Number of blob per chain

If compared to Eq. (9 17) the second equation shows that N/Nr is to be identified with the number of segments n per concentration blob, whereas Nr — Nfn is the number of blobs per chain. The first equation shows that we find a smooth crossover from the dilute limit w — 1 to the semidilute limit w — 0. In the latter limit Eqs. (14 13), (14.14) yield the expected power law... 

Morphological transitions sphere-cylinder-lamella occur, therefore, when the aggregates acquire the crew-eut shape. It is instructive to consider first a lamellar aggregate, i=. Here, the coronal contribution is given by the number of blobs per chain in a planar brush ... 

Electrostatic interactions at the length scales larger than the blob size lead to the elongation of the polyelectrolyte chain into an array of blobs. The size of the polyelectrolyte chain is estimated as the number of blobs per chain N/ge° times the blob size D.°... 

The total electrostatic energy of a chain is equal to the electrostatic energy of a blob times the number of blobs per chain N/ge... 

If the test chain is infinite, the free energy of the polymer solution is invariant by translation of the chain along itself, the effect of excluded volume is, therefore, very different for motions parallel to the chain or transverse to it this is the origin of a reptative motion. The reptation diffusion coefficient is obtained if the entanglement parameter ij/, proportional to the number of blobs per chain, is larger than a critical value at low values of this parameter the self-diffusion constant is given by a Rouse approximation. 

Knowing it is straightforward to obtain the brash thickness L and the free energy per chain Fcorona- L is determined by monomer conservation. Within this picture each grafted chain is a fully stretched string of blobs and the number of monomers per chain N is... 

Nj,=N/f is the number of beads per branch or arm). For larger chains, however, the solvent can penetrate in outer regions of the star and the situation within these regions is more Hke a concentrated solution or a semi-dilute solution. These portions of the arms constitute a series of blobs, whose sizes increase in the direction of the arm end. The surface of a sphere of radius r from the star center is occupied by f blobs. Then the blob size is proportional to rf. Most internal blobs are placed in conditions similar to concentrated solutions and, consequently, their squared size is proportional to the number of polymer units inside them as in an ideal chain. This permits one to obtain the density of units inside the blob, as a function of r ... 

For s 1. the Tit blob is smaller than the whole chain and the blob-concept starts to make sense. For large overlap in view of screening the number of concentration blobs per chain should not be important. Thus iJ should reduce to a function of the blob concentration only. In view of Eq. (9-11) we therefore expect U to become a function of c independent of n. With this assumption the scaling law (9.2) yields... 

Measurement of the temperature dependence of second virial coefficient A2 for polymers with known molar mass M and Kuhn length b allows estimation of the number of thermal blobs per chain A /gj using Eq. (3.109). 

Determine the relation between the chain interaction parameter z [defined in Eq. (3.22)] and the number of thermal blobs per chain N/gr-... 

At the overlap concentration, the correlation length is equal to the dilute coil size because the coils are space-filling at 0 and the correlation blobs are always space-filling. Above the overlap concentration, the correlation length does not depend on the number of monomers in a chain. Correlation blobs behave as shorter chains with g monomers at overlap [compare Eqs (5.19) and (5.21)]. A solution of longer chains (with N > N>g) at the same concentration 0 can be thought of as a melt of correlation blobs with more of these blobs per chain (with Nxjg > Njg). The correlation blobs are... 

The height /f increases linearly with the number of monomers N per chain at constant grafting density. The stretching energy per chain chain in the grafting layer is A T times the number of correlation blobs per chain ... 

In the framework of the scaling theory, the corona of a spherical micelle can be envisioned [53-56] as an array of concentric spherical shells of closely packed blobs. The blob size, (r) = r/grows as a function of the radial distance r from the center of the core. Each blob comprises a segment of the chain within the local correlation length of the monomer density fluctuations [57], and corresponds to a contribution to the free energy of steric repulsion between the coronal chains. After calculating the total number of blobs in the micellar corona, one finds fhe free energy (per coronal chain) as ... 

In the case of crew-cut micelles, //corona < Rcok and the logarithm in (22) can be expanded up to the term linear in //corona// core, to give / corona/ B — Z/corona/i / = //corona/. The thickneSS Of the corona, //corona, scales as //corona = In the framework of the Alexander-de Gennes blob model [51, 52], the micellar corona (the planar brush) can be envisioned as an array of closely packed blobs with size = 5 /, equal to the average distance between the coronal blocks. We note that a constant size of the blobs implies //corona Na. The number of coronal blobs per chain //corona/ is proportional to the free energy of the interchain repulsion that equals fcorona/kBT = ... 

For the corona contribution, some care has to be taken. In order to calculate the free energy of this part, the number of blobs has to be calculated for each morphology, taking into accotmt that the radial dependence of the density changes with the curvatore. For a completely planar surface, de Gennes and Alexander [36,37] showed that the blob size wDl be constant and scale by s, where s is the area available per chain. For a cuiwed micellar core, the surface per corona chain will naturally increase... 

Thus the free energy per monomer of a chain consisting of a number of blobs with connecting parts, each of length Y, is given by... 

It turns out that there is a simple rescaling possible, by which all curves superimpose (at least approximately) on a master curve (Fig. 13b) A b is rescaled by Sbiob. the number of monomers per blob, motivated by the idea that the bottlebmsh polymer is viewed as a pearl-necklace chain of blobs, the blob radius being the cross-sectional radius of the cylindrical brush. From this condition, biob is easily derived numerically from the simulation data. 

Self-diffusion becomes much slower in high-molecular-weight concentrated solutions. A theory for this process has been developed by deGennes. A chain of N monomers in a semidilute or concentrated solution can be represented as a concatenation of blobs of rms length. The chain is constrained in a tube of other chains of length L = (N/g%, where g is the number of monomers per blob. The chain itself has a mean-squared end-to-end length (r = N / g if. 

As usual, we can relate the semi-dilute region to the concentrated regime using blobs as our fundamental units. If, as in Chapter III, we call g = (ca ) the number of monomers per blob, the numter of blobs in the test chain is N/g, and the reptation time is, by a natural extension of eq. (VIII. 13),... 

To verify the scaling hypothesis for the chain size, the plot of the normalized chain size RJ as a function of the number of correlation blobs per chain N/g is shown in Figure 34. AH points for chains with different degrees of polymerization, different fractions of charged monomers, and at different polymer concentrations collapse onto one universal line, with the slope 1/2 as expected for Gaussian chains with N/g, correlation blobs. 

The behavior of PE chains can be analyzed in more detail by the scaling approach based on the concept of thermal and electrostatic blobs. The blob theory assumes that, on small length scales shorter than the correlation length (called also the blob size ), the energy of random thermal motion counterbalances the excluded volume effect of segments, and short parts of the chain behave as ideal chains. Therefore, it holds that where gn is the number of segments per... 

Figure 2 summarises the situation the chain has partially entered into the pore (over a length F >D). It can be pictured as a sequence of blobs, each of size D and of monomer number gD=(D/a)5/3 (theFlory law). The confinement energy per blob is of the order kT [13] and the overall confinement energy is thus kT f/D ( /D being the number of squeezed blobs. The force tending to pull the chain out of the pore is thus kT/D. 

Since each chain is a stretched array of tension blobs, their end-to-end distance / yin an extended state is the product of the tension blob size and the number of these blobs Njg per chain ... 

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