Evidence of a one-dimensional thermodynamic phase diagram for simple glass-formers

Glass formers show motional processes over an extremely broad range of timescales, covering more than ten orders of magnitude, meaning that a full understanding of the glass transition needs to comprise this tremendous range in timescales. Here we report simultaneous dielectric and neutron spectroscopy investigations of three glass-forming liquids, probing in a single experiment the full range of dynamics. For two van der Waals liquids, we locate in the pressure–temperature phase diagram lines of identical dynamics of the molecules on both second and picosecond timescales. This confirms predictions of the isomorph theory and effectively reduces the phase diagram from two to one dimension. The implication is that dynamics on widely different timescales are governed by the same underlying mechanisms.

makes the test much more rigorous. There have been previous indications of connections between short time and long time dynamics, but the present demonstration is a substantial step forward. This is a very important paper and I support it's publication in Nature Communications after the authors have had an opportunity to consider changes in light of my comments below.
1) The introduction makes clear the when the isomorph theory is applicable that there is also a relationship between the structures at the two state points. This is not tested in the current work, as far as I can see. I am surprised that they did not also use neutron scattering to look at structure. The authors should comment on this. 2) I am struggling a bit with the idea that the vibrational dynamics will also have a simple relationship at the two isomorph state points. I accept the authors view that they have shown that this works for some vibrational properties in this manuscript. Molecular systems have internal vibrational modes -are the authors claiming that these high frequency modes also obey the isomorph relationship? If not, could the authors clarify what types of vibrational processes are relevant for isomorph theory and how they know that they are looking at this type of vibrational mode? I am concerned that someone coming to the paper from outside the glass community will see molecules and think that "vibrations" mean something different than the authors intend.
3) There are various mentions of IN6 and IN 16. I wonder if one of these is a typo? I think only one is described in the experimental details.
Reviewer #3 (Remarks to the Author): I enjoyed reading the manuscript and liked the way the Authors wrote it. For the first time simultaneous neutron and dielectric spectroscopy investigations of three different (both van der Waals and H-bonded ) glass-forming liquids in T-P thermodynamic space were performed. It means that in single experiment fast and slow dynamics was simultaneously probed. Based on these measurements, the Authors proved experimentally that isomorph prediction also works for fast dynamics. It is very fundamental finding! In my opinion this manuscript should be definitively published in Nature Communications.

Reply to reviews on manuscript "Evidence of a one-dimensional thermodynamic phase diagram for simple glass-formers"
First of all we would like to thank all referees for their positive reviews and for very useful comments and suggestions. Below are our replies to the specific comments from reviewer 1 and 2 (reviewer 3 had no comments) including details on how we have changed the manuscript.
Replies to reviewer 1, 1. The reader gets stuck at the introduction of the scaled omega units (first paragraph page 5). Did you not achieve the same tau_alpha experimentally? And now you suddenly change the time scale? Here you need to give numbers for the scaling factors. And you need to explain that these scaling factors are practically irrelevant for the determination of the glass temperature.

Changes made
The section about reduced units has been moved from supplementary material into the Methods section, which is part of the main paper in the format of Nature Communications. The first time we mention reduced units we refer to this section. Moreover, the section on reduced units has been extended to include a paragraph on how it affects the alpha relaxation time as determined from the dielectric spectra. In total the section now reads: According to isomorph theory, the relevant scale to look at is in reduced units 11 . The reduced energy units used in Fig. 2 and Supplementary Information Fig. S2 are given bỹ where ω is the energy transfer which is equivalent to the frequency except a factor of . k B is Boltzmann's constant and T is temperature. Here, ρ is the number density and m is the average particle mass, the latter assumed constant. We set all the constants to one since these do not affect the scaling, = m = k B = 1. Effectively, this becomes where ρ is now the volumetric mass density.
Just like reduced energy units, wave vector or momentum transfer Q should also be presented in reduced units: But as the density changes are in the percent range in this study, scaling of Q will be around 1% and will be within the uncertainty of the data and is therefore neglected.
1 When we find the glass transition isochrone, this should also be done from a reduced unit isochrone. However, the difference between the scaling factors from atmospheric pressure to the high pressure state point is for all three samples 1.2 or less, this corresponds to a relative shift of the dielectric curves of less than 0.1 decade. In practice, this is less than our precision in pressure allows us to fix.
Plotting Fig. 3c as a function of the alpha relaxation time on a reduced timescale instead of on an absolute timescale does not change the visual look of the figure. Fig. 3c is therefore plotted on an absolute timescale, because this conveys more information about the state of the system.
2. In the middle of page 6, you conclude that " the superposition observed in the vdW-liquids is a genuine signature of the isomorphs in these liquids." Right, this is the main conclusion of the paper. But here is an opportunity to discuss the physics of the failure of the isomorph theory in dipropylene glycol, a molecule with two hydrogen bonds, which fix two of its three degrees of freedom. This leaves room for one vander-Waals vibration per molecule. It seems that these van-der-Waals vibrations tend to concentrate in the boson peak, similar to the case of selenium (Phillips et al, PRL 63, 2381(1989) making it more pressure-dependent than tau_alpha, which in its turn requires the breaking of hydrogen bonds. If I am right, the Gruneisen parameter of the boson peak should be approximately the same as the one in the two other substances. The authors could check that from the existing data.
This comment has inspired some interesting discussions in our group and we do agree that there is more physics to be extracted. We look forward to continuing the discussion.
One of the aims of the further work in the theoretical part of our group is in fact to work systematically with coarse-graining in order to establish pseudo-isomorphs, which are lines where the dynamics of certain degrees of freedom are invariant while other degrees of freedom are not invariant. This has work has been started in a recent paper [J. Chem. Phys., 145, 241103 (2016)], where it is shown how intramolecular modes can be handled in computer simulations of simple molecular systems.
Regarding the specific idea of looking at the Gruneisen parameter of the boson peak, it is not simple to access. The Boson peak is not well resolved in the liquid, it can barely be separated from relaxations at T g . Therefore one would have to look at the Gruneisen parameter in the glass -however in the glass we do not have PVT-information and we do not know what the density changes of the sample is. Moreover, the while the Gruneisen parameter of one system should be the same for different modes, we do expect it to differ from system to system.

