Classical solution of Navier-Stokes equations with nonslip boundary condition leads to inaccurate predictions of flow characteristics of rarefied gases confined in micro/nanochannels. Therefore, molecular interaction based simulations are often used to properly express velocity and temperature slips at high Knudsen numbers (Kn) seen at dilute gases or narrow channels. In this study, an event-driven molecular dynamics (EDMD) simulation is proposed to estimate properties of hard-sphere gas flows. Considering molecules as hard-spheres, trajectories of the molecules, collision partners, corresponding interaction times, and postcollision velocities are computed deterministically using discrete interaction potentials. On the other hand, boundary interactions are handled stochastically. Added to that, in order to create a pressure gradient along the channel, an implicit treatment for flow boundaries is adapted for EDMD simulations. Shear-Driven (Couette) and Pressure-Driven flows for various channel configurations are simulated to demonstrate the validity of suggested treatment. Results agree well with DSMC method and solution of linearized Boltzmann equation. At low Kn, EDMD produces similar velocity profiles with Navier-Stokes (N-S) equations and slip boundary conditions, but as Kn increases, N-S slip models overestimate slip velocities.
Over the last decades, micro- and nanoelectromechanical systems (MEMS, NEMS) have been rapidly developed. Today, many applications like flow sensors or miniaturized jet engines are taking place in industry. Therefore, investigation of gas flows in micro- and nanochannels is essential for design and operation of microdevices such as micropumps, microvalves, and microturbines [
DSMC method [
Modelling the system as an ensemble of hard particles is an alternative to continuous interaction potential approach. Interaction potential between hard particles is discrete; that is, no force is exerted on the molecules except impacts. Assuming no external force field, resulting trajectories are linear. Intermolecular collisions are instantaneous and binary. When a collision occurs, postcollisional velocities are analytically determined via conservation of energy and momentum. Analytically predicted collision times yield the decomposition of simulation into a series of asynchronous events. This process is event driven since simulation time advances directly from one event to the next. Due to its nature, event-driven approach eases simulation of larger systems for longer periods compared to time-driven one. Since the first introduction of event-driven molecular dynamics (EDMD) simulations [
In order to simulate a confined gas flow using molecular simulations, system boundaries and interactions with the gas molecules must be modelled appropriately. Three types of boundary are usually sufficient to represent a flow in a micro- or nanochannel. These well-known boundaries are periodic, wall, and flow boundaries.
Since the performance of MD simulations is highly affected by the number of simulated molecules, it is desired to represent the computational domain with the smallest possible sample. Periodicity is the most common boundary model for such representations. It also removes surface effects in the simulations of infinite systems. In the presence of periodic boundaries, computational domain is repeated in the direction of periodicity to form an infinite lattice. When a molecule reaches a periodic boundary it continues its course on the opposite face with exactly the same velocity vector.
On the wall boundary, a molecule hits the wall if the distance between the wall and molecular centre is a half molecular diameter. The molecule is reflected back with a new velocity determined according to the wall model. Specular reflection is the most basic wall model in which surface is perfectly smooth at molecular level. When a molecule undergoes specular reflection, normal velocity component is inverted while tangential components remain unchanged, conserving tangential momentum. However, in reality, wall surface is rough and the molecules are reflected at random angles from the wall. Diffuse reflection is the most common model to represent such surfaces. In diffuse model, postreflection velocity of the molecule is substantially independent of the incoming velocity and determined stochastically from a distribution based on wall temperature and velocity.
Several treatments for flow boundaries have been developed to induce a flow inside the channel. These are selective reflection treatment [
Investigation of Shear- and Pressure-Driven flows of a hard-sphere gas in microchannels is beyond the capabilities of MD simulation methods using continuous interaction potentials. Additionally, implicit treatment of flow boundaries is not easily applicable since the method is time driven. Therefore, simulations were carried out with the use of event-driven molecular dynamics simulations in this study. EDMD has similarities in implementation with both classical MD and DSMC methods. In contrast to DSMC method, collisional dynamics such as movements of molecules, determination of collision partners, and calculation of postcollisional velocities are deterministic in the hard-sphere EDMD. All collisions are real and computed in MD simulations. In this study, simulations were conducted with real number of molecules taking advantage of an event scheduling algorithm with
This study is organized as follows: EDMD algorithm and application of boundary treatments are described in Section
In this study, EDMD simulation of monatomic molecules was conducted. Computational domain was a rectangular parallelepiped region. Molecules were modelled as hard-spheres; thus only translational energies were considered. Only binary collisions, as a valid assumption for dilute gases, were employed. Postcollisional velocities were calculated according to mass, diameter, and precollisional velocity information of colliding pair by satisfying conservation equations of energy and momentum.
