gel increases. The degree of the increase in gel volume depends on the cross-linking
of the polymer chains and the polymer–solvent interaction.
The network junctions within a polymer gel swollen in a solvent are moved
away from each other by the solvent molecules. With the movement of the network
junctions, a stress is experienced by the polymer chains attached to the junctions.
This force is caused by the free energy of mixing trying to dilute the polymer
solution. In the meantime, the polymer chains also tend to return to its random
relaxed state with a higher entropy value. Thus, an elastic retractive force in
opposition to the elongation of polymer chains is generated. When the relaxed state
reaches an equilibrium with the strained one, the diffusion of the solvent molecules
from the free solution to the polymer network stops. At this equilibrium state, the
chemical potential of the solvent in the polymer network is equal to that in the free
solution.
For a given gel system comprising polymer chains and solvent molecules, the
swelling degree at the equilibrium state often decreases with the increase of the
number of cross-linking junctions. An increased degree of cross-linking leads to
reduced chain lengths between adjacent junctions and therefore the decreased
extension of the polymer chains.
In practice, the microscopic degree of cross-linking within a polymer gel network can be probed by the degree of swelling in a good solvent. In a typical
swelling experiment, the change in mass or volume for a swollen polymer gel is
measured at a specific temperature. The experimentally measured mass or volume
swelling ratio can be used for the qualitative examination of the level of
cross-linking. At a high level of analysis, a quantitative calculation of the cross-link
density can be conducted by using theoretical methods. Flory–Rehner theory is a
classical method used for the analysis of cross-linking within a polymer according
to the result of a swelling experiment [32]. According to the Flory–Rehner theory,
the average molecular weight between cross-links M c is related to the degree of
swelling of a polymer network at its equilibrium state. In terms of thermodynamics,
a polymer gel can be characterized as an elastic solution. Therefore, the thermodynamics of polymer solutions can be extended to calculate free energy changes in
gel systems. As the swelling of polymer gels is accompanied with visible deformation, the calculation of entropy values is more complicated in comparison to
solution systems.
When placed in a good solvent, the free energy change DG associated with the
swelling of a polymer gel can be written as:
DG ¼ DG m þ DG e
ð5:1Þ
In Eq. (5.1), DG m is the free energy resulting from the mixing of the polymer
network and solvent, and DG e is the free energy caused by the elastic expansion of
the polymer network. The physical processes associated with DG m DG e are presented in Fig. 5.3. When ions are included, the energy contribution from ions
should also be considered.
5.2 Fundamental Aspects
159
of the polymer chains and the polymer–solvent interaction.
The network junctions within a polymer gel swollen in a solvent are moved
away from each other by the solvent molecules. With the movement of the network
junctions, a stress is experienced by the polymer chains attached to the junctions.
This force is caused by the free energy of mixing trying to dilute the polymer
solution. In the meantime, the polymer chains also tend to return to its random
relaxed state with a higher entropy value. Thus, an elastic retractive force in
opposition to the elongation of polymer chains is generated. When the relaxed state
reaches an equilibrium with the strained one, the diffusion of the solvent molecules
from the free solution to the polymer network stops. At this equilibrium state, the
chemical potential of the solvent in the polymer network is equal to that in the free
solution.
For a given gel system comprising polymer chains and solvent molecules, the
swelling degree at the equilibrium state often decreases with the increase of the
number of cross-linking junctions. An increased degree of cross-linking leads to
reduced chain lengths between adjacent junctions and therefore the decreased
extension of the polymer chains.
In practice, the microscopic degree of cross-linking within a polymer gel network can be probed by the degree of swelling in a good solvent. In a typical
swelling experiment, the change in mass or volume for a swollen polymer gel is
measured at a specific temperature. The experimentally measured mass or volume
swelling ratio can be used for the qualitative examination of the level of
cross-linking. At a high level of analysis, a quantitative calculation of the cross-link
density can be conducted by using theoretical methods. Flory–Rehner theory is a
classical method used for the analysis of cross-linking within a polymer according
to the result of a swelling experiment [32]. According to the Flory–Rehner theory,
the average molecular weight between cross-links M c is related to the degree of
swelling of a polymer network at its equilibrium state. In terms of thermodynamics,
a polymer gel can be characterized as an elastic solution. Therefore, the thermodynamics of polymer solutions can be extended to calculate free energy changes in
gel systems. As the swelling of polymer gels is accompanied with visible deformation, the calculation of entropy values is more complicated in comparison to
solution systems.
When placed in a good solvent, the free energy change DG associated with the
swelling of a polymer gel can be written as:
DG ¼ DG m þ DG e
ð5:1Þ
In Eq. (5.1), DG m is the free energy resulting from the mixing of the polymer
network and solvent, and DG e is the free energy caused by the elastic expansion of
the polymer network. The physical processes associated with DG m DG e are presented in Fig. 5.3. When ions are included, the energy contribution from ions
should also be considered.
5.2 Fundamental Aspects
159