Chain root-mean-square

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... 

Figure 14 Measures of disorder m the acyl chains from an MD simulation of a fluid phase DPPC bilayer, (a) Order parameter profile of the C—H bonds (b) root-mean-square fluctuation of the H atoms averaged over 100 ps.

In order to examine whether this sequence gave a fold similar to the template, the corresponding peptide was synthesized and its structure experimentally determined by NMR methods. The result is shown in Figure 17.15 and compared to the design target whose main chain conformation is identical to that of the Zif 268 template. The folds are remarkably similar even though there are some differences in the loop region between the two p strands. The core of the molecule, which comprises seven hydrophobic side chains, is well-ordered whereas the termini are disordered. The root mean square deviation of the main chain atoms are 2.0 A for residues 3 to 26 and 1.0 A for residues 8 to 26. 

The value should be that of single polymer chain elasticity caused by entropic contribution. At first glance, the force data fluctuated a great deal. However, this fluctuation was due to the thermal noise imposed on the cantilever. A simple estimation told us that the root-mean-square (RMS) noise in the force signal (AF-lS-b pN) for an extension length from 300 to 350 nm was almost comparable with the thermal noise, AF= -21.6 pN. 

Figure 3. Partition coefficient of freely jointed chains between the bulk solution and a cylindrical pore. The chains have different numbers of mass-points (n) and different bond lengths, and are characterized by the root-mean-square radius of gyration measured in units of the pore radius. See text for details.

The root-mean-square distance Vr separating the ends of the polymer chain is a convenient measure of its linear dimensions. The dissymmetry coefficient will be unity for (VrV 0< l and will increase as this ratio increases. 

The statistical distribution of r values for long polymer chains and the influence of chain structure and hindrance to rotation about chain bonds on its root-mean-square value will be the topics of primary concern in the present chapter. We thus enter upon the second major application of statistical methods to polymer problems, the first of these having been discussed in the two chapters preceding. Quite apart from whatever intrinsic interest may be attached to the polymer chain configuration problem, its analysis is essential for the interpretation of rubberlike elasticity and of dilute solution properties, both hydrodynamic and thermodynamic, of polymers. These problems will be dealt with in following chapters. The content of the present... 

On the other hand, the correction factor by which W r) is altered through this refined treatment, namely, exp[ —(9n/20)(r/r, ) ] from Eq. (16), depends both on n and on r/Vm If the distance of separation of the ends of the chain lies in the vicinity of its root-mean-square value, i.e., if r / then... 

The foregoing discussion of equivalent chains requires merely that its root-mean-square end-to-end distance shall equal that of the real chain. In order to define completely the equivalent chain, its contour lengths may also be required to coincide with that of the real chain. 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

Thus we may retain the root-mean-square end-to-end distance as a measure of the size of the random-coiling polymer chain, and the parameter jS required to characterize the spatial distribution of polymer segments (not to be confused with the end-to-end distribution) can be calculated from It should be noted that the r used here... 

The evaluation of the elastic free energy AFei rests on the assumption that the root-mean-square distance between the ends of the chain is distorted by the same factor a representing the linear expansion of the spatial distribution. As in the treatment of the swelling of network... 

Equation (10) directs attention to a number of important characteristics of the molecular expansion factor a. In the first place, it is predicted to increase slowly with molecular weight (assuming t/ i(1 — 0/T) >0) and without limit even when the molecular weight becomes very large. Thus, the root-mean-square end-to-end distance of the molecule should increase more rapidly than in proportion to the square root of the molecular weight. This follows from the theory of random chain configuration according to which the unperturbed root-mean-square end- o-end distance is proportional to (Chap. X), whereas /r = ay/rl. 

This remarkably simple treatment suffers one serious deficiency the value of remains quantitatively undefined. More or less intuitively it has been suggested by various investigators that should increase as the root-mean-square end-to-end distance /for a linear chain, or, more generally, as the root-mean-square distance /s2 q beads from the center of any polymer molecule, linear or branched. Accepting this postulate unquestioningly, we should then have/o proportional to and [rj] proportional to These conclusions happen to... 

Here, a is the elongation ratio of the polymer chains in any direction and (r/j1/2 is the root-mean-square, unperturbed, end-to-end distance of the polymer chains between two neighboring crosslinks (Canal and Peppas, 1989). For isotropically swollen hydrogel, the elongation ratio, a, can be related to the swollen polymer volume fraction, u2,j> using Eq. (11). 

Figure 19. The root-mean-square (RMS) position of each segment of both acyl chains of SOPC lipids is plotted as a function of the average position of the segment. The sn tail is given by the closed symbols, and the sn2 tail is given by the open symbols. Various numbers of the tail segments and of the backbone segments are indicated. The lines are drawn to guide the eye. The arrow points to the position of the unsaturated bond, (a) SCF results (conversion from dimensionless units to real units is approximately a factor of 0.2 nm), (b) MD results (the average over the sides of the bilayer is taken)...

The ratio of the root mean square lengths is called the chain expansion factor ... 

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