Argon
Due to the nature of MD simulations, immense computational resources are required. Simulation speed decreases almost linearly as the number of molecules in the system increases. In order to speed up the simulations, computational domain was partitioned into cubical cells using the method described in detail by Kandemir [
Three types of events are possible in event-driven simulations. These are intermolecular collisions, molecule-boundary interactions, and cell crossings. A molecule can be involved in many future events but only the earliest one is registered as the candidate event for the molecule. Thus, the number of candidate events is always equal to the number of simulated molecules. The goals are determination and execution of the earliest event. Following the execution, candidate events of related molecules are invalidated and possible new candidate events are calculated. This process repeats throughout the simulation.
Determination of earliest event by linear search has a complexity of
Macroscopic thermodynamic properties of a simulated system can be estimated by ensemble averaging over particles’ behaviour. However, this would be expensive in terms of computational effort because of the requirement of large number of molecules to obtain smooth profiles. The alternative approach is the time averaging for a smaller number of particles since ensemble average and time average of a phase variable yield equal values in an ergodic system [
From microscopic viewpoint, in addition to bulk velocity, gas molecules are translated by thermal velocity, also. In high speed (hypersonic) flows, the magnitude of the bulk velocity is higher than that of the thermal velocity. For such flows, conventional approach is the use of vacuum boundary at the exit of the channel (when a molecule reaches the vacuum boundary it leaves the computational domain permanently and there is no inward flux at that boundary).
In contrast to hypersonic flows, the magnitude of thermal motion is comparable with bulk velocity in low speed flow and inward flux because thermal motion of molecules should be taken into consideration. In order to cover a wide range of flow regimes, Liou and Fang [
In this approach, the number of incoming molecules and their corresponding velocities are estimated implicitly by the local flow properties. For either upstream or downstream boundary, molecular flux
Tangential velocity components (
If streamwise inlet velocity is zero
Flow properties of both upstream and downstream boundaries must be determined in order to be used in (
Shear-Driven and Pressure-Driven flows in confined microchannels are well-studied problems. They are frequently used as test cases for the robustness and validations of flow simulations. In this section, the theoretical background and the simulation setup for such flows are presented. A new implementation of EDMD simulation including cell partitioning, event scheduling, and implicit boundary treatment was developed in C# with serial processing. Benefiting from object oriented programming (OOP), code can be further extended to cover different molecular types and boundary treatments. On the other hand, considering the computational performance the parallelization of the code is an essential issue in molecular simulations. However, this deserves a detailed study of its own. All simulations were carried out on a fourth-generation Intel i7 4700 CPU computer with 16 GB of RAM.
Shear-Driven flow in a confined channel is obtained by moving the top plate at a constant velocity (Figure
Flow configurations in Shear-Driven flow.
Case A (Kn = 0.1) | Case B (Kn = 0.5) | Case C (Kn = 1.0) | ||||
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Channel height, |
0.050 | 0.263 | 0.010 | 0.053 | 0.005 | 0.026 |
Global average pressure, |
1639 kPa ( | |||||
Global average density, |
26.07 kg/m3 ( | |||||
Boundary conditions | Fully diffusive walls @300 K along |
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Periodicity along |
Flow geometries.
Shear-Driven flow
Pressure-Driven flow
Simulation results were compared with the continuum-based model in which velocity gradient between two walls of the channel is affected by the Knudsen number and has a linear nature in Shear-Driven flows. For a fully diffusive wall, normalized velocity profile is shown as follows, where
Pressure-Driven flow is generated by setting different pressure levels on upstream and downstream boundaries (Figure
Flow configurations in Pressure-Driven flow.
