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Theoretical study and experimental analysis of the scattering efficiency of hollow polymer particles in the dependent light scattering regime.

Abstract We present an extensive theoretical study of the optical properties of hollow polymer particles (HPP) dispersed into a polymer resin as functions of their structure and volume-filling fraction. We show that a dependent light scattering model predicts the occurrence of three different light propagation regimes, the existence of which is consistent with experimental data gathered in the literature. We then analyze the effect of the polymer shell thickness on the magnitude of the near-field and far-field dependent light scattering phenomena. We also discuss and compare the variation of the scattering efficiency of Ropaque[TM], measured in our laboratory, as a function of the pigment volume-filling fraction, with dependent light scattering model. In particular, we show that in the case of HPP, quasi-linear variation of the scattering efficiency as a function of the particle volume-filling fraction is not an undeniable indication of the absence of electromagnetic couplings in the system. Finally, we used a solution of the radiative transfer equation to compare the prediction of the opacity of several simple paint films when a dependent light scattering model is used instead of the independent scattering approximation to calculate the optical properties of the films at a local scale.

Keywords White paints, Opacity, Dependent scattering, Multiple scattering, Hollow polymer particles, Quasi crystalline approximation, Radiative transfer equation

Introduction

Maintaining or increasing the hiding power of white paint films while minimizing cost is a fundamental, technical, and financial issue for coating companies. To reach this goal, one possible approach consists of partially replacing costly rutile titanium dioxide pigments (Ti[O.sub.2]) by hollow polymer particles (HPP) in the paint formulation. In their dry state, these particles are made of spherical air voids encapsulated in a hard polymer shell. The size of the void is designed to optimize light scattering efficiency when it is dispersed into a continuous polymeric medium. The thickness of the hard shell is mostly determined by the requirement for the mechanical properties to be such as to maintain the overall integrity of the particle's structure, as well as by practical constraints required by the method of synthesis. Rohm and Haas (now a subsidiary of Dow Chemical) was one of the first companies to have brought HPP technology to the coating business a few decades ago. Since then other manufacturers have followed suit and now there are several commercial alternatives, such as those provided by Arkema or Dinova Specialties Pvt.

Past theoretical studies have evaluated the scattering efficiency of optimized HPP to be about 8 to 10 times smaller than rutile titanium dioxide pigments (Ti[O.sub.2]). (1,2) Thus, if partial substitutions are technically and financially viable in certain paint formulations, it is believed to be so because the combination of opaque polymers and Ti[O.sub.2] pigments is somehow synergistic. In other words, the light scattering efficiency of the mixture composed of Ti[O.sub.2] and HPP dispersed in a polymeric resin appears to be higher than the light scattering efficiency of both systems taken separately. The phenomenon is generally explained as being due to the influence of the voided particles on the effective index of refraction of the continuous phase of the coating film.

If one considers that the Ti[O.sub.2] pigments are dispersed in an effective homogeneous medium composed of the mixture of the polymeric resin and the HPP, the index of refraction of the mixture is assumed to be lower than that of the sole polymeric phase because of the presence of the air voids (which have a refractive index of 1.0). Nonetheless, this explanation is purely empirical as per our knowledge; existing effective medium theoretical frameworks cannot be applied to such highly scattering systems, whose particle size is of the order of magnitude of the wavelength of the incident radiation.

An alternative explanation that has been sometimes proposed is that the presence of the HPP can improve the spatial dispersion of the Ti[O.sub.2] pigments in the paint film and consequently reduce crowding. However, a recent theoretical study performed on mixtures of fillers and Ti[O.sub.2] pigments tends to definitively refute such a mechanism. (3)

It is clear that for any coating company, the full control of HPP technology, from the optimization of the structure during the synthesis process to the suitable formulation with an optimized package of fillers and Ti[O.sub.2] pigments, could provide significant economical benefits. Therefore, applied studies related to HPP formulation in white paints are usually kept confidential. In the academic sphere, fundamental analyses of HPP optical properties are mostly realized within the approximation of an independent light scattering regime. For these reasons, the aim of this work is to present an extensive and fundamental analysis of the light scattering properties of HPP, taking into account multiple and dependent light scattering effects. To accomplish this task, the article is structured as follows. In the "Theoretical analysis" section, we use the quasi crystalline approximation (QCA) and single-particle T-matrix formalism (STMF) to successively address several major issues related to HPPs' optical properties, such as the identification of the different light propagation regimes and the effect of the polymer shell thickness on HPPs' optical properties. The "Experimental evidence of HPPs' light propagation regimes" section of the article is devoted to the corroboration of the theoretical light propagation regimes previously determined with experimental data gathered in the literature. In the "Analysis of Ropaque Ultra[TM] optical properties" section, we focus on the theoretical and experimental analyses of the optical properties of Ropaque Ultra[TM] because it is one of the most widely used commercial HPPs in the coating industry. In "Interpretation of quasi-linear variation in terms of electromagnetic couplings" section, we discuss the possible reasons why the variation in the scattering efficiency of a hypothetical HPP may be linear with the pigment volume concentration (PVC). Finally, in the "Effect of electromagnetic coupling on macroscopic parameters" section we used the N-Flux model to compare the prediction of the opacity of simple paint films containing HPP, when using independent and dependent light scattering models at a local scale. All numerical optical calculations realized in this study were performed via the Kyolaris Research numerical simulation platform developed by the author.

