WHY THE NEED OF A NEW
RENAISSANCE IN (POLYMER) PHYSICS?
The
French word “renaissance” means “re-birth” in English, but, actually, it does
not need to be translated: everyone uses the original French word to designate
a new paradigm, a new era, a fresh start.
I
used that word, Renaissance, yet added “New” in front of it. Is it not a pleonasm to say: “a new
renaissance”, like “a re-re-birth”?
Not
if you consider that the 1st renaissance in polymer physics occurred
with de Gennes, circa 1971-1979, introducing the reptation model of polymer
dynamics.
The
pre-de Gennes era, in polymer physics, was dominated by physical chemists:
Bueche, Flory are typical examples, Treolar also pops in my mind, but there are
so many others; I would need Boyer’s
legendary memory to be able to turn up a long list of names (see my
previous blog #35).
The
pre-de Gennes era established the concept of macromolecules (Staudinger), and
de Gennes School developed the scaling concepts to adapt the statistical
mechanics of the small molecules to the dynamics of the long macromolecular
chains. This field has become a very sophisticated mathematical model, indeed,
after 40 years of fruitful advances due to ramifications and improvements.
Yet
I maintain that we need to turn the page and start a new renaissance in polymer
physics. Do we need a clean slate?
Let’s
roll back to the time when de Gennes started: how can we consider, differently, the statistics of
interactions of the macromolecular chains? After all, if the concept of macromolecule is
not challenged, do we not need to select a single chain as our statistical
system? Like we always do in physics,
once the properties of the system (here the single chain) is properly
described, after accounting for the presence of the other chains that perturbs
the property of a single isolated chain, we can extrapolate to the whole set of
chains. This is the model that the reptation model proposed and elaborated for
40 years.
The
clean slate that we actually need, in order to reformulate the interactions
between the macromolecules, is not a small matter; in my opinion, it also bears
general consequences in the way we should view interactions in physics, I mean
statistically speaking. The Boltzmann’s
kinetic theory of gases inspired all current statistical models of the steady
state of the interactions between a large set of units, molecules in the case
of Boltzmann. This involves the description of a mean field calculated from the
energy distribution function and it also involves a close statistics, where the
canonical ensemble is well defined and constant.
Should
the Boltzmann’s assumptions be put into question, challenged and perhaps even
considered as THE problem to solve?
If
it is the case, not only do we need a new renaissance in polymer physics, but
perhaps even, more generally, in physics.
You
may not have recognized the man in the picture at the forefront of this blog. He
is not as popular as Feynman, but perhaps should be. He is Professor Ilya
Prigogine (1917-2003), Nobel Laureate in 1977 and author of several statistical
books on the dynamics of “dissipative structures”[1],
[2].
I
happen to use almost the same words: elastic dissipative wave[3],
dissipation energy, vertical structuring due to the minimization of the
dissipative term, etc. in the description of the Grain-Field Statistical model
of the interactions applied to polymers[4]. There is no doubt in my mind that the essence
of my work on dissipation, started independently from Prigogine, has great
resonance with what Prigogine has most brilliantly elaborated. Perhaps luckily, though, I was not inspired by
that work (at the time) and developed a different mathematical formulation of
dissipation, possibly more adapted to the case of interactions between
macromolecules, moreover mathematically simpler to apply, it seems.
I
am not going to elaborate these statements in a blog post, but since I have
been working on those issues for the last 5 years and am ready to send to
publishers a couple of books developing these new ideas on the subject[5],[6],
I propose to “avant-premiere” a
selection of general paragraphs from the books to illustrate what I have
concluded and the questions that remain open.
A. Excerpts from
Prigogine1:
“We
believe that we are only at the beginning of a new development of theoretical
chemistry and physics in which thermodynamics concepts will play an ever
increasing role.”
“In
the classical theory of integrable systems, which has been so important in the
formulation of quantum mechanics, all interactions can be eliminated by an
appropriate canonical transformation. Is
this really the correct prototype of dynamic systems to consider, especially
when situations involving elementary particles and their interactions are
considered? Do we not have first to go to a non-canonical representation which
permits us to disentangle reversible and irreversible processes on the
microscopic level and then only to eliminate the reversible part to obtain well
defined but still interacting units?”
