Mean field critical region

In both eqs. (199) and (200) all prefactors of order unity are omitted. In a system with a large but finite range R of interaction, the nucleation barrier is very high in the mean-field critical region, in which f J(1 —Tj Tz) d 2 1... 

Fig. 38. Schematic plots of the free energy barrier for (a) the mean field critical region, i.e. / 1, and (b) the non-mean field critical region, i.e. R,l( 1 -/ /7 ) 4— V2 i. When AF /Tc is of order unity, a gradual transition from nucleation to spinodal decomposition (in a phase-separating mixture) or spinodal ordering" (in a system undergoing an order-disorder transition with non-conserved order parameter distinct from tj>) occurs. From Binder (1984b).

In addition, the temperatures To, T involved in the quench must lie in the mean-field critical region, which for general d is given by the condition... 

Fig. 11. Schematic plots of the free energy barriers for a d-dimensional system with a large range r of interaction - or a polymer mixture, respectively 1 (1 — T/Tc)2 d/2 then has to be replaced by adNd/2-i(i T/Tc)2-d/2. A) refers to the mean-field critical region, where rd(l - T/Tc) 2 d/2 > 1, B) to the non-mean-field region. In this figure, kg = l, so the gradual transition from nucleation to spinodal decomposition occurs for AF /TC 1. At AF /Tc rd (1 — T/Tc) 2-d,2,l a crossover occurs from classical nucleation (Le., compact spherical droplets) to spinodal nucleation (i.e., diffuse ramified droplets). From Binder [79]...

Now there is an additional complication in the mean-field critical region of polymers there is no finite size scaling of the form postulated in eqs (7.20)-(7.22) L does not scale with but rather with a thermodynamic length e defined by " " " ... 

As a consequence of eqs (7.25) and (7.28), in the mean-field critical region of a polymer mixture eq. (7.20) does not hold, and should rather be replaced... 

While in the mean-field critical region we hence expect for the maximum value of the scattering function = 0) = n m ) - m ) oc tr /N)... 

Fig. 2.44 Phase diagram for a conformationally-symmetric diblock copolymer, calculated using self-consistent mean field theory. Regions of stability of disordered, lamellar, gyroid, hexagonal, BCC and close-packed spherical (CPS), phases are indicated (Matsen and Schick 1994 ). All phase transitions are first order, except for the critical point which is marked by a dot.

The calculated mean field phase diagrams in the coordinates surface defects concentration - dielectric permittivity 82 are reported in the Fig. 4.16 for the same parameters as those in the Fig. 4.15. It is seen that for high defects concentration the exchange integral is larger than the corresponding mean field (the region Tc > T in the Fig. 4.16), the equilibrium curve shifts to the lower defects concentrations with permittivity increase, while effective mass increase leads to the increase of critical (threshold) concentration. 

Figure 13.13 Mean-field phase diagram tor confoimationally symmetric diblock copolymer melts. Phases are labeled L, lamellar H, hexagonal cylinders Qia3d, bicontinuous Ia3d cubic Qimsm, bcc spheres CPS, close-packed spheres and DIS, disordered region. The dashed lines denote extrapolated phase boundaries, and the dot denotes the mean-field critical ODT.

A2.5.7.2 CROSSOVER FROM MEAN-FIELD TO THE CRITICAL REGION... 

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

The transition from a stable steady-state solution observed at large p to the oscillatory regime assumes the existence of the critical value of the parameter pc, which defines the point of the kinetic phase transition as p > pc, the fluctuations of the order parameter are suppressed and the standard chemical kinetics (the mean-field theory) could be safely used. However, if p < pc, these fluctuations are very large and begin to dominate the process. Strictly speaking, the region p pc at p > pc is also fluctuation-controlled one since here the fluctuations of the order parameter are abnormally high. 

The numerical solution of the system (4.86), by a procedure of the Newton-Raphson type with two variables, requires the calculation of the derivatives dtjds and dt /5Wc . The results we obtained for a square lattice are similar to those by Yonezawa and Odagaki180 for a cubic lattice. The most striking feature is the existence, at low concentration, of a gap in the density of states,179 which isolates the zero energy on which a 3 peak builds up. Thus the HCPA produces a forbidden region of energy for the transport the gap and the 3 peak disappear at a critical concentration, analogous to the percolation threshold of the mean-field of resistances. 

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