# Dependent light scattering in white paint films: clarification and application of the theoretical concepts.

Abstract Among the numerous publications analyzing the causes and
consequences of titanium dioxide crowding on the optical properties of
white paint films, one notes some inconsistencies. First, a significant
number of studies are inclined to describe "dependent" and
"multiple" scattering of light as distinct phenomena. Second,
the transition from independent to dependent light-scattering is often
associated with an ill-defined "threshold" concentration. The
aim of this study is to clarify the intricate connections between these
two scattering regimes and in particular to show that for white paint
films loaded with rutile titanium dioxide pigments,
"dependent" light scattering is merely a particular
manifestation of multiple scattering processes. We also clarify that the
transition from independent to dependent scattering is a continuous
process that cannot be formally related to a specific threshold in the
pigment volume concentration. Finally, we propose a simple method based
on the dependent scattering amplitude to assist paint formulators facing
the task of improving the hiding power of a white paint either by
increasing the quantity of pigments or by improving their spatial state
of dispersion.

Keywords Light scattering, Dependent scattering, Rutile titanium dioxide pigments, Multiple scattering

Introduction

Over the last 50 years, the causes and consequences of titanium dioxide crowding effects on the optical properties of white paint films have been the subject of countless technical and fundamental articles. It is now well established that an increasing pigment-filling fraction gives rise to "dependent" light scattering as a fundamental phenomenon leading to a gradual loss in the scattering efficiency. Dependent scattering, as defined by Van de Hulst, (1) represents any regime of light propagation in which the optical properties of the bulk inhomogeneous material depend on the position correlations of the pigment inclusions dispersed in the host medium.

A bibliographical search shows that a major ambiguity remains concerning the identification of the underlying phenomena that govern such regime. In particular, it appears that a series of publications (2) tend to systematically dissociate the origins of dependent scattering from multiple scattering phenomena even in white paint films. This point of view was clearly upheld by Fitzwater and Hook III (3) in a famous, award winning article, which proposed a novel theory for calculating the effect of dependent light scattering in white coatings. In this approach, the loss in scattering efficiency is correlated to the gradual overlapping of finite spatial intrinsic zones, also called scattering volumes, centered on each scatterer. Increasing the pigment volume concentration (PVC) thus augments overlap, resulting in losses in scattering efficiency. Soon after, Drolen and Tien explicitly strengthened this assertion in a renowned contribution to the study of dependent scattering of light (4) among soot particles.

Keeping the above commentary in mind, the first objective of this study is to clarify the relationship between these two scattering regimes and to show that the process of dependent scattering of light in white paint films containing TiO2 pigments is only one particular manifestation of the multiple scattering phenomena described within the Foldy-Lax formalism. We first briefly describe some light scattering theories that model the optical properties of heterogeneous media. Next, we examine the criteria identifying the transitions from single-to-multiple scattering as well as independent-to-dependent scattering regimes. Finally, evidence of the relation between dependent and mutliple scattering of light in a highly scattering medium is corroborated via numerical simulations.

An additional difficulty frequently encountered in the literature resides in the characterization of the dependent light scattering transition as a threshold type process in terms of the PVC. Consequently, an objective of this study is to show that this transition is a continuous function of the PVC that cannot be related to a "threshold." Finally, we provide a simple procedure based on the use of an interaction function coupled with a semi-empirical model to assist the formulator when faced with the difficult task of increasing the hiding power of paints or slurries by either increasing the PVC or improving the spatial state of dispersion of the pigments.

Study of electromagnetic couplings involved in light-scattering processes between Ti02 pigments

Previous clarification

One of the first experimental studies showing evidence of the dependent scattering of light phenomena in white paint films was published by Tinsley et al.5 in 1949. This study became all the more important since it was used by Drolen and Tian to support their assertion that dependent scattering could not originate from multiple scattering. Since Tinsley's study is difficult to obtain (and sometimes erroneously referenced in the literature) it is worth reviewing its content and conclusions.

An aim of Tinsley's study was to present rutile titanium dioxide pigments as an alternative to the anatase crvstalline form in the formulation of white coatings. Opacity was therefore only one of the many properties analyzed. Measurements of hiding power of different paints as a function of the filling fraction, film thickness, and the two types of white pigments were presented. The particular case of printing ink, characterized by thin film thicknesses, and high pigment-filling fractions, applied on a black substrate was also briefly discussed. Based on a series of results that were not presented in their publication, the authors concluded that opacity and degree of dispersion of the pigments were strongly related. A more complete analysis on this matter had been announced for the near future. However, as we have not been able to find it, this study may have remained confidential.

Despite this series of results, the above article never explicitly mentions either dependent or multiple scattering phenomena. Furthermore, the interpretations of the variations in hiding power as a function of the PVC and film thickness arc kept to simple qualitative comparisons between two crystalline forms.

Consequently, it seems plausible that Drolen and Tien may have drawn their own conclusions when asserting that it could be deduced from Tinsley's study that the crowding effect was not due to multiple scattering. In our opinion, there is no clear demonstration or any experimental results in this article that conclusively justifies such a statement.

Review of the tight propagation models in heterogeneous media

Aside from stochastic formalism, such as Monte Carlo algorithms, there are two different analytical approaches to describe the propagation of light through paint films: Analytic Theory (AT) (6) and methods based on the Radiative Transfer equation (RTE). (7)

Analytical scattering theory is generally based on the excitation field concept which leads to Foldy-Lax-lype multiple scattering equations. (8) The main idea is that there is a local excitation field associated with each scatterer, which is defined as the superposition of the incident field with the field scattered by all the other particles in the system. The field scattered by the ith particle not only depends on the properties of the object, but also on its excitation field, and consequently, all N particles are electromagnetically coupled. Since this model takes into account all interactions between the scatterers, it can be evaluated nearly analytically for a limited number of scatterers (9) or, by truncating the hierarchical expansion of the multiple scattering equation of light. (10)

Radiative transfer theory directly characterizes the transport of energy throughout the medium without resorting to field propagative calculations. Since scattered intensities are simply summed in this context, the RTE effectively neglects coherent effects. The heuristic derivation of the RTE supposes the existence of a characteristic volume element within the medium's optical properties which are parameterized by a phase function as well as scattering and absorption efficiencies. The RTE describes the balance of flux that propagates though this elemental volume, and the multiple scattering processes are taken into account by integrating the flux incident from the surrounding medium. From the well-known Kubleka-Munk1 approach and its numerous derivations to more advanced models, (13) the RTE has been able to model with relative success, the reflection coefficients of a variety of decorative paint films as well as a wide range of heterogeneous media from solar panels (14) to biological tissues (15) and cosmetics. (16)

An important recent advance is the microscopic derivation of the RTE presented by Mishchenko et al.17 They demonstrated that the RTE could be derived from the Maxwell and Foldy-Lax equations without the introduction of phenomenological concepts, provided that (a) the observation point is in the far-field zone of the medium under study, (b) all particles are located in the far-field zone of one another (so that the scattered field can be approximated as a spherical outgoing wave), (c) the positions and orientations of the particles are uncorrelated (independent scattering), and (d) multiple scattering events returning to the same particle are neglected (also referred as the Twersky approximation).

