Gel

Polyionic polymers are polymers with an ionic functional group. The ionic charges prevent the formation of tightly coiled polymer chains. This allows them to contribute more to viscosity in their stretched state, because the stretched-out polymer takes up more space. This is also the reason gel hardens. See polyelectrolyte for more information.

A colloidal gel consists of a percolated network of particles in a fluid medium,[5] providing mechanical properties[6] in particular the rise of elastic behaviour.[7] The particles can show attractive interactions through osmotic depletion or through polymeric links.[8]

Colloidal gels have three phases in their lifespan: gelation, aging and collapse.[9][10] The gel is initially formed by the assembly of particles into a space-spanning network, leading to a phase arrest. In the aging phase, the particles slowly rearrange to form thicker strands, increasing the elasticity of the material. Gels can also be collapsed and separated by external fields such as gravity.[11] Colloidal gels show linear response rheology at low amplitudes.[12] These materials have been explored as candidates for a drug release matrix.[13]

A hydrogel is a network of polymer chains that are hydrophilic, sometimes found as a colloidal gel in which water is the dispersion medium. A three-dimensional solid results from the hydrophilic polymer chains being held together by cross-links. Because of the inherent cross-links, the structural integrity of the hydrogel network does not dissolve from the high concentration of water.[14] Hydrogels are highly absorbent (they can contain over 90% water) natural or synthetic polymeric networks. Hydrogels also possess a degree of flexibility very similar to natural tissue, due to their significant water content. As responsive "smart materials," hydrogels can encapsulate chemical systems which upon stimulation by external factors such as a change of pH may cause specific compounds such as glucose to be liberated to the environment, in most cases by a gel-sol transition to the liquid state.[15] Chemomechanical polymers are mostly also hydrogels, which upon stimulation change their volume and can serve as actuators or sensors. The first appearance of the term 'hydrogel' in the literature was in 1894.[16]

An organogel is a non-crystalline, non-glassy thermoreversible (thermoplastic) solid material composed of a liquid organic phase entrapped in a three-dimensionally cross-linked network. The liquid can be, for example, an organic solvent, mineral oil, or vegetable oil. The solubility and particle dimensions of the structurant are important characteristics for the elastic properties and firmness of the organogel. Often, these systems are based on self-assembly of the structurant molecules.[17][18] (An example of formation of an undesired thermoreversible network is the occurrence of wax crystallization in petroleum.[19])

Organogels have potential for use in a number of applications, such as in pharmaceuticals,[20] cosmetics, art conservation,[21] and food.[22]

A xerogel /ˈzɪəroʊˌdʒɛl/ is a solid formed from a gel by drying with unhindered shrinkage. Xerogels usually retain high porosity (15–50%) and enormous surface area (150–900 m²/g), along with very small pore size (1–10 nm). When solvent removal occurs under supercritical conditions, the network does not shrink and a highly porous, low-density material known as an aerogel is produced. Heat treatment of a xerogel at elevated temperature produces viscous sintering (shrinkage of the xerogel due to a small amount of viscous flow) which results in a denser and more robust solid, the density and porosity achieved depend on the sintering conditions.

Nanocomposite hydrogels[23][24] or hybrid hydrogels, are highly hydrated polymeric networks, either physically or covalently crosslinked with each other and/or with nanoparticles or nanostructures.[25] Nanocomposite hydrogels can mimic native tissue properties, structure and microenvironment due to their hydrated and interconnected porous structure. A wide range of nanoparticles, such as carbon-based, polymeric, ceramic, and metallic nanomaterials can be incorporated within the hydrogel structure to obtain nanocomposites with tailored functionality. Nanocomposite hydrogels can be engineered to possess superior physical, chemical, electrical, thermal, and biological properties.[23][26]

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 act 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 ${\displaystyle N}$ monomers of size ${\displaystyle b}$ between crosslinks to an end-to-end distance ${\displaystyle R}$ is approximately given by[27]

