Brownian dynamics of colloidal hard spheres. 3. Extended investigations at the phase transition regime

The phase behavior of hard-sphere colloidal systems in the volume fraction regime 0.46<<0.64 has been studied in detail using a new and efficient algorithm to treat the nonanalytic interaction pair potential. In particular the influence of various initial configurations such as purely random and facecentered cubic (FCC) has been investigated, and former simulations have been extended toward much longer time scales. Thus, in the case of randomly initiated systems, crystallization could be suppressed for a comparably long time (500 _R , where _R is the structural relaxation time) where the system remained in a metastable glassy state. The concentration dependence of the long-time self-diffusion coefficients of these systems has been analyzed according to free volume theory (Doolittle equation). Numerical data fit excellently to the theoretical predictions, and the volume fraction of zero particle mobility was found close to the expected value of random close packing. In case of the FCC initiated systems, samples remained crystalline within the simulated evolution time of 500 _R if their volume fraction was above the predicted freezing transition _F = 0.494, whereas at lower concentrations rapid melting into a fluidlike disordered state is observed. It should be noted that this algorithm, which neglects higher-order correlations, considering only direct pair interactions, nevertheless yields the correct hard-sphere crystallization phase behavior as predicted in the literature.     

A first-order phase transition defines the random close packing of hard spheres

Randomly packing spheres of equal size into a container consistently results in a static configuration with a density of ∼64%. The ubiquity of random close packing (RCP) rather than the optimal crystalline array at 74% begs the question of the physical law behind this empirically deduced state. Indeed, there is no signature of any macroscopic quantity with a discontinuity associated with the observed packing limit. Here we show that RCP can be interpreted as a manifestation of a thermodynamic singularity, which defines it as the “freezing point” in a first-order phase transition between ordered and disordered packing phases. Despite the athermal nature of granular matter, we show the thermodynamic character of the transition in that it is accompanied by sharp discontinuities in volume and entropy. This occurs at a critical compactivity, which is the intensive variable that plays the role of temperature in granular matter. Our results predict the experimental conditions necessary for the formation of a jammed crystal by calculating an analogue of the “entropy of fusion”. This approach is useful since it maps out-of-equilibrium problems in complex systems onto simpler established frameworks in statistical mechanics.     

High-density properties of hard spheres within a modified Percus-Yevick theory: The role of thermodynamic consistency

Using an integral-equation approach based upon an approximation for the tail function, the equilibrium properties of a system of hard spheres are studied with special concern for the behavior in the region of close packing. The closure adopted is such that full, internal consistency is ensured in the thermodynamics of the model with respect to both the two zero-separation theorems as well as to the more standard virial and fluctuation routes to the equation of state. The scheme also makes use of the continuity properties of the tail function and of the cavity distribution function at contact. These properties are explictly tested in the low-density limit up to the fourth derivative. The theory generates an equilibrium branch bounded on the high-density side by a point corresponding to a packing fraction0.78, a value which closely matches Rogers' least upper bound for the densest packing of spheres. The pair structure of the fluid in the state of random close packing is also compared to the type of local order predicted by the theory at similar densities.     

Disks vs. spheres: Contrasting properties of random packings

Collections of random packings of rigid disks and spheres have been generated by computer using a previously described concurrent algorithm. Particles begin as infinitesimal moving points, grow in size at a uniform rate, undergo energy-onconserving collisions, and eventually jam up. Periodic boundary conditions apply, and various numbers of particles have been considered (N2000 for disks,N8000 for spheres). The irregular disk packings thus formed are clearly polycrystalline with mean grain size dependent upon particle growth rate. By contrast, the sphere packings show a homogeneously amorphous texture substantially devoid of crystalline grains. This distinction strongly influences the respective results for packing pair correlation functions and for the distributions of particles by contact number. Rapidly grown disk packings display occasional vacancies within the crystalline grains; no comparable voids of such distinctive size have been found in the random sphere packings. Rattler particles free to move locally but imprisoned by jammed neighbors occur in both the disk and sphere packings.This paper is dedicated to Jerry Percus on the occasion of his 65th birthday.     

On the freezing and structure of hard spheres under spherical confinement

The equation of state and the structure of hard spheres confined in spherical pores have been investigated via molecular dynamics for different pore radii ranging from 5.0 to 10.0?σ, where σ is the particle diameter. The hard boundary is chosen to capture the pure geometric effect of spherical confinement. A discontinuity in the equation of state was observed, indicating the onset of a freezing-like phase transition, which was similar to that of the bulk hard-sphere fluids. The behaviour of confined particles resembles that of the bulk with increase in the pore size, while its deviation from the bulk is found to be larger at the solid-like phase. For the pore radius below 5.0, FCC-like crystal clusters are not formed in spherically confined hard spheres.     

Nonequilibrium Brownian dynamics simulations of shear thinning in concentrated colloidal suspensions

The effect of interparticle forces on shear thinning in concentrated aqueous and nonaqueous colloidal suspensions was studied using nonequilibrium Brownian dynamics. Hydrodynamic interactions among particles were neglected. Systems of 108 particles were studied at volume fractions of 0.2 and 0.4. For the nonaqueous systems, shear thinning could be correlated with the gradual breakup of small flocs present because of the weak, attractive secondary minimum in the interparticle potential. At the highest shear rate for=0.4, the particles were organized into a hexagonally packed array of strings. For the strongly repulsive aqueous systems, the viscosity appeared to be a discontinuous function of the shear rate. For=0.4, this discontinuity coincided with a transition from a disordered state to a lamellar structure for the suspension.     

Note on the fractal dimension of hard sphere trajectories

Using Monte Carlo molecular dynamics, a new, careful study is made of the approach of the trajectory of a typical particle in a hard sphere fluid to that of a Brownian particle, discussed before by Powles and Quirke and Rapaport. The apparent fractal dimension of the trajectory, as a function of reduced length scale,(), characterizes the transition from mechanical to Brownian motion and differs markedly from 2 in all present computer simulations.     

Friction tensor for a pair of Brownian particles: Spurious finite-size effects and molecular dynamics estimates

In this work, we show thatin any finite system, the binary friction tensor for two Brownian particlescannot be directly estimated from an evaluation of the microscopic Green-Kubo formula, involving the time integral of force-force autocorrelation functions. This pitfall is associated with a subtle inversion of the thermodynamic and long-time limits and leads to spurious results for the estimates of the friction matrix based on molecular dynamics simulations. Starting from a careful analysis of the coupled Langevin equations for two interacting Brownian particles, we derive a method to circumvent these effects and extract the binary friction tensor from the correlation function matrix of the instantaneous forces exerted by the bath particles on the fixed Brownian particles, and from the relaxation of the total momentum of the bath in afinite system. The general methodology is applied to the case of two hard or soft Brownian spheres in a bath of light particles. Numerical estimates of the relevant correlation functions and of the resulting self and mutual components of the matrix of friction tensors are obtained by molecular dynamics simulations for various spacings between the Brownian particles. This paper is dedicated to B. Jancovici on the occassion of his 65th birthday.     

Alternative ensemble averages in molecular dynamics simulation of hard spheres

ABSTRACT

We consider application to the hard sphere (HS) model of the mapped-averaging framework for generating alternative ensemble averages for thermodynamic properties. Specifically, we develop and examine new formulas for the pressure, the singlet and pair densities, and the cavity-correlation function inside the HS core; the pressure formula in particular is constructed such that it gives an ensemble average that exactly corrects the second-order virial equation of state. The force plays a central role in mapped-averaging expressions, and we write them in a way that accounts for the impulsive, event-driven nature of the HS dynamics. Comparison between results obtained conventionally versus mapped averaging finds that the latter has some advantage at low density, while both perform equally well (in terms of uncertainties for a given amount of sampling) at higher densities.     

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

The problem of finding the most efficient way to pack spheres has an illustrious history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal in the 1960s. This problem finds applications spanning from the mathematician’s pencil, the processing of granular materials, the jamming and glass transitions, all the way to fruit packing in every grocery. There are presently numerous experiments showing that the loosest way to pack spheres gives a density of ∼55% (named random loose packing, RLP) while filling all the loose voids results in a maximum density of ∼63%-64% (named random close packing, RCP). While those values seem robustly true, to this date there is no well-accepted physical explanation or theoretical prediction for them. Here we develop a common framework for understanding the random packings of monodisperse hard spheres whose limits can be interpreted as the experimentally observed RLP and RCP. The reason for these limits arises from a statistical picture of jammed states in which the RCP can be interpreted as the ground state of the ensemble of jammed matter with zero compactivity, while the RLP arises in the infinite compactivity limit. We combine an extended statistical mechanics approach ‘a la Edwards’ (where the role traditionally played by the energy and temperature in thermal systems is substituted by the volume and compactivity) with a constraint on mechanical stability imposed by the isostatic condition. We show how such approaches can bring results that can be compared to experiments and allow for an exploitation of the statistical mechanics framework. The key result is the use of a relation between the local Voronoi volumes of the constituent grains (denoted the volume function) and the number of neighbors in contact that permits us to simply combine the two approaches to develop a theory of volume fluctuations in jammed matter. Ultimately, our results lead to a phase diagram that provides a unifying view of the disordered hard sphere packing problem and further sheds light on a diverse spectrum of data, including the RLP state. Theoretical results are well reproduced by numerical simulations that confirm the essential role played by friction in determining both the RLP and RCP limits. The RLP values depend on friction, explaining why varied experimental results can be obtained.     

Adsorption of a spherical nanoparticle in polymer brushes: Brownian dynamics investigation

We use Brownian dynamics simulations to study the adsorption behavior of a nanosized particle in polymer brushes. The adsorption process, the dynamic behavior of the nanoparticle in the brush, the penetration depth, the diffusion coefficient of the nanoparticle in different depths of the brush, and the forces exerted on the nanoparticle by the surrounding brush are all investigated for different grafting densities.     

Third virial coefficient for quantum hard spheres: Two-point Padé approximants for direct and exchange parts

The method of two-point Padé approximants is used to interpolate between the low-temperature and high-temperature expansions of the second and third cluster integrals (b ₂ andb ₃) of a quantum hard-sphere gas.b ₂ is used as a test case, since accurate numerical values are available. Using only a limited number of terms from the series (about as many as are available forb ₃), we are able to represent both the direct and exchange parts to better than 1% over the entire temperature range. Forb ₃ there are no accurate values available, but the qualitative similarity of the results to those forb ₂ leads us to believe that we have a reasonably good representation of both the direct and exchange parts ofb ₃.     

Phase diagram of dipolar hard and soft spheres: manipulation of colloidal crystal structures by an external field

Phase diagrams of hard and soft spheres with a fixed dipole moment are determined by calculating the Helmholtz free energy using simulations. The pair potential is given by a dipole-dipole interaction plus a hard-core and a repulsive Yukawa potential for soft spheres. Our system models colloids in an external electric or magnetic field, with hard spheres corresponding to uncharged and soft spheres to charged colloids. The phase diagram of dipolar hard spheres shows fluid, face-centered-cubic (fcc), hexagonal-close-packed (hcp), and body-centered-tetragonal (bct) phases. The phase diagram of dipolar soft spheres exhibits, in addition to the above mentioned phases, a body-centered-orthorhombic (bco) phase, and it agrees well with the experimental phase diagram [Nature (London) 421, 513 (2003)]. Our results show that bulk hcp, bct, and bco crystals can be realized experimentally by applying an external field.     

On the Brownian motion of a massive sphere suspended in a hard-sphere fluid. II. Molecular dynamics estimates of the friction coefficient

The friction coefficient exerted by a hard-sphere fluid on an infinitely massive Brownian sphere is calculated for several size ratios , where and are the diameters of the Brownian and fluid spheres, respectively. The exact microscopic expression derived in part I of this work from kinetic theory is transformed and shown to be proportional to the time integral of the autocorrelation function of the momentum transferred from the fluid to the Brownian sphere during instantaneous collisions. Three different methods are described to extract the friction coefficient from molecular dynamics simulations carried out onfinite systems. The three independent methods lead to estimates of which agree within statisticalerrors (typically 5%). The results are compared to the predictions of Enskog theory and of the hydrodynamic Stokes law. The former breaks down as the size ratio and/or the packing fraction of the fluid increase. Somewhat surprisingly, Stokes' law is found to hold withstick boundary conditions, in the range 1/4.5 explored in the present simulations, with a hydrodynamic diameterd=. The analysis of the moleuclar dynamics data on the basis of Stokes' law withslip boundary conditions is less conclusive, although the right trend is found as / increases.     

On hydraulic permeability of random packs of monodisperse spheres: Direct flow simulations versus correlations

Hydraulic permeability is studied in porous media consisting of randomly distributed monodisperse spheres by means of computational fluid dynamics (CFD) simulations. The packing of spheres is generated by inserting a certain number of nonoverlapping spherical particles inside a cubic box at both low and high packing fractions using proper algorithms. Fluid flow simulations are performed within the interparticulate porous space by solving Navier-Stokes equations in a low-Reynolds laminar flow regime. The hydraulic permeability is calculated from the Darcy equation once the mean values of velocity and pressure gradient are calculated across the particle packing. The simulation results for the pressure drop across the packing are verified by the Ergun equation for the lower range of porosities (<0.75), and the Stokes equation for higher porosities (∼1). Using the results of simulations, the effects of porosity and particle diameters on the hydraulic permeability are investigated. Simulations precisely specified the range of applicability of empirical or semi-empirical correlations for hydraulic permeability, namely the Carman-Kozeny, Rumpf-Gupte, and Howells-Hinch formulas. The number of spheres in the model is gradually decreased from 2000 to 20 to discover the finite-size effect of pores on the hydraulic permeability of spherical packing, which has not been clearly addressed in the literature. In addition, the scale dependence of hydraulic permeability is studied via simulations of the packing of spheres shrunk to lower scales. The results of this work not only reveal the validity range of the aforementioned correlations, but also show the finite-size effect of pores and the scale-independence of direct CFD simulations for hydraulic permeability.     

Effect of discreteness on heterogeneous flames: Propagation limits in regular and random particle arrays

In a system with discrete heat sources distributed in an inert, heat conducting medium, there exists two asymptotic regimes of flame propagation. When the flame thickness is much greater than the inter-particle spacing, the system approximates a homogeneous medium and the flame can be modeled as a continuum. In the other extreme, when the flame is very thin due to rapid reaction of particles, the heterogeneous flame can no longer be treated as a continuum since discrete effects become dominant. The effects of discreteness are characterized by a strong dependence on the spatial distribution of the sources. The present work investigates the effects of discreteness on flame propagation and demonstrates that these effects result in a propagation limit in the absence of losses. For a system of regularly spaced particles, this limit can be found analytically for one-, two-, and three-dimensional systems, although the flame exhibits a complex dynamic of bifurcations as it approaches this limit. Propagation of a flame beyond this limit is only possible through concentration fluctuations in a system with randomly distributed particles. Two-dimensional numerical simulations with randomly distributed particles show a strong dependence of the propagation limit on the size of the computational domain. A consequence of the random particle distribution is that the flammability limit can only be defined as a probabilistic outcome of the flame propagation simulations.     

Electroless formation of silver nanoaggregates: an experimental and molecular dynamics approach

The ability to manipulate matter to create non-conventional structures is one of the key issues of material science. The understanding of assembling mechanism at the nanoscale allows us to engineer new nanomaterials, with physical properties intimately depending on their structure.

This paper describes new strategies to obtain and characterise metal nanostructures via the combination of a top-down method, such as electron beam lithography, and a bottom-up technique, such as the chemical electroless deposition. We realised silver nanoparticle aggregates within well-defined patterned holes created by electron beam lithography on silicon substrates. The quality characteristics of the nanoaggregates were verified by using scanning electron microscopy and atomic force microscopy imaging. Moreover, we compared the experimental findings to molecular dynamics simulations of nanoparticles growth. We observed a very high dependence of the structure characteristics on the pattern nanowell aspect ratio. We found that high-quality metal nanostructures may be obtained in patterns with well aspect ratio close to one, corresponding to a maximum diameter of 50 nm, a limit above which the fabricated structures become less regular and discontinuous. When regular shapes and sizes are necessary, as in nanophotonics, these results suggest the pattern characteristics to obtain isolated, uniform and reproducible metal nanospheres.     

