Rubber Elasticity
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Rubber elasticity refers to a property of crosslinked rubber: it can be stretched by up to a factor of 10 from its original length and, when released, returns very nearly to its original length. This can be repeated many times with no apparent degradation to the rubber. Rubber is a member of a larger class of materials called elastomers and it is difficult to overestimate their economic and technological importance. Elastomers have played a key role in the development of new technologies in the 20th century and make a substantial contribution to the global economy. Rubber elasticity is produced by several complex molecular processes and its explanation requires a knowledge of advanced mathematics, chemistry and statistical physics, particularly the concept of entropy. Entropy may be thought of as a measure of the thermal energy that is stored in a molecule. Common rubbers, such as polybutadiene and polyisoprene (also called natural rubber), are produced by a process called polymerization. Very long molecules (polymers) are built up sequentially by adding short molecular backbone units through chemical reactions. A rubber polymer follows a random, zigzag path in three dimensions, intermingling with many other rubber molecules. An elastomer is created by the addition of a few percent of a cross linking molecule such as sulfur. When heated, the crosslinking molecule causes a reaction that chemically joins (bonds) two of the rubber molecules together at some point (a crosslink). Because the rubber molecules are so long, each one participates in many crosslinks with many other rubber molecules forming a continuous molecular network. As a rubber band is stretched, some of the network chains are forced to become straight and this causes a decrease in their entropy. It is this decrease in entropy that gives rise to the elastic force in the network chains.


History

Following its introduction to Europe from the New World in the late 15th century,
natural rubber Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, and ...
(
polyisoprene Polyisoprene is strictly speaking a collective name for Polymer, polymers that are produced by polymerization of isoprene. In practice polyisoprene is commonly used to refer to synthetic ''cis''-1,4-polyisoprene, made by the industrial polymerisa ...
) was regarded mostly as a fascinating curiosity. Its most useful application was its ability to erase pencil marks on paper by rubbing, hence its name. One of its most peculiar properties is a slight (but detectable) increase in temperature that occurs when a sample of rubber is stretched. If it is allowed to quickly retract, an equal amount of cooling is observed. This phenomenon caught the attention of the English physicist John Gough. In 1805 he published some qualitative observations on this characteristic as well as how the required stretching force increased with temperature.
By the mid nineteenth century, the theory of
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
was being developed and within this framework, the English mathematician and physicist
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
showed that the change in mechanical energy required to stretch a rubber sample should be proportional to the increase in temperature. Later, this would be associated with a change in
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. The connection to thermodynamics was firmly established in 1859 when the English physicist
James Joule James Prescott Joule (; 24 December 1818 11 October 1889) was an English physicist, mathematician and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work (see energy). T ...
published the first careful measurements of the temperature increase that occurred as a rubber sample was stretched. This work confirmed the theoretical predictions of Lord Kelvin.
In 1838 the American inventor
Charles Goodyear Charles Goodyear (December 29, 1800 – July 1, 1860) was an American self-taught chemist and manufacturing engineer who developed vulcanized rubber, for which he received patent number 3633 from the United States Patent Office on June 15, 1844. ...
found that natural rubber's elastic properties could be immensely improved by adding a few percent sulphur. The short sulfur chains produced chemical cross-links between adjacent
polyisoprene Polyisoprene is strictly speaking a collective name for Polymer, polymers that are produced by polymerization of isoprene. In practice polyisoprene is commonly used to refer to synthetic ''cis''-1,4-polyisoprene, made by the industrial polymerisa ...
molecules. Before it is cross-linked, the liquid natural rubber consists of very long polymer molecules, containing thousands of
isoprene Isoprene, or 2-methyl-1,3-butadiene, is a common volatile organic compound with the formula CH2=C(CH3)−CH=CH2. In its pure form it is a colorless volatile liquid. Isoprene is an unsaturated hydrocarbon. It is produced by many plants and animals ...
backbone units, connected head-to-tail (commonly referred to as chains). Every chain follows a random, three dimensional path through the polymer liquid and is in contact with thousands of other nearby chains. When heated to about 150C, reactive cross-linker molecules, such as sulfur or dicumyl peroxide, can decompose and the subsequent chemical reactions produce a
chemical bond A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of ...
between adjacent chains. A crosslink can be visualized as the letter 'X' but with some of its arms pointing out of the plane. The result is a three dimensional molecular network. All of the polyisoprene molecules are connected together at multiple points by these chemical bonds (network nodes) resulting in a single giant molecule and all information about the original long polymers is lost. A rubber band is a single molecule, as is a latex glove! The sections of polyisoprene between two adjacent cross-links are called network chains and can contain up to several hundred isoprene units. In natural rubber, each cross-link produces a network node with four chains emanating from it. It is the network that gives rise to the elastic properties. Because of the enormous economic and technological importance of rubber, predicting how a molecular network responds to mechanical strains has been of enduring interest to scientists and engineers. To understand the elastic properties of rubber, theoretically, it is necessary to know both the physical mechanisms that occur at the molecular level and how the random-walk nature of the
polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
chain defines the network. The physical mechanisms that occur within short sections of the polymer chains produce the elastic forces and the network morphology determines how these forces combine to produce the macroscopic
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
that we observe when a rubber sample is deformed, e.g. subjected to
tensile strain In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
.


