250 JOURNAL OF COSMETIC SCIENCE polymers. This paper attempts to provide a theoretical explanation of the magnitude of this increase and other factors affecting the stiffness of hair. RESULTS AND DISCUSSION EXPERIMENTAL The instrumentation and experimental procedures were described in previous reports (4). We employed a texture analyzer, Model TA-XT2, from Texture Technologies Corp. to measure the mechanical properties of hair under controlled conditions of humidity. Hair tresses were in the form of omega loops prepared by using wet hair and special holders to impart omega loop configuration. The hair was dried on the roller to form a perma- nent set. The diameter of the loop was approximately 16 mm. The mechanical measurements were carried out by oscillating a plastic probe between the fiber surface and the calibration height of 4 cm (Figure 1). After touching the surface of the hair and sensing a 2.0 G force (a trigger value of force as shown in Figure 2a), the probe produces an additional 1-4 mm additional deformation (6.25-25%) of the loop before rising to the calibration height. One millimeter deformation is typically within the elastic limit of both untreated and resin-modified hair. Four millimeter deformation (25%) usually results in irreversible damage to polymer-treated hair and is employed to study the flexibility of styling products. This area is covered in detail in our recent papers but is not discussed in this publication. The raw data from the experiment include the values of force and distance as a function of time. The presentation of data as force as a function of time gives a series of peaks, with each peak corresponding to one deformation cycle of a hair loop. Plotting force as a function of distance provides an immediate test of linearity and allows one to make a judgment about the elasticity of a given treatment. For most of the systems investigated in the range of 0-6% deformation, the mechanical response was a linear plot of force = f (distance) (Figure 2a,b). It is usually reversible, showing a small hysteresis. A typical force corresponding to 1 mm deformation for untreated hair varies from 10 to 15 G, depending on the batch of hair and its type. After treatment with styling solutions, this value increases 10 to 40 times. A parameter to characterize the stiffness of hair after treatment was defined as a ratio of the measured maximum force at 1 mm deformation for treated and untreated hair. For the set of data presented in Figure 2a and 2b, it is equal to 161/12.1 = 13.3. THEORETICAL CONSIDERATIONS The mechanical analysis of deformation of omega loops prepared from hair can be described, in approximation, by the theory of deformation of thin rings (6). The theory gives the following dependence between deformation, By, and the force, P (Figure 3): a• =-•- -• (1) where E is the modulus of the ring material, R is the radius of a ring (or a loop of hair),

DYNAMIC HAIRSPRAY ANALYSIS 251 I-•dr • into 1.4to 1.6 crn fo•2.0 (:3 Figure 1. Scheme illustrating a procedure for measuring the stiffness and the geometry of hair samples shaped into omega loops. and I is the area moment of inertia. For a circular cross section, I is given by the following equation: 4 q-rr i- 4 (2) where r is the radius of a cross section of a hair fiber. The validity of equation 1 was verified by performing the measurements of force at a constant deformation 8y for a series of steel rings with radii ranging from 12.6 to 42.7 mm. Corresponding forces ranged from 100 to 6 Grams. The results are presented in Figure 4. A plot of 1ogP as a function of 1ogR gave a linear dependence with the slope of 2.5, which, given the experimental constraints, is in reasonable agreement with the theoretical predictions of equation 1. One of the reasons for the discrepancy might be the fact that the theoretical value relates to a perfect ring, while the experimental results