**5. Discussion and Conclusions**

While the mainstream studies on the deviation of light scattering in dispersed media from the continuous medium approximation concentrate on the role of correlations between particles (such as their agglutination), we demonstrate that another mechanism exists in automotive paints with "effect pigments" (metallic and pearlescent paints). Here, the light scattering deviates significantly from the continuous medium approximation even in the absence of such correlations, i.e., when the particles of an "effect pigment" are homogeneously, evenly and stochastically distributed. The deviation is caused by correlations between subsequent light scattering events. We prove that this effect is very rough and exists whenever the flakes of "effect pigment" are aligned parallel to the paint surface. This effect is very widespread because such paints are always applied to achieve exactly this alignment.

Meanwhile, correlations between particles are a more subtle effect of chemical physics, usually caused by positive or negative mutual affinity of particles. This may or may not happen depending on many factors including surfactants. It is not always known if this will happen, and it is even a more rare case if we know it quantitatively (i.e., know the potential of interaction). In this case, predictions of its effect on the optical appearance can only be qualitative.

On the contrary, the effect investigated by us is rather rough and it is not related to the subtle interaction potential. It depends on the flake size (and the law of dependence is simple and close to analytical) and on alignment. If alignment is good, we can use the asymptotic. Thus, quantitative prediction is possible and we have provided nearly analytical laws that allow very accurate calculation. Moreover, being mostly simple analytical formulae, they allow you to "intuitively feel" the effect so that you can figure out what is going to happen, even without massive computations. It is noteworthy that at normal incidence and observation, the deviation from LTE is almost twofold even for tiny flakes, i.e., there is no simple convergence as the particle size goes to 0.

We have derived a correction term which makes the paint appearance calculated using LTE very close to that of the accurate model. If we apply this correction to the first scattering-order BRDF of the LTE approach leaving the higher orders unchanged, the sum will already be very close to the full accurate BRDF. The value of the derived analytical corrections is that they are more easy to obtain numerically and even admit some analytical approximation, while the accurate approach calculations (for individual flakes) are more difficult and time consuming.

Simulation and realistic visualization of pearlescent and metallic paints have direct and inverse problems. The direct problem is calculating how the paint with given composition looks under given illumination and observation directions. The inverse problem emerges when we are looking for the paint composition whose appearance is the closest to the target one. The latter problem is more interesting for the automotive industry. It always occurs when repairing a car body. For modern paint formulations, optical characteristics have many degrees of freedom, and the dimension of the paint matching problem increases to dozens of parameters, and it is almost impossible to adjust them by mixing several basic options. This problem is solved by minimizing the discrepancy between the target appearance and the one calculated by the simulation of the direct problem.

There have been attempts to solve the inverse problem basing on LTE, for example, the radiative transfer equation is used in optical tomography [29]. For the paint visualization task, we have demonstrated that BRDF (visual appearance of paint) depends on the flake size. At the same time, LTE does not operate the flake size, and therefore cannot be used for the inverse problem of the automotive paint simulation. However, after application of our correction term, we have a "modified LTE" and can vary all parameters to find the composition for which the calculated BRDF best fits the measured one. Thus, we can solve not only the direct but also the inverse problem. This is a possible direction for future work.

This work continues [7] where the first steps of this research were taken. Herein, the method is generalized and allows the flakes to vary in size and have a transmittance. However, there are still some limitations and assumptions. First of all, we considered only the case of perfectly aligned flakes. It is assumed that for non-aligned flakes, our method should work with less accuracy. Investigation of the influence of flake alignment on the method accuracy is one of the directions of future work. Another possible study is related to the influence of the flakes with a matte/diffuse surface. In our study, they are specular and do not have a diffuse component while in reality there exists partially diffuse flakes.

**Author Contributions:** Conceptualization, S.E. and A.V.; methodology, A.V.; software, S.E.; validation, A.V.; formal analysis, S.E.; investigation, S.E.; resources, V.G.; data curation, V.G.; writing—original draft preparation, S.E. and A.V.; writing—review and editing, A.V. and V.G.; visualization, S.E. and A.V.; supervision, V.G.; project administration, V.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data presented in this study are available within this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


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