Mishchenko (18) has also recently questioned the traditional descriptions of single and multiple scatterings of light phenomena widespread in the literature. He suggests that multiple scattering is a purely mathematical construct rather than a true physical process. He also points out that since in the context of microscopic quantum electrodynamics, photons are not localized particles of light, the "multiple scattering" terminology has been erroneously used during the last 60 years to characterize what he describes rather as the scattering of arbitrary "packets of energy."

Characterization of the single-to-multiple scattering transitions

In analytic scattering theory, the transition from single-to-multiple scattering is assumed to take place when the contributions of the scattered radiation to the excitation fields of the individual scattered are non-negligible with respect to the external incident field. Although this criterion has intuitive appeal, it depends on the interpretation of the term "negligible" that refers to immeasurable "partial field intensities."

In the absence of absorption, it is often more suitable to base a transition criterion on the relative comparison between the relevant medium dimension, L, and the photon scattering mean free path, lsca, representing the average distance traversed by a photon (henceforth called an "energy packet") between two scattering events and which can both be calculated and measured. This latter length is defined, lsca = 1(Np Csca) where Np and Csca represent the number of particles per unit volume and the scattering cross section. The greater the ratio of L compared to lscaj the larger the number of scattering events undergone by each photon. Nevertheless, this approach is limited because of the lack of a rigorous theoretical framework for calculating Csca in dense systems. Consequently, when lsca cannot be measured, it is generally approximated by extrapolating known analytic expressions in dilute media. In practice, one considers that multiple scattering occurs when there is no longer a simple proportionality between the number of particles per unit volume and the total measured scattered intensity, thus implicitly assuming that the transition from single-to-multiple scattering is a continuous process.

Characterization of the independent-to-dependent scattering transition

Since one expects coherent effects to act on the wavelength scale, an oft-mentioned criterion for predicting the onset of dependent scattering at a local scale is when the particle separation distances are of the order of the wavelength. A disadvantage of this criterion is that it does not take into account the influence of the refractive index contrast of the scattering heterogeneities, which, as we discuss in the next section, is a fundamental parameter.

A more practical method to identify the transition from independent-to-dependent scattering at the local scale is to monitor the variation of interaction function, [DELTA], as a function of the pigments volume fraction, [empty set] This function is defined as [DELTA] [empty set] [equivalent to] [S.sub.D] [empty set]/[S.sub.I] [empty set] where [S.sub.D]is the calculated (or measured) scattering efficiency of the system, and Si represents the scattering efficiency of the same system assuming independent scattering conditions. The occurrence of dependent scattering is thereby revealed by any deviation of [DELTA]from unity. When studying the optical properties of dried paint films, the determination of [S.sub.D] is achieved via ASTM,19'20 which is based on the inversion of Kubelka-Munk equation.

Since Radiative Transfer theory has been shown to reasonably model the optical properties of sparse heterogeneous media, the occurrence of dependent scattering regime at the macroscopic scale can sometimes be presumed when such a model significantly fails to fit experimental data collected from dense inhomogeneous media.21,22 Nonetheless, such failures do not permit the conclusion that dependent and multiple scattering phenomena are two distinct phenomena. It only shows that one or more of the assumptions underlying the RTE framework no longer apply. Two plausible causes of such a failure in predicting the bulk scattering properties are (a) the occurrence of far-field coherent scattering in the absence of strong near-field couplings, or (b) the presence of strong near-field electromagnetic couplings due to multiple scattering between neighboring particles. The aim of the next section is to discuss which of those two possible causes is responsible for the TiO2 crowding effect.

Far-field coherent scattering

Single coherent scattering, also referred to as far-field-dependent scattering by Cartigny et al.,23 results from the occurrence of far-field interferences between the scattered electric waves originating from each particle and is a consequence of the gradual spatial ordering of the medium when the PVC is increased. The computational difference with respect to single scattering approximation is that the total scattered intensity is calculated by taking the square module of the total scattered field instead of directly summing the scattered field intensities by each particle. Also, in the absence of long- and medium-range orders, coherent effects can still exist but in the form of random speckle. It is in fact the only the dependent light-scattering regime that is de-correlated from multiple scattering. The simplicity of its theoretical framework makes it widely useable in many optical techniques (24), (25) employed to study the structure and composition of certain types of heterogeneous media.

[FIGURE 1 OMITTED]

The occurrence of such a scattering regime requires the following conditions to be fulfilled: (1) multiple scattering processes must be negligible, (2) the incident radiation must have sufficient spatial and temporal coherence to allow coherent effects, and (3) particle sizes and shapes must be sufficiently uniform, and the filling fraction must be sufficiently high to allow self-structuring of the medium.

In the case of white paint films containing Ti02 pigments, such conditions are rarely observed for the following reasons: (1) under ordinary observation conditions, the incident light originates either from the sun or from an incandescence lamp which are both diffuse as well as temporarily and spatially incoherent; (2) the shape, orientation, and size distributions of the Ti02 particles combined with the presence of fillers decrease the short- and intermediate-length scale ordering of the medium; and (3) near-field interactions between neighboring particles cannot be neslected.

Evidence of far-field coherent scattering vs near-field interaction via numerical simulations

To illustrate the nature of the underlying mechanism responsible for dependent light scattering between Ti02 pigments, we calculated the exact scattered intensity, denoted idep, for a variety of particles' ensembles (see Fig. 1), and compared it with the scattered intensity given by the single coherent scattering approximation, denoted iffa- The exact intensity, idbp. is calculated using a Recursive-Centered T-Matrix Algorithm, (26) which provides quasi-exact solutions of the multiple scattering equation of light in finite aggregates. Numerical simulations were performed as a function of the filling fraction and the relative index of refraction, nY defined as the ratio between the index of refraction of the scatterers and the surrounding media noted n and n0. Description and application procedures of this theoretical framework have been described previously. (27)

The strength of the near-field interactions can be deduced from the analysis of the Pearson correlation coefficient, g, between Ini.p and Iffa-According to its definition (see Appendix), when the near-field interactions are negligible, Idep [right arrow]IfFa and[pounds sterling]1. However, the stronger interactions increase the difference between IDE:p and IFfa with [pounds sterling]usually decreasing toward zero.

[FIGURE 1 OMITTED]

Figure 2 displays the variations of $ as function of nr from 1.1 to 2.1 at the filling fractions of [10.sup.-4], [10.sup.-2], 2.5 x [10.sup.-2], and 5 x [10.sup.-2]. The values nr = 1.1, 1.2, 1.8, and 2.1 are all approximately obtainable in practice via different combinations of scatlerers composed of latex polymers (n = 1.5) or rutile titanium (n = 2.8) dispersed in either water (no = 1.33) or a polymer matrix. One can observe that the numerical framework we applied demonstrates the large influence of the relative index of refraction in the nature of the electromagnetic couplings between the heterogeneities. For low values of nr the light-scattering process is mainly driven by far-field interactions, independently of the filling fraction. Nevertheless, as the relative index of refraction increases, the contribution of the near-field couplings also increases. Higher filling fractions correspond to stronger couplings and consequently to larger drops in the correlation coefficient. The dependent scattering phenomena from latex emulsions and Ti[O.sub.2] suspensions thus involve two different mechanisms: far-field single coherent scattering and near-field coherent multiple scattering. Furthermore, as all near-field couplings were taken into account via the Foldy-Lax equation, it is possible to state that dependent scattering of light among [TiO.sub.2] particles can be seen as only a particular manifestation of multiple scattering processes.

