Equation of state for rubber elasticity

It is well known that the equation of state of Eq. (28) based on the Gaussian statistics is only partially successful in representing experimental relationships tension-extension and fails to fit the experiments over a wide range of strain modes 29-33 34). The deviations from the Gaussian network behaviour may have various sources discussed by Dusek and Prins34). Therefore, phenomenological equations of state are often used. The most often used phenomenological equation of state for rubber elasticity is the Mooney-Rivlin equation 29 ,3-34>... 

A comprehensive consideration of new phenomenological equations of state for rubber elasticity have been carried out lately by Tschoegl et al.41 45One of their equations of state is given by... 

Performing the indicated differentiation on equation (6-51), we obtain the equation of state for rubber elasticity ... 

The equation of state for rubber elasticity, embodied by any of equations (6-53) through (6-60), is important not only because it is historically the first quantitative treatment of molecular theories for elastomers but also because it laid a conceptual foundation for theories for the physical properties of polymers in general. Some of these have been discussed in detail in previous chapters. Perhaps the single most significant contribution is its recognition of the role of... 

This equation is sometimes called the thermodynamic equation of state for rubber elasticity. For an ideal elastomer dUjdl )t = 0 Equations (2.63) and (2.69) then reduce to... 

Above the glass transition lies the rubbery plateau region, region 3. The equations of state for rubber elasticity (see Section 1.5.4) apply here if the material is crosslinked, these equations may apply up to the decomposition temperature (see dashed line. Figure 1.12). 

Statement, surprising but true for noncrystallizing elastomers, follows from the equation of state for rubber elasticity and the strength of the carbon-carbon bond. The failure envelope is indeed a powerful tool in dealing with ultimate properties. 

Figure 9.6 An analysis of the thermodynamic equations of state for rubber elasticity (18b).

The quantity rf represents the i tropiCj unstrained end-to-end distance in the network. Tlie two quantities rf and rf [see equation (9.27)] bear exact comparison. They represent the same chain in the network and umcross-linked states, respectively. Under many circumstances the quantity rflrf approximately equals unity. In fact the simpler derivations of the equation of state for rubber elasticity do not treat this quantity, implicitly assuming it to be unity (42). However, deviations from unity may be caused by swelling, cross-linking while in tension, changes in temperature, and so on, and play an important role in the development of modem theory. 

The classical statistical theory of rubber elasticity1) for a Gaussian polymer network which took into account not only the change of conformational entropy of elastically active chains in the network but also the change of the conformation energy, led to the following equation of state for simple elongation or compression 19-2,1... 

Flory [3] formalized the equation of state for equilibrium swelling of gels. It consists of four terms the term of rubber-like elasticity, the term of mixing entropy, the term of polymer solvent interaction and the term of osmotic pressure due to free counter ions. Therefore, the gel volume is strongly influenced by temperature, the kind of solvent, free ion concentrations and the degree of dissociation of groups on polymer chains. 

Since the quantity (0V/0L) p is negligible, V being nearly constant, we can write the equation of state for elasticity (or for rubber) as... 

Equations (1.9) and (1.10) are basic to the molecular theories of rubber-Uke elasticity and can be used to obtain the elastic equations of state for any type of deformation [1-3], i.e. the equations interrelating the stress, strain, temperature, and number or number density of network chains. Their appUcation is best illustrated... 

The thermodynamic behavior of rubber shows, therefore, a close analogy with that of an ideal gas, the entropy of which decreases during compression. In fact, the elastic equation of state for an ideal rubber is similar to the molecular form of the equation of state for an ideal gas. The stress replaces the pressure, and the number of network chains the number of gas molecules. 

As illustrated in Figure 2, elastomeric networks consist of chains joined by multifunctional junctions. As early as 1934, it was suggested by Guth and Mark and by Kuhn that the elastic retractive force exhibited by rubber upon deformation arises from the entropy decrease associated with the diminished number of conformations available to deformed polymer chains. It is, therefore, of primary interest to study the statistics of a polymer chain and to establish the elastic equation of state for a single chain. 

