# Rubber elasticity **Rubber elasticity** is the ability of a lightly crosslinked [[Polymer|polymer]] network to be stretched to several times its length and to return, almost completely, when released. It is the mechanical opposite of the elasticity of a metal. In a metal the restoring force comes from stretched interatomic bonds and the energy of the strained state is stored as [[Enthalpy|internal energy]]; in an [[Elastomer|elastomer]] the bonds are barely disturbed, and the restoring force comes from [[Entropy|entropy]] — a stretched network has fewer available chain conformations than a relaxed one, and the chains pull back for the same statistical reason that a gas resists compression. In the microsim below the reader stretches a crosslinked network to a ratio *λ* between 1 and 4 and reads the nominal stress from the Gaussian network result `sigma = n·k·T·(lambda - lambda^(-2))`, where *n* is the number of network chains per unit volume set by the crosslink density, *k* is the [[Boltzmann_constant|Boltzmann constant]] and *T* is the absolute [[Temperature|temperature]]. A second control raises *T*, and the entropy spring gets *stiffer* — the stress at fixed *λ* rises in proportion to *T* — while a metal spring drawn beside it softens slightly. That inversion is the Gough–Joule effect, and it is the cleanest evidence that the force is entropic. On the [[Materials_science|Materials science]] flagship this article serves the *Rubber elasticity* section of Part VIII, Industry, sharing its chain model with [[Viscoelasticity|viscoelasticity]]: the same chains, now tied together, so that the network reaches an equilibrium instead of flowing. ## History Natural rubber reached Europe as a curiosity and became an engineering material only when it could be stopped from softening in summer and cracking in winter. [[Vulcanization|Vulcanization]] — heating [[Natural_rubber|natural rubber]] with [[Sulfur|sulfur]] so that sulfur bridges tie neighbouring chains into a permanent network — was patented in the United States by Charles Goodyear in 1844, and the crosslink density set by that reaction is exactly the *n* in the sim's equation.[^goodyear1844][^callister-elast] The physics was discovered earlier and for a long time not believed. In 1805 John Gough reported that a strip of rubber stretched quickly becomes warm, and, more startlingly, that a strip held under a constant load *contracts* when it is heated.[^gough1805][^treloar-thermo] [[James_Prescott_Joule|James Prescott Joule]] repeated and extended the measurements in 1859 with the calorimetric care the subject needed, and put them in a thermodynamic frame.[^joule1859][^treloar-thermo] Both observations are inexplicable for a bond-energy solid, which expands on heating and softens as it expands. The explanation had to wait for the statistical treatment of chain molecules in the 1930s, when the configurational entropy of a long flexible chain was first computed and the network theory built on it. L. R. G. Treloar's *The Physics of Rubber Elasticity* assembled the experiments and the theory into the standard account, and Paul Flory's *Principles of Polymer Chemistry* set the network statistics in their chemical context.[^treloar-network][^flory1953] The form `sigma = n·k·T·(lambda - lambda^(-2))` dates from that period and is still the first thing quoted about a rubber. ## Molecular-level models The classical network model makes three assumptions. Each chain between crosslinks is long enough that its end-to-end vector obeys Gaussian statistics; the network deforms affinely, so that every crosslink moves with the bulk deformation; and the material is incompressible, so a stretch *λ* along one axis is matched by contractions of `lambda^(-1/2)` across the other two.