No relation.

It is actually a debated matter, whether the glass transition is, or is not a "phase transition" -- and generally, what kind of glass transition one talks about (because there are subtle issues and differences between the thermal glass transition, when the molecular motion becomes restricted - non-ergodic, and the dynamic glass transition when the glassy restrictions arise at shorter time scales -not to be mixed with jamming). Also, to be a "phase transition" there has to be an order parameter identified and for the structural glass like SiO2 it's not an agreed matter (certainly no change of symmetry).

The structural glass is actually related more to the topology of bonds (both covalent and physical), rather than their specific strength.

I am no expert of the glass state, but I would clearly refuse the idea of a nano-polycrystalline material. The common idea of a glass is that of a supercooled liquid, which is kinetically hindered to fall into the low energy crystalline state.There is no evidence from diffraction experiments that there is a nanoscale crystalline order. Do you mean that there is a finite amount of bond energies in a crystal due to the fixed bond distances, but a large spectrum of energies in the glass due to the disordered arrangement? E.G., there has been found an atomic near range order in some binary alloy melts (NiZr) with definite coordination polyhedra, but this looks more like a macromolecular flexible arrangement to me. The transition from melt to glass lacks a symmetry breaking, typical for solid-liquid transitions of 1st or 2nd kind.

I know many of the present theories about glass transition deal with the topology of bonds in glass (but that asks the question: how does glass transition occur). In fact, there is a short range order in glasses:the order parameter could be symmetry but at this scale (of the short range order). And the integration over temperature of all these nanoscale phase transitions would lead to a 'smooth' thermal glass transition (that asks the question: why does glass transition occur).

I'd say no. Glass transition occurs in many materials with very different bonding properties - one can have glassy metals, and there is a belief that potentially ANY material can become a glass if cooled down "fast enough". Despite the lack of a deep understanding of the matter, glass transition is usually recognized as a dynamical phenomenon, a "freezing" of the motions which prevents the system from reaching its condition of thermodynamic equilibrium. There are theories about the glass transition being more fundamental than that, and having an ideal 'infinitely slow cooling' thermodynamic limit, but it's not widely accepted yet.

EDIT: I just remembered this, but I am pretty sure there was research which showed the presence of nanoscale ordering in some glasses. The idea was that this could possibly be applied not just to silicate glasses, but have some relevance to the general phenomenology of glass transition. I don't remember the details - the basic idea is that there is evidence for some quantum phenomena in glasses which can't be explained without some small scale crystalline ordering. This paper should be part of the work: http://arxiv.org/abs/1301.7233, Giancarlo Jug is the researcher I know for sure has been working on the topic for quite some time now.

Glass transition is more similar to a dynamical arrest than a phase transition. In a phase transition a change of entropy (continuous or discontinuous) occurs; in a glass transition, according to my knowledge, order state is unchanged and only characteristic time of motion is affected.

I think it is unlikely that the covalent bond strength is an issue as glass transitions occur for many systems and weather they is a phase transition involved or not is actually debated. The position of the glass transition depends on, for example, the cooling history of the material, so it may be considered to the locking in of thermodynamic parameters that did not have time to equilibrate during the cooling process, because of the high viscosity of the glass. But the debate is weather there is a phase transition underlying the glass transition that would be evident if there was an infinite cooling time. Since bulk properties such as viscosity may be dominant with finite cooling times then I would think that it is intermolecular interactions and how the parts of the amorphous structure are interacting with neighboring parts of the structure that influence the glass transition, rather than the covalent bond strength.

All of you talk about how the glass transition occurs. But, (read Simone's post) if there is a nanoscale ordering in glasses, the strength of the covalent bonds could be an explanation of why the glass transition occurs. Indeed, if glasses have a nanoscale ordering, they are polynanocrystalline. And, for each nanoscale ordered domain there would be a phase transition. Thus, for each domain there is a leap in the entropy as a function of temperature. If you integrate over the whole sample of glass and over temperature, all the leaps in the local entropy result in a smooth variation of the overall entropy.

And, that is an explanation for any glass, any "history" of the glass etc.

That is an interesting point you make there. I don't think it can be quite that simple, but it is a starting point. In fact, at least in polymer glass transition study, some models rely on the assumption of the existence of a "Cooperative Rearrangement Region" - a minimum unit volume on which any relaxations or motions in the glass are bound to be correlated in some way, thus implying an existence of a form of local order, if not an actual crystal (which is straightforwardly impossible in some polymers). In general, your argument is very reminiscent of the classic entropic model by Adam and Gibbs, which however is nowadays considered somewhat simplistic, and possibly a bit wrong. Actually I have a couple publications I was co-author of on my profile which relate to an Adam-Gibbs like model developed by M. Pieruccini.

