Ideal rubber statistics

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. 

An equation for the modulus of ideal rubber was derived from statistical theory that can be credited to several scientists, including Flory, and Guth and James (Sperling, 1986). A key assumption in derivation of the eqmtion is that the networks are Gaussian. 

Stein (67) has used the basic theory of Nishijima for developing equations for predicting the values of various ratios of the fluorescence intensities of differing polarization expected from a stretched ideal rubber. Stein s theoretical efforts made use of Roe and Krigbaum s (55) expression for the complete orientation distribution function derived from the Kuhn and Griin (58) statistical segment model. Stein s equations have yet to be verified by experiment. 

Show that, for an ideal rubber obeying Gaussian statistics (and hence eqn 3.38), the nominal tensile stress-strain curve is approximated to a high accuracy at small strains by... 

In Chapter 5 a condition is derived (eqn 5.7) for determining when localized deformation ( necking ) will spontaneously occur during uniaxial extension of a bar of material. Show that this condition is never satisfied during uniaxial extension of an ideal rubber ob ing Gaussian statistics. Such a material therefore extends uniformly. 

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 ... 

A sample of a certain ideal rubber has a number average RMM between crosslinks of 5 000 and a density of 900 kg/m. A block of this rubber, a cube of side 100 mm, is tested at a temperature of 300 K. What is its tensile modulus What is its shear modulus Axes X, Y, and Z are chosen parallel to the edges of the cube. A compressive force Fx is applied in the -direction, to reduce the X-dimension from 100 mm to 75 mm. Calculate F. What do the Y and Z dimensions become A further compressive force Fy is now applied in the Y-direction to reduce the Y-dimension to 75 mm, the X-dimension remaining at 75 mm. What now are the magnitudes of the forces f and F,. What has the Z-dimension now become How much strain energy is now stored in the block Assume the rubber obeys Gaussian statistics and use the approximation... 

A sheet of ideal rubber is 200 mm square and 10 mm thick. Its edges are aligned parallel to axes X aftd Y. Forces F and Fy are applied in X and Y directions respectively to stretch the sheet homogeneously, to bring the -dimension to 400 mm and the Y-dimension to 300 mm. Calculate F and Fy, assuming the rubber to obey Gaussian statistics and to have a tensile modulus of 1.5 MPa. 

This is the equation of state for an ideal rubber obtained from the statistical mechanics derivation. It is consistent with the equation obtained from the thermodynamic relations. Consequently, for an ideal rubber, we have... 

Whether to use the first or the second form of Finger s constitutive equa tion is just a matter of convenience, depending on the expression obtained for the free energy density in terms of the one or the other set of invariants. For the system under discussion, a body of rubbery material, the choice is clear The free energy density of an ideal rubber is most simply expressed when using the invariants of the Finger strain tensor. Equation (7.22), giving the result of the statistical mechanical treatment of the fixed junction model, exactly corresponds to... 

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. 

A spherical rubber balloon is of initial wall thickness 0.5 mm and diameter 100 mm. It is inflated to a final diameter of 500 mm. Calculate (1) the final thickness, (2) the true stress in the plane of the balloon wall, and hence (3) the internal air pressure required. Assume the rubber is ideal and obeys the Gaussian statistics, and take the shear modulus to be 1 MPa. 

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... 

However, most students not broadly exposed to statistical thermodynamics find such calculations difficult to follow at first. For this reason we first derive the ideal gas law via the very same principles that are employed in calculating the stress-elongation relationships in rubber elasticity. 

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