KEVOS® MAKEOS · Quality
Rubber Swelling Test: Crosslink Density, Chain Molar Mass and Network Modulus (Flory–Rehner)
From the masses of a rubber specimen before swelling, swollen in a solvent and dried again, what is its crosslink density, and is the cure within my acceptance range?
Equilibrium swelling test of a vulcanised rubber or gel: the rubber volume fraction in the swollen specimen from the initial, swollen and dried masses, the rubber share of the compound and the densities, the volume and mass change and the soluble fraction, the crosslink density from the Flory–Rehner equation for a network of the stated functionality with the polymer–liquid interaction parameter you supply (and its sensitivity to that parameter), the molar mass between crosslinks, and the network shear and Young’s moduli this crosslink density implies, against your acceptance range.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Weighings | ||
| Initial mass of the specimen | g | Mass of the dry specimen before immersion, weighed on an analytical balance. |
| Swollen mass at equilibrium | g | Mass of the specimen swollen to equilibrium, weighed quickly after blotting off surface liquid. |
| Dried mass after swelling | g | Mass of the specimen after the liquid has been driven off to constant mass. |
| Compound and liquid | ||
| Density of the compound | g/cm³ | Density of the compound, measured by hydrostatic weighing or from the compound sheet. |
| Rubber share of the dried compound mass | % | Share of rubber polymer in the dried compound by mass, from the recipe (rubber ÷ total parts) after allowing for extracted oils. |
| Density of the rubber polymer | g/cm³ | Density of the rubber polymer itself, from the polymer supplier. |
| Density of the swelling liquid | g/cm³ | Density of the swelling liquid at the test temperature. |
| Molar volume of the swelling liquid | cm³/mol | Molar volume of the swelling liquid: molar mass ÷ density. |
| Rubber–liquid interaction parameter χ | Interaction parameter of this rubber with this liquid, from the literature for the pair or from a calibration against a known network. | |
| Crosslink functionality | Number of chains that meet at each crosslink: 4 for usual vulcanisation, 3 for some networks made by end-linking. | |
| Test temperature | °C | Temperature at which the specimen was swollen. |
| Acceptance range | ||
| Lowest acceptable crosslink density | mol/m³ | Lowest crosslink density your specification or control chart accepts. |
| Highest acceptable crosslink density | mol/m³ | Highest crosslink density your specification or control chart accepts. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Weighings
What this means: Initial mass of the specimen
- What it is
- The mass of the dry specimen before it is put into the swelling liquid, the reference for the mass and volume change and for the soluble fraction removed.
- Units and format
- Enter a number in g: grams. The form accepts 0.001 g to 10000 g. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 g as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the volume and mass change and the soluble fraction; the crosslink density uses the dried mass instead.
- Get it right
- Condition the specimen and weigh it to at least 1 mg; small specimens magnify weighing errors.
What this means: Swollen mass at equilibrium
- What it is
- The mass of the specimen when it has stopped taking up liquid, including the liquid absorbed into the network; the absorbed liquid volume comes from it.
- Units and format
- Enter a number in g: grams. The form accepts 0.001 g to 100000 g. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6.2 g as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More absorbed liquid lowers the rubber volume fraction and the crosslink density.
- Get it right
- Blot off surface liquid and weigh within seconds in a closed bottle, because volatile solvents evaporate fast.
What this means: Dried mass after swelling
- What it is
- The mass of the specimen after the liquid has been removed to constant mass, which excludes the soluble material the liquid extracted; the rubber volume comes from it.
- Units and format
- Enter a number in g: grams. The form accepts 0.001 g to 10000 g. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.95 g as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher dried mass raises the rubber volume fraction and the crosslink density.
- Get it right
- Dry in an oven or vacuum to constant mass; residual liquid makes the network look tighter than it is.
Compound and liquid
What this means: Density of the compound
- What it is
- The density of the whole compound, used only to turn the initial and dried masses into specimen volumes for the volume change.
- Units and format
- Enter a number in g/cm³: grams per cubic centimetre (1 g/cm³ = 1000 kg/m³). The form accepts 0.5 g/cm³ to 5 g/cm³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.12 g/cm³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the reported volume change only, not the crosslink density.
- Get it right
- Measure it by weighing in air and in water rather than taking a nominal value.
What this means: Rubber share of the dried compound mass
- What it is
- The share of the dried compound’s mass that is rubber polymer, so that only the rubber, not filler or other insoluble ingredients, counts in the rubber volume fraction.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 1% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A lower rubber share lowers the rubber volume fraction and so the crosslink density.
- Get it right
- Work it out from the recipe in parts per hundred rubber, after deducting oils and other ingredients the liquid extracts.
What this means: Density of the rubber polymer
- What it is
- The density of the rubber polymer alone, which turns the rubber’s mass into its volume and the crosslink density into a molar mass between crosslinks.
- Units and format
- Enter a number in g/cm³: grams per cubic centimetre (1 g/cm³ = 1000 kg/m³). The form accepts 0.5 g/cm³ to 3 g/cm³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.92 g/cm³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the polymer’s density, not the filled compound’s, which is higher.
What this means: Density of the swelling liquid
- What it is
- The density of the swelling liquid, which turns the mass of absorbed liquid into its volume inside the swollen network.
- Units and format
- Enter a number in g/cm³: grams per cubic centimetre (1 g/cm³ = 1000 kg/m³). The form accepts 0.3 g/cm³ to 3 g/cm³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.867 g/cm³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the density at the test temperature.
