Vol.3, No.7, 580-593 (2011) Natural Science
Copyright © 2011 SciRes. OPEN ACCESS
The common physical origin of the glass transition,
macromolecular entanglement and turbulence
Jia-Lin Wu
College of Material Science and Engineering, Donghua University, Shanghai, China; *Corresponding Author: jlwu@dhu.edu.cn
Received 27 March 2011; revised 27 April 2011; accepted 3 May 2011.
The interface excitation (IE) on intermolecular
interface is a common concept connecting the
glass transition (GT), macromolecular entan-
glement (ME), and turbulence. IE has an addi-
tional repulsion energy and extra vacancy vol-
ume that result from the two neighboring mo-
lecules with antiparallel delocalization all in, e.g.,
the z-axial ground state of single-molecule in-
stantaneous polarized dipole at GT. IEs only
occur in the 8 orders of 2D IE loop-flows on lo-
cal x-y projection plane. Theoretical proof of the
3.4 power law of ME viscosity reveals that 1) the
delocalization mode of GT and solid-liquid tran-
sition is solitary wave; wave-particle duality of
solitary wave is ascribed to the equal probabili-
ties between appearing and disappearing of IE
loop-flow in inverse cascade and cascade mode;
2) macromolecular chain-length in ME motion
corresponds to Reynolds number in hydrody-
namics; both the ME motion and the turbulent
flow obey the same scale law. IE is not the ex-
citation of dipole energy level at GT. However,
when IEs are associated with the energy levels
of instantaneous polarized dipole, we predict
that the coherent structure formed by multilevel
8 orders of 2D IE loop-flows is the physical ori-
gin of turbulence based on the universal ran-
dom delocalization transition theory.
Keywords: Glass Transition; Mosaic Stricture;
Reptation; Coherent Structure; Random Transition
Turbulence is well-deservedly often called as “the last
great unsolved problem of the classical physics.” Despite
enormous advances in sciences and many highly impor-
tant new scientific domains emerged in the past century,
the modern status of the turbulence theory is quite ex-
ceptional and differs from that of all other new sciences
[1]. On the other hand, Nobel laureate Philip Anderson
wrote, “The deepest and most interesting unsolved pro-
blem in solid state is probably the nature of glass and
glass transition (GT)” [2]. In physical theory, many
complicated phenomena originate from the global prop-
erties (Berry’s Phase) of simple quantum systems [3] and
the emergent properties of the many-times repeated ap-
plication of quite simple physical laws [4]. It is discov-
ered in this paper that the physical origin of turbulence is
the same as that of GT and macromolecular entangle-
ment (ME) when solving GT and ME in polymer phys-
ics. The three famous problems all attribute to the in-
corporation of many physical ingredients, containing the
maximal Berry’s phase of interface excitation (IE)
loop-flows and the slowest also the maximum Brownian
regression-order potential in random systems. The ap-
pearance of the 8th order IE loop-flow plays the domi-
nant role in inducing molecule-cluster localization - de-
localization transition and flow speedup along one-di-
mensional direction in a reference 3D domain.
The minimum energy mode of the structure (confor-
mational) rearrangements in supercooled liquid could be
deduced from the motion mode of ME in melt liquid
state. This means that we can directly use the GT theory
(also proof-test the GT theory) to prove the 3.4 power
law of viscosity in ME [5,6], which is a well - known
experiment law found more than 60 years ago and yet
failing to have theoretical proof ever since. This power
law is theoretically proved in this paper. The real sur-
prise comes when we deduce the scale law of ME mo-
tion from the power law and find that the macromolecu-
lar chain-length corresponds to the Reynolds number in
hydrodynamics; both the ME motion and the turbulent
flow obey the same scale law. These findings indicate
that both ME and turbulence relate to the same theory
and the solid-to-liquid GT is the paradigm from disorder
to more disorder transition in inverse cascade – cascade
Very recently, Aleiner et al. [7] show theoretically that
the one-dimensional gas of short-range interacting ato-
mic bosons in the presence of disorder can undergo a
J.-L. Wu / Natural Science 3 (2011) 580-593
Copyright © 2011 SciRes. OPEN ACCESS
finite-temperature phase transition between two distinct
states: fluid and insulator, and the Anderson localization
- delocalization transition can be identified in the dy-
namics of expansion of disordered bosonic clouds re-
leased from the superimposed trapping potential. This
study may also provide an existent evidence for univer-
sal random delocalization transitions presented in this
paper, if we regard the energy of finite-temperature as
the Brownian directional regression vibration energy
along one-dimensional direction in a 3D local space and
the transition between two distinct states as the inverse
cascade—cascade mode of localization—delocalization
transition in random systems.
This paper is organized as follows. Section 2 reca-
pitulates the mosaic structure theory of GT, including the
fixed point of self-similar two-body interaction, frustra-
tion-percolation transition; and further proves the inverse
cascade – cascade mode is the fundamental mode of GT
via a validity check of GT theory for the well-known
abnormal exponential function in glass state. In Section
3, we unveil the structure of ME by means of the theo-
retical proof for the 3.4 power law of viscosity; and
show the theoretical contrast between random delocali-
zation transition and phase transition. In Section 4, we
present the theoretical predictions for the coherent struc-
ture of turbulence based on the universal geometric frus-
tration-percolation transition theory.
2.1. Inverse Cascade–Cascade Mode
One of the GT theories is the inverse cascade–cascade
mode along local one direction (e.g. notated in local
z-axial, or on x-y projection plane) in a reference (thaw,
excited) domain. The definition of excited domain in GT,
similar to that in a ferromagnetic material, is a contigu-
ous region in which the direction of thawing mole-
cule-cluster spontaneous delocalization (Section 3.3) is
uniform and different from that in neighboring regions.
This idea comes from the insight for cooperative orien-
tation activation energy, ΔEco, on melt high-speed spin-
ning-line [8]. When the work of the stress on over
5000 M/min spinning-line reaches ΔEco, the structure of
the yarn is stable and reaches full orientation, called as
FOY (Full Orientation Yarn) in current polyester fiber
industry. This phenomenon is called stress-induced GT.
The rate of change of the stress-induced liquid-to-solid
GT is 107 times of that in general quencher from melt
state to frozen glass state. A logical explanation for melt
high-speed spinning 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 (MT) tem-
perature Tm to GT temperature Tg, 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 spin-
ning-line; and finally the mode is frozen in glasses as the
soft matrix. At solid-to-liquid GT, increasing tempera-
ture only increases the number of IE loop-flows per unit
time to augment the number of thawing domains to ac-
celerate GT. According to the fundamental view and
central assumption of de Gennes’ soft matrix on mosaic
structures in glasses [9], an intrinsic 8 orders of transient
2D mosaic geometric structure theory of GT has been
proposed [10], Figure 1.
