Chain vectors

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

The elastic free energy AFe causes difficulty because of its sensitivity to the crystallization model assumed. To estimate AFe for lamellar morphology, consider first an important property of a network, amorphous or crystalline. Network crosslinks are considerably restricted in their fluctuations. Fluctuations of crosslinks several chains removed from a particular chain are therefore inconsequential for that chain. A chain in the interior of a path traced through several sequentially connected chains behaves as if the path ends are securely anchored at fixed positions ( 7). If Gj chain vectors make up the path, then... 

This is possible because the result must conform to the traditional theory of elasticity when crystallinity is zero (w = 0). This happens if r is interpreted as an average chain vector (made up of N links), the components x, y, z of which deform affinely, so that in simple elongation along z... 

Early theories of Guth, Kuhn, Wall and others proceeded on the assumption that the microscopic distribution of end-to-end vectors of the chains should reflect the macroscopic dimensions of the specimen, i.e., that the chain vectors should be affine in the strain. The pivotal theory of James and Guth (1947), put forward subsequently, addressed a network of Gaussian chains free of all interactions with one another, the integrity of the chains which precludes one from the space occupied by another being deliberately left out of account. Hypothetical networks of this kind came to be known later as phantom networks (Flory, 1964,... 

James and Guth showed rigorously that the mean chain vectors in a Gaussian phantom network are affine in the strain. They showed also that the fluctuations about the mean vectors in such a network would be independent of the strain. Hence, the instantaneous distribution of chain vectors, being the convolution of the distribution of mean vectors and their fluctuations, is not affine in the strain. Nearly twenty years elapsed before his fact and its significance came to be recognized (Flory, 1976,... 

A general theory of dichroism induced by strain in polymeric networks Is developed by adaptation of methods developed earlier for treating strain birefringence. It is generally applicable to dichrolc bands associated with any specified conformation involving sequences of one or more consecutive bonds. The transition dipole moment is introduced in the local framework of the skeletal bonds associated therewith. Possible differences in transition moments for various conformations and repeat units are taken into account. Numerical calculations for PE chains show gauche bonds, rather than trans, to be more favorably oriented with respect to the chain vector r. 

Moments of rank 1 -4 formed from the components x,y, and z of the chain vector r and expressed in the coordinate system affixed to the first unit are calculated as functions of chain length n for p-phenylene polyamides of type I, -(NH-CGH4-C0)n-, of type II, -(NH-Cgfy—NH—CO—C6H4—C0)n/2-, and of the corresponding polyesters. 

Such structuring is necessarily an intermolecular effect. The simplest type of an intermolecular effect, which should be treated first, is due to the crosslinks between the chains themselves. Dobson and Gordon (50) have remarked that most crosslinks are actually short chains of one or several links, which upon straining the network, become oriented but cannot be stretched. As a result an additional entropy force should arise, which has not yet been accounted for in the Gaussian theory. This force can be calculated on the basis of the Kuhn and Grun (114) chain vector orientation argument, which yields in extension... 

T The matrix X of order 3x3. r The chain vector connecting the ends of the poly-nucleotide. 

The universal packing fraction, t] = 0.625, for the mers of rubber-like polymer systems corresponds to the random close packing of hard spheres. The existence of this universal value may be motivated as follows Assume first the absence of nonbonded interactions and consider a network of Gaussian chains y with chain vectors R(y) occupying a volume v. The force f(y) required to maintain the chain vector fixed at R(y) is... 

Figure 4. (a) Schematic illustrating an ideal chain with end atoms held in place by external forces, (b) Schematic representation of the force acting on a chain in a stretched network when the chain vector R makes an angle 6 with the stretch axis x. Shown are fR and fy, the axial and transverse components of the chain force f. 

Consider next the case in which the surrounding melt is replaced by the same set of chains in the network mode, that is, with the chain vectors controlled corresponding to an applied deformation, which is a constant volume extension X in the same direction as the tethered chain. Then, as shown in MD simulations,... 

A simulation of a collection of like chains under both melt and network conditions was performed to test this hypothesis. In the melt condition, no restrictions were placed on the chain vectors, whereas in the network condition, chain vectors were controlled and subjected to a constant volume extension X. In both cases, ars was found to be diagonal, as expected from symmetry considerations, whereas the principal values were the same, within computation accuracy, both for the strained network where they are independent of X, and for the melt see Fig. 12. 

To obtain a stress-strain relation with this formalism, it is necessary to keep track of the effect of the deformation as it is performed on each chain y, y = 1,..., Nc, in the system of Nc chains with controlled chain vectors. In the initially generated systems, the chain vectors R(y) have an isotropic distribution and the initial stress jf = Yly=i <7y() ) where ft iy) is the stress contribution of... 

Apply deformation X to all chain vectors R(y) and allow the system to return to equilibrium after each change. 

Figure 4.3 Three-dimensional crystal with three periodic bond chain vectors. The figure shows F-faces (100), (010), and (001) S-faces (110), (101), and (Oil) and the K-face (111).

These observations were made and explained by P. Hartman and W.G. Perdok in their periodic bond chain vector (PBC) theory (see the Chapter 1 references), which we introduced in Section 1.2.1. This theory takes into account the fact that 3D crystals are far more complicated structures, which are full of partial bonds and preferred directions. When the PBC vector is parallel with a crystal face, there is maximum growth along that crystal face. 

The above results can be understood from sterlc considerations. For n even, the meso-group vector is parallel to the extended chain vector. Thus for n = 4, a fully extended aliphatic spacer is just long enough to permit double overlaps at both ends of a dimer without sterlc Interferences of the nltro groups. As n (even) Increases, such... 

Tight-binding band theory is described first for the case of distinct molecules uniformly stacked with equivalent separation, a, between each site, Fig. 1. The potential in which the electron moves, K(r), is periodic, V r + a) — V(r). Here r is the vector coordinate for electrons and a is the vector length between equivalent sites in the chain. Vector notation is used because of the three-dimensional extent of the individual molecules. Assuming the chain axis is parallel to the z-axis, a — az (z is a unit vector along the z-axis). 

In case a), the mean values of the chain end-to-end vectors are displaced affinely with the principal extension ratios (p = x, y, z) specifying the macroscopic strain. The fluctuations about these mean values are independent of the sample deformation. Consequently, in the free-fluctuation limit, the transformation of the actual chain vectors is not affine in the K s. The elastic free energy change for deformation results in the expression... 

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