Rubber elasticity - Knowledge (XXG)

1402:) 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 sulphur 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 sulphur 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 the common experience of stretching a rubber band until it breaks. The elastic response of the rubber after breaking is not noticeably different from the original. 501:

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 experiments 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 the 18 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.

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in the same manner as when it was stretched (i.e. uniformly). Experimental observations by Mrowca et al. suggest that this expectation is inaccurate. To capture the extremely fast retraction dynamics, they utilized an experimental method devised by Exner and Stefan in 1874. 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.

476:, 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 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 1456:

displacement of 0 in) in a little over 6 Ms. The linear behaviour 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

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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. 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. This

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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 into work.

3920: 410: 480:(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) are directly related to the number of states, they may be associated with the entropy of the kink at any end-to-end distance. By numerically differentiating the probability distribution, the change in entropy, and hence 425:) 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. 3384: 1382:(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 488: 1607:. When the chain is stretched, the entropy is reduced by a large margin because there are fewer conformations available. As such 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. 4466: 3915:{\displaystyle {\begin{aligned}\Delta F_{\text{def}}({\vec {R}})&=-{\frac {3k_{\text{B}}T{\vec {R}}^{2}}{2Nb^{2}}}=-{\frac {3k_{\text{B}}T\left(\left(R_{x}^{2}-R_{x0}^{2}\right)+\left(R_{y}^{2}-R_{y0}^{2}\right)+\left(R_{z}^{2}-R_{z0}^{2}\right)\right)}{2Nb^{2}}}\\&=-{\frac {3k_{\text{B}}T\left(\left(\lambda _{x}^{2}-1\right)R_{x0}^{2}+\left(\lambda _{y}^{2}-1\right)R_{y0}^{2}+\left(\lambda _{z}^{2}-1\right)R_{z0}^{2}\right)}{2Nb^{2}}}\end{aligned}}} 4130: 5237: 378:

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.

1416: 284:) was regarded mostly as a 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 1424:

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 nonthermal states such as the distortion of chemical bonds and the rule does not apply. At low to moderate strains, theory predicts that the required stretching force is due to a change in entropy in the network chains.

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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.

1626: 2334: 138: 4461:{\displaystyle {\begin{aligned}\Delta F_{\text{def}}({\vec {R}})&=-{\frac {k_{\text{B}}Tn_{s}\langle R^{2}\rangle \left(\lambda _{x}^{2}+\lambda _{y}^{2}+\lambda _{z}^{2}-3\right)}{2Nb^{2}}}\\&=-{\frac {k_{\text{B}}Tn_{s}\langle R^{2}\rangle \left(\lambda _{x}^{2}+\lambda _{y}^{2}+\lambda _{z}^{2}-3\right)}{2R_{0}^{2}}}\end{aligned}}} 4649: 260:, or joined to each other by the bonds made with the cross-linking molecules. Because each rubber polymer is very long, each one participates in many crosslinks with many other rubber molecules, forming a continuous network. The resulting molecular structure demonstrates elasticity, making rubber a member of the class of elastic 2998: 454:

initial equilibrium contour length by the distortion of the chemical bonds along its backbone. In this case, the restoring force is spring-like and is referred to as regime II. The three force mechanisms are found to roughly correspond to the three regions observed in tensile stress vs. strain experiments, shown in Fig. 1.

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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 model is required to simulate the effects of strain on a representative volume element of a network.

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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 favourable to have one kink transition to an extended conformation in order to stretch the chain further. The applied strain can force a single isoprene

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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

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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

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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. Stretching a rubber band will cause it to release

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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

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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

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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

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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

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When stretching a piece of rubber (e.g. a rubber band) it will deform lengthwise in a uniform manner. When one end of the sample is released, it snaps back to its original length too quickly for the naked eye to resolve the process. An intuitive expectation is that it returns to its original length

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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

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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 4x10 cm, an average chain contains about 116 isoprene units

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There are three distinct molecular mechanisms that produce these forces, two of which arise from changes in entropy that is referred to as the low chain extension regime, Ia and the moderate chain extension regime, Ib. The third mechanism occurs at high chain extension, as it is extended beyond its

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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

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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

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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 in (~24 cm) beyond its unstrained length and then released. The styli returned to their original positions (i.e. a

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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

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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

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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

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It is therefore expected 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

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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

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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 way in which the chains are connected is actually quite complicated. There is a wide variation in the

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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

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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

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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

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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

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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

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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

5232:{\displaystyle \Delta f_{\text{def}}={\frac {\Delta F_{\text{def}}({\vec {R}})}{V}}=-{\frac {k_{\text{B}}Tv_{s}\beta \left(\lambda _{x}^{2}+\lambda _{y}^{2}+\lambda _{z}^{2}-3\right)}{2}}={\frac {k_{\text{B}}Tv_{s}\beta }{2}}\left(\lambda _{z}^{2}+{\frac {2}{\lambda _{z}}}-3\right)} 437:(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 299:

showed that the change in mechanical energy required to stretch a rubber sample should be proportional to the increase in temperature. This would later be associated with a change in entropy. The connection to thermodynamics was firmly established in 1859 when the English physicist

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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 insights into the cross-linking condition of the materials.

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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.

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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 these elastic properties.

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The initial morphology of the network, immediately after chemical cross-linking, is governed by two random processes: (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

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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.

1008:, 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 5248: 2631: 1395:

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

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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:

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The freely joined chain, also called an ideal chain, follows the random walk model. Microscopically, the 3D random walk of a polymer chain assumes the overall end-to-end distance is expressed in terms of the x, y and z directions:

5632:{\displaystyle E={\frac {d(\sigma _{\text{eng}})}{d\lambda _{z}}}=k_{\text{B}}Tv_{s}\beta \left.\left(1+{\frac {2}{\lambda _{z}^{3}}}\right)\right|_{\lambda _{z}=1}=3k_{\text{B}}Tv_{s}\beta ={\frac {3\rho \beta RT}{M_{s}}}} 343:

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

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refers to the ability of solid rubber to be stretched up to a factor of 10 from its original length, and return to close to its original length upon release. This process can be repeated many times with no apparent

601: 2329:{\displaystyle {\begin{aligned}\langle R\rangle &=0\\\langle R^{2}\rangle &=\int _{0}^{\infty }R^{2}4\pi R^{2}P({\vec {R}})dR=Nb^{2}\\\langle R^{2}\rangle ^{\frac {1}{2}}&={\sqrt {N}}b\end{aligned}}} 2635: 2103:

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

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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:

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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).

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is temperature dependent. If rubber temperature increases, the elastic coefficient increases as well. This is the reason why rubber under constant stress shrinks when its temperature increases.

2140: 825: 1435:(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. 4008: 4644:{\displaystyle \Delta f_{\text{def}}={\frac {\Delta F_{\text{def}}({\vec {R}})}{V}}=-{\frac {k_{\text{B}}Tv_{s}\beta \left(\lambda _{x}^{2}+\lambda _{y}^{2}+\lambda _{z}^{2}-3\right)}{2}}} 1888:

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:

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is the number of conformations of the polymer chain. Since the deformation does not involve enthalpy change, the change in free energy can simply be calculated as the change in entropy

484:, 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. 2993:{\displaystyle f={\frac {dF({\vec {R}})}{d{\vec {R}}}}={\frac {d}{d{\vec {R}}}}\left({\frac {3k_{\text{B}}T{\vec {R}}^{2}}{2Nb^{2}}}\right)={\frac {3k_{\text{B}}T}{Nb^{2}}}{\vec {R}}} 5695: 327:

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.

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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.

1350:{\displaystyle P(r,N)\;=\;P(N)P(r|N)\;=\;p_{x}{\left(1-p_{x}\right)}^{N-1}\,4\pi r^{2}\left({\frac {2nb^{2}\pi }{3}}\right)^{-{3}/{2}}\exp \left(-{\frac {3r^{2}}{2nb^{2}}}\right)} 3377: 3340: 3303: 3266: 3229: 3192: 3044: 2128: 6607:

Buche, M.R.; Silberstein, M.N. (2020). "Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble".

5986: 5942:{\displaystyle F\approx {\frac {k_{\text{B}}T}{L_{\text{p}}}}\left({\frac {1}{4\left(1-{\frac {r}{L_{\rm {c}}}}\right)^{2}}}-{\frac {1}{4}}+{\frac {r}{L_{\text{c}}}}\right)} 4724: 491:

Fig. 2 Isoprene backbone unit. composed of carbon atoms (dark grey) and hydrogen atoms (white). Carbon atoms labeled '1' and '6' are in adjacent units on the polymer chain.

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heat, 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

2360: 983: 5380:{\displaystyle \sigma _{\text{eng}}={\frac {d(\Delta f_{\text{def}})}{\lambda _{z}}}=k_{\text{B}}Tv_{s}\beta \left(\lambda _{z}-{\frac {1}{\lambda _{z}^{2}}}\right)} 5742: 5715: 4676: 4003: 3976: 3949: 3128: 3101: 3074: 2443: 2422: 4005:

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:

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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

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found that natural rubber's elastic properties could be immensely improved by adding a small amount of sulfur to produce chemical cross-links between adjacent

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and its discussion about entropy, the force generated from the deformation of a rubber chain from its original unstretched conformation can be derived. The

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Anthony, R.L.; Caston, R.H.; Guth, E. (1942). "Equations of state for natural and synthetic rubber-like materials. I. Unaccelerated natural soft rubber".

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and we may associate an entropy with its end-to-end distance. The probability distribution for the end-to-end distance of a Kuhn length is approximately

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Fig. 3 Probability density for an average network chain vs. end-to-end distance in units of mean cross-link node spacing (2.9 nm); n= 52, b= 0.96 nm.

288:. In 1805 he published some qualitative observations on this characteristic as well as how the required stretching force increased with temperature. 2094:{\displaystyle P({\vec {R}})=P(R_{x})P(R_{y})P(R_{z})=\left({\frac {2nb^{2}\pi }{3}}\right)^{-{3}/{2}}\exp \left({\frac {-3R^{2}}{2Nb^{2}}}\right)} 2338:

The Flory theory of rubber elasticity suggests that rubber elasticity has primarily entropic origins. By using the following basic equations for

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Fig. 1 Stress vs. tensile strain for a natural rubber network. Experimental data by Treloar (solid blue), theoretical simulation (dashed red).

6568: 6092: 530: 156: 148: 5988:), an infinite force is required, which is intuitive. Graphically, the force begins at the origin and initially increases linearly with 789: 1572: 53: 5245:(by definition) is the first derivative of the energy in terms of the extension ratio, which is equivalent to the concept of strain: 6592: 1373: 174: 119: 6539: 4903: 433:. These non-straight regions evoke the concept of "kinks" and are in fact a manifestation of the random-walk nature of the chain. 5753:

The worm-like chain model (WLC) takes the energy required to bend a molecule into account. The variables are the same except that

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L. A. Wood and G. Martin, Journal of Research of the National Bureau of Standards-A. Physics and Chemistry Vol 68A, No. 3 (1964).