Molecular-level models

There are actually several physical mechanisms that produce the elastic forces within the network chains as a rubber sample is stretched. Two of these arise from entropy changes and one is associated with the distortion of the molecular bond angles along the chain backbone. These three mechanisms are immediately apparent when a moderately thick rubber sample is stretched manually. Initially, the rubber feels quite stiff, i.e. the force must be increased at a high rate with respect to the strain. At intermediate strains, the required increase in force is much lower to cause the same amount of stretch. Finally, as the sample approaches the breaking point, its stiffness increases markedly. What the observer is noticing are the changes in the
modulus of elasticity An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
that are due to the different molecular mechanisms. These regions can be seen in Fig. 1, a typical stress vs. strain measurement for natural rubber. The three mechanisms (labelled Ia, Ib and II) predominantly correspond to the regions shown on the plot. The concept of
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
comes to us from the area mathematical physics called
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
which is concerned with the study of large thermal systems, e.g. rubber networks at room temperature. Although the detailed behavior of the constituent chains are random and far too complex to study individually, we can obtain very useful information about their 'average' behavior from a statistical mechanics analysis of a large sample. There are no other examples of how entropy changes can produce a force in our everyday experience. One may regard the entropic forces in polymer chains as arising from the thermal collisions that their constituent atoms experience with the surrounding material. It is this constant jostling that produces a resisting (elastic) force in the chains as they are forced to become straight. While stretching a rubber sample is the most common example of elasticity, it also occurs when rubber is compressed. Compression may be thought of as a two dimensional expansion as when a balloon is inflated. The molecular mechanisms that produce the elastic force are the same for all types of strain. When these elastic force models are combined with the complex morphology of the network, it is not possible to obtain simple analytic formulae to predict the macroscopic stress. It is only via numerical simulations on computers that it is possible to capture the complex interaction between the molecular forces and the network morphology to predict the stress and ultimate failure of a rubber sample as it is strained.


The Molecular Kink Paradigm for rubber elasticityD. E. Hanson and J. L. Barber, Contemporary Physics 56 (3), 319–337 (2015), LAPR-2015-022971