Finally, common knowledge on light-scattering phenomena holds that light-scattering properties depend not only on nr and[empty set], but arc also on a function of the particle size (see, for example, reference 28). In order to keep computations times at a reasonable level, the radius of the scattering objects in this study was simply fixed to 0.11 [micro] since this is characteristic of the average size of commercial [TiO.sub.2] pigments used in white paint formulations.

Partial conclusions

The results of the previous discussions lead to the following conclusions:

(a) In white paint films, quantitative dependent scattering calculations of the electromagnetic near-field interactions between neighboring particles require the use of the exact vector multiple scattering equation of the Foldy-Lax type.

(b) Fizwater and collaborators could only reach semi-qualitative agreements when applying their semi-empirical theory on samples of latex emulsions, since they were using a near-field interaction model elaborated from Ti02 samples on a system that is mainly characterized by far-field coherent interactions. The reverse also holds: i.e., Latex emulsions are popular candidates for systematic experimental studies on account of the excellent control that can be achieved with respect to particle's size and shape distributions. Nevertheless, it appears hazardous, to extrapolate semi-empirical results elaborated from latex emulsions to model the rutile titanium crowding effect.

(c) In semi-empirical models, near-field dependent light scatterings are sometimes included as "corrections" to the far-field single coherent scattering theory.29 If such an approach can lead to acceptable numerical adjustments with experimental data, then it is questionable since far-field and near-field dependent scatterings involve two different fundamental mechanisms.

(d) The modeling of dependent light scattering effects in dense media strongly depends on the value of the relative index of refraction of the system: (1) For low nr, the use of single coherent scattering approximation can lead to a fairly accurate modeling provided the particles' shape can be approximated by spheres and that their spatial correlation is mainly driven by a hard sphere potential. (2) For large nx, dependent scattering effect can be either included in the RTE framework by modifying the local optical properties via theoretical or semi-empirical approaches[sup.30,31] or by using the QCA or DRTF frameworks. [sup.32]

(e) Dependent-light-scattering transition is often represented as a function of the particle radii and the PVC. Their analyses and interpretations of such charts should be handled with care because (1) some of them do not take into account the influence of refractive index contrast; and (2) they sometimes do not precisely indicate whether they solely represent the far-field-dependent scattering phenomenon or the entire dependent scattering process including near-field interactions. Thus, it occasionally happens that far-field coherent scattering charts are incorrectly used to illustrate [TiO.sub.2] crowding effect. (33)

(f) The strong near-field interactions are not readily modeled in realistic colloidal systems within the presently available theoretical frameworks, which are the chief causes of the inability to the retrieve particle size distributions of dense titanium oxide paint films or slurries from light-scattering techniques, such as Frequency Domain Photon Migration. (34)

Study of the dependent light-scattering transition

Description and characterization of dependent scattering transition

In principle, the wavelength dependency of the pigments' scattering efficiency would make it necessary to define a transition range for each wavelength of the visible spectrum. In practice, this complexity can be overcome by evaluating a wavelength-independent scattering efficiency as introduced in ASTM 2805D.

Typical analysis from dried paint films measurements (see Fig. 3) show that at least below the critical pigment volume concentration (CPVC) and in the absence of trapped microvoids into the film, the variation of the wavelength independent scattering efficiency is a monotonically increasing function of the PVC. The linear trend at low filling fraction is characteristic of weakly interacting particles and can be accurately reproduced by independent scattering theory. As the PVC is increased, the proportionality between $ and [empty set] is gradually lost because of the onset of the dependent light-scattering phenomenon.

[FIGURE 3 OMITTED]

Different theoretical studies have also pointed out the continuity of the variation of the scattered near-electric held intensity as a function of the separation distance between [TiO.sup.2] pigments.[SUP.35,36] This lack of discontinuity kept Fitzwater and Hook III from being able to a priori define the spatial boundaries of the scattering volume introduced in their theoretical framework. To overcome this problem, they had to arbitrarily correlate the extensions of the scattering volumes to the boundaries of "associated spheres" defined as "the spheres on neighboring particles that just touch each other."

Despite numerous experimental and theoretical evidences to the contrary, it is still sometimes presumed3 that the [TiO.sup.2] crowding effect is a threshold-type process in terms of the PVC. Such a threshold, hereafter denoted [empet set]t is said to correspond to the loss of proportionality between S and[empt set]. However, the identification of a threshold concentration and a loss of proportionality are (a) physically erroneous because there is no discontinuity in the underlying fundamental process associated with the scattering event, and (b) mathematically ambiguous since no further indication is given on the value of the interaction function, [DELTA], associated with[empet set]t,

Finally, it must be emphasized that if the interaction function does provide information on the magnitude of the dependent light-scattering phenomenon at a local scale, it does not provide quantitative information on its effect on the macroscopic optical parameters such as the reflection and transmission coefficients. The difficulty to elaborate a more practical characterization based on those parameters, which are key indicators for paint formulators, originates from their dependence on various variables such as the film thickness and the substrate's color. Thus, any intent to characerize dependent light scattering transition via macroscopic optimal parameters would require elaborating a standardized framework to define the values of the additional variables.

Characterization of dependent light-scattering amplitude

One can circumvent the above difficulty by characterizing the dependent light-scattering amplitude via a filling fraction-independent parameter, which would unequivocally allow for the determination and comparison of the strengths of [TiO.sup.2] crowding effects between different paint formulations. Such a determination can be made by adjusting the variation of the wavelength-independent scattering efficiencies with a two- or three-parameter mathematical function.

A typical example is given in Fig. 3, which shows wavelength-independent scattering efficiencies as a function of the PVC evaluated by applying ASTM 2805D on the diffuse reflection coefficients of a series of white paint films.

The variation of the scattering efficiencies are accurately fitted up to 20% PVC by a second-order polynomial function given by SD[empty set] = a1[empty set] + [a.sup.2][empty set] [sup.2] Thus, a1 directly provides % which is defined as of the derivative of SD in the limit of [empty set][right arrow] 0, whereas [a.sup.2]can be used as a filling fraction-independent coefficient to characterize the dependent scattering amplitude. Further investigations should be performed to identify the optimum mathematical function for modeling a broad range of white paint films scattering efficiencies in the largest possible range of concentrations.

The samples under study were simple coatings composed by rutile titanium dioxide pigments dispersed in an acrylic polymer with the adequate package of surfactant. Measurements were performed in the visible range with diffuse illumination and D65 illuminant on a spectrophotometer Minolta CM-3700d equipped with an integrating sphere.

Practical application

A complex challenge that paint formulators have to face is to establish whether it is more economically viable to increase the scattering efficiency of a given paint formulation by solely augmenting its PVC or by trying to improve the pigments' spatial state of dispersion, assuming that the latter is technically possible and financially viable.

