Lattice Boltzmann model for simulating flows with multiple phases and components

Wetting dynamics and particle deposition for an evaporating colloidal drop: a lattice Boltzmann study

A three-dimensional lattice Boltzmann method (LBM) has been developed for multiphase (liquid and vapor) flows with solid particles suspended within the liquid phases. The method generalizes our recent two-dimensional model [A. Joshi and Y. Sun, Phys. Rev. E 79, 066703 (2009)] to three dimensions, extends the implicit scheme presented therein to include interparticle forces and introduces an evaporation model to simulate drying of the colloidal drop. The LBM is used to examine the dynamical wetting behavior of drops containing suspended solid particles on homogeneous and patterned substrates. The influence of the particle volume fraction and particle size on the drop spreading dynamics is studied as is the final deposition of suspended particles on the substrate after the carrier liquid evaporates. The final particle deposition can be controlled by substrate patterning, adjusting the substrate surface energies and by the rate of evaporation. Some of the envisioned applications of the model are to develop a fundamental understanding of colloidal drop dynamics, predict particle deposition during inkjet printing of functional materials and to simulate the drying of liquids in porous media.

Volumetric formulation for a class of kinetic models with energy conservation

We analyze a volumetric formulation of lattice Boltzmann for compressible thermal fluid flows. The velocity set is chosen with the desired accuracy, based on the Gauss-Hermite quadrature procedure, and tested against controlled problems in bounded and unbounded fluids. The method allows the simulation of thermohydrodyamical problems without the need to preserve the exact space-filling nature of the velocity set, but still ensuring the exact conservation laws for density, momentum, and energy. Issues related to boundary condition problems and improvements based on grid refinement are also investigated.

Multirelaxation-time interaction-potential-based lattice Boltzmann model for two-phase flow

The application of the lattice Boltzmann method in two-phase flows is often restricted by the numerical instability at low viscosities. In this work, a multirelaxation-time (MRT) lattice Boltzmann model (LBM) is developed using the interaction potential approach. With the MRT collision term and a general force term, the new MRT model is able to significantly enhance the numerical stability at low viscosities, without appreciable increase in computation time or memory use. Advanced force formulation using the multirange potential can also be readily incorporated into the current MRT scheme. Numerical tests are first performed in two dimensions under equilibrium conditions. The MRT model is able to reduce the lowest stable viscosity by an order of magnitude compared to the single relaxation time LBM. In addition, the spurious velocity at the gas-liquid interface can also be significantly decreased by tuning the adjustable relaxation parameters. Then two sets of three-dimensional simulations are conducted to investigate the buoyant rise of a gas bubble in a low-viscosity liquid. In particular, millimeter air bubble in water, which is difficult for traditional two-phase LBM due to both low viscosity and high-surface tension, is successfully simulated using the MRT technique developed in this study. The simulated bubble shape and velocity are compared with the experimental results and empirical correlations in the literature, and the satisfactory agreement proves the validity of the MRT-LBM for two-phase flows.

Dynamic density functional theory versus kinetic theory of simple fluids

By combining methods of kinetic and density functional theory, we present a description of molecular fluids which accounts for their microscopic structure and thermodynamic properties as well as their hydrodynamic behavior. We focus on the evolution of the one-particle phase space distribution, rather than on the evolution of the average particle density which features in dynamic density functional theory. The resulting equation can be studied in two different physical limits: diffusive dynamics, typical of colloidal fluids without hydrodynamic interaction where particles are subject to overdamped motion resulting from coupling with a solvent at rest, and inertial dynamics, typical of molecular fluids. Finally, we propose an algorithm to solve numerically and efficiently the resulting kinetic equation by employing a discretization procedure analogous to the one used in the lattice Boltzmann method.

Simulation of flow of mixtures through anisotropic porous media using a lattice Boltzmann model

We propose a description for transient penetration simulations of miscible and immiscible fluid mixtures into anisotropic porous media, using the lattice Boltzmann (LB) method. Our model incorporates hydrodynamic flow, advection-diffusion, surface tension, and the possibility for global and local viscosity variations to consider various types of hardening fluids. The miscible mixture consists of two fluids, one governed by the hydrodynamic equations and one by advection-diffusion equations. We validate our model on standard problems like Poiseuille flow, the collision of a drop with an impermeable, solid interface and the deformation of the fluid due to surface tension forces. To demonstrate the applicability to complex geometries, we simulate the invasion process of mixtures into wood spruce samples.

Derivation and thermodynamics of a lattice Boltzmann model with soluble amphiphilic surfactant

We derive a lattice Boltzmann model of a ternary fluid mixture, one component of which consists of amphiphilic molecules with a rotational degree of freedom. The model was first introduced elsewhere where it was constructed on the basis of the earlier Shan-Chen type lattice Boltzmann model for binary fluids. We provide a rigorous derivation of the model from an underlying continuum kinetic theory. In particular, we show how the model can be interpreted as a discretization of a Vlasov-Boltzmann type kinetic theory for a fluid composed of two species of oppositely charged monopoles and a composite, dipolar molecule. We also derive a free-energy functional for the model, including the contribution from the amphiphiles, and confirm that the free energy is lowered during phase separation and during the formation of a stable emulsion.

Lattice Boltzmann model for the complex Ginzburg-Landau equation

A lattice Boltzmann model with complex distribution function for the complex Ginzburg-Landau equation (CGLE) is proposed. By using multiscale technique and the Chapman-Enskog expansion on complex variables, we obtain a series of complex partial differential equations. Then, complex equilibrium distribution function and its complex moments are obtained. Based on this model, the rotation and oscillation properties of stable spiral waves and the breaking-up behavior of unstable spiral waves in CGLE are investigated in detail.

SHIFT: an implementation for lattice Boltzmann simulation in low-porosity porous media

The lattice Boltzmann (LB) method has proven to be a promising method for simulating fluid dynamics in porous media. When fluid flow in pores is the only concern, a standard LB implementation, which stores one or two sets of particle distribution functions (PDFs) for both pore and solid cells, wastes a large amount of memory, especially for low-porosity media. This paper proposes a LB implementation scheme that stores a single set of PDFs for pore cells only and therefore makes it possible to simulate flow through larger and more-realistic porous models. A unique feature of this scheme is that it decomposes all PDFs into a set of 1D arrays in such a way that each array corresponds to a set of pore cells that connect one another along a pair of opposite LB velocity directions. This allows LB propagation and a standard bounce-back rule to be realized together as one or two circular shifting operations on every array. For this reason, this scheme is called SHIFT. Although PDFs are not stored in an efficient way for LB collision operation, it is shown that the incurred overhead could be reduced by properly arranging PDF arrays according to the pore structures. A D3Q15 LB implementation of SHIFT using the lattice Bhatnagar-Gross-Krook model is applied to simulate the Stokes flow through models of four natural and synthetic rock samples with porosities ranging from about 10% to 38%. Results show that SHIFT requires 36-82 % less memory than a comparable D3Q15 LB does, which stores a single set of PDF for both pore and solid cells. SHIFT achieves minimum performances of over 11 and 3.8 mega-lattice-updates-per-second (MLUPS) for the combined propagation and bounce-back operation and the collision operation, respectively, and therefore a minimum of 2.8 MLUPS in total on a computer with one AMD Opteron 2218. The performance of the collision operation is significantly improved for all cases when a simple K -mean clustering technique is employed to rearrange PDF arrays. It is argued and shown that the number of PDF arrays per pore cell and the length frequency of PDF arrays are useful measurements on the geometry and topology of the pore structures and these characteristics are able to explain SHIFT performance variations.

Multicomponent lattice Boltzmann model from continuum kinetic theory

We derive from the continuum kinetic theory a multicomponent lattice Boltzmann model with intermolecular interaction. The resulting model is found to be consistent with the model previously derived from a lattice-gas cellular automaton [X. Shan and H. Chen, Phys. Rev. E 47, 1815 (1993)] but applies in a much broader domain. A number of important insights are gained from the kinetic theory perspective. First, it is shown that even in the isothermal case, the energy equipartition principle dictates the form of the equilibrium distribution function. Second, thermal diffusion is shown to exist and the corresponding diffusivities are given in terms of macroscopic parameters. Third, the ordinary diffusion is shown to satisfy the Maxwell-Stefan equation at the ideal-gas limit.

Fully reversible transition from Wenzel to Cassie-Baxter states on corrugated superhydrophobic surfaces

Liquid drops on textured surfaces show different dynamical behaviors depending on their wetting states. They are extremely mobile when they are supported by composite solid-liquid-air interfaces (Cassie-Baxter state) and immobile when they fully wet the textured surfaces (Wenzel state). By reversibly switching between these two states, it will be possible to achieve control over the fluid dynamics. Unfortunately, these wetting transitions are usually prevented by surface energy barriers. We demonstrate here a new, simple design paradigm consisting of parallel grooves with an appropriate aspect ratio that allows for the controlled, barrierless, reversible switching of the wetting states upon application of electrowetting. We report a direct observation of the barrierless dynamical pathway for the reversible transitions between the Wenzel (collapsed) and Cassie-Baxter (suspended) states and present a theory that accounts for these transitions, including detailed lattice Boltzmann simulations.

Taylor Flow in Microchannels: A Review of Experimental and Computational Work

Over the past few decades an enormous interest in two-phase flow in microchannels has developed because of their application in a wide range of new technologies, ranging from lab-on-a-chip devices used in medical and pharmaceutical applications to micro-structured process equipment used in many modern chemical plants. Taylor flow, in which gas bubbles are surrounded by a liquid film and separated by liquid plugs, is the most common flow regime encountered in such applications. This review introduces the important attributes of two phase flow in microchannels and then focuses on the Taylor flow regime. The existing knowledge from both experimental and computational studies is presented. Finally, perspectives for future work are suggested.

Phase-field modeling by the method of lattice Boltzmann equations

In this paper, at first, a lattice Boltzmann method for binary fluids, which is applicable at low viscosity values, is developed. The presented scheme is extension of the free-energy-based approach to a multi-relaxation-time collision model. Various benchmark problems such as the well-known Laplace law for stationary bubbles and capillary-wave test are conducted for validation. As an appealing application, instability of a rising bubble in an enclosed duct is studied and irregular behavior of the bubble is observed at very high Reynolds numbers. In order to highlight its capability to simulate high Reynolds number flows, which is a challenge for many other models, a typical wobbling bubble in the turbulent regime is simulated successfully. Then, in the context of phase-field modeling, a lattice Boltzmann method is proposed for multiphase flows with a density contrast. Unlike most of the previous models based on the phase-field theory, the proposed scheme not only tolerates very low viscosity values but also emerges as a promising method for investigation of two-phase flow problems with moderate density ratios. In addition to comparison to the kinetic-based model, the proposed approach is further verified by judging against the theoretical solutions and experimental data. Various case studies including the rising bubble, droplet splashing on a wet surface, and falling droplet are conducted to show the versatility of the presented lattice Boltzmann model.

Lattice Boltzmann method for oscillatory Stokes flow with applications to micro- and nanodevices

A lattice Boltzmann (LB) method based on the linearized Boltzmann Bhatnagar-Gross-Krook equation for numerical simulation of oscillatory (unsteady) Stokes flow is proposed. Unlike the conventional (nonlinear) LB method that utilizes the time domain exclusively, the proposed method is formulated in the frequency domain to allow for direct access to the complex-valued stress, force, and velocity field--these parameters are of direct interest in characterizing microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). The proposed method circumvents the requirement for time-dependent boundary velocities, as is needed in the conventional LB method, and convergence of the two methods is compared. Validity of the proposed method is assessed using three classical (unsteady) flows: (1) one-dimensional oscillatory Couette flow between two plates; (2) two-dimensional flow generated by an oscillating circular cylinder; (3) three-dimensional flow generated by an oscillating sphere. The observed excellent numerical performance in all three cases demonstrates that this linear lattice Boltzmann method can be used to study the dynamics of micro- and nanoscale devices of any dimensionality. This is particularly relevant to MEMS and NEMS, where the resonance properties of individual nanomechanical components immersed in fluid can underpin overall device performance.

Modeling the coupling of reaction kinetics and hydrodynamics in a collapsing cavity

We introduce a model of cavitation based on the multiphase Lattice Boltzmann method (LBM) that allows for coupling between the hydrodynamics of a collapsing cavity and supported solute chemical species. We demonstrate that this model can also be coupled to deterministic or stochastic chemical reactions. In a two-species model of chemical reactions (with a major and a minor species), the major difference observed between the deterministic and stochastic reactions takes the form of random fluctuations in concentration of the minor species. We demonstrate that advection associated with the hydrodynamics of a collapsing cavity leads to highly inhomogeneous concentration of solutes. In turn these inhomogeneities in concentration may lead to significant increase in concentration-dependent reaction rates and can result in a local enhancement in the production of minor species.

