Dynamics of Rod-Like Macromolecules in Heterogeneous Materials

The localization transition and the critical properties of the Lorentz model in three dimensions are investigated by computer simulations. We give a coherent and quantitative explanation of the dynamics in terms of continuum percolation theory and obtain an excellent matching of the critical density and exponents. Within a dynamic scaling Ansatz incorporating two divergent length scales we achieve data collapse for the mean-square displacements and identify the leading corrections to scaling. We provide evidence for a divergent non-Gaussian parameter close to the transition. Transport in heterogeneous and disordered media has important applications in many fields of science including composite materials, rheology, polymer and colloidal science, and biophysics. Recently, dynamic heterogeneities and growing cooperative length scales in structural glasses have attracted considerable interest (Berthier et al., 2005; Bertin et al., 2005). The physics of gelation, in particular of colloidal particles with short range attraction (Campbell et al., 2005; Manley et al., 2005; Ruzicka et al., 2004; Zaccarelli et al., 2005), is often accompanied by the presence of a fractal cluster generating sub-diffusive dynamics. It is of fundamental interest to demonstrate the relevance of such heterogeneous environments on slow anomalous transport. The minimal model for transport of particles through a random medium of fixed obstacles, is known as Lorentz model, and already incorporates the generic ingredients for slow anomalous transport. Earlier, the Lorentz model has played a significant role as a testing ground for elaborate kinetic theories, 26 LOCALIZATION AND CONTINUUM PERCOLATION shortly after the discovery of long-time tails in auto-correlation functions for simple liquids in the late 1960s (Alder andWainwright , 1970), since the nonanalytic dependence of transport coefficients on frequency, wavenumber, and density predicted for simple liquids (Bedaux and Mazur , 1973; Dorfman and Cohen , 1970; Ernst et al., 1971;Kawasaki , 1971; Tokuyama andOppenheim , 1978) has a close analog in the Lorentz model (Ernst and Weijland , 1971; Weijland and van Leeuwen , 1968). The simplest variant of the Lorentz model consists of a structureless test particle moving according to Newton’s laws in a d-dimensional array of identical obstacles. The latter are distributed randomly and independently in space and interact with the test particle via a hard-sphere repulsion. Consequently, the test particle explores a disordered environment of possibly overlapping regions of excluded volume; see Fig. 2.1. Due to the hard-core repulsion, the magnitude of the particle velocity, v = |v|, is conserved. Then, the only control parameter is the dimensionless obstacle density, n∗ := nσ d , where σ denotes the radius of the hard-core potential. At high densities, the model exhibits a localization transition, i. e., above a critical density, the particle is always trapped by the obstacles. Significant insight into the dynamic properties of the Lorentz model has been achieved by a low-density expansion for the diffusion coefficient by Weijland and van Leeuwen (1968) rigorously demonstrating the non-analytic dependence on n∗. As expected, for low densities the theoretical results compare well with Molecular Dynamics simulations (Bruin , 1974). Elaborate selfconsistent kinetic theories (Götze et al., 1981a,b; Masters and Keyes , 1982) have allowed going much beyond such perturbative approaches. They give a mathematically consistent description of the localization transition, which allows to calculate the critical density within a 20% accuracy and extend the regime of quantitative agreement to intermediate densities. In addition, they have provided a microscopic approach towards anomalous transport and mean-field-like scaling behavior (Götze et al., 1981b). A different line of approach focusing on the localization transition starts from the fractal nature of the void space between the overlapping spheres in the Lorentz model and considers it as a continuum percolation problem (Elam et al., 1984; Halperin et al., 1985; Kertész, 1981; Machta and Moore , 1985; Stenull and Janssen , 2001), which in this context has also been termed “Swiss cheese” model (Halperin et al., 1985). These authors conjectured that the transport properties close to the percolation threshold can be obtained by analyzing an equivalent random resistor network. The equivalence, however, has been shown only for geometric properties close to the percolation point (Ker2 The Localization Transition and Continuum Percolation 27 Figure 