Theoretical analysis

Theoretical framework

The analytical theory of the multiple scattering of light in heterogeneous media is based on the vector form of the Foldy-Lax equation. (4,5) Among the different approached solutions of this Lippman-Schwinger-type equation, (6) at least two of them have been regularly used to study the local optical properties of paint films: the finite difference methods (FDM) and the N-particles-centered T-matrix formalisms (NCTMF).

Finite difference methods, whether they are time domain or frequency domain, (7) have the significant advantage of allowing consideration of arbitrary-shaped particles, up to the limit of resolution of the spatial grid. Despite this advantage these methods are either limited to two-dimensional "small" systems of highly packed particles, or they involve prohibitive computational resources since computational resources are correlated to the size of the cell under study. In addition, the evaluation of the far-field optical parameters requires an intermediary step which involves transformation of the local electromagnetic field information.

The N-particles-centered T-matrix formalisms (8,9) are limited to axisymmetric particles with prohibitive requirements on the particles' relative positions, except when they are spherical. (10) However, because computational time only depends on the individual size parameter and the total number of particles in the cell, they are suitable to model a three-dimensional ensemble of spherical particles in a large range of PVC. (11,12) The latter, noted as X, is defined as the product of the module of the incident wave vector by the characteristic dimension of the scatterer.

Now, regardless of their respective merits and limitations, those formalisms have two major drawbacks which made them inappropriate for our study: (a) they require the implementation of an explicit and time-consuming numerical procedure to evaluate the configuration average optical parameters of the random heterogeneous media under study; (b) they provide resolutions of the multiple scattering equation of light on a fixed N-particle system dispersed in a continuous medium, typically from only 10 to 100, which limits the corroboration with experimental data generated from macroscopic films.

For the aforementioned reasons, it was clear that the QCA (13,14) was the most suitable theoretical framework to conduct this study. First, because it directly provides the evaluation of the configuration average scattering coefficient of a semi-infinite three-dimensional ensemble of particles by taking into account the electromagnetic couplings up to the pair correlation function. Second, because its domain of application and limitations are most appropriate for a system such as HPP dispersed in a polymer resin: i.e., one possessing a quasi-spherical structure, narrow polydispersity in size, and a reasonably low relative index of refraction with the non-absorbing surrounding medium.

The calculations of the optical properties were realized according to the QCA formalism described in Ref. 15 with the difference that we applied a hard sphere potential, instead of a sticky one, to characterize the particles' interactions. Also, unless otherwise specified, the index of refraction of the surrounding medium, noted as [n.sub.0], is set to 1.5 to simulate a typical polymeric material and the wavelength of the incident radiation in vacuum, noted as [[lambda].sub.0], is fixed as 0.545 [micro]m close to the maximum sensitivity of the human eye. The thickness and index of refraction of the polymer shell are noted as [epsilon] and [n.sub.2], respectively while the index of refraction of the air void, noted as [n.sub.1], is set to 1.0.

Identification of the light scattering regimes

In this section we have neglected the presence of the hard polymer shell to constrain the analysis of the results in a two-dimensional space. Note that when the shell thickness is ignored, the void volume concentration (VVC) is equal to the PVC. The radius of the air void, denoted as [r.sub.1], ranged from 0.030 to 0.300 [micro]m while the PVC, denoted as [phi], varied from 0.001 to 0.150.

Figures 1 and 2 represent the correlation functions of the scattering coefficient and scattering efficiency, as functions of [r.sub.1] and [phi]. Those quantities, which are defined in Appendix B, are denoted as [[GAMMA].sub.C](r, [phi]) and [[GAMMA].sub.s] (r, [phi]) respectively. Now, beyond the expected result that, irrespective of their size and in diluted systems, HPPs scatter light practically and independently one from the other, it is possible to observe two major features:

1. As the PVC is gradually increased, the dependent light scattering phenomenon is progressively enhanced, giving rise to three different types of light propagation regimes, characterized by [GAMMA](r, [phi]) < 1, [GAMMA](r, [phi]) [??] 1, and [GAMMA](r, [phi]) > 1, respectively. Those values indicate that HPPs' optical properties are respectively, lower than, approximately equal to, and superior to the HPP optical properties calculated under the independent scattering approximation (ISA).

2. The two propagation regimes characterized by [[GAMMA].sub.c](r, [phi]) [??] 1 and [[GAMMA].sub.s](r, [phi]) [??] 1 take place on two different ranges of radii: from about 0.220 to 0.250 [micro]m for the former and 0.090 to 0.120 [micro]m for the latter. Therefore, the classification of light propagation regimes is not an intrinsic property of the HPP as it may vary depending on which optical parameter is considered. For example, HPPs having a radius near 0.160 [micro]m have simultaneously [[GAMMA].sub.C](r, [phi]) < 1 and [[GAMMA].sub.S](r, [phi]) > 1.

It is also worth recalling that the increase of the dependent light scattering phenomenon as a function of the PVC is a continuous process. Consequently, the identification of ranges of radii for which the amplitudes of dependent light scattering can be said to be "negligible", i.e., [GAMMA](r, [phi]) [??] 1, is somehow arbitrary. In these particular examples, we considered that the electromagnetic couplings between the scatterers could be neglected when the values of the correlation functions ranged from about 0.95 and 1.05. Also, it is still unclear if such an independent-like light propagation regime is the result of either a real absence of correlation among the particles, or arises from the combination of different effects that effectively cancel each other out. This point will be discussed in the third part of this article.

