SciPost Submission Page
Spatial structure of unstable normal modes in a glass-forming liquid
by Masanari Shimada, Daniele Coslovich, Hideyuki Mizuno, Atsushi Ikeda
This is not the current version.
|As Contributors:||Daniele Coslovich · Masanari Shimada|
|Arxiv Link:||https://arxiv.org/abs/2009.07972v1 (pdf)|
|Date submitted:||2020-09-18 03:23|
|Submitted by:||Shimada, Masanari|
|Submitted to:||SciPost Physics|
The phenomenology of glass-forming liquids is often described in terms of their underlying, high-dimensional potential energy surface. In particular, the statistics of stationary points sampled as a function of temperature provides useful insight into the thermodynamics and dynamics of the system. To make contact with the real space physics, however, analysis of the spatial structure of the normal modes is required. In this work, we numerically study the potential energy surface of a glass-forming ternary mixture. Starting from liquid configurations equilibrated over a broad range of temperatures using a swap Monte Carlo method, we locate the nearby stationary points and investigate the spatial architecture and the energetics of the associated unstable modes. Through this spatially-resolved analysis, originally developed to study local minima, we corroborate recent evidence that the nature of the unstable modes changes from delocalized to localized around the mode-coupling temperature. We find that the displacement amplitudes of the delocalized modes have a slowly decaying far field, whereas the localized modes consist of a core with large displacements and a rapidly decaying far field. The fractal dimension of unstable modes around the mobility edge is equal to 1, consistent with the scaling of the participation ratio. Finally, we find that around and below the mode-coupling temperature the unstable modes are localized around structural defects, characterized by a disordered local structure markedly different from the liquid's locally favored structure. These defects are similar to those associated to quasi-localized vibrations in local minima and are good candidates to predict the emergence of localized excitations at low temperature.
Submission & Refereeing History
You are currently on this page
Reports on this Submission
Anonymous Report 3 on 2020-10-19 Invited Report
1) The authors throughly examine the spatial structure of unstable modes and provide clear differences between the unstable and stable modes.
2) The authors were able to study modes below close enough to the mode-coupling temperature to see a change of the modes.
3) The authors tested very clear hypothesis, and thus there are clear conclusions.
1) Some of the specifics were not well defined in the paper. Specifically how the mode-coupling temperature was obtained and it was unclear how the authors defined the mobility edge.
2) The systems may be too small to properly see the full real space decay of the modes.
Shimada, Coslovich, Mizuno, and Ikeda examine the structure of unstable saddle modes in supercooled liquids as a function of temperature, and compared these saddle modes with the stable modes. Since there were able to modify a previous algorithm to more efficiently find these unstable modes, they were able to examine the modes in more depth and for larger systems than in previous studies. To examine the spatial extent it is very important to be able to study as large a system as possible to limit any system size effects. However, since the spatial decay extends over half the box length in some figures, e.g. Figure 5 a, b, and c, these systems may still be too small. The authors do indicate that the system is too small to examine some asymptotic results.
The authors opened up a new pathway for future research in studying the spatial structure of unstable and stable modes, and future work with different systems, bigger systems, and at lower temperatures will eventually follow. It also opens up avenues for more detailed examination of the mode structure. The introduction is somewhat long but detailed and provides the reader with more than enough introduction to the subject, and the conclusion is clear, concise, and includes directions for future work. The description of the methods along with the citations to the relevant literature makes it possible for a researcher familiar with the field to be able to reproduce the results for this system and other systems.
1) It it not clear in the text exactly how the mode coupling temperature was determined. While it is expected to be the same as what was done it Reference 26, it would be helpful to the reader to know specifically how the mode coupling temperature was determined for this work.
2) In the discussion of the mobility edge, the reader is left to assume that the mobility edge is defined the same as in Reference 26. A simple sentence stating that this is the case would be helpful.
3) Can the authors estimate the size of the system needed to observe the asymptotic scaling expected in Figure 3?
4) As a minor point, the introduction is very long and the reader can become a little overwhelmed. The authors may consider making the introduction more focused.
Anonymous Report 2 on 2020-10-18 Invited Report
1) The manuscript is well written with clear figures and all technical details are well explained.
2) I find this article to provide interesting insights into the unstable modes in real space of finite dimensional liquids.
