Statistics of Ideal Rubber Elasticity

A typical rubber consists of long chains connected by short crosslinks every few hundred carbon atoms. The chain segments between crosslinks are known as network chains. (These segments are characterized by Me, the molecular weight between crosslinks.) The change in entropy upon stretching a sample containing N moles of network chains is 

For an ideal rubber, in which the tensile force is given by 

The engineering tensile stress, cr, is defined as the tensile force divided by the initial cross-sectional area of the sample A and is, therefore. 

The modulus is representative of the strength of a material (or how much resistance the material will give) when a force (axial tension, shear stress) is applied. 

Equations 13.19-13.24 point out two important concepts (1) the force (or modulus) in an ideal rubber sample held at a particular strain increases in proportion to the absolute temperature (in agreement withEq. 13.13), and (2) die force is inversely proportional to the molecular weight of the chain segments between crosslinks. Thus, increased crosslinking, which reduces Me, is an effective means of stiffening a mbber. Equation 13.24 is often used to obtain Me from mechanical tests and thereby evaluate the efficiency of various crosslinking procedures. 

Even non-crosslinked polymers exhibit rubbery behaviour above their Tg values for limited periods of time. This is due to mechanical entanglements acting as temporary crosslinks then represents the average length of chain segments between entani ements. 

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The formal thermodynamic analogy existing between an ideal rubber and an ideal gas carries over to the statistical derivation of the force of retraction of stretched rubber, which we undertake in this section. This derivation parallels so closely the statistical-thermodynamic deduction of the pressure of a perfect gas that it seems worth while to set forth the latter briefly here for the purpose of illustrating clearly the subsequent derivation of the basic relations of rubber elasticity theory. 

In analogy to the kinetic theory of ideal gases, the statistical theory of rubber elasticity is often called the kinetic theory of rubber elasticity. Reflect upon the similarities and differences between the basic philosophies of these two theories. 

The discussion of the chain statistics permits one, thus, to have a more quantitative description of a flexible, linear macromolecule. The random coil of a sufficiently long molecule can be compared in mass-density and randomness to an ideal gas at atmospheric pressure. The elastic compression and expansion of gases are caused by changes in entropy. It will be shown below that corresponding behavior exists for the extension and contraction of random-coil macromolecules (entropy or rubber elasticity, see Sect. 5.6.5). Combining many random coils into a... 

The essential concept involved in the statistical theory of rubber elasticity is that a macroscopic deformation of the whole sample leads to a microscopic deformation of individual polymer chains. The microscopic model of an ideal rubber consists of a three-dimensional network with junction points of known functionality greater than 2. An ideal rubber consists of fully covalent junctions between polymer chains. At short times, high-molecular-weight polymer liquids behave like rubber, but the length of the chains needed to describe the observed elastic behavior is independent of molecular weight and is much shorter than the whole chain. The concept of intrinsic entanglements in uncrosslinked polymer liquids is now well established, but the nature of these restrictions to flow is still unresolved. The following discussion focuses on ideal covalent networks. 

The state of the ideal rubber can be specified by the locations of all the junction points, ij, and by fce end-to-end vectors for all tire chains connecting the junction points,. The first postulate of the statistical theory of rubber elasticity is that, in the rest state with no external constraints, the distribution fimction for the set of chain end-to-end vectors is a Gaussian distribution witii a mean-squared end-to-end distance that is proportional to the molecular weight of the chains between jimcnons ... 

One of the most characteristic properties of gel is the rubber elasticity. Rubber elasticity has attracted attention since fire early era of polymer science research and has been developed through statistical mechanics [35, 36]. The basis of rubber elasticity is the micro-Brownian motion of the polymer chains [37]. Specifically, the rubber elasticity originates fixrm entropy and is mechanistically different from the energetic elasticity of fire crystalline solid. The ideal relationship between stress and strain is given by... 

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