Screening of the interaction

The interaction between polymers is given by a repulsive potential of the form 

However, the interaction between two polymers in solutions looks as if it is modified by the presence of the other polymers, and this situation leads to a special kind of screening. Actually, Edwards48 who introduced this idea says that the real interaction can be replaced by an effective interaction bE(r) whose essential property is to be screened in the sense that 

The effective interaction represented by b(q) can be considered as the sum of two terms, a direct interaction and an indirect interaction through one or several interacting polymers. The definition of the structure factor and its diagram representation (see Fig. 13.18) shows that b(q) must be defined by the equality 

For a grand ensemble of polymers, whose concentration is determined by the fugacity f, H(q) is given by (13.2.133). In other words, b(q) is given as a function of concentration by the coupled equations 

In this case, relation (13.2.139) is satisfied since (13.2.143) implies the equality 6(8) = 0. In the general case, the relation remains true provided that oo when q - 0 (and this is likely to be true). 

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The scattering cross-section is considerably different from the Rutherford cross-section, because the distance of closest approach, Ri i , is rather large at low energies. Thus, electronic screening of the interaction between the nuclei is important. The screened scattering potential V(r) reads ... 

If one or both reactants are not charged, it is not necessary to be concerned about ionic screening of the interaction between reactants for the homogeneous distribution. Other interactions between reactants are much smaller and do not screen the coulomb interaction. Furthermore, if the long-range transfer term is unimportant, then Rice and Baird [356] showed that... 

The excellent agreement between the MM, QM/MM and QM PES results directly from the consistent and consequent screening of the interactions for overlapping charge densities. Therefore, as was already shown previously [74,137], our model gives static and response potentials that are as good as SCF calculations (and better if the dispersion is accounted for) which let us separate the total system into different subsystems. 

The vector q represents the set of qi. The second-order coefficient, Jijirij), depends on the distance between the two atoms i andand at large distances should equal l/r,y. At shorter distances, there may be screening of the interactions, just as for the dipole-dipole interactions in the earlier section on Polarizable Point Dipoles. This screened interaction is typically assumed to arise... 

When we warm up the solution at fixed concentration C in the domain IV of the temperature-concentration diagram, the square radius 3X2 and the overlap ratio increase. The chain is then subject to two contradictory effects first the solvent quality increases, second the screening of the interaction becomes stronger. Let us first recall that the average square distance 3X2 varies in the following manner (see Chapter 12, Section 3.3.2)... 

The magnitude of potential of bubbles immersed in solutions of electrolytes are monotonously depressed by increasing the concentration of the mono- or divalent salts dissolved in the solutions. This behavior is attributed to the screening of the interactions between the applied electric field and the charged bubbles, induced by the salts. They also decrease as the acidity of the immersing solution increases. This decrease is all the more pronounced as the salinity is lower. For aU the authors, the isoelectric point ranges (or would range by extrapolation) from pH 1.5 to 3. 

The next step to include electron-electron correlation more precisely historically was the introduction of the (somewhat misleading) so-called local- field correction factor g(q), accounting for statically screening of the Coulomb interaction by modifying the polarizability [4] ... 

Transition in the Single Chain Dynamics owing to the Screening of Hydrodynamic Interactions... 

A model that can take these findings into account is based on the idea that the screening of hydrodynamic interactions is incomplete and that a residual part is still active on distances r > H(c) [40,117]. As a consequence the solvent viscosity r s in the Oseen tensor is replaced by an effective... 

With respect to the screening of hydrodynamic interactions, one is confronted with the occurrence of a multiple-transition behavior. Instead of the expected crossover from ordinary (unscreened) Zimm to enhanced Rouse relaxation, one observes, at increasing concentrations, additional transitions from enhanced Rouse to screened Zimm and from screened Zimm to enhanced Rouse relaxation. This sequence of crossover effects are highly indicative of an incomplete screening of hydrodynamic interactions. 

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