What this means: Molar volume of the swelling liquid
- What it is
- The molar volume of the swelling liquid, its molar mass divided by its density, which appears in the Flory–Rehner equation as the size of one liquid molecule.
- Units and format
- Enter a number in cm³/mol: cm³/mol. The form accepts 10 cm³/mol to 1000 cm³/mol. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 106.3 cm³/mol as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The crosslink density is inversely proportional to the molar volume.
- Get it right
- Compute it at the test temperature from the same density used above.
What this means: Rubber–liquid interaction parameter χ
- What it is
- The Flory–Huggins interaction parameter of this rubber with this liquid, which measures how much the liquid likes the polymer and sets how far the network would swell with no crosslinks at all.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 1.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.39 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher χ lowers the crosslink density calculated from the same swelling, strongly so when the swelling is large.
- Get it right
- Use a published value for the exact polymer and liquid at the test temperature, and keep the same value for every batch you compare.
What this means: Crosslink functionality
- What it is
- The number of network chains that meet at one crosslink: four where two chains are joined at a point, as in most vulcanisation.
- Units and format
- Enter a plain number with no unit. The form accepts 3 to 12. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher functionality reduces the 2v₂ ÷ f term and lowers the crosslink density slightly.
- Get it right
- Use 4 unless the network chemistry clearly gives another number.
What this means: Test temperature
- What it is
- The temperature at which the specimen was swollen, used to work out the network shear modulus RTν from the crosslink density.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -50 °C to 200 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 23 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the modulus in proportion to absolute temperature; the crosslink density itself is unchanged.
- Get it right
- Swell in a temperature-controlled room or bath; χ and the swelling both change with temperature.
Acceptance range
What this means: Lowest acceptable crosslink density
- What it is
- The lowest crosslink density your specification or control chart accepts for this compound and test method.
- Units and format
- Enter a number in mol/m³: mol/m³. The form accepts 0 mol/m³ to 100000 mol/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mol/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the lower line of the acceptance range.
- Get it right
- Set the range from tests of good batches with the same liquid, χ and procedure.
What this means: Highest acceptable crosslink density
- What it is
- The highest crosslink density your specification or control chart accepts for this compound and test method.
- Units and format
- Enter a number in mol/m³: mol/m³. The form accepts 0.001 mol/m³ to 100000 mol/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 mol/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the upper line of the acceptance range.
- Get it right
- A value above the range can mean over-cure or excess filler interaction as well as a recipe error.
Method and formulas
- Rubber volume Vr = md × wR ÷ ρR, absorbed liquid Vs = (ms − md) ÷ ρs, v₂ = Vr ÷ (Vr + Vs).
- Flory–Rehner (Klenin Eq. 3.5-65, liquid in equilibrium with the network): ν = −[ln(1 − v₂) + v₂ + χv₂²] ÷ [V₁(v₂^⅓ − 2v₂ ÷ f)].
- Mc = ρR ÷ ν (Eq. 3.5-67); shear modulus G = ν R T (Eq. 3.5-50) and Young’s modulus E = 3G (Eq. 3.5-58) for the dry network.
- Example: 2.000 g swells to 6.200 g and dries to 1.950 g, 60 % rubber of density 0.92 in a liquid of 0.867 g/cm³ and 106.3 cm³/mol: v₂ = 0.2060; with χ = 0.39 and f = 4, ν = 156.7 mol/m³, Mc = 5,872 g/mol and G = 0.386 MPa at 23 °C: in range.
Limitations
- Affine Flory–Rehner network: other network models (phantom network, Mooney–Rivlin corrections) give different absolute values, so compare results only within one method.
- No Kraus correction for reinforcing filler, which restricts swelling and raises the apparent crosslink density.
- The result is very sensitive to χ, and χ itself varies with concentration; entanglements count as crosslinks.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check the state of cure of a rubber batch against an acceptance range
- Compare cure systems, cure times or ageing in the laboratory
- Estimate the network modulus of a gel or rubber from a swelling test
- Investigate under-cure or reversion complaints
Who it is for
- Rubber technologists and compounders
- Quality and laboratory staff
- Seal and moulded-rubber engineers
What you get
- Rubber volume fraction v₂ and swelling ratio
- Volume change, mass change and soluble fraction
- Crosslink density and its sensitivity to χ
- Molar mass between crosslinks
- Network shear and Young’s moduli
- In or out of the acceptance range
Interpreting the result
- Treat the crosslink density as a comparative quality measure: a batch that swells more than usual has fewer effective crosslinks or more degraded network.
- The modulus shown is the network’s contribution; filled compounds are stiffer than it.
Before relying on the result
- Weigh at intervals until the swollen mass is steady before recording it.
- Run a reference compound of known cure with every batch of tests.
References
- Flory P. J. and Rehner J., J. Chem. Phys. 11, 521 (1943): statistical mechanics of crosslinked polymer networks, swelling.
- Klenin V. J., Thermodynamics of Systems Containing Flexible-Chain Polymers (Elsevier, 1999; ISBN 0-444-82373-5), §3.5.1–3.5.2: network shear modulus (Eq. 3.5-50), E = 3μ (Eq. 3.5-58), chemical potential of the liquid in a swollen network (Eqs. 3.5-64 and 3.5-65) and chains per volume (Eq. 3.5-67). Method reference; no tables or figures are reproduced.
- Kraus G., J. Appl. Polym. Sci. 7, 861 (1963): swelling of filler-reinforced vulcanizates.
- ASTM D6814 (crosslink density of crumb rubber by swelling) and ISO 1817 (effect of liquids on rubber): test practice. Consult the current editions.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Explains the concept · KEVOS® original book
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