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At the heart of the problem lies the fact that a vitrifying material exhibits a spectacular growth of viscosity or relaxation time upon cooling or compression, but at the same time undergoes only minute structural changes. Thus, at the molecular level, the structure of a glass is almost indistinguishable from that of a normal liquid as probed by, e. This is unlike any conventional thermodynamic phase transition, such as the liquid-to-crystal transition, which is marked by the appearance of long-range, periodic structural order Figure 1. Figure 1. Schematic picture of the structure of A a normal liquid, B an amorphous solid, i.

The first panel highlights four green particles which are separated by a distance r from a red reference particle. The right panels illustrate the corresponding radial distribution functions g r , which describe the probability of finding a particle a distance r away from any reference particle, relative to the ideal-gas case. Another major unresolved piece of the glass puzzle is that not all materials vitrify in the same manner. More specifically, the viscosity growth as a function of inverse temperature can differ significantly from one material to another.

It is widely believed that a thorough understanding of the mechanisms underlying fragility will be key to achieving a universal description of the glass transition, but no theory to date has been able to predict a material's degree of fragility from the sole knowledge of its microscopic structure [ 20 ]. Figure 2. So-called strong glass formers such as silica exhibit an Arrhenius-type growth of the viscosity upon cooling, while fragile glass formers such as o -terphenyl show a much steeper temperature dependence close to T g.

Many materials, including colloidal hard spheres and confluent cells, fall in between these two extremes. While the viscosity already gives an important clue about the complex behavior of glass-forming materials, the most detailed information is contained in the microscopic relaxation dynamics, and this will also be the focus of the remainder of this review. A common probe of such dynamics is the time-dependent density-density correlation function or so-called intermediate scattering function, F k, t , which probes correlations in particle density fluctuations over a certain wavenumber k and over a time interval t [ 21 ].

Simply put, F k, t measures to what extent the instantaneous molecular configuration of a material will resemble the new configuration a time t later; the wavenumber k designates the inverse length scale over which this resemblance is measured. The behavior of F k, t upon cooling thus reveals how the microscopic relaxation dynamics changes during the vitrification process [ 22 , 23 ] Figure 3. In a normal high-temperature liquid, F k, t will decay to zero in a rapid and simple exponential fashion, since the particles can move around easily and therefore quickly lose track of their initial positions.

As the temperature decreases toward the glass transition temperature, the plateau in F k, t will extend to increasingly long times, until it finally exceeds the entire time window of observation. Thus, at the glass transition, F k, t fails to decorrelate on any practical time scale—implying that particles always stay reasonably close to their initial positions—, marking the onset of solidity.

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Figure 3. Schematic picture of the structure and dynamics in a normal liquid, supercooled liquid, and glass. A,B depict a typical trajectory of a particle in the normal liquid phase and glassy phase, respectively. In the glassy state, particles become trapped in a cage formed by their neighbors. The dashed red line indicates the typical size of a cage, with a radius of approximately one particle diameter d.

As the temperature is decreased or the packing fraction is increased, the system becomes more glassy and F k, t decays more slowly. There are several other aspects in the dynamics of supercooled liquids that differ markedly from those seen in ordinary liquids, including the emergence of dynamic heterogeneity [ 26 — 29 ] and the breakdown of the Stokes-Einstein relation [ 30 , 31 ].

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Dynamic heterogeneity refers to the fact that structural relaxation does not take place uniformly throughout the entire material—as in a normal liquid—, but rather in clusters of collectively rearranging particles, while the rest of the supercooled liquid remains temporarily frozen Figure 4.

The appearance of such mobile domains will vary both in space and in time, thus giving rise to non-trivial spatiotemporal fluctuations that become more pronounced as the glass transition is approached. Dynamic heterogeneity cannot be seen in F k, t itself, but rather in the fluctuations of F k, t among different particle trajectories [ 32 , 33 ].

This ratio holds generally for normal liquids, but in the supercooled regime the viscosity increase tends to be stronger than the diffusion-constant decrease. This breakdown of Stokes-Einstein behavior is widely believed to be a manifestation of dynamic heterogeneity, but the fundamental origins of both phenomena remain poorly understood.

Figure 4. Illustration of dynamic heterogeneity in supercooled liquids. In the normal liquid phase A , particle motion occurs rather homogeneously across the entire sample. Conversely, in a supercooled liquid B , particle motion occurs heterogeneously in clusters of collectively moving particles, and the appearance of such mobile clusters fluctuates both in space and in time. The figure is based on Weeks et al. Finally, we mention another hallmark of glassy dynamics that is rather general for out-of-equilibrium systems, namely aging [ 2 , 20 , 23 , 34 , 35 ].

