Optical properties of deposit models for paints: full-fields FFT computations and representative volume element

Abstract

International audienceA 3D model of microstructure containing spherical and rhombi-shaped inclusions "falling" along a deposit direction is used to simulate the distribution of nanoscale color pigments in paints. The microstructures anisotropy and length scales, characterized by their covariance functions and representative volume element, follow that of transerseley isotropic or orthotropic media. Full-field computations by means of the fast Fourier method are undertaken to compute the local and effective permittivity function of the mixture, as a function of the wavelength in the visible spectrum. Transverse isotropy is numerically recovered for the effective permittivity of the deposit model of spheres. Furthermore, in the complex plane, the transverse and parallel components of the effective permittivity tensor are very close to the frontiers of the Hashin-Shtrikman's domain, at all frequencies (or color) of the incident wave. The representative volume element for the optical properties of paint deposit models are studied. At fixed accuracy, it is much larger for the imaginary part of the permittivity than for the real part, an effect of the strong variations of the electric displacement field, exhibiting hot-spots, a feature previously described in the context of conductivity

A Mixed Boolean and Deposit Model for the Modeling of Metal Pigments in Paint Layers

“…At the nanometric scale, the effect of pigments on the optical properties has been studied experimentally and compared with effective medium theories (Cummings et al, 1984). More recently, numerical works have been carried out to estimate the effect of the dispersion of pigments in a deposit model (Azzimonti et al, 2013). To represent the material and compute its properties using numerical means, a microstructural model is required.…”

Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields

“…In recent years, Fourier-based methods, originally introduced by Moulinec et al [1], have become ubiquitous for computing numerically the properties of composite materials, with applications in domains ranging from linear elasticity [2], viscoplasticity [instead of 'thermoplasticity'] [3], and crack propagation [4] to thermal and electrical [5,6] and also optical properties [7]. The success of the method resides in its ability to cope with arbitrarily complex and often very large microstructures, supplied as segmented images of real materials, for example, multiscale [instead of 'multistage'] nanocomposites [8], austenitic steel [9], granular media [5] or polycrystals [instead of 'polycrystal'] [10,11,12].…”

Calculation of cancellous bone elastic properties with the polarization‐based FFT iterative scheme

“…The optimal choice of α and β to minimize the number of iterations is a very difficult task that depends on elastic properties, model resolution, and geometry . In general, the convergence rate reduces markedly with the increase of the contrast between the elastic properties of the microstructure phases.…”

“…It is interesting to observe that the above scheme reduces to the Eyre-Milton 35 and augmented Lagrangian 36 schemes when the coefficients α and β are set to α = β = 2 and α = β = 1, respectively. 32,33 The iterative scheme 15 converges when stress σ i μ and strain ε i μ are respectively equilibrated and compatible fields, and the average strain equals the prescribed macroscopic strain ε i μ y ð Þ V ¼ ε: 38 Moulinec and Silva 38 did a comprehensive review and analysis of the conditions for the convergence of the iterative scheme, and they summarized the sufficient conditions for convergence. Unfortunately, they could not demonstrate the convergence of the method for the case of microstructures with voids and rigid inclusions (ie, microstructures with infinite contrast).…”