**1. Introduction**

Nowadays, the use of computer simulation to develop new materials has become a widespread practice. From the point of view of the visual perception of the material, it is important to model the interaction of light with it. Many modern materials are dispersed media, i.e., they consist of optically contrasting particles distributed in the volume of a transparent substance. Such media are used in modern light sources (diffusers in flat sources, luminous linear sources and devices in car interior), and they are also the basis of modern paints, auto glass, plastics, and inks for 3D printers.

Visual appearance is the main characteristic of paint and it manifests itself through the human perception of objective optical properties, such as color, brightness (reflection coefficient), glossiness, texture (spatial heterogeneity), etc. Hence, simulation and visualization of optically complex materials, such as multilayer paints with a complex microstructure (like pearlescent and metallic paints) in the automotive industry, have been developed in recent years. Advanced software allows one to simulate light propagation through a paint composed of clear varnish with pigment particles and flakes (metallic or interference ones) dispersed in it. The color of these paints depends on the observation and illumination directions. Therefore, its visual appearance should be described by a bidirectional reflectance distribution function (BRDF). The primary task is to calculate how the paint with given composition looks under given illumination and observation directions [1].

**Citation:** Ershov, S.; Voloboy, A.; Galaktionov, V. Simulation of Light Scattering in Automotive Paints: Role of Particle Size. *Mathematics* **2023**, *11*, 2429. https://doi.org/10.3390/ math11112429

Academic Editors: Camelia Petrescu and Valeriu David

Received: 29 April 2023 Revised: 18 May 2023 Accepted: 22 May 2023 Published: 24 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

We consider two main approaches of calculating light scattering in dispersed media. The first one is based on the continuous medium model [2]. In this model, any infinitesimal volume scatters light proportionally to that volume. This model leads to the differential light transport equation (LTE) (or Radiative Transfer Equation in [2]). There are several methods within this approach. For example, deterministic methods, the method of discrete ordinates [3], finite difference, and matrix methods like doubling/adding [4]. Additionally, it can be solved via stochastic (Monte Carlo) integration. The continuous medium approximation (i.e., LTE) works pretty well for atmosphere for which it was originally created. Presently, it is quite popular and widely applied to a wide class of turbid media. However, this approach becomes inaccurate when the pigment particles are large or are packed densely in the paint.

The second approach is the simulation of light propagation through an ensemble of paint flakes and particles represented as an explicit geometry. Here, we can solve it by using the Monte Carlo ray tracing (MCRT) [5]. It is possible to either create a huge sample geometry that includes billions of randomly distributed particles or use many random samples of a rather small piece of paint geometry (corresponding to small area of paint layer) and average over them. This approach is straightforward but rather resource and time consuming. However, it does provide accurate solution, and thus we call it the "accurate approach".

Comparing the measurements of real paint samples with simulation, we found that for effect pigments (e.g., metallic flakes), the LTE solution cannot correctly predict the total integral reflectance [6]. Moreover, the measured BRDFs, while generally running more or less closely to the simulated ones, show a strong deviation for nearly normal incidence and observation. The LTE seriously underestimates it, while ray tracing through an ensemble of flakes gives a more accurate result. This is a major problem for the simulation of automotive metallic and pearlescent paints that must look bright. Thus, investigating the reasons of this problem and development of a possible numerical solution is the main motivation of our research.

Our study shows that this effect is due to the correlations between close incident and scattered rays and not the correlations between particles. The mathematical method was borrowed from a thought experiment of perfectly aligned flakes. We consider a ray that goes down to some depth where it is reflected upward by the flake and leaves the layer. We calculate the probability of this event and thus obtain the attenuation (extinction) of the light. It does not follow the usual exponential attenuation in a continuous medium, i.e., it is an anomalous extinction. It is not the one found in the studies of correlated particles. Having the probability of such light path, we calculate the statistical properties of the single scattered light. Then, we take the LTE results as the sum of the scattering orders and replace the first of them with the one given above, leaving the higher orders unchanged. It provides a very good approximation to the results of the accurate model. This allows to obtain near-accurate results much faster than the Monte Carlo ray tracing and without its noise. Additionally, the correction term is almost an analytic function that allows to understand the role and effect of various paint parameters and predict some nontrivial effects. It also predicted that the BRDF could deviate greatly (up to twofold) from the LTE results, even for tiny particles, in the case of near-normal illumination and observation. That is, there is no simple good convergence to the LTE results when the particle size approaches 0.

Initially, these ideas had been applied to the simplified case when all flakes have equal size and are opaque [7]. However, in reality, flakes vary in size and are not always opaque; for example, mica flakes are semi-translucent. This paper aims to remove these limitations and thus considers a general case.

Continuous medium approximation works with the product "area times concentration" and thus do not use the flake size separately. It means that for this approximation, the mean surface luminance is exactly the same for particles n-times larger area in n-times lower concentration. In this paper, we investigated the dependence of the painted surface

luminance (BRDF) on the flake size. We compared LTE with a more reliable and accurate (yet expensive) simulation. This simulation is MCRT in the explicit paint geometry that contains an ensemble of individual flakes, instead of replacing their scattering with the phase function of the continuous medium.

The main contribution of this paper is that it proves that the effect of coarse particles can emerge even in a model where positions of these particles are not correlated. This is different from the mainstream studies which have only concentrated on the role of these correlations to describe the effect of coarse particles. Moreover, we suggest a semi-analytical model of dependence based on particle size which not only allows to calculate a more accurate BRDF but also admits an intuitive comprehension of how various parameters of medium affect the BRDF. In case of the divergence in results of LTE and accurate approaches, we propose a simple approximation that allows to improve the accuracy of the LTE results for coarse particles.

The remainder of the paper is organized as follows. Section 2 describes related studies. The proposed method is introduced in Section 3. Section 4 presents the results calculated using the proposed method and their comparison with other methods. In Section 5, discussion and conclusions are stated.
