Figure 1.4:Our mission. We consider jammed systems of different coordination (directly connected to the inverse packing fraction quantified by the blue axis) and investigate their response to external tension and finite temperatures. In this sense we move up in the plane as marked by the yellow arrows.
1.2 Glasses
The jamming diagram sets out the possible connection between jammed materials and glasses. Yet, it is not purely speculative. The amorphous structure is a strong link between granular material, colloids, foams and glass. The response to thermodynamic and mechanical control parameters gives an additional hunch on how similar their phase transitions are. This section will give more evidence on the connection.
Despite the similarities, atoms of glasses are not spherical and interactions andn be attractive and long-ranged. Hence, the properties of glasses and the jammed soft sphere packings have to be observed closely in order to assess how parallel they are. The focus will be the vibrational modes of glasses and how they relate to the findings of Section 1.1.5.
When a glass is blown in such a way that its cooling is sufficiently quick it moves into a metastable, supercooled state instead of crystallizing. Further decrease of temperature leads to evermore arrested dynamics of the supercooled liquid.
Below a specific temperatureTG the liquid will have rigid mechanical properties on all timescales of relevant length. This is the glass transition temperature which marks the glass transition. The liquid is now in the glass state. The
1 Introduction
cooling process is called vitrification. The glass transition temperature depends on the material and the rate at which the cooling happens [33]. It is believed that a crystalline groundstate, reachable through a second-order phase transition, exists for glass, but is practically unreachable due to the quenched dynamics. In order to arrive at this underlying crystalline groundstate, the cooling would have to be infinitesimally slow. The practical interpretation of this hypothesis can be debated. As a last remark we want to emphasize that the glass transition is marked by dynamics out of equilibrium in which the history of the system is important [34].
Glasses have a higher density of states for low-frequency modes compared to the Debye-behaviour for the density of states of acoustic modes, which are seen in most crystals. This phenomenon, the boson peak [35], is long-known [36]. At low temperatures this results, for instance, in a higher heat capacity and lower thermal conductivity than what is expected from a crystal [37,38]. Glasses store more heat and transport less of it than crystals. Silica,SiO2, exists in crystalline and amorphous, glassy form. Experiments show that for low temperatures around 25mK, the specific heat capacity of silica glass is more than 1000 times larger than that of crystal [38]. Also, the heat capacity for glasses scales linearly with T instead of T3, which is the prediction of Debye’s model [39]. It is widely believed that the atomic motions which are associated with the boson peak are indispensable for understanding this particular behaviour.
Strong theoretical and experimental evidence exists that the boson peak can be understood with the concept of soft modes (1.1.5). Brito and Wyart [40] show that in hard sphere liquids rearrangements happen during vitrification along soft modes. This implies that rearrangements leading to structural relaxation happen as extended collective motions instead of localized events [41].
1.2.1 Silica
Silica is an important material to probe glasses experimentally. Trachenko et al. [42] use a model which describes silica as Si04-tetrahedra which are connected through joints. The tetrahedra are rigid compared to the forces of the rather flexible joints. The joints are modelled as loaded harmonic springs with zero equilibrium length and a spring constant tuned to mimic experimental results.
This leads torigid unit modes, which are rotations of tetrahedra without deform-ing. They are the lowest vibrational energy modes in the system.
We apply Maxwell’s criterion for stability to the tetrahedral structure. In total, each tetrahedron has 6 degrees of freedom for translation and rotation. Then, each bridging oxygen at the corner of a tetrahedron imposes 3 constraints. This gives 6 degrees per tetrahedron as every oxygen is shared by two tetrahedra. As
1.2 Glasses no degree of freedom is left, we conclude that the whole system is isostatic. When the pressure on such a system is increased, so called five-fold defects occur. This means that the number of five-fold Silicon atoms increases. This way the whole system is departing from isostaticity and the effect on vibrational modes can be studied [43]. It shows that the excess of modes shifts away from low frequencies much like in sphere packings [25].
Silica glass [42] and the corresponding crystal [44] with the same chemical com-position show a similar density of states, i.e. with a boson peak. We can draw the conclusion that order cannot be the crucial factor which determines the den-sity of states, for the glass is amorphous and the crystal’s structure is ordered.
Additionally, because it is very similar, the vibrational spectrum alone cannot account for the peculiar behaviour of glasses compared to crystals. Wyart [29,45]
proposes that it is:
1. the coordination instead of positional disorder which matters for the low-frequency spectra [26] and
2. the exact nature of the modes which is decisive and which is affected by disorder [29].
For crystals anomalous modes come as plane waves. For amorphous solids the anomalous modes are very heterogeneous. Hence there is a difference between crystals and glasses and anomalous modes are the decisive factor in their different thermodynamic behaviour [46]. This view is not undisputed and the exact nature of the boson peak in silica is subject to current research and debate [26, 47].
Chen et al. [48] tested in lab-experiments the validity of the results for the vi-brational spectrum of the soft sphere model (see section 1.1.5). They used a bi-disperse mixture of Poly(N-isopropylacrylamide) (NIPA) microgel particles, which swell with decreasing temperature. Consequently, temperature is used to tune the packing fraction of the jammed system. They found that the vi-brational properties are in good agreement with those in athermal soft sphere packings: Firstly, they are in the excess of low frequency modes which extends to lower frequencies when approaching the jamming point is approached. Sec-ondly, the same nature of the modes as seen in the soft spheres was recovered: at low frequency, modes are quasilocalized; at intermediate frequencies, modes are highly disordered and extended; at high frequencies they are localized [48].
The exact process of the melting of a sphere packing is not trivial. One of the questions is how a particle can escape its position if it is fully enclosed by other particles. The collective motions of the neighbouring particles which are needed for this to happen are a subject of ongoing research [49].
1 Introduction