Localization phenomena in models of ion-conducting glass formers
Jürgen Horbach, Thomas Voigtmann, Felix Höfling, Thomas Franosch
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Localization phenomena in models of ion-conductingglass formers
J¨urgen Horbach , a , Thomas Voigtmann , , Felix H¨ofling , , and Thomas Franosch , Institut f¨ur Materialphysik im Weltraum, Deutsches Zentrum f¨ur Luft- und Raumfahrt (DLR), 51170K¨oln, Germany Zukunftskolleg und Fachbereich Physik, Universit¨at Konstanz, 78457 Konstanz, Germany Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, England, UnitedKingdom Max-Planck-Institut f¨ur Metallforschung, Heisenbergstraße 3, 70569 Stuttgart and Institut f¨ur Theore-tische und Angewandte Physik, Universit¨at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany Institut f¨ur Theoretische Physik, Universit¨at Erlangen-N¨urnberg, Staudtstraße 7, 91058 Erlangen,Germany Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS),Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstraße 37, 80333 M¨unchen,Germany
Abstract.
The mass transport in soft-sphere mixtures of small and big particlesas well as in the disordered Lorentz gas (LG) model is studied using moleculardynamics (MD) computer simulations. The soft-sphere mixture shows anoma-lous small-particle diffusion signifying a localization transition separate from thebig-particle glass transition. Switching off small-particle excluded volume con-straints slows down the small-particle dynamics, as indicated by incoherent in-termediate scattering functions. A comparison of logarithmic time derivatives ofthe mean-squared displacements reveals qualitative similarities between the lo-calization transition in the soft-sphere mixture and its counterpart in the LG.Nevertheless, qualitative differences emphasize the need for further research elu-cidating the connection between both models.
Transport in heterogeneous disordered materials and in porous media arises in many diversesubjects of science and engineering such as water resource management, oil recovery, the physicsof crowded biological systems, glass technology or even the geophysical understanding of theeruption of volcanos [1]. Examples are fast ion transport in alkali-silicate melts and otherindustrially relevant glass formers [2], protein motion on cell membranes [3, 4] and within theliving cell [5], among others. Common to all these materials is that they consist of at leasttwo components, one of which (the “fast component”) is responsible for the observed transportphenomena, while the other forms the heterogeneous matrix and is either completely frozen as,e.g., in porous media, or relaxes orders of magnitude slower, as in glass-forming ion conductors.A well-known reference point for diffusion in random environments is the Lorentz gas(LG) [6, 7]: considering a single particle diffusing through a set of fixed obstacles, one essentiallyarrives at an off-lattice model for the transport through porous media or ion-conducting sys-tems. The LG exhibits a localization transition where diffusion of the tracer ceases altogether if a e-mail: [email protected] Will be inserted by the editor -1 t F s ( q , t ) q σ =0.74 (0.18)2.26 (0.44)9.6 (1.95)17.5 (3.96) Fig. 1.
Incoherent intermediatescattering function F s ( q, t ) for thesmall/tracer particles. Solid anddashed lines correspond respectivelyto the BSSM with and withoutinteractions among the small par-ticles, while dotted lines representthe LG model. For the BSSM,the wavenumbers are in the range0 . ≤ qσ ≤ .
5, as indicated (with σ the diameter of the big particles).The wavenumbers in brackets corre-spond to the LG. The number densityof the BSSM is ρ = 3 .
26. For the LG,the packing fraction of the obstaclesis n ∗ = 0 . the density of scatterers exceeds a certain value; this transition is understood as a dynamic crit-ical phenomenon connected with the percolation of void space in the heterogeneous backgroundmedium [8, 9].Obviously, several aspects of ion-conducting systems are described in an oversimplified man-ner by the LG model. In a real ion conductor, the frozen environment is not formed by fixedobstacles that are randomly distributed. Instead, the “obstacle particles” exhibit structuralcorrelations (at least on short length scales) and they are subject to thermal motion. Moreover,in most ion-conducting systems the mobile ions cannot be considered as non-interacting tracerparticles and the correlations between ions have to be taken into account. Realistic modeling forsodium silica melts [10–12] reveals that the silica network can constitute such a quasi-arrestedarray of correlated obstacles through which sodium ions meander on preferential diffusion path-ways. The chemical properties of the silica melt to form a tetrahedral network appears not tobe essential: a similar decoupling of diffusive transport is observed in dense size-asymmetricYukawa melts [13] and binary soft-sphere mixtures [14]. Mode-coupling theory has been usedto predict it for size-disparate hard-sphere systems [15, 16]. As shown recently by moleculardynamics (MD) computer simulations [17], such systems may exhibit a glass transition wherethe fast component remains mobile even for high-density states at which the matrix does notshow relaxation over the entire simulation time window. At these high densities, evidence wasfound that the small particles show anomalous diffusion and approach a localization transi-tion that bears resemblance to the LG transition. Similar findings hold for an entirely frozenmatrix, where non-trivial predictions of a mode-coupling theory developed by Krakoviack [18]have been tested recently [19, 20]. However, it is an open question to what extent the small-particle localization dynamics in binary soft-sphere mixtures can be understood as a dynamiccritical phenomenon that falls into the same universality class as the localization transition ofthe LG [8, 9] and other continuum percolation models [21]. In particular, the question ariseshow the localization of the small particles is affected by interactions between the mobile tracers.In this contribution, we present first steps to address the latter issues. We compare on aqualitative level the anomalous diffusion of a binary soft-sphere mixture with that of a LG. Theeffect of collective interactions among the small particles is investigated by switching off theirinteractions. We performed MD simulations of an equimolar binary mixture of purely repulsive soft spheres(BSSM) with a size ratio of 0.35. Diameters are chosen additively, and nonadditive energeticinteractions further decouple the species. Temperature is unity, and all masses are equal. For ill be inserted by the editor 3 -2.0 0.0 2.0 4.0 6.0 8.0log( t )0.00.51.01.52.0 γ ( t ) BSSM - with interactionBSSM - no interactionLG1 23
Fig. 2.