The principal difficulty resides in the lack of theoretical and semi-empirical models that can accurately predict the maximum scattering efficiency of a given paint formulation, which corresponds to the optimum pigment spatial dispersion state (see typical variations in Fig. 4). Consequently, it is generally solved via trial-and-error processes by varying the engineering dispersion processes and modifying the packages of surfactants of the initial formulation. The aim of this section is to present a straightforward procedure to help the formulators develop their empirical knowledge to progressively decrease the number of necessary samples formulated in the laboratory. The method is the following:

(a) Samples of the original paint formulation are applied at different PVCs on white and black substrates at constant wet thickness. The scattering efficiencies [S.sub.I] and [S.sub.D] of each application are obtained from diffuse reflectance measurements using the procedure proposed in the previous section.

(b) The influence of the pigments' spatial state of dispersion is taken into account by expressing the paint scattering efficiency, denoted [s.sub.d.sub.t]as a linear combination of [S.sub.I] and [S.sub.D] via the introduction of a dispersion coefficient, denoted [OMEGA]such that

[FIGURE 4 OMITTED]

[S.sub.D.sup.T][empty set],[OMEGA]= [OMEGA]S I [empty set] + (1 -[OMEGA]) SD[empty set], 0 [less than or equal to] [OMEGA][less than or equal to] 1 (1)

The limit [OMEGA] = 1 corresponds to the ideal case of independent scattering (curve (a) in Fig. 4), whereas the limit [OMEGA]= 0 corresponds to the paint scattering efficiency of the original formulation (somewhere between curves (b) and (c) in Fig. 4).

(c) Measurements of the variations in the scattering efficiency as a function of the pigment's spatial state of dispersion are realized via the evaluation of a gain function, denoted G, defined as G =[s.sup.d.sub.t] [empty set] [OMEGA]/[S.sub.D.sub.T] [empty set] [OMEGA]= 0).

(d) The Kubelka-Munk theory is employed to evaluate the reflection coefficients of the paint films over white and black substrates, denoted RW(Z, [empty set] [OMEGA])\and R b{Z, [empty set] [OMEGA]) respectively, as functions of the film thickness, Z,[empty set], and [OMEGA].

(e) Defining a contrast ratio as, CR(Z,[empty set] [OMEGA]) = R b (Z, [empty set] [OMEGA])/, one plots a threshold curve corresponding to the thickness required to produce a contrast ratio of CR = 0.98 for each value of\[OMEGA] as function of Z and [empty set].

(f) Finally, the previously identified curves are superimposed in a single chart as shown in Fig. 5,

In the framework of our example, we assume that the formulator is dealing with an original white paint formulation whose hiding power is HP = 10.8 m2/L at a pigment's filling fraction of 17% (see Point A of curve HPi in Fig. 6).

First, we, imagine that one wishes to improve the formulation to reach a hiding power of 22.2 m2/L (curve HP3 in Fig. 6). Inspection of Fig. 6 shows that neither improvements in spatial dispersion nor increases in the PVC can reach this objective.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Next, let us assume that the original formulation has to be modified to reach a hiding power of 15.4 m2/L (curve HP2 in Fig. 6). Inspection of Fig. 6 shows that the formulator has three options:

(1) To increase the filling fraction up to 19.5% (see path AB in Fig. 6) keeping the pigments' spatial state of dispersion unchanged ([ohm]=0.0).

(2) To improve the pigments' spatial state of dispersion up to [ohm]=0.6 (path AE in Fig. 6) keeping the filling fraction to its initial value of 17%.

(3) To combine an improvement of the pigments' spatial state of dispersion with an increase of the filling fraction: see, for example, path ACD in Fig. 6, which involves an increase from [ohm]=0.0 to[ohm]=s 0.2 and an augmentation of the filling fraction from 17% to 18.5%.

The values of the gain function G corresponding to options (1), (2), and (3) are 0.00, 1.33, and 1.11 respectively.

Thus, based on their semi-empirical knowledge and the newly acquired information, formulators must consider which options are viable and which must be discarded. For instance, in the framework of our example, option (3) looks possible and more realistic than option (2), which supposes an improbable increase of 1.33 of the paint-scattering efficiency by solely improving the pigments" spatial state of dispersion.

Conclusion

We have used numerical simulations based on a rigorous theoretical framework to show that contrary to some statements made in the literature, dependent scattering of light in white paint films containing rutile titanium dioxide pigments is a particular manifestation of multiple scattering. More specifically, it results from electromagnetic near-field interactions between neighboring particles which is described by vector electromagnetic treatments of light (like rigorous solutions of the electromagnetic Foldy-Lax equations).

We have also pointed out that the transition from independent-to-dependent scattering, as a function of the PVC is a continuous process that cannot be formally related to a specific concentration threshold. Consequently, attempts to characterize dependent light-scattering transition by either determining the value of [DELTA]t = f([empty set]t) at a particular concentration[empty set] t, or by providing the filling fractions, [empty set] t corresponding to a standardized value of [DELTA]t would prove problematic. The underlying reason is that the absence of a threshold-type transition does not allow any unequivocal and unambiguous relationship between these two quantities. A common consensus among the paint community would be required to justify the choice of their values which would in any case remain questionable. To circumvent this problem, we have to characterize the Ti02 crowding effect amplitude via filling fraction-independent parameters.

Finally, we proposed a method based on the introduction of a dispersion coefficient and the evaluation of a gain function, to support the formulator in the difficult task of evaluating whether the hiding power of a white paint formulation should be improved by increasing the amount of pigment or by improving the spatial dispersions state.

Acknowledgments The authors would like to thank Fernando Zaldo and Centro de Invcstigacion en Polimeros of Grupo Comex who provided the experimental set of data used in this study.

Appendix

Expression of the Pearson correlation coefficient

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

For a determined system defined by the set of parameters (nr,[empty set]), calculations were performed on K = 1000 different random configurations, which were proven to be sufficient for configuration average statistics.

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(25.) Blum, L, Stell, G, "Scattering of Polydisperse Fluids." J. Chem. Phys., 71 (1) 42-46 (1979)

(26.) Auger, JC, Stout, B, "A Recursive Centered T-Matrix Algorithm to Solve the Multiple Scattering Equation: Numerical Validation."./. Quant Spectrosc. Radial Transf., 79-80 533-547 (2003)

(27.) Auger, J-C, Martinez, VA, Stout B, "Theoretical Study of the Scattering Efficiency of Rutile Titanium Pigments as Function of Their Spatial Dispersion." JCT Research., 6 (1) 89-97 (2009)

(28.) Dick, VP, Ivanov, AP, "Extinction of Light in Dispersive Media with High Particle Concentrations: Applicability Limits of the Interference Approximation." /. Opt. Soc. Am. A, 16 (5) 1034-1039 (1999)

(29.) Gobel, G, Kuhn, J, Fricke, J, "Dependent Scattering Effects in Latex-Sphere Suspensions and Scattering Powders." Warns Random Media, 5 413-426 (1995)

