SEMIPERMANENT DYE DIFFUSION IN HAIR 3 Plotting log DT vs. 1/T should yield a straight line, and the activation energy of diffusion can be determined from the slope of the line. MICROSPECTROPHOTOMETRY (5) Dye concentration profiles obtained as a function of dyeing time can be used to calculate not only the diffusion coefficient, but also fractional dye uptake, M•/M•. The latter involves integration of the dye concentration profiles. To calculate the dye content of a fiber from an intensity scan, the fiber cross section is assumed to be circular and is divided into a series of concentric rings of width dr as shown in Figure la. The dye content of the i th ring of a fiber cross section is proportional to the area of the ring times its optical thickness: 2'rrr,dr bi, (6) where bi is the optical thickness of the i th ring which is proportional to the absorbance of the i th ring, Ai = In Io/Ii. (I o and I, are the measured intensities of the beams transmitted through the blank and the i th ring, respectively.) a) •••'• Optical volume of the ß = 2w-ridr b i r•ng b) A=•.n o Cross section of equilibrium dyed fiber c) [ l•7Ai=J•n Iø Ii Cross section of nonequilibrium dyed fiber Figure 1. Diagrams showing optical volume of fiber cross section.

4 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS The optical thickness and hence the absorbance of an equilibrium dyed fiber is the same at any position in the cross section (Figure lb), so the total dye content of the equilibrium dyed fiber is Moo oc • 27rridr b oc• 27rridr A. (7) i=l i=l For a dyed fiber which has not reached equilibrium, the optical thickness is a function of the radius of the fiber as shown in Figure lc. Therefore, the total dye content of the fiber dyed at time t is proportional to Mt oc • 2'n'ridr bi oc • 2'n'ridr Ai. (8) i=l i=l The fractional dye uptake is given by •-] 2'n'ridr Ai • r, Ai Mt •=1 i=l = = (9) Moo •27rridrA A• ri i=l i=l Since the readings of intensity are taken at regular distance intervals (•) from the center of the cross section along the diameter, r, can be written as if, so that eq. 9 becomes • it•Ai • iAi M t i=• •=• A• + 2A2 + . . . +nAn __ -- -- A il• A i A n (n + 1) i=l i=l 2 (lO) which is readily evaluated. PARTITION COEFFICIENT Another important aspect of dyeing behavior is the degree of interaction between dye and fiber at equilibrium for a given fiber-dye-solvent system. A convenient way to evaluate this property is by calculating the partition coefficient of the dye between fiber and solution, and hence the affinity, -Al• ø. The partition coefficient, K, can be determined from the value of the equilibrium dye uptake, Moo, at concentration C of the dyebath at a given temperature: [D]f Moo K '-• = , (11) [D] C where [D]f = the dye concentration in the fiber phase (mol/kg), [D]s = the dye concentration in the solution phase (mol/1) = (mol/kg), Moo = the equilibrium dye uptake (mol/kg), and C = the dye concentration of' the solution at equilibrium (mol/1).