What is the Flory-Rehner equation?

The Flory-Rehner equation relates the equilibrium swelling of a crosslinked polymer network to its crosslink density. It combines elastic (Gaussian chain) and mixing (Flory-Huggins) contributions: Ve = −[ln(1−vr) + vr + χ·vr²] / [Vs·(vr^(1/3) − vr/2)].

What is the polymer volume fraction vr?

vr is the volume fraction of polymer in the swollen gel: vr = Vp / (Vp + Vs_total), where Vp is the dry polymer volume and Vs_total is the absorbed solvent volume. Values typically range from 0.05 (loosely crosslinked) to 0.9 (tightly crosslinked).

Where do I find the Flory-Huggins parameter χ?

χ values are tabulated in polymer handbooks (e.g., Polymer Handbook by Brandrup & Immergut). For rubber in toluene χ ≈ 0.39; for PDMS in toluene χ ≈ 0.46; for polyacrylamide hydrogels in water χ ≈ 0.49. χ can also be estimated from solubility parameters.

What units does the result use?

Crosslink density Ve is expressed in mol/cm³. Multiply by 1000 to convert to mmol/cm³, or divide by the polymer density to express per gram of polymer.

What does a higher crosslink density mean?

A higher Ve means more crosslinks per unit volume — the polymer network is tighter, stiffer, less swellable, and less permeable. Typical values: loose hydrogels ~10⁻⁵ mol/cm³; hard epoxies ~10⁻³ mol/cm³.

Why does my result come out negative?

A negative Ve usually means the input vr is outside the physically valid range (0 < vr < 1), or the χ value is too high for the chosen solvent system. Double-check your masses and densities, and verify that the swollen mass is greater than the dry mass.

Crosslink Density Calculator

Compute polymer crosslink density (Ve) from equilibrium swelling data using the Flory-Rehner equation.

Swelling Experiment Data

All masses in grams, densities in g/cm³.

Sample Masses

Dry mass (g)

Swollen mass (g)

Densities (g/cm³)

Polymer (ρ_p)

Solvent (ρ_s)

Solvent & Interaction

V_s (cm³/mol)

Molar volume of solvent

χ (Flory-Huggins)

Polymer–solvent parameter

Common solvent reference values

Solvent	ρ (g/cm³)	Vs (cm³/mol)
Toluene	0.865	106.3
Hexane	0.659	130.7
Chloroform	1.480	80.7
Acetone	0.791	73.8
Water	1.000	18.0
Dichloromethane	1.325	63.9
THF	0.889	81.7

Enter swelling data on the left and click Calculate.

Crosslink Density

— mol/cm³

Effective crosslink density (V_e)

Intermediate Values

Polymer volume fraction (v_r): —
Absorbed solvent volume (cm³): —
Dry polymer volume (cm³): —
Numerator −[ln(1−v_r) + v_r + χv_r²]: —
Denominator V_s·(v_r^1/3 − v_r/2): —

Interpretation

Typical Ve ranges

Material class	Ve (mol/cm³)
Loose hydrogels	~10⁻⁵ – 10⁻⁴
Lightly vulcanized rubber	~10⁻⁴ – 10⁻³
Thermosets / epoxies	~10⁻³ – 10⁻²

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Summary

This tool calculates the effective crosslink density (Ve, mol/cm³) of a polymer network from equilibrium solvent-swelling experiments. It applies the Flory-Rehner equation: Ve = −[ln(1−vr) + vr + χ·vr²] / [Vs·(vr^(1/3) − vr/2)], where vr is the polymer volume fraction at equilibrium swelling, Vs is the molar volume of the solvent, and χ is the Flory-Huggins polymer–solvent interaction parameter. Crosslink density is a key structural parameter for rubbers, hydrogels, and thermoset resins — it governs mechanical stiffness, swelling resistance, and transport properties.

How it works

Weigh a dry polymer sample to get the dry mass (md).
Immerse the sample in solvent until equilibrium swelling is reached, then weigh the swollen sample (ms).
Enter the density of the polymer (ρp) and the solvent (ρs) in g/cm³.
Enter the molar volume of the solvent (Vs) in cm³/mol — this is molecular weight divided by density.
Enter the Flory-Huggins interaction parameter (χ) for your polymer-solvent pair.
The calculator computes the polymer volume fraction vr and applies the Flory-Rehner equation to yield Ve in mol/cm³.

Use cases

Characterizing crosslink density in vulcanized rubber.
Evaluating network uniformity in epoxy and polyurethane thermosets.
Measuring gel fraction and network density in hydrogels for biomedical applications.
Quality control of crosslinked coatings and adhesives.
Comparing the effect of crosslinker concentration on final network structure.
Academic polymer labs — replicating swelling experiments from published protocols.

Frequently Asked Questions

Last updated: 2026-09-30 · Reviewed by Nham Vu