[^treloar-network] Under those assumptions the free energy of the network is purely entropic, and differentiating it with respect to the stretch gives the nominal stress `sigma = n·k·T·(lambda - lambda^(-2))`, with `G = n·k·T` the shear modulus. For small strain the bracket reduces to `3·eps`, so [[Young's_modulus|Young's modulus]] is `E = 3·G`. The sim's default preset is a lightly crosslinked rubber of density 950 kg/m³ whose average molar mass between crosslinks is 6 kg/mol, giving `n = 9.5 × 10²⁵` chains per cubic metre. At 300 K that is a shear modulus of 0.395 MPa and a Young's modulus of 1.19 MPa (derived). Running the stretch slider then gives 0.42 MPa at λ = 1.5, 0.69 MPa at λ = 2, 1.14 MPa at λ = 3 and 1.56 MPa at λ = 4 (derived). Two things are worth reading off those numbers. The modulus is about 1.7 × 10⁵ times smaller than that of [[Steel|steel]] (derived), which is why a rubber band stretches to four times its length under a load a steel wire would not notice; and the curve is concave, flattening as λ grows, because `lambda^(-2)` dies away and the model becomes nearly linear in λ. Real rubber does the opposite above about λ = 4, turning sharply upward as the chains approach full extension, so the Gaussian result is ILLUSTRATIVE outside the sim's range. ### The Molecular Kink Paradigm for rubber elasticity The Gaussian network treats a chain as a featureless random walk whose only property is the number of freely orienting units in it. A kink-based picture instead takes the chain's stored length to sit in discrete, localized conformational features — kinks — so that extension is the progressive straightening of those features rather than the biasing of a structureless walk.[citation needed] The practical difference appears at large stretch. A random walk has no upper limit built into its Gaussian approximation, while a chain of a finite number of segments does, and once most of the kinks are pulled out the force rises steeply toward an asymptote at the fully extended length. That finite-extensibility upturn is real and is treated in the standard theory by replacing Gaussian statistics with the exact inverse Langevin function for a freely jointed chain, which reproduces both the low-strain Gaussian limit and the high-strain divergence.[^treloar-nongauss] In filled rubbers the upturn is reinforced by strain-induced crystallization, in which stretched natural-rubber chains align well enough to crystallize and stiffen the material further. The sim stops at λ = 4 for this reason: below it the Gaussian form is a fair description, and above it the article would be quoting a model outside its range. ## Experiments Three measurements decide between an energy spring and an entropy spring, and rubber fails the energy test in all three. The stress at fixed stretch rises with temperature rather than falling. Stretching a strip adiabatically warms it, and letting it retract cools it, which anyone can feel by stretching a wide rubber band against the lip and then releasing it. And the measured [[Gibbs_free_energy|internal-energy]] contribution to the retractive force is small — most of the force is `-T·(dS/dL)`, the entropy term.[^treloar-thermo] Each of the three is a demanding experiment in its own right. The stress-against-temperature measurement has to run at fixed length and wait for equilibrium, because a rubber that has just been stretched is still relaxing; the [[Isothermal_process|isothermal]] and [[Adiabatic_process|adiabatic]] responses differ by the amount of that heat; and the strip must be cycled several times before recording, since an unworked rubber softens over its first few extensions. The classical results reported below are equilibrium values taken after those precautions, and the sim reproduces their trend rather than their absolute numbers.[^treloar-thermo] ### Variation of tensile stress with temperature The Gough–Joule effect is the sim's temperature control. Because `sigma` is proportional to *T* at fixed *λ*, heating the sim's preset from 300 K to 360 K raises the stress at λ = 3 from 1.14 MPa to 1.37 MPa, a rise of exactly 20 %, the ratio of the absolute temperatures (derived). Read the other way round: a strip carrying a fixed weight must shorten when heated, until the increased `n·k·T` again balances the load. That is Gough's 1805 observation, and it is the experiment the sim animates beside a metal spring, whose modulus falls by a few percent over the same 60 K.[^gough1805][^treloar-thermo] The comparison is not quite a pure entropy measurement. Heating also expands the rubber, lowering *n*, and the strip must be held at fixed length rather than fixed load for the proportionality to be clean; careful work therefore corrects the raw data for thermal expansion before quoting the entropic fraction.[^treloar-thermo] The sim omits that correction and says so, so its temperature slider is ILLUSTRATIVE of the sign and the order of the effect rather than a reproduction of a calorimetric result. Thermodynamically the sign follows from the fundamental relation `dE = T·dS - P·dV`, which defines temperature as `T = (dE/dS)_V`.