Theoretical study of the scattering efficiency of rutile titaniumdioxide pigments as a function of their spatial dispersion

Jean-Claude Auger, Vincent Arnaud Martinez,

Brian Stout

� FSCT and OCCA 2008

Abstract We propose an original theoretical frame-work to model the scattering efficiency of white paintfilms as a function of the volume fraction and spatialstate of dispersion of rutile titanium dioxide pigments,taking into account electromagnetic couplings. Numer-ical calculations are performed using a multiple Tmatrix formalism on an ‘‘elemental’’ volume extractedfrom the bulk of the paint and which we model aspigments and fillers in a polymer matrix. Qualitativestudies show that, due to the dependent scatteringphenomenon, the size of fillers can modulate themagnitude of loss in scattering efficiency by modifyingthe spatial state of dispersion of the pigments in thepolymer matrix. In particular, fillers whose size iscomparable to the dimension of the pigments improvethe scattering efficiency by impeding crowding. It isalso shown that the optical properties of the bulkmaterial at arbitrary concentration can be approxi-mated by extrapolating the optical properties calcu-lated on a limited number of scatterers.

Keywords Optical property, Opacity, Rutile titaniumdioxide pigment, Dependent scattering, Multiplescattering, Spatial dispersion, T-matrix formalism

Introduction

The opacity of architectural white paint films is dueto the multiple scattering interactions that occurbetween the incident radiation and small particlesdispersed in the polymer matrix. Total covering isachieved when all the photons penetrating the paintundergo a sufficient number of scattering processes tobe scattered back into the incident media before theycan be reflected or absorbed by the substrate orabsorbed within the film. The strength of the singlescattering phenomena is directly related to the contrastbetween the index of refraction of the pigments andthe index of refraction of the surrounding medium. Thelarger the contrast, the stronger is the amplitude of thescattering events. For this reason, rutile titaniumdioxide (TiO2) with an index refraction of �2.8 andnegligible absorption in the visible range is an efficientpigment for use in white coatings.

Now, if the existence of a nonnegligible refractionindex contrast of the particles with the surroundingmedium is the sine qua non condition to generatescattering, there also exists a wide range of otherparameters that can modulate its strength and effi-ciency. Such parameters can be classified as eitherintrinsic or extrinsic with respect to the pigments. Theintrinsic category refers to the chemical nature, size,and shape of the pigments while extrinsic parametersinclude the filling fractions and the spatial dispersionstate of the pigments in the polymer matrix. Throughthe years, paint manufacturers have realized numerousstudies to increase the effectiveness of TiO2

1,2 byoptimizing each of these variables. One major result ofthis optimization is that many commercial TiO2 gradesnow have their size distribution as narrow as techni-cally possible and centered about 0.250 lm in diameterto maximize scattering.

A major issue under constant investigation is theminimization of the loss in scattering efficiency due to

J.-C. Auger (&)Center for Laser Diagnostics, Department of AppliedPhysics, Yale University, New Haven, CT 06520, USAe-mail: jeanclaude_auger@hotmail.com

V. A. MartinezDepartment of Applied Physics, RMIT University,Melbourne, VIC 3000, Australia

B. StoutInstitut Fresnel, UMR 6133, Faculte des Scienceset Techniques, Centre de Saint Jerome,Marseilles Cedex 20 13397, France

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DOI 10.1007/s11998-008-9116-6

89

the pigment packing that unavoidably arises in highlyconcentrated paints. This decrease in light scatteringeffectiveness is attributed to the occurrence of depen-dent light scattering phenomena3,4 and is known tocause significant increase in the cost of the manufac-ture of paints. Indeed, the higher is the filling fraction;the larger the amount of TiO2 pigments that must beadded to the formulation to reach a nonnegligibleenhancement of the overall paint opacity. Semiempir-ical studies have shown that this undesirable phenom-enon can be attenuated by improving the spatialdispersion and limiting the aggregation states of thepigments in the dry film.5–7 To our knowledge, fewstudies7,8 propose the analysis of this intricate problemthrough a theoretical framework that takes intoaccount exact dependent light scattering calculations.