The Molecular Kink Paradigm proceeds from the intuitive notion that molecular chains that make up a
natural rubber Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, and ...
(
polyisoprene Polyisoprene is strictly speaking a collective name for Polymer, polymers that are produced by polymerization of isoprene. In practice polyisoprene is commonly used to refer to synthetic ''cis''-1,4-polyisoprene, made by the industrial polymerisa ...
) network are constrained by surrounding chains to remain within a ‘tube’. Elastic forces produced in a chain, as a result of some applied strain, are propagated along the chain contour within this tube. Fig. 2 shows a representation of a four-carbon isoprene backbone unit with an extra carbon atom at each end to indicate its connections to adjacent units on a chain. It has three single C-C bonds and one double bond. It is principally by rotating about the C-C single bonds that a polyisoprene chain randomly explores its possible conformations. Sections of chain containing between two and three isoprene units have sufficient flexibility that they may be considered statistically de-correlated from one another. That is, there is no directional correlation along the chain for distances greater than this distance, referred to as a
Kuhn length The Kuhn length is a theoretical treatment, developed by Hans Kuhn, in which a real polymer chain is considered as a collection of N Kuhn segments each with a Kuhn length b. Each Kuhn segment can be thought of as if they are freely jointed with ...
. These non-straight regions evoke the concept of ‘kinks’ and are in fact a manifestation of the random-walk nature of the chain. Since a kink is composed of several isoprene units, each having three carbon-carbon single bonds, there are many possible conformations available to a kink, each with a distinct energy and end-to-end distance. Over time scales of seconds to minutes, only these relatively short sections of the chain, i.e. kinks, have sufficient volume to move freely amongst their possible rotational conformations. The thermal interactions tend to keep the kinks in a state of constant flux, as they make transitions between all of their possible rotational conformations. Because the kinks are in thermal equilibrium, the probability that a kink resides in any rotational conformation is given by a
Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability t ...
and we may associate an
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
with its end-to-end distance. The probability distribution for the end-to-end distance of a
Kuhn length The Kuhn length is a theoretical treatment, developed by Hans Kuhn, in which a real polymer chain is considered as a collection of N Kuhn segments each with a Kuhn length b. Each Kuhn segment can be thought of as if they are freely jointed with ...
is approximately
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
and is determined by the Boltzmann probability factors for each state (rotational conformation). As a rubber network is stretched, some kinks are forced into a restricted number of more extended conformations having a greater end-to-end distance and it is the resulting decrease in entropy that produces an elastic force along the chain. There are three distinct molecular mechanisms that produce these forces, two of which arise from changes in entropy that we shall refer to as low chain extension regime, IaD. E. Hanson and R. L. Martin, Journal of Chemical Physics 133, 084903 (084908 pp.) (2010) and moderate chain extension regime, Ib.D. E. Hanson, J. L. Barber and G. Subramanian, Journal of Chemical Physics 139 (2013), LAPR-2014-018991 The third mechanism occurs at high chain extension, as it is extended beyond its initial equilibrium contour length by the distortion of the chemical bonds along its backbone. In this case, the restoring force is spring-like and we shall refer to it as regime II.D. E. Hanson and R. L. Martin, The Journal of Chemical Physics 130, 064903 (2009), LAPR-2009-006764 The three force mechanisms are found to roughly correspond to the three regions observed in tensile stress vs. strain experiments, shown in Fig. 1. The initial morphology of the network, immediately after chemical cross-linking, is governed by two random processes:P. Flory, N. Rabjohn and M. Shaffer, Journal of Polymer Science 4, 435–455 (1949)D. E. Hanson, Journal of Chemical Physics 134, 064906 (064906 pp.) (2011) (1) The probability for a cross-link to occur at any isoprene unit and, (2) the random walk nature of the chain conformation. The end-to-end distance
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
for a fixed chain length, i.e. fixed number of isoprene units, is described by a random walk. It is the joint probability distribution of the network chain lengths and the end-to-end distances between their cross-link nodes that characterizes the network morphology. Because both the molecular physics mechanisms that produce the elastic forces and the complex morphology of the network must be treated simultaneously, simple analytic elasticity models are not possible; an explicit 3-dimensional numerical modelD. E. Hanson, Polymer 45 (3), 1058–1062 (2004)D. E. Hanson, Journal of Chemical Physics 131, 224904 (224905 pp.) (2009)D. E. Hanson and J. L. Barber, Modelling and Simulation in Materials Science and Engineering 21 (2013), LAPR-2013-017962 is required to simulate the effects of strain on a representative volume element of a network.