Aging implies that the behavior of a material depends explicitly on its age, i. These changes are commonly a manifestation of the material's gradual approach to an equilibrium state.

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In the supercooled phase, such aging effects are usually observed after a small temperature quench, but vanish after a sufficiently long equilibration time. Within the glass state, however, ergodicity is broken and the relaxation time to reach equilibrium exceeds—by definition—any practical time scale. Hence, a glass can be regarded as a supercooled liquid that has fallen out of equilibrium, and its properties depend explicitly on its history. There is a large body of literature devoted to aging effects in glasses see e. In this review, we focus on one of several theories that seeks to describe the above complex phenomenology of glass-forming materials, namely Mode-Coupling Theory MCT [ 36 , 37 ].

We outline the key physical ingredients and sketch of the MCT derivation, its predictions, successes, and failures, as well as recent improvements and extensions of the theory. For an overview of the many other existing theories of glass formation, such as free volume theory, Adam-Gibbs theory, Random First Order Transition RFOT theory, dynamic facilitation and kinetically constrained models, energy landscape approaches, and geometric frustration, see e.

A discussion of these alternative theories falls outside the scope of the present work; here we only mention that, of all the other existing frameworks, RFOT theory is directly related to MCT at high temperatures, but is further augmented with thermodynamic, Adam-Gibbs-like concepts at low temperatures. As already noted in the introduction, MCT provides a purely first-principles route toward the description of glassy behavior, making it a unique theory that does not rely on any phenomenological assumptions.

Explicitly, MCT aims to predict the full microscopic relaxation dynamics of a glass-forming material—as a function of time, wavenumber, temperature, and density—, using only knowledge of static, time-independent properties as input. Aside from constants such as the system's temperature and density, the main theory input is the average microscopic structure of the material.

The simplest experimental measure of the latter is the static structure factor S k , which can be obtained directly from scattering experiments. It must be noted that MCT also admits more intricate three-particle correlation functions as additional structural input, but—with the exception of network-forming fluids [ 45 , 46 ]—the sole knowledge of S k generally suffices. Importantly, it is through these structural metrics that MCT knows about the chemical composition of the material under study. That is, the theory is able to distinguish between, say, a glass-forming fluid of silica or Lennard-Jones particles only through their differences in wavevector-dependent structure.

In the standard formulation of MCT, the theory seeks to predict the full dynamics of the intermediate scattering function F k, t of a given material, starting with the exact equation of motion for F k, t. Below we sketch the derivation of this equation, followed by a discussion of the various MCT approximations made to solve it. MCT makes the ad hoc assumption that the latter memory function can be approximated as a product of F k, t functions, thus yielding a closed, self-consistent equation see Figure 5.

As described in section 2. Figure 5. Sketch of the MCT equations. The theory seeks to predict the full dynamics of the intermediate scattering function F k, t for all possible wavevectors k and all times t. Hence, the static structure factor must be given as input to the theory, and dynamical information is given as output. Let us first define our variables of interest, namely the collective density modes,. The intermediate scattering function F k, t probes the time-dependent correlations between these collective density modes,.

In order to obtain an exact equation of motion for F k, t , we make use of the so-called Mori-Zwanzig projection formalism [ 48 , 49 ]. Physically, this idea relies on a separation of time scales in the dynamics, whereby the fast variables are integrated out. Here we will focus mainly on molecular glass-forming fluids, in which case the variables of interest are the collective density modes of Equation 1 and their associated current modes. On smaller length scales, however, i. Importantly, in this notation, time-dependent correlation functions may now be identified simply as scalar products between such vector elements, e.

We define the full matrix of all possible scalar products as C t , with matrix elements. Furthermore, in analogy to ordinary projections in vector space, we can now use these scalar products to define a projection operator P A as. This projection formalism, introduced by Zwanzig and Mori, thus establishes a link between dynamic variables and standard vector algebra. Let us now look explicitly at the time-dependent dynamics of a glass-forming supercooled liquid.

For classical fluids that obeys Newton's equation of motion, the time evolution of A t can always be formally written as. The definition of L can be found in e. Note that for colloidal glass-forming systems undergoing Brownian rather than Newtonian motion, a similar equation applies when considering only the density modes in A and replacing the Liouvillian by the so-called Smoluchowski operator [ 50 ].

