New Mathematical Model for Determining Time-Dependent Adsorption and Diffusion of Dyes into Fibers through Dye Sorption Curves in Combination Shades

Abstract

Dye uptake of textile substrates can be described as time-dependent by a new mathematical model, in which the sorption process is divided into fast and slow subprocesses. The fast subprocess describes the adsorption of the dye onto the fiber surface, and the slow one details the diffusion of the dye into the fiber. In addition, dye desorption is simultaneously considered along with adsorption. Relating this concept to the dyeing process, it is possible to divide the process into two parts—dye adsorption and diffusion. The model is verified by dyeing cotton with direct dyes, but the results are also transferable to other fibers and dye classes. Using this model, optimum dyeing parameters and dye combinations can be determined from the sorption curves, which are easily obtained by UV-VIS spectrophotometry.

A fluorescent extension to the Kubelka–Munk model

“…This requires that the pores of the fiber be large in comparison to the molecular dimensions of the dye, and that the pore network is accessible to the dye molecules. 13 In this case, the amount of dye that can be adsorbed onto the fiber is increased, and it may have a strong effect on the scattering properties of the fiber. An example of this is given in Fig.…”

New Mathematical Model for Determining Time-Dependent Adsorption and Diffusion of Dyes into Fibers Through Dye Sorption Curves in Combination Shades

“…At the end of the column, the absorbance (E) of the dye solution was detected by a uv-/vis diode array spectrophotometer, Hewlett Packard 8452, at 396, 520, and 590 nm. A schematic drawing of the experimental equipment is shown in Part I [2]. All dyeings occurred within a temperature range of 20-80°C under isothermal conditions.…”

“…According to the mathematical model described in Part I of this series [2], the total dye uptake m(t) of the cotton, dye uptakes mit) and met) at the outer and inner surfaces of the fibers (see Figure 2), and the velocity constants k, = k~ds and ki = kjdd can be computed by this model; ke is the quotient of the velocity constants of the sorption (~) and desorption (ds) processes at the outer surface, and ki is the quotient of the velocity constants of the sorption (kd) and desorption (dd) processes at the inner surface [2].…”

“…of the substrate, but cannot exactly distinguish between them. This is because the solutions of k~ and ki are the quotients of ksld.~ and kdldd, respectively [2].…”

“…In Part I of this series, we presented a new mathematical model [2] that describes dye uptake in the noncrystalline parts of the cellulose matrix as time dependent, divided into fast and slow subprocesses. The fast subprocess describes the adsorption of dye onto the easily accessible outer fiber surface, and the slow subprocess describes diffusion of the dye into the inner disordered regions of the fiber [2].…”