Mesh Transport Theory aims to provide a relationship between the diffusivity of solutes in a hydrogel and the hydrogel’s structural properties. The theory first indicates a relationship in terms of the respective mesh size and can be represented as follows:

\begin{align*} \xi = \phi_{s}^{-\frac{1}{3}}\left[\left(1 – \frac{2}{f}\right)\bar{l}^{2}C_\infty\lambda N_{j}\right]^\frac{1}{2} \end{align*}

Mesh transport theory further distinguishes the difference between mesh size and mesh radius as necessary for developing conclusions regarding solute diffusivity. The mesh radius is dependent on the 3D geometry of a hydrogel network due to geometrical changes surrounding junctions. Thus, the mesh radius differs based on the junction functionality of a polymer network. Currently, the case-specific mesh radii due to various junction functionalities can be represented as follows:

\[ \texttip{r_{m}}{Mesh Radius} = \begin{cases} \frac{\sqrt{6}}{3}\xi & f = 4 \\ \frac{1}{2}\xi & f = 6 \\ \frac{\sqrt{2}}{4}\xi & f = 8 \end{cases} \]

Prior to understanding solute diffusion through hydrogels, solute diffusion through water must be addressed. The Stokes-Einstein Equation provides a description for solute diffusion through a solvent with no net fluid flow and is represented as a function of the solute’s effective radius and the solvent’s viscosity. The Stokes-Einstein equation will be used to determine the value for the free diffusion coefficient (D0) in the following equation for diffusivity in hydrogels:

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

Mesh transport theory ultimately relates the solute diffusion coefficient in a hydrogel to the solute’s size, the polymer volume fraction, and the mesh radius. The resulting modified multiscale diffusion model can be predicted entirely from a priori knowledge or predictions of the solute and hydrogel properties.

\begin{align*} \frac{D}{D_{0}} = erf \left(\frac{r_{FVW}}{r_{s}}\right)exp\left[-1^{*}\left(\frac{r_{s}}{r_{FVW}}\right)^{3}\left(\frac{\phi_{s}}{1 – \phi_{s}}\right)\right] + erfc \left(\frac{r_{FVW}}{r_{s}}\right)exp\left[\frac{-\pi}{4}\left(\frac{r_{s} + r_{f}}{r_{m}}\right)^{2}\right] \end{align*}

Mesh Radius and Diffusion Coefficients Calculator

rm = 0 nm

D⁄D0 = 0

D0 = 0 µm2/s

D = 0 µm2/s

Parameter Definitions and Ranges

Mesh Size (ξ, nm)

Swollen Polymer Volume Fraction (φs)

Junction Functionality (f)

Average Bond Length in the Polymer Backbone (l̄, nm)

Flory’s Characteristic Ratio (𝐶∞)

Number of Backbone Bonds in the Polymer Repeating Unit (λ)

Degree of Polymerization Between Junctions (Nj)

Mesh Radius (rm, nm)

Free Solution Diffusion Coefficient (D0, μm2/s)

Boltzmann Constant (kb, J/K)

Temperature (T, °C)

Solution Viscosity (η, Pa⋅s)

Solute Hydrodynamic (Stokes) Radius (rs, nm)

Diffusion Coefficient (in a Hydrogel) (D, μm2/s)

Radius of Free Volume Voids in Water (rFVW, nm)

Polymer Fiber Radius (rf, nm)

Key Assumptions

1. Mean-Field Approach

Like rubberlike elasticity theory and equilibrium swelling theory, mesh transport theory is built on a mean-field approach (Wikipedia Link). The use of average values, such as mesh size, mesh radius, solute hydrodynamic radius, and polymer volume fraction assume multiple levels of homogeneity throughout the hydrogel. In reality, mesh radii are likely to have a broad distribution unless the hydrogel synthesis is precisely controlled, resulting in a variety of network openings that may or may not allow passage of a similarly varying distribution of solute sizes, shapes, and orientations. Nevertheless, the mean-field approach helps to establish relationships between hydrogel properties, solute properties, and diffusion coefficients.

2. No Convection

The modified multiscale diffusion model treats all solute transport within hydrogels as diffusion, therefore assuming that no bulk convection of solutes occurs within hydrogels. This assumption is supported by the overall solid-like state of a hydrogel, with networked polymer chains interrupting bulk mass transport and absorbing momentum. However, hydrogels that contain large polymer-free voids may experience some degree of local convection.