(30.) Auger, J-C, Stout, B, Lafait, J, "Dependent Light Scattering in Dense Heterogeneous Media." Physica B, 279 21-24 (2000)

(31.) Vargas, WE, "Optical Properties of Pigmented Coatings Taking into Account Particle Interactions." J. Quant. Spec-trosc. Radiat. Transf, 78 (2) 187-195 (2003)

(32.) Tsang, L, Kong, JA, Shin, R, Theory of Microwave Remote Sensing. Wiiey-Interscience, New York, 1985

(33.) Liu, L, Gong, R, Huang, D, Nie, Y, Liu, C, "Calculation of Emittance of a Coating Layer with the Kubelka-Munk Theory and the Mie-Scattering Model." J. Opt. Soc. Am. A, 22(11)2424-2429(2005)

(34.) Richter, SM, Sevick-Muraca, EM, "Characterization of Concentrated Colloidal Suspensions Using Time-Dependent Photon Migration Measurements." Colloid Surf. A: Physicochem. Eng. Asp., 172 163-173 (2000)

(35.) McNeil, LE, Hanuska, AR, French, RH, "Orientation Dependence in Near-Field Scattering from Ti03 Particles." Appl. Opt., 40 3726-3736 (2001)

(36.) Auger, J-C, Stout, B, "Local Field Intensity in Aggregates Illuminated by Diffuse Light: T Matrix Approach." Appl. Opt., 47 (16) 2897-2905 (2008)

J.-C.Auger (El)

Kyoralis Research and Consulting, 5 rue du Loing, 75014 Paris, France e-mail: kyolarisresearch@gmail.com

B. Stout

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

DOI 10.1007/s 11998-011-9371-9

Keywords Light scattering, Dependent scattering, Rutile titanium dioxide pigments, Multiple scattering

Introduction

Over the last 50 years, the causes and consequences of titanium dioxide crowding effects on the optical properties of white paint films have been the subject of countless technical and fundamental articles. It is now well established that an increasing pigment-filling fraction gives rise to "dependent" light scattering as a fundamental phenomenon leading to a gradual loss in the scattering efficiency. Dependent scattering, as defined by Van de Hulst, (1) represents any regime of light propagation in which the optical properties of the bulk inhomogeneous material depend on the position correlations of the pigment inclusions dispersed in the host medium.

A bibliographical search shows that a major ambiguity remains concerning the identification of the underlying phenomena that govern such regime. In particular, it appears that a series of publications (2) tend to systematically dissociate the origins of dependent scattering from multiple scattering phenomena even in white paint films. This point of view was clearly upheld by Fitzwater and Hook III (3) in a famous, award winning article, which proposed a novel theory for calculating the effect of dependent light scattering in white coatings. In this approach, the loss in scattering efficiency is correlated to the gradual overlapping of finite spatial intrinsic zones, also called scattering volumes, centered on each scatterer. Increasing the pigment volume concentration (PVC) thus augments overlap, resulting in losses in scattering efficiency. Soon after, Drolen and Tien explicitly strengthened this assertion in a renowned contribution to the study of dependent scattering of light (4) among soot particles.

Keeping the above commentary in mind, the first objective of this study is to clarify the relationship between these two scattering regimes and to show that the process of dependent scattering of light in white paint films containing TiO2 pigments is only one particular manifestation of the multiple scattering phenomena described within the Foldy-Lax formalism. We first briefly describe some light scattering theories that model the optical properties of heterogeneous media. Next, we examine the criteria identifying the transitions from single-to-multiple scattering as well as independent-to-dependent scattering regimes. Finally, evidence of the relation between dependent and mutliple scattering of light in a highly scattering medium is corroborated via numerical simulations.

An additional difficulty frequently encountered in the literature resides in the characterization of the dependent light scattering transition as a threshold type process in terms of the PVC. Consequently, an objective of this study is to show that this transition is a continuous function of the PVC that cannot be related to a "threshold." Finally, we provide a simple procedure based on the use of an interaction function coupled with a semi-empirical model to assist the formulator when faced with the difficult task of increasing the hiding power of paints or slurries by either increasing the PVC or improving the spatial state of dispersion of the pigments.

Study of electromagnetic couplings involved in light-scattering processes between Ti02 pigments

Previous clarification

One of the first experimental studies showing evidence of the dependent scattering of light phenomena in white paint films was published by Tinsley et al.5 in 1949. This study became all the more important since it was used by Drolen and Tian to support their assertion that dependent scattering could not originate from multiple scattering. Since Tinsley's study is difficult to obtain (and sometimes erroneously referenced in the literature) it is worth reviewing its content and conclusions.

An aim of Tinsley's study was to present rutile titanium dioxide pigments as an alternative to the anatase crvstalline form in the formulation of white coatings. Opacity was therefore only one of the many properties analyzed. Measurements of hiding power of different paints as a function of the filling fraction, film thickness, and the two types of white pigments were presented. The particular case of printing ink, characterized by thin film thicknesses, and high pigment-filling fractions, applied on a black substrate was also briefly discussed. Based on a series of results that were not presented in their publication, the authors concluded that opacity and degree of dispersion of the pigments were strongly related. A more complete analysis on this matter had been announced for the near future. However, as we have not been able to find it, this study may have remained confidential.

Despite this series of results, the above article never explicitly mentions either dependent or multiple scattering phenomena. Furthermore, the interpretations of the variations in hiding power as a function of the PVC and film thickness arc kept to simple qualitative comparisons between two crystalline forms.

Consequently, it seems plausible that Drolen and Tien may have drawn their own conclusions when asserting that it could be deduced from Tinsley's study that the crowding effect was not due to multiple scattering. In our opinion, there is no clear demonstration or any experimental results in this article that conclusively justifies such a statement.

Review of the tight propagation models in heterogeneous media

Aside from stochastic formalism, such as Monte Carlo algorithms, there are two different analytical approaches to describe the propagation of light through paint films: Analytic Theory (AT) (6) and methods based on the Radiative Transfer equation (RTE). (7)

Analytical scattering theory is generally based on the excitation field concept which leads to Foldy-Lax-lype multiple scattering equations. (8) The main idea is that there is a local excitation field associated with each scatterer, which is defined as the superposition of the incident field with the field scattered by all the other particles in the system. The field scattered by the ith particle not only depends on the properties of the object, but also on its excitation field, and consequently, all N particles are electromagnetically coupled. Since this model takes into account all interactions between the scatterers, it can be evaluated nearly analytically for a limited number of scatterers (9) or, by truncating the hierarchical expansion of the multiple scattering equation of light. (10)

Radiative transfer theory directly characterizes the transport of energy throughout the medium without resorting to field propagative calculations. Since scattered intensities are simply summed in this context, the RTE effectively neglects coherent effects. The heuristic derivation of the RTE supposes the existence of a characteristic volume element within the medium's optical properties which are parameterized by a phase function as well as scattering and absorption efficiencies. The RTE describes the balance of flux that propagates though this elemental volume, and the multiple scattering processes are taken into account by integrating the flux incident from the surrounding medium. From the well-known Kubleka-Munk1 approach and its numerous derivations to more advanced models, (13) the RTE has been able to model with relative success, the reflection coefficients of a variety of decorative paint films as well as a wide range of heterogeneous media from solar panels (14) to biological tissues (15) and cosmetics. (16)

An important recent advance is the microscopic derivation of the RTE presented by Mishchenko et al.17 They demonstrated that the RTE could be derived from the Maxwell and Foldy-Lax equations without the introduction of phenomenological concepts, provided that (a) the observation point is in the far-field zone of the medium under study, (b) all particles are located in the far-field zone of one another (so that the scattered field can be approximated as a spherical outgoing wave), (c) the positions and orientations of the particles are uncorrelated (independent scattering), and (d) multiple scattering events returning to the same particle are neglected (also referred as the Twersky approximation).

