Turn swelling data into nanoscale network insight fast. Check modulus-based estimates for crosslink consistency easily. See mesh size instantly, then download clean reports now.
1) Elastic modulus method
The network mesh size can be approximated from the plateau shear modulus using: ξ = (kBT / G)^{1/3}. Here kB is the Boltzmann constant, T is absolute temperature, and G is shear modulus.
2) Swelling-based method (Flory–Rehner)
First compute polymer volume fraction in the swollen state: v2s = Vpoly / (Vpoly + Vsolv), with Vpoly = mdry/ρpoly and Vsolv = (mswollen − mdry)/ρsolv.
The molecular weight between crosslinks is obtained from: −[ln(1−v2s)+v2s+χv2s²] = (V1 ρpoly/Mc)(v2s^{1/3}−v2s/2), then ξ = l √(Cn(Mc/Mr)) v2s^{−1/3}.
| Case | T (°C) | G (kPa) | mdry (g) | mswollen (g) | χ | V1 (cm³/mol) | Estimated ξ (nm) |
|---|---|---|---|---|---|---|---|
| A | 25 | 10 | 0.10 | 1.00 | 0.45 | 18 | ~1.6 (elastic), ~6–12 (swelling) |
| B | 25 | 30 | 0.20 | 1.20 | 0.40 | 18 | Smaller mesh than A |
| C | 37 | 5 | 0.08 | 1.10 | 0.50 | 18 | Larger mesh than A |
Mesh size (ξ) is the characteristic spacing between network strands in a swollen hydrogel. It governs diffusion of proteins, drugs, and nanoparticles, and it influences stiffness and degradation in many biomedical devices. When ξ approaches a solute’s hydrodynamic diameter, transport becomes hindered and release slows.
This calculator provides two routes: an elastic-network estimate from shear modulus and a swelling-based estimate combining Flory–Rehner with chain statistics. The modulus route is useful for screening, while the swelling route links ξ to solvent affinity and crosslink density through equilibrium uptake data.
The elastic estimate uses ξ = (kBT/G)^{1/3}. Use a small-strain plateau shear modulus from oscillatory rheology or an equivalent network modulus. If you only have Young’s modulus, convert using G ≈ E/[2(1+ν)] and choose ν near 0.5 for many gels.
The swelling path starts with v2s, the polymer volume fraction in the swollen state. It is computed from dry mass, swollen mass, and densities. For water-swollen gels near room temperature, ρ_solv ≈ 1.00 g/cm³ is often adequate, but cosolvents can shift density.
χ captures polymer–solvent interactions and can be temperature dependent. Small changes in χ can change the inferred Mc and ξ, especially for highly swollen gels (low v2s). Use literature values for your system or estimate χ from swelling series when possible. V1 is the solvent molar volume (water ≈ 18 cm³/mol).
After solving Flory–Rehner for the molecular weight between crosslinks Mc, chain statistics convert Mc into a mesh estimate using ξ = l·sqrt(Cn·Mc/Mr)·v2s^{-1/3}. Mr is repeat unit molar mass, Cn is the characteristic ratio, and l is an effective bond length describing segment geometry.
Many soft, highly swollen hydrogels report ξ from a few nanometers to tens of nanometers, depending on formulation and ionic strength. A stiffer gel (larger G) generally yields a smaller elastic ξ, while stronger swelling (smaller v2s) usually increases the swelling-based ξ. Agreement within a factor of two is often reasonable for screening.
Report temperature, measurement frequency, equilibration time, solvent composition, and whether masses are blotted consistently. Provide the χ source and density assumptions. Export CSV for lab notebooks and generate a PDF snapshot for reports. When comparing batches, keep protocols constant to avoid systematic bias in ξ.
Use both. The modulus method reflects mechanical network response, while the swelling method reflects thermodynamics and uptake. Consistent trends across both are most reliable for comparing formulations.
Non-equilibrium swelling underestimates uptake, increasing v2s and biasing the swelling-based mesh smaller. Swell until mass stabilizes over repeated measurements at the same temperature.
Quite sensitive, especially for highly swollen gels. A small χ change can noticeably shift Mc and ξ. Prefer system-specific χ values or validate with a χ range to bracket uncertainty.
Yes, convert to shear modulus using G ≈ E/[2(1+ν)]. For many water-rich gels, ν is close to 0.5. Use consistent conversion when comparing samples.
Differences can come from frequency-dependent modulus, incomplete equilibration, ionic effects, entanglements, or χ mismatch. Re-check units and ensure the modulus represents the network plateau.
Mr is the molar mass of one repeating monomer unit along the backbone, not the whole polymer chain. It links Mc to the number of statistical segments in a network strand.
Yes. Use the top buttons for a snapshot of current inputs. For a results-focused export, press Calculate first so the results block is included in the PDF and server CSV.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.