While Equation 8 is formally exact, it does not necessarily yield any new physical insight into the complex time-dependent dynamics of supercooled liquids. The memory function K t is given by the time-autocorrelation function of this fluctuating force; physically, K t represents a dissipative term that ultimately breaks the conservation of A. Importantly, Equations 9 and 10 , which are known as the generalized Langevin equation and memory equation, respectively, are both exact.

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By Equation 10 , the difficulty of predicting the full time-dependent dynamics of F k, t is now deferred to the the question of how the memory function K t evolves with time. In general, there is no rigorous solution for this equation, and hence approximations must be made. Approximate the memory function as a four-point density correlation function. Physically, this projection is motivated by the fact that for particles interacting through an arbitrary pair potential, such products of densities emerge naturally in the expression for the fluctuating force [ 22 ].

This may seem rather counterintuitive at first, since the fluctuating force is a fast variable while density modes are slow by definition, but it can be shown by Fourier transformation that, for an n -body interaction potential, f t always contains products of n density modes [ 41 ]. In the standard MCT formulation, it is assumed that the pair densities dominate the fluctuating force entirely, but higher-order generalizations with projections onto an n -density-mode basis have also been considered [ 51 , 52 ]. Mathematically, the projection onto pair densities also corresponds to the first non-vanishing component in density space, i.

Overall, this approximation brings the memory function K t , which is the time-correlation function of f t , into the form of a four-point density correlation function :. Factorize four-point correlation functions into two-point correlation functions. Nonetheless, it can be shown that the factorization is exact for so-called Gaussian variables [ 53 ], but density modes in general do not behave as such. Thus, our full equation of motion for the intermediate scattering function F k, t is now governed by a memory function containing precisely the same function, but for many different wavenumbers.

In arriving at this equation, we have also assumed that S k contains all the relevant microscopic structural information using the so-called convolution approximation [ 22 , 23 , 54 ] , but in general the vertices may also contain higher-order, triplet-density correlations [ 45 , 55 ]. The damping coefficient, on the other hand, appears in the MCT equation in the form of the memory function K MCT t note the first derivative of F k, t in the integrand. Consequently, we may interpret the memory function as a generalized, time-dependent damping, which will ultimately cause the dynamical slowdown in F k, t [ 23 ].

The fact that the true schematic MCT memory function contains the non-linear product x 2 t instead of a simple delta-function, however, has important consequences for the dynamics and gives rise to a strong feedback effect that is absent in an ordinary damped oscillator. Figure 6 shows the solutions x t for a one-dimensional damped harmonic oscillator and for the schematic MCT model as a function of time for different damping factors a.

It may be seen that, as the damping a is increased, the harmonic oscillator undergoes only a moderate change in the dynamics, while the MCT solution develops a plateau and exhibits an orders-of-magnitude dynamical slowdown. The full wavevector-dependent MCT equation predicts a similar scenario, in which the memory function constitutes a non-linear dynamical feedback mechanism with the damping strength implicitly controlled by small changes in the static structure factor as the temperature is decreased , though of course the explicit coupling of density modes at different wavevectors leads to richer and more complicated behavior than that predicted by schematic MCT see section 3.

Despite the simplicity of schematic MCT models, it is now well established that many predictions of schematic MCT—which can often be derived analytically see [ 38 ] —are also preserved in the full wavevector-dependent version, and hence schematic MCT approaches remain widely used to gain better insight into the phenomenology of glassy materials. Figure 6. Note the logarithmic scale for the time axis. While analytic solutions of the full wavevector-dependent MCT equation generally do not exist, it is always possible to solve the equation numerically, namely by iteratively making an ansatz for F k, t for all k , subsequently constructing the memory function, and updating F k, t until convergence is reached.

See Fuchs et al. Moreover, it has also been shown that this equation applies reasonably well to glass-forming polymer chains [ 59 ], suggesting that MCT captures at least some degree of universal dynamical behavior. Finally, we note that MCT-based equations have also been formulated for, e. These MCT extensions and variations, however, will not further be discussed in this review. As a final note, we mention that an alternative and more recent first-principles-based theory has been formulated that is somewhat related to MCT, namely the Self-Consistent Generalized Langevin Equation approximation [ 39 ].

This SCGLE theory also starts from the exact generalized Langevin equation Equation 9 , but employs different and somewhat simpler approximations to obtain a self-consistent equation for the dynamics. Instead of projecting the fluctuating force onto pair density modes, the main assumptions of SCGLE theory are a Vineyard-like approximate relation between the memory function of F k, t and the memory function of the self part F s k, t , and a Gaussian-like approximation for the memory function of F s k, t that relates its dynamics to the Brownian motion of individual particles [ 39 ].