Consequently, the aim of this work is twofold: first,to propose an original approach that allows studyingthe complex relation between the spatial state ofdispersion of rutile TiO2 pigments and their scatteringefficiency; and second, to discuss the effects of suchrelation on the opacity of the global paint film. Ourframework consists in calculating and comparing theoptical properties related to elemental volumes ofdifferent paints under study. The multiple scattering Tmatrix formalism9 is described in the next section. Inthe second section, the formalism is applied to studythe variations of the optical properties of TiO2

pigments randomly and uniformly dispersed in apolymer matrix as a function of the filling fractions.Finally, the last section treats the analysis of thechanges in the optical properties when the spatialdispersion of the pigments is altered by taking intoaccount the presence of fillers.

Theoretical framework

Modeling the dry paint films

White, water-based coatings are complex systems,primarily composed of ‘‘opaque’’ pigments (mostcommonly rutile titanium dioxide), fillers, dispersants,rheology modifiers, and a latex binder. From the opticalpoint of view, the pigments are the only elements thatefficiently scatter light and thereby contribute to theopacity. Rheology modifiers and dispersants are in toosmall proportions to be relevant while most of thefillers, such as clays, have refraction indices rangingfrom 1.5 to 1.7, and have an index contrast too small toeffectively contribute to the scattering efficiency. Nev-ertheless, the fillers whose sizes vary from half to abouttens of microns can indirectly modify the opticalproperties by altering the spatial distribution of thepigments in the dry film. Thus, considering bulkproperties and disregarding the interfaces, white paintscan be modeled as a binary mixture composed of TiO2

pigments and fillers dispersed in a continuous polymermatrix at the volumetric concentrations fp and fc,

respectively. Our approach consists in calculating theoptical properties of an elemental of volume element,V0, extracted from the bulk material, and to compre-hensively take into account both multiple and depen-dent scattering phenomenae. To simplify thetheoretical description, it is assumed that the pigmentsand the fillers have spherical shapes and constant radii,denoted rp and rc, respectively. Furthermore, theelemental cell V0, containing Np pigments and Nc

fillers, is assumed to be spherical.

Multiple T matrix formalism

The incident radiation upon the cell is taken to be amonochromatic plane wave, characterized by its vac-uum wavelength, k0, and its in-medium wave vector,Kinc; such that jKincj ¼ 2pn0=k0 where n0 is the realindex of refraction of the embedding medium. Thepositions of the pigments, of complex refraction indexcontrast, ~np � np=n0; are given in terms of position

vectors RðiÞ ði ¼ 1; . . . ;Np ¼ NÞ defined with respect toa main reference frame noted R0: Combining themultiple scattering equation of light and the superpo-sition principle of electromagnetic theory, the exciting

electric field upon the ith scatterer ðEðiÞexcÞ can beexpressed as the sum of the applied incident electricradiation ðEincÞ and the scattered fields of all the

remaining particles ðEðjÞs Þ such that:

EðiÞexc ¼ Einc þXN

j¼1j 6¼1

EðjÞs i ¼ 1; . . . ;N ð1:1Þ

The multiple scattering T matrix formalism hasalready proved to be an effective approach to solvingthe linear system of couple equations defined in (1.1).10

Assuming that the incident, scattered, and internalelectromagnetic fields are expanded on the sphericalvector wave functions, the multiple-scattering T-matri-ces can be re-written such that they satisfy theequations:

TðiÞN ¼ T

ðiÞ1 Iþ

XN

j¼1j 6¼1

Hði;jÞ

TðjÞN Jðj;iÞ

2

664

3

775 i ¼ 1; . . . ;N ð1:2Þ

The Jði;jÞ

and Hði;jÞ

are matrices11 which translate theincident and scattered fields, respectively, from areference frame centered on the jth particle to areference frame centered on the ith particle. The T

