also scatters less light in the forwarddirection (which tends to decrease ‘*). The onlyway to determine which of these competingeffects dominates is to perform a numerical cal-culation with the Mie theory, as we have donein Figure 7.

We also observe that the effect of interchang-ing the core and shell is actually weakest whenthe volume average of the dielectric constantmatches the dielectric constant of the surround-ing medium (i.e., when the refractive index ofthe variable layer is �1.548). This is consistentwith our interpretation in terms of an effectivesize: when the volume average of the dielectricconstant matches the surrounding medium, thedielectric constants of the core and shell will dif-fer significantly from the surrounding mediumand in either morphology the edges of the par-ticle will scatter relatively strongly. We havefound similar effects in particles with multiplelayers.

Another interesting phenomenon occurs whenwe look at the wavelength dependence of themean free path. In Figure 8 we show a plot of ‘*versus wavelength for core–shell particles with

Figure 8. Transport mean free path versus wavelength for light diffusing througha system of core–shell particles with an outer diameter of 140 nm. The volume frac-tions of the core and shell are 50% in both plots (background medium nb ¼ 1.5).

Figure 7. Transport mean free path for light diffus-ing through a system of core–shell particles with anouter diameter of 140 nm. The volume fractions of thecore and shell are 50% in both plots, and for each plotone of the components (core or shell) is fixed at arefractive index of 1.45 while the refractive index ofthe other component is varied (background mediumnb ¼ 1.5).


an outer diameter of 140 nm and two layers, thelayers having refractive indices 1.45 and 1.52and volume fractions of 50% for each layer. Theonly difference between the two plots in Figure 8is the order of the layers. The difference is strik-ing: interchanging the layers of the particle canchange the red–blue contrast by a factor of 4.

In both cases the transport mean free path islonger for red light (long wavelengths), as wouldbe expected for light diffusing through a systemof small, weakly scattering particles. (Rayleighscattering) However, the ratio ‘*(700 nm)/‘*(400 nm) is 1.64 when the outer layer has arefractive index of 1.45, and 6.24 when the outerlayer has a refractive index of 1.52. When theratio of mean free paths for red and blue light islarge, the appearance of the plastic material isreddish when viewing transmitted light and blu-ish when viewing scattered light. When theratio is closer to unity, both the scattered andtransmitted light will have a more neutral ap-pearance. The parameters used here are onlyexamples, and similar phenomena are observedfor other choices of parameters. The key point isthat by coating the particles one can control thecolor of a material as well as the transparency.

Coatings on Particles Large Compared with k

Next we turn our attention to particles signifi-cantly larger than k. We begin by studying par-ticles with an outer diameter of 4 lm, for whichsize the two types of core materials (ncore

¼ 1.400 and ncore ¼ 1.594) lead to significantlydifferent scattering efficiencies, as seen inFigure 2. We consider the same types of coatingmaterials as for small spheres with single coat-ings (ncore ¼ 1.35, 1.449, and 1.594 for low-indexcores, ncore ¼ 1.449, 1.546, and 1.65 for high-index cores). In Figure 9 we plot Qscatt versuscore volume fraction for the two types of corematerials.

As before, we find that the only significantreduction in Qscatt is achieved when the refrac-tive index of the medium is intermediatebetween the refractive indices of the core andshell. However, there are two major differencesbetween this case and the smaller particles thatwe examined earlier. The first is that Qscatt

varies by only a factor of 3 in Figure 9. On thesmaller particles a coating that reduced the vol-ume-averaged dielectric contrast to zero couldreduce Qscatt by an order of magnitude or more.The second is that the optimal coating thickness

does not bring the volume-averaged dielectriccontrast to zero. In these larger particles Qscatt

is sensitive to interference of waves as well asthe volume average of the dielectric constant, sowe cannot use any simple heuristic to estimatethe optimal coating thickness; only a full Miecalculation can give accurate predictions. Ourcalculation shows that, for the parametersstudied here, a particle with a core volume frac-tion of �20% produces the optimal (minimum)scattering efficiency when the coating materialis chosen so that the refractive index of thematrix is intermediate between the refractiveindices of the core and shell.

