Critical points fluctuations

The form of Eq. (3) illustrates an intimate connection between the problem of self-maintained interfaces and that of critical point fluctuations, for which... 

An exception to this conclusion is found in the innnediate vicinity of critical points, where fluctuations become much more significant, although—with present experimental precision—still not of the order of N.)... 

If the finite size of the system is ignored (after all, A is probably 10 or greater), the compressibility is essentially infinite at the critical point, and then so are the fluctuations. In reality, however, the compressibility diverges more sharply than classical theory allows (the exponent y is significantly greater dian 1), and thus so do the fluctuations. 

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

Now, assume that we are getting closer to the critical point of our transition, i.e., to the point of the second-order transition. In the case of a uniform system the critical region can be described by the divergent correlation length of statistical fluctuations [138]... 

At higher temperatures, other degrees of freedom than the radius R must also be considered in the fluctuation. However, this becomes critical only near the critical point where the system goes through a phase transition of second order. The nucleation arrangement described here is for heterogeneous or two-dimensional nucleation on a flat surface. In the bulk, there is also the formation of a three-dimensional nucleation, but its rate is smaller ... 

As can be seen from Fig. 6, liquid-liquid demixing clearly precedes crystallization in case Cl. Moreover, crystallization in this case occurs at a higher temperature than in cases C2 and C3. Apparently, the crystallization takes place in the dense disordered phase (which has a higher melting temperature than the more dilute solution Fig. 5). In case C2, the crystallization temperature is close to the expected critical point of liquid-liquid demixing, but higher than in case C3. This suggests that even pre-critical density fluctuations enhance the rate of crystal nucleation. 

A similar phenomenon occurs at the liquid-vapour critical point. The system contains relatively large regions of liquid-like and vapour-like density, which fluctuate continuously3. 

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