Editing Gel (section)

=== Thermodynamics of gel deformation ===
A gel is in essence the mixture of a polymer network and a [[solvent]] phase. Upon stretching, the network [[Cross-link|crosslinks]] are moved further apart from each other. Due to the polymer strands between crosslinks acting as [[Entropic force|entropic springs]], gels demonstrate elasticity like [[Rubber elasticity#Polymer chain theories|rubber]] (which is just a polymer network, without solvent). This is so because the [[Thermodynamic free energy|free energy]] penalty to stretch an [[Ideal chain|ideal polymer]] segment <math>N</math> monomers of size <math>b</math> between crosslinks to an [[End-to-end vector|end-to-end distance]] <math>R</math> is approximately given by<ref name=":0">{{Cite book |last1=Rubinstein |first1=Michael |last2=Colby |first2=Ralph H. |url=https://www.worldcat.org/oclc/50339757 |title=Polymer physics |date=2003 |publisher=Oxford University Press |isbn=0-19-852059-X |location=Oxford |oclc=50339757}}</ref>

: <math>F_\text{ela} \sim kT \frac{R^2}{Nb^2}.</math>

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 (physics)|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 (mechanics)|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 <math>V_{0}</math> that is stretched by factors <math>\lambda_1</math>, <math>\lambda_2</math> and <math>\lambda_3</math> in the three orthogonal directions during swelling after being immersed in a solvent phase of initial volume <math>V_{s0}</math>. The final deformed volume of gel is then <math>\lambda_1\lambda_2\lambda_3V_{0}</math> and the total volume of the system is <math>V_{0}+V_{s0}</math>, 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 <math>\lambda_1</math>, <math>\lambda_2</math> and <math>\lambda_3</math> and hence it is of interest to derive the [[Deformation (physics)|deformation free energy]] as a function of them, denoted as <math>f_\text{gel}(\lambda_1,\lambda_2,\lambda_3)</math>. For analogy to the historical treatment of [[rubber elasticity]] and mixing free energy, <math>f_\text{gel}(\lambda_1,\lambda_2,\lambda_3)</math> is most often defined as the free energy difference after and before the swelling normalized by the initial gel volume <math>V_{0}</math>, that is, a free energy difference density. The form of <math>f_\text{gel}(\lambda_1,\lambda_2,\lambda_3)</math> naturally assumes two contributions of radically different physical origins, one associated with the [[Elastic Deformation|elastic deformation]] of the polymer network, and the other with the [[Entropy of mixing|mixing]] of the network with the solvent. Hence, we write<ref name=":1">{{Cite book |last=Doi |first=M. |url=https://www.worldcat.org/oclc/851159840 |title=Soft matter physics. |date=2013 |publisher=Oxford University Press USA |isbn=978-0-19-150350-4 |location=Oxford |oclc=851159840}}</ref>

: <math>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).</math>

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]]:

: <math>f_\text{net}(\lambda_1,\lambda_2,\lambda_3) = \frac{G_0}{2} (\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3),</math>

where <math>G_0</math> denotes the [[shear modulus]] of the initial state. On the other hand, the mixing term <math>f_\text{mix}(\lambda_1,\lambda_2,\lambda_3)</math> is usually treated by the [[Flory–Huggins solution theory|Flory-Huggins free energy]] of [[Polymer solution|concentrated polymer solutions]] <math>f(\phi)</math>, where <math>\phi</math> is polymer volume fraction. Suppose the initial gel has a polymer volume fraction of <math>\phi_0</math>, the polymer volume fraction after swelling would be <math>\phi=\phi_0/\lambda_1\lambda_2\lambda_3</math> since the number of monomers remains the same while the gel volume has increased by a factor of <math>\lambda_1\lambda_2\lambda_3</math>. As the polymer volume fraction decreases from <math>\phi_0</math> to <math>\phi</math>,  a polymer solution of concentration <math>\phi_0</math> and volume <math>V_{0}</math> is mixed with a pure solvent of volume <math>(\lambda_1\lambda_2\lambda_3-1)V_{0}</math> to become a solution with polymer concentration <math>\phi</math> and volume <math>\lambda_1\lambda_2\lambda_3V_{0}</math>. The free energy density change in this mixing step is given as

: <math>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)],</math>

where on the right-hand side, the first term is the [[Flory–Huggins solution theory|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 <math>\phi = \phi_0/\lambda_1\lambda_2\lambda_3</math> leads to

: <math>f_\text{mix}(\lambda_1, \lambda_2, \lambda_3) = \frac{\phi_0}{\phi} [f(\phi) - f(0)] - [f(\phi_0) - f(0)].</math>

Note that the second term is independent of the stretching factors <math>\lambda_1</math>, <math>\lambda_2</math> and <math>\lambda_3</math> and hence can be dropped in subsequent analysis. Now we make use of the [[Flory–Huggins solution theory|Flory-Huggins]] free energy for a polymer-solvent solution that reads<ref>{{Cite book |last=Doi |first=M. |url=https://www.worldcat.org/oclc/59185784 |title=The theory of polymer dynamics |date=1986 |others=S. F. Edwards |isbn=0-19-851976-1 |publisher=Clarendon Press |location=Oxford |oclc=59185784}}</ref>

: <math>f(\phi) = \frac{kT}{v_c} [\frac{\phi}{N} \ln\phi + (1 - \phi) \ln(1 - \phi) + \chi \phi (1 - \phi)],</math>

where <math>v_c</math> is monomer volume, <math>N</math> is polymer strand length and <math>\chi</math> is the [[Flory–Huggins solution theory|Flory-Huggins]] energy parameter. Because in a network, the polymer length is effectively infinite, we can take the limit <math>N\to\infty</math> and <math>f(\phi)</math> reduces to

: <math>f(\phi) = \frac{kT}{v_c} [(1 - \phi) \ln(1 - \phi) + \chi \phi(1 - \phi)].</math>

Substitution of this expression into <math>f_\text{mix}(\lambda_1,\lambda_2,\lambda_3)</math> and addition of the network contribution leads to<ref name=":1" />

: <math>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).</math>

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 (physics)|elasticity]] that favors shrinkage. At equilibrium, the two effects exactly cancel each other in principle and the associated <math>\lambda_1</math>, <math>\lambda_2</math> and <math>\lambda_3</math> define the equilibrium gel volume. In solving the force balance equation, graphical solutions are often preferred.

In an alternative, scaling approach, suppose an [[Isotropy|isotropic]] gel is stretch by a factor of <math>\lambda</math> in all three directions. Under the [[Affine transformation|affine network]] approximation, the mean-square [[End-to-end vector|end-to-end distance]] in the gel increases from initial <math>R_0^2</math> to <math>(\lambda R_0)^2</math> and the elastic energy of one stand can be written as

: <math>F_\text{ela} \sim kT \frac{(\lambda R_0)^2}{R_\text{ref}^2},</math>

where <math>R_\text{ref}</math> 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 <math>\nu=\phi/Nb^3</math> to give<ref name=":0" />

: <math>G(\phi) \sim \frac{kT}{b^3} \frac{\phi}{N} \frac{(\lambda R_0)^2}{R_\text{ref}^2}.</math>

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