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A basic concept in chain-particle cluster-motion, from frozen glassy state to melt state, is the 2D soft nano-scale mosaic structure formed by 8 orders of 2D interface excitation (IE) loop-flows, from small to large in inverse cascade and rearrangement structure in cascade along local one direction. IE has additional repulsive energy and extra vacancy volume. IE results from that the instantaneous synchronal polarized electron charge coupling pair is able to parallel transport on the interface between two neighboring chain-particles with antiparallel delocalization. This structure accords with de Gennes’ mosaic structure picture, from which we can directly deduce glass transition temperature, melt temperature, free volume fraction, critical entangled chain length, and activation energy to break solid lattice. This is also the inherency maximum order-potential structure in random systems.

This letter is an introduction of the motion principle in polymer physics theory. The confluence of both the thermodynamic and the kinetic dimensions of the solid ↔ liquid glass transition (GT) presents one of the most formidable problems in condensed matter physics [^{7} times of that in general quencher from melt state to frozen glass state. A logical explanation is that the macromolecules can complete liquid-to-solid GT with full orientation within the millisecond of time in z-space. Their motion mode, within the entire range from melt transition temperature to GT temperature, allows the direction of inverse cascade-cascade in every “excited domain” is in arbitrary in melt state and all apt to z-axial on melt spinning-line; and finally the mode is frozen in glasses as the soft matrix.

On the other hand, in physical theory, many complicated phenomena originate from the global properties of parallel transport of simple quantum systems (Berry’s Phase) [

Van der Waals interaction includes the contribution of instantaneous induced dipole—induced dipole. Generally, instantaneous polarized dipole electron charges randomly distribute on an interface 1 - 2 forming electron cloud (blue zone) on x-y projection plane in

At GT, an interesting and unexplored corner of Van der Waals interaction theories is that the instantaneous synchronal polarized electron charge coupling pair (two small blue dots) may parallel transport on an interface, e.g., the interface 1-2 between particles and in

interface 2 - 3 between and in

Thus, the site-phase difference between two z-component molecules, or chain-particles, is p: the state of all polarized electron charges in each instantaneous dipole in the two z-component chain-particles, in

The IE loop-flow in

The formation of a central excited particle can be also regarded as the cooperative contribution from its 4 neighboring particle fields, , , , and based on Brownian motion theory

At the instant (local) time in field, once 4th loop-flow is finished, a new loop-flow of a_{0} surrounded by the 12 IEs, 1 ® 5 ® 6 ® 4 ® 7 ® 8 ® 3 ® 9 ® 10 ® 2 ® 11 ® 12 ® 1, with timescale will occur as in _{0} field, which are all denoted as. Whose cycle direction is negative, contrary to that of. The energy of IE with on the loop denoted as. The 4 excited particles are 4 concomitant mosaic cells with. Loop has 12 IEs and is of the loop potential energy of 12. This means that the evolution energy from to is 8. The number of cooperative excitation particles in is 5.

Similarly, the 2nd order of transient 2D IE loop-flow and 2D cluster with, denoted as, is formed at the instant local time when its 4 neighboring particle fields:, , , and finished in field. Whose cycle direction is positive, contrary to that of, in

The energy of IE on loop is denoted as . The number of IEs of loop is 20. The number of IEs inside loop is 12, and equals to the number of the IEs of. This means that the evolution energy from to is 8 . The number of cooperative excitation particles in is 13 surrounded by 20 IEs. Thus, 20 IEs can excite 13 particles to hop randomly along local +z-axial direction.

In the same way, the third order of 2D IE loop-flow and cluster at the instant time in a_{0} field, , can be obtained as in

and deasil interacted with in field. In _{0} field. The number of IEs of loop is 28. The number of IEs inside loop is 20, which equals to the number of the IEs on loop. This means the evolution energy from to is 8. The number of cooperative excitation particles in third order cluster is 25.

Thus, 28 IEs can excite 25 particles to hop randomly along local z-axial direction.

From the results of, and, it is clear that the more the number of cooperatively delocalization particles is, the less the average needful IE energy for each particle is. The key point is what the minimum excited energy is and what the number of cooperatively delocalization particles excited by the energy at the GT is.

