128 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS ANALYTICAL CONSIDERATIONS AND RESULTS The application of linear viscoelastic theory to the set of keratin fibers was originally achieved by Chapman (e.g. 7), who confirmed that the cohesive set of wool is governed by the principles of generalized, linear viscoelasticity. Cohesive set is caused by the breaking and reforming of hydrogen bonds, while permanent set is produced by cleaving and recombining some of the sulfur cross-links in keratin. Although the or- igins of both types of set are different, the mechanistic similarity, involving exchange reactions of either secondary or covalent bonds, lends plausibility to the application of similar theoretical approaches for permanent as well as cohesive set. The application of this theory to the permanent set of human hair has been attempted and described in a previous paper (4). Though a reasonable correlation was found be- tween the theoretical sets calculated from the Young's moduli of reduced and reoxidized fibers and the actually observed sets, the calculated values were systematically lower than the actual values. This means that the actually observed fiber set was greater than would have been expected from the change of the Young's modulus of a fiber during reduction. Denby (8) derived a simple formula for the time-dependent cohesive set of wool fibers that in analogy may be applied to permanent set. Since only the initial set at the moment of release is studied, Denby's equation for fiber recovery, eliminating the time-dependent factor, is given by (4): R c = Bre/Bro (8) giving the calculated value for fiber bending set as S c= 1 - B•e/Bro (9) where the subscript "c" stands for "calculated." Equations 8 and 9 are consistent with the experimental observation that the residual fiber bending stiffness or extensional force, also termed "force of retraction" (3), under a given set of conditions is related to the set the fiber receives (9- 11). To correlate the experimental values of set determined on fiber loops and those calcu- lated on the basis of Equation 9, the knowledge of the value of Bro is required for every single experiment. From the experimental point of view, the determination of the new bending stiffness of the released fiber Bro is quite difficult because the bending deformation acquired by the fiber during reduction and subsequent reoxidation prevents a further application of the balanced fiber method. One possibility to determine Bro is to treat an undeformed fiber and measure its bending stiffness afterwards. To follow this approach would have required, beside the measurement of bending relaxation and set, a third, parallel experiment that in the view of the scope of the investigation excluded itself from systematic application. It was thus only applied to verify the validity of the theoretical approach to the problem described below. During the permanent waving treatment, the value of the bending stiffness is decreased by the reducing treatment and is then restored to a value close to the initial value (4)

BENDING OF HAIR AND PERMANENT WAVING 129 during the reoxidation process. The new bending stiffness of the reoxidized fiber Bro can be related to the initial bending stiffness of the fiber B o as B• = C Bo (10) where C is a variable to describe the reoxidation efficiency for the bending stiffness after a given treatment. Substituting Equation 10 into Equation 9 leads to s c = 1 - Bre/(C Bo) (11) B•e was obtained by determining the bending stiffness remaining at the end of the final rinsing process, and B o was estimated graphically by extrapolating the curve of the time-dependent initial bending stiffness B(t) to t = 0 (see Figure 2). The primary aim of the investigation was to prove the equality of the experimentally determined fiber bending set and the set predicted from bending stiffness changes: S c '- S • (12) implying that a plot of calculated versus expeimental set values should generate a straight line through the origin with a slope of unity, yielding with Equation 11 for the interrelation of set and bending experiment results: s • '-- 1 - B•e/(C B o) (13) where C is unknown. C was treated as a variable that had to be optimized, assuming it to be constant, that is independent of the reducing conditions. This assumption parallels the previous obser- vations (3,4) that the degree of restoration of the extensional modulus after reduction and reoxidation is largely insensitive to the reducing conditions. The relation between the calculated sets for various values of C with the actual sets was examined by means of an origin linear regression method. The value of C was varied to obtain a value as close as feasible to unity for the regression coefficient, i.e., for the slope of the regression line. The confidence limits of the optimized value of C were estimated from the 95% confidence limits of the regression coefficient. The result is given in Table I. In order to check the justification of this approach, C was also estimated experimentally for two reducing conditions. Hair specimens were cut into halves. On one part of the hair the initial stiffness B o was determined. The other half was subjected without being deformed to the reduction/reoxidation sequence and subsequently its new stiffness B•o Table I Values for C = B•o/Bo Obtained Experimentally and Calculated on the Basis of the Linear Regression Between Experimental and Calculated Set Data (_+ 95% confidence limits) Conditions Experimental C Calculated C 0.3 M TGA, pH 9 1 MTGA, pH8.4 0.96 - 0.03 0.94 + 0.06 0.96 -+ 0.05