What is Swelling?
Swelling refers to the change in volume of a gel as it absorbs a compatible solvent, such as water for hydrogels. Swelling is possible because the solid component of the hydrogel, which is typically a polymer network, is both elastic and hydrophilic, enabling enough water to be absorbed to increase the volume, but creating elastic tension within the networked polymer chains that prevents total dissolution.
What is the Relevant Property?
Polymer volume fractions are the standard method for characterizing swelling in a hydrogel. They represent the fraction of volume in a hydrogel that is water, typically under the assumption that the rest of the volume is water or aqueous solution. Three states define the swelling profile of a hydrogel, starting with the initial or relaxed polymer state, which immediately follows network formation, the equilibrium-swollen state, which follows swelling the gel in an excess of water, and the dry state, which can be achieved with heat and vacuum to remove the water. Assuming the dry state is non-porous, the polymer volume fraction in the dry state is equal to one, and the volumes in the initial state and the equilibrium-swollen state can be compared to the dry volume to calculate the initial polymer volume fraction (φ0) and the swollen polymer volume fraction (φs). Additionally, the swelling from the initial state to the swollen state can be evaluated by dividing their respective volumes, resulting in what we describe as the reference ratio (θ).
Alternative Parameters:
Reference Ratio (θ)
The reference ratio (θ) describes the extent of swelling from the initial state to the equilibrium-swollen state. While the reference ratio is a precise descriptor of hydrogel swelling, it cannot entirely summarize a hydrogel’s swelling behavior without also knowing the initial polymer volume fraction. Notably, we have shown with poly(vinyl alcohol) (PVA) and poly(ethylene glycol) diacrylate (PEGDA) hydrogels that the reference ratio changes with the degree of polymerization between junctions but not with the initial polymer volume fraction (Richbourg et al., 2021, Science Advances). This independent relationship suggests that the extent of swelling from the initial state to the equilibrium-swollen state depends primarily on the length of polymer chains between junctions and that swelling corresponds to extending those chains in response to the chemical potential of polymer-and-solvent mixing.
Polymer Mass Fractions (ψ0 or ψs)
Polymer mass fractions (ψ0 or ψs) 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.
Swelling Ratios
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.
How is it Measured?
Three methods are used fairly frequently to measure polymer volume fractions. While buoyancy-based volumetric measurements are the most direct, precise, and reproducible of the three measurements, direct measurement of hydrogel sample dimensions is a viable alternative for sufficiently large hydrogel samples, and mass-based measurements can be used where necessary as long as the polymer density is well-established, and the assumption of additive volumes has been validated, ideally by comparison with buoyancy-based volumetric measurements.
Buoyancy-Based Volumetric Measurements
Buoyancy-based volumetric measurements are the most accurate and reproducible method for measuring polymer volume fractions. In addition to a hydrogel sample in the relaxed state (typically about 0.3 g, best in triplicate), this measurement method requires excess water and a way to dry the hydrogel (for subsequent swollen state and dry state measurements), a scale, a density kit (3D-printed versions available on our Tools page), heptane (or another solution that is immiscible with water and the polymer and has a consistent density), and a fume hood (since you are using heptane). First, you take the just-synthesized, “relaxed state” sample and measure its real mass using the non-submerged upper stage of the density kit. Then, you submerge the sample completely in the non-solvent solution and weight it again while seated on the submerged stage. Note this is not the real weight, but the difference of the sample’s weight and the buoyant force exerted by the solution. ALso, the force-sensitive part of the scale is not supporting the weight of the beaker, only the sample and its hanger. Once the mass and submerged apparent mass have been recorded, you can calculate the hydrogel volume from the difference divided by the density of the non-solvent solution.
The hydrogel should then be swollen to equilibrium, typically by incubation in excess water for at least 24 hours, and its swollen volume should be measured in the same way. Finally, the polymer network should be dried of all water, typically by heat and vacuum to produce a dense, glass-like or rubbery solid. The glassy or rubbery polymer network can be volumetrically measured again for the dry volume. Lyophilization is an alternative drying method, but the dry sample cannot then be submerged in solution for an accurate volumetric measurement. If the polymer density is known, lyophilization and dry mass measurement can be used to estimate the dry volume.
