Computer Simulation of Shock Waves in Condensed Matter. Matthew R. Farrow 2 November 2007
|
|
- Phoebe Dawson
- 6 years ago
- Views:
Transcription
1 Computer Simulation of Shock Waves in Condensed Matter Matthew R. Farrow 2 November 2007
2 Outline of talk Shock wave theory Results Conclusion Computer simulation of shock waves
3 Shock Wave Theory
4 Shock wave theory Shock waves in gases Theory of shock waves has its roots in the field of gas-dynamics. Piston
5 Shock wave theory Shock waves in gases Piston moving at velocity u compresses gas. Piston
6 Shock wave theory Shock waves in gases A continuous solution across the shock front yields a meaningless result: Solutions to the gas-dynamic equations are trivial in which all quantities are constant Piston Discontinuity
7 Shock wave theory Shock waves in gases In order to satisfy the boundary conditions the variables are assumed to be discontinuous. Discontinuity regarded as limiting case of a very large gradient across a region which tends to zero. Shock waves are these discontinuities.
8 Shock wave theory Shock waves in solids Conservation relations are used to create a set of equations that govern shock wave dynamics - The Rankine-Hugoniot equations:
9 Shock wave theory The Hugoniot Hugoniot curves are relations between the thermodynamic state variables that describe the possible states of the system during shock compression. A single curve relating say, the volume and pressure, is called a Hugoniot. Hugoniot Hugoniot Pressure Isentrope Rayleigh Rayleigh line Line Us Hugoniot Isentrope V 0 Specific Volume up
10 Shock wave theory Hugoniots and shocks in solids For moderate temperatures and pressures, Hugoniots are well represented by: Where is usually between 1 and 2.0 for most materials However, some researchers[1] believe that a quadratic term is required: Shock waves in solids are described using the hydrodynamic approximation. If the Shock is sufficiently strong, the effects of yield strength are neglected [1] F. E. Prieto and C. Renero. Equation for shock adiabat. J. Appl. Phys.,41:3876 &, 1970.
11 Shock wave theory Isentrope and Rayleigh Line The isentrope is the curve of constant entropy Extremely difficult to calculate experimentally; Created from cold compression simulations where the system is maintained at zero kelvin; As 0K, and also adiabatic, the cold compression isotherm curve is also the isentrope The Rayleigh line connects the initial state with the final (shocked) state, and obeys the following relation:
12 Shock wave theory Equation of state The Equation of State (EOS) is the goal of shock wave simulations The Equations of State (EOS) gives the all the properties of the material in terms of Pressure, P, Volume, V and Energy, E (or Temperature, T); For example, the ideal gas EOS: PV = nrt However, the full EOS for most materials are very difficult to determine experimentally. Hugoniot is a line on the EOS surface
13 Shock wave theory Advantages of studying shock waves Studying shock waves gives greater understanding in many different areas: Geophysics: Planetary core modelling Earthquakes Astrophysics: Shock waves through stellar material Supernova events Industrial applications: Explosives investigations Laser cutting
14 Computer Simulation of Shock Waves
15 Computer simulation of shock waves Computer Simulation Allows for experimentally inaccessible scenarios to be explored; Computer simulation is a powerful tool if used correctly! Requires an accurate model of the system Interatomic Forces Dynamics of system Constraints of system Two approaches considered for shock waves: Atomistic molecular dynamics Ab-initio molecular dynamics
16 Computer simulation of shock waves Molecular dynamics Molecular dynamics (MD) is a simulation tool used to explore the atomic / molecular interactions of a system. Ideal for the dynamic processes during a shock; Solves Newton s equations of motion Timestep important consideration Macroscopic properties are calculated as space and/or time-averages Forces in MD can be generated by ab-initio or empirical potentials.
