## CALCULATOR for SOLUTE DIFFUSION in HYDROGELS

We have developed a model for the diffusivity of solutes in hydrogels combining three main theoretical frameworks, which we call the Multiscale Diffusion Model (MSDM). This tool predicts the diffusivity of solutes in hydrogels given the various parameters for the system. The calculations are based on the following equation:

$$\frac{D}{D_o} = \left[erf\left(\frac{r_{FV}}{r_s}\right)exp\left( -\left(\frac{r_s}{r_{FVW}}\right)^3 \left(\frac{\phi_p}{1-\phi_p}\right)\right)+ erfc\left(\frac{r_{FV}}{r_s}\right) exp\left(-\pi\left(\frac{r_s+r_f}{\xi+2r_f}\right)^2\right) \right]$$

In the formula above, D is the diffusivity of a solid inside the hydrogel, D0 is the diffusivity of the same solid in the hydrogel liquid, and rFV and rFVW are the free volume void sizes in hydrogel and water, respectively. rs is the hydrodynamic radius of the solute, ɸp is the polymer volume fraction, rf is the radius of the polymer chain, and ξ is the hydrogel mesh size. While this equation is primarily for prediciting solute diffusivity in hydrogels, it can also be used to estimate hydrogel mesh sizes from known solute diffusivities.

The detailed derivation of the Multiscale Diffusion Model (MSDM) can be found in Axpe et al. Macromolecules 2019 (PDF).

## ASSUMPTIONS

• Ideal hydrogel system
• Spherical, inert solute
• System fluid is water
• Free volume void size in water and hydrogel are similar (rFVrFVW = 0.269 nm)
• Radius of the fibers for Poly(ethylene glycol) = 0.51 nm, Alginate = 0.83 nm, and Polyacrylamide = 0.88 nm

## DIFFUSIVITY and MESH SIZE CALCULATOR

 Polymer Polyacrylamide Poly(ethylene glycol) Alginate Temperature (°C) Polymer Volume Fraction (%) Hydrodynamic Radius of Solute (nm) Mesh Size (nm) Diffusivity (µm2/s)

This tool was built by Vincent Wu (vincent.wu@berkeley.edu). The project source code is available here on Github.