Flory exponent percolation

Let us turn our attention to the scaling of a SAW on a disordered lattice with strong disorder, directly at p = ppc- The Flory formula (2), giving a surprisingly good estimation for the critical exponent i/ of SAWs on pure lattice, has tempted a number of authors to search for generalizations to determine the exponent i/p of SAWs on the percolation cluster [16]. The most direct one is ... 

The exponent Vp for a SAW on a percolation cluster in several dimensions. FL Flory-like theories, EE exact enumerations, US, RG real-space and field-theoretic RG. The first line shows i saw for SAW on the regular lattice (d = 2 [13], d = 3 [11]). 

The field theory for SAWs on the percolation cluster developed in Ref. [22] supports an upper critical dimension d p = 6. The calculation of Up was presented to the first order of perturbation theory, however the numerical estimates obtained from this result are in poor agreement with the numbers observed by other means. In particular, they lead to estimate that i/p i/ in d = 3. Recently this investigation has been extended to the second order in perturbation theory [101], which leads to the qualitative estimates of critical exponents in good agreement with numerical studies and Flory-like theories. 

Figure 9. The correlation exponent i/p. Bold line (133), thin line one-loop result [22], filled boxes Flory result i/p = 3/(tipc+2) with dpc from [109]. Exponents for the shortest and longest SAW on percolation cluster [110] are shown by dotted lines.

Thus, this review explains critical exponents and percolation theories and compares these theories, preferred by physicists, with the classical approach (Flory-Stockmayer type theory used by chemists, and summarizes experimental evidence both in favor and against the theoretical predictions. Since most readers are well acquainted with classical theories we emphasize here recent developments of percolation theories. Due to the rapid development of the situation since 1979, some earlier reviews are partly outdated now. We hope that the same can be said soon about the present article, too. 

Specifically for gelation, we will discuss in Sect. C.V. various modifications of the simple percolation model of Fig. 1 and check if the exponents diange. In most cases, they do not in particular, the lattice structure (simple cubic, bcc, fee, spinels ) is not an important parameter since different lattices of the same dimensionality d give the same exponents within narrow error bars. More importantly, percolation on a continuum without any underlying lattice structure has in two and three dimensions the same exponents, within the error bars, as lattice percolation. In the classical Flory-Stockmayer theory which does not employ any periodic lattice structure, the critical exponents are completely independent of the functionality f of the monomers or the space dimensionality d. But if the system is not isotropic or if the gel point is coupled with the consolute point of the binary mixture solvent-monomers , the exponents may change as discussed in Sect. D. 

By adjusting the parametm-s of the function p = p(T) or p = p(T, 0), which corresponds experimentally to a change in the solvent, an interesting situation described by the central part of Fig. 7 results, where the sol-gel boundary meets the phase separation curve exactly at the critical consolute point. In this case, the Bethe lattice theory which corresponds to the Flory-Stockmayer model, gives classical exponents for random-bond percolation along the whole sol-gel boundary. This is true even for the special case where the critical consolute point and the end point of the gelation line coincide then, one has to use the concentration 0 and not the temperature T as a variable to define critical exponents. 

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