2.1 Typical particle trajectories in a two-dimensional Lorentz model slightly below nc over a few thousand collisions each. Colors encode different initial conditions; obstacles have been omitted for clarity. Most trajectories being in the percolating void space have some overlap; a few trajectories are confined to finite clusters. Blowup: a particle squeezes through narrow gaps formed by the obstacles. stein , 1983). As a peculiarity of continuum percolation, differences to lattice percolation may arise due to power law tails in the probability distribution of the conductances (“narrow gaps”). Such random resistor networks have been investigated extensively by means of Monte-Carlo simulations (Derrida et al., 1984;Gingold and Lobb , 1990) and renormalization group techniques (Harris et al., 1984; Lubensky and Tremblay, 1986), providing reliable numeric and analytic results for the critical behavior (Havlin and Ben-Avraham , 2002). In this Letter, we present a direct numerical analysis of the dynamic properties of the Lorentz model without resorting to random resistor networks. By means of extensive Molecular Dynamics simulations, we obtain a quantitative description of the dynamic properties over the full density range, in particular, focusing on both sides of the critical region. This allows for a quantitative test of the conjectured mappings to continuum percolation theory. Furthermore, we explore the range of validity of the dynamic scaling hypothesis for the Lorentz model (Kertész and Metzger , 1983). The probability distribu28 LOCALIZATION AND CONTINUUM PERCOLATION tion of particle displacements, i. e., the van Hove self-correlation function, G(r, t) := 〈 δ(r −1R(t)) 〉 , and its second moment, the mean-square displacement, δr(t) := 〈 |1R(t)|2 〉 , are the appropriate quantities for this purpose; 1R(t) = R(t)− R(0) denotes the displacement of the test particle at time t . Over a wide range of obstacle densities, we have simulated several hundred trajectories in three dimensions, employing an event-oriented Molecular Dynamics algorithm. For each of Nr different realizations of the obstacle disorder, a set of Nt trajectories with different initial conditions is simulated. Below the critical density, we have chosen Nr ≥ 25 and Nt ≥ 4. At very high densities, where the phase space is highly decomposed, these values have been increased up to Nr × Nt = 600. In order to minimize finite-size effects, the size of the simulation box, Lbox, has been chosen significantly larger than the correlation length ξ , Lbox = 200σ ≫ ξ . The results for the mean-square displacement cover a non-trivial time window of more than seven decades for densities close to the transition, see Fig. 2.2. At low densities, one observes only a trivial cross-over from ballistic motion, δr(t) = vt, to diffusion, δr(t) ∼ t , near the mean collision time τ = 1/πnvσ 2 as expected from Boltzmann theory. With increasing density, an intermediate time window opens where motion becomes sub-diffusive, δr(t) ∼ t with z > 2. This time window extends to larger and larger times upon approaching a certain critical density nc . For the density n ∗ = 0.84, the sub-diffusive behavior is obeyed over more than five decades and is compatible with a value of z ≈ 6.25. The power law, δr(t) ∼ t , indicated in Fig. 2.2, discriminates nicely trajectories above and below nc . One also observes a density-dependent length scale l characterizing the end of the subdiffusive regime by δr(t) ≃ l; upon approaching nc this cross-over length scale l is found to diverge. For long times, the dynamics eventually becomes either diffusive or localized for densities below or above nc , respectively. The diffusion coefficient D has been extracted from the long-time limit of δr(t)/6t ; in Fig. 2.4, D is shown in units of the Boltzmann result, D0 = τv/3. With increasing density, D is more and more suppressed until it vanishes at nc as a power law, D ∼ |ε|μ, where ε := (n∗ − nc )/nc defines the separation parameter. Anticipating the exponent μ from percolation theory, a fit to our data yields the critical density, nc = 0.839(4), and the power law 1This relation may be violated for |ε| < 0.01 We checked that our findings are not affected by finite-size effects. A detailed analysis is presented in Section 4.3. 2This value for nc corresponds to a critical volume fraction for the obstacles, φc = 1 − exp(− 4π 3 nc ) = 0.9702(5). 2 The Localization Transition and Continuum Percolation 29 10 -2 10 0 10 2 10 4 10 6 10 0 10 2 10 4 10 6 10 8 0.30 0.40 0.50 0.60 0.65 0.70 0.75 0.78 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.90 0.95 1.00 1.10 Time t/v−1σ δ 2 (t )/ σ 2 Density n∗ L2box