Diffusion

Diffusion means penetration/movement of substance owing the existence of conc. gradient i.e. movement of particles between the two surfaces having different density from higher to lower one.

This is very important because it affects the fastness properties and the color yield.

Diffusion depends on

1. 
   
   Dye size and nature of fibers and dyes
2. 
   
   Structure of fibers (crystallinity and orientation)
3. 
   
   Forces of interaction
4. 
   
   Environment pH, solvent, temperature etc

Diffusion coefficient

The behavior of dye movement from higher concentration to lower one is described in terms of Fick’s law, which states that the no. of particles which diffuse through a cross-section in the x direction (S in moles/cm²) in a time t (seconds), the so called flux F (g moles/unit area/unit time) is proportional to the gradient of conc. dc/dx (in moles/cm⁴)

ds/dt = F = -DA. dc/dx

Where ds/dt = rate of diffusion

A area of cross section (in cm²)

D= diffusion coefficient/ diffusivity (cm²/sec)

The diffusion coefficient, D indicates dye diffusing through unit time through unit cross section area of the fiber under unit concentration gradient. Fick’s law is applicable only in cases in which the concentration gradient dc/dx is independent of time. Thus D is a measure of the diffusion properties of dyes and permeability of fiber.

Methods of measuring diffusion coefficient

Determination of D is important because, firstly to correlate structure of dye and/or fiber

Secondly, to calculate rates of dyeing or rates of desorption which may asses the practical situation

For determination of D, two methods are available

1. 
   
   Non steady method: by varying the concentration gradient
2. 
   
   Steady method: by maintaining a steady concentration gradient in the substrate throughout the process

Non steady method:

During the dyeing process, the system is in a non steady state since the concentration gradient in the substrate decreases as the dye concentration at the center of the fiber increases and also the dye concentration in the dyebath decreases. Hence Fick’s second law needs to be applied; D d²c/dx²=dc/dt

Steady State method¹

In dyeing fiber and films, steady state conditions are only present at the very beginning. Thus measurement of this type are possible only a film substrate.

Theory

The film is interposed between two different concentrated solutions (one of them may be zero) and after allowing a sufficient time for a steady concentration gradient to be set up in the film, the rate at which the dye is transferred from concentrated to dilute is measured. Conditions being so arranged that concentration become that constant. D can be calculated, under this conditions, from equation ds/dt=-DA. dc/dx

[fig: Neale’s apparatus for the diffusing dye through a cellulose sheet]

Procedure

The film is clamped between the flanges of two tubes, dye solution is placed on one side and a blank solution is on the other. Two compartments are stirred to avoid any hydrodynamic complications and the rate at which dye is transferred from the dye solution is measured.

After a sufficient period of time had elapsed for the blank solution to contain a measurable amount of dye, it was replaced by blank solution. This procedure was repeated. As dye diffused through the film, the blank solution was removed at intervals and the quantity of dye estimated colorimetrically so that dye concentration in the compartment was maintained approximately zero throughout the experiment.

Now as dc/dx is constant, ds/dt can be determined directly, so D can be calculated. A is in cm^2, dc concentration difference between two sides of membrane, x thickness of membrane.

Diffusion Model

There are two models for diffusion

1. 
   
   Pore model
2. 
   
   Free Volume Model

Pore Model

The pore model was first proposed for the dyeing of cellulosic fibers in 1935

Details:

This model considers the fiber to be network of interconnecting pores. When filled with water, the latter allow the dye molecules to diffuse and be simultaneously absorbed on the wall of the pore.

The fiber is heterogeneous material and the dye must follow a tortuous/zigzag path in order to avoid the impenetrable crystalline regions. In passing through the amorphous region of the substrate, the large dye molecule must weave its way through a network of chains or along the surfaces of crystalline regions and may even encounter voids. In general the diameter of the pores may be expected to vary from one type of fiber to another. For example direct dye is easily sorped by cotton than viscose rayon due to larger pore size.

For this model, the channels or pores are considered to be zigzag and occupy a fraction of α volume of the substrate.

Defects:

1. 
   
   The pore model can’t be accepted in its entirety. The main defect is that structural parameters of the polymer, namely the size, shape and tortuiosity of the pores can’t be defined with any confidence.
2. 
   
   If the pores are defined, the polymer structure is treated secondary one, the pores merely supplying surfaces for dye adsorption.
3. 
   
   An exert correlation between the dye affinity and the diffusion coefficient can only be accepted if the dye molecule in the pore could be located out of range of the adsorption forces i.e. in the interior of the pore and the solution considered to possess the properties of normal water, which is practically impossible
4. 
   
   The model assumes that the dye molecules diffuse without hindrance down the pore. Such a situation is unlikely
5. 
   
   It is assumed that the channels (pores) are circular in cross sections, but observed dichroism in oriented materials and as dye molecules are linear, planar and rigid structure, a pore with an elliptical cross section is more realistic.

Fig. change in cross section of a fiber after orienting

Free Volume Model

Theory

Increasing the temperature (above T_d)² will result in an increase in the segmental of the polymer, thereby allowing more ‘holes’ to be made available for the diffusing, according to Williams-Landel-Ferry (WLF) equation:

log D_T/D_Td = A (T-T_d) / (B + T – T_d)

D_T, D_Td is measured diffusion coefficients at temperature at T & T_d

A, B semi empirical constant.

Details

The free volume model is that volume in a liquid and solid not occupied by the constituent atoms; in fact it arises from the thermal motion of the actions and hence increases with temperature.

Below T_d, the polymer chains may be regarded frozen into position and they only motions they can undergo a thermal vibrations. When T_d is reached/ sufficient energy is available for bond rotation in the backbone of the polymer chain. An adequate free volume has been created to provide a large energy to accommodate rotating polymer segment. Once this segment has moved, the space it has vacated allows another segment to move. The onset of the segmental motion occurs over a narrow temperature range which includes T_d.

Comparison between pore and free volume model

The controversies between the two models, as which one is more correct, the question is incorrectly formulated as any model is a simplification of reality and is therefore expected to fail in certain cases. Thus it is impossible to improve one is superior to another.

In all dyeing processes on the major fibers used today both models are probably effective simultaneously but in widely varying proportions. 

The diffusion coefficients on the more porous fibers were found to be below and above T_d. Therefore both models exists every dyeing process and thus for most widely used textile fiber, with direct and indirect evidence one can conclude that free volume model is dominant one for polyester dyeing.

1 The steady state methods simply monitor the passage of dye through the material without reference to the internal distribution where non steady states yield a more detailed study of various factors.

2 T_d is glass transition temperature measured under dyeing conditions i.e. in water T_g is measured with dry polymers.