17 Computer simulation of shock waves Ab-initio molecular dynamics In ab-initio MD Schrodinger s equation is solved to find the ground state energy of the system - Forces are then calculated on the ground state energy. Born-Oppenheimer approximation used de-couples motion of electrons and ions; Requires a wavefunction to be stored Computationally expensive to store: 10 points in each dimension on a 3D sample grid scales as 10 3N where N is number of electrons. For example, 2 electrons storage cost is 15Mb RAM whereas 5 electrons storage is ~ 15 Tb RAM! (for complex numbers)
18 Computer simulation of shock waves Empirical potentials Empirical potentials are approximations to the Potential Energy Surface. Good for shock waves as: Using large systems (>1000 s atoms); Simple to code; Computationally cheap to run; Shown to be accurate (enough!) for shock wave calculations.
19 Computer simulation of shock waves Atomistic molecular dynamics The key to successful atomistic MD is the choice of the empirically determined potential that describes the energy surface of the system. Many different potentials available All created for a particular system Created by curve fitting to experimental data, ab-initio data, or sometimes both. Choosing the right potential is very important for meaningful results!
20 Computer simulation of shock waves Lennard-Jones potential(i) Simple two-body potential; Van-der-Waals attraction, with a computationally efficient (but physically meaningless) r 12 repulsive term; Easy to calculate forces; Great for materials such as argon; BUT! Argon is not a very exciting material
21 Computer simulation of shock waves Lennard-Jones potential(ii) Lennard-Jones Potential U(r) Interatomic Spacing (r) U(r) = 0, if r > rcutoff
22 Computer simulation of shock waves BKS potential (I) Where alpha and beta are atomic species, q, their charges and A, b, and C are constants. Named after authors - van Beest, Kramer and van Santen [1] Empirical potential parametrised by ab-initio cluster calculations Good for quartz (SiO 2) and its polymorphs (coesite, stishovite,etc..) [1] Interatomic force fields for silicas,aluminophosphates, and zeolites: Derivation based on ab-initio calculations, G.J.Kramer, N.P.Farringher,B.W.H. van Beest and R.A. van Santen, Phys.Rev B,43,6 (1991)
23 Computer simulation of shock waves BKS potential (II) The BKS Potential 11 O-O Si-O 9 7 Energy Interatomic distance (angstroms) Care has to be taken with simulations to avoid atoms becoming closer than maximum - they would feel an attractive force!
24 Computer simulation of shock waves Periodic Boundary Conditions (PBC) Used to simulate large-scale bulk systems Each simulation cell is part of an infinite array of replica cells Particle numbers are conserved as a particle leaving one cell will be replaced by another entering from the other side. System becomes periodic Minimum Image Convention used with PBC: Ensures forces are evaluated between the smallest pair separation between each atom and neighbours
25 Computer simulation of shock waves Momentum mirror modification to PBC: Shock wave generation PBC during shock compression would result in many shocks! Results would be meaningless as no steady shock would be travelling through the system due to wrap around of atoms Momentum mirror on one of the directions is used: U Momentum Mirror t = 0
26 Computer simulation of shock waves Momentum mirror modification to PBC: Shock wave generation Position and velocity reversed if passing the mirror during an MD timestep Vacuum region required to eliminate repulsive force from periodic images U Vacuum Region Momentum Mirror t = t1 -U
27 Computer simulation of shock waves Non-equilibrium molecular dynamics (NEMD) Equilibrium MD can be used to create the isentrope, isotherm or cold compression curves seen earlier Shock waves are dynamic events and so equilibrium MD is not appropriate to capture dynamics Non-equilibrium MD (NEMD) is used instead. Uses NVE ensemble No equilibration period -all important physics in first few nanoseconds! No thermostat or barostat
28 * When T=0 the system cannot support a steady wave and the results are meaningless. T > 0 to allow for transverse flow as a steadying mechanism Computer simulation of shock waves Simulating a shock wave A shock wave is simulated thus: At time t = 0, all atoms in an equilibrated at finite temperature * and relaxed system are given a velocity up in the direction of the momentum mirror up Vacuum Region Momentum Mirror t = 0
