OIL FILMS AND MOISTURE ABSORPTION ON HAIR 139 part of this plot (for (C/C eq ) -0.5) will be linear and its slope (calculated by linear regression) is related to the diffusion coefficient as D = (1r/l6) (slope)2 cm2/s (3) Diffusion coefficients for hair samples used in this work, calculated by equation 3, are displayed in Figure 3 for sorption and in Figure 4 for desorption as a function of RH. It should be noted that the diffusion coefficients obtained by this procedure are apparent diffusion coefficients because they are determined for a fiber sample and an average value of the radius is used in the calculation. True diffusion coefficients can be determined only by establishing concentration profiles as a function of time for diffusion into a single fiber. In Figure 3, we can see that as relative humidity increases from 0%, the diffusion coefficients increase, reach a maximum, and then decrease. In Figure 4, for desorption, we see the same pattern, although the values of the diffusion coefficients at all humidities are somewhat higher than those found in the sorption mode. This is because sorption takes place in an unswollen fiber in which diffusion is difficult, whereas desorption occurs from a swollen fiber in which diffusion is considerably easier. Swelling of the fiber is believed to open several pathways for diffusion. This is true of most materials. In the case of hair we see anomalous behavior, i.e., diffusion coefficients decrease with an increase in swelling. We propose the following hypothesis to explain the observed diffusion behavior. The overall rate of diffusion into the hair cortex is controlled by two factors: (a) diffusion of water molecules through the cell membrane complexes (CMCs) and (b) diffusion into the matrix of the cortical cells. The latter involves the swelling of the cell, which can constrict the CMCs and reduce the rate of diffusion through these channels. This 1.60E-09 1.40E-09 1.20E-09 1.00E-09 8.00E-10 C 6.00E-10 4.00E-10 2.00E-10 0.00E+00 0 -a- control --+-coconut -..- mineral -+--sunflower 10 20 30 40 50 RH(%) 60 70 80 90 100 Figure 3. Plot of calculated water vapor sorption diffusion rates for various oil-treated hair as a function of RH.

140 JOURNAL OF COSMETIC SCIENCE N.$_ 2.50E-09 ....... ---------------------------------. Untreated 2.00E-09 1.50E-09 1.00E-09 5.00E-10 0.00E+00 +---"T'""-----r-------r------.---.----..-----r-----r------r-----1 0 10 20 30 40 50 RH(%) 60 70 80 90 100 Figure 4. Plot of calculated water vapor desorption diffusion rates for various oil-treated hair as a function of RH. happens mainly because the outward swelling of the cortex is restricted by the cuticle sheath. This directs the swelling pressure inwards, leading to the constriction of CMCs. Based on this realistic picture, we can explain the shape of the diffusion coefficient versus the RH curves seen in Figures 3 and 4. In the sorption mode, at low RH, diffusion through CMCs is fast, but slows down when the molecules cross over into the cell because the matrix is not swollen. As the humidity increases, the diffusion rate increases because the swelling leads to an increase in free volume. Beyond 70% RH, capillary condensation leads to excessive swelling of the cells, which constricts the CMCs, thus reducing the rate of diffusion through the CMCs. Therefore, it is reasonable to assume that during sorption, at low humidities (left side of the maximum), diffusion into the cells is the controlling step. At high humidities (right of the maximum), diffusion into the cells is fast, and therefore the rate-controlling step is the diffusion through the CMCs, which are constricted by the swelling of the cell. During desorption at high humidities (right side of the maximum), the rate-controlling step is the diffusion through the CMCs, which open up as the cells contract due to loss of water. At low humidities (left side of the maximum), as the cells contract, diffusion through these cells slows down and becomes the rate-controlling step. Diffusion through the CMCs is fast. This seems to be a reasonable hypothesis for the shape of diffusion coefficient versus RH curves. This concept is applicable to diffusion into the cuticular sheath also, except that in the cuticle the endocuticle adjacent to the CMC may participate in the diffusion. For both sorption and desorption, the diffusion rates are substantially lower for the oil-treated hair compared to the untreated hair. Two possible reasons can be suggested for the lowering of the regain for the oil-treated hair: (a) the penetrated oil molecules block the sites for the sorption of water and (b) in the case of oil-treated hair, sorption