# Mobile particles in an immobile environment: Molecular Dynamics simulation of a binary Yukawa mixture

###### Abstract

Molecular dynamics computer simulations are used to investigate the dynamics of a binary mixture of charged (Yukawa) particles with a size–ratio of 1:5. We find that the system undergoes a phase transition where the large particles crystallize while the small particles remain in a fluid–like (delocalized) phase. Upon decreasing temperature below the transition, the small particles become increasingly localized on intermediate time scales. This is reflected in the incoherent intermediate scattering functions by the appearance of a plateau with a growing height. At long times, the small particles show a diffusive hopping motion. We find that these transport properties are related to structural correlations and the single–particle potential energy distribution of the small particles.

###### pacs:

61.20.Lc###### pacs:

66.30.Hs###### pacs:

61.43.ErTime-dependent properties; relaxation Self-diffusion and ionic conduction in nonmetals Other amorphous solids

## 1 Introduction

The dynamics of fluid particles in an immobile environment has been of fundamental interest to understand anomalous transport processes in confined geometry and porous media [1]. Examples include different types of Lorentz gas models [2, 3, 4, 5], polymers in quenched disorder [6, 7, 8], hard–sphere mixtures [9] and ion–conducting silicates [10, 11]. Simple model systems that provide a time–scale separation of transport properties among different species are mixtures of small and large particles at high densities. While it is difficult to experimentally realize such systems on an atomistic scale (for an exception see Ref. [12]), it is possible to carry out experiments on colloidal suspensions that contain disparately–sized particles. About a decade ago, Imhof and Dhont [13, 14, 15] performed dynamic light scattering experiments on a binary mixture of colloidal silica particles with a size ratio of 1:9.3. The effective interactions between colloids are hard–sphere–like, hence phase behavior and transport properties are governed by packing effects. An interesting finding of Imhof and Dhont is the existence of different phases where the large particles exhibit a structural arrest, yet the small particles are still mobile. The simplest case of such a phase could be one consisting of mobile small fluid-like particles in a crystalline matrix of large particles. It can be realized, at least, in an intermediate regime below the freezing transition.

In this work, we use molecular dynamics simulations to study a binary mixture of charged particles with a size–ratio of 1:5. Similar to the experiments by Imhof and Dhont [13, 14, 15], the system exhibits a phase transition from a fluid to a mixture of crystalline large particles and fluid–like small particles. When temperature is sufficiently lowered, the small particles are localized and crystallize, eventually, in a sublattice with respect to the large–particle lattice. In the temperature regime between the freezing transition and the final crystallization into a sublattice, the movement of the small particles can be characterized by a hopping motion in an external periodic potential (created by the large particles). Similar to supercooled liquids, a “ relaxation regime” can be identified in the incoherent intermediate scattering functions (with the wavenumber and the time) of the small particles. These functions show a plateau region between the microscopic regime and the long–time decay to zero. At a given wavenumber, the height of the plateau increases with decreasing temperature. This indicates that the particles are more and more localized. Thus, the small particles display a kind of delocalization–to–localization transition around the liquid–to–crystal transition of the large particles. Although the effective packing fraction of the small particles, considered in this work, is relatively small, structural correlations among the small particles occur, associated with a complex distribution of single particle potential energies.

The system considered in this work shows various nontrivial features that might be generic for the dynamics in confined geometry and porous media, especially when Coulomb–like interactions become important. It can be also realized experimentally in charged colloidal suspensions. Therefore, with a similar system as the one used by Imhof and Dhont [13, 14, 15] the predictions of our model could be rationalized experimentally. On the other hand, our model considers essentially the motion of mobile particles in a periodic potential. It should be much simpler to develop analytic theories for this case than for the transport of particles in a disordered potential energy landscape.

## 2 The model and simulation details

Charged particles (colloids) are modelled by an effective screened Coulomb (or Yukawa) potential,

(1) |

where is the distance between particles and . The parameters () denote the distance between two particles at contact, , with the radius of an -particle. The interaction range is characterized by the screening parameter . In the limit , the Yukawa potential corresponds to bare Coulomb interactions.