Mishchenko (18) has also recently questioned the traditional descriptions of single and multiple scatterings of light phenomena widespread in the literature. He suggests that multiple scattering is a purely mathematical construct rather than a true physical process. He also points out that since in the context of microscopic quantum electrodynamics, photons are not localized particles of light, the "multiple scattering" terminology has been erroneously used during the last 60 years to characterize what he describes rather as the scattering of arbitrary "packets of energy."

Characterization of the single-to-multiple scattering transitions

In analytic scattering theory, the transition from single-to-multiple scattering is assumed to take place when the contributions of the scattered radiation to the excitation fields of the individual scattered are non-negligible with respect to the external incident field. Although this criterion has intuitive appeal, it depends on the interpretation of the term "negligible" that refers to immeasurable "partial field intensities."

In the absence of absorption, it is often more suitable to base a transition criterion on the relative comparison between the relevant medium dimension, L, and the photon scattering mean free path, lsca, representing the average distance traversed by a photon (henceforth called an "energy packet") between two scattering events and which can both be calculated and measured. This latter length is defined, lsca = 1(Np Csca) where Np and Csca represent the number of particles per unit volume and the scattering cross section. The greater the ratio of L compared to lscaj the larger the number of scattering events undergone by each photon. Nevertheless, this approach is limited because of the lack of a rigorous theoretical framework for calculating Csca in dense systems. Consequently, when lsca cannot be measured, it is generally approximated by extrapolating known analytic expressions in dilute media. In practice, one considers that multiple scattering occurs when there is no longer a simple proportionality between the number of particles per unit volume and the total measured scattered intensity, thus implicitly assuming that the transition from single-to-multiple scattering is a continuous process.

Characterization of the independent-to-dependent scattering transition

Since one expects coherent effects to act on the wavelength scale, an oft-mentioned criterion for predicting the onset of dependent scattering at a local scale is when the particle separation distances are of the order of the wavelength. A disadvantage of this criterion is that it does not take into account the influence of the refractive index contrast of the scattering heterogeneities, which, as we discuss in the next section, is a fundamental parameter.

A more practical method to identify the transition from independent-to-dependent scattering at the local scale is to monitor the variation of interaction function, [DELTA], as a function of the pigments volume fraction, [empty set] This function is defined as [DELTA] [empty set] [equivalent to] [S.sub.D] [empty set]/[S.sub.I] [empty set] where [S.sub.D]is the calculated (or measured) scattering efficiency of the system, and Si represents the scattering efficiency of the same system assuming independent scattering conditions. The occurrence of dependent scattering is thereby revealed by any deviation of [DELTA]from unity. When studying the optical properties of dried paint films, the determination of [S.sub.D] is achieved via ASTM,19'20 which is based on the inversion of Kubelka-Munk equation.

Since Radiative Transfer theory has been shown to reasonably model the optical properties of sparse heterogeneous media, the occurrence of dependent scattering regime at the macroscopic scale can sometimes be presumed when such a model significantly fails to fit experimental data collected from dense inhomogeneous media.21,22 Nonetheless, such failures do not permit the conclusion that dependent and multiple scattering phenomena are two distinct phenomena. It only shows that one or more of the assumptions underlying the RTE framework no longer apply. Two plausible causes of such a failure in predicting the bulk scattering properties are (a) the occurrence of far-field coherent scattering in the absence of strong near-field couplings, or (b) the presence of strong near-field electromagnetic couplings due to multiple scattering between neighboring particles. The aim of the next section is to discuss which of those two possible causes is responsible for the TiO2 crowding effect.

Far-field coherent scattering

Single coherent scattering, also referred to as far-field-dependent scattering by Cartigny et al.,23 results from the occurrence of far-field interferences between the scattered electric waves originating from each particle and is a consequence of the gradual spatial ordering of the medium when the PVC is increased. The computational difference with respect to single scattering approximation is that the total scattered intensity is calculated by taking the square module of the total scattered field instead of directly summing the scattered field intensities by each particle. Also, in the absence of long- and medium-range orders, coherent effects can still exist but in the form of random speckle. It is in fact the only the dependent light-scattering regime that is de-correlated from multiple scattering. The simplicity of its theoretical framework makes it widely useable in many optical techniques (24), (25) employed to study the structure and composition of certain types of heterogeneous media.

[FIGURE 1 OMITTED]

The occurrence of such a scattering regime requires the following conditions to be fulfilled: (1) multiple scattering processes must be negligible, (2) the incident radiation must have sufficient spatial and temporal coherence to allow coherent effects, and (3) particle sizes and shapes must be sufficiently uniform, and the filling fraction must be sufficiently high to allow self-structuring of the medium.

In the case of white paint films containing Ti02 pigments, such conditions are rarely observed for the following reasons: (1) under ordinary observation conditions, the incident light originates either from the sun or from an incandescence lamp which are both diffuse as well as temporarily and spatially incoherent; (2) the shape, orientation, and size distributions of the Ti02 particles combined with the presence of fillers decrease the short- and intermediate-length scale ordering of the medium; and (3) near-field interactions between neighboring particles cannot be neslected.

Evidence of far-field coherent scattering vs near-field interaction via numerical simulations

To illustrate the nature of the underlying mechanism responsible for dependent light scattering between Ti02 pigments, we calculated the exact scattered intensity, denoted idep, for a variety of particles' ensembles (see Fig. 1), and compared it with the scattered intensity given by the single coherent scattering approximation, denoted iffa- The exact intensity, idbp. is calculated using a Recursive-Centered T-Matrix Algorithm, (26) which provides quasi-exact solutions of the multiple scattering equation of light in finite aggregates. Numerical simulations were performed as a function of the filling fraction and the relative index of refraction, nY defined as the ratio between the index of refraction of the scatterers and the surrounding media noted n and n0. Description and application procedures of this theoretical framework have been described previously. (27)

The strength of the near-field interactions can be deduced from the analysis of the Pearson correlation coefficient, g, between Ini.p and Iffa-According to its definition (see Appendix), when the near-field interactions are negligible, Idep [right arrow]IfFa and[pounds sterling]1. However, the stronger interactions increase the difference between IDE:p and IFfa with [pounds sterling]usually decreasing toward zero.