To establish a useful equation of state for the mechanical behavior of a rubber network, it is necessary to predict the most probable overall dimensions of the molecules under the influence of various externally applied forces. An interesting approach to rubber elasticity consists of simulating network chain configurations (and thus the distribution of end-to-end distances) by the rotational isomeric state technique cited above. Based on the actual chemical structure of the chains, it enables one to circumvent the limitations of the Gaussian distribution function in the high deformation range. Nonetheless, the Gaussian distribution function of the end-to-end distance is very useful. It is obtained from a simple hypothetical model, the so-called freely jointed chain, which can be treated either exactly or at various levels of approximation. 

During the last decade, the classical theory of rubber elasticity has been reconsidered significantly. It has been demonstrated (see, e.g. Ref.53>) that, for the phantom noninteracting network whose chains move freely one through the other, the equations of state of Eqs. (28) and (29) for simple deformation as well as for W, Q and AIJ [Eqs. (30)-(32) and (35)—(37)] are proportional not to v but to q, which is the cycle rank of the network, i.e. the number of independent circuits it contains. For a perfect phantom network of uniform functionality cp( > 2)... 

Starting with the equation of state of rubber elasticity [equation (6-60)], show that for an ideal elastomer... 

Figure 2.29- - An analysis of the thermodynamic equation of state [Eq. (2.69)] for rubber elasticity using a general experimental curve of force versus temperature at constant length. The tangent to the curve at T is extended back to 0°K. For an ideal elastomer, the quantity (dU/df)r is zero, and the tangent goes through the origin. The experimental line is, however, straight in the ideal case. (After Flory, 1953.)...

However, specific new theories, called the equation of state theories, have been worked out in an effort to understand the special conditions of mixing long chains (29-34). At equilibrium, an equation of state is a constitutive equation that relates the thermodynamic variables of pressure, volume, and temperature. A simple example is the ideal gas equation, PV = nRT.The van der Waals equation provides a fundamental correction for molecular volume and attractive forces. Equations of state may also include mechanical terms. Thus rubber elasticity phenomena are also described by equations of state see Chapter 9. 

The integration of Eq. (8.17) between specified limits leads to a relation between the equilibrium tensile force, /eq, and the melting temperature. This is analogous to integrating the Clapeyron equation for vapor-liquid equilibrium. In this case, if the equation of state relating the pressure and volume of the Uquid is known, the dependence of the pressure on temperature is obtained. For the present problem the equation of state relating the apphed force to the length of the network is required. This information can be obtained from the theory of rubber elasticity.(6-9)... 

The early versions of the statistical theory of rubber elasticity assumed an affine displacement of the average positions of the network junctiOQs with the macroscopic strain This is tantamount to the assertion that the network Junctions are firmly embedded in the medium of which they are part. The elastic equation of state derived on this basis for simple extension at constant volume takes the familiar neo-Hookean form, i.e. Eq. (7), with... 

The statistical mechanical theory for rubber elasticity was first qualitatively formulated by Werner Kuhn, Eugene Guth and Herman Mark. The entropy-driven elasticity was explained on the basis of conformational states. The initial theory dealt only with single molecules, but later development by these pioneers and by other scientists formulated the theory also for polymer networks. The first stress—strain equation based on statistical mechanics was formulated by Eugene Guth and Hubert James in 1941. 

Further deduction of the int ral to an ejqilicit form, and the subsequent evaluation of the equation of state of rubber-like elasticity are sra t-forward (2). One can easily show that /( . r/6) apjaroaches a Gaussian function for small r. 

Equation (2.53) is stating that the network modulus is the product of the thermal energy and the number of springs trapped by the entanglements. This is the result that is predicted for covalently crosslinked elastomers from the theory of rubber elasticity that will be discussed in a little more detail below. However, what we should focus on here is that there is a range of frequencies over which a polymer melt behaves as a crosslinked three-dimensional mesh. At low frequencies entanglements... 

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