Rubbers elasticity

Elasticity is the property of materials that can deform reversibly under external forces, or stress. The relative deformation amount is called the strain. For example, Hooke s spring has linear elasticity with the force being proportional to the deformation, F = kx, where k is the spring constant. 

Thomas Young introduced a property of materials, which is now known as the Young modulus or the elasticity modulus, that quantifies their elastic behavior. The Young modulus, E, is defined as 

a is the tensile stress, defined as the applied force, F, over the area, A, 

e is the tensile strain, dehned as the ratio of the amount of deformation AL over the original length, L (see Fig. 11.5) 

Many polymers, such as rubber, are characterized by high elasticity. In the 1950s, Volkestein in Leningrad and Paul Rory at Cornell worked out the details that connect the strain of a polymer to applied stress. Using statistical arguments they showed how the two are linearly related, closely following Hooke s law. 

The theory of rubber elasticity is largely based on thermodynamic considerations. It will be briefly discussed as an example of how thermodynamics can be applied in polymer science. Eor more detailed information the reader is referred to the various textbooks [10-13]. It is assumed that there is a three-dimensional network of chains, that the chain units are flexible and that individual chain segments rotate freely, that no volume change occurs upon deformation, and that the process is reversible (i.e., true elastic behavior). Another usual assumption is that the internal energy U of the system does not change with deformation. Eor this system the first law of thermodynamics can be written as  

If the equilibrium tensile force is F and the displacement dl then the work done by 

The change in U with respect to 1 at a constant temperature and volume is  

the equilibrium tensile force F is determined by the change in internal energy with deformation and the change in entropy with deformation. An ideal rubber is defined as a material for which the change in internal energy with deformation equals zero. Thus, the only contribution to the force F is from the entropy term  

When a rubber is deformed, its entropy S changes. The long chains tend to adopt a most probable configuration this is a highly coiled configuration. When the material is stretched, the chains uncoil, resulting in a less probable chain configuration. When the force is removed, the system wants to return to the more probable coiled- 

An extension of rubber elasticity (i.e. of the description of large, static and incompressible deformations) to nematic elastomers has been given in a large number of papers [52, 61-66]. Abrupt transitions between different orientations of the director under external mechanical stress have been predicted in a model without spatial nonuniformities in the strain field [52,63]. The effect of electric fields on rubber elasticity of nematics has been incorporated [65]. Finally the approach of rubber elasticity was also applied recently to smectic A [67] and to smectic C [68] elastomers. Comparisons with experiments on smectic elastomers do not appear to exist at this time. Recently a rather detailed review of the model of an- 

Let us begin by considering a prismatic piece of rubber with orthogonal edges LzjLx = Ly, and inquire about the force produced by an extension ALz in 2 -direction. We will use as independent variable the extension ratio A, defined as 

Thermodynamics provides the general tool to be applied in order to obtain the force and we first have to select the appropriate thermodynamic potential. In dealing with a rubber, we choose the Helmholtz free energy, considering that one is usually interested in the force under isothermal conditions and, moreover, that rubbers are essentially incompressible. For a piece of rubber which is to be extended in one direction, the Helmholtz free energy is a function of A. Knowing. F, the force f required to obtain an extension A follows by taking the derivative 

As we shall see, ideal rubbers possess properties which reproduce at least qualitatively the main features in the behavior of real rubbers and thus can provide an approximate first description. Equation (7.6) implies that for an ideal rubber, force and temperature are linearly related 

Indeed, a strict proportionality to the absolute temperature is the characteristic signature for all forces of entropic origin. Just remember the ideal gas which is probably the better known system, where we find p T. 

Temperature measurements can be used in order to detect and specify deviations from the ideal case for a given sample. If a piece of rubber is thermally isolated and stretched, its temperature increases. The relation between an elongational step dA and the induced change dT in temperature follows from 

A typical stress-strain curve for a pure gum natural rubber vulcanizate (i.e., without carbon black or other fillers ) is shown in Fig. 83. The stress rises slowly up to an elongation of about 500 percent (length six times initial length), then rises rapidly to a value at break in the neighborhood of 3000 pounds per square inch based on the 

The term elastomer is currently applied to a set of polymeric materials that possess exceptional elastic properties similar to those of natural rubber. These properties can be summarized as (1-3) 

Capability for instantaneous and extremely high extensibility under low mechanical stresses. 

Elastic reversibility, i.e., the capability to recover the initial length when the deforming force is removed. 

