Space-resolved dynamics of a tracer in a disordered solid
Introduction
The structural dynamics in glass-forming liquids slows down by orders of magnitude upon cooling or compression eventually leading to a quasi-arrested state or a disordered solid. Conventionally the dynamics close to the glass transition slows down uniformly as manifested in the divergence of a single time scale characterizing the slowest process in the system, referred to as the α-process. A coherent theoretical picture for a variety of phenomena has emerged by the mode-coupling theory (MCT) of the glass transition [1], developed by Wolfgang Götze and collaborators in the last 25 years. In particular, this approach has explained how a two-step relaxation process with a nontrivial two-time fractal results from the equations of motion close to a bifurcation point. These equations encode a mechanism to yield power laws with exponents that are not simple fractions, which has no precedence in other areas of physics and yet has been identified as generic in nonlinear integro-differential equations.
The mathematical properties of the MCT solutions are well understood for single component liquids [2] and mixtures [3]; in particular, it has been shown that the solutions allow a representation in terms of a continuous distribution of relaxation rates [4]. The success of the mode-coupling theory is based on the fact that it not only explains how the slow dynamics emerges in principle, but that it also has provided a series of testable predictions which challenge the physical intuition obtained so far. To mention just a few, MCT coherently explains the physics of suspensions of hard-sphere colloids [5], [6], [7], the reentrant phenomenon in ‘attractive colloidal glasses’ [8], [9], a pronounced minimum in the light-scattering spectra of supercooled liquids [10], [11], [12], or the composition dependence of the structural relaxation in mixtures [13], [14]. For many years only the standard mode-coupling equations have been discussed and attempts to go beyond have been focusing on extensions for non-spherical molecules [15], [16], [17], [18] and more recently to include shear in colloidal suspensions [19], [20].
The overall success of MCT is encouraging to investigate more complex glass transition scenarios to obtain a deeper insight into the nature and quality of the approximations involved. Dynamic heterogeneities have been identified to play an important role and systems that display strongly heterogeneous behavior could be the key to further elucidate the nature of the glass transition, a particularly interesting candidate are strongly size-disparate mixtures where structural arrest may occur in several steps. There, the majority component of large particles can undergo a glass transition characterized by a frozen disordered structure, whereas the minority species of smaller size meanders through the emerging network of channels. Neutron scattering experiments and molecular dynamics simulations indicate a significant separation of time scales for sodium silicate melts also at finite concentration [21], which could be rationalized within standard MCT calculations [22] later. A similar splitting of relaxation times has been found also for size-disparate soft spheres [23] and Yukawa mixtures [24]. The dynamics within a frozen matrix or nanoporous medium has been studied only recently [25], [26] corroborating an intriguing interplay of several dynamic transitions predicted within an extension of the mode-coupling theory for the dynamics within a disordered matrix [27], [28]. The standard MCT of mixtures appears to qualitatively describe many aspects of the splitting of the dynamics [29], yet ultimately predicts that the structural arrest of the majority and minority particles occurs at the same critical point. The observed peculiarities are rationalized as a precursor phenomenon for the two-step freezing not contained in the standard MCT. Krakoviack's extensions to disordered matrices treat the frozen obstacles and the fluid on unequal footing [27], [30] thereby allowing for multiple transitions. Although both approaches appear to give a satisfactory qualitative picture, theoretical issues remain that are poorly understood. First, the freezing in the disordered solid is accompanied by a divergent length scale [27], [28], [30], [31], [32], [33] suggesting that the transition is driven by an entire hierarchy of length scales rather than the Lindemann length for caging. It appears that MCT in its current form is ill-suited to deal with this phenomenon [1]. This manifests itself in ‘spurious long-time anomalies’ [30] that, if treated correctly, lead to a change in the exponent of the anomalous transport observed in the mean-square displacement [34].
An alternative approach dealing with the dynamics of a single tracer in a disordered matrix is within the framework of critical phenomena. The localization transition of the tracer is due to an underlying percolation transition of the void space as the density of the matrix is altered. As is well known, this geometric problem leads to a self-similar distribution of clusters [35] entailing a series of scaling predictions. It has been suggested that the dynamics of the Lorentz model shares the same universality class with a random resistor network with power law distributed conductances [36], [37], [38]. The critical dynamics of the Lorentz model in the vicinity of the percolation threshold can be described within a scaling ansatz for the van Hove correlation function [39]. Furthermore, corrections to scaling due to irrelevant scaling variables are not negligible and have been included recently [40], [41].
