RHEOLOGIC AGING OF COSMETIC LOTIONS 149 magma was related to the logarithm of time. Subsequently he found (7) that a better linear relation held for a plot of the logarithm of the growth against logarithm of time. In another recent paper, Levy (8) has studied recovery rates from dis- turbances in strongly thixotropic bentonire gels. He reported that the re- covery strength of the gel was related to the logarithm of time in a manner analogous to that noted by Chong (6). Actually, in the range of recoveries observed by him, the logarithm of recovery would fit the data quite as well and for our purposes may therefore be assumed to hold. In the field of metallurgy (9), it has long been an empirical fact that many annealing processes are well depicted by the log-log plot against time. Similarly the exchange reaction of HC1 gas on sodium chloride (10) fol- lows this relationship. Other experiments (11) studying irradiation de- fects have equated a linear function of per cent change to the logarithm of time, while a group (12) studying the polymorphic transition in tin uti- lized a double log function of the fraction transformed as the variable against the logarithm of time. THEORETICAL BASIS FOR THE RELATIONSHIP Avrami (13) has examined the general kinetics of phase transformation in solid systems and has shown that, for conditions of isolated nucleation, a proportionality can exist between transformed volume and a power of the elapsed time. If it is assumed that we are dealing in systems where such isolated nucleation of changes is diffusion inhibited, then thick cosmetic lotions should, in a first approximation, obey this proportionality. If it is then assumed that the extensive property (P) under examination is di- rectly related to the degree of transformation that occurred in time (t),then p= at b and logP = blogtd-c where a, b and c are constants of the systems to be evaluated empirically. For cosmetic lotions, diffusion controlled and isolated nucleation reac- tions are possible in many cases. Certainly, the diffusion control of the solid state might be expected to be less rigorously obeyed for a viscous semi- heterogeneous mixture than for the solid, and indeed this is the general case in the cosmetic lotions. However, the resultant behavior should be expected to be some compromise between the normal kinetics of the homo- geneous system which involve a linear function of time and the limiting solid-state type dependency on the logarithm of time. APPLICATION TO COSMETIC LOTIONS In Fig. 2 are shown logarithmic viscosity-time plots for several different systems. The solid lines represent heat emulsified solid fat systems, the dotted line a hand lotion, and the dashed lines antiperspirant lotion formu-

150 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS )- O•lO 2 I O.I e• e"'•' YIELD VALUE •.0, •• ß • PLASTIC VISCOSITY _ •O/O•O THIXOTROPIC AREA 0 2 0.4 0 6 0,8 I 2 4 6 8 I 0 AGE (W[EKS) Figure 1.--Data of Levy and Rutowski (5) replotted on logarithmic basis, to show the build-up of the rheologic parameters of bentonire with time. 20,000 200 I00 0.04 O, I 0.4 I 4 I 0 40 I00 400 ELAPSED TIME ( DAYS ) Figure 2.--Some aging curves of cosmetic lotions. ( ..... cosmetic hand lotion, "solidified" fat systems from a hot emulsion, ---- antiperspirant lotions).