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assumed to operate along the chain contour. In addition to the chain's contour length, the only other important parameter is its

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The positive linear behaviour of the stress with temperature sometimes leads to the mistaken notion that rubber has a negative

762:{\displaystyle P(r|n)=4\pi r^{2}\left({\frac {2nb^{2}\pi }{3}}\right)^{-{3}/{2}}\exp \left(-{\frac {3r^{2}}{2nb^{2}}}\right)\,} 72: 1050: 57: 4850: 4789: 4729: 2778:{\displaystyle \Delta F({\vec {R}})\approx -T\Delta S_{d}({\vec {R}}^{2})=C+{\frac {3k_{\text{B}}T}{Nb^{2}}}{\vec {R}}^{2}} 79: 6313:

D. E. Hanson and J. L. Barber, Modelling and Simulation in Materials Science and Engineering 21 (2013), LAPR-2013-017962

4120:{\displaystyle \langle R_{x0}^{2}\rangle =\langle R_{y0}^{2}\rangle =\langle R_{z0}^{2}\rangle =\langle R^{2}\rangle /3} 459: 1419:

Fig. 4 Variation of tensile stress with temperature as strain held constant at four values (100%, 200%, 300% and 380%).

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Before it is cross-linked, the liquid natural rubber consists of very long polymer molecules, containing thousands of

226:. A rubber polymer follows a random winding path in three dimensions, intermingling with many other rubber polymers. 6674: 1473:. Numerical simulations, based on the molecular kink paradigm, predict velocities consistent with this experiment. 86: 827:

The factor of two comes about because two isoprene units (one from each chain) participate in the cross-link. The

46: 6008:. The force then plateaus but eventually increases again and approaches infinity as the chain length approaches 409: 828: 68: 1447:

Fig. 5 Displacement of the end and midpoint of a rubber sample vs. time as it snaps back from high extension.

222:. This process builds polymers up by sequentially adding short molecular backbone units to the chain through 6679: 6475:

Guth, Eugene; James, Hubert M. (May 1941). "Elastic and Thermoelastic Properties of Rubber like Materials".

5642: 349: 192: 1575:) than when it is under tension. Thus, when the tension is removed, the reaction is spontaneous, leading Δ 442: 6049: 2339: 985:. The equation can be understood as simply the probability that an isoprene unit is NOT a cross-link (1− 487: 481: 374: 366: 6362:

Treloar, L.R.G. (1944). "Stress-Strain Data for Vulcanized Rubber under Various Types of Deformation".

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D. E. Hanson, J. L. Barber and G. Subramanian, Journal of Chemical Physics 139 (2013), LAPR-2014-018991

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Invoking the theory of rubber elasticity, a polymer chain in a cross-linked network may be seen as an

6626: 6511: 6440: 6044: 1616: 1485: 203: 6465:

G. S. Whitby, "Plantation Rubber and the Testing of Rubber", Longmans and Green, London, 1920. p 461

5958: 417:

The Molecular Kink Paradigm proceeds from the intuitive notion that molecular chains that make up a

6249:

D. E. Hanson and R. L. Martin, The Journal of Chemical Physics 130, 064903 (2009), LAPR-2009-006764

4681: 446: 438: 211: 6161: 6011: 5756: 3268:). So microscopically, the deformed polymer chain can also be expressed with the extension ratio: 3133: 2626:{\displaystyle S=k_{\text{B}}\ln \Omega \,\approx {\frac {-3k_{\text{B}}{\vec {R}}^{2}}{2Nb^{2}}}} 1821: 6684: 6650: 6616: 5242: 477: 345: 207: 5388: 2365: 1011: 93: 6642: 6588: 6564: 6088: 2437: 1498: 516: 223: 2345: 1571:

must be negative, implying that the rubber in its natural state is more entangled (with more

962: 6634: 6519: 6484: 6448: 6401: 6371: 6211:

D. E. Hanson and J. L. Barber, Contemporary Physics 56 (3), 319–337 (2015), LAPR-2015-022971

308: 5720: 5700: 4654: 3981: 3954: 3927: 3106: 3079: 3052: 6543: 6223:

D. E. Hanson and R. L. Martin, Journal of Chemical Physics 133, 084903 (084908 pp.) (2010)

2398: 6630: 6536: 6515: 6444: 6343:

D. E. Hanson and J. L. Barber, Phys. Chem. Chem. Phys. 20, 8460 (2018), LAPR-2018-029488

6202:

Joule JP. On thermodynamic properties of solids. Phil Trans R Soc Lond. 1859;149:91–131.

1848: 6059: 5991: 5783: 5393: 2392: 1871: 1801: 1781: 1761: 1741: 1604: 1383: 1076: 1056: 834: 576: 556: 536: 511: 418: 292: 277: 219: 295:

was being developedand within this framework, the English mathematician and physicist

6668: 6654: 2130:

will be zero. However, the root mean square can be a useful measure of the distance.

1587: 1564: 324: 253: 250: 242: 230: 6137: 6113: 510:

still possible by bond distortions (e.g. bond angle increases), bond stretches, and

3130:, a deformed shape can then be expressed by applying an individual extension ratio 1579:

to be negative. Consequently, the cooling effect must result in a positive ΔH, so Δ

441:, the probability that a kink resides in any rotational conformation is given by a 361:

immediately apparent when a moderately thick rubber sample is stretched manually.

312: 281: 234: 5410:

is defined as derivative of the stress with respect to strain, which measures the

3924:

Assume that the rubber is cross-linked and isotropic, the random walk model gives

6261:

P. Flory, N. Rabjohn and M. Shaffer, Journal of Polymer Science 4, 435–455 (1949)

5955:=0), the force required to do so is zero, and to fully extend the polymer chain ( 17: 6638: 1481: 430: 373:

The concept of entropy comes to us from the area of mathematical physics called

336: 301: 296: 35: 2440:

can be used on cross-linked polymers to predict their stress-strain relations:

2391:. Note that the force equation resembles the behaviour of a spring and follows 4678:

is the number of strands in network, the subscript "def" means "deformation",

3379:. The free energy change due to deformation can then be expressed as follows: 2522:{\displaystyle \Omega =C\exp \left({\frac {-3{\vec {R}}^{2}}{2Nb^{2}}}\right)} 473: 257: 6452: 6431:

Mrowca, B.A.; Dart, S.L.; Guth, E. (1944). "Retraction of stressed rubber".

6080: 5411: 276:

Following its introduction to Europe from America in the late 15th century,

265: 218:, or large, chain-like molecules. Polymers are produced by a process called 6646: 4127:. Plugging in the change of free energy equation above, it is easy to get: 1868:. Above the glass transition temperature, the polymer chain oscillates and 256:

to the cross-linking molecule. These bonds cause rubber polymers to become

1729:{\displaystyle {\vec {R}}=R_{x}{\hat {x}}+R_{y}{\hat {y}}+R_{z}{\hat {z}}} 1415: 245:. During the process, a small amount of a cross-linking molecule, usually 6299:

D. E. Hanson, Journal of Chemical Physics 131, 224904 (224905 pp.) (2009)

6273:

D. E. Hanson, Journal of Chemical Physics 134, 064906 (064906 pp.) (2011)

6054: 1526: 422: 365:

increases markedly. What the observer is noticing are the changes in the

319: 199: 6488: 1443: 323:

peroxide, can decompose and the subsequent chemical reactions produce a

1625: 340: 261: 238: 215: 6523: 6405: 6375: 348:

that is observed when a rubber sample is deformed (e.g. subjected to

246: 1551:. Since stretching is nonspontaneous, as it requires external work, 6621: 6138:"Polymerization | Definition, Classes, & Examples | Britannica" 6114:"Polymerization | Definition, Classes, & Examples | Britannica" 1442: 1414: 1372: 486: 408: 335:

physical mechanisms that occur at the molecular level and how the

6563:, vol. 1, Madison, WI: The University of Wisconsin Press, 1586:

The result is that an elastomer behaves somewhat like an ideal

385:

When these elastic force models are combined with the complex

131: 29: 6561:

Chemical Demonstrations: A Handbook for Teachers of Chemistry

6184:

Proc. Lit. and Phil. Soc., Manchester, 2d ser., 1, 288 (1805)

4962:{\displaystyle \lambda _{x}=\lambda _{y}=\lambda _{z}^{-1/2}} 6502:

Brown, J. B. (May 1963), "Thermodynamics of a Rubber Band",

4726:, which is the number density per volume of polymer chains, 1488:

proposed the entropic origins of rubber elasticity in 1941.

6322:

J. P. Joule, Phil. Trans. R. Soc. London 149, 91–131 (1859)

5500: 1590:, inasmuch as (to good approximation) elastic polymers do 5951:

Therefore, when there is no distance between chain ends (

936:{\displaystyle P(N)=p_{x}{\left(1-p_{x}\right)}^{N-1}\,,} 6079:

Pal, Sanjay; Das, Mithun; Naskar, Kinsuk (2023-05-17),

249:, is added. When heat is applied, sections of rubber's 4969:. So the previous free energy per volume equation is: 4893:{\displaystyle \lambda _{x}\lambda _{y}\lambda _{z}=1} 4832:{\displaystyle \lambda _{x}\lambda _{y}\lambda _{z}=1} 4778:{\displaystyle \beta =\langle R^{2}\rangle /R_{0}^{2}} 6537:

http://scifun.chem.wisc.edu/HomeExpts/rubberband.html

6162:"Polymerization - an overview | ScienceDirect Topics" 6014: 5994: 5961: 5806: 5786: 5759: 5723: 5703: 5645: 5422: 5396: 5251: 4975: 4906: 4853: 4792: 4732: 4684: 4657: 4476: 4133: 4011: 3984: 3957: 3930: 3387: 3348: 3311: 3274: 3237: 3200: 3163: 3136: 3109: 3082: 3055: 3008: 2790: 2638: 2534: 2446: 2401: 2368: 2348: 2138: 2110: 1894: 1874: 1851: 1824: 1818:

is the distance between the fixed and free ends, and

1804: 1784: 1764: 1744: 1638: 1104: 1079: 1059: 1014: 965: 862: 837: 792: 604: 579: 559: 539: 6352:

