Subchain

The next step in the development of a model is to postulate a perfect network. By definition, a perfect network has no free chain ends. An actual network will contain dangling ends, but it is easier to begin with the perfect case and subsequently correct it to a more realistic picture. We define v as the number of subchains contained in this perfect network, a subchain being the portion of chain between the crosslink points. The molecular weight and degree of polymerization of the chain between crosslinks are defined to be Mj, and n, respectively. Note that these same symbols were used in the last chapter with different definitions. 

Next let us apply random walk statistics to the subchain before and after stretching. 

Equation (1.41) gives the probability of finding one end of a chain with degree of polymerization n in a volume element dx dy dz located at x, y, and z if the other end of the chain is located at the origin. We can use this relationship to describe the unstretched chain shown in Fig. 3.2a all that is required is to replace n by n, the degree of polymerization of the subchain. Therefore for the unstretched chain (subscript u) we write... 

In the volume elements describing individual subchains, the x, y, and z dimensions will be different, so Eq. (3.32) must be averaged over all possible values to obtain the average entropy change per subchain. This process is also easily accomplished by using a result from Chap. 1. Equation (1.62) gives the mean-square end-to-end distance of a subchain as n, 1q, and this quantity can also be written as x + y + z therefore... 

This expression gives the average entropy change per chain to get the average for the sample, we multiply by the number v of subchains in the sample. The total entropy change is... 

Comparing this result with Eq. (3.1) shows that the quantity in brackets equals Young s modulus for an ideal elastomer in a perfect network. Since the number of subchains per unit volume, i /V, is also equal to pN /Mj, where M, is the molecular weight of the subchain, the modulus may be written as... 

Suppose the un-cross-linked polymer chain has a molecular weight M which, upon crosslinking, is divided into subchains of molecular weight M, . This means that each subchain is a fraction of the original chain. Since the crosslink... 

Even better agreement between theory and experiment has been obtained in other theories by abandoning the notion of affine deformation and recognizing that shorter subchains experience a greater strain than do longer subchains for a given stress. We shall not pursue this development any further, however, and shall turn next to a consideration of other types of deformation. 

The molecular theory begins by subdividing the polymer molecules into subchains with the following properties ... 

The degree of polymerization of the subchain is n. If the degree of polymerization of the molecule as a whole is n, then there are n/n subchains per molecule. We symbolize the number of subchains per molecule as N. Other properties of the subchain-which, incidentally, should not be confused with the chains between crosslink points in elastomers-will also have the subscript s as they emerge. 

The length of the subchain is sufficient to justify the use of random flight statistics in its description. 

The mass of the subchain is pictured as concentrated in a bead, connected to adjacent beads by Hookean springs which, individually, obey Eq. (3.45). 

The displacement of beads representing subchains is resisted by viscous forces which follow Eq. (2.47). 

Whether the beads representing subchains are imbedded in an array of small molecules or one of other polymer chains changes the friction factor in Eq. (2.47), but otherwise makes no difference in the model. This excludes chain entanglement effects and limits applicability to M < M., the threshold molecular weight for entanglements. 

The subchain of this model is an artifact about which we have no information. After developing expressions for the behavior of the subchains, we must describe the latter in terms of the actual polymer chains. 

We begin the mathematical analysis of the model, by considering the forces acting on one of the beads. If the sample is subject to stress in only one direction, it is sufficient to set up a one-dimensional problem and examine the components of force, velocity, and displacement in the direction of the stress. We assume this to be the z direction. The subchains and their associated beads and springs are indexed from 1 to N we focus attention on the ith. The absolute coordinates of the beads do not concern us, only their displacements. 

Note that the quantity n l is the mean-square end-to-end distance in the subchain according to Eq. (1.62). We shall designate it (r ). ... 

The solutions describe the vibrational modes of the system. As waves, the solutions are characterized by integers p which essentially count the number of nodes along the chain in a particular mode of vibration. The upper limit of p corresponds to the number of subchains in the molecule N, . 

We return to Eq. (3.45) to describe the elasticity of a subchain. According to that equation, the force can be written... 

The remaining step is to eliminate those terms which refer to the subchain by expressing them in terms of the polymer molecule as a whole. Specifically, we recall that (r ) = n l, = n f, and = n/n. Substituting into Eq. (3.97),... 

If we combine Eqs. (3.98) and (3.94), we can eliminate from the latter to obtain an expression for the relaxation time of mode p which is free of any reference to the subchain ... 

If Gc chains crystallize (partially) there will remain G-Gc completely amorphous chains and G + 3Gc/2 total elastic elements (amorphous chains and subchains). This is true only for the model described with 1/2 Gc chains folding once and 1/2 Gc chains not folding at all. If each elastic element is Gaussian in its behavior, the elastic free energy Fg can be written as... 

Condition (3) still holds, but the relationship between r and r must be established. Upon crystallization the 1th chain may iubdivide into two amorphous subchains if it does not fold... 

Note The term subchain may be used to define designated subsets of the constitutional units in a chain. 

Polymer composed of highly branched macromolecules containing mesogenic groups of which any linear subchain generally may lead in either direction, to at least two other subchains. 

---