ðiÞ1

represents the single-particle T matrix of the ithobject,12 which relates the expansion coefficients ofthe excitation field impinging on the particle into theexpansion coefficients of the field radiated by the

particle. Finally, the TðiÞN symbolizes the N-particle T

matrix of the ith objects. The TðiÞN directly relates the

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scattered field to the expansion coefficients the fieldincident on the system while taking into account thepresence of all the other particles in the system. It isindependent of the orientation or polarization of theincident wave and it characterizes the scatteringproperties of the particle while taking into accountthe presence of the others. Once the N-particle Tmatrices are calculated by solving (1.2), the opticalproperties of the whole system can be easily deter-mined for arbitrary incident fields.13 In this study,numerical simulations were performed with the Recur-sive Centered T Matrix Algorithm (RCTMA), whosethorough description and validation have already beenpresented in the literature.14

The number of particles that can be handled in thecalculations is largely limited by the size parameter(defined as jKincjrpÞ of the primary particles, whichplays a dominant role in determining the number ofmultiple components necessary to describe the inter-actions. The larger the dimension of the particles withrespect to the wavelength of the incident radiation, thelarger the number of partial waves that must be takeninto account to accurately represent the electric fieldsexpansions on the spherical wave basis and conse-quently the higher are the computational resourcesrequired to achieve reliable calculations.

Construction of the cell

An essential aspect of our framework is to generate theposition vectors RðiÞ such that the spatial distribution ofthe scatterers within the cell can reasonably approxi-mate an elemental volume of the modeled paint understudy. To fulfill this requirement, we apply the follow-ing procedure: (a) the volume of the cell V0 iscalculated from the knowledge of Np, fp, and rp, (b)the radius of the fillers rc is deduced from fc, Nc, andV0, (c) the Cartesian coordinates, x(i), y(i), and z(i)

related to the ith position vector (of both pigments andfillers) which are randomly generated from a uniformdistribution function and accepted provided that eachadded particle is entirely confined within V0 and thatits volume does not overlap with any previously placedobjects in the cell. When present, the fillers weresystematically placed in the cell before the pigments.

Numerical calculations and statistical treatment

Taking Kinc to be parallel to the Oz axis of the mainreference frame, the RCTMA is applied at eachconfiguration, labeled by k, to compute the scattering

cross sections Cðk;pÞs and asymmetry parameter g(k,p) of

the cell, where p = 1, 2 correspond, respectively, to twoorthogonal linear polarizations of the incident radia-tion. The expressions of the configuration and polar-ization average scattering cross section and scatteringefficiency are calculated from:

hCsiKN �1

2K

Xk

k¼1

X2

p¼1

Cðk;pÞs ð1:3Þ

hsiKN �1

2K

XK

k¼1

X2

p¼1

Cðk;pÞs 1� gðk;pÞ� �

ð1:4Þ

where N, fp, and rp are fixed, K is the total number of

configurations, h iKN stands for configuration and polar-ization average operating. The corresponding opticalproperties per unit pigment volume are simply given by

hCVs i

KN � hCsiKN=Nmp and hSiKN � hsi

KN=Nmp where mp is

the volume of a single pigment. We believe that thisstatistical approach is a nonnegligible improvementcompared to previous works15,16 that proposed themodulation of couplings between the objects by trans-lating the particles along the different axii of aCartesian coordinate system.

Finally, it must be mentioned that multiple T matrixformalism allows direct analytical formulations of theorientation and polarization average scattering cross

section ðCðkÞs Þ (as well as the asymmetry parameter gðkÞÞwith respect to the incident radiation. Such formula-tions provide an alternative scheme to evaluate hCsiKNand hsiKN ; which in principle should involve fewerconfigurations to converge to the same level ofprecision. Nevertheless, in addition to the complexityof the analytical formulations that must be pro-grammed, this approach has the disadvantage ofrequiring the evaluation of numerous translationalmatrices, entailing considerable amount of CPU time.