A shifted Lennard–Jones (WCA) potential is used to model the excluded volume of the particles:

(2) |

The energy parameter is set to . Note that the potential, Eq. (2), is purely repulsive. It can be regarded as an approximation to a hard–sphere interaction for two particles at contact.

The binary Yukawa system of volume fraction consists of small and big particles in three dimensional space. The size ratio between them is . We choose , and , and mass . The Yukawa interactions are short–ranged with a fixed value of the inverse screening parameter (screening length) where potential energies are of the order of . For the energy parameters, the values , , and are chosen. This choice implies that small and large particles have the same surface charge density. In the following, all the physical quantities are measured in units of the large particle’s mass , diameter and energy . The Boltzmann constant is set to . . Note that a cut–off is introduced at

Newton’s equations of motion for the charged particles are integrated using the velocity form of the Verlet algorithm. The time step is chosen to be for , for , and for (note that is measured in units of the time ).

At each temperature, the system was first equilibrated in the ensemble by coupling it to a stochastic heat bath. The equilibration time was sufficiently longer than the time needed to observe a diffusive motion of the small particles (typically, a factor of longer for , and a factor of for .) At the lowest temperatures (), the production runs were over 10 to 20 million time steps. This was not enough to see a diffusive motion of the small particles. Note that for the equilibration at a given temperature, we used a final configuration of the next–higher temperature as an initial configuration. Microcanonical production runs were performed at the temperatures , 2.5, 2.0, 1.0, 0.8, 0.5, 0.4, 0.2, 0.15, 0.1, 0.05, 0.04, 0.03, 0.025, 0.02, 0.015, 0.012, 0.01. At each temperature, eight independent runs were done to improve statistics.

## 3 Results

First, we consider partial pair correlation functions [16] () that measure structural correlations between small (“s” ) and large (“l”) particles. These quantities are proportional to the probability to find a particle of type at a distance from a particle of type .

Upon decreasing temperatures the large particles crystallize around into a fcc lattice structure. Fig. 1 shows the above and slightly below the freezing transition of the large particles at and , respectively. We see that the curve for at shows only tiny differences to that at . This also holds for the functions at the two temperatures. Beyond the first peak the amplitude in and is only slightly different at and at . This indicates a fluid–like behavior in the small–small and small–large correlations. In contrast, the function changes significantly from to . Note that the rather broad peaks at are reminiscent of a pair correlation function of a normal liquid state. However, a closer inspection of other structural quantities such as the static structure factor shows that the large particles form a fcc crystal and the broad peaks in are due to thermal fluctuations at the relatively high temperature . This can be also infered from the snapshot at (see Fig. 1). One can clearly identify the crystalline planes, which are strongly distorted by the thermal motion of the large particles.

By decreasing temperature from to , significant structural changes occur. In Fig. 2, both and the snapshot clearly show the long–range structural order of the fcc crystal at . The crystalline structure of the large particles is also reflected in the correlations among the small particles at . The function displays peaks at and . The second peak at is at the same position as the first peak in , which corresponds to the nearest neighbor distance in the fcc crystal. This indicates the tendency of the small particles to populate a sublattice with respect to the fcc structure of the large particles. An interesting feature is the rather sharp peak of at . This is due to the coordination of a small particle with another small particle. It means that some of the small particles form pairs in a given local site. Note that approximately 75% of the small particles have no nearest neighbor, 20% exhibit a pairing, and 5% of the small particles are two–fold or three–fold coordinated. The occurrence of the small particles with zero neighbour or pairs in local sites is also reflected in . The first peak in this function shows a shoulder around , which is approximately the distance between a large particle and the next–nearest small particle of a small particle pair.

The periodic structure of the large particles’ crystalline matrix is accompanied by fluid–like small particles, at least at high temperatures. To characterize in detail the “disorder” in the small–small correlations, we show in Fig. 3 the small particles’ distribution of single–particle potential energies (denoted by ) at four different temperatures. It is remarkable that the main peak in moves to higher energies while temperature decreases. This means that the formation of the fcc structure by the large particles is not energetically favoured by the small particles as it increases their potential energy on average. Another significant feature of is the emergence of a high–energy tail. This is magnified in the inset of Fig. 3 by a double–logarithmic plot of the data. This high–energy tail tends to dissappear towards low temperatures. This is due to the fact that the kinetic energy sets an upper bound for the accessible potential energies.