FRAP images taken immediately after bleaching reveal convective smearing in solution but not in hydrogels.
3. Polymer-Solute Non-Interaction

The mesh transport models make the practical assumption that the solutes remain within the mobile, aqueous phase of the hydrogel. Any direct interactions with the polymer network would require further calculations in the form of a reaction or adsorption isotherm that would further complicate the solute’s effective diffusion within the hydrogel. The polymer network is therefore treated as volumetric or gate-like obstructions separated by at least a monolayer of water molecules. To maintain this assumption, it is important that the solute is hydrophilic, or it would likely favor the more hydrophobic components of the polymer network.

Strange interactions await there…
4. Solutes are Stable Hydrodynamic Spheres

To use the Stokes-Einstein equation and the modified multiscale diffusion model, which both define a solute by its hydrodynamic radius, one must assume that the solutes in question are effectively hard and spherical. Non-spherical molecules might have non-isotropic motion, especially in hydrogels, where there would be an enhanced tendency to move along the past of least resistance. For example, a rod-like solute would move easily in the direction of the rod but encounter more polymeric obstructions for off-axis movement. If the solutes are significantly non-spherical, then the diffusion coefficient predictions made by the modified multiscale diffusion model are likely to be inaccurate. Similarly, a flexible molecule may result in complex partial entrapment within and around dynamic polymer chains or may even transfer kinetic energy to and from the networked polymer chains, further deviating from model predictions, which assume negligible polymer-solute interactions.

5. Dualism of Obstruction and Free Volume Theories

The multiscale diffusion model [Axpe et al., 2019] was created by combining obstruction and free volume theories. The size range over which the two theories differ is managed with an error function and complementary error function comparing the radius of the solute and the radius of free volume voids. As a result, obstructions and free volumes are established as the two predominant influences on solute diffusion in hydrogels, and other possible interactions are effectively excluded from consideration. Any demonstrated influences on diffusion that cannot be grouped into either the obstruction or free volume components would therefore require reconsideration of the entire model.

Figure from [Axpe et al., 2019] showing how the Multiscale Diffusion Model strikes a balance between the predictions of Free Volume Theory and Obstruction Theory. Reprinted here with permission from ACS for educational purposes only.
6. Free Volume Void Sizes

The radius of free volume voids in water is the reference point for the modified multiscale diffusion model, marking the point of conversion from the free volume theory to the obstruction theory. In the modified multiscale diffusion model, it is assumed that the size of free volume voids in water is constant at 0.269 nm and equivalent to the size of free volume voids within a hydrogel. Practically, it is likely and possible that the structure of the hydrogel affects the size of free volume voids within the hydrogel, as the original creators of the multiscale diffusion model characterized using Positron Annihilation Lifetime Spectroscopy (PALS) [Axpe et al., 2019]. However, PALS requires highly specialized equipment, and the measured free volume voids did not differ greatly from that of free water. Therefore, the radius of free volume voids in water is used in the modified multiscale diffusion model.

7. Network Geometry Matters

From our recent (2021) work modeling how network geometries change with changing junction functionalities, we introduced the concept of mesh radius in solute transport. The rationale for using mesh radius instead of the well-established mesh size is that the network’s geometry gets more restrictive to spherical solutes as the junction functionality increases, even if the mesh size does not change.

Additional Issues to Consider

Polymer Chain Dynamics

Both obstruction theory and free volume theory, which are merged in the multiscale diffusion model, treat polymer networks as a static, unchanging hindrance to solute diffusion. In reality, at the molecular scales relevant to solute diffusion, the solution, solutes, and polymer network are constantly fluctuating in a molecular dance known as Brownian motion (Wikipedia link). As a result, a particular network portal may be small at one millisecond as all the chains shift towards each other and much larger the next millisecond as the chains bounce of each other and spread out again. Rather than the average mesh radius acting as a strict cutoff, it becomes an average influence, and many interesting, dynamic interactions may occur as solutes near the size range of the mesh radius. It’s important to remember that the network is dynamic over time, and what could happen in one moment may be impossible in the next moment. In the current models, this variability is addressed with mean-field averages, but future models might be able to describe and use chain dynamics to control solute diffusion in hydrogels.

While mesh and mesh radii represent average values, actual network portals may change shape dramatically over short time periods, resulting in significant variability over whether a solute can actually pass through a given network portal.
Size Exclusion, Partitioning, and Immobilization

While chain dynamics describe why the mesh radius does not act as a strict size cutoff, it is still quite likely that the network will either exclude or immobilize significantly large or unwieldly (see reptation section below) solutes. Partitioning between the hydrogel and surrounding solution is a critical factor in drug release and can be affected by network structure as well as polymer-solute interactions [Kotsmar et al., 2012]. It is also important for drug delivery applications to understand how loading and release profiles differ in a hydrogel [Liu et al., 2013]. Finally, hydrogel structures and dynamics can result in non-chemical immobilization of solutes, possibly by local entrapment under a low dynamic possibility of release. Robust characterization of size exclusion conditions, structural factors affecting partition coefficients, and solute immobilization dynamics will greatly enhance the predictive capabilities of solute diffusion and release from hydrogels for non-ideal solutes.