Application to the study of the optical propertiesof randomly dispersed TiO2 pigments

Description of the systems

The aim of this section is to apply our method to studythe variations of the configuration average scatteringcross section and scattering efficiency of variousensembles of titanium dioxide pigments randomlydispersed in a polymer resin (see Fig. 1a). Henceforth,the subscripts (superscripts) N and K are omitted forthe sake of clarity. Calculations were performed asfunctions of the filling fraction and the number ofparticles such that fp = 0.01, 0.05, 0.10, 0.15, 0.20 andNp = 1, 3, 4, 5, 6, 9, 12, 15, 18, 21, and 30. In a previouswork17 we have found that decreasing K from 3000 to1000 and then to 200 led to changes in ÆCsæ from about1% to a few percent, respectively. Therefore, weadopted K = 1000 except in the cells containing 30particles where, as a result of the extensive computa-tion times involved, K, the number of configurationswas limited to 200. The wavelength of the incidentradiation was fixed at 0.546 lm, which corresponds tothe middle of the visible range and the corresponding

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indices of refraction for the rutile titanium and thepolymer matrix were set to 2.8 and 1.5, respectively.The radius of the TiO2 particles was fixed to 0.132 lm,which reasonably approximate the maximum of theparticle size distribution of common titanium dioxidepigments available on the market. The variations ofÆCsæ and Æsæ as a function of the number of particles inthe cell and for the different filling fractions aredisplayed in Figs. 2a and 2b. The bold lines correspondto the optical properties of the cell obtained fromindependent single particle scattering calculations,henceforth referred to as Cind and sind, respectively.These quantities are useful to quantify the magnitudeof the electromagnetic couplings by, respectively,evaluating the ratios ÆCsæ/Cind and Æsæ/sind.

Effect of the number of particles in the cell

Inspection of Figs. 2a and 2b shows that independentlyof the concentration, the scattering cross sections andscattering efficiencies monotonously increase as func-tions of the number of particles in the cells. Thisupward trend is readily understood by recalling thatmultiple T-matrix formalism assumes that the incidentradiation is a monochromatic plane wave with aninfinite spatial extension. Thereby, as the size of thecell increases along with the number of particles, theinteraction front with the incident wave is enlarged andthe scattering cross section is enhanced. The monoto-nous character of the rise is due to the configurationaveraging process that makes ÆCsæ and Æsæ intrinsicproperties of the system that do not depend on therelative positions between the scatterers but rather onan average distance, which is only function of the fillingfraction. Therefore, any resonant modes that could beassociated with a specific arrangement of the particlesand which could lead to possible oscillations in thevariations of the optical parameters when increasingthe number of particles from N to N + 1 are canceledout. A further significant feature is that the statisticalvariations of ÆCsæ and Æsæ for the different filling

Fig. 2: (a) Configuration average scattering cross sectionas function of the number of particles in the cell for nofillers (Fig. 1a). (a) fp = 0.01, (b) fp = 0.05, (c) fp = 0.10, (d)fp = 0.15, (e) fp = 0.20, (f) Independent scattering approxi-mation. (b) Configuration average scattering efficiency asfunction of the number of particles in the cell for no fillers(Fig. 1a). (a) fp = 0.01, (b) fp = 0.05, (c) fp = 0.10, (d) fp = 0.15,(e) fp = 0.20, (f) Independent scattering approximation

Fig. 1: Representation of the cells under study. (a) Randomly dispersed pigments without fillers. (b) Randomly dispersedpigments and fillers having identical sizes. (c) Randomly dispersed pigments and fillers having different sizes

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fractions do not overlap with other. Therefore, one canassert that the conclusions of qualitative comparisonsbetween different systems with different concentra-tions are independent of the number of particles in thecell.