The structure is formed by the 8 orders of 2D IE loop-
flows from small to large in inverse cascade and rear-
rangement structure in cascade. The number of IEs on
the 8th order loop is identified by means of the geomet-
ric frustration, i.e., the appearance of the 8th order di-
rectly corresponds to that of boson peak and geometric
frustration in order to satisfy the claims for the theory of
GT [9-12]. The number of IE of the 8th order loop
equals to that of the 7th order, and the directions of the
two loops are opposite. Thus the 8 orders of 2D IE loop-
flows with one external degree of freedom (DoF) may be
also regarded as a domain-scale interacting classical
boson formed by 200 z-component chain-particles (mo-
lecules) [10] in a z-axial thaw domain. Furthermore, this
GT theory shows explicitly that the geometric frustration
also directly corresponds to the geometric frustration-
-percolation transition of the 8th order loop-flow to
balance the potential and the kinetic [10]. The centric 8th
order (V8) loop one by one connects with its neighboring
4 7th order (V7) loops in Figure 1. When the 4 V7 loops,
one by one, all evolve to their V8 loops (each V8 loop has
one external DoF); the central V8 loop obtains 5 inner
DoF and cascades to vanish (that is also the microscopic
melting mode with renewed cluster energy, kTm° (v8)).
This evolving picture holds true for all IE loops that
form the flow-percolation at the GT.
In Figure 1, each (small square) chain-particle has a
z-axial displacement energy
that comes from the
Berry’s phase of parallel transports of its 4 IEs surround
its z-axis. The 8th order loop induces each of 136 parti-
cles to hop in z-axial respectively at different regression
times [10]. The central 5 cavity cells, in accordance with
de Gennes’ simple picture of “a localized ‘vacancy’
among clusters” [9], are also the mosaic cells denoting
the 5 cooperative excited particles [10] delocalized in the
correlation of sharing energy Ec (Section 3). This struc-
ture can deduce three non-integrable energies: kT2°,
kTg°, kTm° and an icosahedral directional ordering at
the GT.
The 8 orders of IE loop-flows also define the z-axial
self-similar 8 orders of transient 2D clusters vi (relaxa-
tion time
i) and 8 orders of transient 3D hard spheres
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Figure 1. Sketch of the 8 orders of transient 2D mosaic
structure, the maximum order potential structure in random
in a thaw domain [10]. The purpose to introduce the IE
concept is to find out the additional 2D transient (
i scale)
regression directional ordering potential formed by the
IE loop-flows occurring respectively at the discrete local
instantaneous times (Brownian regression times) ti in 3D
ideal random system (the Flory disorder [13]). The addi-
tional IE energy Δ
i) with the relaxation time
i (i = 1,
2,, 8) has and only has 8 orders. The number 8 is
based on the limited domain-wall vibration frequencies
in the random first-order transition theories [11] of the
GT. The author also independently proved that there are
only 8 orders of self-similar 2D clusters vi and 3D hard
i at the GT [14].
The two global properties of IEs in z-space are found.
(i) The fixed point Uc* for 8 orders of self-similar Len-
nard-Jones (L-J) potentials [14], Ui = fi /
i) = 4 [(
/qi)12 (
i /qi)6], Uc* = –15/16, Figure 2. Here
i) is
the potential well energy of i-th order cluster with direc-
tional volume vi, or
i and relaxation time
i, so
also notate as
0(vi) or
i) at GT;
i+1), Figure 2 reveals that the
fixed point is an universal constant for all self-similar
molecule-clusters, containing the non-flexible system of
i+1), with inverse cascade - cascade mode in
random systems. (ii) The fixed point for reduced second
Virial coefficients [15], B2(vi) 3/8, in Figure 3. The
existences of the two fixed points have proved that the
IE comes from the balance between the self-similar L-J
potential fluctuations in z-axial and the (non-integrable)
geometric phase induced potential fluctuation on x-y
projection plane. The so-called “tunneling” [16,17] re-
sults from the fact that the generating and transferring of
all additional 320 IEs [10] should pass through the 8
orders of additional self-similar attractive potential cen-
ter of –17/16
i), (i = 1, 2,,8) [14], Figure 2, each of
which is lower than potential well energy
i) of i-th
order cluster vi.
i) is also the energy of one external
DoF for i-th order loop-flow. The theoretical proof of the
well-known WLF experimental equation in polymer
physics [18] validates that the energy of all the 320 IEs
in the 2D mosaic structure is exactly the cooperative
orientational activation energy ΔEco to break a solid do-
main [18]. This result agrees with the idea of destabiliz-
ing the crystalline state [12], and the solid-liquid coexist
state evolves as the particle-flow through 8 orders of
fast-slow mosaic relaxation states.
It should be pointed out that the activation energy
ΔEco also correlates to the icosahedral directional order-
ing at GT. The icosahedral ordering [19-23] is one of the
key concepts to understand the GT and the relaxation
dynamics. All the 320 IEs in a local z-space can be also
regarded as 320 = 20 2 8. Here, 20 = 5 (cooperative
excited particles [10]) 4 (the number of interacting
faces per chain-particle); 2 as the two-body delocalizing
in z-space; 8 as each IE has 8 components [10], 8Δ
the energy of one external DoF that can be experimen-
tally detected at the GT.
The experimental value of IE energy for flexible sys-
tem is Δ
0 (
i) = Δ
5.6 104 eV from WLF equation
[18]; and Δ
5.5 104 eV from ΔEco on melt spin-
ning-line [8,10]. So
i) = 8Δ
4.4 103 eV. This
value is consistent with the experimental results (4 12
103 eV) for Boson peak measured by high-resolution
inelastic neutron scattering [12]. For non-flexible general
Figure 2. Two-body delocalization along 8 orders of geodesic,
the fixed point of 15/16
i) is the universal constant for all
self-similar molecule-clusters
i with inverse cascade–cascade
mode in random systems.
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system, each of 320 IEs may have different IE energy.
Thus, each material has its own two inherent values of
independent of temperature in its own random system:
the geometric frustration—percolation (also the localiza-
tion—delocalization) transition energy, Ec = kTg° (vi),
and the renewed cluster energy, kTm°(vi). kTg°(vi) is also
the non-integrable directional regression vibration (DRV)
kinetic energy of i-th order cluster vi, kTg°(v8) = kTg in
inverse cascade; and kTm°(vi) = kTg°(vi) + 4
0(vi), kTm°(v8)
= kTm, of the 8 orders of clusters in cascade [10].
