Many gels display
thixotropy – they become fluid when agitated, but resolidify when resting. In general, gels are apparently solid, jelly-like materials. It is a type of
non-Newtonian fluid. By replacing the liquid with gas it is possible to prepare
aerogels, materials with exceptional properties including very low density,
high specific surface areas, and excellent thermal insulation properties.
Thermodynamics of gel deformation A gel is in essence the mixture of a polymer network and a
solvent phase. Upon stretching, the network
crosslinks are moved further apart from each other. Due to the polymer strands between crosslinks acting as
entropic springs, gels demonstrate elasticity like
rubber (which is just a polymer network, without solvent). This is so because the
free energy penalty to stretch an
ideal polymer segment N monomers of size b between crosslinks to an
end-to-end distance R is approximately given by : F_\text{ela} \sim kT \frac{R^2}{Nb^2}. This is the origin of both gel and
rubber elasticity. But one key difference is that gel contains an additional solvent phase and hence is capable of having significant volume changes under
deformation by taking in and out solvent. For example, a gel could swell to several times its initial volume after being immersed in a solvent after equilibrium is reached. This is the phenomenon of gel swelling. On the contrary, if we take the swollen gel out and allow the solvent to evaporate, the gel would shrink to roughly its original size. This gel volume change can alternatively be introduced by applying external forces. If a uniaxial compressive
stress is applied to a gel, some solvent contained in the gel would be squeezed out and the gel shrinks in the applied-stress direction. To study the gel mechanical state in equilibrium, a good starting point is to consider a cubic gel of volume V_{0} that is stretched by factors \lambda_1, \lambda_2 and \lambda_3 in the three orthogonal directions during swelling after being immersed in a solvent phase of initial volume V_{s0}. The final deformed volume of gel is then \lambda_1\lambda_2\lambda_3V_{0} and the total volume of the system is V_{0}+V_{s0}, that is assumed constant during the swelling process for simplicity of treatment. The swollen state of the gel is now completely characterized by stretch factors \lambda_1, \lambda_2 and \lambda_3 and hence it is of interest to derive the
deformation free energy as a function of them, denoted as f_\text{gel}(\lambda_1,\lambda_2,\lambda_3). For analogy to the historical treatment of
rubber elasticity and mixing free energy, f_\text{gel}(\lambda_1,\lambda_2,\lambda_3) is most often defined as the free energy difference after and before the swelling normalized by the initial gel volume V_{0}, that is, a free energy difference density. The form of f_\text{gel}(\lambda_1,\lambda_2,\lambda_3) naturally assumes two contributions of radically different physical origins, one associated with the
elastic deformation of the polymer network, and the other with the
mixing of the network with the solvent. Hence, we write : f_\text{gel}(\lambda_1, \lambda_2, \lambda_3) = f_\text{net}(\lambda_1, \lambda_2, \lambda_3) + f_\text{mix}(\lambda_1, \lambda_2, \lambda_3). We now consider the two contributions separately. The polymer elastic deformation term is independent of the solvent phase and has the same expression as a rubber, as derived in the Kuhn's theory of
rubber elasticity: : f_\text{net}(\lambda_1,\lambda_2,\lambda_3) = \frac{G_0}{2} (\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3), where G_0 denotes the
shear modulus of the initial state. On the other hand, the mixing term f_\text{mix}(\lambda_1,\lambda_2,\lambda_3) is usually treated by the
Flory-Huggins free energy of
concentrated polymer solutions f(\phi), where \phi is polymer volume fraction. Suppose the initial gel has a polymer volume fraction of \phi_0, the polymer volume fraction after swelling would be \phi=\phi_0/\lambda_1\lambda_2\lambda_3 since the number of monomers remains the same while the gel volume has increased by a factor of \lambda_1\lambda_2\lambda_3. As the polymer volume fraction decreases from \phi_0 to \phi, a polymer solution of concentration \phi_0 and volume V_{0} is mixed with a pure solvent of volume (\lambda_1\lambda_2\lambda_3-1)V_{0} to become a solution with polymer concentration \phi and volume \lambda_1\lambda_2\lambda_3V_{0}. The free energy density change in this mixing step is given as : V_{g0} f_\text{mix}(\lambda_1 \lambda_2 \lambda_3) = \lambda_1 \lambda_2 \lambda_3 f(\phi) - [V_0 f(\phi_0) + (\lambda_1 \lambda_2\lambda_3 - 1) f(0)], where on the right-hand side, the first term is the
Flory–Huggins energy density of the final swollen gel, the second is associated with the initial gel and the third is of the pure solvent prior to mixing. Substitution of \phi = \phi_0/\lambda_1\lambda_2\lambda_3 leads to : f_\text{mix}(\lambda_1, \lambda_2, \lambda_3) = \frac{\phi_0}{\phi} [f(\phi) - f(0)] - [f(\phi_0) - f(0)]. Note that the second term is independent of the stretching factors \lambda_1, \lambda_2 and \lambda_3 and hence can be dropped in subsequent analysis. Now we make use of the
Flory-Huggins free energy for a polymer-solvent solution that reads : f(\phi) = \frac{kT}{v_c} [\frac{\phi}{N} \ln\phi + (1 - \phi) \ln(1 - \phi) + \chi \phi (1 - \phi)], where v_c is monomer volume, N is polymer strand length and \chi is the
Flory-Huggins energy parameter. Because in a network, the polymer length is effectively infinite, we can take the limit N\to\infty and f(\phi) reduces to : f(\phi) = \frac{kT}{v_c} [(1 - \phi) \ln(1 - \phi) + \chi \phi(1 - \phi)]. Substitution of this expression into f_\text{mix}(\lambda_1,\lambda_2,\lambda_3) and addition of the network contribution leads to The coupling between the ion partitioning and polyelectrolyte ionization degree is only partially by the classical
Donnan theory. As a starting point we can neglect the electrostatic interactions among ions. Then at equilibrium, some of the weak acid sites in the gel would dissociate to form \text{A}^-that electrostatically attracts positive charged \text{H}^+ and salt cations leading to a relatively high concentration of \text{H}^+ and salt cations inside the gel. But because the concentration of \text{H}^+ is locally higher, it suppresses the further ionization of the acid sites. This phenomenon is the prediction of the classical Donnan theory. However, with electrostatic interactions, there are further complications to the picture. Consider the case of two adjacent, initially uncharged acid sites \text{HA} are both dissociated to form \text{A}^-. Since the two sites are both negatively charged, there will be a charge-charge repulsion along the backbone of the polymer than tends to stretch the chain. This energy cost is high both elastically and electrostatically and hence suppress ionization. Even though this ionization suppression is qualitatively similar to that of Donnan prediction, it is absent without electrostatic consideration and present irrespective of ion partitioning. The combination of both effects as well as gel elasticity determines the volume of the gel at equilibrium. == Animal-produced gels ==