396 JOURNAL OF COSMETIC SCIENCE 200 150 1oo 50 2,3 I I I I I I I 0 5 10 15 20 25 30 35 20(degree) Figure 9. X-ray diffraction intensities versus 2 0 curves for the supercontracted hair at 8.4%. Curves 1, 2, and 3 denote the azimuthal angles at q• = 0% 75 ø, and 90 ø (approximately the same as the curve at q• = 75ø), respectively. amorphous diffused ring (as an arc pattern in the case of the latter). Returning to the diffraction curves in Figure 10, the [3-reflection is unlikely to always occur in the supercontracted fibers. No distinct reflection near 2 0 = 19 ø can be recognized on the I•e 1 versus 2 0 curve of the 9.9% contracted fiber. The equatorial reflection in the vicinity of 2 0 = 9 ø includes the components of reflections from both the o•- and [3-crystallites (20). This means that the intensity at 0.98 nm is not proportional to the content of the o•-crystallites in fibers. It can be expected, therefore, that the relative o•-content is less than the maximum intensity value near 2 0 -- 9 ø for the supercontracted fibers shown in Figure 10. Whether the [3-crystallites originate from the deformed o•-helices or from the matrix segments remains unresolved (19,21,22). This makes it difficult to interpret the results of the present x-ray analyses. In any way, permanently straightened hairs with about 5% to 8% supercontraction might provide a variety of forms of protein chains cross-linked with each other by disulfide bonds. Although it is difficult to determine precisely the amount of each material in crystalline and random coil forms, it can be presumed that most of the o•-helices are transformed into random coil chains by super- contraction up to about 10%, as suggested by the L c versus A H m plot. The permanency related to hair straightening seems to be a result of the irreversible transformation of o•-crystal into the amorphous phase. It is of interest to demonstrate the relationship of the length of the o•-crystallites cross-linked with disulfide bonds, L w to the length of random chains in the network after randomization of the o•-helical chains, Li• c. The mean square length, (ro2), and the maximum length, rm, for the random chains are represented in terms of the segment length, 1', and the number of segment, n' as in equations 13 and 14 (23,24):

METHOD FOR PERMANENT HAIR STRAIGHTENING 397 lOO 8o 6o 4o 20- 0 o 35 a 5 10 15 20 25 30 20(degree) Figure 10. Ire • versus 2 0 curves for the untreated and the supercontracted hairs at different extents of contraction in %: , Ife• at •p 0ø Ire • at •p = 90ø (a) untreated (b) 8.4 (c) 9.9 (d) 12.5. 2 ntlt2 r o = (13) r m = n'l' (14) From equations 13 and 14, equation 15 can be derived: LRc/L m = ((ro2))•/2/rm = (n')•/21'/n'l ' = 1/(n')•/2 (15) where L m is the maximum length of the extended or-chain, namely, the [3-chain length. The pitch lengths per amino acid residue in the or-helix and the [3-pleated sheet struc- tures are 1.5 and 3.4 fk, respectively (25). Thus, we obtain equation 16: Lo•/L m = 1/(3.4/1.5) = 0.44 (16) From equations 15 and 16, equation 17 can be obtained: LRc/L = = 1/0.44(n')•/2 (17) When supercontraction is 10% and affine deformation is assumed, LRc/L = -- 0.9, and the number of segments, n', in a random chain of the network cross-linked with disulfide bonds can, therefore, be calculated to be about 6.3. The value of n' for the S-[3-cyanoethylated human hair synthesized by blocking the free thiol groups with acrylonitrile after reduction of the disulfide bonds amounted to about 86% of the total cystine content of the hair (662.5 pmol/g of hair) and has been reported to be about 6.3 for the network chains comprised of the low-sulfur (LS) proteins or the or-helix-forming proteins with a molecular weight of about 50,000 (26,27). The mo- lecular weight of the chain between the disulfide cross-links, Mc, was also determined to be 7,920. Hence, the molecular weight per segment: Mc/n' = 1,250. This value is