Solute Movement via Reptation

Reptation theory describes a unique regime of snake-like motion for polymeric solutes in highly restrictive environments (Wikipedia Link). While small or rigid molecules tend to move as hydrodynamic spheres, polymeric solutes can be forced into a reptation regime by their environment, wherein their shape becomes more linear, and their net movement depends on each section of the polymer finding a space to occupy along a statistically determined path. A solute diffusing via reptation moves more slowly than a sphere-like diffusing molecule since each point along the polymer has a chance to fail to proceed, whereas the spherical molecule moves as a unit. However, reptation enables large polymeric molecules to move through a network that they could not have entered as a spherical globule. While the principle of reptation was described by De Gennes, it is not included in the current modified multiscale diffusion model. A model that includes both spherical hydrodynamic diffusion and reptation would first need to identify the transition conditions between sphere-like solutes and reptation. Such a model would be a significant breakthrough in hydrogel design and modeling for drug delivery and molecular purification applications.

Reptation is much easier to show visually than to describe with words.

Theory History

Mesh transport theory technically begins with the Stokes-Einstein Equation (Wikipedia), which describes the self-diffusion coefficient of a solute in a liquid (equation show at the top of this page). The Stokes-Einstein equation is associated with Einstein’s 1905 publication on Brownian Motion (Wikipedia Link for Brownian Motion). The Stokes-Einstein equation is thoroughly validated for spherical solutes diffusing in non-turbulent liquids.

The next major contribution toward mesh transport theory, apart from work on equilibrium swelling and rubberlike elasticity in swollen polymer networks, was in 1972, with Ogston, Preston, and Wells, “On the transport of compact particles through solutions of chain-polymers” in the Proceedings of the Royal Society of London. Here, Ogston and colleagues treated a polymer solution as an obstructive environment to solute transport, presenting a volume fraction of polymeric rods that act as obstacles to randomly diffusing solutes. Ogston focused on the radius of the solute, the cross-sectional radius of the “fiber” polymer chains and the polymer volume fraction to calculate the frequency of obstructions, noting that the calculation was best suited for small solutes.

\begin{align*} \frac{D}{D_{0}} = exp\left[-\phi^\frac{1}{2}_{s} \frac{r_{s}+r_{f}}{r_{f}}\right] \end{align*}

In 1983 and1984, Peppas and Reinhart, “Solute diffusion in swollen membranes” in Journal of Membrane Science introduced an alternative to the Ogston model focused specifically on solute diffusion in swollen polymer networks. In what they described as Free Volume Theory, the polymer network acts as a collection of connected molecular cages. A solute can jump through one molecular cage to an adjacent one if there is a large enough hole in the cage and if there is not a solute already occupying the next space. Diffusion within a network is therefore a series of jumps with an associated probability of successfully jumping based on the size of the solute, the cage holes, and the resting spaces.

After developing the theory, preliminary validation with albumin diffusing through poly(vinyl alcohol) (PVA) hydrogels led to a fitted free volume theory equation for solute transport in hydrogels (notably, they used the Peppas-Merrill equilibrium swelling equation to calculate the average molecular weight between crosslinks).

\begin{align*} \frac{D}{D_{0}} = k_{1} \left(\frac{M_{c}-M^*_{c}}{M_{n}-M^*_{c}} \right)^2 exp\left[-k_{2} r^2_{s} \frac{\phi_{s}}{1-\phi_{s}}\right] \end{align*}

In the above equation, Mc* describes the cutoff molecular weight between crosslinks, below which the solute can no longer pass through the network. The parameters k1 and k2 are empirical fitting coefficients.

Inspired by the scaling concepts of De Gennes in Scaling Concepts in Polymer Physics, 1979, Lustig and Peppas, “Solute diffusion in Swollen Membranes. IX. Scaling Laws for Solute Diffusion in Gels,” (1988) in Journal of Applied Polymer Science reinvented free volume theory as a simple scaling law based on the concept of mesh size:

\begin{align*} \frac{D}{D_{0}} ≅ \left(1- \frac{r_{s}}{\xi} \right) exp\left[-Y \frac{\phi_{s}}{1-\phi_{s}}\right] \end{align*}

In the above equation, the exponential factor Y incorporates several structural factors, including a square of the solute’s hydrodynamic radius, but the authors argued that can it generally assumed that Y = 1 for most solutes.

