Molecular Dynamics. Timothy O Sullivan Supervised by Naida Lacevic & John Sader The University of Melbourne

Transcription

1 Molecular Dynamics Timothy O Sullivan Supervised by Naida Lacevic & John Sader The University of Melbourne 1

2 1 Introduction Molecular Dynamics (MD) is a method of simulating systems of nanoscale volumes, in ways that are normally unavailable via analytic or statistical methods. Using MD simulations, this unseen world can be probed, allowing for real-time calculations of a physical, molecular system. From these calculations, various properties of the system can be inferred. This gives information of changing behaviours over time and indicates the various effects of different boundary conditions. This report will investigate a ground up approach towards MD simulations, which will then be verified via thermodynamic calculations performed on the system. The algorithm developed will then be used to investigate the effects of varying boundary conditions on a simulated solution to the Navier-Stokes equation; Couette Flow. 2 Principles To simulate the dynamics of a system, one must first have some idea of how the system will behave. This can easily be determined via the system s characteristic equations of motion and an associated potential. For this report, we will assume a system comprising of N particles, obeying Newton s Equations: F i = m i a i (1) Where F i is the force on the i-th particle, and a i, m i and r i are the acceleration, mass and position of the i-th particle respectively. First and foremost, the force on each particle must be calculated. This can be done via the following relation: (2) (3) Where U ij is the potential between particles i and j, and r ij is their separation. With knowledge of (1), (2) and (3), a potential is all that is required to simulate the system. The potential to be used in this report will be the Lennard-Jones potential: 2

3 Figure 1: The Lennard-Jones Potential. Image credit: By Olaf Lenz, Rainald62 [GFDL ( CC-BY-SA-3.0 ( or CC BY-SA ( via Wikimedia Commons The Lennard-Jones potential is a reasonably accurate model of the interatomic potential. The second term represents the attractive forces between atoms, ie dispersion forces, and the first term represents the repulsive forces due to the Pauli-Exclusion principle. Also note the quantities ε and σ in the equation for the potential. The presence of these quantities suggests some sort of characteristic energy and length scale for the system. 2.1 Scaling The system can now be scaled accordingly: V = V ε (4) r = r σ T = k BT ε (5) (6) t = t σ m ε (7) 3

4 v = v ε m (8) F = Fσ ε P = Pσ3 ε (9) (10) From here the * notation will be dropped for convenience. Any other quantities not listed here will be scaled accordingly. 3 Simulating the System Now that the system has been mathematically quantified, the ODE s need to be solved. The ODE s specifically are N versions of (2), which are all coupled to each other. The standard method would be to use the Runge-Kutta algorithm, however Runge-Kutta exhibits poor long-term energy conservation (Frenkel and Smit, 2001). So a different numerical solving algorithm is required. 3.1 The Verlet Algorithm Instead, the Verlet algorithm is put to use. Its formulation is as follows: Take perturbations of a particle s position forward and backward in time: r(t + t) = r(t) + v(t) t + 1 2! m F(t) t2 + 1 d 3 r(t) 3! dt 3 t 3 + O( t 4 ) (11) r(t t) = r(t) v(t) t + 1 2! m F(t) t2 1 d 3 r(t) 3! dt 3 t 3 + O( t 4 ) (12) By adding (11) and (12) together, the following is obtained: r(t + t) = 2r(t) r(t t) + 1 m F(t) t2 + O( t 4 ) (13) So by calculating the forces on the particles at every time step, their positions can be solved for all time. 4

5 One amendment that must be made to this algorithm is the calculation of velocities. The Verlet algorithm does not care about velocities, so they need to be treated separately as follows: r(t + t) r(t t) v(t + t) = 2 t (14) 3.2 Initial Conditions A square packing of particles was chosen as the initial conditions for the system: Figure 2: Square packing initial conditions. It should be noted however, that the initial conditions don t necessarily matter, as long as there isn t severe overlap of any particles. The initial velocities are also chosen such that the system s initial temperature can match the desired temperature. 3.2 Boundary Conditions Periodic boundary conditions were chosen for this project as they eliminate any effects a physical boundary may have on the system. It also allows for a pseudosimulation of an infinite system. The principles of these boundary conditions are as follows: Imagine the unit-cell (the system to be simulated) at the centre of an infinitely large array of identical image cells: 5

