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

Image of FIG. 1.

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FIG. 1.

There are M self-similar L-J potential well energies along one same direction at GT.

Image of FIG. 2.

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FIG. 2.

At the equilibrant point of attraction and repulsion, f c , f c ∈(−1, 0), there are numerous f c points and self-similar L-J potential curves satisfying f c = f (σ i /q i, R ) = f (σ i +1 /q i +1, L ). They have no physical relevance except for curves via the fixed point, f c *, of L-J potentials at GT.

Image of FIG. 3.

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FIG. 3.

Self-similar clusters of two different sizes satisfy f c = f (σ i /q i, R ) = f (σ i +1 /q i +1, L ) along q-axis, the growing direction of thaw clusters. The two σ i respectively in two σ i+ 1 clusters are in ‘balance state’ of attraction and repulsion during the process of σ i augments to σ i+ 1.

Image of FIG. 4.

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FIG. 4.

At the fixed point of GT, f c *, there are 8 sharp-angled points, q i,R , forming the delocalization path of cluster σ 1(a 0) along 8 orders of geodesic.34 The i-th order of geodesic is the shortest line of 2π cycle between q i,R and q i +1,R on the i-th order of potential of cylindrical surface, which is formed by f c * surround axis f = −1 (also the z-axis in the figure) +2π cycle (i as even number) or –2π cycle (i as odd number) of Δɛ (τ i ) on x-y projection plane perpendicular to q-axial once, if σ 0 is promised in +z-axial.

Image of FIG. 5.

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FIG. 5.

During GT, the path of compacting cluster starts from a disturbance Δχ on point (3/8, –15/16), via fluctuation-correlation losing one reduced IE energy, to the minimum point 0’ along slope –1, then via fluctuation-correlation gaining one reduced IE energy, to the maximum point (5/8, –15/16) along slope +1, in which one reduced IE energy, 1/8, transfers from the surface on i-th order cluster to that on (i+1)-th order.

Image of FIG. 6.

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FIG. 6.

The instantaneous 2D mosaic geometric potential picture of transition from inverse cascade to cascade occurs in x-y projection plane on the 8th order of sharp-angled point at GT in Fig. 4.

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/content/aip/journal/adva/2/2/10.1063/1.4704662
2012-04-11
2014-04-17

Abstract

The existence of fixed point in self-similar Lennard-Jones (L-J) potentials has been proved based on the mosaic geometricstructuretheory of glass transition (GT) [J. L. Wu, Soft Nanoscience letters, 1, 3–86 (2011)]. A geometric local-global mode-coupling recursive equation, different from the current Mode-Coupling Theories, has been introduced to find out the non-integrable induced potential structure of boson peak at GT. The recursively defined variable in reduced recursive equation is the potential fluctuation of reduced L-J potentials associated with reduced geometric phase potentials. A series of results have been deduced directly at GT. (i) There are only 8 orders of molecule-clusters. (ii) Two orthogonally fast-slow reduced phase potentials, 3/8 and 5/8, are accompanied with density fluctuation and clusters hop-delocalization along 8 geodesics. (iii) The stability condition of potential fluctuation is the Lindemann ratio. (iv) A new reduced attractive potential of –17/16, lower than reduced potential well energy –1, occurs.

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Scitation: Soft matrix and fixed point of Lennard-Jones potentials for different hard-clusters in size at glass transition
http://aip.metastore.ingenta.com/content/aip/journal/adva/2/2/10.1063/1.4704662
10.1063/1.4704662
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