Crossover exponents

The crossover exponent for the disorder is defined through the homogeneity of the singular part of the free energy in terms of the scaling fields /ui (temperature) and (j,2 (disorder). Under decimation, the free energy behaves... 

What makes this problem so difficult—and indeed most copolymer problems difficult— is the necessity to take the average over all possible 2" distributions of monomers on the polymer. Nevertheless, some exact enumeration studies of this problem have been conducted. Martin [16] estimated the location of the transition using exact enumeration methods. An open question is the value of the crossover exponent, (j>, which describes the shape of the free-energy near the adsorption critical temperature. The conclusion from the above study, and others, is that the difference between this exponent and its homopolymer counterpart, if it exists, is too small to be detectable by any current numerical studies. 

If we start at any point on the boundary between the DS and AS phases, under renormalization, we flow to the symmetrical fixed point S = C = A, D = E, with A, D having the value corresponding to the point 5 in Fig. 14. Linearizing the recursion equations near this fixed point two eigenvalues A = 2.7965 (corresponding to the sw ollen bulk state) and A = 2.1583 greater than one [58] are found. This point is the expected sj mmetrical special absorption point which describes the polymer at the desorption transition.. A simple calculation gives the crossover exponent

surface with the surface t = constant. 

A casual look at Fig. 18(b) may give the impression of the existence of a re-entrant adsorbed phase as u is increased. One should, however, realize that these figures are merely a projection on the w-u plane of three dimensional figures in which the third dimension is given by t. The value of crossover exponent (j> for the 4-simplex lattice is 0.7481 equal to the value found for HB 2,3). 

A, B same as for the DS fixed point. The linearization of recursion relations about this fixed point gives two eigenvalues Ai = 3.1319 and A2 = 2.5858 greater than one. The line 0Jc u, t) is therefore a tricritical line. The crossover exponent (j> = 0.8321. 

The crossover exponent, polymer conformation in the vicinity of the adsorption threshold, kBTa/[wan- Defining a scaled distance, Ts 1 - TJT, from the adsorption transition, one finds for the number of contacts with the substrate a scaling law of the form ... 

Figure 3 Scaling plot of the surface excess Fs versus the scaling variable,-r/V = ( waii/ a-1)/V , using the values >=0.5,ea=0.00938 for the crossover exponent and the adsorption threshold, respectively. Five different chain lengths, N, are considered as indicated in the key. Reprinted with permission from Metzger, S. Muller, M. Binder, K. etal. J Chem. Phys. 2003, 118, 8489-8499. Copyright 2003, American Institute of Physics.

The critical behaviours of higher order satellites near a continuous phase transition are of particular interest, as they provide the opportunity to study the crossover exponents of the respective symmetry breaking perturbations in the spin Hamiltonian. Each order of satellite has an associated order parameter critical exponent given by / = 2 - a -where a is the specific heat exponent, and uniaxial symmetry. For this model, theory predicts that = o P, with... 

Thus, 0Q < 0 < 0j in both 2-d and 3-d. Notice that the relation for the SCF given in Eq. (12) depends on loop probabilities and the corresponding exponents for all loop sizes between and N, As the loop size increases from to Nj the exponent must crossover from 0 to 0q. In this sense 0 plays the role of a crossover exponent with a value intermediate between... 

This simple approach yields the crossover exponent,... 

The above developments had two major consequences. First, the identification of the various hexatic phases led to a detailed classification of the large variety of thermotropic liquid crystal phases. The essential structural details of these phases are illustrated in the Appendix8>9. Second, the diffraction patterns yield novel crossover critical phenomena. In three dimensions, this presented single experiments which yield a simultaneous measurement of many anisotropy crossover exponents, previously obtained from separate complex experiments. For very thin layers, the patterns allow detailed studies of the crossover from two to three dimensions, and provide new quantitative information on the two dimensional hexatic phase. 2 The present paper aims at a brief review of these developments. 

The contact probability for a swollen chain with the surface,/j, can be calculated as follows [33]. In order to force the chain of polymerization index TV to be in contact with the wall, one of the chain ends is pinned to die substrate. The number of monomers which are in contact with the surface can be calculated using field-theoretical methods and is given by V , where 0 is called die surface crossover exponent [6,31]. The fraction of bound monomers follows to be/j N, and dius goes to zero as the polymer length increases, for < 1. Now instead of speaking of... 

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]. 

One sees that the data for xi span the full range from the mean-field result (xi = 1 /4) to the scaling prediction (cf. Eq. (33)), while the data for X3 are compatible with the mean-field result. However, the experimental results for X2 seem to indicate a clear discrepancy with the mean-field result X2 = 1/2 (cf. Eq. (35)). Enders et al. then also allowed for a non-mean-field crossover exponent, writing... 

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