Blob model

In Section 4.2, we will learn about the thermodynamics of semidilute solutions. We will consider linear flexible chains only. The mean-field theory explained in Section 2.2 is assumed to be effective in a whole range of concentrations. The theory, however, fails to explain various experimental results. The failure can be ascribed to the stronger interactions between chain molecules consisting of covalently bonded monomers compared with the mean-field interactions that do not distinguish bonded monomers from nonbonded monomers. Fortunately, the blob model and the scaling theory explain the thermodynamics that characterize semidilute solutions. 

Section 4.3 focuses on dynamics. We will first examine the overall concentration fluctuations of highly entangled chains. We will then look at the motions of each chain through the maze of other chains. 

In this chapter, we attach the subscript 0 to denote the value of the relevant quantity in the dilute solution limit. For instance, is the root-mean-square radius of gyration, R, in solutions at sufficiently low concentrations. It may appear strange that the chain size changes with concentration, but, as we will learn in Section 4.2.2, the chain size diminishes because the excluded volume that swells the chains at low concentrations becomes negligible at higher concentrations. 

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The above-mentioned conformational fluctuations can be easily understood on the basis of the blob model [104-109]. For example, this is illustrated for the transition from I to I. Within the chain swollen by excluded-volume... 

The crossover from 0- to good solvent conditions leads at constant x = (x — 0)/0 to increasing Q(Q,x)/Q3 with decreasing Q. Qualitatively, this effect is well described in the framework of the blob model using the method of the first cumulant, proposed by Akcasu and coworkers... 

The effect of excluded volume on g for linear chains has been calculated, first more qualitatively by Weill and des Cloiseaux193 on the basis of scaling arguments, then by Akcasu and Benmouna202 quantitatively on the basis of the blob-model. The result is as follows... 

Furthermore we should stress that the length scales identified by the blob model concern characteristic features of the segment-segment correlations. Thus the appealing picture of a chain build up from well defined individual blobs should not be taken literally. Rather each segment can be taken as the center of its correlated blob, and translational invariance along the chain is not destroyed. 

The concentration blob model of scaling theory has been introduced in [DCF+75, Far7G], where it has been shown to qualitatively explain a lot of experiments, A comprehensive review is given in [dG79]. 

A 13.2,2 Choice of cq. The parameter cq determines the crossover among the dilute or semidilute regimes. It is thus related to the overlap concentration c introduced in the blob model. We may determine it from an analysis of the osmotic pressure in the excluded volume limit. The scaling law reads... 

Now consider the weak coupling region / < 1. Prom Eq. (13.33 ii) we find that q2 —v g2, z fixed, implies,/ — 0. From Eqs. (13,24), (14.16) we then find the trivial limiting behavior Ja(q) 1/q2, in accord with the temperature blob model. For a system close to the excluded volume limit for large q we first may find some region described by the law (14.17), the trivial behavior developing only for extreme values of the momentum (see Fig. 14.2),... 

Most of the discussion above has been for brushes in a good solvent. Williams [75] has applied the blob model for polymer brushes to grafted chains in a poor solvent. In this case, the blobs no longer repel but attract, leading to a reduction... 

In this chapter we will briefly discuss mechanisms of the positron slowing down, the spatial structure of the end part of the fast positron track, and Ps formation in a liquid phase. Our discussion of the energetics of Ps formation will lead us to conclude that (1) the Ore mechanism is inefficient in the condensed phase, and (2) intratrack electrons created in ionization acts are precursors of Ps. This model, known as the recombination mechanism of Ps formation, is formulated in the framework of the blob model. Finally, as a particular example we consider Ps formation in aqueous solutions containing different types of scavengers. 

There are two models which utilize this mechanism, the spur model [18, 16] and the blob model (diffusion-recombination model) [19, 20]. In spite of the fact that both models answer the question about the Ps precursor in the same way, they differ as to what constitutes the terminal part of the e+ track and how to calculate the probability of the Ps formation. 

The blob model describes reactions (16) in terms of nonhomogeneous kinetics via Eqs. (17-18) on concentrations of the particles. It is an adequate approach to the problem because the number of particles involved is large and local motion of the intrablob electrons and the positron is fast... 

During the last 50 years a great deal of experimental data has been accumulated on the yields of Ps, H2 and e-q in various aqueous solutions. Here, we apply the blob model to calculate these yields in solutions of NOJ, H202, HCIO4, Cl-, Br-, I- and F-. They were selected to embrace the greatest possible variety of solute properties with respect to intratrack reactive species. 

Table 5.1 Reaction parameters of intratrack primary species obtained from application of the blob model to description of Ps, e q and radiolytic hydrogen yields in aqueous solutions.

The theoretical base of the spur process is Onsager s theory of the geminate pair recombination. Contrary to this, the blob model is most appropriate for consideration of early radiation-chemical processes in multiparticle track entities, such as blobs and ionization columns. The main distinction between the spur and blob comes from the large difference in the initial number of ion-electron pairs they contain. 

Several attempts have been made to relate these conductivity exponents tc or Sc with the percolation cluster statistical exponents discussed earlier. For example, in the node-link-blob model, the conductivity E of the network is proportional to the number of parallel links, while the... 

Using the node-link-blob model (see Section 1.2.1(d)) for the percolation cluster, one can in fact easily derive a rigorous bound of the fracture exponent Tf (Chakrabarti 1988, Ray and Chakrabarti 1988). This derivation is given below. One can see there that the above estimate for Tf in (3.12) turns out to be its lower bound. 

Fig. 3 A comparison of different coarse grain lipid models. The Shelley model " of DMPC, and Marrink and Essex models of DPPC are compared to their atomistic equivalents (for ease of comparison, hydrogen atoms of the atomistic models are not shown). Solid lines represent harmonic bonds connecting CG particles, and the CG particle types for the Shelley and Marrink models are labelled (the labels are the same as those used in the main text). The point charges (represented by + and —) and point dipoles (represented by arrows) are shown for the Essex model (the charges and dipoles are located at the centre of their associated CG particle). The Shelley and Marrink models use LJ particles (represented by spheres), while the Essex model uses a combination of LJ particles (spheres) and Gay-Berne particles (ellipsoids). Finally, the blob model proposed by Chao et al is also shown for comparison. This model represents groups of atoms as rigid non-spherical blobs that use interaction potentials based on multipole expansions.

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