Crosslink Density

Two routes to the same number

A cured network has one number that matters more than any other: the molar mass between crosslinks, M_c. It sets the modulus, the swelling, the tear energy and how far the material can stretch before it breaks. There are two ordinary ways to measure it, and they are worth doing both, because where they disagree is informative rather than annoying.

Swelling works because a crosslinked polymer in a good solvent expands until two things balance: the free energy gained by mixing with solvent, and the entropy lost by stretching the network strands. Flory and Rehner wrote that balance down. It needs no instrument beyond a balance and a fume hood, which is why it is still the standard bench measurement on a cured binder.

Modulus works because rubber elasticity is entropic: the stiffness of a network above its T_g comes from strands resisting being straightened, so the plateau modulus counts strands directly.

From equilibrium swelling

From the rubbery plateau

Read the modulus well above T_g, where the curve has flattened. Taking it on the shoulder of the transition is the commonest way to get a number several times too high.

Comparing the two

A worked example

A cured PDMS network, swollen to equilibrium in toluene. The solvent's molar volume is 106.3 cm³/mol, χ for PDMS–toluene is about 0.465, and the dry polymer density is 0.97 g/cm³. Suppose the swollen sample settles at a polymer volume fraction v₂ = 0.25 – it has taken up three times its own volume in solvent.

Putting the same M_c through the modulus expression predicts what a DMA should read on the rubbery plateau: G = ρRT÷M_c = 0.40 MPa, so a tensile modulus near 1.2 MPa. That is an ordinary value for a soft silicone elastomer, which is the first sanity check – if the two routes predict something wildly outside the range the material obviously occupies, one of the inputs is wrong before either result is interesting.

How much the χ you chose is worth

The swelling route has one input nobody measures and everybody looks up, and it dominates the answer. Repeating the calculation above with χ moved by ±0.05 – well inside the spread between literature sources for a single polymer–solvent pair:

A swing of 0.1 in χ moves M_c by more than a factor of two. The sensitivity is worst exactly where swelling experiments are usually run – as χ approaches 0.5 the solvent becomes marginal, the bracketed term heads toward zero, and small errors are amplified without limit. Two practical consequences: quote M_c from swelling as an order-of-magnitude figure unless you have fitted χ for your own system, and prefer a genuinely good solvent (low χ) where the calculation is far better conditioned. Comparing two cures measured the same way in the same solvent is reliable; comparing an absolute M_c against someone else's is not.

When the two routes disagree

They usually do, and the direction of the disagreement is diagnostic rather than a sign that one measurement failed:

• Modulus gives a lower M_c than swelling. The commonest result, and usually real: trapped entanglements act as crosslinks mechanically, so the modulus counts them as strands, while swelling lets some of them slip. The gap is a rough measure of how much of the network's stiffness is topological rather than chemical.
• Swelling gives a much lower M_c than modulus. Suspect incomplete extraction. Soluble chains and unreacted resin left in the sample count as polymer in v₂ but carry no load, and they inflate the apparent crosslink density. Extract to constant mass before swelling, and dry to constant mass afterwards.
• Both are high, and the material is soft. Dangling chain ends are the usual explanation. A strand attached at only one end contributes mass but no elastic restoring force, so the effective network is looser than the stoichiometry suggests. This is most pronounced at low conversion, or where one component was in excess – and it is the reason the calculated M_c from a recipe is an upper bound on network quality, not a prediction. Network Formation computes that recipe figure with dangling ends already removed – a junction only counts if three of its arms reach the network – which leaves intramolecular loops as the main thing it cannot see.
• The modulus number is several times too small. Check where the plateau was read. Taking the modulus on the shoulder of the glass transition rather than well above it is the single commonest error in this measurement, and it goes the other way – it makes the network look far more tightly crosslinked than it is. See thermal analysis for placing the plateau on a DMA trace.

Solvent reference

Molar volume is the solvent's molar mass divided by its density, and is well defined. χ is not: it depends on the polymer, on temperature, and mildly on concentration, and a value quoted without naming the polymer is close to meaningless. The figures below are molar volumes and densities only — take χ from a source for your specific polymer–solvent pair, or fit it from swelling in two solvents.

What to distrust

χ dominates the swelling answer. It enters as χv₂² against a logarithm, and moving it from 0.34 to 0.45 can change ν by a factor of two without anything about the sample changing. If you only have a guess at χ, treat the swelling result as an order of magnitude.

Both routes count only elastically effective strands. Dangling ends, loops that start and finish at the same junction, and unattached sol fraction all carry no load and hold no network. That is why extracting the sol fraction before weighing matters, and why a lightly cured sample reads worse than its chemistry suggests.

The equation shown is the affine model. It assumes junctions move exactly as the bulk does. The phantom model, in which junctions fluctuate, replaces the v₂⁄2 term with 2v₂⁄f and gives a lower ν for the same swelling — typically by a factor of (1 − 2/f). For a tetrafunctional network that is a factor of two, so quote which model you used.

Related tools

Before the network forms, Step-Growth & Gel Point predicts when it will, and Network Formation extends that to mixed functionalities and predicts the M_c to compare against. Flory–Huggins covers what χ actually is, and why 0.5 is where a solvent turns marginal. Thermal Analysis is where the plateau modulus is measured, and Chain Dimensions covers the uncrosslinked chain in solution.