In theory, it is not difficult to associate this elastic behavior with the molecular structure of polymers. The coiled conformation of polymers is responsible for the anomalous macroscopic deformation observed in 

The properties of elastomeric materials are controlled by their molecular structure which has been discussed earlier (Section 4.4.5). They are basically all amorphous polymers above their glass transition and normally cross-linked. Their unique deformation behaviour has fascinated scientists for many years and there are even reports of investigations into the deformation of natural rubber from the beginning of the last century. Rubber elasticity is particularly amenable to analysis using thermodynamics, as an elastomer behaves essentially as an entropy spring . It is even possible to derive the form of the basic stress-strain relationship from first principles by considering the statistical thermodynamic behaviour of the molecular network. 

The first law of thermodynamics establishes the relationship between the change in internal energy of a system AU and the heat dQ absorbed by the system and work dW done by the system as 

The bars indicate that dQ and dW are inexact differentials because Q and W, unlike U, are not macroscopic functions of the system. If the length of 

The deformation of elastomers can be considered as a reversible process and so dQ can be evaluated from the second law of thermodynamics which states that for a reversible process 

Most of the experimental investigation on elastomers have been done under conditions of constant pressure. The thermodynamic function which can be used to describe equilibrium under these conditions is the Gibbs free energy (Equation 4.1), but since elastomers tend to deform at constant volume it is possible to use the Helmholtz free energy, A in the consideration of equilibrium. It is defined as 

Above about 45°C, however, considerable yielding can be observed. Note that the transition between brittle and ductile behavior occurs at a temperature that is significantly below the T. Various theories have been advanced to explain yielding phenomena in polymers, some involving free volume arguments while others involve various types of molecular motion. As far as we can make out, none of these are entirely satisfactory and we won t discuss them here. Instead, we will finish off our discussion of stress/strain behavior by considering rubber elasticity. 

Rubber is an extraordinary material with a rich history, shaped by a collection of intriguing characters and full of trials, tribulations and triumphs. There is so much good stuff that we devoted two of the chapters in our The Incredible World of Polymers CD to the subject. Here we will discuss the molecular basis of rubber elasticity. 

FIGURE 13-45 Stress/strain plot for natural rubber [redrawn from a figure in L. R. G. Treloar. 

The Physics of Rubber Elasticity, Third Edition, Clarendon Press. Oxford (1975)J. 

To understand robber elasticity we have to revisit some simple thermodynamics (the horror. the horror ). Let s start with the Helmholtz free energy of our piece of rubber, by which we mean that we are considering the free energy at constant temperature and volume (go to the review at the start of Chapter 10 if you ve also forgotten this stuff). If E is the internal energy (the sum of the potential and kinetic energies of all the particles in the system) and 5 the entropy, then (Equation 13-26)  

Cross-linked elastomers are a special case. Due to the cross-links this polymer class shows hardly any flow behaviour. 

The kinetic theory of rubber elasticity was developed by Kuhn (1936-1942), Guth, James and Mark (1946), Flory (1944-1946), Gee (1946) and Treloar (1958). It leads, for Young s modulus at low strains, to the following equation  

The term 1 — 2Mc/Mn corrects for the dangling ends, which is important if the crosslink density is small. 

37) shows that the modulus of a rubber increases with temperature this is in contrast with the behaviour of polymers that are not cross-linked. The reason of this behaviour is that rubber elasticity is an entropy elasticity in contrast with the energy elasticity in normal solids the modulus increases with temperature because of the increased thermal or Brownian motion, which causes the stretched molecular segments to tug at their anchor points and try to assume a more probable coiled-up shape. 

The theory of rubber elasticity also leads to the following stress-deformation expression (for unidirectional stretching and compression)  

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S. L. Aggrawal and co-workers, iuj. E. Mark, M.,Mdvances in Elastomer and Rubber Elasticity, Plenum Press, New York, 1968, p. 16. 

Langley, N.R. and Polmanteer, K.E., Role of chain entanglements in rubber elasticity. Polym. Prep. Am. Chem. Soc. Div. Polym. Chem., 13(1), 235-240 (1972). 

For a polymer to exhibit rubber elasticity, it must have two properties ... 

Roasting A metallurgical process in which a sulfide ore is heated in air, forming either the free metal or the metal oxide, 539 Rock candy, 17 Rowland, F. Sherwood, 311 Rubber elasticity, 470 Ruminant, 620 Rusting, 87... 