Graphics processing unit implementation of lattice Boltzmann models for flowing soft systems

A graphic processing unit (GPU) implementation of the multicomponent lattice Boltzmann equation with multirange interactions for soft-glassy materials ["glassy" lattice Boltzmann (LB)] is presented. Performance measurements for flows under shear indicate a GPU/CPU speed up in excess of 10 for 1024(2) grids. Such significant speed up permits to carry out multimillion time-steps simulations of 1024(2) grids within tens of hours of GPU time, thereby considerably expanding the scope of the glassy LB toward the investigation of long-time relaxation properties of soft-flowing glassy materials.

Capillary filling in microchannels with wall corrugations: a comparative study of the Concus-Finn criterion by continuum, kinetic, and atomistic approaches

We study the impact of wall corrugations in microchannels on the process of capillary filling by means of three broadly used methods: computational fluid dynamics (CFD), lattice Boltzmann equations (LBE), and molecular dynamics (MD). The numerical results of these approaches are compared and tested against the Concus-Finn (CF) criterion, which predicts pinning of the contact line at rectangular ridges perpendicular to flow for contact angles of theta > 45 degrees . Whereas for theta = 30, 40 (no flow), and 60 degrees (flow) all methods are found to produce data consistent with the CF criterion, at theta = 50 degrees the numerical experiments provide different results. Whereas the pinning of the liquid front is observed both in the LB and CFD simulations, MD simulations show that molecular fluctuations allow front propagation even above the critical value predicted by the deterministic CF criterion, thereby introducing a sensitivity to the obstacle height.

Lattice Boltzmann scheme for mixture modeling: analysis of the continuum diffusion regimes recovering Maxwell-Stefan model and incompressible Navier-Stokes equations

A finite difference lattice Boltzmann scheme for homogeneous mixture modeling, which recovers Maxwell-Stefan diffusion model in the continuum limit, without the restriction of the mixture-averaged diffusion approximation, was recently proposed [P. Asinari, Phys. Rev. E 77, 056706 (2008)]. The theoretical basis is the Bhatnagar-Gross-Krook-type kinetic model for gas mixtures [P. Andries, K. Aoki, and B. Perthame, J. Stat. Phys. 106, 993 (2002)]. In the present paper, the recovered macroscopic equations in the continuum limit are systematically investigated by varying the ratio between the characteristic diffusion speed and the characteristic barycentric speed. It comes out that the diffusion speed must be at least one order of magnitude (in terms of Knudsen number) smaller than the barycentric speed, in order to recover the Navier-Stokes equations for mixtures in the incompressible limit. Some further numerical tests are also reported. In particular, (1) the solvent and dilute test cases are considered, because they are limiting cases in which the Maxwell-Stefan model reduces automatically to Fickian cases. Moreover, (2) some tests based on the Stefan diffusion tube are reported for proving the complete capabilities of the proposed scheme in solving Maxwell-Stefan diffusion problems. The proposed scheme agrees well with the expected theoretical results.

Multiphase lattice Boltzmann method for particle suspensions

A two-dimensional mass conserving lattice Boltzmann method (LBM) has been developed for multiphase (liquid and vapor) flows with solid particles suspended within the liquid and/or vapor phases. The main modification to previous single-phase particle suspension models is the addition of surface (adhesive) forces between the suspended particle and the surrounding fluid. The multiphase dynamics between fluid phases is simulated via the single-component multiphase model of Shan and Chen [Phys. Rev. E 47, 1815 (1993)]. The combined multiphase particle suspension model is first validated and then used to simulate the dynamics of a single-suspended particle on a planar liquid-vapor interface and the interaction between a single particle and a free-standing liquid drop. It is observed that the dynamics of suspended particles near free-standing liquid droplets is affected by spurious velocity currents although the liquid-vapor interface itself is a local energy minimum for particles. Finally, results are presented for capillary interactions between two suspended particles on a liquid-vapor interface subjected to different external forces and for spinodal decomposition of a liquid-vapor mixture in the presence of suspended particles. Qualitative agreements are reached when compared with results of suspended particles in a binary mixture based on multicomponent LBM models.

