Statistical weight matrix for

In the RIS model for PE described by Abe et al. [143], the conformational partition function, Z, of the unperturbed chain with n bonds is formulated using a well-known statistical weight matrix for bond i, Uj, that is usually written in a 3 x 3 form. 

Collect the statistical weights into statistical weight matrices (of dimension Vj.j X v , where v,- is the number of rotational isomeric states of bond i), one per skeletal bond subject to conformational change, indexing rows and columns of the matrices with the RIS of the bond. The generic statistical weight matrix for bond i is termed U,-. 

The statistical weight matrix for bond i in branch j is denoted by jUj. Matrices are given as... 

In order to calculate the partition function and molecular averages, a 4x4 statistical weight matrix for the fth residue is formulated to correlate the states of residues i-1, i, and / + 1 of the polymer chain ... 

Two matrices that describe conformations that alternate in sequence are used to specify conformation in tactic vinyl polymers. Tacticity introduces asymmetric elements into the chain. One convention used in specifying chain asymmetry consists of distinguishing between right handed chains termed d and left handed or 1 chains. This convention, admittedly arbitrary, is borrowed from the field of optical activity. For a tactic polymer with a threefold potential (Figure 2.13), the statistical weight matrix for a d placement between neighboring bonds is... 

As an example, the statistical weight matrix for polyethylene with its three-fold rotational states (trans, gauche plus, gauche minus) is... 

The results of an illustrative calculation are depicted in Fig. 3.4. The chain has 0 = 112°, i = 3, and < > = 180° and 60°. The statistical weight matrix for all internal bonds is given by Eq. (3.9) with t = if/ = 1. When cr and freely rotating chain with the same bond angle. Imposition of a symmetric torsional potential that penalizes the g states, with tr = 0.4, increases the C . Introduction of a pair-wise interdependence, via cr = 0.4 and w = 0.1, produces a further increase in C . Obviously, the interdependence of the bonds can have a strong effect on the unperturbed dimensions of the chain. 

For polyethylene, the first two points specify dimensions of 3 x 3 for U(. Assuming the order of indexing of rows and columns is f, if ", , the third point specifies the presence of a in all elements in the second and third columns. The fourth point specifies the appearance of in the 2,3 and 3,2 elements. Therefore the statistical weight matrix for polyethylene is the 3x3 matrix in equation (4) (12). 

The 3x3 statistical weight matrix for bond i has rows indexed by the state of bond z — 1, columns indexed by the state of bond i, and the order of indexing is t, g, g in both cases. Each statistical weight matrix is assembled as U, = V D where V, is a 3 x 3 matrix that incorporates the statistical weights for second-order interactions, and D, is a diagonal matrix that incorporates the statistical weights for first-order interactions. Efficient description of the interrelationships between selected pairs of matrices employs a matrix Q with the property = I3. 

Here Ui is the appropriate statistical weight matrix for a specific bond type, J and J are 1 X V and v x 1 row and column vectors respectively. They can be represented as... 

Expressions for the n-alkane chain, including polymethylene (PM), are illustrative of the application of the scheme outlined above. As is well known, the internal C-C bond of an n-alkane chain exists in one of the three staggered conformations trans t) and two gauche forms (g ). Corresponding to equation (9), the statistical weight matrix for 1 [Pg.51]

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