The Single Chain Problem

We begin by a purely geometric discussion of vector orientations in an n-dimensional space and present it so that the calculations are meaningful when n is not a positive integer. Our iq)proach follows the q pendix of Ref. 6 and is due to G. Sarma. 

Let US first define an average over all orientations (equally wei ited) of each spin. This is the analog of the integration over solid angles (for n - 3). We denote this average by () . 

The subscript (o) emphasizes the difference between this type of average (where all states are equally weighted) and a thermal average (where they are weighted by the Boltzmann exponential exp (— /t) thermal averages are written ( ) (without a subscript). The relationship between the two types of averages is, for any fimction G(Si... S ... ) of the spins  

We now focus on one of the vectors S( (which we call S for simplicity). If we perform an e ransion of the partition functionZ following eq. (X. 14) 

19) is not surprising the diagonal terms must all be equal, and their sum is equal to n as can be seen from the normalization condition of eq. (X.ll). The nondiagonal terms vanish by symmetry. 

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Thus, an electron on the inverted band TTF chain corresponds to a hole with a normal band structure, and vice-versa. The interactions are shown in Fig. (1). The intrachain interactions g and gare backward and forward scattering Interactions, as in the single chain problem, and are assumed to be the same on both chains. The interchain interactions are w and These interactions... 

The second difficulty with the Why Tree technique is that it becomes derailed if one of the answers provided is wrong and/or if a critical factor is overlooked or (incorrectly) considered to be unimportant. This concern links to the single chain problem. If the analyst follows the wrong chain, then the results become worthless. 

From the late 1950s, many papers in the literature studied this problem [23], Most of them employed either the matrix method or the generating function method to calculate the chain partition function. However, in order to apply the theoretical method directly to many chain problems in solutions and gels, we here reformulate the single chain problem using the combinatorial counting method. 

On the physical side, there are many complications. As mentioned, single-chain adsorption is never observed. One always reaches a situation where many chains compete for the same portion of surface. Furthermore the single-chain problem may be modified by the existence of long range van der Waals forces between the surface and each monomer. The corresponding potential decreases relatively slowly (as D ) and the attraction energy may not be cast into the form used in eq. (I.S7). 

A similar equation was derived (for the single chain problem) in Chapter VI through the Kubo formula. The present derivation is more direct. 

Any pair a , Um) in this sequence represents one possible formulation of the single-chain problem. 

In practice, the scheme as explained above is not implemented. The consecutive generation of all possible chain conformations is a very expensive step. The reason for this is that there are of the order of ZN number of conformations, where Z is the lattice coordination number. A clever trick is to generate a subset of all possible conformations and to use this set in the SCF scheme. This approach is known in the literature as the single-chain mean-field theory, and has found many applications in surfactant and polymeric systems [96]. The important property of these calculations is that intramolecular excluded-volume correlations are rather accurately accounted for. The intermolecular excluded-volume correlations are of course treated on the mean-field level. The CPU time scales with the size of the set of conformations used. One of the obvious problems of this method is that one should make sure that the relevant conformations are included in the set. Typically, the set of conformations is very large, and, as a consequence, the method remains extremely CPU intensive. 

Show that a stress relaxation modulus of an entangled but non-concate-nated melt of rings on the basis of the single chain dynamic modes described in Problem 9.31 is... 

As mentioned already, distinction of single and multiple helical chains is not easy by the fiber diffraction method. This problem is very easily solved by the C NMR method, if these polymorphic structures can be identified with the aid of sample history and other experimental techniques and mutual conformational conversions among them can be manipulated by a series of physical treatments under a controlled manner, as illustrated in Fig. 24.3 [16-19]. The single chain form can be obtained from a sample of either the single helix by dehydration or the triple helix by lyophilization from DMSO solution. Even a multiple-stranded helix can be completely dispersed as a result of the conformational transition to a random coil form in DMSO... 

Surprising result. However, SOlyom showed that in a single chain if 2g2 > g-j > o the charge density wave (CDW) response is divergent as In the two chain problem... 

We begin with the most rigorous version of self-consistent PRISM based on a Monte Carlo evaluation of the effective single-chain problem. Theoretical predictions of Grayce and co-workers" are compared with many-chain simulation results for the mean-square end-to-end distance of the hard-core chain model as a function of polymer packing fraction in... 

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