29 Computer simulation of shock waves Simulating a shock wave A shock wave is simulated thus: When the atoms reach the mirror they reflect and setup a shock propagating away from the mirror at velocity Us Shock front up Vacuum Region Momentum Mirror t = t1 Us
30 Computer simulation of shock waves Simulating a shock wave A shock wave is simulated thus: Simulation is ended when the shock reaches maximum compression. Shock front Vacuum Region Momentum Mirror t = tmax Us
31 Results
32 Results Simulations with argon Lennard-Jones potential 500,1000, 1800, 2000, 3600, 4000 atom system sizes Piston velocities of 1, 2, 4, 6, 8, 16, 32, 64, and 128 kms -1 Simple system to start with - allowed for analysis tools to be written and tested 4000 atom system in equilibrium
33 Results Static compression (Equilibrum MD) Static Compression Hugoniot or f Argon Lennard-Jones 12-6 Potential Static Compression Experimental data 200 Pressure (GPa) Volume (V / V0)
34 Results Simulations with argon
35 Results Velocity profile along system 2000 Velocity profile along simulation cell 1km/s shock wave after 3ps, adjusted for system frame of reference 1500 Velocity (m/s) 1000 Shock front Position (angstroms)
36 Results Hugoniots of argon Hugoniots of argon for different system sizes NEMD Shock wave calculations atoms 4000 atoms Experimental Data 1800 atoms Specific Volume (V/V0) Pressure (GPa)
37 Results Hugoniots of argon Hugoniots of argon for different system sizes NEMD Shock wave calculations atoms 4000 atoms Experimental data 1800 atoms Specific Volume (V/V0) Log of Pressure (GPa)
38 Results Transients at the momentum mirror Pzz Pressure Plots for Ar Pzz (GPa) Timestep
39 Results Transients at the momentum mirror Transients are an artifact of the momentum mirror Asymmetric, unlike a real shock that has material both sides of the impact region The region directly next to the mirror will have atoms that are trapped; Atoms want to escape but the mirror keeps them in place Heats them up unrealistically Region near mirror should be disregarded when computing averages of system properties
40 Results Shock waves in quartz Quartz (SiO 2) and its polymorphs are the 2nd most abundant substances in the Earth s crust High-temperature, high-pressure polymorphs: Stishovite found in meteorite craters Coesite (at low-temperatures) and Stishovite BKS potential used to model quartz and stishovite Aim to find conditions in which phase change occurs Care will have to be taken under high-compression
41 Results Quartz structure -quartz
42 Results Stishovite structure Stishovite
43 Results Shock waves in quartz using BKS potential Initial BKS simulations show that an SiO 2 trimer has a linear molecule for its lowest energy configuration Promising - as same structure as CO 2 Further testing still required.
44 Conclusions
45 Conclusions General conclusions Studying shock waves has many different fields in which to contribute Simulation is a great tool for predicting experimentally inaccessible states Non-equilibrium molecular dynamics is ideal for computer simulation of shock waves
46 Conclusions Further conclusions Argon simulated and shown that Lennard- Jones potential is good up to moderate pressures After that the unphysical r12 region is probed and simulation will deviate from true behaviour Transients at start of shock simulation overcome by calculations on steady-state region BKS potential to be used to model quartz - stishovite phase transition
47
High-throughput Simulations of Shock-waves Using MD. Jacob Wilkins University of York
High-throughput Simulations of Using MD Jacob Wilkins University of York jacob.wilkins@york.ac.uk Mar 2018 Outline 1 2 3 4 5 What are shock-waves? are pressure-waves moving through material faster than
More informationPhysics of Explosions
Physics of Explosions Instructor: Dr. Henry Tan Pariser/B4 MACE - Explosion Engineering School of Mechanical, Aerospace and Civil Engineering The University of Manchester Introduction Equation of state
More information5.60 Thermodynamics & Kinetics Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 5.60 Thermodynamics & Kinetics Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.60 Spring 2008 Lecture
More informationImperfect Gases. NC State University
Chemistry 431 Lecture 3 Imperfect Gases NC State University The Compression Factor One way to represent the relationship between ideal and real gases is to plot the deviation from ideality as the gas is
More informationWhat is Classical Molecular Dynamics?
