T2 · ThermodynamicsRubber & Elastomer TechnologyGATE 2027 Syllabus (XE5)

Flory-Rehner Vulcanized Rubber Crosslink Density

Calculates network crosslink density (nue in mol/cm³) of vulcanized rubber from equilibrium solvent swelling measurement.

Mathematical Canonical Relation

\nu_e = - \frac{\ln(1 - v_2) + v_2 + \chi \cdot v_2^2}{V_1 \cdot \left( v_2^{1/3} - \frac{v_2}{2} \right)}

Plain Formula: nu_e = -(ln(1 - v2) + v2 + chi*v2^2) / (V1 * (v2^(1/3) - v2/2))

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Rubber Vol Fraction (v₂) (fraction)

Range: 0.05 – 0.6 fraction

Solvent Molar Vol (V₁) (cm³/mol)

Range: 50 – 200 cm³/mol

Flory-Huggins Parameter (χ) (fraction)

chi

Range: 0.1 – 0.9 fraction

Calculated Engineering Result:

1.83 × 10⁻⁴ mol/cm³

Engineering Verification:Adequate crosslinking density for tire treads and automotive elastomeric mounts.

When & Where to Apply

Use when evaluating sulfur/peroxide vulcanization state in tire treads or industrial seals via equilibrium swelling.

Common Engineering Pitfalls

Using un-swollen polymer volume fraction instead of swollen gel fraction v2

Physical Assumptions & Validity Limits

Affine network deformation model
Equilibrium swelling in solvent
Isotropic gel expansion

Variable Symbols, SI Units & Dimensional Specifications

Symbol	Variable Meaning	SI Unit	Typical Industrial Range
\nu_e	Crosslink density	mol/cm³	—
v_2	Volume fraction of rubber in swollen gel	fraction	0.22
V_1	Solvent molar volume (Toluene = 106.3)	cm³/mol	106.3
\chi	Flory-Huggins polymer-solvent parameter	dimensionless	0.39

Worked Industrial Numerical Example: Natural Rubber Tire Tread Vulcanization Density

Problem Statement:

A sulfur-cured Natural Rubber sample swollen in toluene at equilibrium gives v2 = 0.22. Given toluene V1 = 106.3 cm³/mol and NR/toluene interaction chi = 0.39, calculate crosslink density nue.

Step-by-Step Numerical Solution:

Calculate numerator: -[ln(1 - 0.22) + 0.22 + 0.39 * (0.22)^2] = -[-0.24846 + 0.22 + 0.018876] = 0.009584
Calculate denominator: 106.3 * ((0.22)^(1/3) - 0.22/2) = 106.3 * (0.60368 - 0.11) = 52.478
Divide: nue = 0.009584 / 52.478 = 1.826 x 10^-4 mol/cm³

Final Answer: nue = 1.83 x 10⁻⁴ mol/cm³

💡 Engineering Interpretation: Crosslink density of 1.83x10^-4 mol/cm³ confirms adequate sulfur crosslinking for commercial passenger tire treads.

Authoritative Technical Reference & Source Provenance

Statistical Mechanics of Cross-Linked Polymer Networks by P. J. Flory & J. Rehner (J. Chem. Phys., 1943), 11, Page 521.

Connected Curriculum Lesson

Vulcanization Kinetics & Crosslink Density Measurement

Study Full Lesson