“A
general feature of interest is that dissipative structures are very sensitive
to global features which characterize the environment of chemical systems, such
as their size and form…, “
“For
example, the occurrence of dissipative structures generally requires that the
system’s size exceeds some critical value…”
“…It
is precisely because of inequalities (3.2) and (3.4) that d2S is a Lyapounov function. Its existence ensures the damping of all
fluctuations. That is the reason why near equilibrium a macroscopic description
for large systems is sufficient. Fluctuations can only play a subordinate role,
appearing as corrections to the macroscopic laws which can be neglected for
large systems.”
B
Excerpts from my books in footnotes 3, 5 and 6 (polymer interactions):
“…In
our view, “conformers”, the constituents of the macromolecules, gather into
statistical systems which go beyond belonging to individual
macromolecules. A conformer is shown in Figure 7-1, duplicated from Ref.
276. The macromolecules themselves
represent a chain of "covalent conformers" put together as an
entity. The problem is to determine
whether the chain properties, derived from its statistics, control entirely the dynamics of the
collection of chains making up a polymer. This is what has been assumed by all
the other theories, and this is what the Dual-Split kinetics and the
Grain-Field statistics challenge…”
“…to
simplify, one could view the difference between our statistical model and the
classical model to describe the properties of polymers as follows: according to
the classical views, the statistical systems are the macromolecules, i.e. a
network of chains; the properties of the chains are disturbed by the presence
of other chains and by the external conditions (temperature, stress tensor, electrical
field, etc.). This classical definition
of the statistical system contrasts with our approach where the statistical
systems are the “dual-conformers”,
not the macromolecules, assembled as a network of dual-conformers. The
interactive coupling between the dual-conformers is defined by a new
statistics, the Grain-Field Statistics, which explores the correlation between
the local conformational property of the dual-conformers and their collective
behavior as a dissipative network…”
“… the statistics that are used by the classical models
and by our model to describe the RIS (rotational isomeric states) of the
conformers are fundamentally different: the classical molecular dynamic
statistics is the Boltzmann statistics, famous for its kinetic formulation of
the properties of gases. The Dual-Split or Dual-Phase statistics, leading to
the Grain-Field Statistics, is inspired by the classical Boltzmann concept but
departs from it by defining a dissipative term in the equations and assuming
that the Free Energy remains always equal to its minimum value, that of the
equilibrium state, even for transient states. The kinetics created by such
changes in the fundamental equations result in the formation of Free Energy structures,
which we have once called “the Energetic Kinetic Dissipative Network of
conformers (EKNET)” ([265] to [270]) and
more recently, while dealing with rheology “ the Elastic Dissipative Network”
([276], [283a])…”
“…In our analytical formulation of the dynamics of these
“open dissipative systems of
interactions” generated by our two modifications of the classical
formula, we realized that essentially two mechanisms of structuration of the
Free Energy prevail and compete: a “vertical structuring” and a “horizontal”
structuring, each specifically applying its own version of the basic equations.
This distinction increases the complexity of the analytical solution but is, in
our opinion, a fundamental aspect of the way interactions work. The vertical
structuring refers to a split of the units (collectively interacting in the
system) into 2 compensating sub-systems having each a different statistical
partition. The horizontal structuring offers a different split of the
collective set, via the generation of Ns identical sub-systems, each with the same
statistical partition. Each split mechanism generates a dissipative function.
The total dissipative function ought to be minimized (it is 0 at equilibrium),
a condition that creates their compensation, i.e. whether they work
independently, in sequence or together…”
“…the details of the simulations performed using this
model of polymer interactions shows that there is a temperature that we
associate with TLL, that is the dynamic transition temperature beyond
which the classical molecular models based on the Boltzmann statistics and our
open dissipative network model are compatible and coherent. This stipulates
that classical molecular theories of polymeric materials will provide the same
results as our model for T > TLL, a temperature at which the
dissipative function kinetically collapses.
Below TLL, the behavior that results from the interactions is
dominated by the statistics of the Dual-Phase and Cross-Dual-Phases; thus,
below TLL,
the macromolecular aspect does not statistically dictate the properties. The
projections of these macromolecular statistical models (reptation, for
instance) are physically unfounded below TLL, in our opinion, which
explains their failure to describe the experimental results under those
conditions (Ch. 7 of [276], [283b])...”