Low chain extension regime, Ia

The Molecular Kink Paradigm envisions a representative network chain as a series of vectors that follow the chain contour within its tube. Each vector represents the equilibrium end-to-end distance of a kink. The actual 3-dimensional path of the chain is not pertinent, since all elastic forces are assumed to operate along the chain contour. In addition to the chain's contour length, the only other important parameter is its
tortuosity Tortuosity is widely used as a critical parameter to predict transport properties of porous media, such as rocks and soils. But unlike other standard microstructural properties, the concept of tortuosity is vague with multiple definitions and vari ...
, the ratio of its contour length to its end-to-end distance. As the chain is extended, in response to an applied strain, the induced elastic force is assumed to propagate uniformly along its contour. Consider a network chain whose end points (network nodes) are more or less aligned with the tensile strain axis. As the initial strain is applied to the rubber sample, the network nodes at the ends of the chain begin to move apart and all of the kink vectors along the contour are stretched simultaneously. Physically, the applied strain forces the kinks to stretch beyond their
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
end-to-end distances, causing a decrease in their entropy. The increase in free energy associated with this change in entropy, gives rise to a (linear) elastic force that opposes the strain. The force constant for the low strain regime can be estimated by sampling
molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
(MD) trajectories of a kink, i.e. short chains, composed of 2–3 isoprene units, at relevant temperatures, e.g. 300K. By taking many samples of the coordinates over the course of the simulations, the probability distributions of end-to-end distance for a kink can be obtained. Since these distributions (which turn out to be approximately
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
) are directly related to the number of states, we may associate them with the entropy of the kink at any end-to-end distance. By numerically differentiating the probability distribution, the change in entropy, and hence free energy, with respect to the kink end-to-end distance can be found. The force model for this regime is found to be linear and proportional to the temperature divided by the chain tortuosity.


Moderate chain extension regime, Ib

At some point in the low extension regime, i.e. as all of the kinks along the chain are being extended simultaneously, it becomes energetically more favorable to have one kink transition to an extended conformation in order to stretch the chain further. The applied strain can force a single isoprene unit within a kink into an extended conformation, slightly increasing the end-to-end distance of the chain, and the energy required to do this is less than that needed to continue extending all of the kinks simultaneously. Numerous experimentsJ. P. Joule, Phil. Trans. R. Soc. London 149, 91–131 (1859) strongly suggest that stretching a rubber network is accompanied by a decrease in entropy. As shown in Fig. 2, an isoprene unit has three single C-C bonds and there are two or three preferred rotational angles (orientations) about these bonds that have energy minima. Of the18 allowed rotational conformations, only 6 have extended end-to-end distances and forcing the isoprene units in a chain to reside in some subset of the extended states must reduce the number of rotational conformations available for thermal motion. It is this reduction in the number of available states that causes the entropy to decrease. As the chain continues to straighten, all of the isoprene units in the chain are eventually forced into extended conformations and the chain is considered to be ‘taut’. A force constant for chain extension can be estimated from the resulting change in free energy associated with this entropy change. As with regime Ia, the force model for this regime is linear and proportional to the temperature divided by the chain tortuosity.


High chain extension regime, II

When all of the isoprene units in a network chain have been forced to reside in just a few extended rotational conformations, the chain becomes taut. It may be regarded as sensibly straight, except for the zigzag path that the C-C bonds make along the chain contour. However, further extension is still possible by bond distortions, e.g., bond angle increases, bond stretches and
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
rotations. These forces are spring-like and are not associated with entropy changes. A taut chain can be extended by only about 40%. At this point the force along the chain is sufficient to mechanically rupture the C-C covalent bond. This tensile force limit has been calculated via
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
simulations and it is approximately 7 nN, about a factor of a thousand greater than the entropic chain forces at low strain. The angles between adjacent backbone C-C bonds in an isoprene unit vary between about 115–120 degrees and the forces associated with maintaining these angles are quite large, so within each unit, the chain backbone always follows a zigzag path, even at bond rupture. This mechanism accounts for the steep upturn in the elastic stress, observed at high strains (Fig. 1).