When we consider light propagation in a sys-tem of such particles, we find a third differencebetween these large particles and the smallerparticles that we studied above: adding a coat-ing to a particle does not optimize ‘*. In Figure10 we plot ‘* versus the volume fraction of corematerial for light propagating through a systemof particles with various core and coating refrac-

Figure 9. Scattering efficiency Qscatt of a core–shellparticle versus volume fraction of core material. Par-ticles have an outer diameter of 4 lm and coatingswith various refractive indices, and are embedded ina background medium of refractive index nb ¼ 1.50.


tive indices, surrounded by a medium withrefractive index 1.5. For all particle types con-sidered here, ‘* (and hence the amount of trans-mitted light) is maximized when the volumefraction of the core material is either 0 or 1,depending on the choice of parameters. Themodest reduction in Qscatt achieved by the coat-ing cannot compensate for the effect on theangular distribution of scattered light, and so ‘*changes monotonically as a function of the corevolume fraction.

We have also considered a wide variety ofother large particles with single coatings, varyingthe particle size, the volume fraction of the corematerial, and the refractive indices of the coreand shell. In general, it is very difficult to find acombination of parameters that cause ‘* to devi-ate from the trend seen in Figure 10: ‘* is almost

always a monotonically increasing or decreasingfunction of the core volume fraction. The mostfrequent exceptions occur when ‘* is approxi-mately the same for core volume fractions ofeither 0 or 1, and in those cases the maximumvalue of ‘* achievable by coating the particles israrely more than 10% larger than the valueachievable with homogeneous particles.

The implications for particle design are clear:when doping a polymer with large particles, ifthe goal is to minimize scattering then use homo-geneous particles whenever possible. If homoge-neous particles are not feasible for the applica-tion, then use either the thickest or thinnestcoating possible, depending on the circumstances.

Multiple Coatings

Finally, we have also considered the possibility ofreducing scattering from particles in various sizeregimes by incorporating more layers. We havefound that adding multiple coatings to a particlecan indeed increase ‘*. However, in almost allcases that we have examined, particles with mul-tiple coatings scatter more strongly than homo-geneous particles or particles with a single judi-ciously chosen coating. In rare cases where par-ticles with multiple coatings did scatter less thanhomogeneous or single-coated particles, we neverfound a case where adding more coatings re-duced ‘* by more than a factor of 2.

DISCUSSION

Having illustrated the typical ways in whichcoatings can modify the scattering properties ofspherical particles, our next goal is to develop aconceptual understanding of these results. Wewant to be able to formulate design rules thatare supported by our calculations as well as aconceptual understanding of the phenomenon.We will use approximate models to study theproperties of coated particles in two limits: TheRayleigh–Gans approximation (RGA, applicablein the case of low refractive index contrast andsmall particles), and a combination of geometri-cal optics and planar coating theory to studyparticles large compared with the wavelength oflight. From the Rayleigh–Gans model we willdeduce why the volume-averaged dielectric con-trast is the most important parameter to opti-mize when trying to reduce scattering by smallparticles. From the geometrical optics approxi-

Figure 10. Transport mean free path ‘* for lightdiffusing through a system of core–shell particles.Particles have an outer diameter of 4 lm and coatingswith various refractive indices, and are embedded ina background medium of refractive index nb ¼ 1.50.(a) Core refractive index ¼ 1.400, (b) core refractiveindex ¼ 1.594.


mation we will deduce why a coating on a largeparticle rarely produces significant changes in r or g.

Rayleigh–Gans Approximation

In the RGA, the intensity of light scattered froma particle in a given direction is proportional tothe Fourier transform of the particle’s spatiallyinhomogeneous dielectric contrast e(r) � eb. Thespatial frequency and direction at which wecompute the Fourier transform depends on thedirection of scattering. We define the scatteringvector q as the difference between the wavevec-tor of the incident wave and the wavevector of awave scattered in a given direction (h, /):

q � kincident � kscattered ð9Þ

The intensity of the wave scattered in a givendirection is proportional to square of the scatter-ing amplitude f(q), and in the RGA one canshow that18

f ðqÞ /Zparticle volume

ðeðrÞ � ebÞ expðiq rÞdV

ð10Þ

We omit factors of 2, p, k0, and so forth becausewe are primarily interested in qualitative trends.Equation 10 is approximately valid as long as

jeðrÞ=eb � 1j; nbk0rN jeðrÞ=eb � 1j 1 ð11Þ

Even within this domain of applicability, how-ever, eq 10 may not be quantitatively accuratewhen q ¼ 0, especially if it predicts that f(0) � 0.