Overall, SCGLE theory also amounts to a closed, self-consistent dynamical equation that requires only simple static properties as input. An important advantage is that the SCGLE theory can be readily extended to account for non-equilibrium aging effects, such that the dynamics depends explicitly on the waiting time or age of the material [ 69 — 72 ].

This is to be contrasted with standard MCT, which in its standard form applies only to stationary quasi- equilibrium systems and consequently cannot make any predictions of aging phenomena; the extension of MCT to non-equilibrium aging systems is technically rather involved [ 73 ]. The SCGLE equations have also been extended to multi-component systems [ 74 , 75 ] and to non-spherical particles [ 76 , 77 ], and are generally somewhat simpler to use than the MCT equations.

Thus, MCT predicts the full microscopic dynamics given only time-independent information as input. In order to describe the entire vitrification process from liquid to glass, one typically measures S k for a series of temperatures or densities, and performs a separate MCT calculation for every relevant temperature and density. In this section, we summarize the main successes and failures of such MCT predictions. Despite the various approximations made in MCT, the theory gives a remarkable set of accuracte predictions. Firstly, MCT is indeed capable of predicting a glass transition, which is non-trivial considering that the static structure factor S k —the main theory input—changes only very weakly upon vitrification Figure 3C.

As mentioned earlier, the relaxation time of the predicted F k, t is used as an indicator for the glassiness: at the glass transition, the relaxation time diverges and F k, t fails to decay to zero on any time scale. The corresponding non-ergodicity parameter f k is also often in good quantitative agreement with the results of computer simulations and experiments see, e.

Mathematically, MCT's ability to predict a glass follows from the non-linearity of the equation by virtue of the product of two F k, t functions in the memory function , which renders the theory very sensitive to any small change in structural input. This non-linearity leads to a feedback mechanism that ultimately drives the dramatic dynamical slowdown: upon cooling, S k will become slightly larger at certain wavevectors, causing the vertices to increase as well.

Consequently, the memory function will become larger and produce a stronger damping for F k, t. The resulting slower intermediate scattering function will further strengthen the memory function, slowing down the dynamics even more. This non-linear feedback effect explains at least qualititatively why the relaxation dynamics can change so dramatically upon only small changes in the structure and temperature [ 23 ]. A related success of MCT is its prediction of the cage effect as a microscopic mechanism for vitrification Figures 3 , 7. Caging refers to the fact that, in a supercooled liquid, particles become transiently trapped in local cages formed by their neighboring particles, which in turn are trapped in their respective cages, preventing them from moving around as in a normal liquid.

As long as the material is on the supercooled-liquid side of the transition, the particles will eventually manage to escape their cages, but at and below the glass transition, the cage effect keeps them trapped indefinitely. The only motion in the glassy state then corresponds to a vibrational or rattling motion of the particles within their confining cages. More mathematically, the cage effect emerges from MCT by considering that the most prominent change in S k upon supercooling occurs at the main peak at wavenumber k 0 , corresponding to length scales of approximately one particle diameter.

As a consequence, the first intermediate scattering function that falls out of equilibrium at the glass transition is F k 0 , t , which in turn drives the freezing on all other wavevectors. Notably, within MCT, the dominant structural length scale governing vitrification thus remains on the order of only one particle diameter, in stark contrast with conventional critical phenomena that are usually accompanied by diverging, macroscopic length scales. However, as will be described in section 4. Figure 7. At very short times, particles undergo ballistic motion. Regardless of the molecular details of the material, which are contained in S k , MCT also makes several general predictions for the relaxation dynamics [ 22 , 23 , 36 , 38 ].

We will return to this point in the next subsection. This is an entirely non-trivial and remarkable prediction that is fully consistent with experiments and simulations. This is again in excellent agreement with experimental and simulation data, and physically arises from the coupling of multiple density-mode relaxation channels over different length scales, each relaxing on its own time scale.

Among the other celebrated results of MCT, we mention here its qualitative prediction of complex reentrant effects in sticky hard spheres particles with a hard repulsive core and short-ranged attractions [ 80 ] and ultrasoft repulsive particles [ 81 ], which exhibit glass-fluid-glass and fluid-glass-fluid phases upon a monotonic increase in attraction stength and density, respectively. In the case of sticky hard spheres, MCT has also provided a qualitative explanation for the existence of the two distinct glass phases in terms of different dominant length scales [ 80 ].