A. A. Markov, Izv. Peterb. Akad. 4 (1), 61–80 (1907)

60:. Unsourced material may be challenged and removed. 6027: 6000: 5980: 5941: 5792: 5772: 5736: 5709: 5689: 5631: 5402: 5379: 5231: 4961: 4892: 4831: 4777: 4718: 4670: 4643: 4460: 4119: 3997: 3970: 3943: 3914: 3371: 3334: 3297: 3260: 3223: 3186: 3149: 3122: 3095: 3068: 3038: 2992: 2777: 2625: 2521: 2416: 2383: 2354: 2328: 2122: 2093: 1880: 1860: 1837: 1810: 1790: 1770: 1750: 1728: 1349: 1085: 1065: 1029: 977: 935: 843: 819: 761: 585: 565: 545: 820:{\displaystyle p_{x}=2{\frac {\text{}}{\text{}}}} 394:The Molecular Kink Paradigm for rubber elasticity 6285:D. E. Hanson, Polymer 45 (3), 1058–1062 (2004) 390:failure of a rubber sample as it is strained. 6193:Lord Kelvin, Quarterly J. Math., 1, 57 (1857) 291:By the mid-nineteenth century, the theory of 8: 4752: 4739: 4361: 4348: 4218: 4205: 4106: 4093: 4087: 4066: 4060: 4039: 4033: 4012: 2291: 2277: 2179: 2166: 2149: 2143: 2117: 2111: 1411:Variation of tensile stress with temperature 1093:) between its terminating cross-link nodes: 6547: 4470:The free energy change per volume is just: 450:produces an elastic force along the chain. 1477:Historical approaches to elasticity theory 1164: 1160: 1127: 1123: 6620: 6019: 6013: 5993: 5972: 5960: 5926: 5917: 5904: 5892: 5878: 5877: 5868: 5847: 5834: 5820: 5813: 5805: 5800:. Then, the force follows this equation: 5785: 5764: 5758: 5728: 5722: 5702: 5681: 5672: 5666: 5650: 5644: 5621: 5598: 5586: 5573: 5549: 5544: 5527: 5522: 5513: 5489: 5476: 5460: 5442: 5429: 5421: 5414:of the rubber in laboratory experiments. 5395: 5364: 5359: 5350: 5341: 5323: 5310: 5295: 5281: 5265: 5256: 5250: 5210: 5201: 5192: 5187: 5163: 5150: 5143: 5117: 5112: 5099: 5094: 5081: 5076: 5058: 5045: 5038: 5012: 5011: 5002: 4992: 4983: 4974: 4949: 4942: 4937: 4924: 4911: 4905: 4878: 4868: 4858: 4852: 4817: 4807: 4797: 4791: 4769: 4764: 4755: 4746: 4731: 4708: 4702: 4689: 4683: 4662: 4656: 4618: 4613: 4600: 4595: 4582: 4577: 4559: 4546: 4539: 4513: 4512: 4503: 4493: 4484: 4475: 4445: 4440: 4414: 4409: 4396: 4391: 4378: 4373: 4355: 4342: 4329: 4322: 4300: 4271: 4266: 4253: 4248: 4235: 4230: 4212: 4199: 4186: 4179: 4155: 4154: 4145: 4134: 4132: 4109: 4100: 4081: 4073: 4054: 4046: 4027: 4019: 4010: 3989: 3983: 3962: 3956: 3935: 3929: 3899: 3876: 3868: 3847: 3842: 3824: 3816: 3795: 3790: 3772: 3764: 3743: 3738: 3715: 3705: 3683: 3655: 3647: 3634: 3629: 3606: 3598: 3585: 3580: 3557: 3549: 3536: 3531: 3508: 3498: 3483: 3465: 3454: 3453: 3443: 3433: 3409: 3408: 3399: 3388: 3386: 3363: 3353: 3347: 3326: 3316: 3310: 3289: 3279: 3273: 3252: 3242: 3236: 3215: 3205: 3199: 3178: 3168: 3162: 3141: 3135: 3114: 3108: 3087: 3081: 3060: 3054: 3025: 3016: 3007: 2979: 2978: 2969: 2951: 2941: 2925: 2907: 2896: 2895: 2885: 2875: 2857: 2856: 2847: 2830: 2829: 2810: 2809: 2797: 2789: 2769: 2758: 2757: 2747: 2729: 2719: 2701: 2690: 2689: 2679: 2649: 2648: 2637: 2614: 2596: 2585: 2584: 2577: 2564: 2560: 2545: 2533: 2506: 2488: 2477: 2476: 2466: 2445: 2400: 2367: 2347: 2312: 2294: 2284: 2267: 2237: 2236: 2224: 2208: 2198: 2193: 2173: 2139: 2137: 2109: 2078: 2060: 2047: 2030: 2025: 2020: 2016: 1997: 1984: 1967: 1948: 1929: 1902: 1901: 1893: 1873: 1850: 1829: 1823: 1803: 1783: 1763: 1743: 1715: 1714: 1708: 1690: 1689: 1683: 1665: 1664: 1658: 1640: 1639: 1637: 1333: 1315: 1305: 1284: 1279: 1274: 1270: 1251: 1238: 1227: 1216: 1204: 1192: 1176: 1169: 1149: 1103: 1078: 1058: 1013: 996:−1 successive units along a chain. Since 964: 929: 917: 905: 889: 882: 861: 836: 809: 797: 791: 758: 744: 726: 716: 695: 690: 685: 681: 662: 649: 638: 614: 603: 578: 558: 538: 175:Learn how and when to remove this message 120:Learn how and when to remove this message 1624: 531:conditional probability density function 241:and then solidified in a process called 198:Rubber, like all materials, consists of 6071: 4847:In a uniaxial deformed rubber, because 1563:is always positive (it can never reach 237:, are extracted from plants as a fluid 6426: 6424: 6387: 6385: 6339: 6337: 6309: 6307: 6305: 6295: 6293: 6291: 5690:{\displaystyle v_{s}=\rho N_{a}/M_{s}} 6331:L.D. Loan, Pure Appl. Chem. 30 (1972) 6281: 6279: 6269: 6267: 6257: 6255: 6245: 6243: 6233: 6231: 6229: 7: 6219: 6217: 1778:is the number of segments of length 1095: 1053:) relates the network chain length ( 853: 595: 58:adding citations to reliable sources 5780:, the persistence length, replaces 496:Moderate chain extension regime, Ib 5879: 5717:is the mass density of the chain, 5274: 4995: 4976: 4496: 4477: 4138: 3392: 3002:Note that the elastic coefficient 2672: 2639: 2557: 2447: 2375: 2349: 2199: 1758:is the length of a rigid segment, 1391:Numerical network simulation model 147:tone or style may not reflect the 25: 4843:Case study: Uniaxial deformation: 3372:{\displaystyle \lambda _{z}R_{z}} 3335:{\displaystyle \lambda _{y}R_{y}} 3298:{\displaystyle \lambda _{x}R_{x}} 3261:{\displaystyle \lambda _{z}L_{z}} 3224:{\displaystyle \lambda _{y}L_{y}} 3187:{\displaystyle \lambda _{x}L_{x}} 3039:{\displaystyle 3k_{\text{B}}T/Nb} 2123:{\displaystyle \langle R\rangle } 1629:Model of the freely jointed chain 1433:coefficient of thermal expansion 157:guide to writing better articles 136: 34: 6559:Shakhashiri, Bassam Z. (1983), 831:for finding a chain containing 505:High chain extension regime, II 45:needs additional citations for 5981:{\displaystyle r=L_{\text{c}}} 5448: 5435: 5287: 5271: 5023: 5017: 5008: 4524: 4518: 4509: 4166: 4160: 4151: 3459: 3420: 3414: 3405: 2984: 2901: 2862: 2835: 2821: 2815: 2806: 2763: 2707: 2695: 2685: 2660: 2654: 2645: 2590: 2482: 2436:is the distance. Usually, the 2248: 2242: 2233: 1973: 1960: 1954: 1941: 1935: 1922: 1913: 1907: 1898: 1720: 1695: 1670: 1645: 1157: 1150: 1143: 1137: 1131: 1120: 1108: 1051:joint probability distribution 872: 866: 622: 615: 608: 467:Low chain extension regime, Ia 307:In 1838 the American inventor 27:Property of crosslinked rubber 1: 6081:"Origin of Rubber Elasticity" 4719:{\displaystyle v_{s}=n_{s}/V} 6585:Physics of Rubber Elasticity 6028:{\displaystyle L_{\text{c}}} 5773:{\displaystyle L_{\text{p}}} 3150:{\displaystyle \lambda _{i}} 1838:{\displaystyle L_{\text{c}}} 1037:. The product of equations ( 851:isoprene units is given by: 553:in units of the Kuhn length 6639:10.1103/PhysRevE.102.012501 6587:, Oxford University Press, 6504:American Journal of Physics 2432:is the spring constant and 1845:is the "contour length" or 1398: 1073:) and end-to-end distance ( 1045: 1039: 573:to the end-to-end distance 6701: 2384:{\displaystyle -T\Delta S} 1614: 1611:Freely-jointed chain model 1533:is the entropy, we obtain 1030:{\displaystyle N\sim 2.2n} 533:relates the chain length 229:Natural rubbers, such as 214:. Rubber's molecules are 1583:will be positive there. 