J. Cosmet. Sci., 68, 173–182 (March/April 2017) 173 Model-based analysis of the torsional loss modulus in human hair and of the effects of cosmetic processing FRANZ J. WORTMANN, GABRIELE WORTMANN, HANS-MARTIN HAAKE, and WOLF EISFELD, School of Materials, University of Manchester, Manchester M13 9PL, United Kingdom (F.J.W., G.W.) and BASF Personal Care and Nutrition GmbH, 40589 Duesseldorf, Germany (H.-M.H., W.E.). Accepted for publication February 1, 2017. Synopsis Torsional analysis of single human hairs is especially suited to determine the properties of the cuticle and its changes through cosmetic processing. The two primary parameters, which are obtained by free torsional oscillation using the torsional pendulum method, are storage (G′) and loss modulus (G″). Based on previous work on G′, the current investigation focuses on G″. The results show an increase of G″ with a drop of G′ and vice versa, as is expected for a viscoelastic material well below its glass transition. The overall power of G″ to discriminate between samples is quite low. This is attributed to the systematic decrease of the parameter values with increasing fi ber diameter, with a pronounced correlation between G″ and G′. Analyzing this effect on the basis of a core/shell model for the cortex/cuticle structure of hair by nonlinear regression leads to estimates for the loss moduli of cortex (G″co) and cuticle (G″cu). Although the values for G″co turn out to be physically not plausible, due to limitations of the applied model, those for G″cu are considered as generally realistic against relevant literature values. Signifi cant differences between the loss moduli of the cuticle for the different samples provide insight into changes of the torsional energy loss due to the cosmetic processes and products, contributing toward a consistent view of torsional energy storage and loss, namely, in the cuticle of hair. INTRODUCTION The behavior of human hair under torsional stresses and strains is an important contribut- ing factor for the formation and maintenance of a hair style ( 1). Because of the nature of torsional deformation, the results for a fi ber are biased toward contributions from its outer regions ( 2). For human hair, the method is thus especially suited to investigate the properties of the cuticle. In a recent publication ( 3), we presented a set of data from in- vestigations on untreated and cosmetically treated human hair fi bers using the torsional pendulum technique. For that investigation, we concentrated on considerations of the storage modulus G′, which is derived from the frequency of the free torsional oscillation. A basic core/shell model of cortex and cuticle was applied to model the observed decrease Address all correspondence to Franz J. Wortmann at franz.wortmann@manchester.ac.uk.

JOURNAL OF COSMETIC SCIENCE 174 of G′ with fi ber diameter or rather polar moment of inertia. This analysis enabled to obtain estimates for the torsional storage moduli of cuticle and cortex through nonlinear curve fi tting and extrapolation. The results of the analysis supported the hypothesis that the torsional storage modulus of the cuticle is signifi cantly higher than that of the cortex. Though the absolute value for the modulus of the cortex was too low compared to litera- ture values, plausible changes of cuticle and cortex moduli were determined after cos- metic treatments. This part of the investigation now is focused on the logarithmic decrement Λ, as a measure of energy loss in the fi ber and as one of the primary variables from a torsional pendulum experiment. The loss modulus G″, as primary physical variable, is determined indirectly from the logarithmic decrement Λ and the torsional storage modulus G′ for an individual measurement. G′ is proportional to the energy stored and G″ to the energy lost during a torsional oscillation. The objective is to investigate whether the structure-based, basic core/shell model approach for G′ (3) is also applicable for G″. This includes estimates of the loss moduli of cuticle and cortex as well as the effects of cosmetic treatments. The potential as well as the specifi c limitations of the approach are discussed. MATERIALS AND METHODS THEORETICAL BACKGROUND Free torsional oscillation, e.g., of a fi ber in a torsional pendulum apparatus (2,4, 5), yields the complex torsional modulus G* as: G* = G′ + iG″ (1) where G′ and G″ are the storage and loss modulus, respectively. G′ is given by: = ′ 2 2 4π J l G I T (2) where J is the moment of inertia of the pendulum, l the length of the fi ber, I the polar moment of inertia of the fi ber, and T the time taken for one oscillation. The cross-section of a hair fi ber is generally assumed to be best described as elliptical so that the polar moment of inertia is given by: I = (π/4) (a3b + b3a) (3) where a and b are the semiaxes of the ellipse. The use of the polar rather than the torsional moment of inertia (6) assumes the limiting case that no warping of the test specimen occurs (7), which is plausible for small deforma- tions and low resonance frequencies (8), as realized in this study. The situation is certainly different for combinations of high tensile and torsional strains (9). The approach was furthermore chosen to provide better comparability of data with previous investigations