Furthermore, the schematic version of MCT [ 24 , 38 ], which is obtained by ignoring all wavevector dependence in Equation 12 , is rigorously exact for certain classes of spin-glass models with queched disorder so-called p -spin spherical spin glasses , pointing toward a possible deep connection between systems with quenched and self-generated disorder. Even though MCT successfully predicts a glass transition, its most notable failure is that the predicted glass transition temperature T c occurs at much higher temperatures than the true experimental value T g.

Thus, the static structure factor for which MCT predicts a glassy state corresponds in reality to only a mildy supercooled liquid. In practice, the MCT predictions are often rescaled such that T c coincides with T g [ 68 ], but even with such a relative comparison, MCT generally fails to accurate describe the dynamics in the deeply supercooled regime.

This discrepancy is attributed to MCT's lack of ergodicity-restoring relaxation mechanisms that keep the experimental system in the liquid phase well below T c Figure 8. MCT fails to account for such hopping motion and thus strongly overestimates the degree of caging—a feature that is believed to arise from its so-called mean-field nature. In practice, the predicted MCT transition at T c is therefore interpreted as a crossover point where the dynamics changes into an activated form [ 83 ].

In section 4, we will return to this point and address recent efforts to incorporate activated dynamics directly into the theory. We note that activated dynamics may also be incorporated via, e. A description of RFOT falls outside the scope of the present work, but we refer the interested reader to [e. Figure 8. Typical MCT prediction purple curve and simulation result blue curve for the dynamical slowdown of a glass-forming material as a function of the control parameter. The figure is based on Charbonneau et al. As mentioner earlier, MCT's prediction of a power-law divergence of the relaxation time also breaks down in most experimental and simulated glass-forming systems.

More generally, the fact that MCT always yields a power law, regardless of the molecular composition of the material, also implies that MCT has essentially no notion of the concept of fragility. At best, an MCT power law may correctly describe the relaxation dynamics of fragile glass formers, but strong glass formers exhibit a fundamentally different, Arrhenius-type growth of the relaxation time.

Indeed, an accurate first-principles prediction of the fragility of a material on the sole basis of its microscopic structure remains a major open challenge in the field [ 20 ]. Nonetheless, we note that MCT can predict other properties of strong glass formers rather accurately, such as the wavevector-dependent non-ergodicity parameter in the glassy phase [ 45 ]. MCT is also generally unable to account for the breakdown of the Stokes-Einstein relation in the deeply supercooled regime.

This is again attributed to the inherent mean-field character of the theory and the absence of activated hopping dynamics [ 82 ]. Moreover, in its standard formulation, MCT does not offer an explanation for the emergence of dynamic heterogeneity, since MCT only predicts a single F k, t for a given wavevector, density, and temperature, and hence does not give access to correlations in the fluctuations of F k, t. However, as discussed in section 4. Furthermore, despite its mean-field character, it was recently shown that MCT does not become exact in the mean-field limit of infinite dimensions for a system composed of hard spheres [ 87 — 89 ], making it difficult to rationalize the set of standard-MCT approximations in a simple physical manner.

Moreover, MCT assumes that the material is in quasi- equilibrium, and consequently fails to account for non-equilibrium aging and protocol-dependent history effects. The afore-mentioned SCGLE theory does allow for explicit aging predictions [ 72 ], and hence it constitutes an attractive alternative to MCT in this regard. The latter is believed to also play an important role in the process of glass formation, and in particular may point toward an underlying thermodynamic transition that in practice is masked by the dynamic transition.

Nonetheless, it is possible that MCT is implicitly aware of at least some changes in thermodynamic properties through changes in the static structure factor [ 90 , 91 ]. Since standard MCT is not exact, as exemplified by the drawbacks and failures discussed in the previous section, various attempts have been made in the last few decades to improve the theory's predictive power for glassy dynamics. Finally, we also briefly discuss recent generalizations of MCT to a new class of soft condensed-matter systems referred to as active matter. Such active materials are composed of particles that can undergo autonomous motion through the consumption of energy, and are now emerging as a new paradigm to understand collective behavior seen in many living systems.

The recent realization that active particles can also vitrify into a glassy state has spurred the formulation of various MCT frameworks for active matter, the development of which will be reviewed in section 4. Both approaches amount to a perturbative treatment of nonlinear couplings to certain current modes that are neglected in the standard formulation of MCT, and which cut off the sharp MCT transition such that the strict divergence of the relaxation time at T c is removed.

