128 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS sive set of a hair that is induced by the breaking and reformation of hydrogen bonds. Cohesive set is achieved by wetting and subsequent drying of a hair in the bent state. Though the mechanisms of the impartation of permanent and cohesive set are different, involving either covalent or secondary bonds, the analogy of the mechanisms involving bonds that can break and reform suggests an analogous theoretical approach. The laws of the formation and the recovery of the cohesive set are governed by the general principles of linear viscoelasticity. Chapman (7) has shown that the set of a wool fiber can be predicted from its relaxation behavior by a generalized theory of viscoelas- ticity. Denby (8) developed an approximation to calculate the time-dependent recovery of a bent wool fiber from its relaxation behavior that holds well for slow relaxation processes. Denby's equation for the calculation of fiber recovery, R, in its simplest form is given by: and the related set generally as: R = B(t)/B(t-w) (1) s = 1-R (2) B(t) is the time-dependent flexural stiffness of the wool fiber at any time, t, after the initial deformation at time t = 0. The fiber is released at t = w, and B(t-w) is the bending stiffness of the same fiber during the time of release. Omitting the time dependence of recovery, the initial recovery R o of the permanently waved fiber in analogy to equation 1 is given by: Ro = BrdBro (3) Bre is the residual of the initial bending stiffness of the deformed fiber after the treat- ment (the suffix "re" stands for "reduced"), and Bro , the new bending stiffness of the reoxidized fiber (the suffix "ro" stands for "reoxidized"). The implication of equation 3 parallels the practical observation that the set of a curl increases with the degree of softening of the hair, that is, with the progressing decrease of the initial stiffness Bre and with the completeness of the reoxidation, so that the bending stiffness after the treatment Bro is as close as possible to the initial value. The bending and extensional properties of a hair can be interrelated by straightforward mechanical principles. Assuming that extensional and compressional properties are equal, the flexural stiffness, B, of a hair and its Young's modulus, E, are related by: B = El (4) where I is the moment of inertia of the cross-section of a circular fiber given by: I = q-r d4/64 (5) and d is the diameter of the fiber. Since the bending stresses enforcing and those opposing recovery act on the same fiber cross-section, equation 3 can be modified by inserting equation 4 to yield: R o = Ere/Ero (6) Ere is the residual of the initial relaxation modulus proportional to the static stress level

EXTENSION OF PERMED HAIR 129 reached in the strained fiber after reoxidation (t t4). Ero is the initial, i.e., the Young's modulus of the reoxidized fiber. It is important to mention that the step from equation 3 to equation 6 via equation 4 is only justified if the moduli are uniform over the whole cross-section of the fiber. In the view of this approach, any deviation between the calculated and the measured recovery would be a measure of the inhomogeneity of the radial distribution of the modulus which might arise from inhomogeneity of the degree of reduction within the fiber (see Discussion). The stress of a wool fiber in water (20øC) at small strains (½ 2%) relaxes and ap- proaches an equilibrium stress of about 80% of the initial value within 20-30 min (9, 10). This time-dependent fraction of stress in the strained fiber is removed during the course of the chemical treatment of a deformed fiber, so that Ere, the modulus attained by the fiber after physical and chemical relaxation, is expected to be constant with time. E•o is the modulus in equilibrium with the deformed state and hence relates to the unstrained fiber. Under the internal fiber strains, during recovery, E•o is subject to the usual stress relaxation of a wool fiber due to hydrogen bond cleavage that will increase recovery by removing the cohesive fraction of set in the fiber. By analogy to the approach by DeJong (6), this time dependence of the cohesive set can be accounted for by introducing the relaxation function •(t) into equation 6, yielding for the time dependence of recovery: R(t) = E•e/[Ero xlt(t)] (7) Considering the stress relaxation of a wool fiber in water, the relaxation function is expected to decrease towards an equilibrium value of •(oo) = 0.8. Experiments have shown that equation 7 properly accounts for the equilibrium contri- bution of cohesive effects to recovery, though it is unsuitable for the description of the time dependence of these effects. The time dependence of the removal of cohesive set from permanently waved fibers is the object of further investigations. EXPERIMENTAL CONSIDERATIONS AND DATA EVALUATION Tobolsky (4) reviewed some principles of dynamic and static relaxation tests for the investigation of the permanent deformation of rubbers that relax through thiol-disulfide interchange. The relaxation of the stress in a fiber at constant strain is a measure of the cleavage of those bonds that stabilize the undeformed state. The remaining stress at the time of release after fiber deformation, stress relaxation, and fiber treatment is a measure of the fraction of the initial interactions that at time t = w still oppose the deformation and hence enforce recovery. The change of the dynamic stress at constant dynamic strain reflects the net effect of the bonds broken and the bonds reformed during the treatment. The dynamic stress after reoxidation and during the final rinse is related to E•o, the modulus of the fiber after treatment which is in equilibrium with the deformed state. Figure ! shows schematically the force-time diagram produced during the experiment. The initial modulus Eo of the fiber, assuming elastic behavior is given by: