Structural and dynamic properties of colloids near jamming transition

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1 Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) Structural and dynamic properties of colloids near jamming transition Anil Kumar, Jianzhong Wu Department of Chemical and Environmental Engineering, University of California, Brourns Hall Room A317, Riverside, CA 92551, USA Received 2 February 2004; accepted 9 July 2004 Abstract Molecular dynamics simulations were used to study the jamming transitions occurring in colloidal dispersions interacting through a short-ranged van der Waals attraction and a longer-ranged electrostatic repulsion as represented by the classical Derjaguin Landau Verwey Overbeek (DLVO) theory. The structural and dynamics properties near the jamming transition were investigated at various parameters leading to the dynamic arrest. The structural investigations revealed that at low-volume fractions, aggregation of particles lead to chain-like clusters and the gel-like structures were formed due to the caging effects. We demonstrated that following the behavior of any of the dynamical properties was sufficient to deduce the jamming phase diagram as all of them showed a critical-like behavior at the onset of the jamming transition. The simulation results also suggested that a long-range repulsive interaction was essential for the formation of a glass-like structure at low-colloidal volume fractions Elsevier B.V. All rights reserved. Keywords: Molecular dynamics simulation; Derjaguin Landau Verwey Overbeek (DLVO) theory; Jamming transition 1. Introduction The intriguing physical phenomena related to the nonequilibrium phase behavior of colloidal dispersions are of great interest for a variety of applications ranging from material fabrications to protein crystallography [1]. An emerging view from the literature is that a variety of loosely defined fluid solid transitions in colloids may follow essentially the same physical behavior [2,3]. These transitions embrace apparently disparate solidification phenomena as diverse as particulate and polymer gelation, coagulation, aggregation, and glassification. This converging viewpoint is supported by recent advances in understanding the role of attractive forces in glass transition and has been generalized in terms of jamming diagram. However, the exploration of these ideas and the development of a common interpretive scheme for a broad variety of non-equilibrium processes are still at the very early stage. Corresponding author. Tel.: ; fax: address: jwu@engr.ucr.edu (J. Wu). A conventional theoretical approach for describing the offequilibrium solidifications in colloidal systems is provided by the mode-coupling theory (MCT), which was originally developed for atomic systems interacting thorough a spherically symmetric intermolecular potential [4 7]. This theory is based on the caging effect of condensed systems that has been known as the essential feature distinguishing the dynamics of a liquid from that of a glass. Later, MCT was applied to colloidal glass transitions in the framework of a pseudo-onecomponent model where colloidal forces are represented by the simple hard-sphere-like potentials [8]. It has been successfully applied to interpreting some observations associated with glass transitions in colloids. MCT has achieved good numerical agreements in comparison with results from light-scattering and computer simulations [9 14]. In particular, MCT gives an accurate description of the dynamic slowing down near the transition from a super-cooled liquid to an amorphous solid-like state driven by the short-range attractive interactions [15 19]. It predicts that a reentrant liquid to glass transition as recently observed in experiments [20 24]. Most previous simulation studies on the non-equilibrium phase transitions of colloids have been focused on the /$ see front matter 2004 Elsevier B.V. All rights reserved. doi: /j.colsurfa

