Scientific Computing II

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1 Scientific Computing II Molecular Dynamics Simulation Michael Bader SCCS Summer Term 2015 Molecular Dynamics Simulation, Summer Term

2 Continuum Mechanics for Fluid Mechanics? Molecular Dynamics the Physical Model Quantum vs. Classical Mechanics Van der Waals Attraction Lennard Jones Potential Molecular Dynamics the Mathematical Model System of ODE Initial Conditions Computational Domain & Boundary Conditions Molecular Dynamics Simulation, Summer Term

3 Essentials from Continuum Mechanics for Fluids Fluid: notion covering liquids and gases liquids: hardly compressible gases: volume depends on pressure small resistance to changes of form Continuum: space, continuously filled with mass homogeneous subdivision into small fluid voxels with constant physical properties is possible idea valid on micro scale upward (where we consider continuous masses and not discrete particles) Molecular Dynamics Simulation, Summer Term

4 Description of State consideration of a control volume V 0 (Eulerian perspective) description of the fluid s state via the velocity field v( x, t) and two thermodynamical quantities, typically the pressure p( x, t) and the density ρ( x, t) for incompressible fluids, the density ρ is constant (if there are no chemical reactions) Molecular Dynamics Simulation, Summer Term

5 Molecular Dynamics for Fluids? Example: channel flow at decreasing density, Kn = mean free path characteristic length Kn = Kn = ux (y/h)/u LBM Li et al. 0.2 Present LBM: 2nd Order, VA 2nd Order Slip NS Ohwada et al y/h ux (y/h)/u LBM Li et al. 0.5 Present LBM: 2nd Order, VA 2nd Order Slip NS Ohwada et al y/h continuum description only valid on coarse scale other methods (Molecular Dynamics, Direct Simulation Monte Carlo,...) required Molecular Dynamics Simulation, Summer Term

6 Scope of Application Loschmidt number 2, cm 3 : number of molecules in 1 cm 3 of an ideal gas Avogadro constant mol 1 : number of carbon-12 atoms in 12g of carbon-12, or number of molecules in 1 mol (1 mol, under normal conditions, taking a volume of 22.4 litres) Avogadro number: notion being used in different ways for both of the above constants, which depend on each other (2, cm cm 3 mol 1 = mol 1 ) time steps for numerical simulations are typically in the order of femtoseconds (1fs := s) Molecular Dynamics Simulation, Summer Term

7 Pour Me A Glass... assume we want to simulate all molecules in one glass (0.5l) of water for 1sec assume a time step of 1fs assume we only need one floating point operation per molecule in each time step molecules (biggest MD simulation: molecules) timesteps operations Using SuperMUC, this means we need at least years for the computations PB of memory assuming we need to store only 3+3 unknowns per molecule (position+velocity) hence: scope of application is limited to micro- and nanoscale simulations (at least for the near future) Molecular Dynamics Simulation, Summer Term

8 Classical Molecular Dynamics Quantum mechanics: Schrdinger equation, probability distributions, etc. too difficult for N -body seetings, where N is large approximation via classical Molecular Dynamics based on Newton s equations of motion: F i = m i i r molecules are modelled as particles; simplest case: point masses there are interactions (forces) between molecules multibody potential functions describe the potential energy of the system; the velocities of the molecules (kinetic energy) are a composition of the Brownian motion (high velocities, no macroscopic movement), flow velocity (for fluids) total energy is constant energy conservation Molecular Dynamics Simulation, Summer Term

9 Fundamental Interactions Classification of the fundamental interactions: strong nuclear force electromagnetic force weak nuclear force gravity r j r i r k O interaction potential energy the total potential of N particles is the sum of multibody potentials: U := 0<i<N U 1(r i ) + 0<i<N i<j<n U 2(r i, r j ) + 0<i<N i<j<n j<k<n U 3(r i, r j, r k ) +... there are ( N n ) = N! n!(n n)! O(Nn ) n-body potentials U n, particulary N one-body and 1 N(N 1) two-body potentials 2 force F = gradu Molecular Dynamics Simulation, Summer Term

10 Forces vs. Potentials i r ij j some potentials from mechanics: harmonic potential (Hooke s law): U harm (r ij ) = 1 2 k (r ij r 0 ) 2 ; potential energy of a spring with length r 0, stretched/clinched to a length r ij gravitational potential: U grav (r ij ) = g m i m j r ij ; potential energy caused by a mass attraction of two bodies (planets, e.g.) the resulting force is F ij = gradu (r ij ) = U r ij integration of the force over the displacement results in the energy or a potential difference Newton s 3rd law (actio=reactio): F ij = F ji Molecular Dynamics Simulation, Summer Term

11 Intermolecular Two-Body Potentials { r ij d hard sphere potential: U HS (r ij ) = 0 r ij > d Force: Dirac Funktion ( ) n σ soft sphere potential: U SS (r ij ) = ɛ r ij r ij d 1 Square-well potential: U SW (r ij ) = ɛ d 1 < r ij < d 2 0 r ij d 2 r ij d Sutherland potential: U Su (r ij ) = r ij > d Lennard Jones potential van der Waals potential U W (r ij ) = 4ɛσ 6 ( 1 r ij ) 6 Coulomb potential: U C (r ij ) = 1 4πɛ 0 q i q j r ij ɛ = energy parameter σ = length parameter (corresponds to atom diameter, cmp. van der Waals radius) Molecular Dynamics Simulation, Summer Term ɛ r 6 ij