“…If one tests the predictions of the classical approach
under conditions that bring its state above TLL, one may conclude
that those data validate the classical views since they provide correct answers
in the range tested. This is not an easy task, because TLL is rate
dependent, pressure and shear dependent and molecular weight dependent. Thus,
although one will find in the literature convincing experimental evidence of
success for the classical models, which is the reason for their acceptance, we
claim that these successes are due to the use of conditions that bring the
state of the polymer above its TLL transition…”
“… in most experimental set-ups used by the industry, TLL is raised to such high values, due to the high
rates and pressures, that the range of validation of the classical macromolecular
dynamics to predict the properties is in default: the use of such classical models
in such conditions provides the wrong answers….”
“…we have advocated elsewhere (Ch. 7 of [276]), to
abandon the classical interpretation of polymer physics by single chain
macromolecular dynamics because of its inability to describe the full range of
its behavior and other essential properties of polymers such as the dielectric
TSD/TWD responses analyzed in [283f]…”
“…we propose that the mechanism of relaxation, in
polymers, is due to the dynamic coupling of two types of splitting processes of
the total statistical population of conformers in interactions: the creation of
Ns(t)
Energetic Kinetic systems (horizontal splitting) and the modulation of the
conformational structure of these systems by the dissipative function (vertical
splitting)...”
“…It might be more appropriate to categorize TL,L as one of the kinetic manifestations resulting
from the cooperative kinetic process already giving rise to Tb, ,Tg and Tg,ρ. Beyond TL, L,
the organization of the inter-intra molecular interactions between the various
dipoles as a dissipative network is kinetically inefficient, hence has
ended. As we said earlier, a description
of the properties of the polymer by invoking the properties of the individual
macromolecules embedded in a mean field is acceptable from this point on…”
C. Excerpts
from the book in footnote 4 and in Vol. I of the book in footnote 6
(Grain-Field Statistics):
“…The
study of kinetics is a discipline that describes the evolution of the units of
a population of, say, chemical molecules that participate in chemical
reactions. Another example would be to describe the evolution of units of a
population which could occupy different “states”. Many other terms have been
used to describe the same objective: “statistics”, or “dynamics”, for instance,
as shown in the following definitions: the population partition that evolves
with time can be studied with the tools of “statistics”, a transient statistics
in fact, a field also regarded as “dynamics”.
All these definitions are used in our presentation. The important thing
here is to define the terms quantitatively...”
“…Can we modify the set of equations driving the kinetics so the system
Free Energy stays at its minimum value at all times? The Dual Split Kinetics model describes new
sets of kinetic equations fulfilling these conditions. There are two types of
solutions that we have studied, vertical and horizontal splitting, and several
possible hybrid combinations of the two…”
”…In this section we present the assumptions driving the new non-equilibrium
statistics and study the difference between its results and results obtained
classically. The new equations converge to traditional kinetic equations at
long times or under "true" equilibrium conditions. Under
non-isothermal conditions the system becomes self-dissipative, and the duality
is responsible for a structure of the Free Energy…”
“…Note the presence of an additional term, Ln (Nb/Nf), in the expression of the Free
Energy. This function is what we designate the
"dissipative term"… “Its introduction is fundamental in our work
on interactions; it is the source of the originality of the new statistics and
results in the study of a new generation of dynamic open-self-dissipative
systems…”
“…In summary,
simple relationships between Lnux, Δx and Δe exist which are revealed by varying Δe in Eqs. (5) to (7). The vertical splitting
kinetics is, on its own, powerful enough to simulate the effect of activating
the dipoles (permanent and/or induced) at the polarizing temperature Tp, and observing
its thermally activated depolarization as a Debye current...”
“…We now imagine solutions that
combine the Vertical and Horizontal Dual Split Kinetics to simulate the
dynamics of statistical units in interactions… we just want to illustrate one
of the solutions of the Grain-Field Statistics that we have explored
extensively to simulate the thermal properties and the rheology of polymers.
More generally, the description of the several combinations possible and their
simulation constitutes a vast and fascinating program of investigation.
