Diffusion coefficient calculated by complementary error function for the sublimation diffusion of disperse dye

Abstract

In order to find a simple and reliable method for the calculation of the diffusion coefficient, the correlation equation of concentration and distance in the form of complementary error function was derived from solving an ordinary differential equation of the diffusion equation. A disperse dye in paste was treated at 170°C–190°C for 3–4 h for the sublimation diffusion into polyethylene terephthalate using a film roll method. Quadratic regression analysis on the profile of dye concentration–distance was used to determine the surface concentration. The diffusion coefficient of each layer was calculated by obtaining the variable value of the complementary error function from the ratio of the mean dye concentration of each layer to the surface concentration. From linear regression analysis on the Arrhenius plot of the logarithm of the diffusion coefficient versus the reciprocal of absolute temperature, the correlation coefficient for the diffusion of 3 h was 0.9978 and that of 4 h was 0.9991. Thus, it was expected that the diffusion coefficients determined by the equation of complementary error function adopted in this experiment were reliable. The activation energy of diffusion for 3 h was 30.5 kcal mol−1 and that for 4 h was 27.4 kcal mol−1.