${\displaystyle F_{\text{ela}}\sim kT{\frac {R^{2}}{Nb^{2}}}.}$

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 swelled 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 ${\displaystyle V_{0}}$ that is stretched by factors ${\displaystyle \lambda _{1}}$, ${\displaystyle \lambda _{2}}$ and ${\displaystyle \lambda _{3}}$ in the three orthogonal directions during swelling after being immersed in a solvent phase of initial volume ${\displaystyle V_{s0}}$. The final deformed volume of gel is then ${\displaystyle \lambda _{1}\lambda _{2}\lambda _{3}V_{0}}$ and the total volume of the system is ${\displaystyle V_{0}+V_{s0}}$ that is assumed constant during the swelling process for simplicity of treatment. The swelled state of the gel is now completely characterized by stretch factors ${\displaystyle \lambda _{1}}$, ${\displaystyle \lambda _{2}}$ and ${\displaystyle \lambda _{3}}$ and hence it is of interest to derive the deformation free energy as a function of them, denoted as ${\displaystyle f_{\text{gel}}(\lambda _{1},\lambda _{2},\lambda _{3})}$. For analogy to the historical treatment of rubber elasticity and mixing free energy, ${\displaystyle f_{\text{gel}}(\lambda _{1},\lambda _{2},\lambda _{3})}$ is most often defined as the free energy difference after and before the swelling normalized by the initial gel volume ${\displaystyle V_{0}}$, that is, a free energy difference density. The form of ${\displaystyle f_{\text{gel}}(\lambda _{1},\lambda _{2},\lambda _{3})}$ naturally assumes two contributions of radically different physical origins, one is associated with the elastic deformation of the polymer network, and the other with the mixing of the network with the solvent. Hence, we write[28]

${\displaystyle f_{\text{gel}}(\lambda _{1},\lambda _{2},\lambda _{3})=f_{\text{net}}(\lambda _{1},\lambda _{2},\lambda _{3})+f_{\text{mix}}(\lambda _{1},\lambda _{2},\lambda _{3}).}$

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:

${\displaystyle f_{\text{net}}(\lambda _{1},\lambda _{2},\lambda _{3})={\frac {G_{0}}{2}}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}-3),}$

where ${\displaystyle G_{0}}$ denotes the shear modulus of the initial state. On the other hand, the mixing term ${\displaystyle f_{\text{mix}}(\lambda _{1},\lambda _{2},\lambda _{3})}$ is usually treated by the Flory-Huggins free energy of concentrated polymer solutions ${\displaystyle f(\phi )}$, where ${\displaystyle \phi }$ is polymer volume fraction. Suppose the initial gel has a polymer volume fraction of ${\displaystyle \phi _{0}}$, the polymer volume fraction after swelling would be ${\displaystyle \phi =\phi _{0}/\lambda _{1}\lambda _{2}\lambda _{3}}$ since the number of monomers remains the same while the gel volume has increased by a factor of ${\displaystyle \lambda _{1}\lambda _{2}\lambda _{3}}$. As the polymer volume fraction decreases from ${\displaystyle \phi _{0}}$ to ${\displaystyle \phi }$, a polymer solution of concentration ${\displaystyle \phi _{0}}$ and volume ${\displaystyle V_{0}}$ is mixed with a pure solvent of volume ${\displaystyle (\lambda _{1}\lambda _{2}\lambda _{3}-1)V_{0}}$ to become a solution with polymer concentration ${\displaystyle \phi }$ and volume ${\displaystyle \lambda _{1}\lambda _{2}\lambda _{3}V_{0}}$. The free energy density change in this mixing step is given as

${\displaystyle V_{g0}f_{\text{mix}}(\lambda _{1}\lambda _{2}\lambda _{3})=\lambda _{1}\lambda _{2}\lambda _{3}f(\phi )-[V_{0}f(\phi _{0})+(\lambda _{1}\lambda _{2}\lambda _{3}-1)f(0)],}$

where on the right-hand side, the first term is the Flory–Huggins energy density of the final swelled gel, the second is associated with the initial gel and the third is of the pure solvent prior to mixing. Substitution of ${\displaystyle \phi =\phi _{0}/\lambda _{1}\lambda _{2}\lambda _{3}}$ leads to

${\displaystyle f_{\text{mix}}(\lambda _{1},\lambda _{2},\lambda _{3})={\frac {\phi _{0}}{\phi }}[f(\phi )-f(0)]-[f(\phi _{0})-f(0)].}$

Note that the second term is independent of the stretching factors ${\displaystyle \lambda _{1}}$, ${\displaystyle \lambda _{2}}$ and ${\displaystyle \lambda _{3}}$ 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[29]

${\displaystyle f(\phi )={\frac {kT}{v_{c}}}[{\frac {\phi }{N}}\ln \phi +(1-\phi )\ln(1-\phi )+\chi \phi (1-\phi )],}$

where ${\displaystyle v_{c}}$ is monomer volume, ${\displaystyle N}$ is polymer strand length and ${\displaystyle \chi }$ is the Flory-Huggins energy parameter. Because in a network, the polymer length is effectively infinite, we can take the limit ${\displaystyle N\to \infty }$ and ${\displaystyle f(\phi )}$ reduces to

