Root mean square chain length

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

Using the Monte Carlo method, for walks that can intercept, confirms this result. The root mean square length of the 100 step walks in Fig. 3.7 is 14.135. The theoretical distribution in the figure is derived in Section 3.4. It is the product of a 4irr term (the surface area of a sphere of radius r on which the chain end lies) and the Gaussian distribution of Eq. (3.13). 

Calculate the root-mean-square length of a polyethylene chain of M = 250000 g moP assuming that the equivalent random link corresponds to 18.5 C—C bonds. 

As shown in Appendix 9.A, from considering the polymer chain as a Gaussian chain consisting of No segments with the root mean square length b, the functional form for 5L t) is derived as... 

The time-correlation function 5L 0)5L t)) of Eq. (9.3) will be derived by considering the polymer chain as a Gaussian chain consisting of No segments each with the root mean square length b. Let 5 (t) be the contour position of the nth bead relative to a certain reference point on the primitive path. Then the contour length of the primitive chain at time t is given by... 

The theory of cyclization dynamics was first presented by Wileaski and Fixman [WF] (5). A number of curious features of the theory prompted detailed attention by Doi (11), by Perico and Cuniberti (12), and by others (13). The theory is developed in terms of the bead-and-spring Rouse-Zimm [RZ] model (14). Unrealistic in detail, this model is quite useful for describing low frequency, large flmq[>litude chain motions. The RZ model, figure 2, treats the chain as a series of n beads connected by (n-1)harmonic springs of root-mean-squared length b. 

The only parameter, b, is the inverse of the most probable chain end separation and is under the above conditions Z3/n. The extended length of the chain is, of course, equal to nC, whereas the average chain end separation (the root-mean-square length /W equals /n. 

Fig. 6. Freely jointed chain of Kuhn segments and head/spring chain model. For coarse-grain treatments for long time and length scales, the polymer may be modeled by a chain of beads and massless entropic springs. The beads are assumed to have a hydrodynamic radius a. The Kuhn segment length, b, is defined as the root mean squared length of the springs. The root mean squared chain end-to-end distance is called the Flory radius Rp...

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 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. 

At low concentrations, when uncharged polymers are dissolved in a solvent in which they do not crosslink or entangle, they possess a viscoelastic response through hydrodynamic and entropic effects. We can begin by considering an isolated chain in its quiescent state. The chain will be in constant motion. In the absence of any specific interactions, the chain will evolve to its maximum entropy state. We can represent the chain as N links or submolecules each with a length b. These links are formed from a few monomer units of the chain. The root mean square end-to-end length of the chain is... 

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

A typical freely jointed chain will therefore be quite compact since the root-mean-square value of its end-to-end length, J(ft2) = f N b, will be small compared with its length if it were stretched out (i.e., Nb, when N is large). Figure 10.9 shows a simulated molecule of polyethylene, (-CH2-CH2-)/v, which approximates a freely jointed configuration. 

Fig. 10. The root-mean-square layer thickness tms as a function of the square root of chain length r for four values of polymer volume fraction 0 . Hexagonal lattice,...

Flory-type free energy calculations show that the root mean square end-to-end distance of a polyelectrolyte increases linearly with the chain length at infinite dilution and without added salt [40]. Using the above perturbation theory, scaling relations at finite densities are obtained. The influence of the interaction with other polymer chains, counterions, and added salt is captured in the Debye screening length xT1. 

Fig. 4 Root mean square end-to-end distance of flexible polyelectrolyte chains as a function of chain length for lB/b=0.5. The Debye screening length decreases from top to bottom (tc=0.05, 0.1, 0.2, 0.4, 0.8). The slopes of the straight lines are 1 and 3/5, respectively...

Fig. 7 Density dependence of the root mean square end-to-end distance of a flexible polymer for the chain lengths N= 127, 63, 31, 15 (top to bottom) and hlb=0.833. Dots mark results from computer simulations of Refs. [27, 49]...

Figure 7 displays the root mean square end-to-end distance as a function of the monomer density for various chain lengths. As expected, the chains are expanded in dilute solution and they contract with increasing density. The dots mark results from computer simulations [27, 49]. As is obvious from the figure, the above approach yields an excellent agreement with simulation results not only qualitatively but also quantitatively. Hence, we expect to find reliable results also for other quantities by our approach. 

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