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Edited by: Matthieu Micoulaut, Université Pierre et Marie Curie, France

Reviewed by: Roger Jay Loucks, Alfred University, USA; Mathieu Bauchy, University of California Los Angeles, USA; Normand Mousseau, Université de Montréal, Canada

Specialty section: This article was submitted to Glass Science, a section of the journal Frontiers in Materials

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Using rigidity (constraint) theory of glasses, the effects of low-frequency vibrational modes anomalies in the glass transition are studied. It is discussed how the possibility of tailoring by chemical doping the number of low-frequency modes gives clues about how to determine the glass transition temperature and glass formation ability. In particular, we present the effects of floppy modes in the specific heat, entropy, internal energy below glass transition, as well as a discussion of the thermodynamical effects above the glass transition. All the previous results can be extended to include the Boson peak, since it can also be understood from a rigidity point of view as a dilution of bonds in an over-constrained network. Finally, we discuss how a new subject is emerging: floppy modes effects in the electronic properties of flexible systems. Such relationship provides a natural connection with topological insulators and two dimensional materials like graphene.

Although in many senses, we have achieved several milestones in the understanding of glass transition (Jackle,

In particular, there are several important questions to be solved by any suitable theory of glass transition,

What is fast enough? In other words, provide the minimal amount of cooling in order to form a glass.

What chemical factors determine glass transition?

How relaxation is achieved?

What is the origin of the low-frequency vibrational modes (LFVMs) anomalies, like the Boson peak, in glasses?

In a general sense, we have a partial understanding of all these questions (Elliot,

The key provided by rigidity theory lies in the fact that, on the one hand, it gives a definitive answer to the origin of the floppy mode peak (Thorpe,

The outline of this paper is the following. In the first section, we present a brief introduction to rigidity theory (RT) and floppy modes. Then, we devote a section to show how the Boson peak can be included inside the same picture. In the last sections, we discuss the effects on the thermodynamical properties, glass transition, and relaxation of glasses when the Boson and floppy mode peaks are present. Finally, a section on electronic properties and floppy modes is presented, and the conclusions are given.

Rigidity theory was used first in glasses by Phillips (

Why this magical number? The answer lies in a topological phase transition between a flexible to a rigid network (Thorpe,

When _{c}_{c}

Such fraction can be found using a mean-field approximation known as the Maxwell counting (Thorpe, _{c}

Notice that rigidity percolation theory was made for zero temperature (Thorpe, _{g}_{g}

For _{g}_{0} with weight _{j}_{0}/_{B}

To understand this two last points, consider the internal energy per particle (_{0}. Notice that in ideal rigidity, ω_{0} = 0 since floppy modes do not store elastic energy. As explained before, in real systems, residual forces produce a blue shift with a non-zero ω_{0}, as confirmed by neutron scattering (Kamitakahara et al., _{R}_{D}_{f}_{0}_{B}_{B}T_{0}. For low temperatures, i.e., below _{D}^{3} Debye contribution,
_{D}_{f}_{ν} follows the Debye model, but with a smaller value due to the reduced spectral weight of non-floppy modes. For high temperatures _{D}_{ν}(_{B}_{ν}(_{B}

One of the most important questions in the field is why RT works to understand the supercooled liquid and non-equilibrium processes.

Initially, to answer this question, numerical simulations were performed on simple models of association leading to some preliminary conclusions (Huerta and Naumis,

The concept of temperature-broken constraints (Gupta and Mauro,

From a more abstract point of view, the complexity of the energy landscape, determined by a distribution of metastable basins (Flores-Ruiz and Naumis,

All glasses present an excess of low-frequency vibrational modes (LVFMs) relative to the Debye crystal model, like the Boson peak (BP) or floppy modes (FM). They are in the THz range of frequencies (Buchenau et al., _{2} (Nakayama, _{4} tetrahedra units are connected by flexible bonds (Trachenko et al.,

During the last years, it has become clear that the Boson peak can also be explained as a consequence of rigidity. To understand this, suppose that we have an over-constrained network in which _{c}_{c}_{c}_{c}_{c}

_{c}

This question can be answered first by studying random bond dilution in periodic lattices (Flores-Ruiz and Naumis, _{BP}_{D}_{BP}_{D}_{c}

Furthermore, the peak has an almost transverse mode nature and can be related with a Van Hove singularity occurring in the glass (Flores-Ruiz and Naumis,

Thus, a new picture emerges by the process of bond dilution. The Boson peak is due to a diluted connectivity in an over-constrained network. As < _{c}

For example, in SiO_{2} under pressure a blue shift of the LFVM anomalies is observed (Trachenko et al., _{2} reveals an isostatic network due to the connection of rigid unit modes (RUM) made from SiO_{4} tetrahedra with flexible bonds. Pressure leads to the formation of defects with an increased coordination number (Trachenko et al.,

Notice that in real glasses, floppy modes, and the Boson peak can coexist since regions with different elastic properties can coexist (Bhosle et al.,

In the next section, we will discuss the importance of low-frequency modes for the glass transition.

