STIFFNESS OF HUMAN HAIR FIBERS 475 The straight-line relationship in Figure 2 of stiffness index to linear density and hence to approximate cross-sectional area is anticipated from eq 7 in the Appendix since the bending modulus should be nearly a constant. The scatter of points within the 95% confidence limits in Figure 2 is attributed mainly to shape differences among fibers with flatter fibers giving low and rounder fibers high stiffness values. Slight inelastic yielding of thinner fibers during measurement may account for the intercept not pass- ing through the graph origin in Figure 2. To more graphically illustrate the variation in stiffness from fine to coarse fibers, the photographs in Figure 3 were taken of four fibers weighted equally and with the proximal or root ends at the left of the hook. The linear densities correspond ap- proximately to fiber diameters of 107, 83, 73 and 55 •. It is apparent that the weight must be increased for the thickest fiber if the stiffness index D is to be measured. EFFECT OF ATTACHED WEIGHTS Stiffness indices were determined for three fibers with different weights attached to the fiber ends. In accordance with eq 7 and 9 in the Appendix, the stiffness index squared is plotted against the reciprocal of the attached weight for each fiber. Results in Figure 4 show linearity in agreement with theoretical prediction. When fibers such as those shown in Figure 3 are not all measurable at the same weight, stiffness results are more conveniently compared in terms of stiffness coefficients (see x z o 1.50 1.00 .50 I 2 .9 4 5 6 7 B RECIPROCAL OF LOAD (I/g) Figure 4. Effect of load on the stiffness index using functions indicated by eq 3

476 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Appendix eq 9), which compensate for different weight loadings. Weights should be large enough to produce vertical hang without imparting a set to the fiber. As a precau- tion, fibers should be examined after measurement for signs of imparted set. The effect of weight change on the stiffness index and on bending moments at the hook are derived in the Appendix. Briefly, increasing weights fourfold will double the bending moment and halve the distance between fiber legs. EFFECT OF HOOK WIRE DIAMETER (W) The diameter of wire used tq suspend fibers was varied from 0.19 to 3.16 mm to test effects on the "D" measurement. Results in Table I show that D values are inde- pendent of wire size for most of the range covered. Careful handling of fibers is espe- cially important with the fine wire sizes. At the largest wire size, the D measurement increases for all but the stiffest fiber. Empirically we learned that wire size is too large if the fiber tends to orient on the sup- port wire in a plane perpendicular to the wire. On the other hand, if a fiber tends to swivel freely, the support wire has an acceptable diameter. An explanation of this behavior involves the problem of point and arc contacts between fiber and substrate, which is of considerable interest also for frictional studies (21, 22) using capstan methods. With fine wires, curvature of the wire surface is greater than that of the bent fiber and theoretically contact exists only at a "point." As wire size becomes larger, curvature of the wire surface will, at some stage, equal the curvature of the bent fiber. Beyond this stage, contact between fiber and wire becomes an arc which increases in length as wire size is further increased. Equal curvature of fiber and wire is achieved when the wire diameter (W) is equal to half the D measurement (see Appendix). "Point" contact thus exists when the ratio D/W is above two and arc contact below two. The data in Table I provide an interesting empirical test of the D/W criterion. For wire sizes to 1.62 mm, D/W is greater than two and D values do not depend on wire size. For the 3.16-ram wire and all fibers except the stiffest, the D/W ratio is less than two and D measurements change. For the stiffest fiber, D/W = 2.18 (point contact) and the D measurement is unaffected by this wire size. Table I Hook Wire Diameter Wire (cm) 0.019 0.024 0.031 0.038 0.076 0.128 0.162 0.316 Lin. Dens. "D" Distance 46.9•g/cm 0.35 0.35 0.34 0.34 0.33 0.33 0.33 0.45 51.1 0.39 ..... 0.39 0.47 56.7 0.41 0.41 0.41 0.40 0.41 0.41 0.39 0.48 58.0 0.44 ..... 0.44 0.51 63.6 0.48 ..... 0.46 0.52 69.0 0.50 ..... 0.51 0.54 89.8 0.61 ..... 0.60 0.62 89.3 0.68 ..... 0.69 0.69