QUASI-STATIC TORSIONAL DEFORMATION OF SINGLE HAIR FIBERS 389 Given this apparent dominate contribution of the cuticles to the torsional modulus, it may be useful to use a model to separate out the respective contributions of the cuticles from cortex. To this end, we have used an approach based on the work conducted by Wortmann et al. (23). Applying this core/shell model, the following equation for the torsional storage modulus G′ is defi ned (equation 5), where the polar moment of inertia, I, is the addition of the polar moments of inertia of the cortex and the cuticle, Ico and Icu, respectively: a a +G = co co cu G I Icu Ga I (5) Individual cut icle cells are generally accepted to have an approximate thickness of 0.5 μm (24) and the total number of cuticle cells in unaltered European hair is 6–10. Given that the hair used is not directly taken from the scalp, we have assumed that the hair used has six cuticle cell layers and therefore a thickness of 3 μm. Using this information, the theo- retical polar moment of inertia of the cortex and the cuticle can be deduced by equation (6). Figure 4. Plot of the torsional modulus G as a function of the polar moment of inertia, Ip, of 192 virgin European hair fi bers. Line of best fi t was obtained using the equation for the torsional modulus G′ as described previously. Table I Means for the Experimental Torsional Storage Modulus of Virgin European Hair at 50% RH Based on the Analysis of 192 Fibers where 2 R s is the Residual Variance between the Experimental Data and the Fit Data Based on equation (5) and 2 T s is the Total Variance with a Value of 3.089 × 102 Cuticle layers Thickness (μm) G (GPa) q102 R s2 Gcuticle, (GPa) Gcortex (GPa) r2 4 2 1.923 1.201 6.747 0.655 0.611 6 3 1.923 1.205 5.043 0.592 0.610 8 4 1.923 1.209 4.191 0.523 0.609 10 5 1.923 1.213 3.680 3.680 0.607 The coeffi cient of determination, r2, provides the goodness of fi t based on an assumed number of cuticle layers. Refer to (23) for more details on calculations.

JOURNAL OF COSMETIC SCIENCE 390 ʌ 3 3 3 3 + , 4 = – – o i o o i i I a b a b a b a b (6) where the subscri pts o and i denote the outer and inner axis radii, respectively, of the core/ shell. Assuming that the cuticle thickness is constant along the length of the fi ber, theo- retical values of the torsional storage modulus for the cuticle and the cortex at 50% RH have been calculated (Figure 4, Table I). Estimations from our measurements on this par- ticular set of virgin fi bers suggest that the cuticle is 8–10 times more rigid than the cortex. Studies to measure the torsional modulus with the cuticles intact and after removal of the cuticles on virgin hair from the same source should be conducted to verify these theoretical predictions. EFFECT OF MOISTURE AND OXIDATIVE DAMAGE ON TORSION It is well documented that water has the ability to signifi cantly alter the mechanical proper- ties of keratin fi bers (25–28). In hydrated hair, the intermediate fi laments retain the structural integrity of the fi ber which is relatively unaffected by the presence of water, whereas within the matrix proteins, hydrogen bonds are broken leading to radial swelling between the protofi laments as opposed to along the fi ber axis (12). This results in a lower- ing of the elastic modulus in the presence of water. The extent to which the modulus is changed is somewhat dependent on the extent to which the role the matrix plays in the deformation. Consequently, moisture has a much larger infl uence on the torsional modu- lus of fi bers than on the tensile or bending moduli (29). As the humidity level increases, the matrix and the endocuticle layers are increasingly plasticized rending the fi ber more deformable. This is illustrated by a reduction in the Figure 5. Typical plot of force measured by the microbalance (mg) as a function of twist angle (degrees) for bleached European hair at various humidity levels.