Optimum radius

The optimum radius, noted as [r.sub.op], is defined as the radius at which the scattering efficiency of a given material is maximum. Consequently, it is a quantity that depends on the wavelength of the incident radiation. Figure 3 represents the variation of the ratio of the scattering efficiencies and the number of particles by unit volume [[rho].sub.0] of a pure air void as a function of its radius, computed with the QCA and the ISA at 15% PVC. It shows that both theories give close estimations of the optimum radius, about 0.125 and 0.110 [micro]m, respectively. Also, the corresponding values of the scattering efficiencies are very similar as they differ only by 3%. This outcome raises the question of whether the identification of the particle's optimum size, as perceived by human vision, from independent light scattering approximation is sufficiently realistic or if, in certain cases, it requires more precise evaluation.

It is also interesting to note that ISA's calculations overestimate QCA's predictions when the radius of the particle is below the optimum size, while when it is greater than the ISA, they underestimate QCA's predictions. The consequence is that at about [r.sub.1] = 0.300 [micro]m, independent light scattering assumptions predict a fall of 50% of the scattering efficiency from its highest value, whereas QCA only predicts a 20% loss. This difference suggests that strictly controlling the air void size distribution in a narrow range as possible around its optimum value in order to maximize the scattering efficiency is not as stringent a requirement as would be expected based on a simple analysis using the independent light scattering approximations.

Characterization of the type of electromagnetic couplings in the different light scattering regimes

To identify the types of electromagnetic couplings involved in the different light propagation regimes, i.e., far-field coherent scattering or near-field interactions, we compared QCA and Interference Approximations (ITA) predictions. (16)

Figure 4 represents the variation of the ratio [S.sub.QCA]/ [S.sub.ITA] as functions of rx and the PVC. It confirms that when the Rayleigh-Gans criterion holds, i.e., when 2X[absolute value of n - 1] [much less than] 1, dependent scattering mainly originates from far-field coherent effects independent of the PVC. In the latter expression, X is the size parameter and n = [n.sub.1]/[n.sub.0].

It also appears that when the particle size exceeds about 0.1 [micro]m, the contribution of the near-field interaction process to the overall dependent light scattering effect correspondingly intensifies as the PVC and rx are increased. This outcome suggests that in systems composed of optimally sized air voids, far-held coherent scattering dominates the dependent light scattering process. In contrast, it has been proposed (17) that it is the near-held contribution which dominates the dependent light scattering process in systems composed of Ti[O.sub.2] pigment at optimum size.

Effect of the polymer shell's thickness

In this study we have assumed that the value of the refractive index of the polymer shell was identical to the one of the medium of propagation. Consequently, variations in the shell thickness [epsilon] only modify the optical properties of HPPs' systems, either by changing the VVC at constant PVC or by altering the spatial state of dispersion of the microvoids within the paint him. Thin shells increase the possibility of near-held couplings because they enhance the probability of encountering two or more microvoids in near contact. At fixed VVC, they also allow formulating at lower PVC. The resulting higher randomness of the particles' localization reduces the possibility of encountering far-field coherent scattering (see Fig. 5a). On the other hand, thick shells decrease the magnitude of near-held coupling because they prevent air voids from being directly in close contact. However, at fixed VVC, they also lead to the structuration of the medium, which is favorable to the enhancement of far-held coherent effects (see Fig. 5b).

Those structural effects can be simply put in evidence by observing changes in the radial pair correlation functions (RPCF) of different HPP made of increasing shell thickness, as a function of the PVC. The three examples shown in Fig. 6 and calculated via the Percus-Yevick approximation (18) confirm the presence of two simultaneous effects while keeping the VVC constant: (i) an increase in the spatial extension of the first sphere of coordination due to the increase in the minimum distance between two air voids' centers; (ii) an increase in the total number of coordination spheres due to a higher degree of spatial organization.

Now, to study the effect of those structural variations on the magnitude and type of electromagnetic couplings, one can compare, at different VVCs, the optical properties of pure air voids with those of HPPs composed of the same air voids surrounded by a polymer shell of well-defined size. An illustration is given in Fig. 7 on two systems characterized by [r.sub.1] = 0.050 [micro]m and [epsilon] = 0.03 [micro]m. By comparing ITA, QCA, and ISA calculations it is possible to draw a conclusion on the strengthening of the far-held dependent light scattering contribution on the scattering efficiency induced by the presence of the polymer shell: the higher the VVC, the stronger it is. One can also note the weakening of the near-held interactions for the coated system. This is demonstrated by noticing that the average value of the ratios [S.sub.ITA]/[S.sub.QCA] evaluated over the full range of VVC is closer to 1 in the case of the pure air void (i.e., 1.005 for the latter and 0.974 in the absence of the polymer shell).

Now, it is also clear that the dependence of the optical properties on the thickness of the polymer shell is not limited to structural and cooperative effects. It is also strongly correlated with intrinsic optical properties of the scatterer, which largely depend on the size parameter of the system. Thus, this additional dependency makes it difficult to generalize the previous reasoning on systems having very different sizes.

