Equilibrium Swelling Theory aims to estimate a hydrogel’s swelling from its structural properties in terms of the swollen polymer volume fraction. The current model, based on a thermodynamic balance of polymer-solvent mixing energy and elastic strain on polymer chains, is as follows:
\begin{align*} \frac{\ln(1 – \phi_{s}) + \phi_{s} + \chi \phi^{2}_{s}}{\phi^{\frac{1}{3}}_{s}} = -1^{*}\left(1 – \frac{2}{f}\right)\left(1 – \gamma\right) \frac{V_{1}\rho_{d}}{M_{r}N_{j}} \phi^{\frac{2}{3}}_{0} \end{align*}Swollen Polymer Volume Fraction Calculator
Note: This calculator uses a nonlinear optimization algorithm that sometimes fails to calculate the swollen polymer volume fraction. Values should be bound by 0 < φs < 1. If the calculator does not yield a reasonable value, the Solver plug-in for Microsoft Excel may perform better.
Parameter Definitions and Ranges
Swollen Polymer Volume Fraction (φs)
The swollen polymer volume fraction (φs) represents the volumetric concentration of polymer after the hydrogel has reached equilibrium with an excess of water. If the hydrogel’s universal structural parameters and polymer-specific parameters are known, the swollen polymer volume fraction can be calculated using equilibrium swelling theory or it can be measured directly on a swollen hydrogel. Values for swollen polymer volume fractions can be lower or higher than the initial polymer volume fraction [Richbourg et al., 2021]. It is unclear if swollen polymer volume fractions can be below the critical overlap concentration. The broadest limits to swollen polymer volume fractions are therefore 0 < φs < 1. Here again, most of the models described here for hydrogels are designed for φ0 < 0.5, and high polymer volume fractions may disrupt modeling assumptions, such as the use of the phantom-like deformation model.
Polymer-Solvent Interaction Parameter (χ)
The polymer-solvent interaction parameter (χ) acts as a catch-all term describing interactions between the polymer and the solvent (water) of a hydrogel network. Like many of the assumptions used in the Rubberlike Elasticity Theory, the polymer-solvent interaction parameter is based on a Mean-Field Theory approach (Wikipedia link). It is generally assumed, but not yet proven, that the polymer-solvent interaction parameter of a polymer network is the same as that of the free polymer in solution. Also, the polymer-solvent interaction parameter is often treated as unaffected by the polymer concentration, but it has been shown to change with the polymer concentration, and the relationship has been fitted using Taylor polynomials [McKenna and Horkay, Polymer 1994]. Here, for simplicity, we assume that the polymer-solvent interaction parameter is polymer concentration-independent and use polymer-solvent interaction parameters commonly reported for the associated free polymer in aqueous solution. The range for polymer-solvent interaction parameters of polymers in good solvents is generally expected to be 0 < χ <0.5. For PVA, χ = 0.494, and for PEG, χ = 0.426.
Junction Functionality (f)
Junction Functionality (f) is a universal structural parameter of swollen polymer networks. It describes the number of chains that converge at each junction point, assumed to be consistent across all junctions in the network. The lower limit for a network is a junction functionality of three, but non-point-like junctions have achieved junction functionalities exceeding 100 [Beamish et al., 2010]. Junction functionalities of four, six, and eight produce networks with regular polyhedron structures that enable precise calculations of mesh radii from mesh sizes [Richbourg, Ravikumar, Peppas, 2021].
Frequency of Chain-End Defects (γ)
The frequency of chain-end defects (γ) is a universal structural parameter of swollen polymer networks. It describes the fraction of network chains that do not connect to two junctions. The frequency of chain-end defects was first estimated by Flory based on the molecular weight of the pre-network polymer, but his definition does not apply to end-linked and mid-linked networks. The lower limit of the frequency of chain-end defects is zero, which means the network is perfectly interconnected. The maximum value is likely dependent on the system’s integrity and junction functionality, but significant reduction in gel viability is expected around a frequency of chain-end defects of 0.5, which would mean that only half of the network chains are connected to two junctions.
Molar Volume of Water (V1, mL/mol)
The molar volume of water (V1) is the volume of a mole of water, 18 mL/mol at room temperature.
Polymer Dry Density (ρd, g/mL)
The polymer dry density (ρd) is a polymer-specific parameter that can be evaluated independently from hydrogel properties. Typical values include 1.26 g/mL for PVA and 1.18 g/mL for PEG [Richbourg et al., 2021].
Molecular Weight of the Polymer Repeating Unit (Mr, g/mol)
The molecular weight of the polymer repeating unit (Mr) is a polymer-specific parameter than can be evaluated independently from hydrogel properties. The molecular weight of the polymer repeating unit can be calculated based on the chemical structure of the repeating unit. Typical values include 44 g/mol for both PVA and PEG [Richbourg et al., 2021].