2 146 A. Kumar, J. Wu / Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) questions regarding the behavior of colloidal systems with only short-ranged attractive interactions in the very dense regime, where the predictions of the MCT are manifested [21,23,25,26]. The low-volume fraction amorphous gels are considered to be a natural extension of the attractive glass, both being driven by the same underlying mechanism of dynamical arrest [15]. Recent simulation results suggest that another mechanism governed by a weak long-ranged repulsion may also play an important role in the gel formation [27]. In this work, molecular dynamics (MD) simulations are applied to exploring the details of jamming transitions in static colloidal systems where the colloidal forces consist of both shorted-ranged attraction and long-ranged repulsions as involved in most practical colloids. Here we use MD instead of Brownian or Stokesian dynamics because near the jamming transition both Brownian motion and hydrodynamic forces are negligible in comparison with much stronger direct interparticle interactions. 2. Simulation details The inter-particle interaction between colloidal particles is represented by the standard Derjaguin Landau Verwey Overbeek (DLVO) potential that consists of excluded volume and electrostatic repulsions and a van der Waals attraction V (r) = A ( κr) + B exp C r36 r r 6 (1) where V(r) represents the reduced (divided by a unit energy ε) colloidal potential and r is the reduced inter-particle separation (divided by colloidal diameter σ). In Eq. (1), the parameters A = 0.881, B = , C = and κ = 2.0 are selected such that the interaction mimics that between polystyrene particles [28]. We use the soft potential instead of the hard-sphere model for the short-ranged repulsion in favor of the numerical convenience of MD simulations. Fig. 1 depicts a schematic plot of the colloidal potential Fig. 1. The inter-particle pair potential used in this study, given by Eq. (1). as given by Eq. (1), which includes a short-ranged attractive well and a slightly longer-ranged repulsive barrier. Unlike a Lennard Jones-like potential that is commonly used to model simple fluids, Eq. (1) has a long-ranged repulsive part as well as short-ranged van der Waals attraction. As shown later and reported by others [23,24,29], the long-ranged repulsion suppresses the spontaneous equilibrium phase separation as the system approaches the long-lived non-equilibrium states. Systems interacting through the Lennard Jones-type potentials will separate into two completely distinct phases when they are quenched into the coexistence region. These two phases do not interpenetrate to form a percolating network, nor do they completely fill the simulation volume. All dynamic properties are calculated by using standard NVE molecular dynamic simulations [30]. To change the temperature, we first use NVT ensemble simulation to estimate the total energy followed by NVE simulation for calculating the structural and dynamic properties. The simulated system consists of 512 colloidal particles in a cubic box with the periodic boundary conditions in all three dimensions. The equations of motion are integrated using the velocity Verlet algorithm [31]. The simulations have been carried out at different volume fractions and temperatures. The system is equilibrated for sufficiently long-time, during which the velocities are rescaled to obtain the desired temperature. Once the total energy becomes a constant and the fluctuation in temperature is within the desired range of ±0.01 (in reduced units), the production run is carried out for one million time steps and the averages are calculated. The trajectory of the system during the production run is analyzed to obtain different dynamical properties. 3. Structural and dynamical properties 3.1. Structure Fig. 2 shows the radial distribution functions of the colloids at a volume fraction of 0.30 and different temperatures. At higher temperatures, the system is in a liquid state, which is evident from the radial distribution functions. But as the temperature is decreased, the second peak in the radial distribution function splits up. At a very low temperature (T = 0.15), the system was jammed into a glass-like structure, thereby prohibiting the appearance of crystalline structures. The split in the second peak into two sub-peaks can be taken as an indication of transition to an amorphous state [32]. Indeed, the splitting of the second peak in the correlation function has been generally accepted as a feature of an amorphous state as that follows from neutron and X-ray scattering experiments and from molecular dynamics simulations. Gazillo and Della Valle showed that the Ornstein Zernike equation also exhibits the second peak splitting in a Lennard Jones liquid in the case where HNC approximation was applied [33]. They concluded that the metastable liquid is also characterized by the changes in the second peak of the radial distribution func-

3 A. Kumar, J. Wu / Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) Fig. 2. Evolution of the radial distribution functions with decreasing temperature for a volume fraction φ = 0.3. Near the jamming transition the second peak broadens out and eventually splits into two sub-peaks. tion, namely by broadening of the peak, which further transforms into a split. All these features are well reproduced in the simulations as shown in Fig. 2. Fig. 3 depicts some instantaneous snap shots of the jammed state at different volume fractions at T = At higher volume fractions (in the glass region), the system is crowded and the particles cannot move because of the excluded volume effect, i.e. the system gets jammed. However, as the volume fraction is decreased, more free volume becomes available and the particles form clusters. Each cluster is trapped in the cage formed by its neighboring clusters, thereby slowing down the dynamics of the system. A detail inspection of Fig. 3 reveals that these clusters are chain-like, much alike the gelation in an associating fluids. Similar observations have been reported by experiments [34] Dynamical properties The hallmark of off-equilibrium solidifications is the extreme slow dynamics as the temperature is lowered toward the transition state. As the temperature of a super-cooled liquid is further decreased, its viscosity swiftly increases and the particles comprising the system move more and more slowly until the system is arrested into a frozen state at the jamming transition. We have reported the behavior of shear viscosity near jamming transition in an earlier communication [27].