Having discussed static properties of the model, we now turn our attention to transport properties of the small particles in the immobile environment of the large particles. Fig. 4a displays the small–particle self-diffusion constant as a function of temperature in a double–logarithmic plot (note that we determined from the mean–squared displacement of the small particles via the Einstein relation [16]). As temperature is decreased from (i.e. above the freezing transition of the large particles) to , the self-diffusion constant decreases by three and a half orders of magnitude. Note that the functional form of can neither be described by a power law nor by an Arrhenius behavior.

A detailed description of the small–particle dynamics can be obtained from the incoherent intermediate scattering functions , the self–part of the time–dependent density–density correlation function for the small particles [16]. In Fig. 4b this function is shown for different temperatures at . This wavenumber corresponds to the location of the first Bragg peak in the static structure factor for the large–large correlations. Note that for other wavenumbers a similar behavior of is observed. Above the liquid–to–crystal transition of the large particles at , exhibits an exponential one–step decay to zero, as expected for simple liquids. Below , a plateau develops at intermediate times, followed by a non–exponential decay to zero. The emergence of the plateau is associated with a caging of the particles. In this case, the small particles are trapped in local potential basins created by the large particles. The height of the plateaus in (also called Lamb–Mössbauer factor [17]) measures the localization of the particles with respect to the length scale corresponding to the considered wavenumber [17]. We can infer from Fig. 4b that around the small particles undergo a transition from a completely delocalized state (with vanishing plateau) to a more and more localized state towards low temperatures. In the light of the potential energy distribution (see Fig. 3), the delocalization–to–localization transition of the small particles is related to the gradual disappearance of the high–energy tails.

The plateau sets in by an oscillation. With decreasing temperature, this oscillation and the whole microscopic regime, i.e. the decay of the curves onto the plateau, shift to the right on the time axis. This behavior corresponds to a softening of the cage towards lower temperatures and is the subject of a forthcoming publication.

Further information on the long–time regime and the diffusive motion of the small particles can be extracted from the Fourier transform of , the self-part of the van Hove correlation function, [16]. is the probability to find a particle at time at a distance away from the origin at . In Fig. 5, this quantity is shown for different times at and in a linear–logarithmic plot. At the high temperature, we observe a regular behavior, which is similar to that of simple liquids. With increasing time, the location of the peaks moves continuously to larger distances. However, the behavior is totally different at low temperatures: At , the correlation function exhibits a single peak. As time goes on, a second peak develops and if one waits until five peaks can be identified. The distance between the peak maxima is around , which corresponds to the lattice constant of the large particle’s fcc lattice. We can conclude from this that at low temperatures the small particles do not diffuse in a continuous manner but discontinuously in time by hopping along the sublattice sites of the large particle’s fcc lattice. Note that this interpretation has been first given for similar features in the glassy behavior of a soft–sphere model by Roux et al. [18]

## 4 Conclusions

We have presented an extensive MD simulation study of a binary Yukawa mixture with size ratio 1:5. In this system, interesting phase behavior is observed where the crystallization of the large particles is accompanied by a dynamic delocalization–to–localization transition of the small particles. We speculate that this kind of phase behavior is generic for broad classes of complex systems, in particular mixtures of disparate–sized (charged) colloids (see, e.g., the experiments by Imhof and Dhont [13, 14, 15]). The motion of proteins in lipid cubic phases [19] might be also very similar to that of the small particles in our Yukawa mixture. However, we are not aware of any systematic experiments on fluid–like particles in a crystalline matrix. We hope the present work will stimulate new experimental efforts in this direction.

###### Acknowledgements.

Financial support of the DFG (SFB 625, SFB TR6, and the Emmy Noether programme, grant No. HO 2231/2) and the MWFZ Mainz are gratefully acknowledged. We thank the NIC Jülich for a generous grant of computing time on the JUMP.## References

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