Additionally, among the various solutions, the challenge is to recognize what
combination could possibly correctly simulate the specific interactions in a
given field of the physics of interactions, not just polymer physics…”
“…In each of the combinations of
Vertical and Horizontal structuring mentioned above, we are dealing with auto-generated open dissipative systems
driven by the energetic kinetics assumptions, i.e. by solutions of the
structure of the Free energy so that the
minimization of the total Free energy, for the collective set, always remains
equal to the equilibrium value at that temperature. An open system occurs when
its total population is not kept constant. Such a system compensates its
openness by modulating the structures of its Free energy at various levels
(scales)...”
“…Figure B-21(below) provides
another interesting perspective of the open systems dynamics when their total
population, Bo,
is submitted to a fluctuation around
a mean value. The system is again defined by Eq. (12), but
we now suppose that the value of Bo varies with time like a sine wave as it is
cooled at rate q=-1 oK/s:
Bo= A+B sin (Ct)
and T=To-
qt. A=1000, B= 100 are used in Fig.
B-21.
The
problem is more complex to resolve numerically, but is still quite tractable.
The solution for Ns (T) is plotted on the same graph as Bo (T) for comparison. As the temperature
decreases, from right to left on the figure, (Bo(t) is represented by the squares, and
Ns(t) by
the open dots.
We see in
Fig. B-21 that Ns varies periodically but nothing like a sine
wave.
What is
interesting, however, is to follow the period of the oscillation and the
maximum and the minimum of the amplitude of Ns (T), from right to left: The amplitude of Ns has difficulty to rise up beyond its initial
value at high temperature, showing a slow increase of the peak maximum value,
and a strong non-linear oscillating systems, with lots of harmonics. . Beyond
the 4th oscillation of Bo(t), however, the peak maximum value increases
rapidly and levels off to a plateau value, and the oscillation becomes cleaner,
indicating the muting of many harmonics. The
other remarkable feature is the period doubling of the oscillation at lower
temperature. At high temperature, the period of Ns(t) and Bo(t) are the
same, but as we cross the transition temperature on cooling, the period of Ns(t) becomes
twice the period of Bo(t), a phenomenon observed for “time crystals”, for
instance.
This
Horizontal dynamic system is capable to produce a transition separating two
temperature regions presenting very unique characteristics in material physics
[283d]...”
Figure B-21
“…This
challenge of finding the correct combination of horizontal and vertical
splitting to describe interactions requires a dedicated book [283d]… An
important chapter of that book is titled: “The
Dynamics of Open-Dissipative Systems of Interactions: the Question of their
Finitude and Stability”. This
chapter, whose title, obviously, resonates with the work of Prigogine in the
seventies [500a], although it totally differs from it for its mathematical
treatment, should lay down the basic map of what needs to be resolved by the
next generation of research scientists (interested in these solutions) to
better understand how the local and the global interactions structure one
another to generate transient and steady state events in time-space, i.e.
describe what we call reality (Time vs Space, Matter vs Vacuum). This would
offer new perspectives on the description of the different types of
interactions, and of their unification as solutions of the network of open dissipative
cooperative systems. The 1st
Renaissance in physics, from Newton to Einstein for gravitational interactions,
to the Standard model of interactions to describe the infinitely small, may
have exhausted its resources to complete a grand unification. Should the 2nd
Renaissance, inspired by Prigogine [500b], now take the baton?...”
Jean
Pierre Ibar
New
School Polymer Physics
jpibar@eknetcampus.com
September
14, 2020
Blog
Post #36
[1]
Prigogine I., “Time, Structure and Fluctuations’, Nobel
Lecture (1977).
[2] Prigogine
I., Nicolis G., “Self-Organization in Non-Equilibrium Systems”, Wiley (1977)
ISBN 0-471-02401-5. Also:
Glansdorff P., Prigogine I.,”Thermodynamics of
Structure, Stability and Fluctuations”, London, Wiley-Interscience (1971)
[3] Ibar
J.P., “Physics
of Polymer Interactions. A Novel Approach. Application to Rheology and Processing”,
Hanser, (2019).
[4]
Ibar J.P., “Grain
Field Statistics Applied to Polymer Physics”, book in preparation.
[5] Ibar J.P., “Application of the Dual-Phase and Cross-Dual-Phase model of Polymer Interactions to the Rheology of Polymer Melts. Temperature and Molecular Weight Dependence: A New Approach.”, book, submitted for publication.
[6]
Ibar J.P., “Interactive
Coupling in the Amorphous State of Polymers”, books, Vol. I and II, Accepted for publication