However, more recent theoretical studies have argued on general physical grounds that the invoked couplings to currents in EMCT cannot provide a satisfactory explanation of activated dynamics, since these couplings should always become negligible close to a glass transition [ 94 ]. Moreover, Andreanov et al. Another argument that casts doubt on the general applicability of EMCT is the fact that experimental and numerical simulation studies have unambiguously established that materials obeying Newtonian and Brownian stochastic dynamics exhibit the same deviations from standard-MCT behavior, despite their differences in microscopic dynamical details.

This suggests that the physical mechanisms governing activated behavior below T c have a universal origin in both molecular Newtonian fluids and colloidal Brownian systems. Since the current modes introduced in EMCT cannot be properly defined in Brownian systems [ 40 ], the proposed EMCT mechanism may thus only apply to materials undergoing Newtonian dynamics. Hence, it appears likely that EMCT cannot offer a rigorous, universal remedy for the lack of ergodicity-restoring activated dynamics within the standard MCT framework. An alternative route to rigorously improve MCT was put forward by Szamel in [ 96 ].

To this end, a new and formally exact equation of motion is developed for the four-point correlation functions themselves again by applying the Mori-Zwanzig projection formalism of section 2. The new equation is governed by another memory function that, to leading order, is controlled by six-point density correlation functions, which in turn are dominated by eight-point correlations, etc. Hence, by repeatedly developing a new equation of motion for the new memory function, a hierarchy of coupled equations emerges, in which the uncontrolled factorization approximation may be applied at an arbitrary level to close the set of equations.

This GMCT scheme thus allows, in principle, for a systematic delay of the closure approximation and, notably, remains based entirely on first principles see Figure 9. Figure 9. GMCT seeks to systematically avoid the uncontrolled MCT factorization approximation by developing a new, and formally exact, equation of motion for the unknown memory function. This equation in turn is governed by a new memory function, which is controlled by another memory function, etc.

Standard MCT corresponds to the lowest-order self-consistent closure of this hierarchy. These results indicate that the GMCT hierarchy apparently converges and that the theory becomes more quantitatively accurate as the closure level is increased. The figure is adapted from Janssen and Reichman [ 97 ] with permission. Szamel [ 96 ] and Wu and Cao [ 98 ] showed that GMCT hierarchies factorized at the level of six- and eight-point correlation functions, respectively, indeed bring the predicted glass transition density systematically closer to the empirical value for a system of colloidal hard spheres.

More recent work [ 97 ] also established that the full time-dependent microscopic dynamics for a quasi-hard-sphere glass former is systematically improved by GMCT. In fact, fit-parameter-free third-order GMCT calculations could achieve full quantitative agreement for F k, t up to the moderately supercooled regime, at densities where standard MCT would already predict a spurious glass transition [ 97 ]. In particular, it appears that higher-order GMCT captures at least some aspects of activated dynamics to keep the material ergodic at temperatures below T c , consistent with empirical observations.

Importantly, we note that GMCT is applicable to both Newtonian and Brownian systems, and therefore also holds the potential to offer a more universal picture of glassy dynamics. In addition to accounting for some kind of ergodicity-restoring processes below T c , GMCT might also provide a suitable framework to describe fragility.

The work of Mayer et al. In later studies, we demonstrated that other schematic GMCT models may also give rise to other functional forms of relaxation-time growth, ranging from fragile super-Arrhenius to strong sub- Arrhenius behavior, depending on the choice of schematic parameters [ ]. Although these simplified GMCT models inherently lack any wavevector dependence, and therefore cannot make detailed predictions for any structural glass former with a realistic S k , they suggest that higher-order GMCT has at least the mathematical flexibility to account for different fragilities.

This is notably different from standard MCT, which is mathematically only capable of predicting power-law growth close to the transition. It remains to be tested whether the fully microscopic wavevector-dependent version of GMCT will indeed be able to account for different degrees of fragility, given solely the static structure factors S k and possibly higher-order static correlation functions of strong and fragile materials as input. It might be tempting to assume that, with increasing closure level, the GMCT predictions should become more accurate, but let us reiterate that the current formulation of GMCT still relies on several approximations, and it is still unclear how the remaining assumptions ultimately affect the dynamics.

Finally, we note that by construction , higher-order GMCT also makes microscopic predictions for the approximate dynamics of unfactorized four-point density correlations [ 97 ].