An increase in the swelling degree usually results in lowering elastic modulus. According to the rubber elasticity theory [116-118] the shear modulus of the gel G can be expressed as ... 

Substituting Eq. (12) into Eq. (11) permits us to derive the Hookean spring force law, well-known in the classical theory of rubber elasticity ... 

Treloar LRG (1975) The physics of rubber elasticity, 3rd (edn) Oxford University Press, London, chap VI... 

Crosslinked polymers are rather peculiar materials in that they never melt and they exhibit entropic elasticity at elevated temperatures. The present review on the influence of crosslink density is structured around model polymers of uniform composition but with widely varying numbers of crosslinks. The degree of crosslinking in the polymers was verified by use of the theory of rubber elasticity. 

The effective molecular mass Mc of the network strands was determined experimentally from the moduli of the polymers at temperatures above the glass transition (Sect. 3) [11]. lVlc was derived from the theory of rubber elasticity. Mc and the calculated molecular mass MR (Eq. 2.1) of the polymers A to D are compared in Table 3.1. 

Although the basic concept of macromolecular networks and entropic elasticity [18] were expressed more then 50 years ago, work on the physics of rubber elasticity [8, 19, 20, 21] is still active. Moreover, the molecular theories of rubber elasticity are advancing to give increasingly realistic models for polymer networks [7, 22]. 

Small deformations of the polymers will not cause undue stretching of the randomly coiled chains between crosslinks. Therefore, the established theory of rubber elasticity [8, 23, 24, 25] is applicable if the strands are freely fluctuating. At temperatures well above their glass transition, the molecular strands are usually quite mobile. Under these premises the Young s modulus of the rubberlike polymer in thermal equilibrium is given by ... 

The bracket (1 — 2/f) was introduced into the theory of rubber elasticity by Graessley [23], following an idea of Duiser and Staverman [28]. Graessley discussed the statistical mechanics of random coil networks, which he had divided into an ensemble of micronetworks. 

Finkelmann et al. 256 274,2781 have also investigated the synthesis and the characteristics of siloxane based, crosslinked, liquid crystalline polymers. This new type of materials displays both liquid crystallinity and rubber elasticity. The synthesis of these networks is achieved by the hydrosilation of dimethylsiloxane-(hydrogen, methyl)siloxane copolymers and vinyl terminated mesogenic molecules in the presence of low molecular weight a,co-vinyl terminated dimethylsiloxane crosslinking agents156 ... 

The above equations gave reasonably reliable M value of SBS. Another approach to modeling the elastic behavior of SBS triblock copolymer has been developed [202]. The first one, the simple model, is obtained by a modification of classical rubber elasticity theory to account for the filler effect of the domain. The major objection was the simple application of mbber elasticity theory to block copolymers without considering the effect of the domain on the distribution function of the mbber matrix chain. In the derivation of classical equation of rabber elasticity, it is assumed that the chain has Gaussian distribution function. The use of this distribution function considers that aU spaces are accessible to a given chain. However, that is not the case of TPEs because the domain also takes up space in block copolymers. 

Kikuchi Y., Eukui T., Okada T., and Inoue T. Origin of rubber elasticity in thermoplastic elastomers consisting of crossUnked rubber particles and ductile matrix, J. Appl. Polym. Sci., Appl. Polym. Symp., 50, 261, 1992. 

Kubo, R., Rubber Elasticity/The reprint of the first edition (in Japanese), Syokabo, Tokyo, Japan (1996). 

Rubber elasticity has a long-standing history. Ancient Mesoamerican people were processing rubber by 1600 BC [1], which predated development of the vulcanization process by 3500 years. They made solid rubber balls, sofid and hollow rubber human figurines, wide rubber bands to haft stone ax heads to wooden handles, and other items. 

Mineral oils also known as extender oils comprise of a wide range of minimum 1000 different chemical components (Figure 32.6) and are used extensively for reduction of compound costs and improved processing behaviors.They are also used as plastisizers for improved low temperature properties and improved rubber elasticity. Basically they are a mixture of aromatic, naphthanic, paraffinic, and polycyclic aromatic (PCA) materials. Mostly, 75% of extender oils are used in the tread, subtread, and shoulder 10%-15% in the sidewall approximately 5% in the inner Uner and less than 10% in the remaining parts for a typical PCR tire. In total, one passanger tire can contain up to 700 g of oil. 

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