2.2. Origin of Interface Excitation in
Statistical Physics
This theory differs from current mode-coupling theo-
ries [11] of GT. A phenomenological concept of IE is
introduced by the ripplon and the universal Lindemann
distance increment [10] dL in order to obtain the global
properties of IEs in inverse cascade – cascade. The spirit
in mode-coupling scheme deals with the coupling of
fast-slow relaxation modes and two density modes in
structure rearrangements at GT. In our mode-coupling
scheme, we first focus on finding out the three direction
non-integrable energies kT2°, kTg° and kTm° existing in
the coordinates invalidation from i-th order clusters to (i
+ 1)-th order at GT, in spite of the time complications in
all anharmonic frequencies. This means that the crucial
2D 2 closed loop with the maximum non-integrable
Berry’s Phase plays a major role in anharmonic fre-
quency. The mode-coupling trick is that the relaxation
time complications have been beforehand reduced and
replaced by the slow Brownian directional regressions.
Then we deduce from global to local to find out the mi-
croscopic origin of IE (Section 2.3). The mosaic struc-
ture theory of GT reaches a new level by revealing the
microscopic physical origin of IE.
The fluctuation stability condition in (ii) is that the
chemical potentials are always zero in all subsystems
[15]. There is a profound theoretical contrast between
chemical potential and ‘temperature’. The chemical po-
tential is defined as Fermi energy level on the occasion
when the random kinetic energy of temperature in sys-
tem is zero in solid physics; whereas the non-integrable
random DRV energy, kTg°, in a thaw domain, is defined
as the delocalization energy, Ec, on the occasion when
the chemical potentials in all subsystems are always zero
at the solid-to-liquid GT. The key here is that the kTg is
only comprised by the slowest (z-space) DRV energy,
which is exactly the lowest-frequency (
i scale) mode
that has been explored at the GT. The x-axial and y-axial
i scale) DRV energy of all z-component molecules
(chain-particles) encircled by IEs should be zero in order
to minimize the total dominant GT energy.
In Figure 2, there are 8 sharp-angled points, U
c* =
U(qi,R) = U(qi+1, L) [14], forming the delocalization paths
of two z-axial (i.e. q-axial in the figure)
1 clusters along
8 orders of geodesic with IE energy
(τi) = 1/8
i-th order geodesic is the shortest line of 2
cycle be-
tween qi,R and qi+1,R on i-th order of cylindric potential
surface (on x-y projection plane, taking
0 = 1 as
z-coordinate axis). The red arrows denote the paths of
structure rearrangements in cascade: from q8,R q7,R
q1,R q0,R.
2.3. Absence of Attraction in 2D Lattices
Thus, two
1 clusters are in the delocalization state,
absent of attraction in vibration, along 8 orders of geo-
desic. The two molecule-clusters in the absence of Van
der Waals attraction are all through in a repulsion state
on the x-y projection plane during generating their one
IE. Van der Waals interaction includes the contribution
of instantaneous induced dipole - induced dipole.
Generally, instantaneous polarized dipole electron ch-
arges randomly distribute on interface 1-2 forming elec-
tron cloud (blue zone) in Figure 3(a).
At GT, an interesting and unexplored corner in Van
der Waals interaction theories is that the synchronal in-
stantaneous polarized electron charge coupling pair (two
small blue dots) may parallel transport on an interface
1-2 between chain-particles a0 and b0, or the interface
2-3 between a0 and c0 to form IE in Figure 3(b), or 3(c).
Since the site-phase difference on x-y plane between the
two z-component molecules is [10], the state of the all
polarized electron charges in each instantaneous dipole,
in the two z-component molecules in Figure 3(b-1) or
3(c-1), must be in the same state and in the z-axial min-
imum energy state (ground state) of single-molecule in-
stantaneous dipole. In Figure 3, (b-2) and (c-2) is re-
spectively the projection of (b-1) and (c-1) on x-y plane.
Thus, the additional IE energy is the repulsive energy
between the two instantaneous synchrony z-axial ground
state polarized electron charges that parallel transport,
from one end to other end on an interface in Figure
3(b-2), simply denoted by an arrowhead (12) in 3(d)
on local x-y projection plane. Note that the two IE states
of (b-1) and (c-1) in Figure 3 occur at different local
instantaneous times. The directions of the next two par-
allel transports of instantaneous polarized electron
charge of the reference a0 are denoted as two red-broken
line arrowheads in Figure 3(c-1). IE loop-flow 1 2
3 4 1 in Figure 3(d) offers a non-integrable poten-
to induce ion a0 (the center of mass of a0 particle)
z-direction a displacement z. Black-broken line arrow
denotes the delocalizing direction of each ion in Figure
3. Figure 3 explains the origin of 5-particle cooperative
excited field in [10]. The molecules in the absence of the
x-, y-axial (
i scale) vibrations are in the high density
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Figure 3. The microscopic physical origin of IE. (a) A legiti-
mate state; (b), (c) An absent of attraction state in 2D lattice; (d)
delocalizing instantaneous dipole state.
state, which agrees with de Gennes’ picture of the com-
pact primary clusters [9].
2.4. Random Delocalization Transition
A region of space that can be identified by a single
mean field solution is called a mosaic cell [24]. The IE
loop-flow can expediently affirm interfaces of mosaic
cell. Due to the effect of geometric frustration – percola-
tion transition, the number of IEs on the 8th order loop is
corrected as 60 [10]. Based on the corrected value we
can also validate the theory of IE. By single mean field
solution, we directly obtain the percolation (also the de-
localization) energy Ec = 20/3
0 [10]. Since the energy in
inverse cascade – cascade is not dissipated, we also di-
rectly get the 8 orders of non-integrable potentials Ec(
= 20/3
i) to induce 8 orders of clusters vi inverse cas-
cade along local one direction. The balance between
i) and kTg(
i) is realized by the fixed point in statis-
tical physics, the fixed point of reduced second Virial
coefficients B2(T *) for self-similar clusters vi in different
size. That is, B2(T*) = B2(kTg°/
0) 3/8 at the GT, in
which kTg° (
i) 20/3
i) in Figure 4, independent of
temperature and time, and kTg° (v8) kTg° (
8) = kTg.
In Figure 4, T* = kT /
i), is the reduced tempera-
ture in [25], also the reduced non-integrable DRV energy
in our discussion. Graphical method gives the only set of
approximate solution, B2(Tg*) 3/8, Tg* 20/3, satisfies
the self-similar Eq.5 for the curve of B2(T*)LJ. The nu-
merical solution refers to the result in [25].