$15 at five different volume fractions, (a) φ = 0.4, (b) φ = 0.3, (c) φ = 0.2, (d) φ = 0.15 and (e) φ = 0.1. Fig.$

4 148 A. Kumar, J. Wu / Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) Fig. 3. Snapshots of instantaneous configurations of the jammed state at T = 0.15 at five different volume fractions, (a) φ = 0.4, (b) φ = 0.3, (c) φ = 0.2, (d) φ = 0.15 and (e) φ = 0.1. Fig. 4 shows the log log plot of the mean-squared displacement (MSD) of colloidal particles at five different temperatures. In all these temperatures, the colloidal volume fraction is fixed at φ = 0.3. At high temperatures, the mean-squared displacement exhibits the typical liquid-like behavior, i.e. after a short initial regime of ballistic motion (where <r 2 > t 2 ), MSD reaches the long-time diffusive regime (where <r 2 > t). As the system approaches the jamming transition, the strong interactions between particles hinder the particle motion and eventually leading to the jammed state. Then MSD takes longer-time to reach the diffusive regime after the ballistic regime. The transition is manifested in the MSD as a plateau at the intermediate times. This plateau region extends several orders of magnitude in time as the temperature decreased further and the system goes to a jammed state. Next, we consider the self-part of the intermediate scattering function F(k,t) that relates to the diffusion of a particle. We calculated F(k,t) at a wave vector corresponding to the first peak of the static structure factor. Fig. 5a shows F(k,t) for

5 A. Kumar, J. Wu / Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) different temperatures for k = 1.9. Here the colloidal volume fraction remains φ = 0.3. At high-temperatures, the liquid equilibrates quickly and F(k,t) decays exponentially to zero. The relaxation time τ is defined as the time at which F(k,t) decays to 1/e; this is a measure of the α-relaxation time [35,36]. At low-temperatures, however, F(k,t) shows a stretched exponential behavior and the relaxation time τ increases rapidly. Fig. 5b presents the relaxation time at different temperatures for the different volume fractions investigated in this work. It shows that near the jamming transition, the relaxation time diverges as the transition temperature is approached, following a critical-like behavior τ = τ s ( 1 T c 1 T ) υτ (2) Fig. 4. Mean-squared displacements of the colloidal particles near the jamming transition. The slow dynamics is manifested by a plateau at lowertemperatures. where υ 2.07 is obtained by fitting to the simulation results. In order to verify that other dynamical quantities also follow similar critical-type behavior, we also monitored the dif- Fig. 5. (a) Intermediate scattering function at volume fraction φ = 0.30 and different temperatures. As the temperature approaches the transition temperature, F(k,t) shows deviation from the exponential decay behavior. (b) Relaxation time τ vs. inverse temperature T c at five different volume fractions, φ = 0.1 ( ), 0.15 ( ),02( ), 0.3 ( ), 0.4 (+). The solid lines are the fits to the Eq. (2), with the value of the exponent, υ Fig. 6. (a) Diffusion coefficient D vs. inverse temperature T c at different volume fractions, φ = 0.1 ( ), 0.15 ( ), 0.2 ( ), 0.3 ( ), 0.4 (+). The solid lines are the fits to the Eq. (3), with the value of the exponent, υ D (b) Jamming transition temperature T c vs. 1/φ c for different values of κ.for κ = 2.25, there is no dynamic slowing down observed for φ c = 0.1.

