Mean-square distance

Mean-square end-to-end distance Mean-square radius of gyration Mean-square dipole moment Mean-square optical anisotropy Optical configuration parameter... 

One of the most important fiinctions in the application of light scattering is the ability to estimate the object dimensions. As we have discussed earlier for dilute solutions containing large molecules, equation (B 1.9.38) can be used to calculate tire radius of gyration , R, which is defined as the mean square distance from the centre of gravity [12]. The combined use of equation (B 1.9.3 8) equation (B 1.9.39) and equation (B 1.9.40) (tlie Zimm plot) will yield infonnation on R, A2 and molecular weight. 

Thus, by plotting as a function of in tire limit of small q tire mean square of tire end-to-end distance can be... 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

A polymer chain can be approximated by a set of balls connected by springs. The springs account for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

A molecular fitting algorithm requires a numerical measure of the difference between two structures when they are positioned in space. The objective of the fitting procedure is to find the relative orientations of the molecules in which this function is minimised. The most common measure of the fit between two structures is the root mean square distance between pairs of atoms, or RMSD ... 

The three-dimensional radius of gyration of a random coil was discussed in Sec. 1.10 and found to equal one-sixth the mean-square end-to-end distance of the polymer [Eq. (1.59)]. What we need now is a connection between two-and three-dimensional radii of gyration. Since the molecule has spherical symmetry r, r> = V + r + r, = 3r . If only two of these contributions are present, we obtain (2/3)rg 3 = rg2o- this result and Eq. (1.59) to... 

Next we consider a substitution for the mean-square end-to-end distance r. At first glance this seems easy, since Eq. (1.62) gives r = nlo. There are two problems associated with the use of this substitution ... 

In the volume elements describing individual subchains, the x, y, and z dimensions will be different, so Eq. (3.32) must be averaged over all possible values to obtain the average entropy change per subchain. This process is also easily accomplished by using a result from Chap. 1. Equation (1.62) gives the mean-square end-to-end distance of a subchain as n, 1q, and this quantity can also be written as x + y + z therefore... 

The conformational characteristics of PVF are the subject of several studies (53,65). The rotational isomeric state (RIS) model has been used to calculate mean square end-to-end distance, dipole moments, and conformational entropies. C-nmr chemical shifts are in agreement with these predictions (66). The stiffness parameter (5) has been calculated (67) using the relationship between chain stiffness and cross-sectional area (68). In comparison to polyethylene, PVF has greater chain stiffness which decreases melting entropy, ie, (AS ) = 8.58 J/(molK) [2.05 cal/(molK)] versus... 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

A similarity measure is required for quantitative comparison of one strucmre with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dy between confonnation i and conformation j of a given molecule is defined as the minimum of the functional... 

Eollowing the computational protocol of Czerminski and Elber [39,40] a number of restraints and constraints are added to (1) encourage the mean-square distances between adjacent structures to be approximately constant. 

Dipole-dipole (homonuclear) Afjfhomo) (mean-squared local field) Spin-echo NMR Intemuclear distances, number of surrounding nuclei... 

FIG. 14 Mean square displacement (Ar )io after 10 ps as a function of the initial distance p of a water molecule from the pore axis for the two runs with 96% (full line) and 74% filling. (Taken with permission from Ref. 24.)... 

With the total number of monomers and the volume of the system fixed, a number of statistical averages can be sampled in the course of canonical ensemble averaging, like the mean squared end-to-end distance Re), gyration radius R g), bond length (/ ), and mean chain length (L). 

The adsorption transition also shows up in the behavior of the chain linear dimension. Fig. 6(a) shows the mean-square gyration radii parallel, i gl, and perpendicular, to the adsorbing plate. While these components do not differ from each other for e for e > ej i g strongly increases whereas Rh decreases. In the first case (non-adsorbed chain) oc oc N as a dilute solution in a good solvent in the bulk. For adsorbed chains R /N 0 for oo because the thickness is finite it is controlled by the distance e- e from the adsorption threshold, but does not diverge as N oo. The adsorbed chain follows in fact a... 

FIG. 22 (a) Log-log plot of the scaling function vs C = upper curves refer to the mean-square end-to-end distance, R, while the lower ones show results for i g. (b) The same vs scaling variable N "Region I— free chains, II—crossover, and III—renormaUzed free chains. The slope 0.6 corresponds to l cross = 0.78. [20]... 

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