There are 4 neighboring concomitant excitation centers surrounding the referenced particle center. Thus, the excited field in

The forming of i-th order cluster always results from the cooperative contributions of its 4 neighboring (i-1)-th order of clusters around the central particle. This property origins from Brownian motion and is called as the priority of central particle, which is probably the primary reason to generate dynamic heterogeneity [

a) The number of IEs of i-th order of 2D cluster can be calculated as = 4 (2i + 1) = 12, 20, 28, 36, 44, 52, 60, and (68) for i = 1, 2… 8 and = 4. Note that will be corrected as 60 by geometric frustration-percolation transition.

b) The number of excitation particles in i-th order of cluster is respectively: = + 4i = 5, 13, 25, 41, 61, 85, 113, and (145) for i = 1, 2… 8 and = 1. Note that also will be corrected as 136 by geometric frustration-percolation transition.

c) For z-axial i-th order 2D cluster, the evolution energy from to is 8, we emphasize, this evolution is orienting. The energy to excite i-th order cluster orientable evolution thus is the energy of one external degree of freedom (DoF) to i-th order of cluster (loop-flow) orienting, denoted as. So we directly deduce from a)

d) During the time of (,) in the referenced field, each of its 4 neighboring (, , , and) local coordinates can take any direction (fragmentized and atactic lattices) because of fluctuation. However, from c), at the instant time, the direction of i-th order cluster always starts sticking to the direction of the referenced first order cluster. This characteristic in random systems comes from the Brownian regression motion, or the characteristic of Graph of Brownian Motion. This means that each neighboring i-th order of cluster also has one external DoF of energy as. So the i-th order

of cluster in the (i + 1)-th order is of 5 inner DoF and taking any directions during the time of (,) in field.

In the same way, the 8th order of transient 2D cluster, , can be obtained in field, as in

The uncorrected number of IEs of is 68, denoted by blue-black arrows in _{8} in are also the mosaic cells of in the 4 neighboring -clusters (Note: here the 4 -clusters are respectively in the 4 neighboring 5-particle cooperative excited fields) with field. And 4 cells with in the 4 neighboring -clusters, by the color of blue green, are also the mosaic cells in. (A region of space that can be identified by a single mean field solution is called a mosaic cell [

Note that there are 4 inverted thick-black arrows in the corrected loop-flow in

In

Here is similar to the energy of Curie temperature in magnetism, also the energy of a ‘critical temperature’ existing in the GT presumed by Gibbs based on thermodynamics years ago [_{mig} along one direction, e.g., along the direction of external stress at GT, is an intrinsic attractive potential energy, independent of temperature and external stress and response time. This will be one of the key concepts directly prove the WLF equation. It stands to reason that the attractive potential comes of the IE loop-flows in Brownian motion existing in any random system at any temperature.

The 8 orders of loop-flows in _{c} of dipole charge electron coupling pairs at the GT is the fractal dimension d_{f} of Graph of Brownian Motion [

We introduce the concept of i-th order of directional 3D hard-sphere (hard-cluster) for directional i-th order of 2D cluster in order to reflect the number of particles in an orienting and compacted cluster in density fluctuation when the maximum loop- occurs.

a) Compacting cluster and density fluctuation. The first order of hard-sphere contains 5 (the number of particles in) + 12 (the number of interfaces of, each interface corresponds to an excited particle taking in any direction in 3D local space) = 17 chain-particles that are compacted to form a hard-sphere. Which has an internal density larger than average, because the IEs with extra volumes on inside have been compacted and transferred to that on (static electricity screen effect of IE loop-flow).

b) Hard-sphere with finite acting facets. Note that is a vector. The direction of is negative, as same as the direction of, if the direction of is positive. Finite acting facets surround this orienting 3D hard-sphere. The complexity of dynamical hard-sphere here comes down to the finite acting facets in mosaic structures.

c) 8 orders of self-similar z-axial 3D hard-spheres. In the same way, the second order of hard-sphere contains 13 (the number of particles in) + 20 (the number of interfaces of, the number of side-excited particles) = 33 excited particles. The direction of is positive. Therefore, the number, , of particles in i-th order of hard-sphere can be obtained as

In (4), the sign denotes the moving direction along z-axial of i-th order hard-sphere. By i-th order of hardsphere is meant that the interaction-interface energy of its two-body is the IE energy or transferred energy, , independent of the distance of two excited cents of two-body in z-space.