Effect of the filling fraction

Inspection of Figs. 2a and 2b clearly exhibits that at aconstant number of particles, the difference betweenÆCsæ and ÆCindæ (respectively, Æsæ and sind) is enhancedwhen the pigment filling fraction is increased. Thisgradual loss in scattering effectiveness is due to thestrengthening of the near field electromagnetic inter-actions when the average distance between the scat-terers decreases within the cell.

In addition, a comparative analysis between therelative variations of both optical parameters showsthat the effect of dependent scattering seems to bemore pronounced on the scattering cross sections thanon the scattering efficiencies. For example, the ratiosÆCsæ/Cind and Æsæ/sind of the cell composed by 30particles at fp = 0.2 are, respectively, about 0.3 and0.4. This tendency can be understood by consideringthat if an increase in the filling fraction effectivelyenhances the dependent scattering and thereby de-creases the scattering per particle, this effect is partiallycounterbalanced by an enhancement in backscattering.

Extrapolation of the optical properties to largenumber of scatterers and continuous rangeof concentrations

As was previously established, the restriction to rela-tively low numbers of particles in the cell, imposed byrealistic computation resources, does not affect thereliability of qualitative analysis. Nevertheless, it doescurrently prevent calculations on systems composed byhundreds of scatterers that could provide a reasonableestimation of the optical properties of the bulkmaterial. Therefore, we now focus on the possibleextrapolation of the optical properties of cells com-posed by very large numbers of particles from theoptical properties of relatively small ensemble ofscatterers. This analysis is carried out in terms ofhCV

s i � hCsi=Nmp rather than ÆCsæ; it is a more relevantparameter in light scattering analysis (as can beremarked by its units, length–1).

The variations of hCVs i; evaluated from the data set

presented in Fig. 2a, are plotted in Fig. 3 as a functionof the number of particles. The horizontal line repre-sents the scattering cross section per unit volume ðCV

indÞcalculated using the independent scattering approxi-mation. It can be seen that for each filling fraction, theorientation averaged scattering cross sections exhibitsa monotonous decrease as a function of particlenumber characterized by a rapid drop followed by apseudo asymptotic plateau. We also remark that due to

dependent scattering effects, the higher the fillingfraction, the stronger the descent, and the larger thenumber of particles in the system, denoted N*, neces-sary to reach the ‘‘plateau’’. Each data set were readilyfitted using a power law function, which in generalform is given by:

hCVs i N; fp

� �¼ cþ bNa ð1:5Þ

Taking into account that hCVs i fp;N ¼ 1� �

¼ CVind;

allows us to set:

c ¼ 0

b ¼ CVind

a ¼ a fp

� �

8><

>:ð1:6Þ

In view of the poor statistics in the evaluation of theconfiguration average scattering cross section of sys-tems composed of 30 particles, the fits were performedusing the initial data set up to Np = 21. Since thisframework implies that at a given filling fraction N isfixed and V0 is adjusted such that V0 = Nmp/fp, thecondition fp fi 0 implies V0 fi ¥ and hCV

s i ¼ CVind:

Combining this last remark with the fact that thevariation of ln (a) plotted as function of ln (fp) wasperfectly linear (see Fig. 4), relation (1.5) could finallybe expressed as:

hCVs i N; fp

� �¼ CV

indN�a0fa1p ð1:7Þ

where a0 and a1 were found equal to 0.89 and 0.61,respectively.

