This document discusses dyeing kinetics and diffusion models for dye uptake in fibers. It describes the exhaustion curve which defines the rate of dyeing over time. The initial rate is called the strike. Diffusion occurs through three transport zones: flow, absorption, and diffusion into the fiber. Two main diffusion models are described: the pore model which considers diffusion through pores filled with dye liquor, and the free volume model which involves diffusion through amorphous, mobile regions of the polymer chain. The document contrasts these models and discusses how temperature affects dyeing rates.

1. Dyeing Kinetics, (Diffusion, Pore
Model, Free volume model)
1. Exhaustion
2. Slope of a dyeing exhaustion
3. Effects
4. Diffusion
5. Diffusion mechanisms
6. Transport zones
7. Dyeing rate
8. Diffusion models
Md. Abdul Hannan
Associate Prof. DUET
Hannan_tex62@yahoo.com
01913354913

2. Exhaustion
• The degree of dyebath exhaustion as a function of time describes the rate
and extent of the dyeing process.
• For a single dye, the exhaustion is defined as the ratio of
• i)mass of dye taken up by the material and
• Ii)the total initial mass of dye in the bath,
• but for a bath of constant volume:
• where C0 =concentrations of dye in the dyebath initially
• and Cs =concentrations of dye during the process, respectively.
• For many dyeings, a gradual increase of the dyeing temperature controls
the rate of exhaustion

3. Exhaustion
The slope of a dyeing exhaustion
curve (Figure) defines the rate of
dyeing at any instant during the
process
An equilibrium is reached where no
more dye is taken up by the fibres.
There is now a balance between the
rates of dye absorption and
desorption.
Typical exhaustion curves show the
initial rate of dyeing, the gradual
decrease in dyeing rate as the bath
becomes exhausted and the
equilibrium position
Figure : Dye bath exhaustion as a
function of time at a constant
dyeing temperature
Exhaustion curves at different
temperatures show how the dyeing
rate increases with increasing
temperature

4. Effects
• Exhaustion curves characterise the dyeing
properties of a dye and are useful for selecting
compatible dyes.
• They should have very similar rates of
exhaustion when used in mixtures under the
given dyeing conditions
• The adsorption equilibrium is usually rapid,
and the overall rate of dyeing depends on the
rate of diffusion of the dye into the fibres.

5. Diffusion
Molecular movement of a substrate due to its natural intrinsic mobility, i.e.
without the participation of external forces. A substance in solution will
migrate at constant temperature from zones of higher to lower
concentration until a concentration equilibrium is established within the
entire system. Diffusion is thus a general characteristic of the dilution
tendency of a solution which is accelerated by elevation of temperature.
Diffusion coefficient: A
proportionality factor
which gives the number
of moles of a dissolved
substance diffusing
through a cross sectional
area of I cm2 per unit
time for a given
temperature and solvent
when the concentration
gradient is (-l) measurement of concentration changes

6. Diffusion mechanisms in dyeing
• Dyeing processes are very dynamic as they
involve the transport of diverse products from
the dyeing medium into the fibres.
• Seldom reach a genuine state of thermodynamic
equilibrium.
• Only gives a transient picture of the distribution
of dye between the medium and the fibre after
a certain dyeing time.
• Three transport zones which depend on three
different transport carriers and transport effects
have been defined.

7. Three transport zones
1. The zone of transport due to
flow, determined by the
machine or flow pump.
2. The zone of transport due to
absorption, which is mainly
dependent on the chemistry of
the dye and the ancillary agents
on the liquid side.
3. The zone of transport due to
diffusion, which is mainly
dependent on the chemistry of
the dye and the substrate.

8. Dyeing Rate
• If the dye is delivered at a sufficiently high rate through the laminar
boundary layer on the surface of the fibre, then it will get into the
fibre by a process of absorption and slow diffusion.
• Accesses of a dye into the fibre via a transition interface is the key to
understanding the dye interactions in the area of dye kinetics.
• The rate of diffusion of the dye into the fibre interior is controlled by
its concentration at the fibre surface.
• The dye kinetics and resulting concentration profile are determined by
-the rates of diffusion,
-adsorption and
-immobilisation.
• In the case of reactive dyes, for example, the process of immobilisation
(fixation) is a chemical reaction.

9. • The initial rate of dyeing (the initial slope of exhaustion versus time) is
called the strike. Rapid strike by a dye often results in initial unlevelness .
• The strike depends on :
• dyeing temperature, pH and the addition of chemicals.
• Even for dyes of moderate and low strike, the objective of uniform dyeing
of the fibre mass is rarely achieved during the initial stages of the
operation.
• This is because of
a) irregularities
- inthe material’s construction,
- in the fibre packing,
- in the distribution of residual impurities,
b) differences in temperature
c) flow rate of the solution in contact with the fibres.

10. Materials composed of finer fibres
have a much larger specific
surface (m2 /kg) and a higher
dyeing rate, even though the
equilibrium exhaustion may not
be significantly different from that
of a material made of courser
fibres of the same polymer.
If the dyeing rate is proportional
to the fibre surface area per unit
mass, it will be inversely
proportional to the fibre radius, r,
and thus inversely proportional to
the square root of the fibre tex.