${\displaystyle f(\phi )={\frac {kT}{v_{c}}}[(1-\phi )\ln(1-\phi )+\chi \phi (1-\phi )].}$

Substitution of this expression into ${\displaystyle f_{\text{mix}}(\lambda _{1},\lambda _{2},\lambda _{3})}$ and addition of the network contribution leads to[28]

${\displaystyle f_{\text{gel}}(\lambda _{1},\lambda _{2},\lambda _{3})={\frac {G_{0}}{2}}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})+{\frac {\phi _{0}}{\phi }}f(\phi ).}$

This provides the starting point to examining the swelling equilibrium of a gel network immersed in solvent. It can be shown that gel swelling is the competition between two forces, one is the osmotic pressure of the polymer solution that favors the take in of solvent and expansion, the other is the restoring force of the polymer network elasticity that favors shrinkage. At equilibrium, the two effects exactly cancel each other in principle and the associated ${\displaystyle \lambda _{1}}$, ${\displaystyle \lambda _{2}}$ and ${\displaystyle \lambda _{3}}$ define the equilibrium gel volume. In solving the force balance equation, graphical solutions are often preferred.

In an alternative, scaling approach, suppose an isotropic gel is stretch by a factor of ${\displaystyle \lambda }$ in all three directions. Under the affine network approximation, the mean-square end-to-end distance in the gel increases from initial ${\displaystyle R_{0}^{2}}$ to ${\displaystyle (\lambda R_{0})^{2}}$ and the elastic energy of one stand can be written as

${\displaystyle F_{\text{ela}}\sim kT{\frac {(\lambda R_{0})^{2}}{R_{\text{ref}}^{2}}},}$

where ${\displaystyle R_{\text{ref}}}$ is the mean-square fluctuation in end-to-end distance of one strand. The modulus of the gel is then this single-strand elastic energy multiplied by strand number density ${\displaystyle \nu =\phi /Nb^{3}}$ to give[27]

${\displaystyle G(\phi )\sim {\frac {kT}{b^{3}}}{\frac {\phi }{N}}{\frac {(\lambda R_{0})^{2}}{R_{\text{ref}}^{2}}}.}$

This modulus can then be equated to osmotic pressure (through differentiation of the free energy) to give the same equation as we found above.

Consider a hydrogel made of polyelectrolytes decorated with weak acid groups that can ionize according to the reaction

${\displaystyle {\text{HA}}\rightleftharpoons {\text{A}}^{-}+{\text{H}}^{+}}$

is immersed in a salt solution of physiological concentration. The degree of ionization of the polyelectrolytes is then controlled by the ${\displaystyle {\text{pH}}}$ and due to the charged nature of ${\displaystyle {\text{H}}^{+}}$ and ${\displaystyle {\text{A}}^{-}}$, electrostatic interactions with other ions in the systems. This is effectively a reacting system governed by acid-base equilibrium modulated by electrostatic effects, and is relevant in drug delivery, sea water desalination and dialysis technologies. Due to the elastic nature of the gel, the dispersion of ${\displaystyle {\text{A}}^{-}}$ in the system is constrained and hence, there will be a partitioning of salts ions and ${\displaystyle {\text{H}}^{+}}$ inside and outside the gel, which is intimately coupled to the polyelectrolyte degree of ionization. This ion partitioning inside and outside the gel is analogous to the partitioning of ions across a semipemerable membrane in classical Donnan theory, but a membrane is not needed here because the gel volume constraint imposed by network elasticity effectively acts its role, in preventing the macroions to pass through the fictitious membrane while allowing ions to pass.[30]

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 ${\displaystyle {\text{A}}^{-}}$that electrostatically attracts positive charged ${\displaystyle {\text{H}}^{+}}$ and salt cations leading to a relatively high concentration of ${\displaystyle {\text{H}}^{+}}$ and salt cations inside the gel. But because the concentration of ${\displaystyle {\text{H}}^{+}}$ is locally higher, it suppresses the further ionization of the acid sites. This phenomenon is the prediction of the classical Donnan theory.[31] However, with electrostatic interactions, there are further complications to the picture. Consider the case of two adjacent, initially uncharged acid sites ${\displaystyle {\text{HA}}}$ are both dissociated to form ${\displaystyle {\text{A}}^{-}}$. 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.[30] Due to the complexity of the coupled acid-base equilibrium, electrostatics and network elasticity, only recently has such system been correctly recreated in computer simulations.[30][32]