Rupture of a ferrofluid droplet in external magnetic fields using a single-component lattice Boltzmann model for nonideal fluids

A nonisotropic tensorial extension of the single-component Shan-Chen pseudopotential Lattice Boltzmann method for nonideal fluids is presented. Direct comparison with experimental data shows that this extension is able to capture relevant features of ferrofluid behavior, such as the deformation and subsequent rupture of a liquid droplet as a function of an externally applied magnetic field. The present model offers an economic lattice-kinetic pathway to the simulation of complex ferrofluid hydrodynamics.

Boundary slip from the immersed boundary lattice Boltzmann models

We report an interesting and important observation of the velocity fields from immersed boundary lattice Boltzmann methods (IB-LBM). The computed velocity profiles can deviate from theoretical predictions greatly even for very simple flow situations, both in the immersed boundary layer and the bulk region. A rigorous analysis of the IB-LBM simulated velocity for a symmetric shear flow is carried out, and the analytical solutions indicate a strong dependence of velocity on the relaxation parameter (kinetic viscosity). Also our simulations demonstrate that simply increasing the immersed boundary layer thickness is not an efficient approach to reduce such velocity discrepancy. We hope this work will bring the awareness of this essential issue to people using IB-LBM for various flow situations.

Mesoscopic lattice boltzmann modeling of flowing soft systems

A mesoscopic multicomponent lattice Boltzmann model with short-range repulsion between different species and short (midranged) attractive (repulsive) interactions between like molecules is introduced. The interplay between these composite interactions gives rise to a rich configurational dynamics of the density field, exhibiting many features of disordered liquid dispersions (microemulsions) and soft-glassy materials, such as long-time relaxation due to caging effects, anomalous enhanced viscosity, aging effects under moderate shear and flow above a critical shear rate.

Three-dimensional modelling of the human carotid artery using the lattice Boltzmann method: I. model and velocity analysis

Numerical modelling is a powerful tool in the investigation of human blood flow and arterial diseases such as atherosclerosis. It is known that near wall velocity and shear are important in the pathogenesis and progression of atherosclerosis. In this paper results for a simulation of blood flow in a three-dimensional carotid artery geometry using the lattice Boltzmann method are presented. The velocity fields in the body of the fluid are analysed at six times of interest during a physiologically accurate velocity waveform. It is found that the three-dimensional model agrees well with previous literature results for carotid artery flow. Regions of low near wall velocity and circulatory flow are observed near the outer wall of the bifurcation and in the lower regions of the external carotid artery, which are regions that are typically prone to atherosclerosis.

Front pinning in capillary filling of chemically coated channels

The dynamics of capillary filling in the presence of chemically coated heterogeneous boundaries is investigated both theoretically and numerically. In particular, by mapping the equations of front motion onto the dynamics of a dissipative driven oscillator, an analytical criterion for front pinning is derived under the condition of diluteness of the coating spots. The criterion is tested against two-dimensional lattice Boltzmann simulations and found to provide satisfactory agreement as long as the width of the front interface remains much thinner than the typical heterogeneity scale of the chemical coating.

Thermodynamic consistency of liquid-gas lattice Boltzmann methods: interfacial property issues

In the present study we examine the thermodynamic consistency of lattice Boltzmann equation (LBE) models that are based on the forcing method by comparing different numerical treatments of the LBE for van der Waals fluids. The different models are applied for the calculation of bulk and interfacial thermodynamic properties at various temperatures. The effect of the interface density gradient parameter, kappa , that controls surface tension, is related explicitly with the fluid characteristics, including temperature, molecular diameter, and lattice spacing, through the employment of a proper intermolecular interaction potential. A comprehensive analysis of the interfacial properties reveals some important shortcomings of the LBE methods when central finite difference schemes are employed in the directional derivative calculations and proposes a proper treatment that ensures thermodynamically consistent interfacial properties in accord with the van der Waals theory. The results are found to be in excellent quantitative agreement with exact results of the van der Waals theory preserving all the major features of the interfacial characteristics of vapor-liquid systems of different shapes and sizes.