6 150 A. Kumar, J. Wu / Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) fusion coefficient, D, at different volume fractions and temperatures. The diffusion of particles at short-times reveals that the particles are still mobile and the mean-squared displacement (MSD) reaches a plateau asymptotically. This indicates the jammed state forms a glass-like structure. Fig. 6(a) shows the diffusion coefficient D at different volume fractions as the jamming transition temperature is approached. We find that it follows a critical-type behavior similar to that for the shear viscosity: D = D s ( 1 T c 1 T ) νd (3) where υ D 2.09 is obtained by fitting to the simulation details. This value is very close to the critical exponent value υ 2.07 obtained for the intermediate scattering function. The good agreement provides a self-consistent test of the simulation results as the relaxation time is directly related to the diffusion of particles. Based on Eq. (3), we are also able to deduce the jamming transition temperature T c for different volume fractions, as shown in Fig. 6(b). For comparison, the transition temperatures obtained from the shear viscosity data and the intermediate scattering function are plotted in the same figure. It is clear that the values of the transition temperatures obtained from the behaviors of different dynamic properties are very close and within the simulation uncertainty as shown in the figure. This asserts that the jamming transition is kinetic in nature and characterized by the sudden arrest of the dynamics of the constituent particles of the system. To investigate the effect of long-ranged repulsive interactions on the jamming transition, we have carried out a number of molecular dynamics simulations at different volume fractions for different values of κ. The jamming transition temperatures for each volume fractions and for each values of κ have been determined by following the criticallike behavior of the dynamical properties. These transition temperatures are plotted against the volume fraction for different values of κ in Fig. 7. At the low-values of κ, the fluid undergoes a jamming transition at very low-volume fractions of colloids as long as the temperature is sufficiently low. Upon the increase of κ, the transition occurs at lower-temperatures. We note that as κ increases, both the strength and range of the repulsion decline. At κ = 2.25, i.e. when the repulsive peak in Eq. (1) is almost non-existing, we observed the slowing down of dynamics only at relatively high-volume fractions. For φ = 0.1, the system remains fluid-like even at a very low-temperature (T = 0.1). Fig. 7 suggests that a long-ranged repulsive interaction is essential to form a glass-like jammed structure at low-volume fractions. 4. Conclusions We have studied the structural and dynamic properties of a model colloidal system using molecular dynamics simulations. It was observed that the radial distribution function exhibits a split in the second peak as the system approaches the jamming transition. At low-volume fractions, the gelation occurs through the crowding of clusters. It has also been found that the clusters formed are of chainlike, which is in good agreement with experimental observations. The mean-squared displacement shows a plateau near the transition temperature, which is a signature of bond formation between particles. The slowing down of dynamics is observed in the intermediate scattering function as well as viscosity and diffusion coefficient. The relaxation time obtained from the intermediate scattering functions follows a critical-like behavior as the temperature approaches the transition temperature, as in the case of other dynamical properties like diffusion coefficient and shear viscosity. Our simulation results show that different dynamical quantities follow essentially the same physics at the onset of jamming transition. Following any one of the dynamical properties will enable us to construct a jamming phase diagram that unifies various off-equilibrium phase transitions in colloidal dispersions. The simulation studies also suggest that a long-ranged repulsive interaction is necessary for the formation of a glass-like jammed state at low-colloidal volume fractions. These results offer a convincing proof for unifying different non-equilibrium transitions in colloids under the concept of jamming. Acknowledgments Fig. 7. The two-dimensional jamming phase diagram obtained from the intermediate scattering function. Values of transition temperatures obtained from shear viscosity data and diffusion coefficient data are also plotted for comparison. Jamming transition temperature T c vs. 1/φ c for different values of κ.forκ = 2.25, there is no dynamic slowing down observed for φ c = 0.1. This work is in part sponsored by the University of California Research and Development Program (Grant No ), by Lawrence Livermore National Laboratory (Grant No. MSI ), and by the National Science Foundation (Grant No. CTS ).

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