[FIGURE 1 OMITTED]

Figure 2 displays the variations of $ as function of nr from 1.1 to 2.1 at the filling fractions of [10.sup.-4], [10.sup.-2], 2.5 x [10.sup.-2], and 5 x [10.sup.-2]. The values nr = 1.1, 1.2, 1.8, and 2.1 are all approximately obtainable in practice via different combinations of scatlerers composed of latex polymers (n = 1.5) or rutile titanium (n = 2.8) dispersed in either water (no = 1.33) or a polymer matrix. One can observe that the numerical framework we applied demonstrates the large influence of the relative index of refraction in the nature of the electromagnetic couplings between the heterogeneities. For low values of nr the light-scattering process is mainly driven by far-field interactions, independently of the filling fraction. Nevertheless, as the relative index of refraction increases, the contribution of the near-field couplings also increases. Higher filling fractions correspond to stronger couplings and consequently to larger drops in the correlation coefficient. The dependent scattering phenomena from latex emulsions and Ti[O.sub.2] suspensions thus involve two different mechanisms: far-field single coherent scattering and near-field coherent multiple scattering. Furthermore, as all near-field couplings were taken into account via the Foldy-Lax equation, it is possible to state that dependent scattering of light among [TiO.sub.2] particles can be seen as only a particular manifestation of multiple scattering processes.

Finally, common knowledge on light-scattering phenomena holds that light-scattering properties depend not only on nr and[empty set], but arc also on a function of the particle size (see, for example, reference 28). In order to keep computations times at a reasonable level, the radius of the scattering objects in this study was simply fixed to 0.11 [micro] since this is characteristic of the average size of commercial [TiO.sub.2] pigments used in white paint formulations.

Partial conclusions

The results of the previous discussions lead to the following conclusions:

(a) In white paint films, quantitative dependent scattering calculations of the electromagnetic near-field interactions between neighboring particles require the use of the exact vector multiple scattering equation of the Foldy-Lax type.

(b) Fizwater and collaborators could only reach semi-qualitative agreements when applying their semi-empirical theory on samples of latex emulsions, since they were using a near-field interaction model elaborated from Ti02 samples on a system that is mainly characterized by far-field coherent interactions. The reverse also holds: i.e., Latex emulsions are popular candidates for systematic experimental studies on account of the excellent control that can be achieved with respect to particle's size and shape distributions. Nevertheless, it appears hazardous, to extrapolate semi-empirical results elaborated from latex emulsions to model the rutile titanium crowding effect.

(c) In semi-empirical models, near-field dependent light scatterings are sometimes included as "corrections" to the far-field single coherent scattering theory.29 If such an approach can lead to acceptable numerical adjustments with experimental data, then it is questionable since far-field and near-field dependent scatterings involve two different fundamental mechanisms.

(d) The modeling of dependent light scattering effects in dense media strongly depends on the value of the relative index of refraction of the system: (1) For low nr, the use of single coherent scattering approximation can lead to a fairly accurate modeling provided the particles' shape can be approximated by spheres and that their spatial correlation is mainly driven by a hard sphere potential. (2) For large nx, dependent scattering effect can be either included in the RTE framework by modifying the local optical properties via theoretical or semi-empirical approaches[sup.30,31] or by using the QCA or DRTF frameworks. [sup.32]

(e) Dependent-light-scattering transition is often represented as a function of the particle radii and the PVC. Their analyses and interpretations of such charts should be handled with care because (1) some of them do not take into account the influence of refractive index contrast; and (2) they sometimes do not precisely indicate whether they solely represent the far-field-dependent scattering phenomenon or the entire dependent scattering process including near-field interactions. Thus, it occasionally happens that far-field coherent scattering charts are incorrectly used to illustrate [TiO.sub.2] crowding effect. (33)

(f) The strong near-field interactions are not readily modeled in realistic colloidal systems within the presently available theoretical frameworks, which are the chief causes of the inability to the retrieve particle size distributions of dense titanium oxide paint films or slurries from light-scattering techniques, such as Frequency Domain Photon Migration. (34)

Study of the dependent light-scattering transition

Description and characterization of dependent scattering transition

In principle, the wavelength dependency of the pigments' scattering efficiency would make it necessary to define a transition range for each wavelength of the visible spectrum. In practice, this complexity can be overcome by evaluating a wavelength-independent scattering efficiency as introduced in ASTM 2805D.

Typical analysis from dried paint films measurements (see Fig. 3) show that at least below the critical pigment volume concentration (CPVC) and in the absence of trapped microvoids into the film, the variation of the wavelength independent scattering efficiency is a monotonically increasing function of the PVC. The linear trend at low filling fraction is characteristic of weakly interacting particles and can be accurately reproduced by independent scattering theory. As the PVC is increased, the proportionality between $ and [empty set] is gradually lost because of the onset of the dependent light-scattering phenomenon.

[FIGURE 3 OMITTED]

Different theoretical studies have also pointed out the continuity of the variation of the scattered near-electric held intensity as a function of the separation distance between [TiO.sup.2] pigments.[SUP.35,36] This lack of discontinuity kept Fitzwater and Hook III from being able to a priori define the spatial boundaries of the scattering volume introduced in their theoretical framework. To overcome this problem, they had to arbitrarily correlate the extensions of the scattering volumes to the boundaries of "associated spheres" defined as "the spheres on neighboring particles that just touch each other."

Despite numerous experimental and theoretical evidences to the contrary, it is still sometimes presumed3 that the [TiO.sup.2] crowding effect is a threshold-type process in terms of the PVC. Such a threshold, hereafter denoted [empet set]t is said to correspond to the loss of proportionality between S and[empt set]. However, the identification of a threshold concentration and a loss of proportionality are (a) physically erroneous because there is no discontinuity in the underlying fundamental process associated with the scattering event, and (b) mathematically ambiguous since no further indication is given on the value of the interaction function, [DELTA], associated with[empet set]t,

Finally, it must be emphasized that if the interaction function does provide information on the magnitude of the dependent light-scattering phenomenon at a local scale, it does not provide quantitative information on its effect on the macroscopic optical parameters such as the reflection and transmission coefficients. The difficulty to elaborate a more practical characterization based on those parameters, which are key indicators for paint formulators, originates from their dependence on various variables such as the film thickness and the substrate's color. Thus, any intent to characerize dependent light scattering transition via macroscopic optimal parameters would require elaborating a standardized framework to define the values of the additional variables.

Characterization of dependent light-scattering amplitude

One can circumvent the above difficulty by characterizing the dependent light-scattering amplitude via a filling fraction-independent parameter, which would unequivocally allow for the determination and comparison of the strengths of [TiO.sup.2] crowding effects between different paint formulations. Such a determination can be made by adjusting the variation of the wavelength-independent scattering efficiencies with a two- or three-parameter mathematical function.

A typical example is given in Fig. 3, which shows wavelength-independent scattering efficiencies as a function of the PVC evaluated by applying ASTM 2805D on the diffuse reflection coefficients of a series of white paint films.

The variation of the scattering efficiencies are accurately fitted up to 20% PVC by a second-order polynomial function given by SD[empty set] = a1[empty set] + [a.sup.2][empty set] [sup.2] Thus, a1 directly provides % which is defined as of the derivative of SD in the limit of [empty set][right arrow] 0, whereas [a.sup.2]can be used as a filling fraction-independent coefficient to characterize the dependent scattering amplitude. Further investigations should be performed to identify the optimum mathematical function for modeling a broad range of white paint films scattering efficiencies in the largest possible range of concentrations.