Experimental evidence of HPPs' light propagation regimes

Origin of the experimental data

To corroborate the existence of the three different types of light propagation regimes predicted by the QCA analysis, we have used as a reference the two studies conducted by Park at the beginning of the last decade. The first study focused on the synthesis processes and characterization of the structures of eight types of HPP. (19) The second study provided the measurements and discussed the variations of their optical properties in the visible range when dispersed in a polymeric resin. (20)

Scattering efficiency analysis procedure

Ideally, the fitting procedure of the "measured" scattering efficiency as a function of the PVC should meet the following criteria:

(a) The analytical form of the mathematical function should reproduce, as accurately as possible, the larger panels of systems in the largest range of PVC. It should also allow the physical interpretation for each of the fitting coefficients.

(b) The number of experimental data must be sufficient and adequately distributed in the range of PVC under study.

(c) The determination of the independent light scattering trend should not involve any subjective assessment. This is especially important because it is used as a reference to characterize the amplitude of the dependent light scattering phenomenon.

The simple framework we have been using consists of expressing the scattering efficiency in terms of a third-order polynomial function such that:

[S.sup.T.sub.DEP] ([phi]) = [a.sub.0] + [a.sub.1][phi] + [a.sub.2][[phi].sup.2] + [a.sub.3][[phi].sup.3], (1)

where:

(i) [a.sub.0] is set to zero to guarantee that there is no light scattering in the absence of heterogeneities. In other words: [S.sup.T.sub.DEP] ([phi] = 0) = 0.

(ii) The coefficient [a.sub.1] characterizes the variation of the independent light scattering trend. In the limit of infinitely diluted systems, the condition [phi] << 1 holds, and equation (1) can be simplified into:

[S.sup.T.sub.DEP] ([phi]) [approximately equal to] [S.sup.T.sub.IND]([phi]) = [a.sub.1][phi] (2)

(iii) The absolute value of [a.sub.2] defines the magnitude of the dependent light scattering phenomenon while its sign indicates the major effect of the electromagnetic couplings: either overall receded (negative sign) or globally enhanced (positive sign), when compared with the independent scattering trend.

(iv) The parameter [a.sub.3] provides an indication of the rate of changes of the amplitude of the electromagnetic couplings. Using the definition of the correlation function, it is straightforward that:

[[GAMMA].sub.s] ([phi]) = 1 + [[a.sub.2]/[a.sub.1]] [phi] + [[a.sub.3]/[a.sub.1]][[phi].sup.2] (3)

Thus, when [a.sub.3] = 0, the correlation function follows a linear decay as a function of the PVC. When [a.sub.3] [not equal to] 0, the strength of the electromagnetic coupling decreases either slower or faster than the linear decay, depending on the relative sign between [a.sub.2] and [a.sub.3].

We displayed in Figs. 8a and 8b the examples of variation of the scattering efficiency and correlation function as defined by equations (1) and (3), respectively. It is clear that not all variations provided by these two equations can be related to physical cases.

It is also worth mentioning that equation (1) gives the variation of the scattering efficiency as a function of the PVC only. However, it also contains implicit information related to the spatial state of dispersion of the pigments. (21) Therefore, at fixed PVC, the scattering efficiency can take not only one, but several values. In Ref. 21 it was proposed to explicitly introduce the latter, in the analytical formulation of the scattering efficiency, via a second variable referred to the "dispersion coefficient." However, it remains difficult to obtain a practical model from such an approach.

Clarification of the condition at zero PVC

The fitting method proposed here assumes that the scattering efficiency must vanish at zero PVC. More than just an assumption, it is a necessary condition that is fully coherent with a fundamental physical principle, i.e., in the absence of heterogeneities, there is no scattering.

To justify this approach we use the proof by contradiction: considering the meaning of the scattering efficiency, any argument that would be used to justify a non-zero value at zero PVC would be questionable. In other words, it would appear problematic to validate any physical interpretation from the analysis of a mathematical function representing the variation of the scattering efficiency as a function of the PVC, which does not vanish at PVC = 0.

Our approach is also fully consistent with the representation used by Ross (22) and Tsang et al., (23) which are among the most cited references in the study of the optical properties of inhomogeneous materials.

Selection of the experimental set of data

Park's work is essential because it provides valuable experimental data covering a wide range of HPPs' structure, PVC, and VVC. This is especially important because as commercial products are designed to maximize the scattering efficiency in the visible spectrum, they are limited to a narrow range of sizes. Therefore, while they are easy to procure, they cannot be used to confirm or invalidate the existence of different light propagation regimes.

Also, corroboration between the QCA calculations and Park's set of data was difficult because the indices of polydispersity of the air void radius and polymer shell thickness cover a wide range of values, whereas QCA formalism assumes that all scatterers have identical size. Therefore, to guarantee the reliability of the comparison, we restricted the analysis to HPPs' structures that had the lowest polydispersity indices.

An additional drawback was that the scattering efficiency of each HPP's structure was measured at only three different PVCs. Therefore, taking into account the additional point at PVC = 0, the mathematical adjustments were fixed to a second-degree polynomial function.

