() () 0 20 4 60 8 1 1e+9 2e+9 3e+9 4e+9 5e+9 Virginhair Young’smodulusasa f ti of RH 3x 9% bl h Relati humidit(%) Torsion Testing:Sh ea r Modulus (G Pa ) as Functi of Humidity % RH 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Virgin 3X Bleached Figure 3. Influence of RH on hair’s (A) extensional and (B) torsional moduli. 675 THE MATRIX REVISITED suludom s'gnuoY )aPG( suludoM raehS

676 JOURNAL OF COSMETIC SCIENCE matrix, in Figure 1). The mathematical model describing the viscoelastic mechanics is given by the Kelvin–Voigt equation (6), which represents the fiber stress as the sum of the elastic (Hooke’s law) and viscous (Newton’s law) responses to the applied stress σ ε(t) η ( ) t E t = ⋅ + ⋅ dε d (1) In Equation 1, σ(t) represents the stress applied to the fiber during time t, ε(t) is the recorded strain at the applied stress, E is the elastic modulus of the fiber, and η represents the viscosity of the material. An evaluation of the viscosity of keratin materials, for wool fibers at 20°C and 65% RH, produced values in the range of 1014–1019 Pa, which suggests that the viscosity of the fibers, which is attributed to the matrix is at the limit of highly viscous fluids and soft solids (12). Chemically, the crystalline phase (KIFs) is made of low-sulfur proteins, and the matrix consists of KAPs, classified as high-sulfur, ultrahigh-sulfur, and glycine–tyrosine-rich proteins (13). For the modeling of the mechanics of hair, the matrix was considered to be a polymer cross- linked mainly by disulfide bonds and stabilized by a network of hydrogen bonds (14). The effect of the relative humidity on the fiber mechanics (see Figure 3A and B) is therefore understood as the result of water solvating the hydrogen bonds. One may assume, then, that the matrix behavior is strongly dependent on the abundance of electrostatic bonds and postulate that the complex Young’s modulus of the fiber, E, is the sum of the contribution from the disulfide bonds, E S , and that due to hydrogen bonds, E H (15) E E E S H = + (2) The contributions of the disulfide and hydrogen bonds to this complex modulus are proportional to the numbers of the respective bonds. For the case of hydrogen bonds, this proportionality is assumed to follow Nissan’s model (16) E k N1/3 H N = (3) where N is the number of effective hydrogen bonds per unit volume of keratin fibers and k N is the constant of proportionality in Nissan’s model. An increase in relative humidity leads to a higher water content, which breaks hydrogen bonds and decreases the contribution of the second term, E H , to the overall complex Young’s modulus. This can be seen experimentally in Figure 3A and B. At 100% RH, the contribution of hydrogen bonds to the Young’s modulus is nil, and the mechanics of the fiber are a reflection of only the disulfide bridges E E S 100%RH = (4) These disulfide bonds can be cleaved by chemical treatments, such as oxidative bleaching, to form cysteic acid. In accordance with Equation 4, this should result in a decrease of the elastic modulus of the fiber when measured at 100% RH (see Figure 2A). The magnitude