Finite chain

We now discuss the effects of finite chain length. The difficulties arise from the definition of a bulk free energy term, when the very nature of the chains constrains the crystal thickness to be finite. There are two different approaches to this problem the first to be considered is due to Hoffman et al. [31] and is a simple modification of the infinite chain case, but is somewhat lacking in theoretical justification the second, due to Buckley and Kovacs [23], aims to correct this deficiency and suggests that the interpretation of experimental data given by Hoffman s approach is misleading. 

A different approach [23, 32] considers AF in Eq. (2.2) to consist of a part which must be defined with respect to infinite chain length plus an entropy of localization [33] due to the pairing of chain ends which becomes important in the case of closely stacked lamellar crystals [34]. It amounts to — Kin (Cp) per molecule, where C is a constant related to the flexibility of the chains in the melt, and which arises due to the conformations of the finite chain. Hence ... 

The post-gel Miller-Macosko derivation determines network properties by first calculating the probability that looking out from a A group is a finite chain, P(F ° ). This probability is equal to the probability that A has not reacted (1-a) plus the probability that A has reacted times the probability that looking in to a B group is finite ... 

Once P(F ° ) and P(Fg° ) have been calculated, it is possible to calculate a number of network structure parameters including the weight fraction of sol, wg, and the "effective" crosslink density. A given polymer or crosslinker will be part of the sol only if all of its groups are attached to finite chains. Thus, the weight of the sol is given by... 

Equation (7.19) is a self-consistent equation for AEn, in the form of a sum of a pair of continued fractions (CFs). Although numerical solutions to (7.19) are feasible, we are only concerned with its qualitative features. In particular, we note that an exact WSL occurs when AEn = 0, which happens only if both CFs contain the same number of terms (apart from the trivial case 0 = 0). For the infinite chain, this situation is the case for every allowed energy, so an exact WSL is indeed found. But, for a finite chain, AEn = 0 only for the center state, which thus possesses the exact WSL energy. Therefore, the set of energies for a finite chain form only an approximate WSL. 

In this section, we construct the GF for a finite chain with an applied held (Davison et al 1997), by using the CF elements of the recursion method (Haydock 1980) and thereby build the GF atom-by-atom, in a similar way as the causal-surface GF approach (Pendry et al 1991). 

Regardless of how we may choose to represent the GF, we should note that it is for a finite chain, so that there is a discrete spectrum of m states, whose energies are given by the poles of GijTO(1, 1). 

The energies for the finite chain are given approximately by those in (7.13) for the infinite case, so as the potential strength T increases, the FL... 

Scaling theories are restricted to long polymer chains in good solvents and do not include finite chain effects and polymer-solvent interactions. These models should be complemented by more detailed mean-field calculations and molecular simulations. 

The orientation of the primary alcohol group is gauche-trans. The hydrogen bonding consists of finite chains which intersect at four-coordinated water molecules on two-fold, crystallographic-symmetry axes. 

The comparison shows that both data sets are identical at comparable Q-values, i.e. from this observation any influence of finite-chain-length effects on the measured Schain(Q)t)/Schain(Q) for M =36 kg/mol can be excluded. Taking as reference the very good agreement between NSE results and theoretical predictions, the same comparison to the simulations is rather poor, pointing towards some problems in the simulation of such large chain ensembles. 

In the third model (finite chain with different terminal groups) no reflection symmetry element exists in the Fischer projection. The individual macromolecules are, therefore, chiral and all the tertiary atoms are asymmetric and different. The stereochemical notation for a single chain, depending on the priority order of the end groups, can be R, R2, R. . . R -2, R -i, Rn or R, R2, R3... 

Teramoto et al. (14) have proposed a method of analyzing in the helix-coil transition region. Actually, it is a refinement of Ptitsyn s method (46). In the latter, / in Eq. (C-29) is replaced by fN and the resulting expression is applied to finite chains. Teramoto et al. recast Eq. (C-26) in the form ... 

The present work demonstrates that (1) conclusions reached earlier for model chains also apply to realistic chains, (2) effects seen with finite chains may survive in extremely long chains, (3) the limit at large n for (o, 2 - 1) / (ar2 - 1) provides little information about the effect of excluded volume on the dipole moment of infinitely long chains, and (4) an alternative relationship between a 2 and ar2 may provide useful information on the relationship of and 0 for long chains. 

The quantities K0 and Kt thus define the solution. As indicated in Appendix A, the result, Eqs. (5)-(9), is identical with the familiar statistical mechanical solution for the case of nearest-neighbor interactions, summarized for example, by Schwarz.2 We note the ease with which the results have been obtained here. The procedure could be extended to other cases, for example, a copolymer (i.e., a linear lattice with two types of sites) distributed in a prescribed manner and undergoing a transition to two other types of sites. For the finite chain, however, the use of nearest-neighbor conditional probabilities and detailed balancing will not yield the complete solution.3... 

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