In inverse cascade at the GT, in the percolation evolu-
tion fields from cluster vi-1 to vi+1, we may furthoer rewrite
the reduced Virial expansion as the form of Eq.1 in [15].
 
PV vkT vBvBv
 (1)
Where, potential takes cluster volume vi+1 as variable
which denotes that the induced potential is fast motion
whereas the molecule-clusters moving is slow motion.
The phase difference is invariable as between kinetic
and potential. The result of two-body interaction always
gets evolvement volume to vi+1 from vi. Two-body inter-
action is slower than three-body. In other words, three
-body is always firstly compacted in order to minimize
the totally IE energy. The two-body interaction is in fact
the interaction of two three-bodies in fluctuation at the
GT [15].
In statistical mechanics, the abnormal thermal capac-
ity occurs in self-similar system. From enthalpy H = E +
PV, the definition of Joule-Thomson coefficient [25]
 
 
Figure 4. The fixed point (B2*, T*) (1/3, 20/3) of self-similar
reduced second Virial coefficients validates for all directional
molecule-clusters. T* here is the reduced non-integrable DRV
energy in random systems.
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Assume the state of
J 0 corresponds to GT. When
J 0
VT VT (3)
Rewrite as Eq.4 for vi cluster
 
 
ii ii
Vv kTvVv kTv  (4)
Here kT in (2) (4) should be also regarded as the
non-integrable random DRV energy of i-th order cluster
vi in z-space. The energy kT also is a function of the
self-similar cluster volume v
i, when outside pressure
(stress) remains constant. As long as the condition
J 0
in (2) is satisfied and (4) also holds, CP in the case of (4)
may also show an abnormal change. We see the abno r-
mal thermal capacity may occur in the following two
cases. One is in the polymer GT, CP occurs at the tem-
perature Tg. In this case, the kTg (= Ec) is the percolation
energy. The other is in the low-temperature GT in local
domain region, the abnormal thermal capacity occurs in
the manner of the energy of Boson Perk.
The hard-sphere (square well potential with IE energy)
model at the GT can be also deduced from the 8 orders
of self-similar L-J potentials along the clusters inverse
cascade direction [14]. It can be strictly proved that the
reduced third Virial coefficient for hard-sphere system is
constant, B3 5/8, [25,26] independent of temperature
and cluster volume. Thus, in z-space of the clusters in-
verse cascade, we can also get B3(vi-1) 5/8. Therefore,
in the percolation evolving field from cluster v
i-1 to v
from the interaction of two three-bodies, we have
  
2ii i
PV vkT vBv
in z-space fluctuation. In z-space, from (1),
 
Bv Bv
Vv kTv Vv
From (4), we obtain an important self-similar Eq.5 at
the GT in [15].
Bv Bv
kTv kTv
It Figure 4, if we regard the energy of T
g* as the
z-space reduced DRV kinetic energy of 8 orders of
self-similar clusters, the point (B2, kTg°) (3/8, 20/3
is exactly the universal geometric frustration percola-
tion transition energy for any cluster volume v in random
delocalization transitions. Moreover, the B2 3/8 at the
GT is directly proved by scaling theoretical approach [15].
  
3858 1
PV vBv Bv
kT v
  (6)
where i = 1, 2,, 7, and when i = 8, the induced poten-
tial Ec directly equals to kTg in the geometric frustration
– percolation transition at GT. Eq.6 holds true on all
subsystems (flow-percolation fields), which means that
the kinetic energy always keeps balances with the poten-
tial energy, in the manner of the maximum Berry’s phase
of , in the mode of 8 orders of coupling 2body-3body
fluctuation clusters at GT.
2.5. Order-Induced Molecule Delocalization
The concept of connecting Anderson transition and
GT is the percolation limit model. In the theory of An-
derson disorder-induced (electron) localization [13], the
competition between kinetic energy and potential energy
influences on the electron states can reside in the ratio
W/B, W, the magnitude of the random potential, and B,
the (crystal) bandwidth in the absence of disorder. A
classical method of demonstrating consequences of dis-
order is the percolation model [13]. At the GT, the per-
colation transition energy is Ec(
8) = Ec. ΔEco is the ori-
entation activation energy with 320 IEs on the 8-order
2D mosaic structure in local z-space [10]. The ratio of Ec
c(Ec) 1/6, its physical meaning is that
specifies the occupied fraction of z-space 320 IE states
that allow flow of energy Ec to occupy. The reason of the
cooperative molecules obtaining the probability of delo-
calization is that they share energy Ec. The ratio is con-
sistent with the result (the classical analog for electron
delocalization) of Zallen [13], who suggests
c (Ec)
0.16 1/6, for the percolation limit on a continuum in
3D space. Here, the occupied fraction of all IE states
allowed to flow of energy Ec, takes the place of the oc-
cupied fraction of space allowed to molecules of energy
Ec. This means that there is the inherent theoretical con-
trast between molecule delocalization and electron lo-
calization on the concept of percolation limit.
That is, from the viewpoint of percolation, there are
two classical percolation limits. One is the limit of the
maximum disord er for the potential in random corre-
sponds to the Anderson transition, that is the Zallen’s
viewpoint [13]; whereas, the other is the limit of the
maximum order for the potential in random to the GT. At
the GT, all molecules encircled by 4 IEs on x-y projec-
tion plane are in the minimum excited energy state
(states), i.e., at the z-axial ground state of single - mole-
cule dipole in the absence of x-, y-axial (
i scale) vibra-
tions. The DRV induced potential energy in 5-particle
cooperative excited field [10] can be also regarded as the
overall order potential limit (in contrast to the maximum
disorder potential limit in Anderson random) in the ran-
dom Brownian motion. That is the reason we call the GT
as the order-induced molecule (molecule-cluster) delo-
calization transition in random systems.
In other words, the delocalization energy Ec at the GT
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Copyright © 2011 SciRes. OPEN ACCESS
comes from the maximum order energy in random and
the minimum additional ground-state repulsive energy.
Thus, the percolation limit mode (the crossover from the
site- or bond- percolation in solid to the flow-percolation
in liquid) of the GT can probably explain why the 8 or-
ders of 2D mosaic geometric structure (Figure 1) asso-
ciated with percolation transition so simple. That is, this
nano-scale soft mosaic structure reflects the nature of
random delocalization transition in random systems.
