PERMANENT WAVING AND PERM CHEMISTRY 113 gives rise to considerably slower reaction rates with both solutions. Similar differences in SFTK rates as a function of hair type were also noted by Wickett (17). These fi ndings suggest a hypothesis whereby the poor response of “resistant hair” to the perming process is a consequence of slower transformation rates which result in an insuffi cient number of disulfi de bonds being cleaved during a given exposure time to the thiol solution. F igure 9 shows the result of further investigation into the infl uence of hair type on the rate of transformation with a common perm treatment. Hair was procured from nine different donors, and SFTK experiments were performed using a 0.42 M, pH 9 cysteamine solution. Clearly, there are signifi cant (and sizable) differences in reaction rates associated with these women’s hair. The donors were asked to fi ll out a questionnaire documenting typical habits and practices, and some basic laboratory characterization measurements were also performed (e.g., fi ber dimensions). However, no simple correlations were seen between the hair properties and transformation rates. T he one exception to this statement seems to involve the initial “health” of the hair. The donors in the previous experiment had not used any chemical treatments, and consequently, their hair was considered in relatively good condition. An additional experiment involved comparing the reaction rates for single-source virgin hair with a subbatch of the same hair that was exposed to a standard bleaching treatment. Results showed considerably faster re- action rates associated with the chemically damaged hair. This fi nding is in line with the popular opinion that extreme care is needed in attempting to perm, especially damaged hair, because of the danger of overprocessing. Again, this highlights the “art” of the perm process because stylists attempt to judge the reactivity of hair and obtain the desired result. Figure 9. Halftimes for panelists’ hair when exposed to 0.42 M, pH 9 cysteamine.

JOURNAL OF COSMETIC SCIENCE 114 A DDITIONAL ANALYSIS OF SFTK DATA F urther analysis of SFTK data can be performed to examine the mechanism (i.e., the mathematical expression or model) that describes the progression of the process. When performing this modeling, it is again recalled that progress is dependent on two dis- tinctly different steps: the fi rst involves the rate of the chemical reaction between the re- actants, but the second comprises the readiness by which the perm active can diffuse throughout the hair and allow the reactants to come together. The slower of these two processes will become a bottleneck to the overall transformation, which can only proceed at a rate commensurate with this limiting condition. If the reaction rate is fast, but dif- fusion is slow (i.e., diffusion-limited conditions), an advancing reaction “front” would be anticipated within hair fi bers, whereupon unreacted cystine bonds lie ahead of the inter- face, although reduced bonds would be present behind. Conversely, fast diffusion and slow reaction (i.e., reaction-limited conditions) allow the active to readily penetrate throughout the hair before appreciable reaction occurs. In this case, there would be no well-defi ned interface. Visualization of these behaviors can be attained through micro- scopic means in combination with staining techniques, that is, freshly permed hair is treated with reagents that specifi cally adhere to free thiol sites (27,28) and indicate where the disulfi de bonds have been broken. Figures 10 and 11 show examples of these two oc- currences as identifi ed by fl uorescence microscopy (24). In these specifi c experiments, treatment with a 0.42 M, pH 9.2 ATG solution resulted in an advancing front (i.e., dif- fusion-controlled behavior), whereas treatment with a cysteamine solution under the same conditions did not (i.e., reaction-controlled behavior). T hese two conditions become central in deriving mathematical expressions for describing the progression of the perm reaction. Reese and Eyring (14) proposed a pseudo–fi rst-order mechanism to describe the reaction-controlled condition see equation (11), and Wickett (17) later derived a diffusion-based expression that describes a moving boundary advanc- ing through a cylinder see equation (12). 0 exp o F t F kC t (11)  ¬ ­t3 ž ® 2 0 2KC 0 expž 3T F t F (12) It is again noted that chapters on reaction kinetics in chemistry textbooks tend to deal with homogeneous gaseous and liquid systems, but there is a less well-known literature dealing with the rates of heterogeneous reactions (25) (i.e., solid–solid, solid–gas, and solid–liquid interactions). In this section, we borrow well-established ideas from this fi eld and adapt them to studying the perm process. The rate of a heterogeneous process is given by the following equation: ( ) ( ), = . dα k T f dt α (1 3) wher e α is the fraction of transformation (in our case, the percentage of bonds broken at any time, t), k (T) is the temperature-dependent specifi c reaction rate constant, and f (α) is the mathematical expression that describes the overall progression. The sim- plest reaction mechanism is a conventional fi rst order equation, where the rate is