Hyperscaling relations

Muthukumar [44] further investigated the effects of polydispersity, which are important for crosslinking systems. He used a hyperscaling relation from percolation theory to obtain his results. If the excluded volume is not screened, n is related to df by... 

This condition leads to the hyperscaling relation between critical exponents ... 

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) ... 

Again one concludes that the scaling relations eqs. (80), (90) and (91) are satisfied, while the hyperscaling relation [eq. (93)] would only be satisfied at d — 8. Indeed, using the Lifshitz exponents in the Ginzburg criterion [eqs. (52)—(55)] one does find that the Landau description of Lifshitz points becomes self-consistent only for d > 8. Thus it is no surprise that the behavior at physical dimensionalities (d = 2, 3) is very different from the above predictions. In fact, in d = 2 one does not have Lifshitz points at non-zero temperature (Selke, 1992). 

We note from Table 5.5.1 that v is anisotropic for all the materials, v, — Vj being about 0.13. In no case is v, = 2vj. Also, all the materials satisfy the anisotropic hyperscaling relation (5.5.28) to within experimental limits. For two compounds, 407 and 8S5, a agrees with the predicted value of — 0.007, but the vs do not. Brisbin et al/ have suggested that the A-N transition for some of the compounds, e.g., 10S5 and 9CB, is near a tricritical point as the values of high for these compounds. This... 

However, since f AT , where AT = T — Td/T, we can express AT as For a sharp transition the local fluctuations in T, is required to be much smaller than this critical temperature interval AT, and consequently we need AT, AT, or d/2 > or (2 — dv) < 0. Using the hyperscaling relation a = 2 — dv, the above condition becomes a < 0 indicating nontrivial effect (a new different sharp transition with negative value of a, or a smeared transition) due to quenched disorder for (pure) systems with a > 0 [8]. 

This hyperscaling relation is valid for space dimensions... 

Exponent relations involving the dimensionality d explicitly are called hyperscaling relations. They only hold below the upper critical dimension Above they are destroyed by dangerously irrelevant variables. 

The hyperscaling relation which links exponents to dimensionality of the space, means that density of the gel phase G is equal to that of the largest finite polymer cluster... 

The clusters are no longer trees, but of a more complicated self-similar structure with many loops. The hyperscaling relation, as discussed further below, means that clusters of any chosen size are roughly at the overlap concentration with one another. 

Note that the hyperscaling relation dv = y +ip (in d = 3 dimensions) [34] as well as the Widom [38] scaling relation for the interfacial tension exponent ii = d — )v hold here, but are not obeyed for the mean-field exponents [34]. 

Such hyperscaling relations are also known from other phase transitions a short introduction to scaling in the case of thermal phase transitions is given in the appendix of Ref. 7. In contrast to scaling relations (Eq. (8)), the hyperscaling relation (9) involving the dimensionality d cannot be used in Flory-Stockmayer theories and similar approaches. 

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