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

677 THE MATRIX REVISITED of this effect will be proportional to the severity of the treatment and the resulting number of cleaved disulfide bonds. Although tensile measurements provide some hints to the matrix chemistry and how the balance of the different bonds contribute to its mechanics, deeper investigation of the matrix requires other approaches. An especially precise alternative involves the use of AFM. This technique allows for the evaluation of the mechanical properties of the matrix with superior resolution, limited only by the size of the indenter tip. When this technique was investigated for examining the matrix mechanics, it was noticed that performing indentation on cross sections could not distinguish the effect of matrix filling the space between KIFs. This is because the resolution of the measurement, about 50 nm, is larger than the mean space between KIFs, which is about 10 nm (17). To overcome this difficulty, an elegant solution was employed, whereby information was acquired on the matrix by performing indentation on the longitudinal sections of the fiber (18). Figure 4 shows results obtained by using AFM to investigate the properties of bleached and virgin fibers by this approach. This figure emphasizes the importance of using longitudinal sections during examination. Whereas the modulus values extracted from indentation of the cross sections of bleached and virgin fibers are not very different, the data for the longitudinal sections show the effects of bleaching on fiber mechanics, especially at low RH. The two-phase model (see Figure 1) suggests that hair fibers consist of a composite material of rods embedded in a matrix. This allows the equations of Voigt (19) and Reuss (20) to be used to describe the elastic moduli of composite materials at axial and transverse stresses of E Axial and E Trans , respectively, in terms of the contributions of the moduli of the rods, E KIF, and the matrix, E Matrix . In this way, the modulus of the matrix can be evaluated from the AFM measurements on cross sections and longitudinal sections as (3) E E E E E E Matrix Trans KIF Axial KIF Trans = - - (5) By using Equations 2–4, one can evaluate the modulus of the matrix from the contribution of the disulfide and hydrogen bonds for both virgin and bleached fibers as (3) E k n E Virgin Matrix N S EWC)1/3 = ⋅ + ( 1- (6) and E k n E′ Bleached Matrix N S EWC)1/3 = + ⋅ + -0.022RH 4 0e . (1- (7) where n is the number of water molecules required to break a hydrogen bond EWC is the equilibrium water content of the fiber, which depends on RH and E S and E′ S are the elastic contributions of disulfide bonds to the virgin fibers and after the breaking of some fibers by bleaching, respectively. Based on the AFM measurements and Equations 6 and 7, the Young’s modulus of virgin matrix at 0% RH was estimated to be about 1.8 GPa, and that of bleached matrix at 0% RH was estimated to be about 5.5 GPa (3). The difference is interpreted as being due to the