The blob model

Let us summarize the most important properties of the blob, i.e., the end part of the positron track [3]. 

Size and energy release. The terminal blob consists of about n0 30 overlapped ion-electron pairs. Spatial distribution of these species may be described by Gaussian function where the blob radius  

Quasi-neutrality condition. Because of strong Coulombic interaction, intrablob electrons adjust their motion to the distribution of the primary positive ions and screen them. As a result, the width of the spatial distribution of the electrons is only slightly larger than that of ions. 

Expansion of the blob. Because of attraction between ions and electrons, expansion of the blob is governed by the law of ambipolar diffusion. As a result out-diffusion of electrons is almost completely suppressed, but the diffusion coefficient of ions is increased by a factor of two. Thus, blob expansion proceeds very slowly and may usually be neglected in the problem of Ps formation. 

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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. 

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... 

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. 

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.

Fig. 9.3. The scaling function R (N,Cp)/R (A, Cj, — 0) (excluded volume limit, monodisperse) hh function of s = CpRg N,Cp = 0). The full line is the renormalization group result. (See Chap. 18.) The broken lijies give approximations as motivated bv the blob model...

The collapse transition in the framework of the blob model of the polyelectrolyte macromolecule was considered in Ref. 17. It was proposed that the structure of the polyelectrolyte macromolecule as sequence of blobs remains valid also in poor solvent. However, in this case the space size D of... 

The complexity observed in polymer fluorescence decay kinetics is further exacerbated when fluorescent polyelectrolytes are dissolved in aqueous media [29,30,33,35,37,43,120,122,128-132] segregation of the macromolecular structure into hydrophobic and hydrophilic-rich domains results in differing degrees of water penetration which further complicates the time-resolved fluorescence [26]. Within this context, more recent attempts to describe time-resolved polymer photophysical data include use of the blob model [133,134], which accounts for the range of environments encountered in heterogeneous systems by invoking a distribution of rate constants for excimer formation. 

As far as the excimer decay kinetics of PAA in aqueous media is concerned, de Melo and coworkers [122,130,131] have investigated the time-resolved fluorescence from a series of samples modified with various amounts of pyrene and naphthalene, respectively. Even when the aromatic content was as low as 2mol%, excimer formation was evident in the steady-state spectra. The fluorescence decays were complex irrespective of the label and were best modeled by a triple-exponential function (as in Eq. 2.8) both when emission was sampled in the monomer and excimer regions. In contrast to the distribution of rate constants in the blob model [133,134], the authors favored a scheme that describes the decay kinetics in terms of discrete rate constants. The data were also consistent with previous schemes [124-127] that account for the presence of two distinct types of monomer in addition to that of excimer in macromolecular systems one monomer enjoys kinetic isolation and is unable to form excimers, whereas the second is able to participate in excimer formation within its fluorescence lifetime. The authors [130] concluded from both steady-state and time-resolved data that PAA undergoes a conformational change from a compact form in acidic solution to an open coil at high pH. Furthermore, as the... 

The problems of penetration of polymers with arbitrary connectivity into confined spaces (for examples, pores or slits) have been discussed [62]. It has been noted that the Flory theory, the blob model [62] and the scaling concept provide identical results for linear chains however, in the case of branched polymers, the Flory theory and scaling can lead to contradictory results when applied without invoking additional information. 

Dondos, A. A new relation between the intrinsic viscosity and the molecular mass of polymers derives from the blob model determination of the statistical segment length of flexible polymers. Polymer, 2001, 42(4), 897-901. 

The first theories that implemented a proper balance of intramolecular interactions and conformational elasticity of the branches were developed by Daoud and Cotton [21] and by Zhulina and Birshtein [22-24]. These theories use scaling concepts (the blob model), originally developed by de Gennes and Alexander to describe the structure of semidilute polymer solutions [64] and planar polymer brushes [65, 66]. Here, the monomer-monomer interactions were incorporated on the level of binary or ternary contacts (corresponding to good and theta-solvent conditions, respectively), and both dilute and semidilute solutions of star polymers were considered. Depending on the solvent quality and the intrinsic stiffness of the arms, the branches of a star could be locally swollen, or exhibit Gaussian statistics [22-24]. 

According to the blob model, a flexible neutral star polymer can be envisioned as an array of concentric shells of closely packed blobs. For a visualization of the blobs, see Fig. la. The chain ends are assumed to be localized at the edge (i.e., within the outermost blobs), and each chain contributes one blob to each shell. The chain segment inside a blob remains unperturbed by the interactions with other branches and, therefore, exhibits Gaussian or excluded-volume statistics under theta- or good solvent conditions, respectively. For transparency, we consider first athermal, u = a, and theta-solvent, i = 0, conditions. The blob size at distance r from the star center is equal to the average interchain separation = which... 

In the absence of charges, or at a low degree of ionization of the arms, the star conformation is controlled by a balance between two first terms in (7) (i.e., the short-range interarm repulsions and the conformational entropy of stretched arms). As a result, the star size is given by (4). That is, the power law dependencies, obtained on the basis of the blob model, are recovered. The physical reasons why there is a match of the star size as obtained by the scaling and in the mean field approximations are discussed in details in [23]. 

Finally, the coronal contribution, F ona Rcore), to the free energy of an aggregate of morphology / is calculated as the free energy of a planar (i = 1) or curved (i = 2,3) polymer brush. This is attained by a generalization of the blob model for the case of an arbitrary (finite) curvature of the grafting surface ... 

The structure and the basic thermodynamics of micelles formed by amphiphilic block copolymers with a PE coronal block A can be analyzed using the blob model. However, the ionization of block A in a polymeric amphiphile introduces long-ranged repulsive interactions in the corona of a micelle. As a result, the blob picture for the micellar corona has to be modified, as explained in this section. 

In the low salt limit, atCp > the coronal contribution to the free energy is dominated by the translational entropy of counterions entrapped inside the corona, int = k TabNAi riCp - 1). In this case, all results of the blob model are recovered both for osmotic starlike and crew-cut spherical micelles (59), (61), and (62). 

The Weill-des Cloizeaux theory assumes that no expansion takes place within subchains shorter than aN. On this assumption we may consider a chain model which consists of Gaussian subcoils, each being made of Nc beads and interacting with one another. With such a subcoil viewed as a blob, this model is often referred to as the blob model. However, it is important to recognize that the Weill-des Cloizeaux theory concerns interactions between beads but not those between blobs. Therefore, contrary to many other authors, the author does not consider it relevant to call it the blob theory. 

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