Fisher exponent

The critical exponent t is the same above and below the gel point and is called the Fisher exponent. The number of monomers N in the characteristic branched polymer increases as the gel point is approached (from either ide) and diverges as a powetM )f the distance from the gel point, char-... 

Since the Fisher exponent r for percolation in any dimension is limited to the interval 2 < r < 5/2, the first two moments of the distribution mo and mi do not diverge ahthe gel point. These two moments are dominated by the smaller polymers with a small contribution from the larger ones. Below the gel point, the sol fraction is unity [Psoiip) = mi = 1, see Eq. (6.42)] and the gel fraction is zero [Fgei(/ ) = 1 - P oi(p) = Oj- Above the gel point, we can approximate the cutoff function as a step function that goes to zero... 

Hyperscaling relates the fractal dimension of randomly branched molecules in the gelation reaction with the Fisher exponent and the space dimension (see Table 6.4) ... 

Below the gel point, the system is self-similar on length scales smaller than the correlation length with a power law distribution of molar masses with Fisher exponent r = 5/2 [Eq. (6.78)]. Each branched molecule is a self-similar fractal with fractal dimension 27 = 4 for ideal branched mole-cules in the mean-field theory. The lower limit of this critical behaviour is the average distance between branch points (= ). There are very few... 

The characteristic degree of polymerization N is chosen for each sample so that the n p, N) data for that sample can be superimposed onto the universal curve. For both of the curves in Fig. 6.26, more than 10 samples with different extents of reaction were superimposed in this fashion. Although the Fisher exponent t is not very different for the two classes of percolation, r is sufficiently different not to allow the data from one class to be superimposed with the wrong exponent. 

Use hyperscaling and the Flory-de Gennes calculation of fractal dimension to derive the following approximate relation for the Fisher exponent t in any space dimension and compare with the results in Table 6.4 for 3,4, 5 and 6 ... 

Fig. 22 The coefficient L2 of the term of the structure factor versus correlation length in double logarithmic presentation. According to theory the slope gives the Fisher exponent r] = 0.034 which is found a factor of two larger than predicted. The crossover transition at T has no effect on r] as the critical exponents y and v are renormalized in a same manner...

Table 1. Comparison of the critical exponents of NIPA gel with some known Ising systems. The number in the parentheses indicates the error of the corresponding exponent. The row next to the last is obtained directly by using an = —0.05. The last row is the Fisher renormalized results...

Returning to 3D lattice models, one may note that sine-Gordon field theory of the Coulomb gas should enable an RG (e — 4 — D) expansion [15], but this path has obviously not yet followed up. An attempt to establish the universality class of the RPM by a sine-Gordon-based field theory was made by Khodolenko and Beyerlein [105]. However, these authors did not present a scheme for calculating the critical exponents. Rather they argued that the grand partition function can be mapped onto that of the spherical model of Kac and Berlin [106, 297] which predicts a parabolic coexistence curve, i.e. fi — 1/2. This analysis was severely criticized by Fisher [298]. Actually, the spherical model has some unpleasant thermodynamic features, never observed in real fluids. In particular, it is associated with a divergence of the compressibility KTas the coexistence curve (rather than the spinodal line) is approached. By a determination of the exponent y, this possibility could also be ruled out experimentally [95, 97]. 

The size distribution of sixfold-ordered regions, n, also resembles that of the 2D WCA liquid, being well described by the Fisher droplet model. We fit rtj to Eq. (3.26) and obtained a power law exponent of = 1.35 0.02 and a correlation size of = (3 6) x 10 (the large uncertainty in is due to poor statistics in the tail of n ). This value of is similar to the values obtained for the dense time-averaged WCA liquid. The small s part of is shown in Fig. 71, together with the fit to Eq. (3.26). A extended tail (out to 5-600) is observed in the large 5... 

After these caveats, fig. 17 shows qualitatively the dimensionality dependence of the order parameter exponent /5, the response function exponent y, and correlation length exponent v. Although only integer dimensionalities d = 1,2, 3 are of physical interest (lattices with dimensionalities d = 4,5, 6 etc. can be studied by computer simulation, see e.g. Binder, 1981a, 1985), in the renormalization group framework it has turned out useful to continue d from integer values to the real axis, in order to derive expansions for critical exponents in terms of variables = du — d or e1 = d — dg, respectively (Fisher, 1974 Domb and Green, 1976 Amit, 1984). As an example, we quote the results for r) and v (Wilson and Fisher, 1972)... 

The exponent 6 is directly related to v by Fisher s theorem (see Chapter 3, Section 3.3)... 

The chains being rather short, the exponents vN(w) fluctuate strongly with the parity of N. Fisher and Hiley8 remedied this defect by using the values... 

Fig. 4.7. Variation of the mean exponent with the interaction parameter, (a) for the square lattice (d — 2) (b) for the simple cubic lattice (d = 3). The dashed line gives the uncertainty range. (From Fisher and Hiley.8)...

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