Segmental hard-core

For a binary blend of thread homopolymers the structure of each chain is specified by its statistical segment length, ctm, segmental hard core diameter, Am = d, degree of polymerization, Nm, and the attractive tail potentials, VMM (r) The structural asymmetry on an equal volume basis is given by the ratio... 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

More recentiy, melt-spun biconstituent sheath—core elastic fibers have been commercialized. They normally consist of a hard fiber sheath (polyamide or polyester) along with a segmented polyurethane core polymer (11,12). Kanebo Ltd. in Japan currentiy produces a biconstituent fiber for hosiery end uses called Sideria. 

Fig. 2. Sketch of the interaction potential between segments m and n. The potential can be decomposed into a hard core repulsive potential Unm (hard) and a weak attractive potential Unn, (attr)...

Although the observed undulations are consistent with the scaling picture, it is somewhat in contradiction to the axial contraction. On the other hand, axial contraction in molecular brushes can be limited, first because the backbone segments have limited flexibility and second because of hard core repulsion near the backbone or cylinder axis. In contrast to loose comb-Hke polymers [153], an... 

The DNA-nucleosome interaction parameters are not known at present. In most of the theoretical work it is deemed negligible compared to the DNA-DNA and nucleosome-nucleosome interaction, except for a hard-core excluded volume interaction. Nevertheless, recent work on the mechanism of nucleosome repositioning [67] assumes that the DNA can dynamically detach from the nucleosome surface and reattach in different conformations, such that it is conceivable that distant DNA segments may also transiently bind to open regions of the DNA-binding surface of the nucleosome. 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

We may have to consider that the segment distribution fluctuates in the cylindrical domain in order to formulate the effects of entanglement and jamming in a solution as illustrated in Fig. 13b. In other words, we may no longer be permitted to consider the fuzzy cylinder a hard-core cylinder of the geometry specified by Eq. (43), but have to make its periphery fluctuate. 

As a first approximation we postulate the existence of an equally sized hard core volume located in the center of each cylinder that defines the excluded volume per segment, according to the degree of interpenetration of neighboring cylinders. This simple construction includes complicated local intersegmental configurations, but it imposes a unique local orientation correlation of segments within the domains of a PE melt The shape of those impenetrable correlation cylinders is assumed also to be cylindrical. 

Mixtures of hydrocarbons are assumed to be athermal by UNIFAC, meaning there is no residual contribution to the activity coefficient. The free volume contribution is considered significant only for mixtures containing polymers and is equal to zero for liquid mixtures. The combinatorial activity coefficient contribution is calculated from the volume and surface area fractions of the molecule or polymer segment. The molecule structural parameters needed to do this are the van der Waals or hard core volumes and surface areas of the molecule relative to those of a standardized polyethylene methylene CH2 segment. UNIFAC for polymers (UNIFAC-FV) calculates in terms of activity (a,-) instead of the activity coefficient and uses weight fractions... 

As previously described for binary sterns, volume fractions may be replaced by segment fractions which deal with the hard core volume of the components. 

For high-molecular polymers, a mole fraction is not an appropriate unit to characterize composition. However, for ohgomeric products with rather low molar masses, mole fractions were sometimes used. In the common case of a distribution function for the molar mass, Mb = M is to be chosen. Mass fraction and volume fraction can be considered as special cases of segment fractions depending on the way by which the segment size is actually determined rjr = MjM or rjvA = VJVa = tMi P, respectively. Classical segment fractions are calculated by applying fJfA = where hard-core van der Waals volumes,, are taken into account. 

Milpi)l MAlpA), respectively. Classical segment fractions are calcrrlated by applying rJrA = ratios where hard-core van der Waals volumes, are taken into accormt. Their special values depend on the chosen equation of state or simply some group contribution schemes, e.g., (1968BON, 1990KRE) and have to be specified. 

With the necessary care, all thermodynamic expressions given above can be formulated with mass or volume or segment fractions as concentration variables instead of mole fractions. This is the common practice within polymer solution thermodynamics. Applying characteristic/hard-core volumes is the usual approach within most thermodynamic models for polymer solutions. Mass fraction based activity coefficients are widely used in Equations [4.4.7 and 4.4.8] which are related to activity by ... 

The segment number r is, in general, different from the degree of polymerization or from the number of repeating units of a polymer chain but proportional to it. One should note that Equations [4.4.12 and 4.4.13] can be used on any segmentation basis, i.e., also with r = on a hard-core volume segmented basis and segment fractions instead of... 

To extend the Flory-Orwoll-Vrij model to mixtures, one has to use two assumptions (i) the hard-core volumes n of the segments of all components are additive and (ii) the intermolecular energy depends in a simple way on the surface areas of contact between solvent molecules and/or polymer segments. Without any derivation, the final result for the residual solvent activity in a binary polymer solution reads ... 

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