FORCE ENERGY. only if F = F(r). 1 Nano-scale (10 9 m) 2 Nano to Micro-scale (10 6 m) 3 Nano to Meso-scale (10 3 m)

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1 What is the force? CEE 618 Scientific Parallel Computing (Lecture 12) Dissipative Hydrodynamics (DHD) Albert S. Kim Department of Civil and Environmental Engineering University of Hawai i at Manoa 2540 Dole Street, Holmes 383, Honolulu, Hawaii FORCE ENERGY A push or pull that can cause an object with mass to accelerate Newton s second law: Acceleration: F = ma a = dv dt = d2 r dt 2 A scalar physical quantity that is a property of objects and systems which is conserved by nature The ability to do work: r2 E = F dr r 1 only if F = F(r). 1 / 26 4 / 26 Outline Statistical Mechanical Approaches Nano-scale (10 9 m) MD (Molecular Dynamics) = Deterministic simulation of solving Newton s second law for ion species 2 Nano to Micro-scale (10 6 m) BD () = Updated simulation protocol of MD for ions in a fluid medium, but more applied to volumeless (point) colloidal/nano-particles: Random Forces/Torques DPD (Dissipative ) = Simulation method for Brownian motion of multiple particles using (approximate) pair-wise hydrodynamics. 3 Nano to Meso-scale (10 3 m) SD () = Accurate simulation method for micro-hydrodynamics of spherical particles DHD = General simulation method for micro-hydrodynamics of Brownian and non-brownian particles 2 / 26 5 / 26 What is? : Langevin s Equation The Langevin equations for the system of N Brownian particles: for particle i interacting with j s ṗ i = m i v i = F i (r)+ j ( )ξ ij v j + j α ij f j A study of motion of multiple particles, influenced by forces and torques 3 / 26 1 Molecular Dynamics for conservative forces/torques 2 for hydrodynamic forces/torques 3 Dissipative for stochastic forces/torques * On the average hydrodynamic stochastic p i = m i v i is the momentum, ξ ij is the hydrodynamic friction tensor, F i is the sum of inter-particle and external forces, and j α ijf j represents the randomly fluctuating force exerted on a particle by the surrounding fluid: negligible if particles are much bigger than 1.0 µm. 6 / 26

2 Properties of Random Fluctuating Force, f i 1 Time average is zero: f i = 0 (1) 2 Independently exerted on i and j particle of different positions (i.e., r i and r j ) and at different times (i.e., t and t ) e.g., a falling body in liquid with x(0) = 0 & v(0) = 0 ma = mg βv +f B (t) f i (t)f j ( t ) = 2δ ij δ ( t t ) (2) 3 δ is the Dirac-delta function: δ ij = 0 if i j; and δ ij = 1 if i = j; δ(t t ) = 0 if t t ; and δ(t t ) = 1 if t = t. 4 Related to the friction coefficient indicating α ξ. ξ ij = 1 k B T α ik α jk (3) k 7 / / 26 Integration of the Langevin equation gives the time evolution equation: r i (t+ t) = r i (t)+ j D ij (t) k B T F j t+( D) t+ r G i (4) where the components of r G i are random displacements selected from 3N variate Gaussian distribution with zero means and covariance matrix r G i = 0 and r G i r G j = 2D ij t (5) The Oseen tensor (crude approximation) is given by D ij = k BT 1, for i = j (6a) 6πηa = k BT 8πηr ij ( 1+ r ijr ij r 2 ij ), for i j (6b) and one calculates D = 0. If F j 0, the random motion is dominant in multi-particle dynamics: r G i t. 8 / 26 : Langevin s Equation The Langevin equations for the system of N force-free, non-brownian particles ṗ i = m i v i = ξ ij (v j U) F H j F H is the hydrodynamic forces/torques, p i = m i v i is the momentum, and ξ ij is the hydrodynamic friction tensor. If particles are at rest, U = M F H (8) F H = R U (9) R = (M ) 1 (10) where U is the translational/rotational velocity vector, and M and R are the grand mobility and grand resistance matrixes, respectively. R is dependent on particle positions and calculated as an inverse matrix of M. 11 / 26 (BD) (SD) Langevin equation 1 with inter-particle (conservative) forces f P, drag forces f H = ξv, and random Brownian forces f B m dv = f P +f H +f B (t) (7a) dt f H = ξv (7b) f B (t) = 0 (7c) f B (0) f B (t) = 6ξk B Tδ(t) (7d) Particles translate and rotate in a fluid field of V = U +r Ω +E : r where U is the uni-directional flow; and the vorticity Ω and rate of strain E are represented as Ω = 1 2 V (r) Eij = 1 ( Vi + V ) j = 1 2 x j x 2 ( jv i + i V j ) = E ji i respectively. If no shear, E = 0 1 Ermak and McCammon, J. Chem. Phys. 69 (1978) ; Langevin, C. R. Acad. Sci. (Paris) 146 (1908) / / 26

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4 Directions of force/torque: F x,f y,f z,t x,t y,t z Exerted on each particle with upflow, U = +1( ) / 26 1 is a set of tools for generating high quality raster images of proteins or other molecules. 2 The core program renders spheres, triangles, cylinders, and quadric surfaces with specular highlighting, Phong shading, and shadowing. 22 / 26 Lab work Example 1 SD code code for hydrodynamic force/torque calculation is in /opt/cee618s13/class12/hasonjee/ 1 Copy all the files from /opt/cee618s13/class12/raster3d/example1/ to your own directory. 2 Type and enter: qsub raster_ex1.pbs 3 This pbs script will execute example1h.script and generate an image file, example1h.tff 20 / / 26 Outline Sphere configuration: array 1 2 Under /mnt/home/albertsk/uhtraining/cee618-sp2012/class09/dhd 1 In shsnj_obsd_fts_64.f To rotate image change Euler angles ofalpha0,beta0, and gamma0. To change the distance between the center and your eyes, control distance shsnj_obsd_fts_64.f. 2 spheres.f is included in the main code shsnj_obsd_fts_64.f. 3 There will be three output files from this serial run: 1 sforcefts.dat stores force/torque calculation data. 2 scoordxyz.dat includes (x, y, z) coordinates of N p particles. 3 scoordxyz.r3d contains format coordinate data, translated to the center of mass. 21 / / 26

5 How to generate an image 1 Copy all the files in /opt/cee618s13/class12/dhd-raster3d/ to your own directory. 2 Execute $ make $ make run 3 Then, a file like scoordxyz.tff will be generated. 4 Download the.tff file and view it. 25 / 26 Raster file: scoordxyz.r3d, x, y, z, a, and 3 more 1 Example of material properties and file indirection tiles in x,y pixels (x,y) per tile 4 4 3x3 virtual pixels -> 2x2 pixels background colour 6 T cast shadows 7 25 Phong power secondary light contribution ambient light contribution specular reflection component eye position main light source position E E E E E E E E E E E E E E E E mixed objects 18 * 19 * 20 * 21 # Draw a bunch of spheres 22 # 23 # 24 # E E E E E E E E E E E E E E E E E E E E E / 26