JOURNAL OF COSMETIC SCIENCE 182 their respective swollen states, corresponding to a densely cross-linked matrix microdo- main phase of KAP and a continuous lightly cross-linked rubbery phase of IF. The rub- bery network originates from IF proteins by hydrogen bond scission. When such a non-uniform structure is stretched, microdomains of KAP serve as reinforcing fi ller par- ticles in rubber (14). As shown in Figure 2, when a cube of a swollen fi ber sample was stretched by a force, F, along the z-axis and no deformation of KAP domains and volume change of the rubbery phase were assumed, the volume fraction of the domains in the swollen sample, ϕd, intro- duced to characterize the swollen network, can be represented by equation 1: I , d d d rs V V V (1) where Vd and Vrs are the effective volumes of the domain and the rubbery phase, respec- tively. The extension ratio of the rubber network chain, α, can be represented by the ex- tension ratio of the swollen sample, λ, in equation 2: D OI Id. 1 d (2) When the swollen sample was stretched, non-Gaussian chain statistics were applied and a theoretical relationship between the equilibrium force, F, and the extension ratio of the rubber phase, α, was derived as equation 3 (14,22): ^ ` D D Dn, -1 -3 2 -1 3 1 F G n L n L (3) where F is the equilibrium stress referred to the swollen and unstrained fi ber cross- sectional area, n is the number of segments in the network chain, L-1(x) is the inverse Langevin function, and G is the shear modulus, which can be calculated by equation 4 (14,22,23): ^ ` U I I J. 1 3 1 1 2M c 2 d d c G RT M v M (4) In equation 4, v2 is the volume fraction of hair in the swollen sample, ρ is the density of the unswollen sample (1.30 g/cm3), Mc is the number average molecular weight between cross-links in a rubbery chain (IF), M is the molecular weight of the primary chain (as- sumed to be 5.0 × 104, which is the same as the average molecular weight of IF proteins in wool and hair keratins (24,25)), γ is the fi ller effect of matrix domains in the swollen network, and ϕd is the volume fraction of globular matrix proteins (KAP) in the swollen Figure 2. A simple model for extension of a swollen cube containing microdomains and elastic network chains with effective volume, Vd and the effective volume of rubbery chains, Vrs.

TGA-INDUCED STRUCTURAL CHANGES IN HAIR 183 sample. The fi ller effect, γ, is further represented by equation 5, using ϕd and a shape fac- tor, κ, as the length:breadth ratio for the rod-like fi ller (14,23): J NI N Id 2 2 1 2.5 14.1 . d (5) This semi-empirical equation is applicable to spherical or near-spherical fi ller particles in rubber, with the range of κ values assumed to be 1 ≤ κ ≤ 2. DETERMINATION OF STRUCTURAL PARAMETERS The shear modulus, G, of the swollen fi ber was defi ned as the slope of the initial straight line obtained from plots of the relationship between F and λ − λ-2, because the paren- thetical term in the non-Gaussian expression in equation 3 may be reduced to λ − λ-2 for small deformation of the network (14,15,22). The unknown four parameters, n, Mc, ϕd, and κ, can be determined by fi tting the stress–strain data to equation 3. Here the seg- ment length characteristic in hair proteins, Mc/n (= nr), was determined by fi tting equa- tion 3 to the stress–strain data for a tri-n-butylphosphine-reduced hair sample, which would be expected to have a ϕd value of zero. The value of nr = 1250 obtained was used as one of the known parameters (14,23). The unknown four parameters were thus reduced to three. Therefore, experimentally obtainable parameters were F, G, v2, and λ, and un- known parameters were Mc, ϕd, and κ. By fi tting the experimental data F, G, λ, and v2 for equation 3 with a suitable choice of parameters, ϕd, Mc, and κ, we can evaluate the values of these parameters. We attempted to fi t the equation to the experimental damping Gauss method of nonlinear least squares. These parameters can be obtained using the data over the range of extensions to the in- fl exion point observed at a higher extension range of the stress–strain curve. The least squares refi nement was executed by repeating cycles on the three parameters under the condition that one parameter was fi xed by the software. RESULTS FOR CROSS-LINKED STRUCTURE OF WOOL IF There are two types of IF protein species in wool and hair keratin: Type I and Type II proteins. These form heterodimers with α-helical rod domains and non-helical N and C terminals (25–27). These two types in wool keratin include four protein species each, which have approximately equimolar concentrations (28). The SS contents in the IF of wool and hair are also approximately the same (25,29). The number and location of the cystine residues in wool IF chains sequenced for each of the four species was originally published by Fraser et al. (30), and the average number of cysteines (1/2 cystine) in resi- dues per IF chain was calculated to be seven in the rod domain and 15 in the terminal domain, as given in Table I. When the average molecular weight of IF proteins was as- sumed to be 5.0 × 104, the total cystine content in IF proteins could be calculated as 220 μmol/g (= 22 × 106/2 × 5.0 × 104), which includes 70 μmol/g in the rod domain and 150 μmol/g in the terminal domain (Table I). In 1996, the number, type, and location of cystine cross-links in the wool IF protein were demonstrated by Naito et al. (15) and Arai et al. (16), who investigated the permanent