2.6. Percolation Limit Picture at GT
The theory of IE loop-flow directly and definitely
shows Emig = kT2 = 17/3
0. Emig is defined as the average
energy of cooperative migration along one direction in a
percolation field [10], independent of temperature. Dur-
ing [ti, ti + ti] in a reference domain, the entire transient
i scale displacement energy in z-axial is (136/8)
0 =
0, Figure 1, and the
i scale vibration energy in x-,
y-axial is zero; thus, the average DRV kinetic energy is
also 17/3
0 = kT2. On the other hand, during (ti + ti, ti+1),
the average random energy in the domain is as kT2.
However, the average DRV kinetic energy would occur
in other domains in the system during (ti + ti, ti+1) in the
reference domain at the GT. This is the dual-role of kT2
in kTg, reflecting the balance between random DRV en-
ergy and random thermal motion energy at the condition
of kTg°. If take Emig = kT2 as the average DRV kinetic
energy in dual-role, kT2 is in fact the deferred action
DRV kinetic energy until the appearance of the 8th order
2D loop (the appearance of kTg). In this case, kT2 will be
of the DRV kinetic energy of the 8th order cluster in the
absent of energy
8) in an excited domain in Figure 1.
Vice versa, when the temperature is T2, the reference
domain will also be of the probability of regression order
potential of kT2 if the domain can obtain the energy
8). Thus, the physical meaning of kT2 is that in a 3D
solid-domain, taking any reference direction, as long as
superadded the energy
8) on the domain, the sponta-
neous molecules delocalization would be occur in the
reference direction.
Its singularity different from phase transition is the
non-ergodicity that the adding energy may be either a
few energies only on several domains, or a large energies
one by one on neighboring domains to form a flow-
-percolation field. In any case, if the adding energy
denoted by temperature, it always satisfies kT2 +
0 = kTg.
In other words, here
0 =
8) is a reduced energy to
reduce the magnitude of its action region or its action
time in the system. The noticeable contrast between
phase transition and random transition in physical the-
ory (Table 1) is as follow. At Curie temperature point,
kTc, in magnetism, so as decrease a little of rando m en-
ergy (kT) in whole system, the spontaneous magnetiza-
tion will occur along one direction in whole system.
Whereas, at kT2 of the GT, so as increas e a little of di-
rectional ordering energy
0 in a local region, the mole-
cules spontaneous delocalization will occur along the
direction in the local region. It is in this sense T2 corre-
sponds to Curie temperature in magnetism. The numeri-
cal value of 17/3
0 is also validated by the WLF experi-
mental equation [18], 17/3
0 +
0 = kTg = 20/3
0 = Ec.
There are two critical delocalization energies Ec and kT2.
Ec is the critical percolation transition energy, corre-
sponding to the spontaneous delocalized solitary wave
(Section 3) in percolation field (the “ocean” in percola-
tion theory) at GT; and kT2 is the critical energy of parti-
cle-clusters delocalization at low-temperature GT in lo-
cal domain scale (the “lake” in percolation theory) in
percolation limit model. This percolation limit picture
with separate ‘lakes’ and ‘ocean’ is distinctly different
from the general percolation model. Note: the delocaliz-
ing step-size in glassy is far less than the amplitude of
thermal vibration, Section 3.3, thus the glass state in
random delocalizing is still in stable state.
2.7. Soft Matrix and Emergent Property
The de Gennes’ central assumption of the soft matrix
[9] is also validated. The soft matrix is exactly i-th (i 7)
order 2D loop-flows frozen in glass state. From which
the low-temperature GT would occur through a longer
Brownian regression time to inverse cascade until the
8th order and to cascade at low-temperature. The lower
the frozen, the longer is the required time of inverse
cascade. Therefore, the abnormal exponential function
[27] in glass state can be proved directly.
Generally, the physical quantity
(t) in a system will
return according to the physical law of Eq.7 when the
system deviates from its equilibrium state.
~exp t
However, if the glassy system is driven (or normally
fluctuates) out of equilibrium, it returns according to the
formula [28]
exp t
Where t is the (system) time and
are parame-
ters. Unfortunately this is not a mathematical expression
that is frequently encountered in physics. So little idea
exists of what the underlying mechanisms are [28].
Since the inverse cascade–cascade motions only occur
in some discrete “lakes” in glass state when T Tg. Eq.7
still holds true in these “lakes” regions as long as the t in
(7) is the local domain time. One of the key concepts is
J.-L. Wu / Natural Science 3 (2011) 580-593
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that the equilibrium state of glass state is the equilibrium
state between the random thermo motion energy kT and
the slowest DRV energy. DRV energy always dominates
the number of the excited domains at temperature T.
Assume the glass state we observed is the nonequilib-
rium state that comes from the equilibrium state at the
temperature T1 suddenly drops to the temperature T2 at
the time t = 0, and T2 T1 Tg. During the relaxation
time of t, the entire DRV relaxation energy is
= k(T1
T2) = kT. From the famous Kolmogorov law in cascade
 
. (8)
Where li is the length scale of i-th order of loop (clus-
ter) and i
is the cascade energy mobility; ti is the local
domain time and t is the relaxation time in system. From
Substituting ti in (9) for t of local domain time in the
right term on (7)
~exp exp
The physical meaning of
is very definitude,
denotes the cascade motion in ideal glass state in the
case of the small fluctuation value of kT. The reason of
the deviation from 1/3 for
may be that (a) the influence
of the competition between the change ratio of tempera-
ture dT/dt and the 8 orders of relaxation times on IEs; (b)
the larger value of kT conduces
augment because of
the inverse cascade always lower than cascade. Similarly,
if the nonequilibrium glass state is arose by outside
stress (or electromagnetism and other factors) work W,
the relaxation of W can only realize through the inverse
cascade – cascade in the excited domains as same as that
in the proof of WLF equation [18]. In this case, we only
need replace kT by W in (9). This means that the ab-
normal mathematical expression (10) in glass state is
also only the emergent property of domains in system, as
same as the abnormal expression of WLF equation in the
GT (the many-times repeated applications of Clape-
yron equation governing first order phase transition on
the subsystems will educe WLF equation) [18]. The
general physical relaxation law (7) still holds true in
domain scale in glass state. The theoretical proof also
confirms that inverse cascade–cascade mode is the fun-
damental mode for glass state.
3.1. Theoretical Proof of the 3.4 Power
The viscosity in ME melt is
N3.4 for all linear en-
tangled polymers. Since N is a large number, the ex-
perimental 3.4 power law of viscosity has sensitivity for
any modified theory to de Gennes reptation model. In
concordance with the experimental 3.4 power law of
viscosity, within the experimental error range of viscos-
ity, we first roughly estimate the theoretical error toler-
ance by adopting a fine theory described ME viscosity.
The critical entanglement chain length Nc = 200 [11,30].