Network morphology

Although the network is completely described by only two parameters (the number of network nodes per unit volume and the statistical de-correlation length of the polymer, the
Kuhn length The Kuhn length is a theoretical treatment, developed by Hans Kuhn, in which a real polymer chain is considered as a collection of N Kuhn segments each with a Kuhn length b. Each Kuhn segment can be thought of as if they are freely jointed with ...
), the way in which the chains are connected is actually quite complicated. There is a wide variation in the lengths of the chains and most of them are not connected to the nearest neighbor network node. Both the chain length and its end-to-end distance are described by probability distributions. The term ‘morphology’ refers to this complexity. If the cross-linking agent is thoroughly mixed, there is an equal probability for any isoprene unit to become a network node. For dicumyl peroxide, the cross linking efficiency in natural rubber is unity, but this is not the case for sulfur.D. E. Hanson and J. L. Barber, Phys. Chem. Chem. Phys. 20, 8460 (2018), LAPR-2018-029488 The initial morphology of the network is dictated by two random processes: the probability for a cross-link to occur at any isoprene unit and the Markov random walk nature of a chain conformation. The probability distribution function for how far one end of a chain end can ‘wander’ from the other is generated by a Markov sequence.A. A. Markov, Izv. Peterb. Akad. 4 (1), 61–80 (1907) This conditional probability density function relates the chain length n in units of the Kuhn length b to the end-to-end distance r: The probability that any isoprene unit becomes part of a cross-link node is proportional to the ratio of the concentrations of the cross-linker molecules (e.g., dicumyl-peroxide) to the isoprene units: p_x = 2 \frac \text \text The factor of two comes about because two isoprene units (one from each chain) participate in the cross-link. The
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
for finding a chain containing N isoprene units is given by: where N\geq 1. The equation can be understood as simply the probability that an isoprene unit is NOT a cross-link (1−''px'') in ''N''−1 successive units along a chain. Since ''P''(''N'') decreases with ''N'', shorter chains are more probable than longer ones. Note that the number of statistically independent backbone segments is not the same as the number of isoprene units. For natural rubber networks, the Kuhn length contains about 2.2 isoprene units, so N \sim 2.2 n. It is the product of equations () and () (the
joint probability distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
) that relates the network chain length (N) and end-to-end distance (r) between its terminating cross-link nodes: The complex morphology of a natural rubber network can be seen in Fig. 3, which shows the probability density vs. end-to-end distance (in units of mean node spacing) for an ‘average’ chain. For the common experimental cross-link density of 4x1019 cm−3, an average chain contains about 116 isoprene units (52 Kuhn lengths) and has a contour length of about 50 nm. Fig. 3 shows that a significant fraction of chains span several node spacings, i.e., the chain ends overlap other network chains. Natural rubber, cross-linked with dicumyl peroxide, has tetra-functional cross-links, i.e. each cross-link node has 4 network chains emanating from it. Depending on their initial tortuosity and the orientation of their endpoints with respect to the strain axis, each chain associated with an active cross-link node can have a different elastic
force constant In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
as it resists the applied strain. To preserve force equilibrium (zero net force) on each cross-link node, a node may be forced to move in tandem with the chain having the highest force constant for chain extension. It is this complex node motion, arising from the random nature of the network morphology, that makes the study of the mechanical properties of rubber networks so difficult. As the network is strained, paths composed of these more extended chains emerge that span the entire sample, and it is these paths that carry most of the stress at high strains.


Numerical network simulation model

To calculate the elastic response of a rubber sample, the three chain force models (regimes Ia, Ib and II) and the network morphology must be combined in a micro-mechanical network model. Using the joint probability distribution in equation () and the force extension models, it is possible to devise numerical algorithms to both construct a faithful representative volume element of a network and to simulate the resulting mechanical stress as it is subjected to strain. An iterative relaxation algorithm is used to maintain approximate force equilibrium at each network node as strain is imposed. When the force constant obtained for kinks having 2 or 3 isoprene units (approximately one Kuhn length) is used in numerical simulations, the predicted stress is found to be consistent with experiments. The results of such a calculation are shown in Fig. 1 (dashed red line) for sulfur cross-linked natural rubber and compared with experimental data (solid blue line). These simulations also predict a steep upturn in the stress as network chains become taut and, ultimately, material failure due to bond rupture. In the case of sulfur cross-linked natural rubber, the S-S bonds in the cross-link are much weaker than the C-C bonds on the chain backbone and are the network failure points. The plateau in the simulated stress, starting at a strain of about 7, is the limiting value for the network. Stresses greater than about 7 MPa cannot be supported and the network fails. Near this stress limit, the simulations predict that less than 10% of the chains are taut, i.e. in the high chain extension regime and less than 0.1% of the chains have ruptured. While the very low rupture fraction may seem surprising, it is not inconsistent with our experience of stretching a rubber band until it breaks. The elastic response of the rubber after breaking is not noticeably different from the original.