Degree of Polymerization Between Junctions (Nj)
The degree of polymerization between junctions (Nj) is a universal structural parameter of swollen polymer networks. It describes the average number of repeating units in a chain between two network junctions. Low degrees of polymerization between junctions lead to chains that no longer behave like entropic springs, and high degrees can lead to highly entangled networks [Richbourg, Ravikumar, Peppas, 2021]. For PVA hydrogels, we observed deviations from linear swelling relationships at a lower bound of Nj = 20 [Richbourg et al., 2021]. Upper limits are system-dependent, but intact PEGDA hydrogels were formed with Nj = 589 [Richbourg et al., 2021].
Initial Polymer Volume Fraction (φ0)
The initial polymer volume fraction (φ0) is a universal structural parameter of swollen polymer networks. It describes the volumetric concentration of polymer in water when the hydrogel is formed. As we recently demonstrated, the initial polymer volume fraction is usually equivalent to the relaxed polymer volume fraction previously discussed by Peppas and colleagues [Richbourg et al., 2021]. The lower bound to initial polymer volume fraction is the critical overlap concentration, which varies by polymer, but for four-armed 10 kDa PEG it is around φ0 = 0.048 [Sakai et al., 2008]. The clear upper bound is φ0 = 1, which represents a dry polymer network, but most of the models described here for hydrogels are designed for φ0 < 0.5, and high polymer volume fractions may disrupt modeling assumptions, such as the use of the phantom-like deformation model.
Key Assumptions
1. Inherited Assumptions from Rubberlike Elasticity Theory
Equilibrium swelling theory inherits all the assumptions of Rubberlike Elasticity Theory since it is the basis of the elastic free energy component of the model.
2. Free Energy & Chemical Potential Balances
Equilibrium swelling theory is based on a balance of free energies. In the original form, it is the balance of the entropy gain resulting from polymer and water mixing against the loss of entropy from polymer chains extending and reducing their number of possible conformations. When the hydrogel has absorbed or expelled enough water that it is no longer entropically favorable to shrink or swell further, it has reached swelling equilibrium.
\begin{align*} \Delta G_{tot} = \Delta G_{el} + \Delta G_{mix} \end{align*}The free energy calculation is difficult to use because it is dependent on the total volume of the hydrogel. A partial differential by the number of water molecules converts free energy to chemical potential with respect to the number of water molecules. This approach proved useful, since it could be used to find a local minimum in the free energy based on the amount of water added.
\begin{align*} \frac{\partial \Delta G_{tot}}{\partial n_{1}} = \Delta \mu_{tot} = \Delta \mu_{el} + \Delta \mu_{mix} \end{align*}The potential problems with this approach are what make it complicated. As a hydrogel swells, its volume changes. Should the free energy balance be volume-normalized to a specific volume? Is the starting point still important by the time you differentiate to a chemical potential? What is the zero differential of the chemical potential is actually a local maximum or if there is an alternative local minimum from a different starting point? Most importantly, what if the entropies of mixing and chain elasticity are not the only two relevant forces in equilibrium swelling? Further clarification and definition of the fundamental derivation of the equilibrium swelling model may lead to new conclusions about how best to represent and calculate hydrogel swelling.
3. Consistent Temperature
The equilibrium swelling equation does not include a term for temperature since it is assumed that the swelling is performed at a consistent temperature. From the separated mixing and elastic chemical potentials, it is clear that both scale with temperature, but the ideal gas constant and temperature cancel out when they chemical potentials are balanced against each other at equilibrium. However, it is possible that 1) the relaxed state and swollen state were taken at different temperatures or 2) the mixing chemical potential and/or the elastic chemical potential are not linearly dependent on the temperature. In either case, the equilibrium-swollen polymer volume fraction may be different at different temperatures. This may be especially important for biomedical applications since the insides of human bodies are warmer than room temperature.
Additional Issues to Consider
Additional Chemical Potential Terms
While the default assumption is that only mixing and elastic chemical potentials are significant enough affect hydrogel swelling at equilibrium, it is possible that other chemical potentials could influence hydrogel swelling, as we have shown with the ionic chemical potential [Richbourg and Peppas, 2020]. Full discussion of the ionic chemical potential term here is excluded since it requires further conceptual evaluation and experimental validation. As an secondary example, polymers that respond to magnetic fields may require an additional chemical potential term. For further reading on stimuli-responsive hydrogels, we suggest recent work by Drozdov and & Christiansen in the Journal of Mechanical Behavior of Biomedical Materials, 2021, Huang et al., in International Journal of Applied Mechanics 2020, and the critical papers of Horkay et al., Biomacromolecules, 2000 & 2001.
