Soft matrix and fixed point of Lennard-Jones potentials for different hard-clusters in size at glass transition

Abstract

The existence of fixed point in self-similar Lennard-Jones (L-J) potentials has been proved based on the mosaic geometric structure theory of glass transition (GT) [J. L. Wu, Soft Nanoscience letters, 1, 3, 86 (2011)

The order parameter of glass transition: Spontaneously delocalized nanoscale solitary wave with transverse ripplon-like soft wave

“…The molecule carrying the positive charge is slow motion, its centroid position is almost constant), it seems to be an ordinary vector changing direction, a component which constitutes the L-J potential, but when it makes the synchronous coupling with b 0 and the phase difference between the two is maintained as π , the new characters will appear and constitute an additional and transient geometric phase potential field for L-J potential. 10 In order to introduce the concept of instantaneous spin system of de Gennes n = 0, the IE closed loop state of molecule (or chain-particle) surrounded by its 4 IEs on x-y projection plane has been defined, for example in Fig. 1, as z-direction interface parallel transport state (z-DIPTS), and the molecule (or particle) as z-direction IE molecule (z-DIEM, or z-DIEP), which has been left out of the independent-particle picture of the one-electron theory.…”

“…The most important concept here is the instantaneous maximal 8th order of 2D IE loop-flow in random systems. 9 In the previous paper, 10 the author proved that there were only 8 orders (or referred to as 8 generations) transient direction 2D self-similar molecule-clusters (8 orders of selfsimilar segments) in a thaw domain at GT. The validity of the maximal 8th generation cluster size has been proved theoretically from the following four aspects.…”

“…. , 10 which is the Lindeman criterion 10 destroying solid-versus-liquid lattice structure; 17 (iii) From the size, the number of molecules to rearrange structure: 200; the critical entangled chain length: 200; and the theoretical value of free volume fraction: 0.025, in line with the experimental values at GT, can directly be obtained; 9 (iv) This critical size reveals a finished and instantaneously maximal 2D polarization cluster carried with mosaic positive charges existing in all random systems. The sum of the fast-polarization number density (3/8) and the coupling slow-polarization number density (5/8) is 1, which denotes one reduced and maximal 2D dipole moment cluster in many-electron theory.…”

“…The validity of the maximal 8th generation cluster size has been proved theoretically from the following four aspects. (i) This size neither more nor less satisfies the critical condition of 'delocalization transition' of Van der Waals equation; 10 (ii) This size directly yields the universal Lindeman ratio 0.01047. . .…”

“…This model can be deduced directly based on the 2D soft nanoscale mosaic structure theory of GT. 9,10 In other words, two of the most important puzzles of polymer physics will narrow down to the aspects of a single delocalization mechanism: GT and entanglement chain movement are all attributed to the 2D mosaic geometric structure with inverse cascade/cascade mode (Sec. IV), which is also an instantaneously accompanying spin system with de Gennes n = 0 (Sec.…”