Anisotropic behavior of gyration tensor components

Anisotropic behavior of gyration tensor components parallel and perpendicular to the substrate and their fluctuations as functions of the temperature T tor a 179-mer ats=1. For comparison, we have also plotted the associated heat capacity curve. From [302]. 

One of the most interesting structural quantities in studies of polymer phase transitions is the gyration tensor (13.5). For the hybrid system considered here, it is expected that the respective components parallel (13.6) and perpendicular (13.7) to the substrate will behave differently when the polymer passes transition lines. This anisotropy is obvious 

At very low temperatures, i.e., in pseudophase ACl, we have argued in the previous section that the dominant polymer conformation is the most compact single-layer film. This is confirmed by the behavior of R and Rj ), the latter being zero in this phase. A simple argument that the structure is indeed maximally compact is as follows. It is well known that the most compact shape in the two-dimensional continuous space is the circle. For n monomers residing in it, n nr, where r is the (dimensionless) radius of this circle. The usual squared gyration radius is 

Near rseOJ, the strong layering transition from ACl to AGe is accompanied by an immediate decrease of R ), while R ) rapidly increases from zero to about 0.5 which is exactly the gyration radius (perpendicular to the layers) of a two-layer system, where both layers cover approximately the same area. Note that the single layers are stiU compact, but not maximally. Applying the same approximation as in Eq. (13.10), the planar gyration radius for each of the two layers is now (with n N/2) while we 

It was shown in Section 13.4.2 that the contact numbers ns and are unique system parameters for the pseudophase identification of the hybrid system. We define the restricted partition sum for a macrostate with ns surface contacts and monomer-monomer contacts by 

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