Results and interpretation

Figures 9a-c show the mathematical fits performed on the C-525, C-604, and C-605 samples synthesized by Park. The structural characteristics of each sample and the fitting coefficients are reported in Tables 1 and 2, respectively. The results suggest the occurrence of three light propagation regimes as predicated by our QCA calculations. In Fig. 9d we have reproduced the linear fits performed by Park. They intersect the Oy axis at a positive value for the smallest size (C-525), at about the origin for the (C-604), and in the negative portion of the Oy axis for the largest size (C-605). The exact correspondence between each of the intersection points with the Oy axis, and the size of the particle, reinforces the appropriateness of our fitting procedure.

It is also important to mention that the existence of these three light propagation regimes has been experimentally pointed out in polymer emulsions (24,25) and theoretically predicted in a numerical study covering the light scattering properties of clusters of spheres. (26) It is therefore not unlikely that similar properties in HPP systems could be found (Table 2).

Finally, it is worth mentioning that Park concluded the analysis on HPPs' optical properties by stating "HPPs' scattering efficiencies increased linearly relative to the VVC within a range up to 20% indicating that there is not significant dependent scattering within the given range." We believe that this misjudgment has originated from the use of an inadequate fitting procedure, which, as shown in Fig. 9d, does not fulfill the necessary condition that the scattering efficiency must vanish when the PVC tends to zero.

Analysis of Ropaque Ultra[TM] optical properties

Description and analysis of the experimental data

The light scattering efficiencies of several paint films composed of Ropaque Ultra[TM] embedded in a polymer binder at different PVC were experimentally evaluated in AkzoNobel Decorative Paint Research facilities (see Fig. 10). The method of determination is based on ASTM D8005.

The residual analysis on the linear adjustment, including the condition S = 0 at PVC = 0, does not show a fairly random pattern, implying that a linear model is not the most appropriate. Visual assessment on the linear fit, which excludes the condition S = 0 at PVC = 0, can lead to the conclusion that the variation of S is quasi-linear with respect to the PVC and that the dependent light scattering effects are negligible.

Nonetheless, the use of second- and third-order polynomial functions, as described earlier, suggests that [[GAMMA].sub.S]([phi]) is greater than translucide 1 in the range of PVC under study (see Fig. 11; Table 3). These nonlinear variations imply the occurrence of enhanced dependent light scattering phenomenon.

The discrepancy between these approaches points out the necessity to standardize a method for determining the independent light scattering trend from experimental data. In addition, it appears that for the range of PVC under study, the major difference between the second- and the third-order polynomial function's adjustments is related to the prediction of the independent scattering trend rather than the determination of S([phi]). This poses an additional challenge to simultaneously provide accurate predictions of dependent and independent variations in the same analysis procedure.

Finally, a significant ambiguity, which to our knowledge is only marginally addressed in the literature, resides in the practicality of the experimental uncertainties associated with the scattering efficiency at very low PVC. The latter does not and cannot include the inability of the Kubelka-Munk model to precisely describe the process of light propagation in weakly scattering media. This is a major issue, because obtaining accurate data in such condition is essential to correctly conclude on the evaluation of the ISA trend. However, the hypothesis on which the KM theory is based does not hold when S [much less than] 1. To circumvent these difficulties one could simply consider the KM equations as purely mathematical functions. However, it would then be difficult to justify any physical interpretation of the acquired data.

Analysis of the experimental data via the QCA

Examination of scanning electron microscopy images of Ropaque Ultra shows that the average dimensions of the air void radius and the polymer shell thickness are about 0.180 and 0.050 [micro]m, respectively. These images also reveal that both have non-negligible polydispersity in size. Based on these observations, we have used the QCA formalism to estimate the variation of the scattering efficiency of Ropaque-like particles as a function of the PVC. Due to the limits of the model as well as the uncertainties on the values of some experimental data, the objective was not to find a quantitative fit but rather to seek if in a range of structures similar in sizes to those of Ropaque Ultra, dependent scattering calculations could support our interpretation of the experimental data.

Results are displayed in Figs. 12 and 13 for different refractive indices of the polymer shell and the binder: [n.sub.0] = (1.50, 1.60) and [n.sub.2] = (1.50, 1.55, 1.60). They indicate that the QCA calculations are invariably showing an enhancement of the scattering efficiency (i.e., [[GAMMA].sub.s]([phi]) > 1). It also appears that variations in [n.sub.0] lead to greater deviations of the scattering efficiency than variations in [n.sub.2]. Variations on the scattering coefficients were also systematically larger than variations on the scattering efficiency.

Interpretation of quasi-linear variation in terms of electromagnetic couplings

For the purpose of this discussion we assume the existence of an HPP structure, for which the variation of its scattering efficiency is almost linear in a large range of PVC and vanishes at PVC = 0. In what follows, we propose three possible reasons for such a phenomenon.

1. The linear variation reflects the actual absence of dependent scattering, even when the PVC reaches very high values. This situation may occur when the thickness of the polymer layer is of the same order of magnitude as the radius of the air void. We provide an example with [r.sub.1] = 0.100 [micro]m and [epsilon] = 0.150 [micro]m in Fig. 14. In this particular case, at 0.500 PVC, the value of the VVC is only 0.032. With such a type of structure, the amount of scattering material (i.e., air) per unit volume is always low and the effects of independent scattering are indeed negligible. Note that this scenario is independent of the absolute size of the HPP, as its occurrence depends only on the relative size of the polymer shell and the air void. This is a plausible explanation to justify the quasi-linear trends observed for commercial HPP.