In a sense I feel like an excessively 'thermodynamical' explanation can get a bit too removed from physical reality, since there intuitively are dynamical phenomena at work in glasses - but it surely would be nice and aesthetically pleasing to have a proper model to describe glass transition in a general way instead of just admitting that, well, it kind of happens!

When I talk about polynanocrystalline glass, the 'nano'scale is about 2 to 4 Si-O bonds length. How would it be possible to use statistical physics or thermodynamics at such small scale? I make the assumption that glasses are 'crystalline' at this nanoscale, inhomogeneous at the mesoscale (different crystalline domains with different crystalline geometry linked to each others) and homogeneous at macroscopical scale. So, Simone and Weixiang, I agree with you. Raouf: as I said, average values of number of bonds and stretching force constant do not bring an explanation to the problem.

SiO2 glasses are indeed polynanocrystals. But nobody studied the Si-O covalent bonds at the scale of 1-2nm. I think that a study like that could explain that a liquid remains amorphous depending on the cooling rate.

As each one of you have addresses, the trouble stayes in the very definition of glass transition. No one believes that glass transition is a true thermodynamic transition (even if a large number of scientists propose that it could be a second order phase transition). The only safe claim that one can propose is phenomenological: when we assert that a system is experiencing a glass transition we are generically talking about a change from an equilibrium (metastable) fluid state to a non-equilibrium disordered solid-like (arrested) phase. The trouble with the locution "glass transition" comes from the hystorical work from Kauzmann which, comparing the entropy of the undercooled liquid with that of the stable solid phase, pointed out the existence of a temperature, Tk, at which the entropy of the liquid becomes lower than that of the solid. The extrapolation of this result towards absolute zero temperature leads to a thermodynamic paradox because the second law of thermodynamic seems to be violated. To solve tha paradox it was proposed that a phase transition takes place at the temperature Tk. Many theories of undercooled liquid and glass transition are based on this argument, including the famous and widely adopted theory from Adam and Gibbs. Unfortunately, the Kauzmann paradox is a false paradox becaouse Kauzmann completely disregarded henthalpy besides entropy. The solution stayes in the trivial observation that no isothermal transformation can bring your undercooled liquid to the solid phase. This is quite obvious because the coexistence of liquid and solid phase can occur only along the coexistence line in the phase diagram. Practically, when the undercooled liquid escapes from the metastable phase to the stable (crystalline) phase, the change takes place on so short time that it can be assumed an isoenthalpic one. As a consequence, the stable phase towards which the underccoled liquid will evolve is a mixture of liquid and solid at the coexistence temperature. If one compare the entropy of the metastable liquid and of the stable mixture one immediately see that the system ever proceeds from a lower entropy state to an higher entropy one. No paradox exists, so the main argument at the basis of the idea of glass transition are cancelled. In summary, there is no way for defining a glass as a metastable state: it is just a non-equilibrium state.

This long discussion is just an introduction to the problem. I will stop here. I simply would suggest to give a look to the paper from Hoffmann, "Energy and entropy of crystas, melts and glasses or what is wrong in Kauzmann's paradox?", Mat.-wiss. u. Werkstofftech. 2012, 43, No.6. Probably also a recent paper from myself and coworkers can be of some help: F. Aliotta et al. J. Chem. Phys. 138, 184504 (2013).

A recent and fine review of the state of art with glass transition can be found in G. Biroli et al., J. Chem. Phys. 138, 12A301 (2013). You will see that glass transition is observed in hard sphere systems too, which should give an answer to your primary question

The discussion about the glass transition is very interesting and open. I am a heretic who believes that the glass transition is a true thermodynamic transition. It is due to instability of the liquid which is changing its free volume with temperature. We have fragile-liquid-to-less-fragile-liquid or fragile –to-strong –liquid and strong-to-stronger-liquid transition as in the case of SiO2. The equilibrium transition would produce a latent heat in the first case and a transition without latent heat in strong liquids. The specific heat is equal in many fragile liquids to 1.5×Sm where Sm is the fusion entropy. In strong glasses, the numerical coefficient tends to zero when the free volume disappearance temperature decreases ( T0VFT) and Cp(Tg) is always proportional to Sm. The enthalpy change, and the relaxed enthalpy can be predicted and Cp(Tg) can be calculated from the derivative of the enthalpy. The agreement with many experimental results tends to prove that the reversible transition ( see papers of Cernosek and Holubova who have shown 12 years ago that the reversible vitreous transition only depends on the material composition) occurs even if the latent heat at the transition depends on the relaxation temperature preceding the measurement by heating.