Capillary filling with pseudo-potential binary Lattice-Boltzmann model

We present a systematic study of capillary filling for a binary fluid by using a mesoscopic lattice Boltzmann model for immiscible fluids describing a diffusive interface moving at a given contact angle with respect to the walls. The phenomenological way to impose a given contact angle is analysed. Particular attention is given to the case of complete wetting, that is contact angle equal to zero. Numerical results yield quantitative agreement with the theoretical Washburn's law, provided that the correct ratio of the dynamic viscosities between the two fluids is used. Finally, the presence of precursor films is experienced and it is shown that these films advance in time with a square-root law but with a different prefactor with respect to the bulk interface.

Slip flow over structured surfaces with entrapped microbubbles

On hydrophobic surfaces, roughness may lead to a transition to a superhydrophobic state, where gas bubbles at the surface can have a strong impact on a detected slip. We present two-phase lattice Boltzmann simulations of a Couette flow over structured surfaces with attached gas bubbles. Even though the bubbles add slippery surfaces to the channel, they can cause negative slip to appear due to the increased roughness. The simulation method used allows the bubbles to deform due to viscous stresses. We find a decrease of the detected slip with increasing shear rate which is in contrast to some recent experimental results implicating that bubble deformation cannot account for these experiments. Possible applications of bubble surfaces in microfluidic devices are discussed.

Pressure tensor calculation in a class of nonideal gas lattice Boltzmann models

In nonideal gas lattice Boltzmann (LB) models, obtaining the correct form of the pressure tensor is essential in determining many of the statistical mechanical properties such as the surface tension and the density profile. Here we outline a general approach for calculating the pressure tensor in LB models with interactions beyond nearest neighbors. The statistical mechanical properties calculated from such a pressure tensor are shown to agree very well with those measured from numerical experiments. Comparisons with alternative theories are also made.

Pore-scale studies of spontaneous imbibition into oil-saturated porous media

Spontaneous imbibition phenomenon was modeled using the lattice Boltzmann method (LBM). The model was validated using silicon-etched micromodel and sandpack visualization experiments. The strongly water-wet model saturated with oil (kerosene or mineral oil) was exposed to distilled water in order for a capillary interaction to take place under static conditions. These experiments mimic the transfer between rock matrix and fracture during any wetting phase flow in fractures while the matrix contains a nonwetting phase as encountered in oil, gas, and geothermal reservoirs as well as during the application of subsurface CO2 sequestration or waste disposal reservoirs. Despite the vast amount of research work on this common process, pore-scale investigations and modeling are limited especially at small time scales. The results showed that the LBM captures the physics of the process at pore scale for low viscosity values of nonwetting phase for any type of (cocurrent or countercurrent imbibition) interaction.

Thermal lattice Boltzmann model for gases with internal degrees of freedom

We rigorously derive a dramatically simplified kinetic model for fluids with internal degrees of freedom. With proper discretization in velocity space, the model leads to a lattice Boltzmann model for polyatomic gases. The macroscopic recovery of correct hydrodynamics is theoretically shown and numerically validated.

Lattice Boltzmann models for nonideal fluids with arrested phase-separation

The effects of midrange repulsion in lattice Boltzmann models on the coalescence and/or breakup behavior of single-component, nonideal fluids are investigated. It is found that midrange repulsive interactions allow the formation of spraylike, multidroplet configurations, with droplet size directly related to the strength of the repulsive interaction. The simulations show that just a tiny 10% of midrange repulsive pseudoenergy can boost the surface:volume ratio of the phase-separated fluid by nearly two orders of magnitude. Drawing upon a formal analogy with magnetic Ising systems, a pseudopotential energy is defined, which is found to behave similar to a quasiconserved quantity for most of the time evolution. This offers a useful quantitative indicator of the stability of the various configurations, thus helping the task of their interpretation and classification. The present approach appears to be a promising tool for the computational modeling of complex flow phenomena, such as atomization, spray formation, microemulsions, breakup phenomena, and possibly glassylike systems as well.

Singular forces and pointlike colloids in lattice Boltzmann hydrodynamics

We present an accurate method to include arbitrary singular distributions of forces in the lattice Boltzmann formulation of hydrodynamics. We validate our method with several examples involving Stokeslet, stresslet, and rotlet singularities, finding excellent agreement with analytical results. A minimal model for sedimenting particles is presented using the method. In the dilute limit, this model has accuracy comparable to, but computational efficiency much greater than, algorithms that explicitly resolve the size of the particles.