Unperturbed chain reference

Using [4f) as the unperturbed chain reference and allowing for expansion and contraction,... 

N, indicating that the solvent quality is substantially reduced by addition of salt, as in Wallach s study. If the extrapolation to M = 0 is not sufficient to eliminate all excluded volume contribution, then the estimated unperturbed dimensions could be affected. The samples used in reference 3 were also of varying polydispersity, and the intrinsic viscosity molecular weight relation showed more uncertainty than we have observed in this study, which is also sufficient to explain the difference in estimated unperturbed chain dimensions. 

The initial configuration for aU single unperturbed chain simulations was all-trans, from which the conformation quickly departed. After permitting the chain to explore a sufficient number ( 200 million) of its configurations, statistics were accumulated for the properties of interest We will refer to the chains sampled by this method as continuous unperturbed chains (CUC). Due to the simplicity of the sampling method and the local nature of the unperturbed... 

The Monte Carlo sample then reflects the number of conformations of polymer molecules. This means that observable parameters describing the solution behavior of polysaccharides are averages of the properties of individual conformations. This approach yields properties corresponding to the equilibrium state of the chain. Results refer to a model for an unperturbed chain that ignores the consequences of the long range excluded volume effect, because only nearest-neighbor interactions are accounted for in the computation of the, 4 surfaces. 

Blobs are chain segments exhibiting the behavior of a single unperturbed chain . The choice of the unperturbed reference state depends on the circumstances solvent quality, spatial dimensionality, role of surface interactions etc. For flexible chains the primary choice is between two models A random walk (RW), and a self avoiding random walk (SARW). The definition of a particular blob involves two ingredients Size, and the number of constituting monomers, g. The two are related via w g Cb where a is the monomer size and uj, is determined by the reference state. For a KW Ufj = 1/2 while for three and two dimensional SARW 3/5 and 1/5 3/4 respectively. 

Let us consider a polymer chain with N->oo identical skeletal atoms, either in solution or in the melt, representing our polymer system. Our reference temperature is T0, i.e., the temperature above which no bundles may effectively contribute to crystallization. At T = T0 the chain is assumed to be unperturbed and its configurational partition function is ZN(T0) = kN (N -> oo) [107] for simplicity we use a reduced form Zn = Z /kN (henceforth simply the partition function) so that Zn(T0) = 1. Only at T < To effective bundles may form, see Fig. 1, and we have ZN(T) = 1 + AZN(T - T0) note that the unit term corresponds to the bundle-free infinite-chain configuration. Each bundle with n chain atoms in -c N) will contribute to AZn... 

Following Flory (1969), a 0 solvent is a thermodynamically poor solvent where the effect of the physically occupied volume of the chain is exactly compensated by mutual attractions of the chain segments. Consequently, the excluded volume effect becomes vanishingly small, and the chains should behave as predicted by mathematical models based on chains of zero volume. Chain dimensions under 0 conditions are referred to as unperturbed. The analogy between the temperature 0 and the Boyle temperature of a gas should be appreciated. 

The exponent can vary from v=0.33 for hard spheres up to v=1.0 for rigid rods. For linear chains v=0.5 refers to unperturbed coil dimensions in -solvents and v=0.588 [6] to good solvent conditions. Equation (37) maybe re-writ-ten by expressing the molar mass as a function of the radius of gyration, i.e.. 

Before proceeding to a review of both scaled particle theory and fuzzy cylinder model theory, it would be useful to mention briefly the unperturbed wormlike (sphero)cylinder model which is the basis of these theories. Usually the intramolecular excluded volume effect can be ignored in stiff-chain polymers even in good solvents, because the distant segments of such polymers have little chance of collision. Therefore, in the subsequent reference to wormlike chains, we always mean that they are unperturbed . 

It should be commented that two distinct readjustments of the reference 0 state are implicit in Eqn. (2.2.4), both relating to the change of the lower limit of integration from k to pN. When applied to the integral of the screened interactions [last integral in Eqn. (2.2.4)], this change reflects the adoption of the unperturbed real chain instead of the phantom chain. When applied to the three-body integral [see Eqn. (2.2.7)], it implies a small shift of the 0 temperature from the phantom chain value... 

Theory The swelling behavior of polymer networks is described by several network parameters Xg> a polymer network-solvent interaction parameter u, the concentration of elastically effective network chains and qQ, a reference degree of swelling which is related to the unperturbed end-to-end distance of the polymer chains during network formation. 

The dimension obtained from random flight calculation, which includes the effects of bond angles and hindrances to rotation about bonds, is referred to as the unperturbed dimension of the polymer chain. It is represented by the symbol (rg ). The subscript zero is used to emphasize the condition that the molecule is subject only to local constraints involving the geometrical character of the bond structure and restricted rotation. 

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