2. The quasi-linear variation is the result of different light scattering phenomena that cancel each other. However, the absence of dependent scattering is only observable for the scattering efficiency. The scattering coefficient shows non-linear variation as a function of the PVC, indicating the occurrence of dependent light scattering phenomenon. This issue was highlighted in the analysis of the optical properties of pure air voids in the "Origin of the experimental data" section. It was observed that, for a given radius, [[GAMMA].sub.C]([phi]) and [[GAMMA].sub.s]([phi]) could simultaneously take different values. For example, at [r.sub.1] = 0.160 [micro]m, it appears that [[GAMMA].sub.C]([phi]) < 1 and [[GAMMA].sub.s]([phi]) > 1.

Here, we provide another example on a coated air void with physical dimensions of [r.sub.1] = 0.099 [micro]m and [epsilon] = 0.033 [micro]m. Figure 15 indicates that the scattering efficiency follows a quasi-linear variation whereas the scattering coefficient does not. It seems that the non-linear variation of the latter is counterbalanced by the decrease of the asymmetry parameter in such a way that the scattering efficiency exhibits an apparent linear variation. This decrease is probably due to the increase of multiple scattering that enhances the redistribution of the scattered field in the back hemisphere.

3. The linear variation reflects the real absence (or extreme weakness) of the dependent light scattering phenomenon, even when the VVC is nonnegligible. Nonetheless, for the reason mentioned above, to validate such a statement, one must simultaneously monitor quasi-linear variations on the scattering coefficient and scattering efficiency. An illustration is given in Fig. 16, which shows three different domains in the space of representation ([r.sub.1], [phi]), for which the correlation functions [[GAMMA].sub.C] and [[GAMMA].sub.s] extend over the same range of values: (a) from 0.95 to 1.05, (b) from 0.90 to 1.10, and (c) from 0.80 to 1.20. The thickness of the polymer shell was set such that [r.sub.1] - 3[epsilon]. Note that there is not a common domain for which [[GAMMA].sub.C] and [[GAMMA].sub.S] simultaneously span from 0.98 to 1.02. In the framework of our example, results suggest that the scattering coefficient and scattering efficiency can only exhibit independent scattering simultaneously up to very low values of the PVC, i.e., a few percent, and therefore it is unlikely to happen.

Effect of electromagnetic coupling on macroscopic parameters

Problem

Reaching accurate theoretical or semi-empirical predictions of white paint film's hiding power as a function of the coatings' composition (type of fillers and pigments, PVCs) is a significant subject of interest in the coating industry. The major stage involves the direct modeling of the diffuse reflection coefficient of the paint film via one of the possible solutions of the radiative transfer equation (RTE). (27) The origins of the discrepancies that can be found between experimental data and theoretical predictions are multiple. Beyond the limitations related to the RTE itself, one can blame the imprecision of several of the input parameters such as the pigments' index of refraction, their size, and shape distributions or the reflection coefficients at the air/film and film/substrate interfaces.

Nonetheless, one of the most obvious sources of inaccuracy is certainly the use of independent light scattering approximation to calculate the phase function as well as the scattering and absorption coefficients of the bulk system. The reason this approach is used is that theoretical dependent light scattering models are complex to implement and require prohibitive time of calculation. Thus, if their use can be valuable to improve our knowledge of the basic mechanisms related to the propagation of light in heterogeneous media, they are not very efficient for making systematic studies. Also, such types of fundamental analysis are usually applied for the prediction of the diffuse reflectance of white paint films containing Ti[O.sub.2] pigments. (28)

Here, we focus on the study of the variation of the contrast ratio of white paint films composed of HPP when QCA is used instead of ISA to calculate the local optical properties. We then determine if, and in which conditions, the use of a higher degree of complexity in the theoretical framework is required. We have limited the analysis to three homogeneous HPP whose radii are 0.050, 0.100, and 0.180 [micro]m, respectively. The PVC ranged from 0.01 to 0.30 while the physical thickness of the paint film, noted as Z, spans from 20 to 300 [micro]m.

Procedure

Assuming fixed values of the triplet ([r.sub.1], [phi], Z), we have applied the following four-step procedure:

1. We calculated the scattering coefficient and phase function of the particles in the visible wavelength range (from 0.400 to 0.700 [micro]m) via:

(a) The STMF (independent scattering)

(b) The QCA formalism (dependent scattering).

2. Using the previous calculations as input data, we used the N-Flux formalism (29) to evaluate the diffuse reflectance of paint films over black and white backgrounds. The latter are noted as [R.sup.w.sub.DEP] ([lambda], [phi], Z), [R.sup.b.sub.DEP] ([lambda], [phi], Z), [R.sup.w.sub.IND] ([lambda], [phi], Z), and [R.sup.b.sub.IND] ([lambda], [phi], Z). The subscripts IND and DEP stand for independent and dependent scattering theories, whereas superscripts w and b stand for white and black substrates, respectively.

3. We evaluated the paint films' contrast ratios, noted as [CR.sub.DEP]([phi], Z) and [CR.sub.IND]([phi], Z) that are defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

where [y.sub.10](2) and [D.sub.65]([lambda]) represent the tri-stimulus function and [D.sub.65] illuminant. (30) A similar expression has been used to evaluate [CR.sub.IND] ([phi], Z).

