Chain size, mean-square

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... 

Fig. 20 Mean-square end-to-end distance of chains vs. chain length in a 32 (or above)-sized cubic lattice. The data are those of the polymer volume fractions (Hu and Frenkel, unpublished results)...

Fig. 21 Mean-square displacement vs. evolution time for 16-mers with an occupation density of 0.9375 in a 32-sized cubic lattice. The triangles are for four middle chain units, the circles are for the mass center, and the crosses are for the chain units relative to the center of mass. The lines with slopes of 1.0 and 0.5 indicate the scaling expected according to the Rouse model of polymer chains [56]... 

Parameter characterizing the effect of long-chain branches on the size of a branched molecule in solution and defined as the ratio of the mean-square radius of gyration of a branched molecule, si), to that of an otherwise identical linear molecule si), with the... 

A comparison of Eq. (25) with Eq. (23) shows that the two expressions for xp are identical, since nmean square end-to-end length of the entire chain. This is true for an arbitrary choice of the size of a Rouse subchain. 

Schwarz31) has modified the equation of state for the Gaussian polymer network by introducing a coefficient taking into account the influence of the packing of chains and their sizes on the mean square end-to-end distance of deformed chains. 

In the previous sections, we have dealt only with telechelic oligomers containing linear non-, mono- and bifunctional molecules. In practice, however, oligomer systems containing components of higher functionality are often encountered. If the mean square distance between functional groups in the chain is greater than the pore size of the stationary phase, then... 

Hesselink39,40 derived relations between the root-mean-square thickness and the loop or tail size. For the case in which no intrasegment interaction exists, he derived the segment distribution p4(z) for a single loop of size i by considering all posable configurations of the chain that starts at the interface and returns at its end to the interface. His expression for p4(z) reads... 

One such model is the ideal freely jointed segments chain (Fig. 6.14). In this model the polymer is considered to consist of a chain of n links. We call each chain link a subunit . Each subunit has a length l. This parameter / can correspond to the length of a monomer but it can also be shorter or longer. The angle between adjacent chain links is taken to be arbitrary. The chain forms a random coil. To characterize the size and volume of such a coil we use the mean square of the end-to-end distance R2. The square-root of this value — we call it the size of a polymer chain — is given by... 

Any linear polymer molecule, even a reasonably stiff rod, will coil randomly, provided the chain is sufficiently long. In addition to the size of the monomer units, the tendency for coiling is affected by the forces between the units as well as the interaction between the polymer and the solvent. One measure of the stiffness of a polymer molecule is the end-to-end distance (R). For a polydisperse polymer the root-mean-square average of R (R2)v2 is used. R is affected by the properties of the polymer itself as well as by the interaction of the solvent. The better the solvent the more the polymer... 

A basic characteristic of a single polymer is its spatial dimensions, such as the radius of gyration. The average size of the ideal chain is identical to the mean square displacement of the random walker J(R fd) — IN1/2 (the bracket... 

The values of mean-square length of monomer unit and distance between polymer chain ends are presented in Table 1. It is important to pay attention to the fact that sizes of macromolecule ball in vitrificated low-molecular solvents and in solid non-marked polymer are differed insignificantly. 

When the salt of PMAA is formed the rigidity of polymer chain is increased a little that leads to little increase of values of mean-square length of monomer unit and distance between polymer chain ends. The result obtained under addition to polymer salt solvent in methanol (40 vol. % of water, at large content of water the solution is not vitrificated, but is crystallized) represents special interest at that the macromolecule in solid state keeps conformation of Gauss ball (Figure 2), however, as we should expect, rigidity (parameter a) and sizes of ball are increased (Table 1). 

A radius of gyration in general is the distance from the center of mass of a body at which the whole mass could be concentrated without changing its moment of rotational inertia about an axis through the center of mass. For a polymer chain, this is also the root-mean-square distance of the segments of the molecule from its center of mass. The radius of gyration is one measure of the size of the random coil shape which many synthetic polymers adopt in solution or in the amorphous bulk state. (The radius of gyration and other measures of macromolecular size and shape are considered in more detail in Chapter 4.)... 

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