The samples under study were simple coatings composed by rutile titanium dioxide pigments dispersed in an acrylic polymer with the adequate package of surfactant. Measurements were performed in the visible range with diffuse illumination and D65 illuminant on a spectrophotometer Minolta CM-3700d equipped with an integrating sphere.

Practical application

A complex challenge that paint formulators have to face is to establish whether it is more economically viable to increase the scattering efficiency of a given paint formulation by solely augmenting its PVC or by trying to improve the pigments' spatial state of dispersion, assuming that the latter is technically possible and financially viable.

The principal difficulty resides in the lack of theoretical and semi-empirical models that can accurately predict the maximum scattering efficiency of a given paint formulation, which corresponds to the optimum pigment spatial dispersion state (see typical variations in Fig. 4). Consequently, it is generally solved via trial-and-error processes by varying the engineering dispersion processes and modifying the packages of surfactants of the initial formulation. The aim of this section is to present a straightforward procedure to help the formulators develop their empirical knowledge to progressively decrease the number of necessary samples formulated in the laboratory. The method is the following:

(a) Samples of the original paint formulation are applied at different PVCs on white and black substrates at constant wet thickness. The scattering efficiencies [S.sub.I] and [S.sub.D] of each application are obtained from diffuse reflectance measurements using the procedure proposed in the previous section.

(b) The influence of the pigments' spatial state of dispersion is taken into account by expressing the paint scattering efficiency, denoted [s.sub.d.sub.t]as a linear combination of [S.sub.I] and [S.sub.D] via the introduction of a dispersion coefficient, denoted [OMEGA]such that

[FIGURE 4 OMITTED]

[S.sub.D.sup.T][empty set],[OMEGA]= [OMEGA]S I [empty set] + (1 -[OMEGA]) SD[empty set], 0 [less than or equal to] [OMEGA][less than or equal to] 1 (1)

The limit [OMEGA] = 1 corresponds to the ideal case of independent scattering (curve (a) in Fig. 4), whereas the limit [OMEGA]= 0 corresponds to the paint scattering efficiency of the original formulation (somewhere between curves (b) and (c) in Fig. 4).

(c) Measurements of the variations in the scattering efficiency as a function of the pigment's spatial state of dispersion are realized via the evaluation of a gain function, denoted G, defined as G =[s.sup.d.sub.t] [empty set] [OMEGA]/[S.sub.D.sub.T] [empty set] [OMEGA]= 0).

(d) The Kubelka-Munk theory is employed to evaluate the reflection coefficients of the paint films over white and black substrates, denoted RW(Z, [empty set] [OMEGA])\and R b{Z, [empty set] [OMEGA]) respectively, as functions of the film thickness, Z,[empty set], and [OMEGA].

(e) Defining a contrast ratio as, CR(Z,[empty set] [OMEGA]) = R b (Z, [empty set] [OMEGA])/, one plots a threshold curve corresponding to the thickness required to produce a contrast ratio of CR = 0.98 for each value of\[OMEGA] as function of Z and [empty set].

(f) Finally, the previously identified curves are superimposed in a single chart as shown in Fig. 5,

In the framework of our example, we assume that the formulator is dealing with an original white paint formulation whose hiding power is HP = 10.8 m2/L at a pigment's filling fraction of 17% (see Point A of curve HPi in Fig. 6).

First, we, imagine that one wishes to improve the formulation to reach a hiding power of 22.2 m2/L (curve HP3 in Fig. 6). Inspection of Fig. 6 shows that neither improvements in spatial dispersion nor increases in the PVC can reach this objective.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Next, let us assume that the original formulation has to be modified to reach a hiding power of 15.4 m2/L (curve HP2 in Fig. 6). Inspection of Fig. 6 shows that the formulator has three options:

(1) To increase the filling fraction up to 19.5% (see path AB in Fig. 6) keeping the pigments' spatial state of dispersion unchanged ([ohm]=0.0).

(2) To improve the pigments' spatial state of dispersion up to [ohm]=0.6 (path AE in Fig. 6) keeping the filling fraction to its initial value of 17%.

(3) To combine an improvement of the pigments' spatial state of dispersion with an increase of the filling fraction: see, for example, path ACD in Fig. 6, which involves an increase from [ohm]=0.0 to[ohm]=s 0.2 and an augmentation of the filling fraction from 17% to 18.5%.

The values of the gain function G corresponding to options (1), (2), and (3) are 0.00, 1.33, and 1.11 respectively.

Thus, based on their semi-empirical knowledge and the newly acquired information, formulators must consider which options are viable and which must be discarded. For instance, in the framework of our example, option (3) looks possible and more realistic than option (2), which supposes an improbable increase of 1.33 of the paint-scattering efficiency by solely improving the pigments" spatial state of dispersion.

Conclusion

We have used numerical simulations based on a rigorous theoretical framework to show that contrary to some statements made in the literature, dependent scattering of light in white paint films containing rutile titanium dioxide pigments is a particular manifestation of multiple scattering. More specifically, it results from electromagnetic near-field interactions between neighboring particles which is described by vector electromagnetic treatments of light (like rigorous solutions of the electromagnetic Foldy-Lax equations).

We have also pointed out that the transition from independent-to-dependent scattering, as a function of the PVC is a continuous process that cannot be formally related to a specific concentration threshold. Consequently, attempts to characterize dependent light-scattering transition by either determining the value of [DELTA]t = f([empty set]t) at a particular concentration[empty set] t, or by providing the filling fractions, [empty set] t corresponding to a standardized value of [DELTA]t would prove problematic. The underlying reason is that the absence of a threshold-type transition does not allow any unequivocal and unambiguous relationship between these two quantities. A common consensus among the paint community would be required to justify the choice of their values which would in any case remain questionable. To circumvent this problem, we have to characterize the Ti02 crowding effect amplitude via filling fraction-independent parameters.

Finally, we proposed a method based on the introduction of a dispersion coefficient and the evaluation of a gain function, to support the formulator in the difficult task of evaluating whether the hiding power of a white paint formulation should be improved by increasing the amount of pigment or by improving the spatial dispersions state.

Acknowledgments The authors would like to thank Fernando Zaldo and Centro de Invcstigacion en Polimeros of Grupo Comex who provided the experimental set of data used in this study.

Appendix

Expression of the Pearson correlation coefficient

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

For a determined system defined by the set of parameters (nr,[empty set]), calculations were performed on K = 1000 different random configurations, which were proven to be sufficient for configuration average statistics.

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J.-C.Auger (El)

Kyoralis Research and Consulting, 5 rue du Loing, 75014 Paris, France e-mail: kyolarisresearch@gmail.com

B. Stout

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

DOI 10.1007/s 11998-011-9371-9

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Author: | Auger, J.-C.; Stout, Brian |
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Publication: | JCT Research |

Article Type: | Report |

Date: | May 1, 2012 |

Words: | 6033 |

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