1559:must be negative. Since 460:probability distribution 6583:L.R.G. Treloar (1975), 6535:Rubber Bands and Heat, 6453:10.1103/PhysRev.66.30.2 6085:Elasticity of Materials 2355:{\displaystyle \Omega } 978:{\displaystyle N\geq 1} 151:used on Knowledge (XXG) 6029: 6002: 5982: 5943: 5794: 5774: 5738: 5711: 5691: 5633: 5404: 5381: 5233: 4963: 4894: 4833: 4779: 4720: 4672: 4645: 4462: 4121: 3999: 3972: 3945: 3916: 3373: 3336: 3299: 3262: 3225: 3188: 3151: 3124: 3097: 3070: 3040: 2994: 2779: 2627: 2523: 2418: 2385: 2356: 2330: 2124: 2095: 1882: 1862: 1839: 1812: 1792: 1772: 1752: 1730: 1630: 1599:Polymer chain theories 1448: 1420: 1378: 1351: 1087: 1067: 1031: 979: 937: 845: 821: 763: 587: 567: 547: 492: 443:Boltzmann distribution 414: 356:Molecular-level models 210:that occur due to its 155:See Knowledge (XXG)'s 6166:www.sciencedirect.com 6050:Hyperelastic material 6030: 6003: 5983: 5944: 5795: 5775: 5749:Worm-like chain model 5739: 5737:{\displaystyle M_{s}} 5712: 5710:{\displaystyle \rho } 5692: 5634: 5405: 5382: 5234: 4964: 4895: 4834: 4780: 4721: 4673: 4671:{\displaystyle n_{s}} 4646: 4463: 4122: 4000: 3998:{\displaystyle R_{z}} 3973: 3971:{\displaystyle R_{y}} 3946: 3944:{\displaystyle R_{x}} 3917: 3374: 3337: 3300: 3263: 3226: 3189: 3152: 3125: 3123:{\displaystyle L_{z}} 3098: 3096:{\displaystyle L_{y}} 3071: 3069:{\displaystyle L_{x}} 3041: 2995: 2780: 2628: 2524: 2419: 2386: 2357: 2340:Helmholtz free energy 2331: 2125: 2096: 1883: 1863: 1840: 1813: 1793: 1773: 1753: 1731: 1628: 1446: 1418: 1376: 1352: 1088: 1068: 1032: 980: 938: 846: 822: 764: 588: 568: 548: 490: 412: 375:statistical mechanics 367:modulus of elasticity 6045:Elasticity (physics) 6012: 5992: 5959: 5804: 5784: 5757: 5721: 5701: 5643: 5420: 5394: 5249: 4973: 4904: 4851: 4790: 4730: 4682: 4655: 4474: 4131: 4009: 3982: 3955: 3928: 3385: 3346: 3309: 3272: 3235: 3198: 3161: 3134: 3107: 3080: 3053: 3006: 2788: 2636: 2532: 2444: 2417:{\displaystyle F=kx} 2399: 2366: 2346: 2136: 2108: 1892: 1872: 1849: 1822: 1802: 1782: 1762: 1742: 1636: 1617:Freely Jointed Chain 1521:is the free energy, 1102: 1077: 1057: 1012: 963: 860: 835: 790: 602: 577: 557: 537: 54:improve this article 6631:2020PhRvE.102a2501B 6516:1963AmJPh..31..397T 6489:10.1021/ie50377a017 6445:1944PhRv...66...30M 5532: 5369: 5197: 5122: 5104: 5086: 4958: 4900:it is assumed that 4774: 4623: 4605: 4587: 4450: 4419: 4401: 4383: 4276: 4258: 4240: 4086: 4059: 4032: 3881: 3852: 3829: 3800: 3777: 3748: 3660: 3639: 3611: 3590: 3562: 3541: 2203: 439:thermal equilibrium 212:molecular structure 208:molecular processes 69:"Rubber elasticity" 6548:Shakhashiri (1983) 6542:2019-06-13 at the 6364:Trans. Faraday Soc 6142:www.britannica.com 6118:www.britannica.com 6025: 5998: 5978: 5939: 5790: 5770: 5734: 5707: 5687: 5629: 5518: 5400: 5377: 5355: 5243:engineering stress 5229: 5183: 5108: 5090: 5072: 4959: 4933: 4890: 4829: 4775: 4760: 4716: 4668: 4641: 4609: 4591: 4573: 4458: 4456: 4436: 4405: 4387: 4369: 4262: 4244: 4226: 4117: 4069: 4042: 4015: 3995: 3968: 3941: 3912: 3910: 3864: 3838: 3812: 3786: 3760: 3734: 3643: 3625: 3594: 3576: 3545: 3527: 3369: 3332: 3295: 3258: 3221: 3184: 3147: 3120: 3093: 3066: 3036: 2990: 2775: 2623: 2519: 2414: 2381: 2352: 2326: 2324: 2189: 2120: 2091: 1878: 1861:{\displaystyle Nb} 1858: 1835: 1808: 1788: 1768: 1748: 1726: 1631: 1449: 1439:Snap-back velocity 1421: 1379: 1347: 1083: 1063: 1027: 975: 933: 841: 817: 759: 583: 563: 543: 524:Network morphology 493: 478:molecular dynamics 415: 224:chemical reactions 6675:Rubber properties 6570:978-0-299-08890-3 6524:10.1119/1.1969535 6406:10.5254/1.3540117 6376:10.5254/1.3546701 6094:978-1-83969-961-0 6022: 6001:{\displaystyle r} 5975: 5932: 5929: 5912: 5899: 5885: 5840: 5837: 5823: 5793:{\displaystyle b} 5767: 5627: 5576: 5533: 5479: 5467: 5445: 5403:{\displaystyle E} 5370: 5313: 5301: 5284: 5259: 5216: 5176: 5153: 5138: 5048: 5030: 5020: 5005: 4986: 4639: 4549: 4531: 4521: 4506: 4487: 4452: 4332: 4307: 4189: 4163: 4148: 3906: 3718: 3690: 3511: 3490: 3462: 3446: 3417: 3402: 3019: 2987: 2976: 2954: 2932: 2904: 2888: 2869: 2865: 2842: 2838: 2818: 2766: 2754: 2732: 2698: 2657: 2621: 2593: 2580: 2548: 2513: 2485: 2438:neo-Hookean model 2317: 2302: 2245: 2085: 2010: 1910: 1881:{\displaystyle r} 1832: 1811:{\displaystyle R} 1791:{\displaystyle b} 1771:{\displaystyle N} 1751:{\displaystyle b} 1723: 1698: 1673: 1648: 1499:Gibbs free energy 1371: 1370: 1340: 1264: 1086:{\displaystyle r} 1066:{\displaystyle N} 1004:) decreases with 957: 956: 844:{\displaystyle N} 815: 814: 812: 783: 782: 751: 675: 586:{\displaystyle r} 566:{\displaystyle b} 546:{\displaystyle n} 517:quantum chemistry 188:Rubber elasticity 185: 184: 177: 149:encyclopedic tone 130: 129: 122: 104: 18:Rubber Elasticity 16:(Redirected from 6692: 6659: 6658: 6624: 6604: 6598: 6597: 6580: 6574: 6573: 6556: 6550: 6533: 6527: 6526: 6499: 6493: 6492: 6472: 6466: 6463: 6457: 6456: 6428: 6419: 6416: 6410: 6409: 6389: 6380: 6379: 6359: 6353: 6350: 6344: 6341: 6332: 6329: 6323: 6320: 6314: 6311: 6300: 6297: 6286: 6283: 6274: 6271: 6262: 6259: 6250: 6247: 6238: 6235: 6224: 6221: 6212: 6209: 6203: 6200: 6194: 6191: 6185: 6182: 6176: 6175: 6173: 6172: 6158: 6152: 6151: 6149: 6148: 6134: 6128: 6127: 6125: 6124: 6110: 6104: 6103: 6102: 6101: 6076: 6034: 6032: 6031: 6026: 6024: 6023: 6020: 6007: 6005: 6004: 5999: 5987: 5985: 5984: 5979: 5977: 5976: 5973: 5948: 5946: 5945: 5940: 5938: 5934: 5933: 5931: 5930: 5927: 5918: 5913: 5905: 5900: 5898: 5897: 5896: 5891: 5887: 5886: 5884: 5883: 5882: 5869: 5848: 5841: 5839: 5838: 5835: 5829: 5825: 5824: 5821: 5814: 5799: 5797: 5796: 5791: 5779: 5777: 5776: 5771: 5769: 5768: 5765: 5743: 5741: 5740: 5735: 5733: 5732: 5716: 5714: 5713: 5708: 5696: 5694: 5693: 5688: 5686: 5685: 5676: 5671: 5670: 5655: 5654: 5638: 5636: 5635: 5630: 5628: 5626: 5625: 5616: 5599: 5591: 5590: 5578: 5577: 5574: 5562: 5561: 5554: 5553: 5543: 5539: 5535: 5534: 5531: 5526: 5514: 5494: 5493: 5481: 5480: 5477: 5468: 5466: 5465: 5464: 5451: 5447: 5446: 5443: 5430: 5409: 5407: 5406: 5401: 5386: 5384: 5383: 5378: 5376: 5372: 5371: 5368: 5363: 5351: 5346: 5345: 5328: 5327: 5315: 5314: 5311: 5302: 5300: 5299: 5290: 5286: 5285: 5282: 5266: 5261: 5260: 5257: 5238: 5236: 5235: 5230: 5228: 5224: 5217: 5215: 5214: 5202: 5196: 5191: 5177: 5172: 5168: 5167: 5155: 5154: 5151: 5144: 5139: 5134: 5133: 5129: 5121: 5116: 5103: 5098: 5085: 5080: 5063: 5062: 5050: 5049: 5046: 5039: 5031: 5026: 5022: 5021: 5013: 5007: 5006: 5003: 4993: 4988: 4987: 4984: 4968: 4966: 4965: 4960: 4957: 4953: 4941: 4929: 4928: 4916: 4915: 4899: 4897: 4896: 4891: 4883: 4882: 4873: 4872: 4863: 4862: 4838: 4836: 4835: 4830: 4822: 4821: 4812: 4811: 4802: 4801: 4784: 4782: 4781: 4776: 4773: 4768: 4759: 4751: 4750: 4725: 4723: 4722: 4717: 4712: 4707: 4706: 4694: 4693: 4677: 4675: 4674: 4669: 4667: 4666: 4650: 4648: 4647: 4642: 4640: 4635: 4634: 4630: 4622: 4617: 4604: 4599: 4586: 4581: 4564: 4563: 4551: 4550: 4547: 4540: 4532: 4527: 4523: 4522: 4514: 4508: 4507: 4504: 4494: 4489: 4488: 4485: 4467: 4465: 4464: 4459: 4457: 4453: 4451: 4449: 4444: 4431: 4430: 4426: 4418: 4413: 4400: 4395: 4382: 4377: 4360: 4359: 4347: 4346: 4334: 4333: 4330: 4323: 4312: 4308: 4306: 4305: 4304: 4288: 4287: 4283: 4275: 4270: 4257: 4252: 4239: 4234: 4217: 4216: 4204: 4203: 4191: 4190: 4187: 4180: 4165: 4164: 4156: 4150: 4149: 4146: 4126: 4124: 4123: 4118: 4113: 4105: 4104: 4085: 4080: 4058: 4053: 4031: 4026: 4004: 4002: 4001: 3996: 3994: 3993: 3977: 3975: 3974: 3969: 3967: 3966: 3950: 3948: 3947: 3942: 3940: 3939: 3921: 3919: 3918: 3913: 3911: 3907: 3905: 