The Swollen Polymer Network Model

The initial polymer volume fraction (φ₀) describes the volumetric concentration of polymer in water when the hydrogel is formed. As we recently demonstrated, the initial polymer volume fraction is 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 maximal 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.

The degree of polymerization between junctions (N_j) 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 N_j = 20 [Richbourg et al., 2021]. Upper limits are system-dependent, but intact PEGDA hydrogels were formed with N_j = 589 [Richbourg et al., 2021].

Junction Functionality (f) 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].

Crystal lattice rendering of a tetrafunctional (f = 4) polymer network portal. Created by Akhila Ravikumar and animated by Ezekiel Eliamani.

Crystal lattice rendering of a hexafunctional (f = 6) polymer network portal. Created by Akhila Ravikumar and animated by Ezekiel Eliamani.

Crystal lattice rendering of a octafunctional (f = 8) polymer network portal. Created by Akhila Ravikumar and animated by Ezekiel Eliamani.

The frequency of chain-end defects (γ) 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.

Crosslinking Density (ρ_x, mol/mL) describes the concentration of junctions in the network. Like polymer volume fractions, it is state-dependent, but the state is often unspecified. Most often, it seems that references to crosslinking density are based on the equilibrium-swollen state. Furthermore, the term “crosslinking” can be misapplied and misunderstood. For an end-linked network such as radically polymerized PEGDA, is the crosslinking density calculated from the concentration of acrylate groups? For multi-arm PEG hydrogels, is it based on the concentration of multi-arm cores or the concentration of bifunctional “crosslinking” groups that react with the end-groups of each multi-arm PEG to form a network?

While the crosslinking density alone is a structural parameter that can give some useful information about a hydrogel, it does not address other important factors. For example, it does not indicate how many chains are attached at each junction (the junction functionality), the overall concentration of polymer (or polymer volume fraction), the fidelity of the network (frequency of chain-end defects), or even the length of each polymer chain (or degree of polymerization), each of which can be manipulated independently of the crosslinking density.

The crosslinking density alone, as it is often discussed, is not a robust or strongly predictive structural parameter.

The Average Molecular Weight Between Crosslinks (M_c, g/mol) describes the average molecular weight polymer chains (without any intermediate crosslinks) within a hydrogel. The molecular weight between crosslinks is well-known as the parameter estimated by the Peppas-Merrill equation. In our work, we have converted the average molecular weight between crosslinks to the Degree of Polymerization Between Junctions (N_j) for several reasons:

1. “Between Crosslinks” leads to all the issues with the definition of a crosslink discussed in the “Crosslinking Density” section. “Between Junctions” makes it clear that any junction where three or more chains converge ends the chain of interest. If there is a bifunctional crosslink attached to ends of two linear polymers, it is simply a continuation of the chain.
2. The emphasis on “molecular weight” limits practical comparison between two different polymers. If one polymer has a large side-chain, and the other has a small side-chain, then the repeating units will have very different “molecular weights,” and chains of each polymer with the same number of repeating units will have very different molecular weights. Structurally, we expect hydrogels with similar degrees of polymerization between junctions to behave more similarly than hydrogels with similar molecular weights between crosslinks.

Finally, we note that the molecular weight between crosslinks and the degree of polymerization between junctions are both structural values that are challenging to measure directly. Instead, their primarily utility is to help estimate physical properties of the hydrogel. For this reason, we have made an effort to de-emphasize their role as the end-point of the Peppas-Merrill equation. Instead, we have rearranged the equilibrium swelling equation to focus on how the four structural parameters affect the swollen polymer volume fraction, which is an easily measured physical property.

Swelling ratios are the most frequently reported characterizations for hydrogels. Swelling ratios describe how much a hydrogel swells from one state to another. Unfortunately, since 1) swelling ratios are dimensionless and 2) there are several states of interest for a hydrogel, several different standards and definitions for swelling ratios have emerged, with varying measurements techniques and calculation methods. For clarity, we adopt the use of ‘Q’ for volumetric swelling ratios and ‘q’ for mass swelling ratios.

The most consistently used swelling ratio divides the hydrogel’s swollen volume by its dry volume:

\begin{align*} Q = \frac{V_{s}}{V_{d}} \end{align*}

The equivalent equation for the mass swelling ratio:

\begin{align*} q = \frac{m_{s}}{m_{d}} \end{align*}

However, Peppas and colleagues have also used the “swelling ratio” to describe swelling from the initial, ‘relaxed’ state to the swollen state:

\begin{align*} Q = \frac{V_{s}}{V_{r}} \end{align*}

Other groups frequently calculate the mass swelling ratio (q) and the volume swelling ratio (Q) based on the amount of water added per polymer, therefore subtracting the polymer volume or mass:

\begin{align*} Q = \frac{V_{s}-V_{d}}{V_{d}} \end{align*} \begin{align*} q = \frac{m_{s}-m_{d}}{m_{d}} \end{align*}

Fortunately, if the equation used to calculate the swelling ratio is provided, the swollen polymer volume fraction can be calculated from all the above equations except for the swelling ratio normalized to the relaxed state. Other equations, such as ones used to convert from mass swelling ratio to volumetric swelling ratio based on the density of the polymer and solvent are not listed here.

Polymer mass fractions are the mass-based equivalents to polymer volume fractions. They are not frequently used, but they are less practical for comparison between different hydrogel systems because mass fractions provide less spatial/structural information than volume fractions, and the density of the polymer must be known to convert from mass fractions to volume fractions.

Many biomaterials researchers describe their hydrogel formulations in terms of mass concentrations, weight fractions, and/or molar concentrations. While these descriptions are practical for reproducible methods, they do not provide the physical insight associated with structural parameters. We recommend using and reporting both sets of information and using the structural parameters to help understand if the component concentrations are well-balanced to create desirable network structures. For example, a 5% wt. fraction of poly(vinyl alcohol) and a 0.5% wt. fraction of glutaraldehyde may not seem problematic, but the resulting degree of polymerization between junctions of 23 will probably create a brittle, non-elastic network.

Similarly, reporting hydrogel formulations in terms of ratios, such as the polymer-to-water ratio and the polymer-to-crosslinker ratio may be useful concepts for the designer, but they are not as accessible to others who are not experienced with the relevant system but wish to understand the results. This can be seen in a 2021 Science publication by Kim, Zhang, Shi, and Suo where they brilliantly control the number of entanglements within a hydrogel, but discuss their hydrogel synthesis in ratios of W (water-to-monomer molar ratio), I (initiator-to-monomer molar ratio), and C (cross-linker-to-monomer molar ratio). These terms, while precise, can have non-intuitive and case-specific effects on the resulting hydrogel structure, whereas the four independent structural parameters discussed above aim to clearly link hydrogel structure to physical properties.