Changes made
We have expanded and restructured the discussion inspired by the comments of the reviewers.
3. From there, it would be appropriate to point out that the application of the isomorph theory to glass transition experiments requires the same Gruneisen parameters for all relevant degrees of freedom. The example of the two van der Waals substances shows that the rather harmonic high frequency molecular vibrations are irrelevant.
This comment also instigated a series of discussions in our group in Roskilde. It touches upon some of the very fundamental aspects of isomorph theory. Yet, we believe that entering a discussion of the Gruneisen parameter will be too specialized for this paper. Especially because this point has not really been discussed in any previous theory papers that we can refer to.
Regarding the high frequency molecular vibrations, we agree that the scaling we find, at least the interpretation in terms of isomorph theory, implies that the boson peak and the fast relaxation we present are unaffected by intra-molecular modes. In fact, we believe that the fast dynamics of the van der Waals bonded liquids is governed by the shape of the potential energy surface which also governs alpha relaxation. This again relates to the recent JCP paper mentioned above [J. Chem. Phys., 145, 241103 (2016)] and we have added some discussion of this in the paper.

Changes made
We have expanded and restructured the discussion inspired by the comments of the reviewers.

Replies to reviewer 2
1. The introduction makes clear the when the isomorph theory is applicable that there is also a relationship between the structures at the two state points. This is not tested in the current work, as far as I can see. I am surprised that they did not also use neutron scattering to look at structure. The authors should comment on this.
This is a very good point. The challenge is that the structure factor changes little with the pressure and temperature changes that correspond to changes the alpha-relaxation with orders of magnitude from one end of the dielectric range to the other. This is particularly true when recalling that isomorph predictions concern the behavior measured in reduced units, meaning that the trivial change in structure factor due to density has to be scaled out before comparing state points. The dynamics measured in S(q, ω), on the other hand, changes very clearly in the same P, T-range. Showing that structure is invariant along the isomorph is only interesting if it is NOT invariant in the rest of the phase diagram. This requires very high precision on the S(q) measurements. Nevertheless this is something we hope to pursue in the future. A technicality is that the cell will also need to be modified -as one would ideally have more sample in the beam for S(q) measurements. An alternative route that we are also considering is moving to X-rays. However, working in the relevant temperature with heavy cooling equipment and with pressure cells is easier with neutrons. In either case, the structure studies are beyond the scope of this work.

Changes made
We have added a comment regarding this point in the method section.
Dynamics studied with neutron spectroscopy is well suited for a test of isomorph theory as there are known to be large changes in the dynamics on sub-nanosecond timescales for the same range of density and temperature changes as those that change the alpha relaxation time with some orders of magnitude. The static structure factor on the other hand changes very little, especially when plotted in reduced units.
2. I am struggling a bit with the idea that the vibrational dynamics will also have a simple relationship at the two isomorph state points. I accept the authors view that they have shown that this works for some vibrational properties in this manuscript. Molecular systems have internal vibrational modes -are the authors claiming that these high frequency modes also obey the isomorph relationship? If not, could the authors clarify what types of vibrational processes are relevant for isomorph theory and how they know that they are looking at this type of vibrational mode? I am concerned that someone coming to the paper from outside the glass community will see molecules and think that "vibrations" mean something different than the authors intend.
This is an important point and it relates to the comment made by reviewer 1, that more information could be extracted. It is intermolecular vibrations we have in mind, phonons, local modes, or what is some times referred to as cage rattling.

Changes made
We have expanded and restructured the discussion inspired by the reviewers comments.
3. There are various mentions of IN6 and IN 16. I wonder if one of these is a typo? I think only one is described in the experimental details.
We thank the reviewer for pointing out that this was not clear. In fact data from IN16 are only used in Fig. 1 Fig. 1a and b for PPE. The center panels of (a) and (b) are sketches of the incoherent intermediate scattering function, I(Q, t), while the top and bottom panel show raw data. At T g (Fig. 1a), no broadening is observed on nanosecond timescales (IN16) corresponding to a plateau in I(Q, t), on picosecond timescales from IN5 we observe contributions from fast relaxational processes and vibrations, whereas the alpha relaxation is seen in DS at much longer timescales, a difference of more than 10 orders of magnitude. As the temperature is increased, the processes merge (Fig. 1b), and relaxation dominates the signal in all three spectrometers.
The focus in this paper is on the picosecond dynamics measured on IN5 and IN6, while IN16 data are only used as an illustration in Fig. 1.