BRIEF COMMUNICATION

Discussion on dependent light scattering phenomenon in whitepaint films

Jean-Claude Auger, Brian Stout

� American Coatings Association & Oil and Colour Chemists’ Association 2013

For clarity purpose, this document was redacted usingsimilar structure as Fitzwater and Hook communication.1

Relation between dependent and multiplescattering

In reference (2) we discussed the relationship betweenmultiple and dependent scattering in white paint film.Based on theoretical considerations as well as exper-imental knowledge, we concluded that: ‘‘Dependentscattering of light in white paint films containing rutiletitanium dioxide pigments was a particular manifesta-tion of multiple scattering’’ since light propagation insuch systems should be described by the full vectorFoldy-Lax type multiple scattering equation of light.

We deliberately limited the scope of our conclusion tostrongly scattering particles on account of the possibilityto observe dependent scattering in the absence ofmultiple scattering in a wide range of highly concen-trated colloidal systems like latex emulsions. In suchweakly scattering systems, optical properties are knownto be remarkably well modeled by the InterferenceApproximation (ITA) when the Rayleigh–Gans criteriaholds 2x|n � 1| � 1, where x is the size parameter of thesystem and n is the contrast in index of refraction.

This conclusion does not contradict the possibleobservation of multiple scattering in the absence ofdependent scattering in a large variety of heteroge-neous media excluding white paint films loaded withTiO2 pigments. In such cases, light propagation can beeffectively described by the Radiative Transfer Equa-tion, which is, as formally demonstrated by Mish-chenko et al.,3 a simplification of the Foldy-Lax vectormultiple scattering equation of light.

Consequently, we do not see many major dissimi-larities and/or contradictions between our statementsin reference (2) and the explanations given by Fitz-water and Hook in their ‘‘Response.’’1

We felt that the only difference resided in theunderstanding of the scattering efficiency vs PVC curveobserved on paint films pigmented with RopaqueTM

Ultra, which is mentioned. In this case, our interpre-tation is the following:

• Due to the core–shell structure, 40% PVC ofRopaqueTM corresponds to only about 12% VoidVolume Concentration (VVC) which is far frombeing ‘‘highly concentrated.’’ Thus, the combina-tion of relatively low contrast index of refractionwith moderate volume filling fraction could be theorigin of the absence of dependent light scatteringeffects.

• It is premature to conclude a complete absence ofparticle correlations from sole observation of quasi-linear variation of the scattering efficiency as afunction of the PVC. The validation of such astatement would also require showing linearity ofthe scattering mean free path vs PVC since a quasi-linear variation of S can result from a combinationof nonlinear variation of the extinction coefficients,originating from particle correlations, compensatedby a decrease of the asymmetry parameter causedby multiple scattering.

J.-C. Auger (&)Expert Capability Group RD&I AkzoNobel Eco City,1788 West Nan Jing Road, 200040 Shanghai, Chinae-mail: jean-claude.auger@akzonobel.com

B. StoutInstitut Fresnel, UMR 6133, Aix-Marseille Universite’,D.U. de St Jerome, 13397 Marseille Cedex 20, France

J. Coat. Technol. Res., 10 (6) 929–931, 2013

DOI 10.1007/s11998-013-9539-6

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Independent/dependent scattering transition

The discrepancy between the rise of particle correla-tions observed in NCTMA numerical simulations4 and‘‘measured’’ values of S originates from the impossi-bility of completely averaging far field coherent effectsin finite particle-number simulations with reasonablecalculation times. Such a discrepancy is not problem-atic as long as this model, like any other, is used withinits area of validity.

Based on the analysis of experimental data gatheredin our laboratory and in the literature as well as onfundamental theoretical considerations, but not onNCTMA calculations presented in reference (4), weconcluded that the notion of a concentration thresholdwas inadequate to characterize the independent todependent light scattering transition.

The difficulty resides in the ambiguous definition ofthe threshold as ‘‘the particle concentration belowwhich dependent scattering effect are negligible’’ or‘‘above which dependent scattering effects are likely tobe significant,’’ given, for instance, in references (1)and (5). There is a clear ambiguity in the quantificationof ‘‘negligible’’ and ‘‘significant.’’ Furthermore, differ-ent systems can show very different variations of S as afunction of the PVC. Consequently, the identificationof a PVC threshold is partly subjective and for thisreason different authors have chosen different valuesof the interaction function as a reference.

Now, this inadequacy does not prevent the paintcommunity from defining, for standardization pur-poses, ‘‘a threshold-type transition’’ at a given value ofthe interaction function, as long as it is clear that thechosen value is not based on any fundamental changesin the scattering regime and that it is somehowarbitrary.

We believe that the use of the term ‘‘thresholdconcentration’’ would be more appropriate to charac-terize the PVC at which decrease of the scatteringefficiency can be observed in certain systems. In such acase, not only can the PVC be unequivocally deter-mined (max of the curve and change of sign of thederivative) but also it reflects fundamental changes inthe light propagation regime.

Practical semiempirical hiding models for paints

Semiempirical models reported in the literature,including the one described in references (5) and (1)principally focus on the reproduction and/or predictionof paint film’s scattering efficiency as a function of thePVC (see for example, path OA¢ in Fig. 1). Inreference (2) our intention was clearly different as wefocused on the study of the variation of the scatteringefficiency as a function of the pigment spatial state ofdispersion, at fixed PVC (see path BB¢ in Fig. 1).

