Diffusivity

Partition Coefficients (K) describe the partitioning of solutes between a hydrogel and the surrounding bulk solution at equilibrium. Partition coefficients are always context-specific, as they can be affected by the concentration in the bulk solution, the geometry and size of the hydrogel, and other factors, but under consistent conditions they can provide some insight about the interactions between the solute and the hydrogel. Typically, a partition coefficient greater than one indicates that the solute has a chemical affinity for the polymer used in the hydrogel. Partition coefficients less than one could be affected by chemical repulsion or the structure of the hydrogel limiting the movement of solute into the hydrogel. Large solutes can be systematically excluded by highly crosslinked hydrogels, which will greatly lower their partition coefficients.

Partition coefficients are most simply defined for hydrogels as the concentration within the hydrogel divided by the concentration outside the hydrogel (equation below).

\begin{align*} K = \frac{C_{gel}}{C_{sol}} \end{align*}

A common method for calculating partition coefficients within a hydrogel is to measure the concentration within the solution before introducing the hydrogel and measure it after equilibrium to calculate the reduction in concentration and assume the associated mass of solute was transferred into the gel. The equation below makes that calculation given the volume of the solution (V_sol) and the volume of the gel (V_gel).

\begin{align*} K = \frac{V_{sol} \left(C_{0} – C_{eq}\right)}{V_{gel} C_{eq}} \end{align*}

However, this approach assumes that there is no fouling, or accumulation of the solute at the surface of the hydrogel. Our experiments have shown that there is usually solute fouling at the surface of a hydrogel, so we instead use a confocal microscope to measure the average intensity within the hydrogel and correlate that to a concentration based on a standard curve.

Additionally, if you assume all of the solute stays within the aqueous phase of the hydrogel, it is reasonable to calculate an “Enhancement factor” [Kotsmar et al., 2012] from the partition coefficient, which estimates the solute concentration within the aqueous phase of the hydrogel by deducting the swollen polymer volume fraction (equation below).

\begin{align*} E = \frac{K}{1-\phi_{s}} \end{align*}

Currently, the relationship between a solute’s self-diffusion coefficient within a hydrogel and the partition coefficients is not well-defined, but it is likely important since diffusion coefficients are sensitive to concentration.

The Solute Hydrodynamic Radius (r_s) is the estimated radius of the solute within an aqueous solution assuming it is a hard sphere. Hydrodynamic radii are typically calculated from a solute’s self diffusion coefficient in aqueous solution via the Stokes-Einstein equation.

\begin{align*} D_{0} = \frac{k_{b} T}{6\pi \eta r_{s}} \end{align*}

Hydrodynamic radius is often treated as the defining parameter for a solute’s size, but it does not account for flexibility or non-sphericality, which play a more significant role for diffusion within a hydrogel than diffusion within free solution.

Mesh Size (ξ) is the hydrogel structural parameter most closely associated with solute diffusion in a hydrogel, although the swollen polymer volume fraction is also considered a strong influence. Definitions of mesh size have differed between groups and over time, but the most conceptually consistent definition of mesh size is the average distance between h=network junctions that are connected by a single chain. Since mesh size is a distance, it does not correspond perfectly to degree of polymerization between junctions (N_j) or the frequently discussed molecular weight between crosslinks (M_c). Two chains with the same degree of polymerization between junctions can have different mesh sizes if they are extended to different lengths. Because mesh sizes are typically on the order of nanometers in a synthetic hydrogel and must be measured in a hydrated state to be accurate, they are most often estimated based on more readily measured properties such as the swollen polymer volume fraction. As shown on the Mesh Transport Theory page, we use a modified Canal-Peppas equation to calculate mesh size [Richbourg & Peppas, 2020].

Mesh Radius (r_m) is a concept we introduced recently to address the effect of variable junction functionality on solute transport within hydrogels [Richbourg, Ravikumar, and Peppas, 2021]. A hydrogel’s mesh radius is the maximum radius of a solute that can pass through the hydrogel network’s portals. It is important to dinstinguish between mesh size and mesh radius because mesh radius has a more direct influence on solute transport within a hydrogel, and the same mesh size can result in different mesh radii based on the junction functionality and geometry of the hydrogel. For example, a perfect hydrogel with a junction functionality of 6 would have cubical cells with square portals, so the mesh radius would be half of the mesh size. But for a hydrogel with a junction functionality of 4, or tetrahedral junctions, the mesh radius would be a larger fraction of the mesh size due to the more open geometry. Mesh radius calculations are also provided on the Mesh Transport Theory page.

Solute loading and release kinetics differ from self-diffusion in that the solute is diffusing into a solute-free hydrogel or from the hydrogel into an empty reservoir. Both cases are driven by chemical gradients and are therefore a little more complicated than self-diffusion conditions. One major factor that is relevant in loading and release but not in self-diffusion is the boundary between the hydrogel and the surrounding solution, that can create unique interactions such as solute accumulation at the surface, known as “fouling.” Fouling can slow transport into the hydrogel and cause initial burst releases when unloading from the hydrogel. It is currently unclear whether solute loading and release can be described using the same Mesh Transport Theory equations, but the consistent use of diffusion coefficients to describe loading and release behavior and self-diffusion suggests there could be comparable relationships.