Elastic Force Between Chain Ends

In order to determine the magnitude of the force between the chain ends, let us still keep one end at the origin but apply an equal and opposite (external) force / on the other end so that the distance between the two ends increases from x to X + dx. The work done on the chain in this process is 

If chain stretching is done in a reversible manner, a combination of the first and second laws of thermodynamics yields 

From statistical thermodynamics, the entropy of a system is related to the probability distribution through the following equation  

The positive force/in Eq. (10.3.7) is externally applied and is balanced by an inward-acting internal force, which, in the absence of the external force, tends to make the end-to-end distance go to zero. This, however, does not happen in practice because the spring force is not the only one acting on the chain the equilibrium end-to-end distance is given by a balance of all the forces acting on the polymer molecule. This aspect of the behavior of isolated polymer molecules will be covered in greater detail in the discussion of constitutive equations for dilute polymer solutions in Chapter 14. 

A closer examination of Eq. (10.3.3) reveals a significant difference between the nature of rubbers and the nature of crystalline solids. In general, / includes contributions due to changes in entropy as well as changes in internal energy. In crystalline solids, the change in entropy on deformation is small and all 

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Several general requirements must be met for a polymer to be elastic—that is, to stretch under the application of force but return to its original shape when the force is released. The polymer should be predominantly amorphous so that its Tg is below room temperature. The individual molecules of an amorphous polymer are not in fully extended, anti conformations instead, they have random, coiled conformations. When a force that pulls on opposite ends of the molecules is applied, the molecules assume an anti conformation about more bonds and thus they become longer that is, they stretch. Although the stretching tends to arrange the molecules in extended, zigzag conformations that are favorable for crystallization, the overall shapes of the elastomer molecules are such that crystallization does not readily occur. Furthermore, most elastomers Eire nonpolar, so only weak attractive forces exist between chains. Therefore, when the force is removed, the molecules tend to return to their initial random conformations because these random shapes are favored by entropy (disorder). 

The aim of this section is to find the relation between the elastic force and the deformation for a polymer network. For that purpose the change in entropy associated with deformation of the chains in the network must be evaluated. Figure 3.7 shows the distribution of the chain end-to-end vectors in the deformed (stretched) and undeformed (unstrained) states. The distribution has spherical symmetry in the undeformed state, and when the... 

While the interlamellar forces in n-alkanes are weak and their contribution to the LAM frequencies is small, in systems with strongly interacting chain ends the effect can be very pronounced.44 For example, while in n-alkanes, the ratio v Jv is around 2.8, quite close to 3 as expected from eq 1, vjjv is only 2.1—2.3 for monodisperse oligo(oxyethylenes) and PEO fractions.45-46 In PEO fractions, this effect makes it difficult to distinguish between LAM-3 of the extended chain conformation and LAM-1 of the once folded chain, and peak assignment must be made with caution. The helical conformation of PEO accounts for its low elastic modulus (25 GPa), as obtained from LAM frequencies.47... 

Formally speaking, the reaction of polymerization seems most effective at P -> oo, and on the T vs V2 state diagram (Figure 3.59), the asymptote T —> oo corresponds to v n responding to P ,c. At lower uj < V2,n, the curve A of the solution-gel transition heis a positive first derivative. The specific shape of the curve A depends on the model s details. J his curve ends on the binodal curve of the two-phase gel state due to the elasticity forces of the network chains and the interaction between polymer and LMWL (see above). The numerical values of g have been determined for different types of lattice. It has also been established that the inequality f J/j < V2,c holds true (de Gennes, 1979). 

Using the analogy that can be drawn between the system of end-grafted polymer chains and the chain part of a surfactant monolayer, Milner and Witten [41] obtained simple analytical expressions for the bending elastic quantities. For the case of a melt brush," i.e., one in which the density is forced to be uniform, they obtain... 

When the chains are extended, their conformations may be considered as being determined by equilibrium between the forces of expansion due to excluded volume and the forces of contraction due to chain segments expanding into less probable conformations. Based on random flight statistics, the chains are extended linearly by a factor a over their dimensions. The acmal root-mean-square end-to-end distance is equal to The change in the elastic part of free energy is... 

An elastic solid of parallelepipedic shape, attached on a support by one of its faces, is submitted to the action of a force on the opposite face. Naturally, this system constitutes a dipole because it has two opposed and parallel faces and the force exists on each of these two faces. As in case study A8 Spring End (Chapter 4), the elastic wire is divided into a chain of elementary segments, each being considered a spatial pole, and each intermediary plane between the two active faces constitutes a spatial pole for an elastic solid. Here, too, one of the ends is considered to be representative of a pole. 

In experiments concerning the relationships between length, temperatnre, and force, usually the change in force with temperatnre at constant length is recorded (53,98-101). It is therefore necessary to extend the thermodynamic treatment of the elasticity. Moreover, the force is not purely entropic, and the energetic contribution carries useful information on the dependence on temperature of the average end-to-end distance of the network chains in the unstrained state (21,102). It is therefore important to know how to deduce these quantities from a thermoelastic experiment. 

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