6 Figure 3: Unit cell in an infinite, bulk system. By doing this, any particles that leave the middle cell through the right re-enter through the left and similarly for top and bottom. This gives a system that conserves particle number. The forces on the particles due to the particle images are also calculated. To maintain symmetry and computational simplicity, a cut-off radius is chosen for the potential. This radius will be chosen to be r c = 2.5 for the remainder of the report unless otherwise specified. U(r c ) = 0.016, so the error associated with introducing this cut-off is small enough such that it will not affect the physics of the system. The box width is also chosen to be at least double that of the cut-off radius. By doing so, the particles will only interact with one of each image at most, and more importantly, particles will not be able to interact with their own images. 3.3 Flow Chart The code flow chart has been added to ensure absolute clarity of the project thus far: 6

7 Figure 4: MD code flow-chart. 4 Couette Flow The second aim of the simulation is to simulate an analytic solution to the Navier- Stokes equation: Couette Flow. If we take the Navier Stokes equation: u P + u u = t ρ + ν 2 u (15) Where u is the velocity of the fluid, P is the fluid pressure and ν is the kinematic viscosity. If this equation is then applied to a Couette flow system, ie: Figure 5: Fluid between shearing plates, ie Couette flow. 7

8 And by making the appropriate assumptions, the Navier-Stokes equation reduces to: d 2 u dy 2 = 0 (16) If a no-slip boundary condition is assumed (ie relative velocity between fluid and solid is zero at the fluid-solid interface), the following solution is found: u(y) = U y h (17) So the Navier-stokes solution states that the steady-flow solution to Couette flow is a linear velocity profile, matching the wall velocities at the boundaries (assuming no slip). Now that this solution has been found, it is natural to then check how this solution compares to an MD simulation of the same system. To do this, some additions need to be made to the pre-existing algorithm. It s rather simple and only requires two things. 4.1 The Walls To simulate Couette flow with Molecular Dynamics, the walls need to be added in. The walls will comprise of similar Lennard-Jones particles, like the fluid. However, ε wf, the interaction strength between wall and fluid particles will be varied. It is also assumed that the walls are not affected by this interaction, ie they are completely rigid (a spring potential can be added later, however this not completely necessary and this assumption of rigidity saves a lot of computation time). The walls will also be initialized with a hexagonal packing: Figure 6: Hexagonal Packing of particles. This allows for the most realistic simulation of a solid at very high density, which may be considered at some point in this simulation. The walls will also obey periodic boundary conditions. 4.2 The Thermostat 8

9 The last thing needed to accurately simulate Couette flow is a way to keep the systems temperature constant. This is necessary because the temperature of a physical Couette flow channel stays constant because any kinetic energy added to the system by the walls is constantly being carried downstream to a point of lower energy. However, this doesn t occur in the MD system, due to the periodic boundary conditions. Therefore, any kinetic energy added to the fluid by the moving wall will accumulate. A thermostat is necessary to dissipate this extra heat so that the periodic system can be treated as an infinite, planar channel. There are many ways to do this, with some being better than others. A very quick investigation was conducted into various thermostats to see which one was the most appropriate for the purposes of this report Velocity Rescale A velocity rescale is by far the simplest thermostat to apply. It works by checking a system s temperature, and if it is above or below the desired temperature, the particle velocities are adjusted accordingly. Due to the simple nature of this thermostat, it can be applied in systems of all densities and temperatures. This simplicity comes with a cost however, especially with the Verlet algorithm. Since the Verlet algorithm does not consider velocities, a position rescale must instead be considered. This is extremely risky to do, as it only takes one particle s position to be rescaled too close to that of another particle, and the solver will fail Nose-Hoover Thermostat The Nose-Hoover Thermostat involves adding a friction term to the equations of motion of the particles. However, this thermostat is better suited to a Hamiltonian formulation, as opposed to the Newtonian formulation used in this report. Any resources that were found that did treat the Newtonian formulation with a Nose- Hoover thermostat used different solving algorithms to that of the Verlet Algorithm. Attempts were made to try and resolve the Verlet algorithm with the Nose-Hoover thermostat, however this caused the temperature to oscillate at a fixed frequency instead of being held constant Langevin Thermostat This was the thermostat most commonly found amongst the literature when simulating Couette flow with MD. With some effort, this thermostat did keep the system temperature constant, however the parameters used to achieve this would not work for any other fluid density. Thus any adjustment to the fluid properties meant a complete adjustment of the Langevin thermostat parameters. 9