The experimental value of chain length N is generally
less than 103 (seeing the Fig. 9.5 in [5]). If taking the
range of N as 200 1000, N3.4 /N3.3 1.7 (N = 200) 2
(N = 1000). This indicates the error between theoretical
value and experimental result of viscosity will be about
70% 100% if the theoretical value as 3.3; and N3.4/
N3.35 1.3 1.4. This means that a fine theory should be
able to give the theoretical exponential value range of
viscosity as 3.4 0.05 for flexible polymers. At the same
time, we expect this theory must also be able to predict
the exponential of viscosity that is in line with the ex-
perimental results for non-flexible polymers. This is also
a fine way to check up the GT theory. The existing vari-
ous modified theories [5] cannot match for the 3.4 power
law. This exponent is significantly large than the predic-
tion of 3 by de Gennes reptation model [5,30] based on
the assumption that the chain of length N is a “free
chain”. The reason of the deviation is the required num-
ber of DoF of one-step-walk along z-axial for each chain
particle depends on Nz the z-component of chain N. The
key of the theoretical proof is to find out the number of
DoF, N*, for chain-length N. Only through substitution of
the reference chain of length N in single-chain reptation
model with an equivalent particle-chain of length N*, can
the random diffuse motion of the reference chain N be
entirely free in tube model. Thus, in the ‘equivalent mul-
tichain’ de Gennes reptation model, we have
The one-step-walk along +z-axial of a reference parti-
cle a0 on chain Nz results from the induced action of the
8 orders of 2D loop-flows on the x-y projection plane in
a0 (local excited) field. Only when the 8th order appears
and acts on the particle a0, should one “particle-cavity”
in +z-direction appear in a0 field and bring a0 to move
one step along +z-axial. Statistically, when Nz Nc = 200
(Nc is also the number of the chain-particles of structure
rearrangements [10,11,30,31] in the 8 orders of 2D mo-
saic structure), the moving of the 200 cooperative parti-
cles along z-space on x-y projection plane in a0 field
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Copyright © 2011 SciRes. OPEN ACCESS
should share the energy Ec with a0. Due to the sharing
energy effect, even if a0 has obtained a cavity with
one-step-walk, the motion of a0 should still be correlated
with the 200 particles (molecules) in shared energy Ec.
In other words, the cooperative particles still “drag” the
one-step-walk of a0.
The external DoF of the 8th order 2D loop-flow en-
circled a0 is 1, taking
8) as the energy unit of DoF. In
order to eliminate the correlation of share energy to
make a0 move freely in z-axial, let Ec /
8) = Lg, where
Lg is the equivalent number of particles. Ec thus is as the
“loop-flow” energy with an equivalent chain of length Lg
circling a0. On the one hand, from the viewpoint of par-
ticle a0 on chain Nz, statistically, the Nc chain-particles
on Nz also share the energy Ec. Thus, the probability that
a0 is possessed of one unit DoF on the chain Nz is 1/N.
On the other hand, from the viewpoint of the Lg equiva-
lent particles, statistically, each equivalent particle is
located at its own long-chain with length Nz, and the
probability that each equivalent particle is possessed of
one unit DoF on its own long-chain is also 1/N.
When particle a0 on chain Nz is substituted by an
equivalent chain of length Lg, the probability of the event
that it obtains one unit DoF and freely moves one step
along z-axial, denoted as p+(a0). p+(a0) is namely the
probability of the event that all the Lg equivalent parti-
cles simultaneity move one step along z-direction in the
co-Brownian motion by 200Nz chain-particles in a0 field.
So, p+(a0) is given as
pa N
Statistically, the probability of each particle freely
moving one step in z-space should be all equal to p+(a0).
To maintain the balance of motion between inverse cas-
cade and cascade in z-space, the probability that one
reference 8th 2D loop-flow disappears in cascade in
co-Brownian motion by 200Nz chain-particles is denoted
as p(a0) and obtained as follows.
Let a0 have nz DoFs with
8) as the unit in cascade
motion. nz is actually the number of DoF to make con-
formational rearrangement of the chain Nz in z-space,
which results from the contribution of the reference 8th
order 2D loop-flow transferred from the particle a0 field
to the other particle field on the chain. Similar to (12),
we have
where Lm = kTm /
8), Lm is the number of ‘equivalent
particles’ to eliminate the reference 8th order 2D
loop-flow of a0 in cascade.
In the complex inverse cascade–cascade motion, the
equilibrium condition is
pa pap
 (14)
 (15)
The number of DoF of Nz chain-particles on chain of
length Nz to make conformational rearrangement in
z-space is Nz
*, Nz
* = nz Nz,
In (16), Tg° has been approximately replaced by Tg.
8) H, here H is enthalpy in (2), the constant en-
ergy for rearrange conformation (that also the energy of
z-space solitary wave in Section 3.3) in random system.
So, during the reptation of long-chain N in the 3D space,
the number of DoF Nx
* in the x-space and Ny
* in the
y-space have Nz
* = Ny
* = Nx
*, or
From (11) and (17), we obtain the general power law
expression of viscosity in ME
For flexible chain polymers, kTm = kTg + 4
8), kTg =
8) [10]. We have
NN (19)
83.375 3.4
ηNN N
This theoretical result conforms well with the experi-
mental data. For non-flexible chain polymer, Eq.18 can
be verified. For example, for the polypropylene (PP), the
Tg(PP) is 10C = 263k [32]; the Tm(PP) is 176C =
449k [33]; Eq.18 given
pp(theory) N3.73, conforms
well with the experimental data
pp(experiment) N3.72
The theoretical proof further confirms that inverse
cascade–cascade mode is the fundamental mode in the
solid-to-liquid transition whether in macromolecular or
in small molecular.
3.2. Degrees of Freedom of Chain-Length N
Since cascade is also the mode in turbulent flow, we
discuss the total number of DoF, NL, in the cube with
edge length of L (L N*) in ME melts. The unit for the
length of L here is the loop scale, ls, of the 8th order 2D
loop-flow. Due to the properties of flow-percolation [10],
an 8th order 2D loop-flow may be in arbitrary direction.
In other words, in 3D space, we can consider any one
J.-L. Wu / Natural Science 3 (2011) 580-593
Copyright © 2011 SciRes. OPEN ACCESS
“random-walk chain” with “chain-length” NL and each
“chain-particle” with one unit length l
s. If the “end-to-
-end distance” of the “random-walk chain” is always
equal to L, L is then the root-mean-square of end-to-end
“chain-length” NL, that is, NL = L2, in Figure 5.