Your question about the covalent bonds is very important because the entropy Sm strongly depends on the presence of bonds. The entropy of many normal metallic liquid elements such as copper, gold and silver is equal to 9 J/K/mole. In many glasses, Sm is much smaller because the bonds are still acting at the melting temperature. In SiO2, Sm per gram atom is very low and equal to 1.49 J/K using the measurement of Richet and Bottinga. The bonds are very important.

They certainly have an influence on the viscosity of the melt and could change the homogeneous nucleation temperature of the vitreous phase if you admit my point of view of the existence of a phase transition at Tg.

I agree with you that there are nanoclusters ( superclusters and inter penetrating clusters with magic number of atoms from recent papers) which probably act as growth nuclei of the vitreous phase.

I can agree with Robert Turnier that the question if the so called glass transition is a true (second order) thermodynamic transition or it has just a kinetic character is still an open question. I can agree also that the entropy of a glass and/or an undercooled liquid depends of the existence and nature of inter-molecular bonds. But I insist that the simple comparison of the entropies of the two states is misleading, because enthalpy is different too. We have to remember that the glass transition temperature is a vague concept. Never a well defined temperature is revealed. At best, one can detect that the transition takes place within a less or more defined range of temperature, depending on the way the transition is approached. It you imagine to estrapolate the entropy of the undercooled liquid below the glass transition you observe that the entropy of the glass is ever high that the entropy of the undercooled liquid. This is not surprising, because the glass is the undercooled liquid where the configurational entropy has been frozen (within the time scale of any accessible experiment) at the glass transition. Under further cooling only vibravional entropy decreases (with a slope close to that corresponding to the crystalline ordered stable phase). And, at a given undercooling temperature, the entropy of both the extrapolated supercooled liquid and of the glass will be both lower of the only one corresponding stable phase (which is a mixture of liquid and solid at the coexistence temperature, never it can be the solid at the undercooling temperature!). You have to imagine the system close inside an adiabatic container and you immediately will see that any spontaneous transition have to be an isoenthalpic transition). We have to realize that we are talking about transiotion from metastable state (hypothesized metastable in the case of the glass) to a stable one. So the transition must occur spontaneously (within a given time, it does not mutter how long it is) with no need of any exchage of heat with a reservoir. This immediately shows why it is wrong to talk ebout equilibrium transition. The processes we are talking about are intrinsecally irreversible, out of equilibrium processess.

Besides this, which can be argument of discussions since it is quite clear that the problem is far to be well defined, the answer to the question from Nathalie is clear and it is NOT. There is no relation between the energy of the covalent bond and the fact that glass transition is not (or is) a phase transition. Any process we are dealing with is driven from phenomena that are started in the undercooled liquid regime, before the glass transition is reached. The undercooled liquid is by itself deeply different from the normal liquid because undercooling results in a large increase of the local structural fluctuation. As the result lowering the temperature produce a huge broadneing of the local structure distribution and of the distribution of the corresponding relaxation times, which eventually leads to the break.down of the Stokes-Einstein relation. So it becomes clear that the relevant phenomena takes place on increasing correlation distances and correlation times, so becoming meso-scopic (or macroscopic) in character. In this perspective, should be clear that the details of the local potential become not-relevant because the general behaviour can be explained only at a coarse-grained level. The details of the local potential will play a role for sure in determining some of the peculiar properties of the system (e.g. the temperature at which the glass transition is observed) but the glass transition will be observed (given suitable pressure and temperature conditions) in systems characterized by hugely different local potentials, i.e. it is an emerging behaviour.

So, Francesco, the question which is arising is: can we model the glass transition with thermodynamics or statistical physics?