3904: 3903: 3887: 3886: 3882: 3880: 3875: 3863: 3859: 3851: 3846: 3828: 3823: 3811: 3807: 3799: 3794: 3776: 3771: 3759: 3755: 3747: 3742: 3720: 3719: 3716: 3706: 3695: 3691: 3689: 3688: 3687: 3671: 3670: 3666: 3665: 3661: 3659: 3654: 3638: 3633: 3616: 3612: 3610: 3605: 3589: 3584: 3567: 3563: 3561: 3556: 3540: 3535: 3513: 3512: 3509: 3499: 3491: 3489: 3488: 3487: 3471: 3470: 3469: 3464: 3463: 3455: 3448: 3447: 3444: 3434: 3419: 3418: 3410: 3404: 3403: 3400: 3378: 3376: 3375: 3370: 3368: 3367: 3358: 3357: 3341: 3339: 3338: 3333: 3331: 3330: 3321: 3320: 3304: 3302: 3301: 3296: 3294: 3293: 3284: 3283: 3267: 3265: 3264: 3259: 3257: 3256: 3247: 3246: 3230: 3228: 3227: 3222: 3220: 3219: 3210: 3209: 3193: 3191: 3190: 3185: 3183: 3182: 3173: 3172: 3156: 3154: 3153: 3148: 3146: 3145: 3129: 3127: 3126: 3121: 3119: 3118: 3102: 3100: 3099: 3094: 3092: 3091: 3075: 3073: 3072: 3067: 3065: 3064: 3045: 3043: 3042: 3037: 3029: 3021: 3020: 3017: 2999: 2997: 2996: 2991: 2989: 2988: 2980: 2977: 2975: 2974: 2973: 2960: 2956: 2955: 2952: 2942: 2937: 2933: 2931: 2930: 2929: 2913: 2912: 2911: 2906: 2905: 2897: 2890: 2889: 2886: 2876: 2870: 2868: 2867: 2866: 2858: 2848: 2843: 2841: 2840: 2839: 2831: 2824: 2820: 2819: 2811: 2798: 2784: 2782: 2781: 2776: 2774: 2773: 2768: 2767: 2759: 2755: 2753: 2752: 2751: 2738: 2734: 2733: 2730: 2720: 2706: 2705: 2700: 2699: 2691: 2684: 2683: 2659: 2658: 2650: 2632: 2630: 2629: 2624: 2622: 2620: 2619: 2618: 2602: 2601: 2600: 2595: 2594: 2586: 2582: 2581: 2578: 2565: 2550: 2549: 2546: 2528: 2526: 2525: 2520: 2518: 2514: 2512: 2511: 2510: 2494: 2493: 2492: 2487: 2486: 2478: 2467: 2423: 2421: 2420: 2415: 2390: 2388: 2387: 2382: 2361: 2359: 2358: 2353: 2335: 2333: 2332: 2327: 2325: 2318: 2313: 2304: 2303: 2295: 2289: 2288: 2272: 2271: 2247: 2246: 2238: 2229: 2228: 2213: 2212: 2202: 2197: 2178: 2177: 2129: 2127: 2126: 2121: 2100: 2098: 2097: 2092: 2090: 2086: 2084: 2083: 2082: 2066: 2065: 2064: 2048: 2036: 2035: 2034: 2029: 2024: 2015: 2011: 2006: 2002: 2001: 1985: 1972: 1971: 1953: 1952: 1934: 1933: 1912: 1911: 1903: 1887: 1885: 1884: 1879: 1867: 1865: 1864: 1859: 1844: 1842: 1841: 1836: 1834: 1833: 1830: 1817: 1815: 1814: 1809: 1797: 1795: 1794: 1789: 1777: 1775: 1774: 1769: 1757: 1755: 1754: 1749: 1735: 1733: 1732: 1727: 1725: 1724: 1716: 1713: 1712: 1700: 1699: 1691: 1688: 1687: 1675: 1674: 1666: 1663: 1662: 1650: 1649: 1641: 1550: 1469: 1468: 1464: 1365: 1356: 1354: 1353: 1348: 1346: 1342: 1341: 1339: 1338: 1337: 1321: 1320: 1319: 1306: 1290: 1289: 1288: 1283: 1278: 1269: 1265: 1260: 1256: 1255: 1239: 1232: 1231: 1215: 1214: 1203: 1202: 1198: 1197: 1196: 1174: 1173: 1153: 1096: 1092: 1090: 1089: 1084: 1072: 1070: 1069: 1064: 1036: 1034: 1033: 1028: 984: 982: 981: 976: 951: 942: 940: 939: 934: 928: 927: 916: 915: 911: 910: 909: 887: 886: 854: 850: 848: 847: 842: 826: 824: 823: 818: 816: 813: 811: 810: 802: 801: 777: 768: 766: 765: 760: 757: 753: 752: 750: 749: 748: 732: 731: 730: 717: 701: 700: 699: 694: 689: 680: 676: 671: 667: 666: 650: 643: 642: 618: 596: 592: 590: 589: 584: 572: 570: 569: 564: 552: 550: 549: 544: 406: 405: 401: 309:Charles Goodyear 180: 173: 169: 166: 160: 159:for suggestions. 140: 139: 132: 125: 118: 114: 111: 105: 103: 62: 38: 30: 21: 6700: 6699: 6695: 6694: 6693: 6691: 6690: 6689: 6665: 6664: 6663: 6662: 6606: 6605: 6601: 6595: 6582: 6581: 6577: 6571: 6558: 6557: 6553: 6544:Wayback Machine 6534: 6530: 6501: 6500: 6496: 6474: 6473: 6469: 6464: 6460: 6430: 6429: 6422: 6417: 6413: 6391: 6390: 6383: 6361: 6360: 6356: 6351: 6347: 6342: 6335: 6330: 6326: 6321: 6317: 6312: 6303: 6298: 6289: 6284: 6277: 6272: 6265: 6260: 6253: 6248: 6241: 6236: 6227: 6222: 6215: 6210: 6206: 6201: 6197: 6192: 6188: 6183: 6179: 6170: 6168: 6160: 6159: 6155: 6146: 6144: 6136: 6135: 6131: 6122: 6120: 6112: 6111: 6107: 6099: 6097: 6095: 6078: 6077: 6073: 6068: 6041: 6015: 6010: 6009: 5990: 5989: 5968: 5957: 5956: 5922: 5873: 5861: 5857: 5856: 5852: 5846: 5842: 5830: 5816: 5815: 5802: 5801: 5782: 5781: 5760: 5755: 5754: 5751: 5724: 5719: 5718: 5699: 5698: 5677: 5662: 5646: 5641: 5640: 5617: 5600: 5582: 5569: 5545: 5506: 5502: 5499: 5498: 5485: 5472: 5456: 5452: 5438: 5431: 5418: 5417: 5392: 5391: 5389:Young's Modulus 5337: 5336: 5332: 5319: 5306: 5291: 5277: 5267: 5252: 5247: 5246: 5206: 5182: 5178: 5159: 5146: 5145: 5071: 5067: 5054: 5041: 5040: 4998: 4994: 4979: 4971: 4970: 4920: 4907: 4902: 4901: 4874: 4864: 4854: 4849: 4848: 4813: 4803: 4793: 4788: 4787: 4742: 4728: 4727: 4698: 4685: 4680: 4679: 4658: 4653: 4652: 4572: 4568: 4555: 4542: 4541: 4499: 4495: 4480: 4472: 4471: 4455: 4454: 4432: 4368: 4364: 4351: 4338: 4325: 4324: 4310: 4309: 4296: 4289: 4225: 4221: 4208: 4195: 4182: 4181: 4169: 4141: 4129: 4128: 4096: 4007: 4006: 3985: 3980: 3979: 3958: 3953: 3952: 3931: 3926: 3925: 3909: 3908: 3895: 3888: 3837: 3833: 3785: 3781: 3733: 3729: 3728: 3724: 3711: 3707: 3693: 3692: 3679: 3672: 3624: 3620: 3575: 3571: 3526: 3522: 3521: 3517: 3504: 3500: 3479: 3472: 3452: 3439: 3435: 3423: 3395: 3383: 3382: 3359: 3349: 3344: 3343: 3322: 3312: 3307: 3306: 3285: 3275: 3270: 3269: 3248: 3238: 3233: 3232: 3211: 3201: 3196: 3195: 3174: 3164: 3159: 3158: 3157:to the length ( 3137: 3132: 3131: 3110: 3105: 3104: 3083: 3078: 3077: 3056: 3051: 3050: 3012: 3004: 3003: 2965: 2961: 2947: 2943: 2921: 2914: 2894: 2881: 2877: 2871: 2852: 2825: 2799: 2786: 2785: 2756: 2743: 2739: 2725: 2721: 2688: 2675: 2634: 2633: 2610: 2603: 2583: 2573: 2566: 2541: 2530: 2529: 2502: 2495: 2475: 2468: 2462: 2442: 2441: 2397: 2396: 2364: 2363: 2344: 2343: 2323: 2322: 2305: 2290: 2280: 2274: 2273: 2263: 2220: 2204: 2182: 2169: 2163: 2162: 2152: 2134: 2133: 2106: 2105: 2074: 2067: 2056: 2049: 2043: 1993: 1986: 1980: 1979: 1963: 1944: 1925: 1890: 1889: 1870: 1869: 1847: 1846: 1825: 1820: 1819: 1800: 1799: 1780: 1779: 1760: 1759: 1740: 1739: 1704: 1679: 1654: 1634: 1633: 1619: 1613: 1605:entropic spring 1601: 1534: 1501:. Rearranging Δ 1494: 1486:Hubert M. James 1479: 1466: 1462: 1461: 1441: 1413: 1408: 1393: 1363: 1329: 1322: 1311: 1307: 1301: 1297: 1247: 1240: 1234: 1233: 1223: 1188: 1181: 1177: 1175: 1165: 1100: 1099: 1075: 1074: 1055: 1054: 1010: 1009: 990: 961: 960: 949: 901: 894: 890: 888: 878: 858: 857: 833: 832: 793: 788: 787: 775: 740: 733: 722: 718: 712: 708: 658: 651: 645: 644: 634: 600: 599: 575: 574: 555: 554: 535: 534: 526: 507: 498: 469: 407: 403: 399: 397: 396: 358: 274: 254:chemically bond 206:is produced by 195:to the rubber. 181: 170: 164: 161: 154: 145:This article's 141: 137: 126: 115: 109: 106: 63: 61: 51: 39: 28: 23: 22: 15: 12: 11: 5: 6698: 6696: 6688: 6687: 6682: 6680:Thermodynamics 6677: 6667: 6666: 6661: 6660: 6599: 6593: 6575: 6569: 6551: 6528: 6494: 6483:(5): 624–629. 6477:Ind. Eng. Chem 6467: 6458: 6439:(1–2): 30–32. 6420: 6411: 6381: 6370:(4): 813–825. 