In 1989, Canal and Peppas, “Correlation between mesh size and equilibrium degree of swelling of polymeric networks” in the Journal of Biomedical Materials Research developed a quantitative definition of mesh size based on the statistical mechanics calculations of Flory. This definition of mesh size was widely adopted since previous discussions of mesh size were vague or depended on non-specific scaling laws as discussed by De Gennes.

\begin{align*} \xi = \phi^{-\frac{1}{3}}_{s} \left(\frac{2M_{c} C_{\infty} l^2}{M_{r}}\right)^\frac{1}{2} \end{align*}

From the Canal-Peppas mesh size model, we step forward a decade to look at a coordinating review by Amsden, (1998) “Solute Diffusion within Hydrogels. Mechanisms and Models,” in Macromolecules. Amsden described three major approaches to solute diffusion in hydrogels, the free volume theory, hydrodynamic theory, and obstruction theory, and described several mathematical models and the associated concepts of each. His computational analysis of the models and their relationship to actual diffusion in hydrogel data highlighted many of the limitations of the models and provided a template for further modeling analysis of solute diffusion in hydrogels. It also acts as a concise reference point for investigating alternative solute diffusion in hydrogels models if the modified multiscale diffusion model we focus on here proves inappropriate for or irreconcilable with experimental data.

Our next conceptual leap is based on the work of Axpe et al., (2019), “A Multiscale Model for Solute Diffusion in Hydrogels,” in Macromolecules, which aimed to coordinate free volume theory and obstruction theory over the broad ranges of solute sizes that could transition from one theory dominating to the other. Specifically, they argued that free volume theory is more appropriate at the mid-small range (~0.5-3 nm), obstruction theory is more appropriate at the mid-large range (~3-10 nm), the theories are equivalent at extremes, and an error function scaled to the size of free volume voids mediates the transition. The unmodified multiscale diffusion model follows:

\begin{align*} \frac{D}{D_{0}} = erf \left(\frac{r_{FV}}{r_{s}}\right)exp\left[-1^{*}\left(\frac{r_{s}}{r_{FVW}}\right)^{3}\left(\frac{\phi_{s}}{1 – \phi_{s}}\right)\right] + erfc \left(\frac{r_{FV}}{r_{s}}\right)exp\left[-\pi\left(\frac{r_{s} + r_{f}}{\xi + 2r_{f}}\right)^{2}\right] \end{align*}

In the above equation, rFV represents the radius of free volumes, which we note may differ from the radius of free volumes in water (rFVW) and can be measured using PALS. We also note that the inclusion of the fiber radius in both the numerator and denominator of the last term seems to double-count the effect of the fiber radius. For this reason, the modified multiscale diffusion model at the top of this page does not include the “+2rf” term.

Richbourg and Peppas, (2020) in “The swollen polymer network hypothesis,” Progress in Polymer Science, coordinated the mesh transport theory described above with similar fundamental concepts in equilibrium swelling theory and rubberlike elasticity theory. The analysis here also aimed to make the Canal-Peppas equation more broadly applicable, introducing the weighted average length of backbone chains to include polymers such as PEG with nonvinyl backbones as well as standardizing the terms for cross-polymer comparison and introducing the phantom-like (1-2/f) correction term based on developments in rubberlike elasticity theory applied to hydrogels. At this point, the model for mesh size and the modified diffusion model were in place, but we had yet to introduce the concept of mesh radius.

Richbourg, Ravikumar, and Peppas, (2021) used a short perspective paper, “Solute Transport Dependence on 3D Geometry of Hydrogel Networks,” in Macromolecular Chemistry and Physics, to introduce mesh radius (rm) as a more geometrically precise term for relating hydrogel network structure to solute transport in a hydrogel. In this paper, mesh radius was calculated as a function of mesh size for three perfect polyhedron structures, though it is much more difficult to calculate for imperfect 3D network structures. The significance of mesh radius is in its response to changing junction functionality. With all other structural parameters held the same, changing junction functionality leads to a slight increase and mesh size and a sharp decrease in mesh radius. The resulting question is this: Does increasing junction functionality increase or decrease solute diffusivity? The answer will determine the use of mesh radius in later iterations of mesh transport theory.

Prediction of mesh size and radius changing junction functionality but not changing other structural parameters (initial polymer volume fraction, degree of polymerization between junctions, and frequency of chain-end defects). Reprinted from Richbourg, Ravikumar, and Peppas, 2021.