10 Despite showing great temperature control, the Langevin thermostat showed complete inconvenience when it was used in generality Berendsen Thermostat The Berendsen thermostat is implemented by weakly coupling the system to a heat-bath of the desired temperature. This is done in reality by adding a friction term to the equations of motion, much like the Nose-Hoover thermostat. However it was found that unlike the Nose-Hoover thermostat, the Berendsen thermostat kept temperature constant with no issue when it was coupled with the Verlet algorithm. It was also found to work in complete generality, unlike the Langevin thermostat. No issues of solver failure were ever encountered as long as the Berendsen parameters were given reasonable numbers, unlike the velocity rescale method. However it has been shown that the Berendsen thermostat does not keep the system in the correct thermodynamic ensemble (Hünenburg, 2005). Despite this, the Berendsen thermostat was chosen for this report due to its ability to be applied across a broad array of systems. Any effects this thermostat has on the thermodynamics of MD need further systematic investigation. Since the Berendsen thermostat has been chosen as the thermostat for Couette flow, its equation of motion is specified: d 2 r dt 2 = 1 m {F + 1 2τ [ T 0 T(t) 1] v} (18) Where T 0 is the desired temperature and T(t) is the current temperature. The parameter τ is a relaxation parameter that determines how quickly the system returns to the desired temperature. The Verlet algorithm is then adjusted accordingly. 5 Calculations Various measurements of the system will be made via calculations using the evolving positions of the particles in time. Some will be specified just to ensure that the system is working properly, others will be genuine attempts at understanding the dynamics of the system. 5.1 Energy The first quantity of interest is energy. From these calculations, the systems energy conservation can be confirmed, the temperature can be determined and pressure can start to be calculated. 10

11 5.1.1 Kinetic Energy Kinetic energy is calculated in the usual way: N K = 1 2 v i 2 i=1 (19) That is, a sum over the squares of all particle velocities Potential Energy Imagine each interacting pair of particles as two balls connected by a spring. The sum of the energies in each of these springs is the systems potential, or more explicitly: N 1 N U = U ij (r ij ) i=1 j=i+1 (20) Total Energy The systems total energy is calculated in the usual way: Temperature E = K + U (21) The systems temperature is calculated depending on whether the system is 2- dimensional or 3-dimensional. For 2D: T = K N (22) For 3D: T = 2K 3N (23) 5.2 Pressure The pressure of the system can be calculated in two ways: P = ρt + 1 3V r ij F ij i j<i (23) 11

12 P = ρt ρ2 6 du(r) dr rg(r) dr 0 0 = ρt 2 3 πρ2 dr du(r) r3 g(r) dr (24) (25) Where g(r) is the pair correlation function, a function that describes the overall structure of the fluid. Both methods are equally valid, however (23) is used for convenience. The calculation is not complete however. A correction term must be added due to the presence of the cut-off radius. This is done by using (25). Since the integral from 0 to r c has already been accounted for, the correction term is just the remainder of the integral in (25), ie: P rc = 2 3 πρ2 dr du(r) dr r3 g(r) (26) r c It is then assumed that g(r) = 1 (error associated in doing so is 0.05 at most) (Verlet, 1967), it is found that: Thus, the expression for pressure is: P = ρt + 1 3V P rc = 16πρ 2 ( 2 9r c 9 1 3r c 3 ) (27) i j<i r ij F ij + 16πρ 2 ( 2 9r c 9 1 3r c 3 ) (28) 5.3 Diffusion Another class of properties that are of interest are transport properties of the fluid. Firstly, diffusion is considered. Diffusion is the process by which a fluid will move from a region of high concentration to a region of low concentration. Since there is only one fluid to consider in this simulation, self-diffusion is considered; the process by which individual fluid particles will move throughout their parent fluid. This process can be mathematically quantified via a combination of the continuity equation: Fick s law: c(r, t) t + j(r, t) = 0 (29) j = D c (30) 12