Fig. 3: Configuration average scattering cross section byunit volume as function of the number of particles in thecell (no fillers Fig. 1a). (a) fp = 0.01, (b) fp = 0.05, (c)fp = 0.10, (d) fp = 0.15, (e) fp = 0.20, (f) Independent scatter-ing approximation, (g) fit fp = 0.01, (h) fit fp = 0.05, (i) fitfp = 0.10, (j) fit fp = 0.15, (k) fit = 0.20, (l) interpolationfp = 0.25

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The accuracy of the adjustments obtained fromequation 1.7, superimpose on the original set of data inFig. 3, shows that: (a) It seems possible to extrapolatethe optical properties of cells containing large numberof scatterers in a wide range of filling fractions from theknowledge of the optical properties of cell made byrelatively small number of particles. (b) The loss inscattering efficiency, represented by the ratiohCV

s i=CVind; only depends on the two parameters a0

and a1, which are functions of the intrinsic propertiesnp, rp, and n0 as well as the probability function foreach point of space to be occupied by a scatterer. (c)Accurate extrapolations of hCV

s i to large number ofparticles and high filling fractions rely on the precisedeterminations of a0 and a1. In principle, such deter-minations require a prior series of calculations on largeensemble of particles until at least N fi N* in a largerange of filling fractions. Nonetheless, the linearvariation of ln (a) plotted as a function of ln (fp)suggests that the calculations could be limited to twodifferent low concentrations. (d) As a consequence ofthe previous statement, extrapolation of hCV

s i to theentire range of concentration can be accurately per-formed. An example is given in Fig. 3, which displaysthe variation of hCV

s i as a function of N calculated fromequation 1.7 setting fp to 0.25. It is important to pointout that the variation of the average scattering crosssection given in equation 1.7 by a double power law isconsistent with the relation proposed by Hottel et al.21

obtained from experimental data. Finally, the study inthis paper was carried out on hCV

s i; but similar analysiscan be performed on ÆSæ.

Effect of the presence of fillers

The aim of this final section is to study the changesin scattering efficiency of the cell when the spatial stateof dispersion of the pigment is modified by the

introduction of fillers. To analyze such variations, wecompared the scattering efficiencies of two completelydiffering systems that correspond to a uniform andpacked spatial distribution of the pigments. The formersituation was obtained by assuming that the pigmentsand fillers had similar size (rp » rc) while the latter wasachieved by using fillers whose size was much largerthan the size of the pigments (rp � rc). A comprehen-sive interpretation of this study requires the knowledgeof the concept of optimum scattering efficiency. Thisconcept is introduced in the following section prior tothe discussion.

Notion of optimized scattering efficiency

Opacity is generally given in terms of hiding power(HP) of which units are square meters per liter [m2 L–

1]. Its description requires introducing the notion ofcontrast ratio18 defined as:

CR ¼ 100RY

b

RYw

¼ 100

RRb kð ÞD65 kð Þy10 kð ÞdkRRw kð ÞD65 kð Þy10 kð Þdk

ð1:8Þ

where Rb and Rw are the reflection coefficients of thepaint film measured on, respectively, black and whitesubstrates, y10 is the tristimulus function and D65 is theilluminant. It is assumed that when the CR reaches98%, the human eye cannot distinguish any furtherchanges in opacity. Thereby, HP represents the areathat a unit volume of wet paint can cover at a filmthickness Z sufficient to produce a contrast ratio 98%when the paint has dried. The higher the hiding power,the larger is the area a fixed volume of paint can cover.

Thus, one can define the optimal scattering effi-ciency, noted So, as the scattering efficiency that a paintfilm must reach to fulfill a giving hiding power withoutwasting raw materials. This notion can be illustratedwith the use of the Kubelka–Munk theory.19 Withinthe framework of our example, the theoretical contrastratios of three hypothetical paints whose scatteringefficiencies are 0.25, 0.20, and 0.15 lm–1 have beencalculated as a function of the film thickness anddisplayed in Fig. 5. Results show that to reach a hidingpower of about 14.2 square meters per liter, corre-sponding to Z = 72 lm, the paint’s scattering efficiencymust be at least superior to 0.20 lm–1. For this samepaint thickness, pigments scattering efficiencies of 0.15and 0.25 lm–1, respectively, lead to contrast ratios of96.5 and 98.7%. Thus, while the 0.15 lm–1 efficiencycannot provide total covering at the required thickness,the 0.25 lm–1 efficiency exceeds the constraint, imply-ing a waste of raw materials.