6354: 6345: 6333: 6324: 6315: 6301: 6287: 6275: 6263: 6251: 6239: 6225: 6213: 6204: 6195: 6186: 6177: 6153: 6129: 6105: 6093: 6087:, IntechOpen, 6070: 6069: 6067: 6064: 6063: 6062: 6060:Thermodynamics 6057: 6052: 6047: 6040: 6037: 6018: 5997: 5971: 5967: 5964: 5937: 5925: 5921: 5916: 5911: 5908: 5903: 5895: 5890: 5881: 5876: 5872: 5867: 5864: 5860: 5855: 5851: 5845: 5833: 5828: 5819: 5812: 5809: 5789: 5763: 5750: 5747: 5731: 5727: 5706: 5684: 5680: 5675: 5669: 5665: 5661: 5658: 5653: 5649: 5624: 5620: 5615: 5612: 5609: 5606: 5603: 5597: 5594: 5589: 5585: 5581: 5572: 5568: 5565: 5560: 5557: 5552: 5548: 5542: 5538: 5530: 5525: 5521: 5517: 5512: 5509: 5505: 5501: 5497: 5492: 5488: 5484: 5475: 5471: 5463: 5459: 5455: 5450: 5441: 5437: 5434: 5428: 5425: 5399: 5375: 5367: 5362: 5358: 5354: 5349: 5344: 5340: 5335: 5331: 5326: 5322: 5318: 5309: 5305: 5298: 5294: 5289: 5280: 5276: 5273: 5270: 5264: 5255: 5227: 5223: 5220: 5213: 5209: 5205: 5200: 5195: 5190: 5186: 5181: 5175: 5171: 5166: 5162: 5158: 5149: 5142: 5137: 5132: 5128: 5125: 5120: 5115: 5111: 5107: 5102: 5097: 5093: 5089: 5084: 5079: 5075: 5070: 5066: 5061: 5057: 5053: 5044: 5037: 5034: 5029: 5025: 5019: 5016: 5010: 5001: 4997: 4991: 4982: 4978: 4956: 4952: 4948: 4945: 4940: 4936: 4932: 4927: 4923: 4919: 4914: 4910: 4889: 4886: 4881: 4877: 4871: 4867: 4861: 4857: 4828: 4825: 4820: 4816: 4810: 4806: 4800: 4796: 4772: 4767: 4763: 4758: 4754: 4749: 4745: 4741: 4738: 4735: 4715: 4711: 4705: 4701: 4697: 4692: 4688: 4665: 4661: 4638: 4633: 4629: 4626: 4621: 4616: 4612: 4608: 4603: 4598: 4594: 4590: 4585: 4580: 4576: 4571: 4567: 4562: 4558: 4554: 4545: 4538: 4535: 4530: 4526: 4520: 4517: 4511: 4502: 4498: 4492: 4483: 4479: 4448: 4443: 4439: 4435: 4429: 4425: 4422: 4417: 4412: 4408: 4404: 4399: 4394: 4390: 4386: 4381: 4376: 4372: 4367: 4363: 4358: 4354: 4350: 4345: 4341: 4337: 4328: 4321: 4318: 4315: 4313: 4311: 4303: 4299: 4295: 4292: 4286: 4282: 4279: 4274: 4269: 4265: 4261: 4256: 4251: 4247: 4243: 4238: 4233: 4229: 4224: 4220: 4215: 4211: 4207: 4202: 4198: 4194: 4185: 4178: 4175: 4172: 4170: 4168: 4162: 4159: 4153: 4144: 4140: 4137: 4136: 4116: 4112: 4108: 4103: 4099: 4095: 4092: 4089: 4084: 4079: 4076: 4072: 4068: 4065: 4062: 4057: 4052: 4049: 4045: 4041: 4038: 4035: 4030: 4025: 4022: 4018: 4014: 3992: 3988: 3965: 3961: 3938: 3934: 3902: 3898: 3894: 3891: 3885: 3879: 3874: 3871: 3867: 3862: 3858: 3855: 3850: 3845: 3841: 3836: 3832: 3827: 3822: 3819: 3815: 3810: 3806: 3803: 3798: 3793: 3789: 3784: 3780: 3775: 3770: 3767: 3763: 3758: 3754: 3751: 3746: 3741: 3737: 3732: 3727: 3723: 3714: 3710: 3704: 3701: 3698: 3696: 3694: 3686: 3682: 3678: 3675: 3669: 3664: 3658: 3653: 3650: 3646: 3642: 3637: 3632: 3628: 3623: 3619: 3615: 3609: 3604: 3601: 3597: 3593: 3588: 3583: 3579: 3574: 3570: 3566: 3560: 3555: 3552: 3548: 3544: 3539: 3534: 3530: 3525: 3520: 3516: 3507: 3503: 3497: 3494: 3486: 3482: 3478: 3475: 3468: 3461: 3458: 3451: 3442: 3438: 3432: 3429: 3426: 3424: 3422: 3416: 3413: 3407: 3398: 3394: 3391: 3390: 3366: 3362: 3356: 3352: 3329: 3325: 3319: 3315: 3292: 3288: 3282: 3278: 3255: 3251: 3245: 3241: 3218: 3214: 3208: 3204: 3181: 3177: 3171: 3167: 3144: 3140: 3117: 3113: 3090: 3086: 3063: 3059: 3035: 3032: 3028: 3024: 3015: 3011: 2986: 2983: 2972: 2968: 2964: 2959: 2950: 2946: 2940: 2936: 2928: 2924: 2920: 2917: 2910: 2903: 2900: 2893: 2884: 2880: 2874: 2864: 2861: 2855: 2851: 2846: 2837: 2834: 2828: 2823: 2817: 2814: 2808: 2805: 2802: 2796: 2793: 2772: 2765: 2762: 2750: 2746: 2742: 2737: 2728: 2724: 2718: 2715: 2712: 2709: 2704: 2697: 2694: 2687: 2682: 2678: 2674: 2671: 2668: 2665: 2662: 2656: 2653: 2647: 2644: 2641: 2617: 2613: 2609: 2606: 2599: 2592: 2589: 2576: 2572: 2569: 2563: 2559: 2556: 2553: 2544: 2540: 2537: 2517: 2509: 2505: 2501: 2498: 2491: 2484: 2481: 2474: 2471: 2465: 2461: 2458: 2455: 2452: 2449: 2428:is the force, 2413: 2410: 2407: 2404: 2380: 2377: 2374: 2371: 2351: 2321: 2316: 2311: 2308: 2306: 2301: 2298: 2293: 2287: 2283: 2279: 2276: 2275: 2270: 2266: 2262: 2259: 2256: 2253: 2250: 2244: 2241: 2235: 2232: 2227: 2223: 2219: 2216: 2211: 2207: 2201: 2196: 2192: 2188: 2185: 2183: 2181: 2176: 2172: 2168: 2165: 2164: 2161: 2158: 2155: 2153: 2151: 2148: 2145: 2142: 2141: 2119: 2116: 2113: 2089: 2081: 2077: 2073: 2070: 2063: 2059: 2055: 2052: 2046: 2042: 2039: 2033: 2028: 2023: 2019: 2014: 2009: 2005: 2000: 1996: 1992: 1989: 1983: 1978: 1975: 1970: 1966: 1962: 1959: 1956: 1951: 1947: 1943: 1940: 1937: 1932: 1928: 1924: 1921: 1918: 1915: 1909: 1906: 1900: 1897: 1877: 1857: 1854: 1828: 1807: 1787: 1767: 1747: 1738:In the model, 1722: 1719: 1711: 1707: 1703: 1697: 1694: 1686: 1682: 1678: 1672: 1669: 1661: 1657: 1653: 1647: 1644: 1615:Main article: 1612: 1609: 1600: 1597: 1493: 1492:Thermodynamics 1490: 1478: 1475: 1440: 1437: 1412: 1409: 1407: 1404: 1392: 1389: 1384:force constant 1369: 1368: 1359: 1357: 1345: 1336: 1332: 1328: 1325: 1318: 1314: 1310: 1304: 1300: 1296: 1293: 1287: 1282: 1277: 1273: 1268: 1263: 1259: 1254: 1250: 1246: 1243: 1237: 1230: 1226: 1222: 1219: 1213: 1210: 1207: 1201: 1195: 1191: 1187: 1184: 1180: 1172: 1168: 1163: 1159: 1156: 1152: 1148: 1145: 1142: 1139: 1136: 1133: 1130: 1126: 1122: 1119: 1116: 1113: 1110: 1107: 1082: 1062: 1026: 1023: 1020: 1017: 988: 974: 971: 968: 955: 954: 945: 943: 932: 926: 923: 920: 914: 908: 904: 900: 897: 893: 885: 881: 877: 874: 871: 868: 865: 840: 808: 805: 800: 796: 781: 780: 771: 769: 756: 747: 743: 739: 736: 729: 725: 721: 715: 711: 707: 704: 698: 693: 688: 684: 679: 674: 670: 665: 661: 657: 654: 648: 641: 637: 633: 630: 627: 624: 621: 617: 613: 610: 607: 582: 562: 542: 525: 522: 512:dihedral angle 506: 503: 497: 494: 468: 465: 419:natural rubber 395: 392: 357: 354: 350:tensile strain 339:nature of the 293:thermodynamics 278:natural rubber 273: 270: 251:polymer chains 220:polymerization 183: 182: 144: 142: 135: 128: 127: 42: 40: 33: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 6697: 6686: 6683: 6681: 6678: 6676: 6673: 6672: 6670: 6656: 6652: 6648: 6644: 6640: 6636: 6632: 6628: 6623: 6618: 6615:(1): 012501. 6614: 6610: 6603: 6600: 6596: 6594:9780198570271 6590: 6586: 6579: 6576: 6572: 6566: 6562: 6555: 6552: 6549: 6545: 6541: 6538: 6532: 6529: 6525: 6521: 6517: 6513: 6509: 6505: 6498: 6495: 6490: 6486: 6482: 6478: 6471: 6468: 6462: 6459: 6454: 6450: 6446: 6442: 6438: 6434: 6427: 6425: 6421: 6415: 6412: 6407: 6403: 6399: 6395: 6394:J. Phys. Chem 6388: 6386: 6382: 6377: 6373: 6369: 6365: 6358: 6355: 6349: 6346: 6340: 6338: 6334: 6328: 6325: 6319: 6316: 6310: 6308: 6306: 6302: 6296: 6294: 6292: 6288: 6282: 6280: 6276: 6270: 6268: 6264: 6258: 6256: 6252: 6246: 6244: 6240: 6234: 6232: 6230: 6226: 6220: 6218: 6214: 6208: 6205: 6199: 6196: 6190: 6187: 6181: 6178: 6167: 6163: 6157: 6154: 6143: 6139: 6133: 6130: 6119: 6115: 6109: 6106: 6096: 6090: 6086: 6082: 6075: 6072: 6065: 6061: 6058: 6056: 6053: 6051: 6048: 6046: 6043: 6042: 6038: 6036: 6016: 5995: 5969: 5965: 5962: 5954: 5949: 5935: 5923: 5919: 5914: 5909: 5906: 5901: 5893: 5888: 5874: 5870: 5865: 5862: 5858: 5853: 5849: 5843: 5831: 5826: 5817: 5810: 5807: 5787: 5761: 5748: 5746: 5729: 5725: 5704: 5682: 5678: 5673: 5667: 5663: 5659: 5656: 5651: 5647: 5622: 5618: 5613: 5610: 5607: 5604: 5601: 5595: 5592: 5587: 5583: 5579: 5570: 5566: 5563: 5558: 5555: 5550: 5546: 5540: 5536: 5528: 5523: 5519: 5515: 5510: 5507: 5503: 5495: 5490: 5486: 5482: 5473: 5469: 5461: 5457: 5453: 5439: 5432: 5426: 5423: 5415: 5413: 5397: 5390: 5373: 5365: 5360: 5356: 5352: 5347: 5342: 5338: 5333: 5329: 5324: 5320: 5316: 5307: 5303: 5296: 5292: 