3) It is moreover shown that the localized unstable modes converge towards the quasi-localized soft modes commonly observed for minima as the temperature of the liquid decreases.
I therefore find that this manuscript provides a clear and interesting analysis to answer some questions raised by the complex dynamics of supercooled liquids.
1) My only downside concerns the structural analysis around the eigenmode cores. The analysis is conducted from a description of the polyhedra obtained by a Voronoï construction. I found that the latter only shows fairly small differences. It would have been interesting to carry out other structural analysis such as, for instance, the one carried out in [H. Tong and H. Tanaka Phys. Rev. X 8, 011041 (2018)] which has shown an excellent correlation with the dynamics of supercooled liquids.
This article examines numerically the spatial structure of unstable vibrational modes in glass-forming liquids. It can be seen as an extension of a first study [D. Coslovich, A. Ninarello and L. Berthier, SciPost Phys. 7, 077 (2019)] in which it has been shown that, for a variety of fragile liquids, the fraction of delocalized unstable modes goes to zero as temperature is decreased, and equal to zero below to the mode coupling temperature T_MCT. Here, the spatial structure of the unstable modes and the relations with the structure is studied in more depth.
This work uses a three-dimensional ternary system with system sizes ranging from 1000 to 3000 atoms. Sampling of dynamical configuration of liquids is performed for different temperatures using a new SWAP Monte Carlo algorithm to thermodynamically equilibrate systems below the mode coupling temperature. The eigenmodes of vibration are calculated from the diagonalization of the Hessian matrix. The analyzed configurations are obtained from the minimization of the sum of the squares of the forces in order to find the stationary points of the potential energy landscape. The minimization procedure is followed by an eigenvector following search method in order to obtain properly (i.e. order of saddle points) converged stationary points. Several indicators are studied: the participation ratio, to look at the degree of localization of modes, and the average spatial profiles of displacement and energy of each mode. The local structure is also analyzed (radial distribution functions and Voronoï analysis) and comparisons are carried out between the soft quasilocalized modes of the minima and the saddle points.
The manuscript is well written with clear figures and all technical details are well explained. I find this article to provide interesting insights into the unstable modes in real space of finite dimensional liquids. It reinforces the conclusion of [D. Coslovich, A. Ninarello and L. Berthier, SciPost Phys. 7, 077 (2019)] by showing that the saddle modes localize as the temperature drops and shows that the delocalized modes disappear below T_MCT. It is moreover shown that the localized unstable modes converge towards the quasi-localized soft modes commonly observed for minima as the temperature of the liquid decreases. I therefore find that this manuscript provides a clear and interesting analysis to answer some questions raised by the complex dynamics of supercooled liquids and deserves to be published in SciPost.
My only downside concerns the structural analysis around the eigenmode cores. The analysis is conducted from a description of the polyhedra obtained by a Voronoï construction. I found that the latter only shows fairly small differences. It would have been interesting to carry out other structural analysis such as, for instance, the one carried out in [H. Tong and H. Tanaka Phys. Rev. X 8, 011041 (2018)] which has shown an excellent correlation with the dynamics of supercooled liquids.
Beside this question, I think the authors should also answer the following remarks and questions:
- In equation 6, why use the median and not the mean. Does the presence of outliers question the analysis?
- I find the paragraph starting with
”To briefly summarize the results of Ref.  using our data, we show the scatter plots of the participation ratio in Fig. 1. We show the results at (a) T = 0.45, (b) T = 0.35, (c) T = 0.32, (d) T = 0.30, and (e) T = 0.28. ”
is not very clear and not summarizing well reference . The authors should explicitly explain that, above lamdba_e, the participation ratio tends towards zero as the temperature decreases. Additionally, I found the introduction and definition of mobility edge clearer in .
- The authors write:
”Nevertheless, we can also see a difference between the delocalized unstable modes and the QLVs”
I find this not very clear in Figure 4.
1 -In section II, part A, concerning the sample preparation, the authors explain that they have improved the convergence of the sampling method. Could the authors provide arguments to explain that the states sampled in this way are statistically representative of equilibrium dynamics.
