MIE-SCATTERING MONTE CARLO SIMULATIONS 195 enables a wide range of outputs to be calculated, including total transm1ss10n and reflection, haze, diffuse reflectance, angular intensity distribution of transmitted/ reflected light, and spatial intensity distribution of transmitted/reflected light. The number of photons propagated through the film affects the amount of noise seen in the final values: for all the simulations presented in this paper, 10 million photons were propagated per simulation. ANGULAR INTENSITY DISTRIBUTION The angular intensity distribution of the transmitted and reflected light is calculated by monitoring the angle of each photon as it exits the film, either from the top surface (reflection) or the bottom surface (transmission). The angular intensity distribution is affected both by the roughness of the film surface and by the volumetric scattering properties within the film. The reflected light has two components. Direct reflection off the top surface of the film, from photons which do not enter the film at all, maintains the color of the incident light. The roughness of the top surface affects the appearance of gloss. Backscattered reflection from within the film involves photons being scattered back through the top surface as a result of interacting with particles within the film absorption by pigments within the film causes color changes as determined by the model. The simulation distinguishes between these two components of reflection. Transmitted light cannot be separated into these two components, although its behavior is affected by both the roughness of the film surface and the volumetric scattering within the film. Haze and diffuse reflectance are scalar quantities derived from the full angular intensity distributions calculated above. Haze, an indicator of angular scattering, is the percentage of transmitted light that is deflected by more than 2.5 degrees from the normal. In this paper we define diffuse reflectance to be the reflective counterpart to haze, i.e., the percentage of reflected light that is deflected by more than 2. 5 degrees from the normal. SPATIAL INTENSITY DISTRIBUTION Volumetric scattering within a turbid medium contributes to its characteristic appear- ance (4). As a narrow beam of light passes through a turbid material, the beam spreads spatially due to the diffusion of the photons. The simulation tracks the trajectories of individual photons as they propagate through the coating and calculates the intensity of reflected light as a function of distance from the source. Our incident light source is an infinitesimally narrow collimated beam. It is convenient to focus on the horizontal, not vertical, spreading of the light because the thickness dimension of a film is essentially negligible compared to the area dimensions. The intensity of the transmitted and reflected light is calculated as a function of (horizontal) distance from the beam. Scalar quantities derived from the full spatial intensity distribution function conveniently include the percentage of total photons that travel a distance x from the source, and the distance from the source at which the light intensity drops to a certain percentage of the initial value. RANDOM ROUGH SURFACE MODEL The effect of a rough surface on the scattered light distribution 1s modeled via a

196 JOURNAL OF COSMETIC SCIENCE single-scattering Gaussian slope distribution model (11) where the RMS slope specifies the degree of roughness. The assumption is that particles within the film distribute themselves in a way such that they create a randomly rough surface, and that multiple- scattering effects at the surface can be ignored. This model cannot be used for surfaces with highly structured topologies or surfaces with slope distributions that are known not to be Gaussian. It also cannot take into account RI differences across the surface that may arise due to the particles actually penetrating the film and exposing themselves without a layer of matrix covering them. However, it can capture the main effects of varying the surface roughness of a film. PIGMENT EFFECTS The imaginary component of the complex RI of a material gives rise to absorption. Titanium dioxide has a negligible imaginary component over a large part of the visible light range, contributing to its white appearance, but pigments selectively absorb cer- tain wavelengths of light. By running simulations over a range of wavelengths, respec- tive! y changing the complex RI of the scattering materials as a function of wavelength, the transmittance, reflectance, and absorption curves of pigment mixtures can be cal- culated. The transmittance and reflectance curves can in turn be multiplied by an incident light source such as CIE standard white illuminant D65 to give the intensity distribution (spectra) of the transmitted and reflected light under ambient conditions. RES UL TS AND DISCUSSION In the simulations discussed in this paper, the angle of incidence of the light was specified to be normal to the surface of the film, and the RI of the media above and below the coating was defined to be air (RI = 1.0). The light source is an infinitesimally narrow, monochromatic beam. Ten million photons were propagated per simulation in order to ensure good statistical resolution. EFFECT OF PARTICLE SIZE DISTRIBUTION In a physical system, it is difficult to create sub-micron scale particles with truly monodisperse size distributions. We therefore modeled the polydispersity of a material by explicitly defining a mixture of particles of different sizes. We simulated six hypothetical formulations of titanium dioxide particles dispersed in oil to observe the effect of particle size distribution on optical qualities. In this particle size distribution, we defined five discrete particle sizes. In all six systems, the mean particle diameter was 600 nm and the total loading of titanium dioxide was 2% by volume. The particle size distributions are shown in Figure 2. For the systems represented in Figure 2, the particles were randomly dispersed in a 25-micron-thick silicone oil film, and the wavelength of the incident light was 600 nm. The complex RI for titanium dioxide at wavelength 600 nm was defined to be 2.76 i0.0 (interpolated using values published by Almaz Optics (12)). The real RI for the silicone oil was defined to be 1.4 for all visible wavelengths. The transmission and haze values are shown in Figure 3.