Results and discussion

For the study in this section, the number of pigmentwas set to 21 while the filling fraction of the fillers is

Fig. 4: (a, b) Raw data and corresponding adjustment ofa = a(fp) (left and bottom axis). (c, d) Raw data andcorresponding adjustment of ln (|a|) = ln (fp) (right and topaxis)

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kept constant at fc = 0.1. The configuration averagescattering efficiency per unit volume is calculated as afunction of various filling fractions (fp = 0.01, 0.05,0.10, 0.15, and 0.20) considering: (b) Nc = 21 and (c)Nc = 2 as shown in Figs. 1b and 1c. Results displayed inFig. 6 illustrate that the loss in scattering efficiency dueto dependent scattering phenomena is strongly corre-lated to the spatial dispersion state of the pigmentswithin the cell. Independent of the filling fraction, thehighest scattering efficiency is always reached when thesize of the fillers is similar to the size of the pigments(see Fig. 1b) whereas the strongest loss corresponds tothe system that promotes crowding (see Fig. 1c). Also,as electromagnetic couplings are amplified in moredense systems, the higher the filling fraction, the largerthe differences. Consequently in the range of highfilling fractions, variations in the size of the fillers leadto equal scattering efficiencies with different quantitiesof pigments.

Considering the hiding power that the paint filmmust attain implies an optimal scattering efficiencyaround So = 0.92 lm–1. Figure 6 shows that such valueis attained for fp equal to 0.20 and 0.16 for the systems(c) and (b), respectively. Thus, within the framework ofthis illustration, optimizing the spatial dispersion stateof the pigments in the paint film allows reaching this So

with 4% less titanium dioxide. In practice, the savingsin pigment concentration will need to be compared tothe extra cost related to the use of smaller fillers and tothe modification in mechanical processes of dispersionit can imply during production. Also, in this study, weconsidered that opacity was the only optical propertythat needed to be optimized, but it is important tomention that gloss can also be modified by the changein the fillers size distribution.

Finally, for comparison, the scattering efficiency ofthe cells composed of uniformly dispersed TiO2 (seeFig. 1a) was also included in Fig. 6. Results show that it

lay in between the two latter systems and that at highfeeling fraction it is closer to the results given bysystem (b). Nevertheless, when considering the resultobtained on the scattering cross section alone (notshown in the article), it tends to be closer to the valueobtained by system (c). One can conclude that directcomparison between systems (a) and systems (b) and(c) should be handled with care. Indeed, one couldexpect the scattering efficiency of system (a) to besuperior to the one given by the TiO2 pigments mixedwith charges of comparable size if the dispersing effectis counterbalanced by the lack of real accessiblevolume to the pigments. Such differences might comefrom the random generation process and should be theobject of further studies.

Variation of the standard deviation as a functionof the concentration

In the previous sections, we have studied the variationof the configuration average scattering efficiencieswithout mentioning the standard deviation (rc) relatedto those quantities. Nevertheless, the spreads in opticalproperties are interesting parameters, because theyprovide additional information on the complex rela-tions between the spatial dispersion of the particles andthe electromagnetic couplings. In Fig. 7, we display thevariations of rc as functions of the filling fractions(fp = 0.0001, 0.001, 0.01, 0.05, 0.10, 0.15, 0.20, and 0.29)evaluated from the scattering cross sections of a monodispersed (type Fig. 1b) and poly dispersed ensembleof TiO2 pigments (type Fig. 1c) with the correspondingpigment radii given in Table 1. The particle sizes werechosen to keep N and the total pigment volume, Nmp,constant during the calculations.