12 Intermolecular Two-Body Potentials: Hard Vs. Soft Spheres hard sphere potentials hard sphere Square well Sutherland soft sphere potentials soft sphere Lennard Jones van der Waals potential U 0 σ potential U 0 σ distance r distance r Molecular Dynamics Simulation, Summer Term

13 Van der Waals Attraction: So Small, So Important... made by Robert Michniewicz Molecular Dynamics Simulation, Summer Term

14 Van der Waals Attraction intermolecular, electrostatic interactions electron motion in the atomic hull may result in a temporary asymmetric charge distribution in the atom (i.e. more electrons (or negative charge, resp.) on one side of the atom than on the opposite one) charge displacement temporary dipole a temporary dipole attracts another temporary dipole induces an opposite dipole moment for a non-dipole atom and attracts it dipole moments are very small and the resulting electric attraction forces (van der Waals or London dispersion forces) are weak and act in a short range only atoms have to be very close to attract each other, for a long distance the two dipole partial charges cancel each other high temperature (kinetic energy) breaks van der Waals bonds Molecular Dynamics Simulation, Summer Term

15 Lennard Jones Potential general Lennard Jones potential: (( ) n ( ) m ) σ U LJ (r ij ) = αɛ r σ ij r ij ( ) with n > m and α = 1 1 n n n m n m m m LJ 12-6 potential ( ( ) 12 ( ) ) 6 σ U LJ (r ij ) = 4ɛ r σ ij r ij r ij r j r i O m = 6: van der Waals attraction (van der Waals potential) n = 12: Pauli repulsion (soft sphere potential): heuristic application: simulation of inert gases (e.g. Argon) force between 2 molecules: F ij = U(r ij) r ij = 24ɛ r ij (2 ( σ ) 12 ( ) ) 6 r σ ij r ij very fast decay short range (m = 6 > 3 = d dimension) Molecular Dynamics Simulation, Summer Term

16 LJ Atom-Interaction Parameters atom ɛ σ [ J] a [10 1 nm] b H He C N O F Ne S Cl Ar Br Kr ɛ = energy parameter σ = length parameter (cmp. van der Waals radius) parameter fitting to real world experiments a Boltzmann-constant: k B := J K b 10 1 nm = 1Å (Ångstöm) Molecular Dynamics Simulation, Summer Term

17 Dimensionless Formulation using reference values such as σ, ɛ, reduced forms of the equations can be derived and implemented transformation of the problem position, distance r := 1 σ r (1a) time velocity t := 1 σ ε m t v := t σ v (1b) (1c) Molecular Dynamics Simulation, Summer Term

18 Dimensionless Formulation (2) potential (atom-interaction parameters are eliminated!): U := U ɛ force U LJ (r ij ) := U LJ (r ij ) ɛ F ij U kin := U kin ɛ := F ij σ ɛ = 1 ɛ = 4 ( ( = 24 2 ( ( r ij 2 ) 6 (r ij mv 2 2 = v 2 2 t 2 r ij 2 ) 3 ) (2a) (2b) ) 6 2 (r ) ) ij 3 2 r ij (2c) 2 r ij Molecular Dynamics Simulation, Summer Term

19 Outlook: Multi-Centered Molecules C B1 F B1A1 F B1A2 C B C B2 F B2A2 F B2A1 F BA F AB molecules composed of multiple LJ-centers rigid bodies without internal degrees of freedom additionally: orientation (quarternions), angular velocity additionally: moment of inertia (principal axes transformation) calculation of the interactions between each center of one molecule to each center of the other resulting force (sum) acts at the center of gravity, additional calculation of torque F A1B1 F A1B2 F A2B1 C A F A2B2 C A1 C A2 Molecular Dynamics Simulation, Summer Term

20 Molecular Dynamics the Mathematical Model System of ODEs resulting force acting on a molecule: F i = j i acceleration of a molecule (Newton s 2nd law): F r i = i = m i j i F ij m i = F ij U( r i, r j ) j i r ij (3) m i system of dn coupled ordinary differential equations of 2nd order transferable (as compared to Hamilton formalism) to 2dN coupled ordinary differential equations of 1st order (N: number of molecules, d: dimension), e.g. independent variables q := r and p with p i := m i ri p i = F i (4a) (4b) Molecular Dynamics Simulation, Summer Term

21 Initial Conditions Initial Value Problem: molecule positions and velocities have to be given; initial configuration e.g.: molecules in crystal lattice (body-/face-centered cell) absolute value of initial velocity dependent of the temperature (normal distribution or uniform), e.g. d 2 N k BT = 1 2 N i=1 mv 2 i with v i := v 0 dkb T v 0 := m resp. v 0 := dt t random direction for initial velocity Molecular Dynamics Simulation, Summer Term

22 NVT-Ensemble, Thermostat statistical (thermodynamics) ensemble set of possible states a system might be in: for the simulation of a (canonical) NVT-ensemble, the following values have to be kept constant: N: number of molecules V : volume T : temperature a thermostat regulates and controls the temperature (the kinetic energy), which is fluctuating in a simulation the kinetic energy is specified by the velocity of the molecules: E kin = 1 2 i m i v i 2 the temperature is defined by T = 2 E dnk kin B (N: number of molecules, k B : Boltzmann-constant) simple method: isokinetic (velocity) scaling v corr := βv act mit β = Tref T act further methods e.g. Berendsen-, Nosé-Hoover-thermostat Molecular Dynamics Simulation, Summer Term

23 Periodic Boundary Conditions b a a b replicate simulated box periodically in all dimensions models an infinite space, built from identical cells domain with torus topology Molecular Dynamics Simulation, Summer Term

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