The significance of f(0) is two-fold: we can seefrom eq 10 that in the RGA f(0) is the volume-averaged dielectric contrast �e� eb, and in theprevious section we saw that coated particlesscatter more weakly when a coating reduces�e� eb. The optical theorem, which is discussedin books on light scattering and electromagnet-ism,18,25 states that r is proportional to f(0). Thefact that f(0) and hence r is proportional to�e� eb explains why scattering from small par-ticles is minimized when the dielectric contrastis minimized.

We can also understand why a single coatingwill, in general, be more effective at reducingscattering than can multiple coatings. Supposethat, for a given material, we have the ability tosynthesize several different types of coatings.�e� eb will be minimized if we pick the coating

material with the refractive index that is closestto that of the matrix and make the thickest possi-ble layer. Any other coating materials will haverefractive indices that are closer to that of theparticle, and will hence increase the dielectriccontrast compared with the single-coating design.

Large Particles

On particles too large to be modeled by the RGA,we have found that coatings rarely reduce Qscatt

(or r) by more than a factor of 2, and that theeffect becomes weaker as the particle sizeincreases. Here we seek to explain why the effectof a coating becomes negligible in the limit of verylarge particles. If a spherical particle is signifi-cantly larger than the wavelength of light, thenwe can apply traditional notions and methodsused to study scattering of light from objects largecompared with the wavelength of light. Weassume that scattering can be attributed to threedistinct mechanisms, as illustrated in Figure 11:

1. Reflection from the surface of the particle.2. Any light not reflected from the surface is

refracted in accordance with Snell’s lawupon entering the particle. Regardless ofwhether the light undergoes internal reflec-tions before exiting the particle, refractionupon entering the particle has alreadychanged the direction of propagation, andso the light is considered scattered.

3. Diffraction at the edges of the particle.

Figure 11. Illustration of mechanisms for scatteringfrom a large particle. Two rays of light are shown hit-ting the particle. One is split into refracted andreflected rays; the other is incident on the edge of theparticle and is diffracted. The angle of incidence forthe top ray is h.


Of these mechanisms, only the first one,reflection at the surface, can be enhanced ordiminished by a coating. The angular distribu-tion of light that scatters after entering the par-ticle may also be affected by a coating (due tothe effect on internal reflections), but any lightthat enters the particle is by definition scatteredby refraction, regardless of how many (if any)internal reflections it undergoes. We thereforededuce that a coating on a large particle cannotsignificantly reduce Qscatt, but it may bias scatter-ing in the forward direction (by slightly reducingscattering at the rear surface) and bring g closerto 1. This will have the effect of increasing ‘* forlight diffusing through a system of coated par-ticles, somewhat decreasing the opacity.

To verify our deduction we calculated theangular distribution of light reflected from theback surface of a large spherical particle withand without a thin coating. We used standardformulas for off-normal reflection from planarcoatings to calculate the amount of light fromeach point on the surface, and integrated overthe back surface of the particle to obtain theaverage cosine of the scattering angle.9 We aver-age the reflectance over the two possible polar-izations and integrate the reflected light overthe back surface of the particle to obtain areflectance efficiency which we call QreflectQreflect

is the portion of incident light reflected by theback surface of the particle, and is given by:

Qreflect ¼Z 90�

�90�RðhÞ sin h cos h dh ð12Þ

where h is the angle of incidence in Figure 11,R(h) is the reflectance for light incident at anangle h, the factor of sin h accounts for solidangle in the integration, and the factor of cos haccounts for energy flux at off-normal incidence.

To model a coating’s effect on the angular dis-tribution of scattered radiation we also considerthe contribution of reflected light in the calcula-tion of g, and define a parameter gpartial:

gpartial ¼ 1

2

Z 90�

�90�RðhÞ cosð180� 2hÞ sin h cos h dh

ð13Þ

where cos (180 � 2h) is the cosine of the scatter-ing angle in Figure 11, the factor of 1/2 is a nor-malization, and the other factors are the sameas in eq 12.

In Figure 12 we plot Qreflect versus coatingthickness for high-index (ncore ¼ 1.594) and low-index (ncore ¼ 1.4) particles in a matrix of back-ground refractive index 1.5, and coated withmaterials that would minimize reflection from aplanar surface (ncore ¼ 1.547 for high-indexspheres and 1.45 for low-index spheres). Thecoating thickness is normalized to the wave-length of light inside the coating. For light hit-ting a planar system at normal incidence theoptimum coating thickness would be k/4.