4. We calculated the ratio between the dependent and independent contrast ratios, noted as [[GAMMA].sub.CR] ([phi], Z) such that:

[[GAMMA].sub.CR] ([phi], Z) = [CR.sub.DEP] ([phi], Z)/[CR.sub.IND] ([phi], Z) (5)

The diffuse incident illumination was taken perpendicular to the film. The white and black substrates were assumed to be Lambertian with a reflection coefficient of 0.82 and 0.04, respectively. The air/film interface was assumed to be planar. The corresponding external and internal reflection coefficients were calculated according to Fresnel's law.

Results and discussion

To illustrate the effects of dependent light scattering on the angular redistribution of the scattered field, we have plotted in Fig. 17 the phase functions of the 0.050 [micro]m air void at three different values of the PVC: 0.01, 0.10, and 0.30. Taking the independent light scattering prediction as a reference, one can clearly see the consequence of the far-held coherent effect, which in this case is to increase the backscattering as the PVC is progressively increased. Also, because the size of the particle is relatively small compared to the wavelength of the incident radiation, the phase function does not show a strong peak in the forward direction even when evaluated using the independent light scattering approximation.

Figures 18 and 19 represent variations of the contrast ratios as functions of the him thickness and the PVC for [r.sub.1] = 0.180 [micro]m using ISA and QCA, respectively. We have extended the calculations to very large paint thicknesses to allow indication of the hiding power. It is indeed noticeable that most of the simulated coatings are translucent as only few of them reach a contrast ratio of 0.98 as required by the international standard to indicate full coverage of the substrate. The lack of opacity is the consequence of the relatively low scattering efficiency of HPP as compared to Ti[O.sub.2] pigments. In the framework of our example, it would require a paint him with 300 micrometers thickness loaded at 8% HPP in volume to completely cover the substrate.

Now, the increase in opacity can be directly correlated to the enhancement of the optical thickness of the film. In a first approximation, the latter, noted as [tau], is given by the relation:

[tau]([phi], Z) = [S([phi]) + K([phi])]Z, (6)

where represents the absorption coefficient of the paint film which originates from the pigments and/or the continuous phase. For white coalings it is assumed that K([phi]) [much less than] S([phi]). Therefore, the larger the film's physical thickness, the lower the effect of variation of the bulk scattering efficiency on the opacity of the paint film.

This feature is shown in Figs. 20, 21, and 22 which represent the variation of [[GAMMA].sub.CR] (z,[pi]) as defined by equation (2), for air void radii of 0.180, 0.100, and 0.050 [micro]m, respectively. The results confirm that, independent of the HPP size and the PVC, the larger the film thickness, the greater the value of [[GAMMA].sub.CR]. They also indicate that at low PVC, [[GAMMA].sub.CR] approaches 1, independent of the HPP size. However, it is noticeable that when the PVC is progressively increased, the amplitudes and the sign (positive or negative) of the variations of [[GAMMA].sub.CR] are strongly correlated to the HPP size. For [r.sub.1] = 0.050, [micro]m [[GAMMA].sub.CR] drops to 0.3, whereas it increases up to 1.10 for an HPP radius of 0.180 [micro]m.

Consequently, when evaluating the hiding power of a white paint film loaded with HPP, the use of a dependent light scattering framework is not always justified. For film thickness greater than 50 micrometers, and HPP size larger than about 0.100 micrometers, the difference between the two approaches is only about 2%.

Conclusions

This work leads to the following conclusions:

--The relation between dependent light scattering effects and the thickness of the polymer shell which surrounds the air void is complex. Unlike the case of Ti[O.sub.2] pigments, far-field dependent light scattering effects are much more important among HPP. The most obvious reasons are their quasi-spherical shapes and a narrower particle size distribution which promotes short and long scale organization in the medium and large PVC. The relative weakness of the near-field interaction, at least when the size of the particle is small with respect to the wavelength of the incident radiation, is related to the relatively low contrast in indices of refraction in the system air/ polymer when compared to the system Ti[O.sub.2]/polymer.

--HPPs exhibit three types of light propagation regimes, whose existences were validated by experimental data taken from the literature. A direct and important consequence is that above a certain size, which appear to be close to HPP's optimum one, dependent light scattering enhances the scattering efficiency when compared to the independent scattering case. As a reminder, in paint films containing Ti[O.sub.2] pigments, the scattering efficiency is decreased by the effect of dependent light scattering. This phenomenon is usually called the "crowding effect."

--This is an important result that could explain the unexpected large scattering efficiency of some heterogeneous media. For example, in a recent paper, (31) the authors were intrigued by the low values of the transport mean free path, (32) defined as the inverse of the scattering efficiency; they had measured on the scales of two different beetles. Those values were clearly surprising for such simple heterogeneous media composed by materials having a low index of refraction (about 1.5). Because they assumed that the effect of dependent light scattering could only be lower than the independent scattering value, as for Ti[O.sub.2] pigment, they related the magnitude of the scattering efficiency to the anisotropy of the scale microstructures. Even if this is indeed a plausible explanation, one could now consider that it could also originate from enhanced dependent light scattering phenomena.