Solute Fractions
High concentrations of salt or another solute, such as a drug or protein, could disrupt the assumption of binary polymer and solvent volume fractions. A significant third volume fraction would also require a more thorough consideration of interaction parameters, as there would be interactions between the polymer and solvent, the solvent and the solute, the solute and the solvent, and possibly emergent interactions based on all three components. Addressing the effects of large solute fractions may be especially important for drug delivery applications or for cell encapsulation systems.
Theory History
Equilibrium swelling theory begins with Flory and is well-documented in his text, Principles of Polymer Chemistry, 1953. Here, the free energy of polymer and solvent mixing (described first for a solution of solvent and linear polymers and later applied to polymer networks) is described in terms of the polymer-solvent interaction parameter (χ) [Wikipedia Link for Flory-Huggins Solution Theory].
\begin{align*} \Delta G_{mix} = k_{b} T \left[n_{1} \ln \phi_{1} + n_{2} \ln \phi_{2} + \chi n_{1}\phi_{2} \right] \end{align*}Conversion to chemical potential with respect to the number of solvent molecules (n1) yields an equation that is not dependent on absolute number of solvent or polymer molecules:
\begin{align*} \Delta \mu_{mix} = R T \left[\ln\left(1- \phi_{s}\right) + \phi_{s} + \chi\phi^{2}_{s} \right] \end{align*}As can be seen on the left side of the equilibrium swelling equation at the top of this page, the mixing term of the equilibrium swelling equation has remained unchanged since Flory’s work. Further iterations of equilibrium swelling theory have focused on the elastic chemical potential term.
Progress on the elastic chemical potential derives directly from developments in rubberlike elasticity theory. In brief, Bray, Peppas, and Merrill introduced the relaxed state for hydrogels formed in the presence of water [Bray and Merrill, 1973; Peppas and Merrill, 1977].
\begin{align*} \frac{1}{M_{c}} = \frac{2}{M_{n}} – \frac{\frac{1}{\rho_{d} V_{1}} \left[\ln\left(1-\phi_{s}\right) + \phi_{s} + \chi \phi^2_{s} \right]}{\phi_{r} \left[ \left(\frac{\phi_{s}}{\phi_{r}} \right)^\frac{1}{3} – \frac{1}{2} \left( \frac{\phi_{s}}{\phi_{r}} \right) \right]} \end{align*}Erman and Mark’s Structures and Properties of Rubberlike Networks, (1997) provided an argument for treating hydrogels formed in solution as phantom-like networks, thereby introducing the relevance of the network’s junction functionality.
\begin{align*} \Delta A_{el, ph} = \frac{1}{2}\left(1 – \frac{2}{f}\right)\nu_{e}k_{b}T\left(I_1 – 3\right) \end{align*}Richbourg and Peppas, in “The swollen polymer network hypothesis,” Progress in Polymer Science, (2020) aimed to coordinate previous iterations of equilibrium swelling equations into a comprehensive equation with a generalized term for the frequency of chain-end defects (that removes the assumption that polymer networks are formed by crosslinking linear polymers) and inclusion of the phantom-like deformation.
\begin{align*} \frac{1}{M_{c}} = \frac{ \ln\left(1-\phi_{s}\right) + \phi_{s} + \chi \phi^2_{s} – 2 V_{1} \left[ I^2 + \left( \frac{i \phi_{s} \rho_{d}}{2 M_{r}}\right)^2 \right]^\frac{1}{2} +2 V_{1} I}{ -1^{*}\left(1 – \frac{2}{f}\right)\left(1 – \gamma\right) V_{1}\rho_{d} \phi^{\frac{2}{3}}_{r} \phi^\frac{1}{3}_{s}} \end{align*}In a follow-up experimental validation of hydrogel swelling, Richbourg, et al. in “Precise control of synthetic hydrogel network structure via linear, independent synthesis-swelling relationships, “ Science Advances, (2021) demonstrated the equivalence of the initial polymer volume fraction and the relaxed polymer volume fraction. We therefore updated all reference to the relaxed polymer volume fraction to the initial polymer volume fraction to reduce the number of polymer volume fractions under consideration. We also converted molecular weight between crosslinks to degree of polymerization between crosslinks to facilitate comparison between hydrogels made with different polymers. The results of the swelling study led us to replace “crosslinks” with “junctions,” as discussed in Richbourg, Ravikumar, and Peppas, “Solute Transport Dependence on 3D Geometry of Hydrogel Networks,” Macromolecular Chemistry and Physics, (2021). Finally, we reorganized the equation to prioritize isolating the swollen polymer volume fraction instead of the degree of polymerization between junctions. The resulting equation is displayed at the top of this page.