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

TORSIONAL PERFORMANCE OF HUMAN HAIR 65 treatment makes the hair fi ber overall stiffer, at least at this low humidity. Effects at higher humidities may be quite different and even reverse (3). The decrease of G′ with the subsequent shampoo treatment (WBS), which may be attributed to a positive action of the product in terms of “softening,” is in this context not signifi cant on the 95% confi - dence level (p = 0.29). The latter observation corroborates the observation that data sets for samples WB and WBS in Figure 1 would strongly overlap, if plotted together. This seems to indicate that the discriminative power of the torsion method may thus be satis- factory for harsh hair treatments, but may not be adequate for more subtle ones, in appar- ent agreement with previous observations on a comparable set of samples (3). The values for G′ as such, as given in Table I, are in good general agreement with previous observa- tions by Bogaty (24) and Harper and Kamath (3) for virgin as well as cosmetically pro- cessed hair. ESTIMATING THE TORSIONAL STORAGE MODULI OF CORTEX AND CUTICLE The errors signifi ed by the boxes and whiskers in Figure 2 are implicitly assumed to be random. Figure 1 shows for all samples, however, that the errors, e.g., given by the total variances 2 T s (see Table I), contain in fact a signifi cant systematic component, namely the drop of G′ with the moment of inertia. Fitting Equation 4 to the data sets in Figure 1, this drop of G′ is used to separate the torsional storage moduli of cortex (G′co) and cuticle (G′cu). These two moduli values are given in Table I, together with the coeffi cient of de- termination (r2) of each fi t. The fi ts account in each case for a major fraction of the overall data variance. Figure 3 gives a graphical presentation of the data in Table I. It emphasizes that G′cu is much larger than G′co, supporting the view that the torsional properties of a hair fi ber, similar as in bending (25), are controlled to a signifi cant extent by cuticle properties (2). The large difference between the moduli values for the two morphological components is in agreement with observations by Parbhu et al. (23) and with a priori expectations, re- garding the high degree of cross-linking, namely, in the exo-region of the cuticle (13). The storage modulus of the cortex G′co drops from 0.61 GPa for virgin hair by about 1/3 to 0.40 GPa with permanent waving and bleaching. The relative drop as such is largely Table I. Arithmetic means of the storage modulus G′ for samples V, WB, WBS (see text) at 22°C and 22% RH. N is the number of fi bers tested in each case. 2 T s is the total variance of the data and 2 R s the residual variance after subtracting the amount accounted for by the fi t of the data using Equation 4. G′co and G′cu are the storage moduli for cortex and cuticle determined on the basis of those fi ts. The goodness of fi t is characterized by the coeffi cient of determination r2, giving the fraction of data variance explained by the fi tted regression lines (Equation 7). Fits are conducted on the assumption of a diameter-invariant thickness of the cuticle of 3 μm, thus assuming six layers of cuticle cells (see text). All values are based on the cross-sectional dimensions of the fi bers at 22°C and 55% RH. Sample (N) G′, GPa 2 T s × 102 2 R s × 102 G′co, GPa G′cu, GPa r2 V (69) 1.58 3.81 2.07 0.61 3.60 0.457 WB (56) 1.81 4.47 1.39 0.40 4.84 0.689 WBS (23) 1.76 3.65 0.88 0.37 4.63 0.758