d) 8 orders of self-similar chain-segments. In

e) Number of structure rearrangements. The number of particles in the 8th order of hard-sphere, , or the 8th order of chain-segment sizes, l8, also is the number of particles in structure rearrangement in and, can be easily found out from

The numerical value is consistent with the conjectural results of encompassing rearrangements [

f) Evolution direction of cluster growth transition. From = 200, » 5.8 (chain-particle units), which is the size of ‘cage’ at GT. According to the definitions of 2D clusters and 3D hard-sphere, an interaction of two z-space i-th order of 3D hard-spheres is always equal to the IE interaction with. This means that the cluster in [

The steady excited energy is exactly the flow-percolation energy in percolation on a continuum of classical model in condensed matter physics. The energy of steadily ‘excited state energy flow’ in the process of vanishing and reoccurring is defined as the localization-delocalizetion (percolation) transition energy, named as (the same denotation of as Zallen [

a) Macroscopic melt temperature. It is a very important consequence that the external DoF of an energy flow is 5 in the melting state phase transition; and the renewed energy of a microscopic i-th order cluster (i < 8), , being in renewed cluster within (i + 1)- th order cluster zone, is numerically equal to the energy of the macroscopic melting state of. Namely, , i < 8, corresponds to 5 inner DoF, in which, (10); denotes the energy of one external DoF of i-th order loop-flow in field. The energy of macroscopic melting state can be denoted as,

b) Percolation transition energy. The step (interface) number of -loop is = 60., is less than 60 because of the regression state energy effect of IE and the dynamical mosaic structures of IE loop-flows. The IE energies on -loop in

The key concepts are that a few of interfaces, named as L_{inver}, on the -loop will be excited by the interfaces with relaxation time of in new local fields after. This is the effect of mosaic structures, or, L_{inver} is the step number of mosaic in slow inverse cascade. The others, named as L_{cas}, on the -loop will one by one vanish and their excited energy will rebuild many new -interfaces in the new local fields in fast cascade.

Thus, in the flow-percolation on a continuum, the 60 interfaces of a reference -loop in _{cas} interfaces that occur at the local time of in field and the L_{inver} interfaces that are the mosaic structures of energy flow and occur at a time after. Formula (7) is obtained

The energy of is also the fast-process cascade energy of rebuilding new loop-flows when the L_{cas} interfaces vanish. The energy of L_{inver} is the slow-process inverse cascade energy of all loop-flows from to in new local fields.

The balance excited energy of inverse cascade and cascade in flow-percolation (contained many local fields) is exactly the localized energy. Therefore, in fastprocess cascade, , in which is used as timescale of vanishing, here is the cascade potential energy in flow-percolation.

Since the excited energy in inverse cascade is not dissipated, the evolution energy of each order closed loop is, (1), namely, the ‘singular point energy’ of any i-th order IE closed cycle (Gauss theorem).

Each mosaic step (either the inverted arrow, or the shared interface by –) in _{0} local field. Thus, in the slow-process inverse cascade from to in flow-percolation, the number of the inverse cascade energy forming mosaic step is. For flexible polymer,. Therefore, formula (8) is obtained

Or (9)

Equation 8 is a representative mean field formula. It can be seen that the physics meaning of the right term containing left term on equation is the contribution of mosaic structure. Equation 9 denotes that the delocalized energy (the localized energy, the percolation transfer energy, the transfer energy from inverse cascade to cascade) at the GT has 8 components, , i = 1, 2…8 and the numerical value of each component energy is the same, namely, 20/3. This is also one of the singularities at GT. is a characteristic invariable with 8 order of relaxation times at the GT, independent of GT temperature.

c) Glass transition temperature. We see also comes from the directional non-integrable additional energy at GT, instead of general thermo-random motion energy of molecules. Similar to (2), z-axial non-integrable random regression vibration energy, , of i-th order clusters with can be used to denote the energy of, and it can be also balanced by random thermal vibration (-scale) energy of for general unexcited particles. That is

The numerical value of that can be called the fixed point of clusters from small (,scale) to large (,scale) in inverse cascade at GT, it is independent of GT temperature. is traditionally accepted as GT temperature, which is obtained by slow heating rate. Therefore, is a measurable by experiments. From (2), numerical relationship is:

d) Geometric frustration and high-density percolation. It can be seen that Equation 9 is based on the result of corrected value and this correction from to in

The 8 evolution IE states together with the 4 IEs on a reference particle a_{0} will evolve a new first order of IE loop-flow being of 12 IEs when 8th order cluster appears.