When L N*, NL (N*)2, from (19), we have
NN (20)
Comparing to the famous Kolmogorov relationship
[29] between the number of DoF NR and the Reynolds
numbers Re in (a uniform grid) turbulence
It can be seen that the two terms in two different sub-
jects: turbulence and ME both describe the same com-
plex motion phenomena about inverse cascade – cascade.
Both of them obey the same scaling law, which indicates
there is a universal theory behind them. Reynolds num-
ber is actually the ratio of the inertia force to the drag
force, and the macromolecular chain-length reflects the
ratio of the number of the particles “inertially” one by
one sharing the delocalization energy Ec to the one parti-
cle entangled with them in motion.
3.3. Solitary Wave
Each physical quantity in (12) (18) is the inherent
invariable in random system for every material, inde-
pendent of temperature and time. Eqs.15 and 16 reveal a
z-space solitary wave moving N
z steps along a random
long-chain Nz from one end to other end. A distinct cha-
racteristic of the solitary wave is that the “particle” en-
ergy of each one-step-walk is “quantized”: nz
i) = nz
for flexible system. While nz
i) generally have 8 or-
ders of components, 8 orders of potential well energies
for general non-flexible system. The step-size is only the
nz particles of size. The number of step of its traveling
wave is Nz = N. The reptation in 3D space of a ME chain
is assembled in time by the three solitary waves on the
chain in x-, y- and z-space, which in fact attributes to the
co-Brownian regression motion by 200N chain-particles
respectively in x-, y- and z-space.
Accordingly, the ME structure about viscosity is in
fact the 8 orders of transient 2D mosaic structures on x-y
projection plane and the delocalized solitary wave in
z-space. nz is a small numerical value, e.g. if taking N =
Nc = 200, nz 0.036. However, Eq.15 reveals that nz
connects with the length-boundaries Nz of solitary wave
in its each one-step-walk. The velocity of traveling wave
will be determined by other factors including, e.g., the
generating ratio of IEs by temperature, time or outside
stress work, and the magnitudes of material relaxation
times in IEs. This special characteristic of solitary wave
shows that it differs from the mode of conventional quan-
Figure 5. The relationship between “chain-length” NL
and number of degrees of freedom.
tum-mechanical wave; the latter is always obtained from
the combination of both the wave equation and the bou-
ndary conditions.
3.4. Spontaneous Symmetry Breaking at GT
The paradigm for the phase transition theory is the
ferromagnetic transition. de Gennes [30] emphasized
that polymer solutions should compare with magnets.
However, difficulty occurs in the order parameter for a
polymer solution [30]. de Gennes proposed that the or-
der parameter is the magnetization of a spin system with
a number of spin components n = 0 [30]. Edwards’ more
concrete statement is that the order parameter
is simi-
lar to a quantum mechanical creation (or destruction)
operator [30]. The proof for 3.4 power law predicts a
definitude result for the remaining problem that the order
parameter is the spontaneous delocalized solitary wave
of a classical spin system (8 orders of 2D IE loops)
with a number of additional instantaneous spin compo-
nents n = 0, whether in macromolecules or in small
molecules, within the entire range from solid to liquid
transition. The physical meaning of n = 0 is here that the
additional “spin” of a reference chain-particle only
comes from the contributions of its 4 neighbouring par-
ticle fields, instead of its own inner-rotation change. In
addition, n = 0 also corresponding to self-avoiding ran-
dom walk [30] of 8th order 2D loop-flows.
Contrary to phase transition, we have unveiled the
step by step way from disorder to more disorder in Table
1. In which we have implied that the disordered degree
(measurement) depends on the maximum 2D IE
loop-size and the number of the loops in a random sys-
tem. The conjugate variables of solid-to-liquid transition
in Table 1 are deduced from the results of de Gennes in
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Table 1. The contrast between delocalization transition and phase transition in 3D space.
Critical point Ferromagnetic phase transition Solid-to-liquid delocalization transition
Number of spin components n n = 3 n = 0
Cell Magnetic moment S displacement energy
circled by 4 IEs
Spontaneity Magnetization M Delocalizing solitary wave
“Curie” point Tc kT2: critical delocalization energy
Reduced energy increment
= k (T
Tc) / kTc ,
0, M = 0 in system
0, M = S 0 in system
= k(T
T2) /
0: potential well energy
= 1, GT:
0 in subsystems
1, Low-temperate GT:
0 in domain
= 5, MT: Reptation in subsystems
5, Viscoelasticity (Intermittency)
Symmetry breaking M chooses one-space, breaking the isotropic M
GT: Solitary wave chooses one space, breaking the
isotropic thermal motion
MT: Three solitary waves in x, y, z-space restore the
isotropic thermal motion
Self-similar cluster cluster size
v Finite 8th order cluster
Correlation length
=Nz, macromolecular chain-length
=Nc, for small molecule systems
Order parameter M Solitary wave
Replica symmetry Spin wave 8 orders of 2D IE loop-flows
Conjugate variables Extensive
quantity Intensive quantity
Nz (or Nc), step number of solitary wave
0, one-step-energy of solitary wave
Generalized rigidity Magnetic hysteresis 8th order 2D IE loop
Defect Domain boundary Extra vacancy volume in each IE on 8th order loop
4.1. The Way to Turbulence
Thus, we have ascribed the GT problem to the syste-
matics of the special parallel transport state of bonding
on intermolecular interface. This is precisely in line with
the insight in [28] that crucial to the endeavor of GT
theory is a deeper understanding of the systematics of
bonding in condensed matter within a framework going
considerable beyond the current GT picture. All that we
have done is to replace the GT problem by the bonding
problem of parallel transport of instantaneous polarized
electron charges containing all atoms (atom-clusters) in
a chain-particle at the GT. However, we have progressed
and can now discuss turbulence.
The IE and the 8 orders of 2D mosaic structures at GT
may be called as the ground state IE and the first level of
8 orders of 2D mosaic structures. For small molecule
system, due to the energy transfer of rearrangement
conformation between two z-axial connecting molecules,
Nc z-component molecules respectively in absence of x-,
y-axial vibrations can also be connected one by one to
form a z-space random “long-chain with chain-section as
a molecule” walking Nc steps in the correlation of shar-
ing energy Ec.
Solid-to-liquid transition does not invoke the energy
level of molecule (molecule-cluster) instantaneous in-
duced dipole. However, for each energy level of mole-
cule (molecule-cluster) instantaneous polarized dipole,
e.g., for the first energy level of instantaneous polarized
electron charges of molecule (of all atoms and molecules
in a molecule-cluster), two neighboring molecules with
antiparallel delocalization may be all in the z-axial first
energy level state of single-molecule instantaneous po-
larized electron charges on an intermolecular interface.
