TORSIONAL PERFORMANCE OF HUMAN HAIR 61 where J is the moment of inertia of the pendulum, l the length of the fi ber, I the polar moment of inertia of the fi ber, and ω the frequency of oscillation, with: X 2Q T (2) where T is the time taken for one oscillation. Equation 1 applies if the damping of the torsional oscillation, i.e. , the dissipation of torsional oscillation as frictional energy, is low. Though hair is inherently a viscoelastic material, it is consistently below its glass transition temperature for low to medium relative humidities (9), showing in conse- quence long relaxation times and thus little damping (7,10,11). The specifi c cross-sectional shape of a human hair, though irregular along its length, is generally assumed to be elliptical, so that the polar moment of inertia is given by: Q¬  ž ­ ž ⋅ ( ) 3 3 4® I a b ab (3) where a is the long and b the short semi-axis of the ellipse, respectively. Arithmetic means for G′ were determined from individual values for T, as determined along the oscillation curve and according to Equations 1 and 2. Five repeat measurements were conducted for each fi ber, averaged and taken as the G′-value for that fi ber. Values for fi bers tested for a sample were further summarized by their arithmetic mean, variance, standard error (S.E.), and the limiting value for the 95% confi dence range (1.96 S.E.) (4). Data sets were compared using the Fisher least signifi cant difference (LSD) test (12). This is essentially a multiple t-test and as such very non-conservative (4). It therefore appears generally well suited for testing in the context of cosmetic products and processes. Also separate t-tests for specifi c pairs of samples were conducted. CHARACTERIZATION OF HAIR GEOMETRY All tests and treatments were conducted on dark brown, commercial, Caucasian hair (In- ternational Hair Importers & Products Inc., Glendale, NY). The fi bers were washed with 3% sodium lauryl sulfate (BASF, Duesseldorf, Germany), rinsed thoroughly with warm water, and allowed to air-dry under ambient conditions. To determine the cross-sectional shape of a fi ber, the assumption of general ellipticity was made. For each fi ber, prepared for torsional testing, the smallest and largest diameters were determined at fi ve equidistant positions of a 3-cm-long fi ber through a 360° rota- tion by means of a Laser Scan Micrometer (LSM-500, Mitutoyo, Kanagawa, Japan), as implemented by Dia-Stron Ltd., United Kingdom. At each point along the fi ber, the smallest and largest diameter was determined. The arithmetic means of the measurement data were used to calculate the overall short and the long axis of the elliptical fi ber cross section. Diameter measurements were conducted under conditions of constant, but ambi- ent laboratory climate (approximately 22°C, 55% RH). The diameter data thus obtained

JOURNAL OF COSMETIC SCIENCE 62 were taken as basis for the determination of the moment of inertia of a fi ber according to Equation 3. No corrections were applied for the limited changes of cross-sectional shape at the conditions of torsional measurement (22 ± 2°C, 22 ± 2% RH). HAIR TREATMENTS Hair tresses were taken from the collective of virgin hair, to be referred to as V in what fol- lows, and submitted to a permanent waving treatment with 7% thioglycolic acid at pH 9.5, adjusted with ammonia. The solution was applied in excess to the tresses for 30 min. This was followed by extensive rinsing and reoxidation with 2.2% hydrogen peroxide solution, adjusted to pH 4.0 with citric acid for 30 min. Finally, the tresses were again extensively rinsed and then dried with a hair drier at ambient temperature. To further sensitize the hair, it was subsequently subjected to a strong bleaching treatment with an 8% hydrogen peroxide/persulfate combination at pH 9.4 for 30 min. Subsequently, the hairs were extensively rinsed and dried with a hair drier at ambient temperature. In what follows, hair from tresses, which have undergone this treatment sequence will be referred to as WB (permanently waved and bleached). To further refl ect the general context of hair cosmetic treatments as well as to target more subtle changes of hair, a commercial shampoo was applied, which claims to be able to repair the hair surface. The shampoo was applied on WB-type hair fi bers already fi tted with crimps and thus ready for torsional testing. The hairs were wetted and then covered with the shampoo for 30 min. Subsequently, the shampoo was rinsed off for 1 min and then the hair dried with an air drier for 15 min at ambient temperature. The long contact time was chosen to maximize the potential effect of the shampoo as well as to refl ect the effects of repeated applications. In what follows hairs, which have un- dergone all treatments will be referred to as WBS (permanently waved, bleached, and shampoo treated). THE CORE/SHELL MODEL FOR HAIR To analyze the dependence of G′ on the moment of inertia, as documented for all samples in Figure 1, a core/shell model for hair is applied. For an individual human hair, the cor- tex is surrounded by a layer of cuticle cells, which are arranged in a tile-like fashion with each cuticle cell being in contact with the cortex as well as being visible on the fi ber sur- face (13). The resulting tilt angle away from the fi ber surface is 2–3° (14,15). In the cross section of a hair fi ber, this arrangement presents itself as a sequence of concentric layers of cuticle cells around the cortex. The thickness of each layer is that of an individual cuticle cell and for human hair generally determined as approximately 0.5 μ m. The total number of cuticle layers for undamaged hair near the scalp is expected to be 6–10, leading to an initial cross-sectional thickness of the cuticle layer of 3–5 μm (13). For the hair fi ber collective used for these investigations, the overall cuticle thickness was found microscopically in cross sections to be largely independent of diameter and about 3 μm, equivalent to six layers. In view of the relatively short length of a sample fi ber (3 cm effective length) and its probable origin with respect to the scalp (approximately 10 cm for