Do low-frequency modes anomalies influence glass transition? There are several arguments against this. For example, within the energy landscape picture (Debenedetti and Stillinger,

At the same time, there is a clear indication that low-frequency modes are paramount to the thermodynamical stability of a system. At this point, we will use an opposite approach to that used by most people. Instead of looking at melt cooling, here _{g}. Then the glass is heated, until reach T_{g}

Mechanical rigidity provides a clue on the thermodynamical stability of a system that contains a delicate balance between dimensionality and low-frequency vibrational modes. In particular, stability can be tested by looking at long-range correlations of the quadratic displacement ^{2}(_{j}_{j}_{ij}_{ij}_{j}_{i}^{2} inside the integral in equation (^{2}(^{d}^{–1}. Only for

Summarizing, the stability of a solid is mainly contained in the factor ρ(ω)/ω^{2}. Any excess of low-frequency modes will decrease the stability of the system.

Let us now further develop the previous arguments to show how LFVM can determine _{g}_{0} is the typical value of the melt viscosity, _{0}^{2}(^{2}(_{g}_{0} ≈10^{13} Poise, from equation (^{y}_{0} < 0.5σ (where σ is the atomic size). Thus, we obtain the following bounds,
_{g}_{m}

Using equation (_{g}_{g}_{R}_{0},
_{g}

Here, we discuss recent progress in two fields: strain and thermal relaxation in glasses. In glasses, quenching produces defects with some residual internal stress. Such stress relaxation is known to be described experimentally by stretched exponentials (Phillips,

In real glasses, the accurate predictions of SER theory have been confirmed in a beautiful experiment by using a big homogeneous glass plate (Welch et al.,

Finally, we will discuss how thermal relaxation is modified by the rigid or flexible nature of the system. This is important for glass formation ability, since, for example, a fast relaxation will enhance crystallization. As an example, it has been observed that for metallic glasses, _{g}

The thermal conductivity for low temperatures of diluted flexible systems has been studied under the harmonic approximation, revealing also anomalies (Romero-Arias et al.,

In fact, the study of thermal relaxation started with the well known Fermi–Pasta–Ulam (FPU) problem (Fermi et al.,

The FPU model can be modified to include rigidity (Romero-Arias and Naumis,

The relaxation from an initial temperature

The study of relaxation in two and three dimensional systems using non-linear terms is still an open field since even for the original FPU model the work is scarce.

One of the less studied problems is how rigidity theory enters inside the electronic properties picture. At first sight, all vibrational modes affects electron scattering, and thus their effects are hidden inside the thermal noise. However, after the discovery of graphene and other pure bi-dimensional materials (Geim and Novoselov,

Let us consider a simple example of the effects of floppy modes. For simplicity, we focus on a system described with a one-orbital nearest-neighbor tight-binding Hamiltonian, as happens in graphene (Geim and Novoselov,

Suppose the system is distorted by a floppy mode. Since bond lengths are not modified, the parameters

Floppy modes in graphene are not only important for electron scattering. When boundaries or impurities appear, a fraction of the electronic modes have zero energy (with respect to the Fermi level). Such states are known as Dirac modes in graphene (Barrios-Vargas and Naumis,

These topological “floppy modes” are, in fact, behind the remarkable metal-insulator transition in doped graphene, which transforms graphene in a narrow-gap semiconductor with an enormous technological potential (Naumis,

Rigidity theory allows one to understand, via chemical modification, how low-frequency modes anomalies play an important role in glass transition. The Boson peak can also be included in the theory by considering bond dilution of over-constrained networks. When relaxation is combined with topology, stretched exponentials with magical β exponents are obtained. The ideas of rigidity can also be extended to study the electronic properties of solids, providing bridges between glasses and new topics in material science like graphene and quantum topological phases.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank DGAPA-UNAM project IN102513. This work was finished under the help of a sabatical leave program Estancias Sabaticas en el Extranjero, given by PASPA program in DGAPA-UNAM.

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