Unperturbed radius of gyration

Fig. 4.8. Mean square displacement of the center of mass during the simulation of C3i6H634 [167]. The horizontal arrow denotes the mean square unperturbed radius of gyration...

Fig. 43 Plot of normalized lamellar long periods, Dn/Rg,m of (PS) -arm-(PI)M miktoarm-star copolymers (n = 1, 2, 4 and 16) divided by corresponding diblocks of same series, Di/Rg,u against respective star functionality, n. Normalization factor PgiM represents unperturbed radius of gyration of diblock consisting of one PS block, one PI block and average number of bonds linking these two arms through core. From [121]. Copyright 2003 American Chemical Society...

RIS theory, in the form appropriate for branched molecules, is used to calculate the mean-square unperturbed radius of gyration, < s2>0, for cross-linked polyglycine, poly(L-alanine),poly(L-proline),poly(i-alanyl-D-alanine),poly(i.-prolyl-L-pro lylglycine),poly(L-prolyl-i.-alanylglycine ,poly(glycyl-L-alanyl-L-pro line), andpoly(L-aianyl-L-alanylgIycine).Thecentral amino acid residue in each polypeptide chain is replaced by the L-cysteinyl residue involved in cross-link formation. Each cross-linked molecule is considered to contain two trifunctional branch points, the a-carbon atoms of the two... 

The equilibrium thickness found in these experiments is in very good agreement with the equilibrium results [128,129]. The largest thickness at which surface forces come into play (hw ca. 80 nm) is about 15 times the unperturbed radius of gyration of the copolymer molecule. This figure is consistent with those reported for grafted polymer layers studied with SFA [242,243]. 

It was seen above that jjj may be correlated with chain length Z. Fox and Allen ( 07) noted that a dependence of on polymer coO dimensions, arising in a natural way in some of the fteoretical calculations, is characterized by the parameter X = Z (s yjM) where Vg is the polymer specific volume, is the unperturbed radius of gyration, and (pg is the volume fraction of pwlymer. Expressing data for ijj in terms of X, they noted that rj ocX for X <, jjj oc X - for X X, ... 

An important length-scale associated with polymer solutions is the root-mean-square radius of the polymer coil in the solution. For an unperturbed coil, this is known as the unperturbed radius of gyration, / g, and is given by the following (1) ... 

The mean square unperturbed radius of gyration, (5 )0, is accessible by similar methods that rapidly evaluate and sum the mean square end-to-end distances for all of the subchains, denoted by where i and j identify the chain... 

B requires knowledge of the size of the chains in addition to X and Ni. The unperturbed radius of gyration of a polymer chain, / g, is given by... 

The onset of the steric exclusion force depends on the means of attaching the polymer chains to the substrates. For physically adsorbed polymer chains covering bodh surfaces, the steric exclusion force becomes detectable around 6Rg, where Rg is the unperturbed radius of gyration of the random polymer coil in solution (12), For terminally attached polymer chains, the repulsion commences around 12Rg (14), These values are approximate and depend on a number of factors including solvent quality, temperature, surface concentration and type of polymer chains attached to the surfaces. 

We now examine the equilibration of a system containing 10 chains each of 78 units ( polyethylene ) at 450 K and 0.1 MPa. The mean-squared displacement of the chains vs cycles is linear as well (see Fig. 5) however, after almost 31000 cycles the mean-squared displacement is only 41.1 (not shown in Fig. S). Therdbre, the chains have displaced by only about one-third their unperturbed radius of gyration. Figure 6 shows Mi and M2 vs cycles and Fig. 7 shows the running average of for the two chains that have the largest and smallest initial values in the polyethylene system. We see that at the end of the... 

The most frequently calculated property is the mean square unperturbed end-to-end distance, (r )o. Other properties susceptible to rapid computation include the average of the end-to-end vector, (r)o, and the mean square unperturbed radius of gyration, 5 )0. The viscosity of a dilute solution in a solvent can be estimated from 5 )0 or (r )o via the equivalent sphere model for hydrodynamic properties. Several higher even moments, (r )o and (s )o, p = 2,3,..., provide information about the shape (width, skewness) of the distribution functions for and When combined with information about the electronic charge distribution within individual rigid units, the RIS model can calculate the mean square dipole moment, (m )o-Also accessible are optical properties that depend on the anisotropy of the... 

Since the wall separation of 50[Pg.469]

The gradual increase of Rq ) in Fig. 8.22 corresponds to a gradual evolution of the molecular shape from nearly two-dimensional structures near the walls to three-dimensional Gaussian random-coils at a distance from the walls close to the unperturbed radius of gyration. The molecular shape of chains with center of mass close to the walls, when examined in details, can be described as an apparently random combination of train and loop sequences. The average length of train sequences (that is, a sequence of beads located in the first layer of thickness a near a wall) is 4.1 beads for chains of 100 beads in the absence of a bending potential, and increases to 4.6 beads for the polyethylene-like bead chains simulated in system E, which corresponds to approximately 16 methylene units. Hence, one should take... 

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