SET RELAXATION OF HAIR 207 RESULTS AND DISCUSSION A MATHEMATICAL MODEL FOR SET RELAXATION A hair swatch can be set into a helical coil with either heat or water. When the coil is allowed to relax by suspending freely in air, it becomes more extended with time. Generally, the relaxation is fast at the start, but then becomes more gradual later on. Some typical set-retention curves are shown in Figure 1. The slopes of these curves clearly reflect non-linear, time-related changes which might be portrayed by one of the following functions: (i) Exponential: Y = A EXP (BT) (1) (ii) Power: Y = AT B (2) (iii) Polynomial: Y = A + BT + CT 2 (3) (iv) Hyperbolic: Y = T/(A + BT) (4) Where Y is the percent set retention T is the time after the set is released and A, B, C are constants of the equation. To find out which of these equations would have the best fit to experimental data, we have resorted to the least square analysis. Basically, with this statistical method, one can calculate, from experimental data, the "best values" for the constants of the equation, and from these constants, the correlation coefficient for the equation. The equation with a coefficient closest to unity is the one that fits the data best. Set-retention data, obtained under various conditions, have been fitted to each of the equations and the correlation coefficients obtained are summarized in Table I: Table I Correlation Coefficients for Different Equations Correlation Coefficients Nature Product R.H. in which of Set on Hair Set is Released Exponential Power Polynomial Hyperbolic Water set None 90% 0.67 0.97 0.60 0.72 Water set Hairspray 90% 0.95 0.98 0.98 0.75 Heat set None 65% 0.87 0.95 0.70 0.80 Heat set Hairspray 90% 0.83 0.98 0.98 0.85 Comparison of correlation coefficients in Table I would suggest that the power equation is the most consistent descriptor of the relaxation data. It predicts well the relaxation behavior of either heat set or water set hair over a range of humidities with a correlation coefficient of at least 0.95. The other equations seem to be adequate for some situations, but unsatisfactory for others. The exponential equation, for example, has a reasonably good fit to data, but seems to fail when the set relaxes very fast, as is the case of water set being released at 90% relative humidity. It is not clear why the power equation is more universal. In fact, since the stress decay of keratin fibers held at constant strain follows quite rigorously the exponential path (5-6), one would expect that a similar pattern should be observed for the set relaxation process. We thought that the reason was perhaps due to the difference in the experimental procedure. The stress relaxation experiments are usually carried out under conditions of equilibrium

208 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS moisture content, while in the set holding evaluations, continuous increase in water content by the set tress is concurrent with the set decay. This difference could conceivably account for the failure of the exponential function in some cases. Additional data, however, have sufficiently refuted this theory. For example, by exposing the water set hair (while the hair is still on the rod) at 90% Relative Humidity for 45 minutes prior to release, we should minimize the effect of this "humidity shock." However, we have found that the relaxation of the water set is still not following the exponential law as rigorously as the power function. This intriguing aspect of relaxation of human hair needs clarification. The form of the power function suggests that its behavior would depend on the magnitude as well as the sign of the two constants A and B. It is therefore instructive to examine these constants more closely. In Table II, some typical values of A and B are Table II Some Typical Values of the Constants A and B Values* of Constants Calculated from Least Square Analysis Humidity in Which Nature of Set Set is Relaxed A B Water Set 90% 15 --0.457 Water Set (with Hairspray) 90% 47 -0.139 Heat Set 65% 26 -0.349 Heat Set (with Hairspray) 90% 46 -0.144 *The unit of time, T, is in hours. summarized. The negative values of B require that Y should increase inversely with the variable T and in the limiting case, become infinite at T = 0. Apparently, at the initial point of set relaxation, and in some cases, the very short moment immediately after the relaxation has proceeded, the power function would fail to relate to one of our physical constraints because, by definition, Y cannot exceed 100%. This situation arises mainly because in the process of least square analysis, all the experimental data are weighed equally. Unless the initial condition, namely that Y equals 100% at T = 0, is artificially imposed in the analysis, the resulting function would reflect only the trend set by the majority of the experimental data. This apparent flaw of the power function poses little problem to its utility, however. In practice, one is not particularly concerned about what happens to the set relaxation during the seconds after the relaxation begins, unless the set relaxes extremely fast. In most cases involving water set or even heat set of hair swatches, especially aided by hair spray, the relaxation in the first few seconds is insignificant. Even though the physical meaning of the two constants A and B in the power function is not immediately obvious, these constants, however, can be combined to form parameters, whose physical significance becomes more apparent. Indeed, two such parameters can be easily derived. One is the "half-life" of the initial set of the hair, which is time elapsed in order for the set to lose 50% of its value. The "half-life," designated by T•0, is readily calculated: