Diffusion coefficient calculated by time-lag using the film-roll method

Abstract

A method for determining the diffusion coefficient by time-lag using the film-roll method for the sublimation diffusion of disperse dye was proposed. A polyethylene terephthalate film-roll coated with dye paste was treated at 170–190°C for various times. A solution consisting of the sum of a steady-state solution and a transient solution was obtained by the homogeneous boundary value problem from a trigonometrical series. The boundary conditions of the steady-state first layer and the steady-state first layer amount of dye were determined from the steady-state concentration distribution. For various diffusion times, the steady-state first layer-passed total amounts of dye that passed through the first layer in the steady-state condition were obtained by subtracting the steady-state first layer amounts from the total amounts. The time-lag was calculated from the linear regression line for the plot of the steady-state first layer-passed total number(X) of positive values against time. The diffusion coefficient was calculated by the boundary conditions of the steady-state first layer and the time-lag. For diffusion at 170°C, 180°C, and 190°C, the correlations of the steady-state first layer-passed total amounts with respect to time were very linear and the reliability of the diffusion coefficients obtained by the time-lag was proved by the good linearity of the Arrhenius plot. The activation energy obtained was 36.8 kcal[Formula: see text]mol−1.

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

“…For various diffusion times, the steady-state first layer-passed total amounts ( normalQnormalT) that passed through the first layer under the steady-state condition which is deduced from equation (42) is given by: where normalQnormalT is normalMnormalF−normalQ1normalT, normalQ1normalT is the steady-state first layer amount, and ε(0.004 normalcm) is the thickness of the first layer. 15…”

“…where Q T is M F À Q T 1 , Q T 1 is the steady-state first layer amount, and eð0:004 cmÞ is the thickness of the first layer. 15 In the other study, the mean diffusion coefficient determined by averaging the inter-layer diffusion coefficients D E were calculated by the variable x= ffiffiffiffiffiffiffiffiffiffi ffi 4D E t p of the complementary error function erfc x= ffiffiffiffiffiffiffiffiffiffi ffi 4D E t p from the ratio of C x; t ð Þ to the surface concentration C E 1 at a specified time as follows:…”

“…The film roll coated by the paste was dried at 100 C for 10 min in a dryer. [13][14][15] Figure 1 shows the schematic image of the PET film roll.…”

“…The correlations of the steady-state first layer-passed total amounts with respect to time were very linear, and the reliability of the diffusion coefficients obtained by the time lag was proved by the good linearity of the Arrhenius plot. 15 In this study on the same experiment as in the previous three studies, assuming that the second layer surface is the starting point for diffusion and the second layer surface concentrations are the surface concentrations that vary over time, the functional relationship between the second layer surface concentrations and the treatment times was examined. As the application of the Laplace transform to the diffusion equation removes the time variables to leave an ordinary differential equation, of which the solution yields the Laplace transformation of the concentration as a function of the distance, the solution of the diffusion equation for the boundary condition of the time-varying surface concentration was deduced by applying the Laplace transform.…”

“…The correlations of the steady-state first layer-passed total amounts with respect to time were very linear, and the reliability of the diffusion coefficients obtained by the time lag was proved by the good linearity of the Arrhenius plot. 15…”