2- Although this is a subject already treated in [D. Coslovich, A. Ninarello and L. Berthier, SciPost Phys. 7, 077 (2019)], could the authors specify what is the fraction of the delocalized unstable modes as a function of temperature?
3- In section II.B.3, concerning the fractal dimension, it took me a while to understand that the authors were considering the integer part of the participation ratio. I must admit that I was not familiar with these signs. This is explained further:
“In each box, we only show the dP (λ α) e particles having the largest norms, where [. . . ] denotes the integer part. "
I suggest putting this explanation before at the first occurrence in the text.
4- The authors introduce “e^⊥_e α,ij” as the transverse relative displacement. I think the authors have to take the square root for it to be a displacement.
5- Still regarding formulation, the authors introduce the energy profile, but shouldn't we rather speak here of the energy variation profile?
6- Table I shows the absolute values of the eigenvalues. I find this to be a bit misleading. Also, I didn't understand whether it was an interval or not.
7- The scaling laws discussed in figures 3 and 4 are based (this is also the case in figure 12) on observations covering less than half a decade on the abscissa due to the smallness of systems (as discussed in the article). This aspect of things, giving a more qualitative picture, should be discussed more in the article.
8- ”We can clearly see the cores of the localized modes modes in Fig. 2 (a) and (b), while it is difficult to identify similar cores in the delocalized modes, at least by visual inspection. ”
There is a typo in the repetition of the word “modes”.
9- For more clarity, the authors should add “saddle” and “minima” in the legend of figure 8, as in figure 7.
10- It is written :
”We found that Voronoi signatures in saddles and minima have markedly different statistics at T = 0.45.”
I do not agree. I don't find Voroinoi's signature statistics to be different so markedly. Or, the authors should explain why this can be seen as a significant deviation.
11- It is explained that:
“We restrict the calculation of the radial distribution function g (r) to central particles of species 3 that form the cores of the unstable modes, i.e., the particles whose index is i e †.”
Could the authors justify this choice further? (I read the footnote).
12- It is written :
“Overall, the structure around core particles is almost featureless and resembles the one of the fluid at higher temperature”
To show this, it would have been useful to compare these radial distribution functions with that of a liquid at high temperature. Here the reader is forced to compare with Figure 7 and I find it a pity for such an interesting result.
Anonymous Report 1 on 2020-10-16 Invited Report
No major weaknesses were identified.
In their work “Spatial structure of unstable normal modes in a glass-forming liquid” the authors revisit the potential energy landscape picture of glassy liquids. Their study highlights the difference in the spatial distribution between unstable and stable modes revealing how the unstable modes change and become more localized as the temperature is decreased.
The article is very well written and clearly explained. I recommend publication but still invite the authors to consider the following remarks:
1. In section IIA the authors state that a key difference between the present work and Ref26 is that using a larger tolerance improves the convergence, at the price of some occasional (unphysical?) configurations with exceedingly large energies. Could the authors expand on this point and attempt some (eventually heuristic) explanation of the meaning of the increased tolerance? What are the trade-offs? Why does it work?
2. I am a bit confused by Fig 4: I suppose that the specific values of lambda are unimportant, and what matters is the asymptotic slope of the ration N(r)/N. As the temperature decreases, one is left only with modes below the mobility edge. It is not clear that these modes (the blue lines) are very sensitive to the temperature. The result seems to be that the localized saddles have very low fractal dimensions: can one infer from that they are “compact”? Isn’t it more appropriate to say that they are more point-like, as very few particles (or pairs of particles) for them?
3. Along this line, it would be interesting if the authors could contrast their measure of populations of unstable cores with the excitations predicated in dynamical facilitation [Keyes PRX 2011]. Are the two notions incompatible?
4. Regarding sections IIIB and C, it seems that the local environment (as probed by the radial distribution function or Voronoi tesselations) of saddles and local minima becomes more similar as the temperature is decreased. One then wonders how much of a ovarlap there is between the population of particles participating in the stable and unstable cores. Could one quantify this, for example with joint and conditional probabilities?
1. page 3, last paragraph of section IIA: please clarify the definition of the rust radius
2. Fig 1, both the two vertical lines are dashed (with different dash length) and not dashed and dotted as reported in the caption
3. Fig 4, two values are needed to know the scale of for the x-axis. I suppose that the largest r value is around 7sigma?