5278: 5268: 5262: 5253: 5244: 5239: 5225: 5221: 5218: 5211: 5207: 5203: 5198: 5193: 5188: 5184: 5179: 5173: 5169: 5164: 5160: 5156: 5147: 5140: 5135: 5130: 5126: 5123: 5118: 5113: 5109: 5105: 5100: 5095: 5091: 5087: 5082: 5077: 5073: 5068: 5064: 5059: 5055: 5051: 5042: 5035: 5032: 5027: 5014: 4999: 4989: 4980: 4954: 4950: 4946: 4943: 4938: 4934: 4930: 4925: 4921: 4917: 4912: 4908: 4887: 4884: 4879: 4875: 4869: 4865: 4859: 4855: 4845: 4844: 4840: 4826: 4823: 4818: 4814: 4808: 4804: 4798: 4794: 4770: 4765: 4761: 4756: 4747: 4743: 4736: 4733: 4713: 4709: 4703: 4699: 4695: 4690: 4686: 4663: 4659: 4636: 4631: 4627: 4624: 4619: 4614: 4610: 4606: 4601: 4596: 4592: 4588: 4583: 4578: 4574: 4569: 4565: 4560: 4556: 4552: 4543: 4536: 4533: 4528: 4515: 4500: 4490: 4481: 4468: 4446: 4441: 4437: 4433: 4427: 4423: 4420: 4415: 4410: 4406: 4402: 4397: 4392: 4388: 4384: 4379: 4374: 4370: 4365: 4356: 4352: 4343: 4339: 4335: 4326: 4319: 4316: 4314: 4301: 4297: 4293: 4290: 4284: 4280: 4277: 4272: 4267: 4263: 4259: 4254: 4249: 4245: 4241: 4236: 4231: 4227: 4222: 4213: 4209: 4200: 4196: 4192: 4183: 4176: 4173: 4171: 4157: 4142: 4114: 4110: 4101: 4097: 4090: 4082: 4077: 4074: 4070: 4063: 4055: 4050: 4047: 4043: 4036: 4028: 4023: 4020: 4016: 3990: 3986: 3963: 3959: 3936: 3932: 3922: 3900: 3896: 3892: 3889: 3883: 3877: 3872: 3869: 3865: 3860: 3856: 3853: 3848: 3843: 3839: 3834: 3830: 3825: 3820: 3817: 3813: 3808: 3804: 3801: 3796: 3791: 3787: 3782: 3778: 3773: 3768: 3765: 3761: 3756: 3752: 3749: 3744: 3739: 3735: 3730: 3725: 3721: 3712: 3708: 3702: 3699: 3697: 3684: 3680: 3676: 3673: 3667: 3662: 3656: 3651: 3648: 3644: 3640: 3635: 3630: 3626: 3621: 3617: 3613: 3607: 3602: 3599: 3595: 3591: 3586: 3581: 3577: 3572: 3568: 3564: 3558: 3553: 3550: 3546: 3542: 3537: 3532: 3528: 3523: 3518: 3514: 3505: 3501: 3495: 3492: 3484: 3480: 3476: 3473: 3466: 3456: 3449: 3440: 3436: 3430: 3427: 3425: 3411: 3396: 3380: 3364: 3360: 3354: 3350: 3327: 3323: 3317: 3313: 3290: 3286: 3280: 3276: 3253: 3249: 3243: 3239: 3216: 3212: 3206: 3202: 3179: 3175: 3169: 3165: 3142: 3138: 3115: 3111: 3088: 3084: 3061: 3057: 3047: 3033: 3030: 3026: 3022: 3013: 3009: 3000: 2981: 2970: 2966: 2962: 2957: 2948: 2944: 2938: 2934: 2926: 2922: 2918: 2915: 2908: 2898: 2891: 2882: 2878: 2872: 2859: 2853: 2849: 2844: 2832: 2826: 2812: 2803: 2800: 2794: 2791: 2770: 2760: 2748: 2744: 2740: 2735: 2726: 2722: 2716: 2713: 2710: 2702: 2692: 2680: 2676: 2669: 2666: 2663: 2651: 2642: 2615: 2611: 2607: 2604: 2597: 2587: 2574: 2570: 2567: 2561: 2554: 2551: 2542: 2538: 2535: 2515: 2507: 2503: 2499: 2496: 2489: 2479: 2472: 2469: 2463: 2459: 2456: 2453: 2450: 2439: 2435: 2431: 2427: 2411: 2408: 2405: 2402: 2394: 2378: 2372: 2369: 2341: 2336: 2319: 2314: 2309: 2307: 2299: 2296: 2285: 2281: 2268: 2264: 2260: 2257: 2254: 2251: 2239: 2230: 2225: 2221: 2217: 2214: 2209: 2205: 2194: 2190: 2186: 2184: 2174: 2170: 2159: 2156: 2154: 2146: 2131: 2114: 2101: 2087: 2079: 2075: 2071: 2068: 2061: 2057: 2053: 2050: 2044: 2040: 2037: 2031: 2026: 2021: 2017: 2012: 2007: 2003: 1998: 1994: 1990: 1987: 1981: 1976: 1968: 1964: 1957: 1949: 1945: 1938: 1930: 1926: 1919: 1916: 1904: 1895: 1875: 1855: 1852: 1826: 1805: 1785: 1765: 1745: 1736: 1717: 1709: 1705: 1701: 1692: 1684: 1680: 1676: 1667: 1659: 1655: 1651: 1642: 1627: 1623: 1618: 1610: 1608: 1606: 1598: 1596: 1593: 1589: 1588:monatomic gas 1584: 1582: 1578: 1574: 1570: 1566: 1565:absolute zero 1562: 1558: 1554: 1549: 1545: 1541: 1537: 1532: 1528: 1524: 1520: 1516: 1512: 1508: 1504: 1500: 1491: 1489: 1487: 1483: 1476: 1474: 1472: 1459: 1453: 1445: 1438: 1436: 1434: 1429: 1425: 1417: 1410: 1405: 1403: 1401: 1400: 1390: 1388: 1385: 1375: 1367: 1360: 1358: 1343: 1334: 1330: 1326: 1323: 1316: 1312: 1308: 1302: 1298: 1294: 1291: 1285: 1280: 1275: 1271: 1266: 1261: 1257: 1252: 1248: 1244: 1241: 1235: 1228: 1224: 1220: 1217: 1211: 1208: 1205: 1199: 1193: 1189: 1185: 1182: 1178: 1170: 1166: 1161: 1154: 1146: 1140: 1134: 1128: 1124: 1117: 1114: 1111: 1105: 1098: 1097: 1094: 1080: 1060: 1052: 1048: 1047: 1042: 1041: 1024: 1021: 1018: 1015: 1007: 1003: 999: 995: 991: 972: 969: 966: 953: 946: 944: 930: 924: 921: 918: 912: 906: 902: 898: 895: 891: 883: 879: 875: 869: 863: 856: 855: 852: 838: 830: 806: 803: 798: 794: 779: 772: 770: 754: 745: 741: 737: 734: 727: 723: 719: 713: 709: 705: 702: 696: 691: 686: 682: 677: 672: 668: 663: 659: 655: 652: 646: 639: 635: 631: 628: 625: 619: 611: 605: 598: 597: 594: 580: 560: 540: 532: 523: 521: 518: 513: 504: 502: 495: 489: 485: 483: 479: 475: 466: 464: 461: 455: 451: 448: 444: 440: 434: 432: 426: 424: 420: 411: 402: 393: 391: 388: 383: 379: 376: 371: 368: 362: 355: 353: 351: 347: 342: 338: 332: 328: 326: 325:chemical bond 321: 316: 314: 310: 305: 303: 298: 294: 289: 287: 283: 279: 271: 269: 267: 263: 259: 255: 252: 248: 244: 243:Vulcanization 240: 236: 232: 231:polybutadiene 227: 225: 221: 217: 213: 209: 205: 201: 196: 194: 189: 179: 176: 168: 158: 152: 150: 143: 134: 133: 124: 121: 113: 102: 99: 95: 92: 88: 85: 81: 78: 74: 71: – 70: 66: 65:Find sources: 59: 55: 49: 48: 43:This article 41: 37: 32: 31: 19: 6612: 6609:Phys. Rev. E 6608: 6602: 6584: 6578: 6560: 6554: 6531: 6507: 6503: 6497: 6480: 6476: 6470: 6461: 6436: 6432: 6414: 6397: 6393: 6367: 6363: 6357: 6348: 6327: 6318: 6207: 6198: 6189: 6180: 6169:. Retrieved 6165: 6156: 6145:. Retrieved 6141: 6132: 6121:. Retrieved 6117: 6108: 6098:, retrieved 6084: 6074: 5952: 5950: 5752: 5416: 5240: 4846: 4842: 4841: 4469: 3923: 3381: 3048: 3001: 2433: 2429: 2425: 2337: 2132: 2102: 1737: 1632: 1620: 1602: 1591: 1585: 1580: 1576: 1568: 1560: 1556: 1552: 1547: 1543: 1539: 1535: 1530: 1522: 1518: 1514: 1510: 1506: 1502: 1495: 1480: 1470: 1457: 1454: 1450: 1430: 1426: 1422: 1397: 1394: 1380: 1361: 1044: 1038: 1005: 1001: 997: 993: 986: 958: 947: 784: 773: 527: 508: 499: 470: 456: 452: 435: 427: 423:polyisoprene 416: 384: 380: 372: 363: 359: 333: 329: 317: 313:polyisoprene 306: 290: 282:polyisoprene 275: 258:cross-linked 235:polyisoprene 228: 197: 187: 186: 171: 162: 146: 116: 107: 97: 90: 83: 76: 64: 52:Please help 47:verification 44: 6400:: 826–840. 2393:Hooke's law 1573:microstates 1482:Eugene Guth 1406:Experiments 829:probability 482:free energy 431:Kuhn length 337:random-walk 315:molecules. 302:James Joule 297:Lord Kelvin 202:. Rubber's 193:degradation 6669:Categories 6622:2004.07874 6510:(5): 397, 6171:2024-07-25 6147:2024-07-25 6123:2024-07-25 6100:2024-07-25 6066:References 474:tortuosity 387:morphology 286:John Gough 266:elastomers 204:elasticity 80:newspapers 6685:Mechanics 6655:215814600 6546:, citing 6433:Phys. Rev 5902:− 5866:− 5811:≈ 5705:ρ 5660:ρ 5608:β 5605:ρ 5593:β 5547:λ 5520:λ 5496:β 5458:λ 5440:σ 5412:stiffness 5357:λ 5348:− 5339:λ 5330:β 5293:λ 5275:Δ 5254:σ 5219:− 5208:λ 5185:λ 5170:β 5124:− 5110:λ 5092:λ 5074:λ 5065:β 5036:− 5018:→ 4996:Δ 4977:Δ 4944:− 4935:λ 4922:λ 4909:λ 4876:λ 4866:λ 4856:λ 4815:λ 4805:λ 4795:λ 4753:⟩ 4740:⟨ 4734:β 4625:− 4611:λ 4593:λ 4575:λ 4566:β 4537:− 4519:→ 4497:Δ 4478:Δ 4421:− 4407:λ 4389:λ 4371:λ 4362:⟩ 4349:⟨ 4320:− 4278:− 4264:λ 4246:λ 4228:λ 4219:⟩ 4206:⟨ 4177:− 4161:→ 4139:Δ 4107:⟩ 4094:⟨ 4088:⟩ 4067:⟨ 4061:⟩ 4040:⟨ 4034:⟩ 4013:⟨ 3854:− 3840:λ 3802:− 3788:λ 3750:− 3736:λ 3703:− 3641:− 3592:− 3543:− 3496:− 3460:→ 3431:− 3415:→ 3393:Δ 3351:λ 3314:λ 3277:λ 3240:λ 3203:λ 3166:λ 3139:λ 2985:→ 2902:→ 2863:→ 2836:→ 2816:→ 2764:→ 2696:→ 2673:Δ 2667:− 2664:≈ 2655:→ 2640:Δ 2591:→ 2568:− 2562:≈ 2558:Ω 2555:⁡ 2483:→ 2470:− 2460:⁡ 2448:Ω 2376:Δ 2370:− 2350:Ω 2292:⟩ 2278:⟨ 2243:→ 2218:π 2200:∞ 2191:∫ 2180:⟩ 2167:⟨ 2150:⟩ 2144:⟨ 2118:⟩ 2112:⟨ 2051:− 2041:⁡ 2018:− 2004:π 1908:→ 1721:^ 1696:^ 1671:^ 1646:→ 1303:− 1295:⁡ 1272:− 1258:π 1221:π 1209:− 1186:− 1019:∼ 970:≥ 922:− 899:− 714:− 706:⁡ 683:− 669:π 632:π 200:molecules 165:July 2024 110:July 2024 6647:32794915 6540:Archived 6055:Polymers 6039:See also 5387:and the 2424:, where 1567:), the Δ 1527:enthalpy 1517:, where 447:Gaussian 382:strain. 