Because of this possible distinction, we find thedescription of dependent scattering given by van delHulst as ‘‘variation of scattering properties as func-tion of the position correlation of particle’’ moresuitable than the definition which limits occurrenceof dependent light scattering to the only change inPVC.

In the framework of our study, we assumed that fora given paint formulation, the scattering efficiencycould be expressed in terms of two variables: the PVCand the ‘‘dispersion coefficient,’’ noted as / and X,respectively. The latter characterizes the spatial stateof the pigment’s dispersion within the paint film. Itdepends on the package of surfactant as well as theengineering or drying processes.

Ideally, and assuming simple relation of additivity,the dispersion coefficient should have been correlatedto the worst (X = 0) and optimum (X = 1) pigmentspatial states of dispersion such that:

S /;Xð Þ ¼ XSmax /ð Þ þ 1� Xð ÞSmin /ð Þ ð1Þ

In practice, it is unfortunately impossible to preciselyidentify Smax (/) and Smin (/) before carrying out aminimum series of measurements, varying either thepackage of dispersants and/or the engineering mixingprocess.

Therefore, at the first trial, the only known infor-mation is: the set of experimental data SE (/i), whichvariations lies anywhere between the unknown curveSmax (/) and Smin (/) and the independent scatteringfunction SI (/) deduced from the former (see Fig. 1).Consequently, we had to restrict the definition of thedispersion coefficient to:

SI

Smax

SE

SminB

B′

A′

PVC (unitless)0

S (

μm–1

)

Fig. 1: Scattering efficiency as function of the ParticleVolume Concentration for a given paint formulation

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S /;Xð Þ ¼ XSI /ð Þ þ 1� Xð ÞSE /ð Þ ð2Þ

where now, X = 0 and X = 1 are related to the currentmeasurements and independent scattering efficiencies.As a result, X has to be readjusted in the process if anew series of measurements shows lower values of S.This feature has not been problematic for the purposeof our work described in reference (2).

The dependent light scattering model

Inspection of references (2) and (5) as well as hopefullythe clarifications we added in the previous section,clearly should show that the semiempirical method weintroduced and the Dependent Scattering Theoryproposed by Fitzwater and Hook are not only basedon completely different approaches but also havedifferent purposes.

Nevertheless, since reference (1) largely emphasizedthe outstanding reliability of the Dependent ScatteringTheory developed in reference (5) we simply wouldlike to make the following comment:

This dependent scattering theory uses the scalarapproximation of the more general and rigorous Miescattering theory which is based on the vector waveequations of light.

The key relation in this model involves evaluating aseries of Al(rA) coefficients which represent the weightof the lth partial wave contribution to the totalscattering cross section, as a function of the radius ofthe associated sphere, noted rA. Such coefficients aregiven by equation (7) in reference (5) which is:

Al rAð Þ ¼I rAð Þ

D¼R rA

0 jl krð Þj j2r2drR1

0 jl krð Þj j2r2dr: ð3Þ

The denominator of this expression is an impropernonconvergent integral. As a result, equation (3) ismathematically erroneous and within the framework

presented in references (1) and (5) the Al(rA) coefficientscannot be evaluated either analytically or numerically.

A well-known consequence of this lack of conver-gence is found in the impossibility to fix a finitenormalization of the Vector Spherical Wave Functions,which consequently are only normalized on theirangular components.6

In practice, replacing the ill-defined denominator ofequation (3) with a truncated integral D¢(rm) like thatdefined in equation (4) would have two consequences:(i) it would be mathematically and physically incorrect,(ii) regardless of the fixed value of rA, practically anydesired value of A0lðkrÞ could be obtained by adjustingrm, which has now become a free parameter.

A0l krð Þ ¼ I rAð ÞD0 rmð Þ

with; D0 rmð Þ ¼Zrm

0

jl krð Þj j2r2dr: ð4Þ

References

1. Fitzwater, S, Hook, JW, III, Response to ‘‘Dependent lightscattering in white paint films: clarification and application ofthe theoretical concepts’’ [Auger, J-C., Stout, B., J. Coat.Technol. Res., DOI 10.1007/s11998-011-9731-9]. J. Coat.Technol. Res., 10 (6) (2013). doi:10.1007/s11998-013-9508-0

2. Auger, J-C, Stout, B, ‘‘Dependent Light Scattering in WhitePaint Films: Clarification and Application of the TheoreticalConcepts.’’ J. Coat. Technol. Res., 9 (3) 287–295 (2012)

3. Mishchenko, MI, Travias, LD, Lacis, AA, Multiple Scatteringof Light by Particles. Radiative Transfer and CoherentBackscatttering. Cambridge University Press, Cambridge,2006

4. Auger, J-C, Martinez, VA, Stout, B, ‘‘Theoretical Study of theScattering Efficiency of Rutile Titanium Dioxide Pigments asa Function of Their Spatial Dispersion.’’ J. Coat. Technol.Res., 6 (1) 89–97 (2009)

5. Fitzwater, S, Hook, JW, III, ‘‘Dependent Scattering Theory:A New Approach to Predicting Scattering in Paints.’’ J. Coat.Technol., 57 (721) 39–47 (1985)

6. Tsang, L, Kong, JA, Shin, R, Theory of Microwave RemoteSensing. Wiley-Interscience, New York, 1985

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