Swelling exponent

For the swelling exponent 0.588 the expression for the chain size is R b Note that this scaling result reduces to the prediction... 

Tricritical swelling exponent in two dimensions estimations and exact values... 

In this case, if condition (14.7.2) is not fulfilled, the p-body interaction is not relevant then the chains are asymptotically Brownian in other words, the swelling exponent has the value 1/2. On the contrary, if (14.7.2) is fulfilled, the p-body interaction is relevant then, there exists a p-critical exponent, which we denote by the symbol vp(l > vp > 1/2). Thus, for an isolated p-critical chain, we have... 

In this approximation, vG(z) < v(z) the effective gyration swelling exponent... 

Two distinct swelling exponents, i.e. s for Type 1 systems and. y for Type 2 systems, are needed. The... 

Figure 16.33. Variation of the swelling exponents of prolate and oblate ellipsoids (of revolution) as a function of the axial ratio (the dotted curves represent the average shape parameters)...

Swelling exponents between 0.33 and 0.45 (depending on the anisotropy of the micelle) are expected for prolate micelles, intermediate to the exponents for spherical and cylindrical columnar micelles. Oblate ellipsoidal micelles exhibit swelling exponents between 0.33 and 1. The upper bound - for very flat tablet-shaped micelles - overlaps with those of mesh and lamellar phases. 

Table 16.2. Shape parameters (equations (16.1) and (16.4)) and approximate swelling exponents s and s, cf. equation (16.8)) for known lyotropic mesophases. The (variable) constants /i and depend on the specific symmetry of the phase ( /i is the homogeneity index, ideal equal to 3/4 (cf. Table 16.1) / is the interstitial packing fraction for dense sphere and circle packings, equal to unity for ideal homogeneous packings)...

The interaction parameter B, proportional to the intrinsic viscosity, is a function of the distance to the gelation threshold. This is an evidence of the swelling of the clusters. If in the expression of B(Eqs. 8 and 16) we insert the D and o (= (3-x)/y = 0.47 + 0.04) values determined experimentally, we find D = 2.42 + 0.15 in perfect agreement with the percolation theory (D = 2.5). This fractal dimension, which corresponds to the unswollen state is larger than that measured in the swollen state (D = 1.98 + 0.03). The e independence of the ratio [ql/B, means that the fractal dimension D of clusters of polyurethane is identical in the two solvents used (THF and dioxane). Due to this swelling, exponent v linking size to the distance to the gel point cannot be directly compared with the percolation exponent value v = 0.88. 

For ideal chains, one has (p = v= 1/2, and thus we recover the prediction from the transfer-matrix calculations, Eq. (7). For nonideal chains, the crossover exponent

different from the swelling exponent v. However, extensive Monte Carlo eomputer simulations point to a value for (p very close to v, such that the adsorption exponent v/q> appearing in Eq. (11) is very close to unity for polymers embedded in three-dimensional space [31]. 

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