13 And the definition of the mean of a continuous variable, specifically the meansquared displacement: r 2 (t) drc(r, t)r 2 (31) From (29), (30) and (31), it can be derived that: r 2 (t) = 2dD t (32) ie the Diffusion coefficient of a fluid can be found from the gradient of the meansquared displacement versus time graph. The full derivation of (32) can be found in Understanding Molecular Simulation: From Algorithms to Applications by Frenkel and Smit. To calculate the Diffusion coefficient from these calculations, the squared displacement of each particles is recorded and averaged for every time step of the Verlet algorithm. To smooth the data set, this is repeated for every second time step, every third time step etc. 5.3 Viscosity Once a result has been calculated for the Diffusion coefficient for the LJ fluid, that result can then immediately be used to calculate the Viscosity of the fluid. Viscosity is a measure of a fluids resistance to a shear or tensile stress. Via use of the Stokes- Einstein relation: Dμ = k BT 6πr H (33) Where μ is the fluid viscosity and r H is the hydrodynamic radius, which will be taken to be the molecular radius in the case of Argon.The viscosity can then be immediately be solved for. 5.4 Velocity Profiles Once Couette flow has been successfully simulated, the continuum approximation will be compared with the velocity profile obtained from the MD simulation. To obtain these velocity profiles, the Couette flow channel is divided into bins and the x-velocity of the particles in each bin is summed. These sums are then divided by the number of particles in each bin. These results can then be plotted as a histogram. 13

14 5.5 Slip length Upon obtaining these velocity profiles, it is suspected that there will exist some conditions where the MD result does not agree with the no-slip solution to the Navier-Stokes equation. In these cases, it is said that the fluid is slipping over the solid boundary. A slip-length is a quantisation of this slipping. The slip length is calculated by extrapolating the linear section of the velocity profile until the wallvelocity has been reached, the slip-length is then given by the extrapolation distance, ie: Figure 7: Slip-length definition. Where the blue-dotted line is the no-slip solution to the NS equation and the redline is the MD result (or the NS solution with slip assumed). In this particular case, the slip-length is Results 6.1 Energy Conservation One of the most important aspects of an MD system is that it conserves energy in systems where there are no body or boundary forces applied, ie the first set of simulations. This is checked by running free simulations, that is, simulations with no thermostats applied to the system. It was found that the simulations were conserving energy properly, which is demonstrated by one set of data from a large set that demonstrate such results: 14

15 Figure 8: Energy graphs for 100 particle, 2D system. Immediately it is seen that the simulation is conserving energy. Also note that the kinetic and potential energies take some time to get to the equilibrium states. This time must be accounted for upon taking later calculations. Now that it is known that the system is conserving energy, further calculations can be submitted. 6.2 Pressure Pressure calculations were made on a 3D system of 729 particles, initialised in a cube with 9 particles for its side length. The pressure was calculated via (28), and the simulation was repeated 30 times to gain the error readings of 2σ. The system temperature and density were varied and the pressure results were compared to those of Lentz (2013): Expected Results Results Lentz Results ρ T P P Error P Error % % % % % % Figure 9: Pressure results. The results show that the simulation pressure is in very good agreeance with the expected results. It is also seen that the best result occurs for very low system 15