Such IE is named as the first-energy-level IE. A new 8
orders of 2D mosaic structure can be formed, which is
called the second level of 8 orders of 2D coherent (mo-
saic) structure.
For small molecule liquid, only about 4/15 (8kTm /
Eco = 4/15) interfaces have been excited. As disorder
increases, more ground state IE will be excited to accel-
erate liquid flow, which corresponds to the ordinary flow
As disorder keeps increasing and flow continues
speedup, each ground state IE on the 8th order 2D loop
can update to be the first-energy-level IE with new 8
orders of relaxation times, new additional IE energy and
vacancy volume. All other IEs on x-y projection plane
are still in the ground state of IE. All these z-component
molecules in the absence of x-, y-axial vibrations can
form a z-space random “long-chain” with “chain-section”
as 136 molecules encircled by 60 first-energy-level IEs.
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Copyright © 2011 SciRes. OPEN ACCESS
60 first-energy-level IEs can form a 0-th “cluster-par-
ticle” in z-space in the second level of 8 orders of 2D
loop-flow, denoted as v0
(1) (a0), the figure in the super-
script parentheses denotes the first-energy-level IE of
single-molecule polarized electron charges. The 0-th
“cluster-particle” is self-similar to the z-component mo-
lecule a0 [named as v0(a0)] in Figure 1. The size of the
0-th “cluster-particle” is that of the 8th order cluster
v8(a0) in the first level of mosaic structures. The 0-th
“cluster-particle” has also 4 “interfaces” with the side
length of 15 IEs (one thick-black inverted arrow plus the
14 sky-blue arrows in Figure 1) of the first energy level.
Thus, the second level of 8 orders of 2D mosaic struc-
tures is self-similar to the first level. To drive the 136
molecules cooperatively move one step along +z-space
only needs the energy nz
(1). The mode is the solitary
wave with the first-energy-level IE in z-space.
4.2. Multilevel of Random Localization -
Delocalization Transitions
Inverse cascade is the accumulation of the directional
Brownian regression potential energy in loop-flows, and
cascade is the transfer of energy from potential to kinetic
to drive liquid cluster flow. The larger loop is formed in
inverse cascade, the bigger liquid cluster is driven in
cascade, and the higher efficiency the system has to ex-
cite liquid flowing, and the faster the velocity of flow is.
In the same way, when the flow keeps on speedup, the
third, the forth…the l-th level of 8 orders of 2D coherent
structures can be constructed one by one, in order to
augment the loop scale and the loop-potential so that the
maximu m 2D loop-potential reaches the value of the
potential forced by environment to the liquid flow. At the
moment, the l-th level geometric frustration – percola-
tion transition (seeing the universal pictures of Figures 1
and 2) appears and reaches potential balances with ki-
netic, and the cascade phenomenon of the maximum
loop-flow begins. This is the turbulent flow. The origin
of the intermittency turbulence [35] is that the high
non-ergodic state of flowing arises suddenly to make the
reduced energy increment,
(l), of each level with a value
slightly more than 1, similar to the viscoelasticity in Ta-
ble 1. The heteroclinic orbits [36] in turbulence may be
the flow-lattice edges of the l-th level 2D coherent struc-
The complicated turbulence can come down to the l-th
level solitary wave in z-space and the l 1 reptations
(snake-walks) of l 1 various levels (self-similar sub-
systems) simultaneously occurring in 3D space.
At critical phase transition, the self-similar cluster size
v , while it is only Nc in GT, Table 1, which is
also the localization-delocalization scale of the first level
in universal random delocalization transitions. In other
words, on the way from disorder to more disorder, the
cluster size of
v in critical phase transition will be
here divided up by l levels, Nc
(l), corresponding to the
cooperative localization – delocalization scale of l-th
level, i.e. it is also divided into l levels of geometric fru-
stration—percolation transitions, similar to Figures 1-5,
in random localization—delocalization transitions on the
way from disorder to more disorder. This is the way to
A theoretical perspective on the mosaic structure the-
ory of GT has been proposed. The theoretical approach
of GT connects closely the origin of turbulence. A new
and central concept is the instantaneous parallel trans-
port repulsive state absent of attraction in 2D lattices in
current Van der Waals interaction theories. The solid to
liquid GT is the paradigm of the inverse cascade – cas-
cade mode and the first level of the way to disorder and
turbulence. Nine physical ingredients, random, self-
similar, two-body interaction, fluctuation, frustration,
percolation, delocalization, Berry’s phase (parallel trans-
port) and Brownian regression potential, have been in-
corporated on IE loop-flows at the GT and the universal
random delocalization transition.
The theoretical proofs for the abnormal exponential
function in glass state and the 3.4 power law of viscosity
confirm that the inverse cascade–cascade mode is the
fundamental mode whether in solid to liquid transition or
in macromolecular motion. Due to the effect of incorpo-
ration of physical ingredients, the different terms in dif-
ferent subjects have been also incorporated. The inverse
cascade–cascade mode with mosaic structure reflects the
confluence of both the kinetic dimensions and the ther-
modynamics at the GT. It can be also denoted by both
the delocalization solitary wave (that is also the melt
mode, the conformational rearrangement mode, structure
rearrangement mode) along a local one-dimensional di-
rection and the 8 orders of 2D IE loop-flows (its scale or
size is also a domain scale, cage scale, local scale, cor-
relation scale of two-body interaction, random localiza-
tion-delocalization transition scale, maximum step-scale
of random walk, geometric frustration scale and critical
percolation scale in random systems) on the local pro-
jection plane.
Three different structure terms, the mosaic structure in
GT, the entanglement structure in ME and the coherent
structure in turbulence, are all in fact the transient 2D
geometric structure formed by IE loop-flows in their
own random system. The most important physical quan-
tity correlating GT, ME and turbulence is the directional
ordering induced potential of a maximum 2D IE loop-
flow in random system. Once the loop-flow is formed,
J.-L. Wu / Natural Science 3 (2011) 580-593
Copyright © 2011 SciRes. OPEN ACCESS
the GT appears, the critical ME phenomenon arises, and
the cascade in turbulence begins.
The author is grateful to all colleagues he had the pleasure to col-
laborate and interact, especially when he found the fundamental phys-
ics origin for the orientation activation energy obtained experimentally
on melt high- speed spinning-line in 1986. 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. Support
from the Academia Sinica (Taiwan), and the State Key Lab of Chemi-
cal Fibers and Polymer Materials, Donghua University (Shanghai) is
gratefully acknowledged.
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