Fig. 6: Variation of the scattering efficiency as a functionof the filling fraction corresponding to the following cell: (a)21 pigments Fig. 1a, (b) 21 pigments and 21 fillers Fig. 1b,(c) 21 pigments and 2 fillers Fig. 1c

Fig. 5: Variation of the contrast ratio as a function of thefilm thickness from the Kubelka–Munk theory. (a)S = 0.25 lm–1, (b) S = 0.20 lm–1, (c) S = 0.15 lm–1

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One can see that independently of the system, thespreads in cross sections undergo a strong increasefrom their initial vanishing value, reach a maximum ata given filling fraction noted f m

p , and then slowlydecrease for higher volume fractions. Such variationswere mentioned in a previous work,20 can be explainedby first considering that in very dilute systems,electromagnetic couplings are negligible and the par-ticles scatter light quasi independently one from theother, therefore rc fi 0. An increase of the fillingfraction amplifies the strength of the electromagneticcouplings and consequently increases the probability ofobtaining different values of hCV

scai for each newconfiguration. As the filling fraction is furtherincreased, the incompressibility of the scatterers leadsto the appearance of correlations in the relativepositions of the particles. As each point of space doesnot have the same probability to be occupied, there areless available configurations to the systems and rc

decreases.The drop off in rc when fp > fp

m is much slower thanthe increasing phase at fp < fp

m because even if thereare less available configurations to the system, thestrength of the coupling is stronger at shorter distances,which increases the probability of having large varia-tions hCV

scai from one configuration to another. A finalremark is that at high concentration, rc is larger for thepoly-disperse system because the number of accessible

configurations is larger than for the mono-dispersesystem. In principle, when the mono disperse ensembleof spherical particles reaches the maximum compactarrangement (fp = 0.72), there is only one configura-tion available to the system and rc fi 0.

Conclusion

We have presented an original theoretical frameworkpermitting the study of the variations of the opticalproperties of an ensemble of dielectric particles asfunctions of their spatial state of dispersion in anonabsorbing medium. We have applied this formal-ism to compare the loss of scattering efficiencies ofvarious ensembles of TiO2 pigments with their spatialdispersion states modified by the presence of fillers.The results of our calculations agreed with experi-mental knowledge, that the opacity of white architec-tural coatings can be improved using fillers whosesizes are similar to the size of the pigments. Thisstudy focused on the opacity of white paint films;however, the framework presented here can be easilyapplied to the study of dependent absorption thatoccurs in colored paints by calculating the configura-tion average absorption cross section of absorbingpigments.

It is clear that such an approach cannot be asubstitute for semiempirical and experimental worksbased on direct measurements of optical properties ofpaint films in the laboratory. Nevertheless, we believethat the continued development and use of suchtheoretical frameworks can play a vital role in under-standing the fundamental mechanisms that bind theintrinsic and extrinsic properties of the pigments to theopacity or color of paint films.

Finally, we point out that the elemental volume V0

presented in this framework cannot be easily corre-lated to the volume element of a heterogeneousmedium as introduced in the heuristic derivation ofthe radiative transfer equation. Indeed, even if thisstudy has shown that one can partially work around thelimitations imposed by the truncation of the excitedfield impinging on each particles by extrapolating theconfiguration average optical cross sections to largerensemble of scatterers, two major problems stillremain: (a) to determine the size of the cell at whichthe extrapolation must be performed (b) to accuratelyextrapolate the phase function of a highly concentratedsystem, whose knowledge is required to solve theradiative transfer equation.

Fig. 7: Standard deviation calculated from the configura-tion average cross section of eight TiO2 pigments per cellas a function of the filling fraction. (a) Mono-disperseensemble of particles. (b) Poly-disperse ensemble ofparticles

Table 1: Radii in micrometers of the eight spheres composing systems (a) and (b)

Particle 1 2 3 4 5 6 7 8

(a) 0.110 0.110 0.110 0.110 0.110 0.110 0.110 0.110(b) 0.138 0.138 0.110 0.110 0.087 0.087 0.087 0.087

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Acknowledgments The authors would like to thankEduardo Nahmad for supporting the opening phase ofthis work and Richard K. Chang for permitting itsconclusion.

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