In this paper we investigate the Lorentz model by extensive computer simulations and show that the mean-square displacement becomes anomalous with an exponent differing from a simple fraction. The vanishing of the diffusion coefficient upon approaching the localization transition is connected to the subdiffusive motion and the divergence of the correlation length or mean cluster size. We discuss, in particular the motion of tracers that are confined to the percolating cluster in terms of the mean-square displacement, as well as the time-dependent diffusion coefficient averaged over all initial conditions of the tracers. We then elucidate the transport as response to an alternating external field in terms of the frequency-dependent conductivity and susceptibility. Furthermore, we characterize the space-resolved dynamics in terms of the intermediate scattering function, which is accessible in principle by scattering experiments in principle. Its long-time limit, known as the Lamb-Mößbauer factor or non-ergodicity factor, reveals the trapping of particles inside the finite clusters.
Section snippets
The Lorentz model
A simple model for transport in disordered solids capturing all the relevant ingredients for complex transport was introduced by Lorentz already in 1905 [42]. There a single tracer or equivalently a system of particles which do not interact among themselves traverses a course of immobilized obstacles. The position of the tracer is excluded from the space occupied by the obstacles thus confined to the void space. In the simplest case hard spherical obstacles are distributed independently and
Motion on the infinite cluster
Let us discuss first the dynamics of tracer particles that move on the percolating cluster only. Transport on the percolating cluster is expected to become fractal precisely at the critical obstacle density. After an elapsed time t a particle has moved typically a distance t1/dw, where dw is referred to as the walk dimension and attains the value of dw ≈ 4.81 [41] in the three-dimensional Lorentz model. For the mean-square displacement δr∞2(t): = 〈[R(t) − R(0)]2〉∞ this implies δr∞2(t) ~ t2/dw, where
All-cluster-averaged motion
Conventionally, the Lorentz model considers averages over all allowed starting positions of the tracer, i.e., both on any of the finite clusters as well as on the infinite cluster. Long-range transport occurs only on the percolating cluster, but close to the threshold the finite clusters become arbitrarily large and contribute to transport at all scales. The motion of an ensemble of tracers in such a fractal landscape is also self-similar in the time-domain introducing a new exponent z
Frequency-dependent conductivity
In the context of ion conductors, one is interested in the frequency-dependent complex conductivity σ(ω) which quantifies the alternating electric current density j(ω) as response to a frequency-dependent homogeneous electric field E(ω) in the linear regime as j(ω) = σ(ω)E(ω). For disordered materials that are statistically isotropic, the current is parallel to the electric field and the conductivity σ(ω) transforms as a scalar. A decomposition into real and imaginary parts, σ(ω) = Re[σ(ω)] + i Im[σ(ω
Self-intermediate scattering function
Spatio-temporal information on the dynamics of the tracer can be extracted by considering the self-intermediate scattering function (ISF) defined by
Thus Fs(q,t) is the characteristic function of the displacements ΔR(t): = R(t)–R(0) considered as the random variable. We have anticipated already that due to statistical isotropy Fs(q,t) does not depend on the direction of the wavevector q, but only on its magnitude q = |q|. The intermediate scattering function encodes all
Lamb–Mössbauer factor
The long-time limit of the intermediate scattering functionis referred to as the Lamb–Mössbauer factor or non-ergodicity parameter, also known as the Edwards-Anderson parameter in the spin glass community [1]. A non-vanishing indicates that dynamic correlations are persistent forever implying that the dynamics is non-ergodic. Roughly speaking measures the fraction of particles that are trapped on length scale 2π / q. For the Lorentz model, the presence of finite clusters
Summary and conclusion
Transport in disordered material as studied in terms of the Lorentz model displays many facets that go beyond the simple subdiffusive increase of the mean-square displacement. We have elucidated some of these aspects as they become manifest in the mean-square displacement for tracers that are confined to the percolating cluster only, the time-dependent diffusion coefficient for an all-cluster average, the frequency-dependent conductivity and susceptibility in response to an AC electric field,
Acknowledgments
Thomas Franosch is indebted to Wolfgang Götze for many years of discussions and his priceless insight into complex transport connected to the glass and localization transition.
Financial support from the Deutsche Forschungsgemeinschaft via contract No. FR 850/6-1 is gratefully acknowledged. This project is supported by the German Excellence Initiative via the program “Nanosystems Initiative Munich (NIM).”
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