The energy summation of the 320 different IE states is named the cooperative orientation activation energy to break solid lattice, denoted as. The reason to call it activation energy is that although in microscopical, the energy to break solid lattice is seemly only, <, in microcosmic, the energy of 320 IE space-time states in every solid domain is needed. Thus,

Using , thus,(15)

can be measured on melt high-speed spinningline. In fact, the author [

Moreover,.

is the energy of one DoF of -loop. This value is consistent with the experimental results (4 ~ 12 x 10^{-3} eV) for Boson peak measured by high-resolution inelastic neutron scattering [

The classical free volume ideas have been questioned because of the misfits with pressure effects, but here the pressure effects should be indeed minor [

In the inverse cascade at GT, if each IE appears in the way of one arrow after another according to the appearance probability as mentioned in Figures 2-5, the probability is rather low. In fact, the more probable situation is the evolutive mode of self-similar 2body-3body cluster coupling in the soft matrix of IE loop-flows in

In

The mode of 2body-3body cluster coupling is in statistically advantage (soft matter concept): e.g., per 4 IEs needs 5 instantaneous polarized ions in Figures 1(d) and 6(a). The parameter of polarized ion numberdensity of per IE (the density of particles occupy additional vacancy volume in IEs) in soft matrix, = 5/4 = 1.25. However, 21 IEs needs 24 polarized ions, the ion number-density = 24/21 » 1.14 in

The minimum value of the polarized ion number density = 200/320 = 5/8, in 8th order 2D cluster in

Moreover, there is still 16 (= 136 - 120) + z-axial remainder ions inside the 8-th order of loop-flow, which can offer a static charge repulsive force to facilitate the first excited + z-axial ion a_{0} delocalization in _{0} to leave its coordinates needs 320 IEs, 16/320 = 1/20, this leads to each directional delocalization uncompacted particle needs 20 z-directional excited interfaces with in its 8-th order of loop-flow. On the other hand, disappearing the 8-th order of loop-flow of reference a_{0 }particle also needs its 4 concomitant excited particle, , , , and, one by one finish respective 8th order of loopflows in inverse cascade and delocalize along z-axial. This leads to 5 compacted conformational rearrangement particles in cascade precisely need 20 z-directional excited interfaces disappearing with. That is another explanation for icosahedral directional ordering at the GT.

Distinctly, the mode-coupling scheme based on mosaic geometric structure adopted in this paper is different from the mathematical expressions in the current modecoupling theory of GT. However, the spirit in both modecoupling schemes is the same that deals with the coupling of fast-slow relaxation modes and two density modes in structure rearrangements at the GT. In our scheme, we focus on finding out the three direction nonintegrable energies, and existing in the coordinate invalidation from i-th order clusters to (i + 1)-th order in solid-to-liquid transition, whatever the time complications of anharmonic frequencies may be. The mode-coupling trick is that the relaxation time complications have been beforehand reduced and replaced by the long time Brownian directional regressions with the 2p closed loop-flows of IEs in

It is proposed that the soft matrix surrounding clusters is 2D soft nano-scale 8 orders of IE loop-flows; the filling factor of 2D lattices is 1. The key of the theory is that the random thermo-motion kinetic energy, from glass transition to melt transition, should be balanced by the nonintegrable random regression vibration energy in random systems. That is, entire molecule-cluster delocalization energy (also the GT energy) origins form the maximum order potential energy in random systems. This 2D structure corresponds to the appearance of boson peak and geometric frustration at the GT, and can directly deduce three non-integrable directional energies:, , and along one local space direction, respectively corresponding to Gibbs temperature, glass transition temperature, and melt transition temperature.

The author is grateful to all colleagues he had the pleasure to collaborate and interact, especially when he found the fundamental physics origin for the orientation activation energy obtained experimentally on melt high-speed spinning-line. In particular, the author would like to thank, in random order, Yuan Tseh Lee and Sheng Hsien Lin of Academia Sinica (Taiwan), Yun Huang of Beijing University, Da-Cheng Wu of Sichuan University for useful discussions.