Your question is a good question! But the answer is not so simple. I would be attempted to suggest a statistical Physics approach. But you should take care of the ensemble that you adopt in your calculation. I means, usually people adopts isothermal approaches at fixed volume. Practically they reduce to the Adam and Gibbs approach (or to more modern modification of that approach). But this kind of approaches can be misleading. The trouble is that any isothermal description assumes that any transition (including the transition from the metastable to the stable phase) takes place isothermally. This is intrinscally wrong because (being the physical process intrinsecally non-isothermal) implicitely assume that any exchange of heat from the resevoir and the sample volume takes place istantaneously, which implies an infinite thermal conductivity of your sample. This means that the most correct description should be an isoenthalpic process at constant pressure. I am not a theorician, but I am afraid that this would face you with the trouble of counting the states. Probably a statistical physical approach which describes how the kinetic reaches the structural arrest can be adequate. But you have to take into account that, if you wish to controll if the system reaches the structural arrest or, on the contrary, escapes to the stable face you will be faced with higly not linear equations. This means that you have to well define the specific question you are interested in, to make some assumptions (probably arbitrary) and than to carefully define the limit of the validity of the outcomes from your calculation. As I said, I am an experimentalist but I imagine that if you use as a reference the paper by Biroli et al. that I suggested you in my first answer (and many of the references therein) you can compare the advantages and the drawbacks from the different proposed approaches and select the more appropriate for your aims. Any thermodynamic approach, even if more intriguing, seems to me hugely improbable. The trouble is that you have to compare states at different entropies (the glass, the supercooled liquid and the stable mixture of liquid and solid) which are states at different free energies. Really, each of these states belongs to a different free energy hypersurface so any comparison of the entropy or of the free energy alone is not enough to extract any conclusion. The situation can be even more complicated if you takes into account that the definition of the glass phase as a thermodynamic metastable phase is not clear. I do not believe that a glass is a metastable phase. It should be considered more as an out of equilibrium state. But even if you would assume glass metastability (in absence of any well stated theory this can be just question of opinion) you should define respect to which state it is metastable. In summary, I would suggest to use a statistical Physics approach, using thermodynamic only for the definition of the true stable equilibrium state at each temperature. I hope this can be a useful hint for you.

Francesco, thanks for the hint! but I will not study glass transition. My question was only due to scientific curiosity. If I would (study glass transition), I would use AbInitio simulations on a large scale -geometrical and temporal- (presently impossible): I still believe that quantum mechanics rules glass transition (otherwise, why would so many different covalent bonds exist in glassy states that do not exist in crystalline states)...

Ok! I can agree with you that quantum mechanics can play a role in the specific case. Or if we talk about spin glasses. But from a more general perspective, it is hard to imagine the role of quantum mechanics when glass transition is observed in hard sphere systems. So it is clear that there should be pursued a coarse grain description which describes the overall scenario of the glass transition. When this will become clear the details of the microscopic interaction can be of help in describing the specific properties of a given system. But, approaching the glass transition the system (often, not ever, this can be a further interesting point of discussion) exihibits so large fluctuations that it becomes macroscopically correlated (both in time and in distance, in spite of the apparent disorder) and that is emerging is a collective behaviour. So the problem face us with chaos and complexity. Except for those systems for which the glass transition can be observed after gently cooling, in all the other cases you are bringing the system at the boundary of the metastable basin (and over). So, a crucial point is the comparison between the temperature at which the metastability limit of the underccoled liquid is reached and that at which the diffusion becomes so slow to appear (over the time scale of the experiment) arrested. In summary, a further question exists. Is the glass transition temperature a parameter which gives us information about some property of the system or is it affected by an information about the experiment?

Well, Francesco, you can mimic liquids with granular matter, but the fundamentals of each of these materials are not the same. As for glass transition.

To answer your question, I think (my own opinion) that Tg is affected by any information about the experiment.

The crystals have not glass transition. The glass transition is a cooperative movement of several molecules that at a certain temperature have enought free volumen to start that movement. This explanation regards the free volumen theory. There are other theories about the Tg that relies on the entropy and configurational entropy. The most importtant to understand is that in a crystal there is no posibility for the molecules to reach this cooperative movement. there is no freedon at all to do that. In inorganic materials like SiO2, there are two posibilities, in general, to have the SiO2 in the amorphos state or in a crystaline state. In the crystaline state, there is no Tg. But in the amorphous state, there is a Tg. The value of the Tg, the shape of the Tg and the change in Cp in the Tg depends on the movility of the chains/molecules and this depends of course of the energy of the covalent bonds. But it is more important the 3D amorphous structure of th SiO2. This structure will drive the value and behaviour of the Tg and the relaxation in the Tg.

I wrote a paper about spin glasses.

Spin glasses are paramagnetic ordering of finite block spins.

In this manuscript it is corrected from ferromagnetic ordering of finite block spins to paramagnetic ordering.

I attach my paper such as

Glass transition is also applied by similar model such as

Percolation is also applied by similar analogy such as

Percolation: https://www.researchgate.net/publication/225567464_Glass_formation_in_amorphous_SiO2_as_a_percolation_phase_transition_in_a_system_of_network_defects 

Article Glass formation in amorphous SiO2 as a percolation phase tra...