Finally, the dry volume, relaxed volume, and swollen volume can be used to calculate the relaxed and swollen polymer volume fractions. This method has the added bonus of being able to calculate the density in each state by dividing the sample mass by the sample volume.
Measure Dimensions
An alternative to buoyancy-based volumetric measurements that is reasonably viable with large, stiff hydrogels is to measure their physical dimensions, such as height and diameter for discs or cylinders, in each state and use that to calculate volumes. This method is typically less precise than the buoyancy-based method since hydrogels are often small, soft samples that are easily deformed. Their dimensions can also be slightly distorted by surface-adsorbed water and may be more difficult to measure upon equilibrium swelling or drying.
Measure Masses and Estimate from Densities
Hydrogel sample volumes can be estimated in each state by measuring the mass with a scale and using known densities of the polymer (ρp) and water (ρw) to estimate the polymer volume fraction via the following equation:
\begin{align*} \phi_{s} = \left[1+\frac{\rho_{p}}{\rho_{w}}\left( \frac{m_{s}}{m_{d}}-1 \right) \right]^{-1} \end{align*}While this approach bypasses the need for a density kit and submersion in a non-solvent solution, it requires confident knowledge of the polymer density and lowers the overall precision with respect to volume. This method also assumes that the polymer and water have additive volumes, which may not hold true for systems with complex polymer-solvent interactions, such as biopolymers. When in doubt, the buoyancy-based method can be used to validate the use of mass-based estimates.
Related Properties
In addition to polymer volume fractions in various states, two additional physical properties are closely related to swelling. These properties have their own important uses in hydrogel design, but they are not currently addressed by the swollen polymer network model.
Swelling Rate
Swelling rate is the rate at which a hydrogel absorbs water and swells. Swelling rate is an extrinsic property that depends on the surface area to volume ratio of a hydrogel, but it is also affected by the initial swelling status of the hydrogel as well as the structure of the hydrogel. Since the thermodynamic theories focus on initial states and equilibrium end-states, the swelling and deswelling rates are only measured for our studies to shown when the hydrogel has reacted swelling or drying equilibrium. However, focused control of hydrogel swelling rates based on hydrogel structure has potential for application in both drug delivery and biomechanical actuation.
Hydrophilicity
Hydrophilicity of a material, often measured via contact angle measurements with a goniometer, describes how readily water adheres to the material. Obviously, hydrogels are exceptionally hydrophilic, and the extent of their hydrophilicity affects equilibrium swelling as well as the swelling rate. It will be interesting to study how a polymer’s inherent hydrophilicity interacts with hydrogel structure to affect swelling behavior.
Our Results on Structural Design of Hydrogel Swelling
Our published work in Science Advances highlights our results on structural design of hydrogel swelling in poly(vinyl alcohol) (PVA) hydrogels, poly(ethylene glycol) diacrylate (PEGDA) hydrogels, and gelatin methacrylate (GelMA) hydrogels. Briefly, we controlled the initial polymer volume fraction (φ0) and the degree of polymerization between junctions (Nj) in all three systems, leading to 18 PVA hydrogel formulations, 15 PEGDA hydrogel formulations, and 9 GelMA hydrogel formulations. Major swelling-specific results from that work are summarized below:
Polymer volume fractions calculated from mass-based measurements match buoyancy-based measurements in PVA and PEGDA hydrogels.
Initial polymer volume fraction is equivalent to the relaxed polymer volume fraction in PVA and PEGDA hydrogels.
Degree of polymerization between junctions affects reference ratio (θ) in PVA and PEGDA hydrogels independent of initial polymer volume fractions.
The linear relationship between degree of polymerization between junctions and reference ratio is broadly applicable for synthetic hydrogels.
Biopolymer-based GelMA hydrogels have much less reliable swelling behavior than synthetic PVA and PEGDA hydrogels.