1. Introduction

Hence, GMCT may also offer a suitable starting point to study dynamic heterogeneity, as well as the breakdown of the Stokes-Einstein relation in supercooled liquids, from a strictly first-principles perspective. We expect this avenue of research to be explored in the coming years. Biroli et al. However, it should be noted that the predictions of IMCT are not generally in quantitative agreement with empirical results.

The question to what extent, and under which conditions, IMCT can offer an accurate description of dynamic heterogeneity, and how the IMCT predictions relate to, e. We end this review with a very recent development in the field, namely the study of active matter. Active materials consist of particles that can convert energy into autonomous motion, rendering them out of thermodynamic equilibrium at the single-particle level [ ]. Such particle activity can lead to rich self-organizing behavior, as exemplified in nature by, e.

During the last decade, numerous synthetic active systems have also become available [ ], spurring the development of theoretical approaches to describe the emergent behavior in these non-equilibrium materials.

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In particular, it was found that dense active matter can also exhibit properties of supercooled liquids and vitrifying colloidal suspensions [ 6 , 8 , 10 , 11 , — ], including slow structural relaxation, dynamic heterogeneity, varying degrees of fragility, and the ultimate formation of a kinetically arrested, amorphous solid state. Here we briefly discuss recent extensions of standard MCT to describe the glassy dynamics in active materials. Since many synthetic active particles are composed of colloids undergoing active Brownian motion, all active versions of MCT to date are based on the Smoluchowski formalism for Brownian systems, rather than the Newtonian description for molecular fluids discussed in section 2.

We note, however, that continuum descriptions of active matter, such as those for active liquid crystals, are usually derived from Newtonian-based fluid mechanics [ ]. In this work, they considered so-called active Brownian particles ABPs that move with a constant self-propulsion speed in a random direction, subject to translational and rotational Brownian motion. The authors assumed that a single , non-interacting ABP behaves effectively as a passive colloid, but with a higher effective diffusion constant.

This approximation was subsequently used to derive an effective Smoluchowski operator for the collective dynamics of a dense ensemble of active particles. In essence, this effective-diffusion approach amounts to the removal of explicit rotational degrees of freedom. The resulting MCT approach yields a modified version of Equation 12 , in which both the frequency term and the memory function acquire an activity-dependent prefactor. The main outcome of this MCT study is that the addition of particle activity can soften i.

Frontiers | Mode-Coupling Theory of the Glass Transition: A Primer | Physics

These findings are also in qualitative agreement with computer simulations of a similar active material composed of self-propelling Brownian hard particles [ , ]. A different and more extensive active-matter study was performed by Szamel et al. Here, the authors modeled active particles by an Ornstein-Uhlenbeck stochastic process, characterized by an effective temperature that quantifies the strength of the active forces, and a persistence time that describes the duration of persistent self-propelled motion.

In this model, particle motion is thus described as a persistent random walk. Within their framework, the self-propulsion is first integrated out before applying the projection-operator method and MCT-like approximation; this approach essentially assumes that particle positions evolve on a time scale much larger than the time scale needed for reorientation of the activity direction, somewhat akin to the effective-diffusion assumption of Farage and Brader [ ].

An important difference between the active MCT of Szamel et al. Contrary to the behavior of ABPs, it was found that the incorporation of activity can both enhance and suppress glass formation: for small persistence times, the active fluid relaxes faster than a passive system at the same effective temperature, but for large persistent times the active material becomes more glass-like compared to the passive reference system. This non-monotonic dependence of the relaxation time was observed both in the MCT analysis and in computer simulations, and was attributed to the competition between increasing velocity correlations which speed up the dynamics and increasing structural correlations which slow down the dynamics [ ].

For sufficiently large persistence times, it was found that the fitted MCT glass transition temperature increases monotonically with increasing persistence time, suggesting that—at least within this active-matter model—vitrification occurs more easily as the material becomes more active. An MCT-based scaling analysis for this type of active-matter system was later performed by Nandi and Gov [ ]. Feng and Hou [ ] subsequently studied a quasi-equilibrium thermal version of the active Ornstein-Uhlenbeck model of Szamel and co-workers, which additionally accounts for thermal translational noise.

Their MCT derivation differs from the approach taken by Szamel [ ], however: it is valid only for sufficiently small persistence times since it relies on a perturbative expansion , and does not require explicit velocity correlation functions to be given as input. It was found that the critical density at which the glass transition takes place shifts to larger values with increasing magnitude of the self-propulsion force or effective temperature, and that the critical effective glass temperature increases with the persistence time. In the limit of a vanishing persistence time, the theory naturally yields the expected result for a simple passive Brownian system [ ].

