J. Cosmet. Sci., 61, 439–455 (November/December 2010) 439 A statistical analysis of hair breakage. II. Repeated grooming experiments TREFOR A. EVANS and KIMUN PARK, TRI/Princeton, 601 Prospect Avenue, Princeton, NJ 08540. Accepted for publication July 30, 2010. Synopsis The objective of this work was to introduce the idea of analyzing data from repeated brushing and combing experiments on hair in accordance with standard fatigue testing approaches. In mechanical testing terms, the brushing and combing of hair represents a fatiguing process wherein individual strands experience repeated exposure to an external stimulus. Therefore, in accordance with fatiguing principles, one expects a gradual propagation of fl aws within individual fi bers until, ultimately, catastrophic failure (breakage) results. A pre- vious paper in this series described the modeling of single-fi ber fatigue data using the Weibull approach, and, in so doing, introduced the idea of treating fi ber breakage as a statistical variable. Here, a grouped Weibull methodology was used to analyze breakage data from repeated brushing and combing experiments. At a top level, the generation of the two Weibull parameters provides a means of characterizing these exper- iments. However, the real strength of the approach involves the ability to generate survival probability plots that provide predictions as to the likelihood of fi ber breakage under different conditions. Therefore, assuming laboratory experiments are a reasonable representation of real-life conditions, it becomes possible to predict breakage rates on actual heads as a function of different habits and practices. It is also shown how the two Weibull parameters, together with information about the number of fi bers in the test tresses, allow for the modeling of repeated brushing and combing tests and allow anyone to re-create the experimental outcome for comparison to their own experiences. These principles have been demonstrated using experiments that compare breakage in virgin and chemically damaged hair, while also showing how conditioning treatments provide considerable retardation. INTRODUCTION The generation of stress-strain curves is often used as a means of quantifying the strength of single hair fi bers. It is well-known that the tensile properties of hair can decrease sig- nifi cantly as a result of exposure to the sun’s UV rays or certain deleterious cosmetic treatments (1) (e.g., bleaching, coloring, perming, or relaxing), and extracting various parameters from such experiments can convincingly quantify these effects. However, for some time there has been concern that the break force for a typical fi ber, as measured by such experiments, is higher than that required to pluck hair from a follicle (2). There- fore, while this approach provides a means of characterization, there may be questions regarding the relevance of experimental results to real-life occurrences. The presence of

JOURNAL OF COSMETIC SCIENCE 440 split ends is evidence enough that fi ber breakage does indeed occur on the head—and so another mechanism must explain their formation. It has been suggested that groom- ing gives rise to a combination of bending, torsion, and interfi ber friction that can re- sult in localized stresses suffi cient to cause breakage (3–7) or lead to the weakening of fi bers such that breakage occurs more readily. Accordingly, some have suggested that repeated grooming experiments may represent a better simulation of how consumers assess the strength of their hair. In these experiments, hair tresses are repeatedly brushed or combed a given number of times, with subsequent counting of the number of broken fi bers that result. This type of testing dramatically demonstrates benefi ts associated with conventional conditioning products, in that surface lubrication reduces grooming forces, snagging, and tangling—thus leading to considerably less breakage. Indeed, antibreakage claims on commercial conditioning products are generally substantiated by this method. In the fi rst paper of this series (8), single-fi ber fatigue experiments were studied, and it was demonstrated how fi bers will break in a predictable manner under repeated appli- cation of forces considerably lower than those required to cause failure from a one-time application. The conventional explanation for this occurrence involves repeated exter- nal stimuli causing propagation of fl aws within a fi ber, which ultimately results in catastrophic failure. Results were able to demonstrate the infl uence of a number of ex- perimental variables on the tendency for breakage. In particular, the likelihood of breakage depends strongly on the extent of the repeating stress (i.e., stress = force/unit area), with changes in the magnitude of the applied force and/or fi ber dimensions yield- ing predictable outcomes. Results showed an exponential relationship between the re- peated stress and the number of cycles required to induce breakage, a fi nding that helps explain the benefi cial effect of conditioner treatments in the afore-mentioned repeated grooming experiments. Specifi cally, surface lubrication reduces grooming stresses, which subsequently leads to a considerably higher number of the repeated stimuli (grooming strokes) required to induce breakage. Results from these single-fi ber fatigue experiments also suggested substantially larger differences as a function of hair type and various external factors compared to those obtained from conventional stress-strain testing. As already suggested, repeated grooming experiments represent a version of a fatigue test, as individual fi bers experience a repeated external stimulus as a consequence of the comb or brush passing through the hair. Thus, one may envisage treating results from these experiments according to mathematical approaches commonly utilized in the fi eld of fatigue testing. As shown previously (8), applying Weibull statistics to single-fi ber fatigue experiments allows breakage to be treated as a statistical variable and produces predictions for the likelihood of hair breakage under specifi c conditions. In this work, we describe such an approach for performing comparable analyses in repeated grooming experiments and show how it can be used to model failure rates. BACKGROUND In general, fatigue testing involves the repeated application of an external stimulus, with subsequent evaluation of the cycles before failure. In the fi rst paper in this series, single- fi ber fatigue experiments were performed using commercially available equipment