64 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS force displacement curve is reproduced in Figure 3. The curve is composed of two linear sections: Segment 1 represents the elastic deformation of the foam, while segment 2 is due to the viscous flow. The intersection of these two linear sections is the yield point of the foam, i.e., the threshold force and displacement at which the foam rheology changes from an elastic solid to that of a viscoelastic fluid. The theoretical basis for measuring the viscoelastic quantities of fluids by means of the annular pumping arrangement was developed by Smith et al (10). Accordinly, the force, which is due to the viscous drag of the fluid and acts on the plunger, can be expressed f-- L(t)br/v (4) where v and r/denote the cross-head velocity and the foam viscosity, respectively. The value of b is again given by equation 2 and, as before, L(t) represents the immersion depth of the plunger. Equation (4) predicts a linear relationship between the force and L(t). The validity of this relationship is borne out by the linearity of segment 2 of the displacement vs force E 40 ß Slope 1 [] Slope 2 300 E 200 o 100 • 0 25 50 75 Cross-Head Speed, Cm per Minute Figure 4. The values of slopes ! and 2 of the compression curve as a function of penetration speed. Penetration depth: 3.05 cm. plunger diameter: 2.77 cm. cup diameter: 7 cm. Each point represents a fresh and separate foam sample.

SHAVING FOAM VISCOELASTIC PROPERTIES 65 curve (Figure 3). On the other hand, the functional dependence of the force on q does not agree with the value predicted by equation (2). The reasons for this discrepancy are not clear. A plausible explanation is that equation (2) was derived on the assumption that the fluid in the container does not have a yield point. In the case of materials with yield points, the effective diameter of the container can be less than the real diameter, especially if the vessel is considerably wider than the plunger. Under laminar flow conditions, the shear force in the fluid will gradually diminish from its maximum value at the plunger-fluid interface, towards the periphery of the cylindrical container. If, therefore, at any point along the container radius, the shear force falls below the value of the yield-force, the fluid beyond that point will not flow, but behaves as a solid the surface at that point will act as an effective wall of the vessel. From the force displacement curve, we measured the slopes of segments 1 and 2 at various plunger speeds. The value of slope 1 remained essentially constant within experimental error. Slope 2, however, increased with plunger speed and approached an asymptotic value (Figure 4). We interpreted these results to mean that slope 1 represented the elastic modulus and slope 2 the viscosity of the foam. The viscosity increases with the rate of shearing (the plunger speed) and approaches asymptotically a steady value. This is the usual behavior of dispersed systems in which a degree of structure exists in the unperturbed state and which lose their structure in a shear field (2). MEASUREMENT USING THE DYNAMIC OSCILLATORY MODE As pointed out before, in order to characterize completely the theology of a fluid, it is also necessary to determine two dynamic parameters: the storage (elastic) and the loss (viscous) moduli, desirably on a time scale comparable to the rates of deformation imposed on spreading of the foam over the face. In our experimental arrangement, under oscillatory dynamic movement of the plunger, the axially acting force can be expressed as: f* = f' + if" = bL(t)r/*v = bL (t)(r/'-ir/") v _- bL(t)Bve -iø (5) where v, f', r/' and f", r/" denote the plunger velocity and the respective in and out of phase components of complex force f* and of the complex viscosity r/* L(t) is a periodic function describing the displacement of the cross-head as a function of time. ("In" and "ut" of phase describes the two components of the force that respectively follow the displacement of the plunger instantaneously and lag behind it owing to the foam viscosity. The phase angle describes the magnitude of the lag of the force behind the displacement. These concepts are analogous to those used in the theory of alternating electric currents.) The quantities B and 0 are defined as: B--(r/" + r/"') •/'-- --1 (G" + G"') •/2 (6)