468 JOURNAL OF COSMETIC SCIENCE The total transmittance, (Ts)x, of the continuous sunscreen film, will be calculated by integrating equation 7 along the whole unit area: (Ts)x = 10 -h(v)XAx x dF (7) Integration of equation 7 can be mathematically or numerically achieved, taking into account the continuous film model choice. Note that Ax is considered as a simple numerical coefficient in the integral. The total transmittance (Ts)x is transformed into total absorbance (As)x through equation 8: (As)x = -log (Ts)x (8) Therefore, according to integration of equation 7, the model can achieve an interesting relationship between the parent film absorbance Ax and the resulting model absorbance (As)x. This will be shown further (see Figure 6). PRACTICAL APPROACH The same graphic approach can be realized with experimental in vitro data, (As)x being replaced by measured absorbance (Aexp)x. The same relationship is involved, whatever the wavelength, and so absorbance data (Aexp, A)x, from different sunscreen products, can be.plotted on the same graph (see Figures 9 and 10). Optimization of the continuous film model ß Absorbtivities between 290 and 400 nm for all UV filters suitable for a sunscreen formulation are determined accoMing to the experimental protocol. ß Experimental versus parent uniform film absorbance data, from different sunscreen preparations, are first plotted: (Aexp)x/Ax. Ax data are calculated according to equa- tion 4 with the UV filter absorbtivities previously determined. ß Resulting model absorbance data (As)x is calculated through the thickness function h(v), according to the same Ax data. Optimization of the function parameters is achieved by using an optimization modeling system, with the final fit to the experi- mental data being determined by least-square error assessment, through equations 5 to 8. Method to calculate UV absorption spectrum and specific UV indices of a sunscreen product (UV filter composition should be known) ß Parent film absorbance Ax data between 290 and 400 nm are calculated by using equation 4, in which surface density application and sunscreen UV filter composition are considered. ß Parent film absorbance Ax data are transformed into model absorbance data (As)x through equations 7 and 8 by using the optimized thickness function h(v ). ß Specific UV indices from (Ts)x or (As)x data are calculated. RESULTS AND DISCUSSION CONTINUOUS MODELS OF IRREGULAR SUNSCREEN FILMS Elementary film thickness function h F = 2 x F was tested first. This simple function, which is the simplest mathematical representation of an irregular sunscreen film,

HEIGHT DISTRIBUTION MODEL IN SUNSCREENS 469 achieves a prism-shaped profile. However, instead of being a simple profile description, h v function is also characteristic of a type of sunscreen distribution. As in surface profile analysis, irregular film models can be also described through a height distribution diagram. Weight of any peculiar height in the distribution is a function of its total area (13). Therefore, variable F, previously defined as being a cumulative fraction of the unit area, is also representative of the cumulative height distribution of the sunscreen film. Film profile (cumulative height distribution function) and its associated probability function (height distribution) are plotted on the same graph, according to the same disposal already adopted in surface metrology. The film profile becomes equivalent to the bearing area curve of Abbot and Firestone (13). This kind of graph was adopted for all examples of height distribution studied in that paper. The prism-shaped profile is reported in Figure 2. This basic example corresponds to a very simple distribution where every elementary class of film thickness is equally rep- resented in the total height distribution. Thus, any height is privileged, and the result- ing cumulative distribution is a continuous straight line. The prism model can be seen as a schematic representation of a sunscreen spreading, with equilibrium between de- pleted and covered areas. However, such basic function has obviously little relevance in achieving a realistic model of sunscreen film. Therefore, function hv would be better deduced using classical statistical laws. Among possible probability functions, gamma law, which is associated with asymmetri- cal distributions, seems to be a good candidate to represent the resulting height distri- bution associated with a sunscreen film spread on an irregular substrate. We can assume the height distribution to be skewed (13), friction strain being expected to be more intense in the upper regions of the roughened substrate surface, where depletion domi- nates, than in the deeper valleys, where accumulation is free to occur. According to reference 14, gamma density function is defined by equation 9, h being the random variable "relative thickness" and f the associated probability density function: f= x x 3 x F(c) (9) where I•(c) is the gamma function, b is the scale parameter, and c is the shape parameter. In our approach, the shape parameter c was imposed, the scale parameter b being deduced from the film normalization, according to equation 6. To realize the film thickness profile, the inverse of gamma function is used, h being deduced from its cumulative distribution F. In a first approach, three different-shape parameters were tested, in order to explore the main effects of different realistic film height distributions on the thickness profiles. These kind of graphs are reported in Figures 3-5. In the first model (Figure 3), a 0.5 value was attributed to the shape parameter. In this gamma distribution, the thinnest film fractions are overrepresented. As shown in the graph, the sunscreen film is constituted of large depleted areas, and the model is representative of a bad sunscreen spreading.