Experiments


Variation of tensile stress with temperature

For molecular systems in thermal equilibrium, the addition of energy. e. g. by mechanical work, can cause a change in entropy. This is known from the theories of thermodynamics and statistical mechanics. Specifically, both theories assert that the change in energy must be proportional to the entropy change times the absolute temperature. This rule is only valid so long as the energy is restricted to thermal states of molecules. If a rubber sample is stretched far enough, energy may reside in non-thermal states such as the distortion of chemical bonds and the rule doesn't apply. At low to moderate strains, theory predicts that the required stretching force is due to a change in entropy in the network chains. If this is correct, then we expect that the force necessary to stretch a sample to some value of strain should be proportional to the temperature of the sample. Measurements showing how the tensile stress in a stretched rubber sample varies with temperature are shown in Fig. 4. In these experiments, the strain of a stretched rubber sample was held fixed as the temperature was varied between 10 and 70 degrees Celsius. For each value of fixed strain, it is seen that the tensile stress varied linearly (to within experimental error). These experiments provide the most compelling evidence that entropy changes are the fundamental mechanism for rubber elasticity. The positive linear behavior of the stress with temperature sometimes leads to the mistaken notion that rubber has a negative
coefficient of thermal expansion Thermal expansion is the tendency of matter to change its shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other pro ...
, i.e. the length of a sample shrinks when heated. Experiments have shown conclusively that, like almost all other materials, the coefficient of thermal expansion natural rubber is positive.


Snap-back velocity

When we stretch a piece of rubber, e.g. a rubber band, we notice that it deforms uniformly, lengthwise. Every element along its length experiences the same extension factor as the entire sample. If we release one end, the sample snaps back to its original length very rapidly, too fast for our eye to resolve the process. Our intuitive expectation is that it returns to its original length in the same manner as when it was stretched, i. e. uniformly. However, this is not what happens. Experimental observations by Mrowca et al. show a surprising behavior. To capture the extremely fast retraction dynamics, they utilized a clever experimental method devised by Exner and Stefan in 1874, well before high-speed electronic measuring devices were invented. Their method consisted of a rapidly rotating glass cylinder which, after being coated with lamp black, was placed next to the stretched rubber sample. Styli, attached to the mid-point and free end of the rubber sample, were held in contact with the glass cylinder. Then, as the free end of the rubber snapped back, the styli traced out helical paths in the lamp black coating of the rotating cylinder. By adjusting the rotation speed of the cylinder, they could record the position of the styli in less than one complete rotation. The trajectories were transferred to a graph by rolling the cylinder on a piece of damp blotter paper. The mark left by a stylus appeared as a white line (no lamp black) on the paper. Their data, plotted as the graph in Fig. 5, shows the position of end and midpoint styli as the sample rapidly retracts to its original length. The sample was initially stretched 9.5” beyond its unstrained length and then released. The styli returned to their original positions (displacement of 0”) in a little over 6 ms. The linear behavior of the displacement vs. time indicates that, after a brief acceleration, both the end and the midpoint of the sample snapped back at a constant velocity of about 50 m/s or 112 mph. However, the midpoint stylus did not start to move until about 3 ms after the end was released. Evidently, the retraction process travels as a wave, starting at the free end. At high extensions some of the energy stored in the stretched network chain is due to a change in its entropy, but most of the energy is stored in bond distortions (regime II, above) which do not involve an entropy change. If one assumes that all of the stored energy is converted to kinetic energy, the retraction velocity may be calculated directly from the familiar conservation equation ''E'' = ''mv''2. Numerical simulations, based on the Molecular Kink paradigm, predict velocities consistent with this experiment.


Historical approaches to elasticity theory

Eugene Guth Eugene Guth (August 21, 1905 – July 5, 1990) was a Hungarian American physicist who made contributions to polymer physics and to nuclear and solid state physics. He was awarded a Ph.D. in Theoretical Physics by the University of Vienna in 19 ...
and Hubert M. James proposed the entropic origins of rubber elasticity in 1941.