--It is premature to assume an absence of electromagnetic couplings based on the sole observation of a linear variation of the scattering efficiency as a function of the PVC. To validate such nonexistence, it is necessary to concurrently observe linear variation of the extinction coefficient as well. This finding still has to be corroborated with experimental data.

--The effect of using a dependent light scattering model in lieu of the independent scattering approximation in the prediction of opacity strongly depends on the dimension of the particles but also on the PVC. Interestingly, for sizes comparable with commercial products, ISA predictions underestimate the hiding power, whereas in the case of paint films loaded with Ti[O.sub.2] pigments, the hiding power is usually overestimated.

Acknowledgments The authors would like to acknowledge the AkzoNobel Decorative Paint Business area and Simon Davies from AkzoNobel Expert Capability Group for supporting this work. We also greatly thank Fraser Robertson and Bogdan Ibanescu of AkzoNobel Decorative Research Group for realizing the experimental work on Ropaque Ultra. The authors would also like to thank the anonymous reviewers for their valuable comments and suggestions to improve the content of this manuscript.

Appendix

Appendix A: Optical parameters

The scattering coefficient denoted as 5 is defined as the product of the number of particles by unit volume, noted as [[rho].sub.0], with the scattering cross section [C.sub.sca].

s([phi]) = [[rho].sub.0][C.sub.sca]([phi]) (7)

The scattering efficiency noted as S, also called reduced scattering coefficient in transport theory, is given by the product of the scattering coefficient and the term (1 - g), where g represents the asymmetry parameter.

s([phi]) = s([phi])[1 - g ([phi])] (8)

The asymmetry parameter represents the average cosine of the scattering angle. It characterizes the asymmetry of the phase function p([theta], [phi]) and it ranges from -1 to 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

When g is close to zero, the scattering process is mainly isotropic. When g tends to 1, the phase function is strongly anisotropic with a strong maximum in the forward direction near [theta] = 1 angle. A negative value of g indicates that more light is scattered in the backward hemisphere than in the forward hemisphere.

In the diffusion theory of light, the scattering efficiency is related to the inverse of the transport mean free path. (32) The latter is defined as the length scale on which the direction of propagation is randomized, i.e., when the scattered direction is totally de-correlated from the initial incident direction.

Appendix B: Correlation function

The correlation function denoted as T is defined as the ratio between the values of the optical parameters calculated via the dependent and independent light scattering frameworks, respectively. Thereby

[[GAMMA].sub.c]([phi]) = s([phi])/[S.sub.IND], (10)

[[GAMMA].sub.S]([phi]) = S([phi])/[S.sub.IND], (11)

where [S.sub.IND] and [S.sub.IND] represent the scattering coefficient and scattering efficiency calculated via the independent light scattering assumption. The expression [GAMMA]([phi]) refers to either [[GAMMA].sub.S]([phi]) or [[GAMMA].sub.C]([phi]). The correlation function provides information on the strength of the dependent light scattering phenomenon in the overall light scattering process. The greater the value of [GAMMA]([phi]) is different from unity, the stronger are the dependent light scattering effects.

In, (26) the notion of radius of transition was introduced to characterize the dimension at which the correlation function of a given system transits from [GAMMA]([phi]) < 1 to [GAMMA]([phi]) > 1. It was assumed that the radius of transition was a function of the indices of refraction of the scattering particles and the medium of propagation.

DOI 10.1007/s11998-015-9677-0

J.-C. Auger

Expert Capability Group - RD&I, AkzoNobel, Shanghai, China

D. McLoughlin ([mail])

Decorative Paint Research Group, AkzoNobel Decorative

Paints, Slough, UK

e-mail: daragh.mcloughlin@akzonobel.com

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Table 1: Dimensions of the opaque polymer particles
under study: taken from Ref. 20

        [r.sub.p]       PDI       [r.sub.v]       PDI
          (nm)      ([r.sub.p])     (nm)      ([r.sub.v])    X

C-525      151         1.05          67          1.19       1.1
C-604      291         1.02          192         1.06       3.2
C-605      535         1.20          332         1.05       5.6

PDI stands for polydispersity index, [r.sub.p] is the average radius
of the particle, [r.sub.v is the average radius of the void, and X
represents the size parameter

Table 2: Adjustments on Park's experimental data

                        C-525    C-604    C-605

[a.sub.0]               0.0      0.0       0.0
[a.sub.1]               1.36     6.83      1.31
[a.sub.2]               -64.16   7.89     17.34
[a.sub.3]
[[GAMMA].sub.S](0.04)   0.67     1.05      1.51
[[GAMMA].sub.S](0.10)   --       1.12      2.28

Values of the fitting coefficients and correlation function
according to equation (1) (limited to the second order) and
equation (3)

Table 3: Adjustments on Ropaque experimental

                          A       B        C       D

[a.sub.0]               0.000   -0.004   0.000   0.000
[a.sub.1]               0.176   0.196    0.139   0.118
[a.sub.2]                                0.182   0.455
[a.sub.3]                                        -0.783
[[GAMMA].sub.S](0.04)                    1.052   1.144
[[GAMMA].sub.S](0.10)                    1.131   1.320
[[GAMMA].sub.S](0.20)                    1.262   1.507

Values of the fitting coefficients and correlation function
according to equations (1) and (3): (A) linear fit, (B) linear
fit without the condition S = 0 at PVC = 0, (C) second-order
polynomial function, (D) third-order polynomial function


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