Entropy and enthalpy calculations from perturbation and integration thermodynamics methods using molecular dynamics simulations: applications to the calculation of hydration and association thermodynamic properties

Calculation of association thermodynamic properties using molecular simulation is essential in computational chemistry. In the case of good agreement with the experimental thermodynamic binding properties, this type of calculation may complement experimental works by providing a microscopic view of the association process. Whereas the calculation of the free energy of association is nowadays well controlled, the calculation of the enthalpy and entropy of association remains difficult in most cases, especially as the association involves hosts and guests of biological interest. A novel method for calculating the entropy change from a molecular dynamics simulation is described. Within the theoretical framework, we discuss the different approximations leading to the final stage of the operational expressions of ΔG and ΔH in the NpT ensemble and we establish an expression for ΔS using the Free Energy Perturbation (FEP) formalism in this statistical ensemble. Finally, we illustrate the theoretical considerations by calculations of the hydration entropy changes between cations of different masses and charges. We extend the study by calculating the changes in the thermodynamic properties of association of inorganic cations with a macrocycle of biological interest.     

A combined ab initio and atomistic simulation study of the surface and interfacial structures and energies of hydrated scheelite: introducing a CaWO4 potential model

Density functional theory (DFT) calculations of the calcium tungstate material scheelite CaWO₄ have shown that water introduced into the bulk material remains undissociated and leads to swelling and layering of the structure, a behaviour which may resemble silicate clays more than three-dimensional poly-anionic materials, but which results in a structure that is even more similar to a rare hydrous calcium carbonate phase--a finding which suggests the existence of semi-crystalline hydrous pre-cursor phases to the dehydrated scheelite material. An interatomic potential model was derived for CaWO₄ which adequately reproduces structural and physical properties of the material and is in good agreement with the DFT calculations in respect of the structure and energy of hydration (DFT: 85 kJ mol⁻¹, atomistic: 105 kJ mol⁻¹). Atomistic simulations of a range of scheelite surfaces confirm the dominance of the experimental {1 0 1} and {0 0 1} cleavage planes in the morphology of the dry crystal and the presence of the experimentally found {1 0 3} and {1 0 1} surfaces in the hydrated morphology. Hydration of the surfaces shows non-Langmuir behaviour, where the interactions between surface calciums and oxygen atoms of the water molecules outweigh hydrogen-bonding to the surface oxygen atoms or intermolecularly within the water layer. The hydration energies indicate physisorption of water, ranging from 22 kJ mol⁻¹ on the {0 0 1} surface to 78 kJ mol⁻¹ on the more reactive {1 0 3} surface.     

Stochastic resonance on paced genetic regulatory small-world networks: effects of asymmetric potentials

We study the phenomenon of stochastic resonance on small-world networks consisting of bistable genetic regulatory units, whereby the external subthreshold periodic forcing is introduced as a pacemaker trying to impose its rhythm on the whole network through the single unit to which it is introduced. Without the addition of additive spatiotemporal noise, however, the whole network remains forever trapped in one of the two stable steady states of the local dynamics. We show that the correlation between the frequency of subthreshold pacemaker activity and the response of the network is resonantly dependent on the intensity of additive noise. The reported pacemaker driven stochastic resonance depends significantly on the asymmetry of the two potential wells characterizing the bistable dynamics, which can be tuned via a single system parameter. In particular, we show that the ratio between the clustering coefficient and the characteristic path length is a suitable quantity defining the ability of a small-world network to facilitate the outreach of the pacemaker-emitted subthreshold rhythm, but only if the asymmetry between the potentials is practically negligible. In case of substantially asymmetric potentials the impact of the small-world topology is less profound and cannot warrant an enhancement of stochastic resonance by units that are located far from the pacemaker.