As we would expect, the low-index particlereflects more light than a high-index particle,because of total internal reflection. That neitherparticle’s scattering efficiency is minimizedwhen the coating thickness is k/4 may seemdubious, but keep in mind that not all of thelight hitting the particle is coming in at normalincidence. Reflection at off-normal incidence isminimized by thicker coatings, and so this resultis not surprising.

In Figure 13 we plot gpartial versus coatingthickness for the same parameters as in Figure12. In both cases, the reflection contribution to gis small compared with g (g is of order 0.98 forlarge particles). The reflectance contribution isonly weakly modulated by varying the coating

Figure 12. Calculated reflectance efficiency as afunction of coating thickness for large-core shell par-ticles in a medium with refractive index 1.5. (a) ncore

¼ 1.4, nshell ¼ 1.45, (b) ncore ¼ 1.594, nshell ¼ 1.547.


thickness (modulation of order 0.001 or smaller),so a coating should have only a negligible effecton 1 � g and hence l*. Also, the reflection con-tribution to g is positive, so most reflected lightis scattered in the forward direction. This mayseem counterintuitive, but light that hits aspherical particle near the edges (and hence at aglancing angle) will undergo a glancing anglereflection into the forward direction.

If we add additional layers, it is still possibleto calculate the reflectance at each point on thecoated sphere, but the calculation is much moretedious. Fortunately, most multiple-layer coat-ings share a universal feature: the overall reflec-tivity tends toward 1 irrespective of angle as thenumber of layers increase. At specific anglesand frequencies resonant transmission may hap-pen, but these resonances become quite narrow(both as a function of angle and frequency) asthe number of layers increases, and so integrat-ing the reflected light over the back surface ofthe particle will tend to ‘‘average out’’ the effectsof such resonances. We therefore conclude that,in general, adding additional coatings to a largespherical particle will not reduce reflectance,

but will rather increase the amount of lightreflected in the backward direction. The netresult is to decrease g and hence decrease ‘*,making a system of such particles less transpar-ent rather than more transparent.

We can make this analysis more rigorous bylooking at qualitative features of the boundaryvalue problem for waves incident on sphericalparticles. In eqs 7 and 8 we described the fieldinside each layer as a superposition of outwardpropagating (Nmp

(3)* (k0nr,h,/)) and inward con-verging (Nmp

(3) (k0nr,h,/)) waves. We know thatthe asymptotic form of these waves at large dis-tances from the origin (r � k) is expð6ik0nrÞ

k0nr, and

within a layer of thickness much smaller than rthe factor of r�1 will be approximately constantand the waves will be approximately sinusoidal.The problem for waves of a given order m andpolarization p is then formally identical to asinusoidal plane wave incident on a multilayerdielectric coating. Waves incident on a planardielectric coating of many layers tend to be morestrongly reflected as we increase the number oflayers.

We are therefore confident of our predictionthat multilayer dielectric coatings on sphereswill rarely or never produce a significant reduc-tion in r (or, equivalently, Qscatt). This result hasbeen demonstrated by exact calculations basedon the wave equation, and supported by approx-imate calculations based on a simple model ofscattering from large coated particles.

CONCLUSIONS

Adding a coating to a polymer particle onlyleads to a significant reduction in the particle’sscattering cross section if the coating reducesthe volume-averaged dielectric contrast betweenthe particle and the surrounding matrix. Thescattering cross section and angular distributionof scattered radiation are relatively insensitiveto morphological details unless the volume-aver-aged dielectric constant of the core–shell particleexactly or nearly matches that of the surround-ing matrix. In addition, for large diameter par-ticles, homogenous particles provide the leastscattering than particles with multiple layers.These computational findings can be explainedby simple theoretical models, confirming thatour findings are broadly applicable rather thana relic of the parameters that we investigated inthis work. Finally, the current work also pro-

Figure 13. Calculated reflection contribution toangular distribution of radiation (gpartial) as a functionof coating thickness for large-core shell particles in amedium with refractive index 1.5. (a) ncore ¼ 1.4, nshell

¼ 1.45, (b) ncore ¼ 1.594, nshell ¼ 1.547.


vides a tool for predicting the optical behavior ofsystems filled with homogenous or multi-layeredspherical particles.

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