16 density, whereas this same system is Lentz worst result. The reason for this is unknown and requires further investigation. 6.2 Diffusion & Viscosity Diffusion was calculated via an algorithm based on section 5.3. The calculation was repeated 64 times to gain error estimates and the result was then compared to Rahman (1967): Figure 10: Mean-square displacement versus time. From figure 10, the result is descaled for Argon (σ = 3.4E 10, ε = 120k B ) and it is found that for T = 94.4 K and ρ = 1.374g cm 3, D = (3.31 ± 1.88) 10 9 m 2 /s. Using this value for D, the viscosity μ is calculated using (33) and it is found that μ = (5.82 ± 3.31) 10 5 m 2 /s. For the specified temperatures, the expected results are D = m 2 /s and the range for viscosity is μ = m 2 /s. So it is seen that the results somewhat line up with the literature, however the errors involved are large, indicating that this method, or at least the algorithm used, needs refining. 6.2 Velocity Profiles & Slip lengths. Several Couette flow simulations were run for various wall densities and wall-fluid interaction strengths. The trends in the literature (Thompson and Robbins) show that a systems slip length will increase with increasing wall density and decreasing wall-fluid interacting strength. This is due to changes in the corrugation the fluid particles see in the wall potential, essentially changing the friction that the fluid particles experience. For a low wall density, the wall potential is very corrugated and the fluid particles can get stuck in these corrugations and be dragged along by the wall, decreasing slip. Varying the wall-fluid interaction strength simply changes how hard the fluid particles stick to the wall. The simulations were run with D LJ particles and 225 wall particles. The system temperature was held constant at T = 1.3 and the system density was 16

17 ρ f = The simulation was allowed to run for integration steps with r c = steps were used for the system to come to a steady flow, the remaining steps were used to sample and average the velocity profile. The wall density was varied between ρ w = ρ f and ρ w = 2.52ρ f. The channel had dimensions of length l = 63 and width w = The results were as follows: With associated slip lengths: Figure 11: Couette flow velocity profile. 17

18 Wall-Density ( ρ w ρ ) Wall-Fluid Interaction strength (ε wf ) Slip Length (x/σ) Figure 12: Slip lengths. The results show that the trend of the literature has been matched. It is observed that an increase in slip length occurs with an increasing wall density and a decreasing wall-fluid interaction strength. It is also observed that a negative sliplength occurs for certain values of the parameters. This is what the literature refers to as a lock-length. It occurs when the fluid s velocity profile matches the wall velocity before the fluid has reached the wall. This occurs because multiple layers of the fluid have stuck to the wall and are now essentially acting as wall particles themselves. This explains the strange non-linearities of the second and third velocity profiles, which occur due to the remaining fluid particles interacting with both wall-like fluid particles and wall particles. However, speaking in quantitative terms, the positive slip lengths calculated are not correct in reference to the literature, in fact they are way off. Unfortunately the reason for this has not been systematically investigated, due to the large number of possible causes (ie, thermostat, 2D simulation instead of 3D, differing shear rates to that of the literature). 7 Visuals On top of the results presented, some more visually aesthetic results were also obtained in the form of videos. Links can be found below: 2D 100 particle system (x0.5 speed): 3D 1000 particle system (x0.75 speed): 18

19 Couette flow system: 8 Conclusion An overview of Molecular Dynamics at a basic level and results obtained from the simulations have been presented. The algorithms written conserve energy and produce thermodynamic calculations which agree reasonably well with the expected results from the literature, indicating the simulations are giving a physically accurate representation of the mechanisms occurring within nanoscalevolume, molecular systems. Couette flow, an analytic solution to the Navier-stokes solution, was also compared to the results from MD simulations, and it was found that a slip condition is relevant, and slip does increase for increasing wall density and decreasing wall-fluid interaction strength, as suggested by the literature. However, there were some results that did not agree with literature. These results will require further investigation to resolve. 9 Acknowledgements Special thanks go to Thomas Tommy Tomlinson Nollan for his assistance early on in this project with learning MATLAB. References Frenkel, D, & Smit, B 2001, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, Amsterdam, NLD. Available from: ProQuest ebrary. [24 February 2016]. Hünenberger, P.H., Thermostat algorithms for molecular dynamics simulations. In Advanced computer simulation (pp ). Springer Berlin Heidelberg. Verlet, L., Computer" experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physical review,159(1), p.98. Lentz, L., Final Project: MD Simulationof Liquid Argon, Department of Mechanical Engineering, Massachusetts Institute of Technology, viewed 25/02/2016 < Rahman, A., Correlations in the motion of atoms in liquid argon. Physical Review, 136(2A), p.a405. Thompson, P.A. and Robbins, M.O., Shear flow near solids: Epitaxial order and flow boundary conditions. Physical review A, 41(12), p

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