Diffusion Mechanism
Diffusion behavior follows mathematical models where the solute movement is proportional to the concentration gradient. Classical Fickian transport describes the migration of dye molecules from the dye bath into the solid polymer phase under steady-state conditions. This behavior assumes that polymer relaxation occurs much faster than penetrant diffusion.
Concentration Gradient
Molecular flux moves from high density to lower density until equilibrium is established. Standard Fickian transport models this behavior using a constant diffusion coefficient to predict dye penetration. This simplifies the bath calculation.
Fiber Geometry
Cylindrical boundary conditions require specific solutions to the diffusion equation for accurate textile modeling. When Fickian transport applies to synthetic filaments, the rate of dye uptake is inversely proportional to the square of the fiber radius. Fine denier yarns therefore dye much faster than coarse fibers under identical bath conditions.
Transport Boundary
Polymer structural transitions or temperature shifts can cause deviations from this classical linear behavior. Non-Fickian transport occurs when the dye temperature sits close to the glass transition point of the synthetic substrate. Under these specific conditions, the diffusion rate is controlled by the rate of polymer chain relaxation instead of the concentration gradient, altering the expected exhaustion curve.