[^likharev-ch1] A system whose restoring force comes from entropy has a force proportional to *T*, exactly as the pressure of an ideal gas does; a rubber band is, in this one respect, a one-dimensional gas. ### Snap-back velocity A released rubber band does not recover instantly. The recovery travels as a tensile wave along the strip, and the wave speed is set by the modulus and the density through `c = sqrt(E/rho)`. For the sim's preset — E = 1.19 MPa, ρ = 950 kg/m³ — that is about 35 m/s (derived), against about 5,000 m/s for a steel wire (derived). The five-order-of-magnitude gap in modulus becomes a two-order-of-magnitude gap in wave speed, because the speed depends on the square root. Two corrections matter in practice. The strip is highly stretched when released, so the relevant modulus is the tangent modulus at that stretch, not the small-strain value, and the recovery is faster than the naive estimate. And the retraction is not reversible: some energy is lost as [[Heat|heat]] through the material's [[Viscoelasticity|viscoelastic]] dissipation, which is why a released band does not oscillate for long, and why repeated fast cycling warms it.[^treloar-network] ## Historical approaches to elasticity theory Two traditions meet in rubber. Continuum elasticity describes a solid by moduli measured on it and asks nothing about its constituents; [[Statistical_mechanics|statistical mechanics]] builds the moduli from a model of the constituents. Rubber was the first solid for which the second route gave a modulus with no adjustable constant other than a chain count, which is why it occupies so much space in statistical-mechanics texts relative to its industrial share. The order in which the two traditions arrived matters for how the subject reads. The thermodynamic account came first and is model-free: it says only that the force is mostly entropic, without saying what the entropy belongs to. The chain account came second and identified the entropy with the conformations of the molecule, turning a measured modulus into a count of network chains. Both are set out below in that order, because the second inherits the first's bookkeeping unchanged. ### Thermodynamics The thermodynamic argument does not need a chain model. For a strip held at temperature *T* and stretched by `dL`, the retractive force splits into `f = (dU/dL)_T - T·(dS/dL)_T`, an internal-energy term and an entropy term.[^treloar-thermo] Measuring `f` at several temperatures and fixed length separates them, since the second term is linear in *T* and the first is nearly independent of it: the slope of `f` against *T* gives `-(dS/dL)_T` and the intercept gives `(dU/dL)_T`. For a well-behaved rubber the intercept is small and the slope carries almost all the force, which is the quantitative version of "the force is entropic". For a metal the split is the other way round. Statistical mechanics then supplies the missing piece, the entropy itself, through the relation between entropy and the number of accessible microstates and the fundamental relation `dE = T·dS - P·dV`.[^likharev-ch1][^likharev-ch2] Because stretching a chain reduces the number of conformations compatible with its end-to-end distance, `(dS/dL)` is negative and the force is a restoring one. ### Polymer chain theories The chain model is the [[Ideal_chain|freely jointed chain]]: *N* rigid links of length *b*, each free to point anywhere. Its end-to-end vector performs a [[Random_walk|random walk]], so for `N` large the probability of a given end-to-end distance *R* is Gaussian with a mean square `N·b²`. The entropy of that distribution falls as `R²`, so the free energy rises as `R²` and the chain behaves as a linear spring of stiffness `3·k·T/(N·b²)` — a spring whose stiffness is proportional to temperature and inversely proportional to chain length.[^treloar-network][^flory1953] Assembling such chains into a network, assuming affine deformation and summing over the chains, gives the `n·k·T·(lambda - lambda^(-2))` result directly; the chain length has cancelled, leaving only the number density of chains. Refinements relax the assumptions one at a time. The phantom-network model lets crosslinks fluctuate instead of moving affinely and lowers the modulus by a factor that depends on crosslink functionality. Entanglement models add trapped topological constraints that act as extra crosslinks. Phenomenological strain-energy functions — the Mooney–Rivlin form and its successors — fit the measured stress with two or more constants where the Gaussian theory offers one, and are what [[Finite_element_method|finite-element]] codes use for [[Hydrogel|hydrogels]] and elastomer parts.