Effective exponent, as obtained from thelogarithmic derivative γ ( t ) of the mobile parti-cles for the binary mixture with and without in-teractions between the small particles as well asfor the LG. Results are shown for different den-sities: the three sets marked with number 1 cor-respond to ρ = 2 .
296 and n ∗ = 0 .
40, those withnumber 2 to ρ = 3 .
257 and n ∗ = 0 .
75 and thosewith number 3 to ρ = 4 .
215 and n ∗ = 0 .
82 (here, ρ and n ∗ are the densities of the binary mixturesand the LG, respectively). details, see Ref. 17. MD simulations of the LG were carried out using an event-driven algo-rithm and randomly distributed, possibly overlapping hard sphere obstacles. The dimensionlessobstacle density n ∗ is the only control parameter of the model; details can be found in Ref. 9.Figure 1 shows the time dependence of the incoherent intermediate scattering function, F s ( q, t ), of the small particles for different values of the wavenumber q , as indicated. For theBSSM, at a number density ρ = 3 .
26, the big particles form a glass and the transport of the smallparticles is characterized by anomalous subdiffusive motion crossing over to ordinary diffusionat long times. Two kinds of BSSM systems are considered in Fig. 1: the solid lines correspondto the system with interacting small particles, while the dashed lines correspond to a systemwhere the small–small interaction potential is set to zero, keeping all other parameters fixed.Structure and dynamics of the big particles as measured through the partial static structurefactor are unaffected by this [17]. Surprisingly, at fixed q the curves for the fully interactingcase decay more rapidly than those for the non-interacting case. In Ref. 17, we have proposedthat this difference arises because small-particle interactions result in a more directional andthus more effective diffusive motion.Also shown in Fig. 1 are results for the LG at a packing fraction of the obstacles n ∗ = 0 . n ∗ , the decay of F s ( q, t ) takes place on a similar time scale provided thatone reduces the wavenumbers by about a factor of 4 (as done in Fig. 1) to match the lengthscales of the LG and the BSSM systems. The curves for the LG demonstrate the absence ofa “glassy” plateau at intermediate times. At long times, they decay to a plateau of differentorigin: the presence of closed, finite pockets in the obstacle matrix leads to a localization oftracer particles. However, apart from that the decay of the LG correlators is similar to that forthe BSSM systems. So the question arises whether the transport as seen for the non-interactingparticles in the binary mixture is similar to that seen in the LG.To address this question, we compare the logarithmic derivative γ ( t ) = d[log δr ( t )] / d(log t )for the two binary mixtures and for the LG in Fig. 2; γ ( t ) is calculated from the mean-squareddisplacements δr ( t ) of the mobile particles in both systems. The quantity γ ( t ) has the meaningof an effective exponent that crosses over from γ (0) = 2 (ballistic short-time motion) to γ ( ∞ ) =1 for diffusive particles in the liquid. For localized particles, γ ( ∞ ) = 0 is expected.The selected obstacle densities for the LG in Fig. 2, n ∗ = 0 .
40, 0.75 and 0.82, are below thecritical density of the localization transition, n ∗ c = 0 .
837 [9]. At criticality n ∗ = n ∗ c , the exponent γ ( t ) ≈ .
32 is approached at long times [8], thus confirming the universal subdiffusive behaviorexpected from renormalization group arguments (refer to Ref. 9 for a detailed discussion).Below n ∗ c , such anomalous diffusion is seen in a finite, intermediate time window and universalcorrections to scaling [22] lead to the observation of an effective exponent that is larger than theuniversal critical exponent. At n ∗ = 0 .
82 for example, an effective exponent γ ≈ . Will be inserted by the editor
At low densities (set of curves “1” in Fig. 2), the binary mixtures with and without interac-tions between the small particles show almost identical behavior as the LG with respect to γ ( t ).Nevertheless, there are quantitative differences: At higher densities the effective exponent ofthe BSSM without interaction becomes rather small, the corresponding mean-square displace-ment remains of the order of the interparticle distance, and the cage effect interferes stronglywith the regime of anomalous diffusion. Turning on the interaction, a window of subdiffusionemerges, yet the effective exponent drifts gradually from 0.5 to 0.6. In contrast, the LG displaysclear subdiffusive behavior over several orders of magnitude in time as manifested in an almostconstant γ ( t ), as mentioned above.Clearly, a simplistic mapping between the BSSM and the overlapping LG is not obviousand may not exist due to the highly simplified character of the LG. Thus, the investigationof LG variants with increasing complexity is desirable, which close the gap to the BSSM. Forexample, a better approximation of the matrix structure would be obtained by introducingcorrelated obstacles in the LG (“non-overlapping LG”). Another issue is raised by the hardinteraction potentials in the LG. While a tracer with constant velocity between soft obstacles iseasily mapped to the hard sphere model, the percolation transition is already smeared out forthermal tracers (or tracers in contact with a heat bath) due to the average over the Maxwelldistribution. All these issues are the subject of forthcoming studies.This collaborative work has been started as part of the Research Unit FOR 1394 of DeutscheForschungsgemeinschaft (DFG). We acknowledge a substantial grant of computer time at theNIC J¨ulich. Th. V. acknowledges funding from the Helmholtz-Gemeinschaft (VH-NG 406) andthe Zukunftskolleg Konstanz. References
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