Very recently, Liluashvili, et al. That is, rather than seeking to reduce the active material to a near-equilibrium system, the rotational degrees of freedom governing the reorientation of the active forces are now explicitly coupled to the translational motion. This approach thus avoids the effective-diffusion assumption which in principle may be valid only at low densities and sufficiently long times , and the resulting dynamics now also depends non-trivially on the rotational diffusion constant. The only required material-dependent input for this active MCT is the passive-equilibrium static structure factor.

An important outcome of this study is the three-dimensional fluid-glass phase diagram for hard ABPs as a function of packing fraction, self-propulsion speed, and rotational diffusion constant. It was shown that this surface cannot be collapsed onto a single line in the two-dimensional plane, highlighting the importance of treating the rotational degrees of freedom explicitly. Indeed, depending on the density of the active material, separate regimes could be identified that are dominated either by translational or reorientational motion. As in the study of Farage and Brader [ ], and in agreement with computer simulations [ , ], it was also found that activity generally makes hard-sphere systems more fluid-like and consequently shifts the glass transition to higher packing fractions.

Notably, this active fluidization effect grows monotonously with increasing persistence time or inverse rotational diffusion constant, in contrast with the findings of Szamel and co-workers [ ]. This difference is attributed to the absence of thermal Brownian noise in the model of Szamel et al. Finally, we mention another class of non-equilibrium materials that is closely related to active fluids, namely driven granular matter. Such systems can be realized experimentally by placing granular particles on, e. Kranz, Sperl, and Zippelius [ — ] developed an MCT for driven granular spheres, focusing on the role of energy dissipation due to inelastic particle collisions on the dynamics.

Furthermore, the increasing dissipation was found to have three noticeable effects in the non-ergodicity parameter at and above the glass transition density: i correlations at small wave numbers are enhanced, ii oscillations reflecting the local structure become less pronounced, and iii the localization length measured by the inverse of the width of the non-ergodicity-parameter peak decreases. The last finding is a consequence of the glass transition taking place at a higher density.

This review has sought to provide a brief overview of the main phenomenology of glassy dynamics, and of its theoretical description using Mode-Coupling Theory—arguably the most successful theory of the glass transition that is based entirely on first principles. We have focused mainly on the behavior of the density correlation function F k, t as a probe of the microscopic dynamics associated with vitrification. In the normal liquid phase, this correlation function rapidly decays to zero, but at the glass transition it fails to decay on any practical time scale, marking the onset of rigidity and providing an order parameter for the transition.

Upon approaching the glass transition temperature, several complex features become visible in the dynamics, such as a transient plateau and stretched exponential behavior in F k, t , a breakdown of the Stokes-Einstein relation, and the emergence of dynamical heterogeneity—the latter being associated with increasingly large fluctuations in F k, t.

Remarkably, during the process of glass formation, the microscopic structure of the material, as probed by e. It is this seemingly paradoxical discrepancy between structure and dynamics that makes the glass transition a notoriously difficult problem in theoretical physics. MCT offers a first-principles-based framework to account for at least some aspects of glassy dynamics. Its starting point is the exact equation of motion for F k, t ; through a series of partly uncontrolled approximations, MCT subsequently provides a self-consistent equation for F k, t that can be solved numerically using only the static structure factor as input.

As such, the theory makes a set of detailed predictions for the full microscopic relaxation dynamics of a glass-forming material as a function of time, wavevector, temperature, and density, on the sole basis of simple structural information. Among its notable successes is the qualitative prediction of a glass transition, a physically intuitive picture for glass formation in terms of the cage effect, and the correct prediction of several highly non-trivial scaling behaviors in F k, t.

However, MCT is generally not quantitatively accurate, and cannot account properly for the concept of fragility, the violation of the Stokes-Einstein relation, and the emergence of dynamic heterogeneity. The first studies in this direction show that GMCT can indeed offer a more quantitative description of the F k, t dynamics and can potentially describe fragility, while IMCT offers a framework to qualitatively account for dynamic heterogeneity.

However, GMCT still relies on several approximations such as the neglect of certain wavevector-dependent density correlations, and IMCT provides—just like standard MCT—only a mean-field description of glassy dynamics. Melvil Dewey invented his Dewey Decimal System in , and early versions of his system are in the public domain. Home Groups Talk Zeitgeist. I Agree This site uses cookies to deliver our services, improve performance, for analytics, and if not signed in for advertising. Your use of the site and services is subject to these policies and terms.

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