Correlation functions persistence length

Experimentally, one can use two complementary ways to evaluate the persistence length from the images of the chains. The bond-correlation function... 

Some important properties of polymer chains in dilute solutions [steric hindrance parameter, characteristic ratio, persistence length, radius of gyration, statistical chain segment length (introduced earlier, in Chapter 11), intrinsic viscosity, and viscosity at small but finite concentrations] will be discussed, and new correlations will be presented for the steric hindrance parameter and the molar stiffness function, in Chapter 12. 

Result 1 contradicts the preceding result displayed in Fig. 15.4, where q2H(q) tends towards a constant. The contradiction is only apparent, because the model chain of Yoon and Flory and the real polystyrene chain are different. The explanation has been found thanks to a very precise experiment carried out by Rawiso and Picot.11 These experimentalists prepared their samples of atactic polystyrene in the following way the chains were made of partially deuterated monomers, in which only the protons directly bound to the skeleton carbons are replaced by deuterium atoms. The labelled chains are immersed in a non-labelled polystyrene melt. Therefore, only the correlations between points close to the skeleton contribute to the form function. The result of the experiment (Fig. 15.5, curve a) is qualitatively in agreement with the predictions of Yoon and Flory, who in fact accounted only for the carbon skeleton in their calculation. The persistence length associated with the form function of Fig. 15.5, curve a, is... 

The normal-normal correlation function can be used to define the persistence length of the membrane as the distance over which the normal becomes decorrelated via the thermal undulations the distance r at which gnir) is of order unity. The persistence length, is defined as... 

Find the tangent-tangent correlation function and persistence length for a polymeric chain with curvature elasticity — i.e., a one-dimensional membrane embedded in a three-dimensional space. This is applicable to the physics of a flexible rod undergoing thermal fluctuations. Contrast the results with those of a two-dimensional membrane. 

An alternative measure of chain extension is the persistent length, P , which behaves in a similar asymptotic dependence on n as observed for C in Figure 12.2.11. It is meant as the capacity of the chain to preserve the direction of the first residue (vector) of the chain. Therefore, as the directional persistence dissipates with increasing chain length and the direction of the terminal residue vector L loses correlation with that of the initial vector L, P approaches an asymptotic limit for sufficiently long chains. Both the and P functions in Figure 12.2.11 reveal the [a-D-(l-3)-gle] chain to be much more extended flian that of [P-D-(l-3)-gle]n. 

Figure 12.2.11. Characteristic ratio, correlation function and persistent length as a function of the degree of polymerization, n, for [a-D-(l-3)-glc] (pseudonigeran) and (b) the [P-D-(l-3)-glc] (curdlan) calculated on the in vacuo energy maps.

In the part devoted to neutral polymers, we mentioned that semiflexible and stiff chains do not obey the behavior predicted by the Kuhn model. Restricted flexibility of the chain can be caused by the presence of stiff units with multiple bonds or bulky pendant groups, but it can be a result of external conditions or stimuli. In the preceding part, it was explained in detail that repulsive interactions together with entropic forces increase the stiffness of PE chains. Hence, a sudden pH change can be used as a stimulus affecting the stiffness of annealed PE chains. The properties of semiflexible polymers are usually treated at the level of the wormlike chain (WLC) model developed by Kratky and Porod [31]. The persistence length, /p, is an important parameter strongly related to the WLC model and has been used as the most common characteristic of chain flexibility—in both theoretical and experimental studies. It is used to describe orientational correlations between successive bond vectors in a polymer chain in terms of the normalized orientation correlation function, C(s) = (r,.r,+j). For the WRC model, this function decays exponentially ... 

Fig. 2.6. Schematic representation of the orientational correlation function of a chain. The integral width determines the persistence length Ips...

The basic structure of the persistent chain has, indeed, already been introduced in Fig. 2.5 at the beginning of Sect. 2.3. It shows a chain with varying curvature being represented by a curve of length l t, which possesses at each point a well-defined tangent vector, e(/). In order to describe the chain structure, statistics was employed and the orientational correlation function iFor(A/) introduced by Eq. (2.5)... 

Combining all the information collected so far, for the single chain pair distribution function in a semidilute solution we can predict an overall shape as indicated in Fig. 3.8. For the presentation we choose a plot of 4nr g r) versus r. The curve is a composite of different functions in four ranges, with cross-overs at the persistence length Ips, the thermic correlation length and the screening length Up to r we find the properties of the expanded chain, which... 

---