TORSIONAL PERFORMANCE OF HUMAN HAIR 63 commercial hair), this value is assumed to be constant along the fi ber, in good agreement with observations of cuticle-wear patterns by Garcia et al. (16). Implementing the ring/core-structure of cuticle and cortex, Equation 1 yields: ′ ′ ′ ( co co cu cu G I G I) G I (4) with the total moment of inertia of a fi ber given by: co cu I I I (5) where subscripts co and cu relate to cortex and cuticle, respectively. Treating the cuticle, based on the argument made above and for the current hair sample, as a hollow shaft with a constant wall thickness of 3 μm, Icu is calculated by a suitably modifi ed version of Equation 3 (17). Given the principal objectives of the investigation, no correction was applied at this stage to correct for potential changes of cuticle thickness through cosmetic processing. Figure 1. G′-values for individual fi bers from samples of virgin (V), perm-waved and bleached (WB), plus shampoo-treated (WBS) hair at 22°C and 22% RH. The values are plotted against moments of inertia (Equation 3). The solid lines through the data are fi ts according to Equation 4 with the parameter values given in Table I.

JOURNAL OF COSMETIC SCIENCE 64 Equation 4 was fi tted to the data in Figure 1 using the Excel solver function and targeting the minimum of the residual sum of the squared errors for G′. This yields the residual variance 2 R s as: œ ′ − − 2 , ,th ( 1 i exp i i R G G)′ s2 N (6) G′i,exp are the individual experimental values for the storage modulus, while G′i,th are the related values on the fi tted curve for the same moment of inertia. The overall quality of the fi ts of the curves is given, in the usual way (4) by the coeffi cient of determination r2, which gives the fraction of data variance explained by the curve fi t, so that: − 2 2 1 R T s2 r s (7) where 2 R s and 2 T s are the residual and the total variance, respectively. RESULTS AND DISCUSSION BASIC RESULTS Figure 1 shows the individual results for G′ for virgin (V) and treated fi bers (WB & WBS), plotted against the moment of inertia of the individual fi bers (Equation 3). The lines through the data are theoretical fi ts on the basis of the core/shell model (Equation 4). Despite substantial scatter, the data for G′ show in all cases a pronounced overall drop with increasing moment of inertia and thus with overall fi ber diameter. The data confi rm that the torsional storage modulus of hair is not a material constant. This drop with mo- ment of inertia is in agreement with observations by Persaud and Kamath (2) at 50% RH, but in contrast to earlier observation by Wolfram and Albrecht (18), who found such a change only for hair fi bers in water but not at 65% RH. The assumption, backed by our experimental observations, that the thickness of the cu- ticle is independent of fi ber diameter, implies that the cross-sectional fraction of the cu- ticle decreases with increasing fi ber diameter. In view of Figure 1 this, fi rstly, leads to the qualitative conclusion that G′ for the cuticle has to be substantially different from that of the cortex, in agreement with earlier considerations (2,19,20) and in contrast to more recent observations (21,22). In fact, the results, secondly, imply that the modulus of the cuticle has to be larger than that of the cortex, in agreement with and supporting the observations by Parbhu et al. (23). Table I gives the arithmetic means of the storage moduli of the three samples and the related total data variances (s2 T ). Figure 2 graphically summarizes the experimental re- sults for G′ in the form of a box-and-whisker plot. In this presentation G′ shows a sub- stantial and signifi cant increase (LSD test, p 0.001) with the perm and bleach treatment (WB). From a materials point of view, this leads to the conclusion that the harsh chemical