Thermodynamics

Temperature affects the elasticity of elastomers in an unusual way. When the elastomer is assumed to be in a stretched state, heating causes them to contract. Vice versa, cooling can cause expansion. This can be observed with an ordinary
rubber band A rubber band (also known as an elastic band, gum band or lacky band) is a loop of rubber, usually ring or oval shaped, and commonly used to hold multiple objects together. The rubber band was patented in England on March 17, 1845 by Stephen P ...
. Stretching a rubber band will cause it to release heat (press it against your lips), while releasing it after it has been stretched will lead it to absorb heat, causing its surroundings to become cooler. This phenomenon can be explained with the
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...
. Rearranging Δ''G''=Δ''H''−''T''Δ''S'', where ''G'' is the free energy, ''H'' is the
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
, and ''S'' is the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, we get . Since stretching is nonspontaneous, as it requires external work, ''T''Δ''S'' must be negative. Since ''T'' is always positive (it can never reach
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...
), the Δ''S'' must be negative, implying that the rubber in its natural state is more entangled (with more
microstates A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...
) than when it is under tension. Thus, when the tension is removed, the reaction is spontaneous, leading Δ''G'' to be negative. Consequently, the cooling effect must result in a positive ΔH, so Δ''S'' will be positive there. The result is that an elastomer behaves somewhat like an ideal monatomic gas, inasmuch as (to good approximation) elastic polymers do ''not'' store any potential energy in stretched chemical bonds or elastic work done in stretching molecules, when work is done upon them. Instead, all work done on the rubber is "released" (not stored) and appears immediately in the polymer as thermal energy. In the same way, all work that the elastic does on the surroundings results in the disappearance of thermal energy in order to do the work (the elastic band grows cooler, like an expanding gas). This last phenomenon is the critical clue that the ability of an elastomer to do work depends (as with an ideal gas) only on entropy-change considerations, and not on any stored (i.e., potential) energy within the polymer bonds. Instead, the energy to do work comes entirely from thermal energy, and (as in the case of an expanding ideal gas) only the positive entropy change of the polymer allows its internal thermal energy to be converted efficiently (100% in theory) into work.


Polymer chain theories

Invoking the theory of rubber elasticity, one considers a polymer chain in a cross-linked network as an entropic spring. When the chain is stretched, the entropy is reduced by a large margin because there are fewer conformations available. Therefore, there is a restoring force, which causes the polymer chain to return to its equilibrium or unstretched state, such as a high entropy random coil configuration, once the external force is removed. This is the reason why rubber bands return to their original state. Two common models for rubber elasticity are the freely-jointed chain model and the worm-like chain model.