11. • The rate of dyeing is of greater practical
importance than dyeing equilibrium because few
commercial dyeing processes reach equilibrium.
• Dyeing rates are quite sensitive to :
• -The degree of agitation of the bath,
• - The liquor ratio,
• - The type and construction of the material,
• -The dyebath temperature and pH,
• -The concentration of dyeing assistants,
• -The dye substantivity.

12. • Practical dyeing rates depend upon the way dye
bath exhaustion changes as a function of time at
a constant temperature for most practical
purposes, the temperature may vary useful for
selecting compatible dyes that exhaust at about
the same rate under the same conditions.
• Typical exhaustion curves show the initial rate of
• dyeing, the gradual decrease in dyeing rate as the
bath becomes exhausted and the equilibrium
position (Figure 10.2).

13. Relation between surface area &pick-up
This equation gives a linear relationship
between the mass-specific surface
area of the fibre and the dye pick-
up.
The dyeing speed, therefore, increases
with increasing fibre fineness.
The even distribution of the dye over
the fibre cross section among other
things may be regarded as a
measurement standard for successful
dyeing.
After adsorption onto the fibre surface,
the dye must migrate into the
interior through the fibre mass or
through existing channels and
distribute itself there in the optimum
manner.

14. • At the same time, the diffusion medium has a
vital influence on the rate of diffusion. This
becomes evident in the variable order of
magnitude of the true diffusion coefficient.
• The diffusion coefficient is significantly greater
in the gas phase than in liquid or solid phases.
• The main driving force behind diffusion is the
difference in concentration.

17. Diffusion Models
• Theories of dye
kinetics arc concerned
with the nature of
dye diffusion in solid
polymers.
• Essentially they are
based on two
important
fundamentally
different models of
dye diffusion in fibres
namely:
1) The pore diffusion
model and
2) The free-volume
or mobile segment
model

18. The Pore Model
• The pore model represents the fibre as a solid structure with a network of
interconnected channels or pores which are filled with the dyeing liquid,
which is normally water.
• The dissolved dye diffuses through these pores, where it can be
simultaneously adsorbed on the walls of the pores. For quantitative
expressions of the rate of diffusion, the porosity P is of primary
importance as well as the adsorption equilibrium.
• P is the proportion of pores in relation to the total volume of the fibre
available under the conditions of dyeing.
• In I966, Weisz developed a mathematical model for diffusion processes
which is overlaid by an adsorption/desorption equilibrium of the dye
liquor at the external and internal boundary layers of the fibre. The pore
model presupposes, of course, that the pores are connected to each other
as well as to the external dye bath and that their diameter is sufficiently
large for the dye molecules to find rooms in them.

19. • Model concepts for dye uptake on cellulosic fibres are generally
based on the pore model. According to this, a network of pores
swollen and filled with water is present in the fibres within
which dye diffusion and sorption takes place followed, if
applicable, by chemical reactions as in the case of reactive dyes.
• The first mechanistic hypotheses on the uptake of anionic direct
dyes soon after their discovery were based on the assumption
that these dyes form colloidal particles in the fibre voids.
• It was concluded that substantivity is largely based on the fact
that the dye molecules which have diffused into the fibre form
aggregates in the cellulose pores.
• Likewise assisted by experimental findings, the mechanism of
monomolecular adsorption in the pores has found support for
low to medium depth dyeings at least.

20. The Free Volume Model
• The free volume model describes the dyeing process as diffusion
of the dye through the less ordered ("amorphous") regions of the
polymer matrix .
• The rate of diffusion is therefore determined by the mobility of
the polymer chain segments.
• The most important support : the temperature dependence of
the rates of dyeing for a particular type of fibre is less above a
certain temperature.
• The resistance of the solid structure of the fibre to the
penetration of dye is much lower above this temperature. This
is referred to as the glass transition temperature of the fibre in
question (Tg), or more precisely, the glass transition temperature
under dyeing conditions or the dyeing transition point (Td), since
the classical glass transition temp. is a parameter which is
measured in the dry state.

21. • Both parameters, Tg and Td, correspond to those temperatures at
which, from a microscopic perspective.
• the less ordered component of the polymer is converted from a
glass-like state into the viscoelastic state, or. at a molecular level,
at which the less ordered segments of the macromolecule move
against each other, i.e.. become, in effect, like "micro fluids".
• The glass transition temperature plays an important role here. It
corresponds. At a molecular level, to the temperature at which
the amorphous regions of a polymer are converted from a glass-
like state to a gummy (i.e., viscoelastic) state.
• Above this temperature, parts of the polymer chain (thread)
molecule become mobile.

22. • This segment change in the spatial arrangement of the chain
moleculesmobility causes an uninterrupted in these regions.
• "Holes“ are formed above Tg and disappear again or occur at
neighbouling sites of the polymer segment involved.
• In the viscoelastic state. therefore. The polymer structure cannot
be conceived in static terms: the structure changes constantly.
• The possibility for the diffusion of relatively small molecules
through such a structure is a problem of probability (or expressed
in physical terms, it is a question of entropy) as to whether
"holes", "channels“ and adsorption sites for small molecules are
formed by segment mobility.