JOURNAL OF COSMETIC SCIENCE 34 for tA=const, where t=0 is the start of the recovery experiment. For the analysis of the experimental data, equation 7 is combined with equation 6 and m=0.28 to yield: 0.28 (t) 1/{1 K exp[ (t/ ) ]} W R (8) where K = ΔB/B∞ is the ratio between the elastic bending rigidities of the fi laments and the matrix in the composite, respectively. The initial recovery at t=0 is accordingly given by: 0 . R (9) Fitting equation 8 to all experimental curves yielded the values for the rigidity ratio, K, and for the characteristic relaxation time, τ. The mean values for K and the related values for R0 for the various relative humidities are summarized in Table I. The water content of hair for the various humidities was deduced from sorption isotherms for similar hair material (19). Consideration of the τ-data versus aging time, tA, on a log/log-scale (1) showed that log(τ) shifts synchronously with log(tA), confi rming the value of the expected (14) aging rate, μ, for hair (11) as: A log / logt 1 P W d d (10) To correct for the effects of aging at a given water content, the parameter of the reduced, characteristic relaxation time, τ r , is introduced: W W /t r A (11) so that r log log log W W A t (12) The results for the arithmetic means of log τ r are summarized in Table I and discussed in detail in reference 1. Fibers can be regarded as non-aging as long as the experimental time is small against the aging time, such as t 0.1t A . For longer times, aging reduces the relaxation rate and in- duces deviations from the curve expected for the non-aging material (14). For a known aging rate this effect can be compensated by introducing the concept of effective time, λ, given for μ=1 by: O ) A t A ln(1 t/t (13) On the scale of effective time, the experimental data represent the recovery performance of the non-aging fi ber. Experimentally, the validity of the concept underlying equations 10–13 has been shown by Chapman, namely, for wool (20).
VISCOELASTIC BENDING RECOVERY OF HAIR 35 MODEL CALCULATIONS To demonstrate the two principal effects of aging on hair fi ber recovery, Figure 1 shows the calculated recovery curves at 65% RH for different initial aging times and experi- mental times well beyond t=tA. The curves for the hypothetical non-aging hair start close to the time-independent, com- mon value of R0=0.18 at 65% RH (see Table I). The curves follow the path that is pre- scribed by equation 8 and are shifted on the log λ scale with respect to the aging time according to equation 10. If the material is non-aging, recovery will be relatively fast and half of the initial set will be lost after about ten times the initial aging time. Deviations between the non-aging and the real, aging material occur at time t 1/10tA, as marked in Figure 1. Comparison of the curves shows that recovery is drastically slowed through aging. Though small differences are observed with respect to the individual aging time, all recovery curves approach a narrow range of fi nal recovery values around Rf 0.6 for very long recovery times (log t 8, t 3 years). This is in agreement with practical Table I Arithmetic Means for K=ΔB/B∞ and for the Reduced, Characteristics Relaxation Times, as log τr RH (%) w (%) K log τr R0 15 4.0 5.9 0.53 0.14 33 7.8 6.2 0.15 0.14 45 9.5 6.2 0.22 0.14 65 13.0 4.7 0.031 0.18 74 15.1 3.3 −0.13 0.23 82 17.3 2.4 −0.01 0.29 Taken from reference 1. RH is the relative humidity and w the water content of hair. R0 is the value of the initial recovery, given by equation 9. Figure 1. Recovery curves for various, initial aging times, as indicated. Broken curves (---) represent the recovery performance of the hypothetical, non-aging material and relate to the log λ scale. Solid curves show the behavior of the real, aging material, relating to the log t scale. Deviations between the two groups of curves become apparent at t1/10 tA, as marked.
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