[^mooney1940][^rivlin1948] None of them changes the central result that the modulus is `n·k·T` to within a factor of order one. ## See also - [[Elastomer]] - [[Vulcanization]] - [[Natural_rubber]] - [[Hydrogel]] - [[Viscoelasticity]] - [[Polymer]] - [[Ideal_chain]] - [[Entropy]] ## References [^gough1805]: Gough, John (1805). "A description of a property of caoutchouc, or Indian rubber." *Memoirs of the Literary and Philosophical Society of Manchester*. (Volume, pages and any DOI to pin against the society's record; the observations are also reported in Treloar's historical chapter.) [^joule1859]: Joule, J. P. (1859). "On some thermo-dynamic properties of solids." *Philosophical Transactions of the Royal Society of London* 149. (Pages and DOI to pin against the journal record.) [^goodyear1844]: Goodyear, Charles. "Improvement in India-Rubber Fabrics." U.S. Patent 3,633 (1844). (Patent number, filing and grant dates to pin against the USPTO record; no URL asserted here.) [^treloar-thermo]: Treloar, L. R. G. (1975). *The Physics of Rubber Elasticity*, 3rd ed. Oxford: Clarendon Press. Chapters on the thermodynamic analysis of rubber elasticity: the Gough–Joule effect, the separation of the retractive force into internal-energy and entropy terms, and the thermal-expansion correction. Not a Portal Book; pages to pin. [^treloar-network]: Treloar, L. R. G. (1975). *The Physics of Rubber Elasticity*, 3rd ed. Chapters on the statistical theory of the network: Gaussian chain statistics, the affine and incompressibility assumptions, and the derivation of the nominal stress in simple extension. Not a Portal Book; pages to pin. [^treloar-nongauss]: Treloar, L. R. G. (1975). *The Physics of Rubber Elasticity*, 3rd ed. Chapter on non-Gaussian chain statistics: the inverse Langevin function, finite extensibility and the upturn of the stress–strain curve at large extension. Not a Portal Book; pages to pin. [^flory1953]: Flory, P. J. (1953). *Principles of Polymer Chemistry*. Ithaca, New York: Cornell University Press. Chapters on the configurational statistics of polymer chains and on network structure and rubber elasticity. Not a Portal Book; pages to pin. [^mooney1940]: Mooney, M. (1940). "A theory of large elastic deformation." *Journal of Applied Physics* 11. (Pages and DOI to pin against the journal record.) [^rivlin1948]: Rivlin, R. S. (1948). "Large elastic deformations of isotropic materials." *Philosophical Transactions of the Royal Society A* 240. (Part, pages and DOI to pin against the journal record.) [^callister-elast]: Callister, W. D.; Rethwisch, D. G. (2010). *Materials Science and Engineering: An Introduction*, 8th ed. Hoboken: Wiley. Chapter 15, Characteristics, Applications and Processing of Polymers — elastomers, the vulcanization reaction and the effect of crosslink density on modulus. Not a Portal Book; page to pin. [^likharev-ch1]: Likharev, K. K. (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics* (Portal Book 075). Chapter 1, Review of Thermodynamics, pp. 5–28; the fundamental relation `dE = T·dS − P·dV` and the definition `T = (dE/dS)_V` at pp. 10–12. https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^likharev-ch2]: Likharev, K. K. (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics* (Portal Book 075). Chapter 2, pp. 29–72; the statistical definition of entropy and its relation to the number of accessible microstates (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics ### Bibliography - Likharev, K. K. (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Portal Book 075. https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics - Treloar, L. R. G. (1975). *The Physics of Rubber Elasticity*, 3rd ed. Clarendon Press. Not a Portal Book. - Flory, P. J. (1953). *Principles of Polymer Chemistry*. Cornell University Press. Not a Portal Book. - Callister, W. D.; Rethwisch, D. G. (2010). *Materials Science and Engineering: An Introduction*, 8th ed. Wiley. 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