Density Fluctuation within the Phases

1 Density Fluctuation within the Phases In a real two-phase specimen, there are inhomogeneities within the individual phases, due to the atomic nature of the material and to the density fluctuations at all size scales arising from thermal motions of atoms (or frozen-in thermal motions, in the case of glasses). Thus the scattering from such a specimen consists of four components  

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Another effect of the power law behavior of these dense systems is that they deviate positively from Porod s law (/ cx which may be considered as a general rule for this kind of solid. This is induced by density fluctuations within the phases due to their internal microstructure of them. 

The perspective upon which the concept of a superheat limit emerges is based on random molecular density fluctuations within the hulk of a liquid producing hole-like regions of moleculardimensions that act as bubbles. A phase transition occurs when abubble, formed by these molecular processes, grows to a size such that it is in unstable equilibrium (i.e.. a critical size nucleus ) with the surrounding liquid. The "nucleation rate ,/, refers to the mean rate of forming such vapor nuclei in units of nuclei/(volume time). 

Reference has also been made to clean or purified water. In fact, all the numbers shown in Table 1 and Figure 7 are based on the homogeneous nucleation model, according to which the generation of nuclei is solely the result of random density fluctuations within the liquid phase. It is, however, well known that nucleation can be catalysed by internal and external factors, in which case the growth and persistence, possibly also the dimensions and structures, of critical clusters are... 

In the region of a first-order transition ip has equal minima at volumes V and V2, in line with the Maxwell construction. The mixed phase is the preferred state in the volume range between V and V2. It follows that the transition from vapour to liquid does not occur by an unlikely fluctuation in which the system contracts from vapour to liquid at uniform density, as would be required by the maximum in the Van der Waals function. Maxwell construction allows the nudeation of a liquid droplet by local fluctuation within the vapour, and subsequent growth of the liquid phase. 

The mathematical treatment described so far applies to ideal two-phase systems with infinitesimal sharp density steps and perfectly constant densities within the two types of domains. While this situation can be approximated in very good approximation for certain types of samples, for example, porous silica prepared by calcination of a polymer template, in the majority of two-phase systems formed by polymers, the interface between the two domain is not sharp but fuzzy on a certain length scale, see Figure 2 for semicrystalline polymers or Figure 4 for block copolymers, and there are also density fluctuations within one or both of the two domains, deviating from the assumed constant density. Both types of effeas lead to deviations from the ideal asymptotic Porod behavior that may need to be taken into account in an SAXS analysis. 

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