Case D | Case E | |
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Inlet pressure, |
250 kPa | 75 kPa |
Pressure ratio, PR | 2.5 | 5 |
Initial global average density ( |
4.00 kg/m3 | 1.20 kg/m3 |
Initial Kn | 0.080 | 0.268 |
Channel height, |
0.4 | |
Aspect ratio ( |
5 | |
Boundary conditions | Fully diffusive walls @300 K along |
|
Periodicity along |
For Pressure-Driven flows, Arkilic et al. defined the pressure distribution along the fully diffusive channel as a function of Knudsen number at the exit of the channel (
In this section, we present simulation results of Shear-Driven and Pressure-Driven flows for aforementioned cases. Results were compared with analytical solutions derived from Navier-Stokes equations with slip flow boundary conditions and DSMC results. Simulation configuration (number of simulated molecules, spacing ratio, boundary types, etc.) greatly affects computational speed. As an example, simulation of one million argon molecules in a cubical region when the spacing ratio is 5.0 takes 450 seconds to reach one collision per particle on the aforementioned PC configuration.
In Cases A–C, six Couette flow configurations at different Knudsen numbers were simulated. Comparison of the predicted slip velocities between simulation results and the analytical solution is given in Table
Comparison of slip velocities.
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Case A (Kn = 0.1) | 8.33 | 7.75 | 7.56 |
Case B (Kn = 0.2) | 25.00 | 18.59 | 19.29 |
Case C (Kn = 0.5) | 33.33 | 25.46 | 26.22 |
Predicted velocity profiles and deviations from the analytical solution are presented in Figures
Shear-Driven flow, Case A
Shear-Driven flow, Case B
Shear-Driven flow, Case C
Another finding is that nearly identical velocity profiles were obtained for different spacing ratios at the same Knudsen number. As the spacing ratio reduces from 7.0 to 4.0, computational speed quadruples. Utilization of this piece of information can ease the computational cost of larger systems; that is, EDMD performs faster in smaller spacing ratios [
Pressure distribution along the channel is given in Figures
Pressure distribution in Pressure-Driven flow, Case D: (a) comparison of pressure distribution along
Pressure distribution in Pressure-Driven flow, Case E: (a) comparison of pressure distribution along
Local mean free path increases along the channel due to pressure gradient. This yields an increase in local Knudsen number (Figure
Local Kn distribution in Pressure-Driven flow.
Ratio of the slip velocity to the local average velocity is given in Figure
Normalized slip velocity for slip-to-transition region.
Normalized velocity profiles (
Velocity profiles in
Velocity profiles in
This paper presents an event-driven molecular dynamics (EDMD) simulation of monatomic gas flows in microchannels. In this study, molecules are considered as hard-spheres with discrete interaction potentials and interaction times are analytically predictable. Therefore, each collision is real and takes place in its calculated position. Postcollisional velocities are also analytically calculated according to the conservation of momentum and energy. By its deterministic nature, EDMD differs from DSMC method in which simulated system is represented by a smaller group of molecules and collision pairs are selected stochastically. Additionally, by using cell partitioning technique and use of priority queues for event sorting, EDMD is now a strong alternative to DSMC method in terms of not only accuracy but also computational speed.
In this study, an implicit treatment for flow boundaries is adapted from DSMC method to EDMD for the first time in order to induce Shear- and Pressure-Driven flows in confined channels. The number of introduced molecules through upstream and downstream boundaries and their corresponding velocities and locations are determined implicitly from local number density, mean flow velocity, and temperature.
Velocity slip in Pressure-Driven flows is compared with DSMC results and solution of linearized Boltzmann equation. The good agreement confirms the robustness of the treatment and its applicability to EDMD simulations. For both Shear- and Pressure-Driven flows, velocity slip is also compared with the one derived from Navier-Stokes equations of slip boundary conditions. For low Knudsen number
Additionally, simulations of two flows having the same
The authors declare that there is no conflict of interests regarding the publication of this paper.