Diffusion coefficient for diffusion of time-varying surface concentration by the film-roll method

Self Cite

In this study, an equation to obtain reliable diffusion coefficients for the sublimation diffusion of disperse dye in paste into polyethylene terephthalate film using the film-roll method was derived from the solution of the diffusion equation with the constant surface concentration. The constant surface concentration in the equation, where the total amount of dye passed through the first-layer surface is a linear function of the square root of time, was obtained by the steady-state straight line for a sufficiently long heat treatment time. The diffusion coefficients were obtained by the constant surface concentration and the slope of the linear regression line for the plot of the total amount against the square root of time. Their reliability was proved by the good linearity of linear regression for the Arrhenius plot. The activation energy for the diffusion coefficients obtained by this equation was similar to that obtained by another equation derived from the solution with the time-varying second-layer surface concentration. The diffusion coefficients for data containing [Formula: see text] in the plot of the total amount against the square root of time were judged to be more reliable than those for data not containing [Formula: see text].

Application of an equation derived from the solution of diffusion equation with constant surface concentration to the film-roll method

Park

2022

Self Cite

Determination of constant diffusion coefficient for error function solution of diffusion equation using film-roll method

Park

2023

Self Cite

The error function solution of the diffusion equation with the constant surface concentration was derived by the heat kernel to determine the constant diffusion coefficient for diffusion satisfying the infinite dye-bath condition. For the sublimation diffusion of disperse dye in paste into PET film using the film-roll method, the constant surface concentration was determined from the concentration distribution, and the diffusion coefficients of each layer were obtained by the constant surface concentration from the error function solution. When comparing the diffusion coefficients between the layers and comparing the mean diffusion coefficients for different times at a specific temperature, the constant surface concentrations determined from the quadratic regression curves for concentration–distance plots were more appropriate than those determined from the steady-state concentration distributions, which was also confirmed by the plots of concentrations obtained from the error function solution. At a specific temperature, the average of the mean diffusion coefficients obtained by the constant surface concentration of the quadratic regression curve at three specific times matched well with the constant total-amount diffusion coefficient obtained by the slope of the linear regression line for the plot of total amount against square root of time, which was confirmed by their Arrhenius plots.

Determining the constant diffusion coefficient by Fourier series solution of the diffusion equation

Park

2024

The purpose of this study was to obtain the diffusion coefficient by calculating directly from the Fourier series solution of the diffusion equation, which has not been studied so far. The Fourier series solution is the sum of a steady-state solution and a transient solution. The steady-state solution is a linear function of distance. The transient solution is the product of a distance function and a time function. The Fourier sine series solution, a function of distance only, is negative for the steady-state solution. For the sublimation diffusion of disperse dye in paste into polyethylene terephthalate (PET) film using the film-roll method, the constant surface concentration and the diffusion distance were determined from the quadratic regression curve for the concentration distribution at a long time. The concentration distribution of the transient solution is a downwardly convex curve with negative values. The concentration distribution of the Fourier series solution is also a downwardly convex curve. The diffusion coefficient in the exponential term, a function of time only, was determined from the value of the exponential term obtained by the surface concentration and the diffusion distance. The constant diffusion coefficients determined from the Fourier series solution are reliable because they are very similar to those obtained from the error function solution and have excellent linearity of the linear regression lines for their Arrhenius plots.