596 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS The diffusion constant can be derived from the familiar Stokes-Einstein relationship RT D • -- ($) 6=•r•N where v is the viscosity of the medium, and r• is the molecular radius. This has been shown to hold for spherical particles. Hence it is possible to determine the effective radius of a spherical particle from a knowledge of the diffusion constant, and the viscosity of the solute. Dyestuffs will, in general, not be spherical. Calculations (2) have been made which account for the diffusion of ellipsoids of revolution, which perhaps provides a more satis- factory analogy. For an elongated ellipsoid of axial ratio e (e 1), we have f• (1 -- e•) '• fs e '• In 1 d- (1 --e•) • (4) where f• is the determined frictional coefficient and fs is the frictional coeffi- cient of a sphere of the same volume, and for an oblate ellipsoid (• 1), we have f• (e • -- 1)• f• = e• + tan-• (e •-- 1)• (•) A typical small dye such as acid alizarin black R may be imagined to fulfil either condition. The formula is OH SOaH OH NOs If one allows free rotation about the azo-bond, the molecule approximates to an ellipsoid of axial ratio 1.5. Substituting in the formula above, one has f• --= 1.01. This is sufficiently close to 1 to disregard the effect. Alterna- f• tively one can imagine that the axes are the complete cross section of the molecule and also the depth. Then e --• 4, and we have f• = 1.16. In this f• case, there is a possibility of error which may be as much as 20%. If aggregation of the dye molecules occurs, one would reasonably expect an apparent increase in the radius and hence a lower diffusion constant. If one imagines dimerization with the plain faces of the molecule parallel and close

DIFFUSION PROCESSES IN HUMAN HAIR 597 to each other, we probably have no change in the effective size if the molecule is regarded as an elongated ellipsoid, the extra bulk being compensated for by the decreased freedom of movement. With an oblate ellipsoid, however, f• the axial ratio will decrease by a factor of 2, so we have-• = 1.04 thus, it is a better approximation to a sphere. If this is the case and two planar molecules aggregate in this way, the effective radius will increase by a factor of 1.13, not 1.26 which will occur with the doubling of a sphere. It is doubtful if this could be detected by our present methods, but since the diffusion constant is only reduced by some 10%, it is probably of minor importance in simple diffusion into hair. It may, however, be important if size is a limiting factor. Because of doubt about the behaviour of dyestuffs, the effect of shape will be ignored in subsequent discussion. If the same principles apply to diffusion in hair as to diffusion in water, it should be possible to determine values of D in both media, and hence determine the apparent viscosity of the water in hair. If a barrier is suddenly encountered at a particular molecular size, the apparent viscosity should suddenly increase. Many years ago, Hill (3) solved Fick's second law for infinitely long uniform cylinders in an infinite bath, and obtained a solution of the form Ct Coo -- 1 -- Ae-BK __ Ce-EK (6) Dt where A, B, C and E are constants of known value, K -- a2, and C t and Coo are uptake at times, t and oo. Hill calculated values of cC---•tm for varying values of K, so it is possible to determine values of D from a know- ledge of the uptake/time curve and the radius. This equation gives reason- ably constant values for D, with certain dyes taken up on to cellulose. Crank has developed these ideas for the state where the molecule diffusing into the fibre takes part in rapid irreversible reaction (4), computing values of (•Dt') • for three values of Ct and four values Ofc•. He showed that X, a2,/ Coo a 2 -- 4 C• 1 -• 2 In r--t-- 1 (Z) a where S is the number of reactive sites per unit volume within the fibre, and C B is the number of molecules of solute per unit volume of solution, and r t gives the position of the advancing front at time, t, so that it is possible to