Freely-jointed chain model

The freely joined chain, also called an ideal chain, follows the
random walk model In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space. An elementary example of a random walk is the random walk on the integer n ...
. Microscopically, the 3-D random walk of a polymer chain assumes the overall end-to-end distance is expressed in terms of the x, y and z directions: \vec = R_x\hat + R_y\hat + R_z\hat In the model, b is the length of a rigid segment, N is the number of segments of length b , R is the distance between the fixed and free ends, and L_\text is the "contour length" or Nb. Above the glass transition temperature, the polymer chain oscillates and r changes over time. The probability distribution of the chain is the product of the probability distributions of the individual components, given by the following Gaussian distribution: P(\vec) = P(R_x) P(R_y) P(R_z) = \left( \frac\right)^ \exp \left( \frac \right) Therefore, the ensemble average end-to-end distance is simply the standard integral of the probability distribution over all space. Note that the movement could be backwards or forwards, so the net average \langle R\rangle will be zero. However, one can use the root mean square as a useful measure of the distance. \begin \langle R\rangle &= 0 \\ \langle R^2\rangle &= \int_0^\infty R^24\pi R^2 P(\vec)dR = Nb^2 \\ \langle R^2\rangle^\frac &= \sqrt b \end The Flory theory of rubber elasticity has pointed out the rubber elasticity has primarily entropic origins. By using the following basic equations for
Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal In thermodynamics, an isotherma ...
and its discussion about entropy, the force generated from the deformation of a rubber chain from its original un-stretched conformation can be derived. The \Omega is the number of conformations of the polymer chain. Since the deformation does not involve enthalpy change, the change in free energy can just be calculated as the change in entropy-T\Delta S. It can be observed that the force equation resembles the behavior of a spring and follows
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
: F = kx, where ''F'' is the force, ''k'' is the spring constant and ''x'' is the distance. Usually, neo-Hookean model can be used on cross-linked polymers to predict their stress-strain relations: \Omega = C \exp \left ( \frac \right ) S = k_\text \ln \Omega \, \approx \frac \Delta F(\vec) \approx -T\Delta S_d(\vec^2) = C+\frac \vec^2 f =\frac = \frac\left(\frac\right) = \frac \vec Note that the elastic coefficient 3 k_\text T/N b is temperature dependent. If we increase the rubber temperature, the elastic coefficient also rises. This is the reason why rubber under constant stress shrinks when its temperature increases. We can further expand the Flory theory into a macroscopic view, where bulk rubber material is discussed. Assume the original dimension of the rubber material is L_x, L_y and L_z, a deformed shape can then be expressed by applying an individual extension ratio \lambda_i to the length (\lambda_x L_x, \lambda_y L_y, \lambda_z L_z). So microscopically, the deformed polymer chain can also be expressed with the extension ratio: \lambda_x R_x, \lambda_y R_y, \lambda_z R_z. The free energy change due to deformation can then be expressed as follows: \begin \Delta F_\text (\vec) &= - \frac = -\frac \\ &=-\frac \end Assume that the rubber is cross-linked and isotropic, the random walk model gives R_x, R_y and R_z are distributed according to a normal distribution. Therefore, they are equal in space, and all of them are 1/3 of the overall end-to-end distance of the chain: \langle R_^2 \rangle=\langle R_^2 \rangle = \langle R_^2 \rangle = \langle R^2 \rangle/3. Plugging in the change of free energy equation above, it is easy to get: \begin \Delta F_\text (\vec)&=-\frac \\ &=-\frac \end The free energy change per volume is just: \Delta f_\text = \frac = -\frac where n_s is the number of strands in network, the subscript "def" means "deformation", v_s = n_s / V, which is the number density per volume of polymer chains, \beta = \langle R^2\rangle / R_0^2 which is the ratio between the end-to-end distance of the chain and the theoretical distance that obey random walk statistics. If we assume incompressibility, the product of extension ratios is 1, implying no change in the volume: \lambda_x \lambda_y \lambda_z = 1. Case study: Uniaxial deformation: In a uniaxial deformed rubber, because \lambda_x \lambda_y \lambda_z = 1 we assume \lambda_x = \lambda_y = \lambda_z^. So the previous free energy per volume equation is: \Delta f_\text = \frac = -\frac = \frac \left(\lambda_z^2 + \frac-3\right) The
engineering stress In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strai ...
(by definition) is the first derivative of the energy in terms of the extension ratio, which is equivalent to the concept of strain: \sigma_\text=\frac = k_\textT v_s \beta\left(\lambda_z-\frac\right) and the
Young's Modulus Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
E is defined as derivative of the stress with respect to strain, which measures the
stiffness Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a b ...
of the rubber in laboratory experiments. E=\frac=k_\textT v_s \beta \left.\left(1+ \frac\right)\_ = 3 k_\textT v_s\beta = \frac where v_s = \rho N_a / M_s, \rho is the mass density of the chain, M_s is the number average molecular weight of a network strand between crosslinks. Here, this type of analysis links the thermodynamic theory of rubber elasticity to experimentally measurable parameters. In addition, it gives in sights into the cross-linking condition of the materials.


Worm-like chain model

The worm-like chain model (WLC) takes the energy required to bend a molecule into account. The variables are the same except that L_\text, the persistence length, replaces b . Then, the force follows this equation: F \approx \frac \left ( \frac - \frac + \frac \right ) Therefore, when there is no distance between chain ends (''r''=0), the force required to do so is zero, and to fully extend the polymer chain ( r = L_\text ), an infinite force is required, which is intuitive. Graphically, the force begins at the origin and initially increases linearly with r. The force then plateaus but eventually increases again and approaches infinity as the chain length approaches L_\text.


See also

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Elasticity (physics) In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ap ...
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Hyperelastic material A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density ...
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Polymers A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic an ...
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Thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...


References

{{Reflist Rubber properties Thermodynamics Mechanics