320:isoprene 262:polymers 216:polymers 6627:Bibcode 6512:Bibcode 6441:Bibcode 1525:is the 1509:− 1465:⁄ 1049:) (the 1043:) and ( 341:polymer 272:History 264:called 239:colloid 94:scholar 6653: 6645: 6591: 6567: 6091: 5639:where 4651:where 1529:, and 959:where 398:": --> 346:stress 247:sulfur 96: 89: 82: 75: 67: 6651:S2CID 6617:arXiv 992:) in 101:JSTOR 87:books 6643:PMID 6589:ISBN 6565:ISBN 6089:ISBN 5241:The 3978:and 3103:and 1484:and 400:edit 233:and 73:news 6635:doi 6613:102 6520:doi 6485:doi 6449:doi 6402:doi 6372:doi 5444:eng 5283:def 5258:eng 5004:def 4985:def 4505:def 4486:def 4147:def 3401:def 2457:exp 2038:exp 1592:not 1546:− Δ 1542:= Δ 1292:exp 1022:2.2 703:exp 352:). 56:by 6671:: 6649:. 6641:. 6633:. 6625:. 6611:. 6518:, 6508:31 6506:, 6481:33 6479:. 6447:. 6437:66 6435:. 6423:^ 6398:46 6396:. 6384:^ 6368:40 6366:. 6336:^ 6304:^ 6290:^ 6278:^ 6266:^ 6254:^ 6242:^ 6228:^ 6216:^ 6164:. 6140:. 6116:. 6083:, 6035:. 5697:, 4839:. 3951:, 3342:, 3305:, 3231:, 3194:, 3076:, 2552:ln 2395:: 1798:, 1505:=Δ 1471:mv 1460:= 593:: 268:. 6657:. 6637:: 6629:: 6619:: 6522:: 6514:: 6491:. 6487:: 6455:. 6451:: 6443:: 6408:. 6404:: 6378:. 6374:: 6174:. 6150:. 6126:. 6021:c 6017:L 5996:r 5974:c 5970:L 5966:= 5963:r 5953:r 5936:) 5928:c 5924:L 5920:r 5915:+ 5910:4 5907:1 5894:2 5889:) 5880:c 5875:L 5871:r 5863:1 5859:( 5854:4 5850:1 5844:( 5836:p 5832:L 5827:T 5822:B 5818:k 5808:F 5788:b 5766:p 5762:L 5730:s 5726:M 5683:s 5679:M 5674:/ 5668:a 5664:N 5657:= 5652:s 5648:v 5623:s 5619:M 5614:T 5611:R 5602:3 5596:= 5588:s 5584:v 5580:T 5575:B 5571:k 5567:3 5564:= 5559:1 5556:= 5551:z 5541:| 5537:) 5529:3 5524:z 5516:2 5511:+ 5508:1 5504:( 5491:s 5487:v 5483:T 5478:B 5474:k 5470:= 5462:z 5454:d 5449:) 5436:( 5433:d 5427:= 5424:E 5398:E 5374:) 5366:2 5361:z 5353:1 5343:z 5334:( 5325:s 5321:v 5317:T 5312:B 5308:k 5304:= 5297:z 5288:) 5279:f 5272:( 5269:d 5263:= 5226:) 5222:3 5212:z 5204:2 5199:+ 5194:2 5189:z 5180:( 5174:2 5165:s 5161:v 5157:T 5152:B 5148:k 5141:= 5136:2 5131:) 5127:3 5119:2 5114:z 5106:+ 5101:2 5096:y 5088:+ 5083:2 5078:x 5069:( 5060:s 5056:v 5052:T 5047:B 5043:k 5033:= 5028:V 5024:) 5015:R 5009:( 5000:F 4990:= 4981:f 4955:2 4951:/ 4947:1 4939:z 4931:= 4926:y 4918:= 4913:x 4888:1 4885:= 4880:z 4870:y 4860:x 4827:1 4824:= 4819:z 4809:y 4799:x 4771:2 4766:0 4762:R 4757:/ 4748:2 4744:R 4737:= 4714:V 4710:/ 4704:s 4700:n 4696:= 4691:s 4687:v 4664:s 4660:n 4637:2 4632:) 4628:3 4620:2 4615:z 4607:+ 4602:2 4597:y 4589:+ 4584:2 4579:x 4570:( 4561:s 4557:v 4553:T 4548:B 4544:k 4534:= 4529:V 4525:) 4516:R 4510:( 4501:F 4491:= 4482:f 4447:2 4442:0 4438:R 4434:2 4428:) 4424:3 4416:2 4411:z 4403:+ 4398:2 4393:y 4385:+ 4380:2 4375:x 4366:( 4357:2 4353:R 4344:s 4340:n 4336:T 4331:B 4327:k 4317:= 4302:2 4298:b 4294:N 4291:2 4285:) 4281:3 4273:2 4268:z 4260:+ 4255:2 4250:y 4242:+ 4237:2 4232:x 4223:( 4214:2 4210:R 4201:s 4197:n 4193:T 4188:B 4184:k 4174:= 4167:) 4158:R 4152:( 4143:F 4115:3 4111:/ 4102:2 4098:R 4091:= 4083:2 4078:0 4075:z 4071:R 4064:= 4056:2 4051:0 4048:y 4044:R 4037:= 4029:2 4024:0 4021:x 4017:R 3991:z 3987:R 3964:y 3960:R 3937:x 3933:R 3901:2 3897:b 3893:N 3890:2 3884:) 3878:2 3873:0 3870:z 3866:R 3861:) 3857:1 3849:2 3844:z 3835:( 3831:+ 3826:2 3821:0 3818:y 3814:R 3809:) 3805:1 3797:2 3792:y 3783:( 3779:+ 3774:2 3769:0 3766:x 3762:R 3757:) 3753:1 3745:2 3740:x 3731:( 3726:( 3722:T 3717:B 3713:k 3709:3 3700:= 3685:2 3681:b 3677:N 3674:2 3668:) 3663:) 3657:2 3652:0 3649:z 3645:R 3636:2 3631:z 3627:R 3622:( 3618:+ 3614:) 3608:2 3603:0 3600:y 3596:R 3587:2 3582:y 3578:R 3573:( 3569:+ 3565:) 3559:2 3554:0 3551:x 3547:R 3538:2 3533:x 3529:R 3524:( 3519:( 3515:T 3510:B 3506:k 3502:3 3493:= 3485:2 3481:b 3477:N 3474:2 3467:2 3457:R 3450:T 3445:B 3441:k 3437:3 3428:= 3421:) 3412:R 3406:( 3397:F 3365:z 3361:R 3355:z 3328:y 3324:R 3318:y 3291:x 3287:R 3281:x 3254:z 3250:L 3244:z 3217:y 3213:L 3207:y 3180:x 3176:L 3170:x 3143:i 3116:z 3112:L 3089:y 3085:L 3062:x 3058:L 3034:b 3031:N 3027:/ 3023:T 3018:B 3014:k 3010:3 2982:R 2971:2 2967:b 2963:N 2958:T 2953:B 2949:k 2945:3 2939:= 2935:) 2927:2 2923:b 2919:N 2916:2 2909:2 2899:R 2892:T 2887:B 2883:k 2879:3 2873:( 2860:R 2854:d 2850:d 2845:= 2833:R 2827:d 2822:) 2813:R 2807:( 2804:F 2801:d 2795:= 2792:f 2771:2 2761:R 2749:2 2745:b 2741:N 2736:T 2731:B 2727:k 2723:3 2717:+ 2714:C 2711:= 2708:) 2703:2 2693:R 2686:( 2681:d 2677:S 2670:T 2661:) 2652:R 2646:( 2643:F 2616:2 2612:b 2608:N 2605:2 2598:2 2588:R 2579:B 2575:k 2571:3 2547:B 2543:k 2539:= 2536:S 2516:) 2508:2 2504:b 2500:N 2497:2 2490:2 2480:R 2473:3 2464:( 2454:C 2451:= 2434:x 2430:k 2426:F 2412:x 2409:k 2406:= 2403:F 2379:S 2373:T 2320:b 2315:N 2310:= 2300:2 2297:1 2286:2 2282:R 2269:2 2265:b 2261:N 2258:= 2255:R 2252:d 2249:) 2240:R 2234:( 2231:P 2226:2 2222:R 2215:4 2210:2 2206:R 2195:0 2187:= 2175:2 2171:R 2160:0 2157:= 2147:R 2115:R 2088:) 2080:2 2076:b 2072:N 2069:2 2062:2 2058:R 2054:3 2045:( 2032:2 2027:/ 2022:3 2013:) 2008:3 1999:2 1995:b 1991:n 1988:2 1982:( 1977:= 1974:) 1969:z 1965:R 1961:( 1958:P 1955:) 1950:y 1946:R 1942:( 1939:P 1936:) 1931:x 1927:R 1923:( 1920:P 1917:= 1914:) 1905:R 1899:( 1896:P 1876:r 1856:b 1853:N 1831:c 1827:L 1806:R 1786:b 1766:N 1746:b 1718:z 1710:z 1706:R 1702:+ 1693:y 1685:y 1681:R 1677:+ 1668:x 1660:x 1656:R 1652:= 1643:R 1581:S 1577:G 1569:S 1561:T 1557:S 1555:Δ 1553:T 1548:G 1544:H 1540:S 1538:Δ 1536:T 1531:S 1523:H 1519:G 1515:S 1513:Δ 1511:T 1507:H 1503:G 1467:2 1463:1 1458:E 1399:4 1396:( 1366:) 1364:4 1362:( 1344:) 1335:2 1331:b 1327:n 1324:2 1317:2 1313:r 1309:3 1299:( 1286:2 1281:/ 1276:3 1267:) 1262:3 1253:2 1249:b 1245:n 1242:2 1236:( 1229:2 1225:r 1218:4 1212:1 1206:N 1200:) 1194:x 1190:p 1183:1 1179:( 1171:x 1167:p 1162:= 1158:) 1155:N 1151:| 1147:r 1144:( 1141:P 1138:) 1135:N 1132:( 1129:P 1125:= 1121:) 1118:N 1115:, 1112:r 1109:( 1106:P 1081:r 1061:N 1046:3 1040:1 1025:n 1016:N 1006:N 1002:N 1000:( 998:P 994:N 989:x 987:p 973:1 967:N 952:) 950:3 948:( 931:, 925:1 919:N 913:) 907:x 903:p 896:1 892:( 884:x 880:p 876:= 873:) 870:N 867:( 864:P 839:N 807:2 804:= 799:x 795:p 778:) 776:1 774:( 755:) 746:2 742:b 738:n 735:2 728:2 724:r 720:3 710:( 697:2 692:/ 687:3 678:) 673:3 664:2 660:b 656:n 653:2 647:( 640:2 636:r 629:4 626:= 623:) 620:n 616:| 612:r 609:( 606:P 581:r 561:b 541:n 421:( 404:] 280:( 178:) 172:( 167:) 163:( 153:. 123:) 117:( 112:) 108:( 98:· 91:· 84:· 77:· 50:. 20:)

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