What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential functions Newton s equations of motion are integrated
More informationIntroduction to molecular dynamics
1 Introduction to molecular dynamics Yves Lansac Université François Rabelais, Tours, France Visiting MSE, GIST for the summer Molecular Simulation 2 Molecular simulation is a computational experiment.
More informationExample questions for Molecular modelling (Level 4) Dr. Adrian Mulholland
Example questions for Molecular modelling (Level 4) Dr. Adrian Mulholland 1) Question. Two methods which are widely used for the optimization of molecular geometies are the Steepest descents and Newton-Raphson
More informationPHYS 643 Week 4: Compressible fluids Sound waves and shocks
PHYS 643 Week 4: Compressible fluids Sound waves and shocks Sound waves Compressions in a gas propagate as sound waves. The simplest case to consider is a gas at uniform density and at rest. Small perturbations
More informationChapter 1. The Properties of Gases Fall Semester Physical Chemistry 1 (CHM2201)
Chapter 1. The Properties of Gases 2011 Fall Semester Physical Chemistry 1 (CHM2201) Contents The Perfect Gas 1.1 The states of gases 1.2 The gas laws Real Gases 1.3 Molecular interactions 1.4 The van
More informationConcepts of Thermodynamics
Thermodynamics Industrial Revolution 1700-1800 Science of Thermodynamics Concepts of Thermodynamics Heavy Duty Work Horses Heat Engine Chapter 1 Relationship of Heat and Temperature to Energy and Work
More information1. What is the value of the quantity PV for one mole of an ideal gas at 25.0 C and one atm?
Real Gases Thought Question: How does the volume of one mole of methane gas (CH4) at 300 Torr and 298 K compare to the volume of one mole of an ideal gas at 300 Torr and 298 K? a) the volume of methane
More informationSurvey of Thermodynamic Processes and First and Second Laws
Survey of Thermodynamic Processes and First and Second Laws Please select only one of the five choices, (a)-(e) for each of the 33 questions. All temperatures T are absolute temperatures. All experiments
More informationShock Waves. 1 Steepening of sound waves. We have the result that the velocity of a sound wave in an arbitrary reference frame is given by: kˆ.
Shock Waves Steepening of sound waves We have the result that the velocity of a sound wave in an arbitrary reference frame is given by: v u kˆ c s kˆ where u is the velocity of the fluid and k is the wave
More informationAnswers to Problem Set Number 04 for MIT (Spring 2008)
Answers to Problem Set Number 04 for 18.311 MIT (Spring 008) Rodolfo R. Rosales (MIT, Math. Dept., room -337, Cambridge, MA 0139). March 17, 008. Course TA: Timothy Nguyen, MIT, Dept. of Mathematics, Cambridge,
More informationCE 530 Molecular Simulation
1 CE 530 Molecular Simulation Lecture 14 Molecular Models David A. Kofke Department of Chemical Engineering SUNY Buffalo kofke@eng.buffalo.edu 2 Review Monte Carlo ensemble averaging, no dynamics easy
More informationarxiv:cond-mat/ v2 [cond-mat.mtrl-sci] 26 Nov 1999
arxiv:cond-mat/9911407v2 [cond-mat.mtrl-sci] 26 Nov 1999 Equations of state for solids at high pressures and temperatures from shock-wave data Valentin Gospodinov Bulgarian Academy of Sciences, Space Research
More informationIdeal Gas Behavior. NC State University
Chemistry 331 Lecture 6 Ideal Gas Behavior NC State University Macroscopic variables P, T Pressure is a force per unit area (P= F/A) The force arises from the change in momentum as particles hit an object
More informationA Nobel Prize for Molecular Dynamics and QM/MM What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential
More informationIntroduction to thermodynamics
Chapter 6 Introduction to thermodynamics Topics First law of thermodynamics Definitions of internal energy and work done, leading to du = dq + dw Heat capacities, C p = C V + R Reversible and irreversible
More informationIntroduction to Molecular Dynamics
Introduction to Molecular Dynamics Dr. Kasra Momeni www.knanosys.com Overview of the MD Classical Dynamics Outline Basics and Terminology Pairwise interacting objects Interatomic potentials (short-range
More informationA Study of the Thermal Properties of a One. Dimensional Lennard-Jones System
A Study of the Thermal Properties of a One Dimensional Lennard-Jones System Abstract In this study, the behavior of a one dimensional (1D) Lennard-Jones (LJ) system is simulated. As part of this research,
More informationvan der Waals Isotherms near T c
van der Waals Isotherms near T c v d W loops are not physical. Why? Patch up with Maxwell construction van der Waals Isotherms, T/T c van der Waals Isotherms near T c Look at one of the van der Waals isotherms
More informationIntroduction to Thermodynamics And Applications. Physics 420 Patrick Lawrence
Introduction to Thermodynamics And Applications Physics 420 Patrick Lawrence Topics Confusion about Heat, Internal Energy and Temperature Methods of heat transfer The Ideal Gas Law Compression Applications
More informationAn introduction to Molecular Dynamics. EMBO, June 2016
An introduction to Molecular Dynamics EMBO, June 2016 What is MD? everything that living things do can be understood in terms of the jiggling and wiggling of atoms. The Feynman Lectures in Physics vol.
More informationInteraction between atoms
Interaction between atoms MICHA SCHILLING HAUPTSEMINAR: PHYSIK DER KALTEN GASE INSTITUT FÜR THEORETISCHE PHYSIK III UNIVERSITÄT STUTTGART 23.04.2013 Outline 2 Scattering theory slow particles / s-wave
More informationRate of Heating and Cooling
Rate of Heating and Cooling 35 T [ o C] Example: Heating and cooling of Water E 30 Cooling S 25 Heating exponential decay 20 0 100 200 300 400 t [sec] Newton s Law of Cooling T S > T E : System S cools
More informationGodunov methods in GANDALF
Godunov methods in GANDALF Stefan Heigl David Hubber Judith Ngoumou USM, LMU, München 28th October 2015 Why not just stick with SPH? SPH is perfectly adequate in many scenarios but can fail, or at least
More informationReal Gases. Sections (Atkins 6th Ed.), (Atkins 7-9th Eds.)
Real Gases Sections 1.4-1.6 (Atkins 6th Ed.), 1.3-1.5 (Atkins 7-9th Eds.) Molecular Interactions Compression factor Virial coefficients Condensation Critical Constants Van der Waals Equation Corresponding
More informationReview of Fluid Mechanics
Chapter 3 Review of Fluid Mechanics 3.1 Units and Basic Definitions Newton s Second law forms the basis of all units of measurement. For a particle of mass m subjected to a resultant force F the law may
More informationPressure Volume Temperature Relationship of Pure Fluids
Pressure Volume Temperature Relationship of Pure Fluids Volumetric data of substances are needed to calculate the thermodynamic properties such as internal energy and work, from which the heat requirements
More informationA thermodynamic system is taken from an initial state X along the path XYZX as shown in the PV-diagram.
AP Physics Multiple Choice Practice Thermodynamics 1. The maximum efficiency of a heat engine that operates between temperatures of 1500 K in the firing chamber and 600 K in the exhaust chamber is most
More information1.3 Molecular Level Presentation
1.3.1 Introduction A molecule is the smallest chemical unit of a substance that is capable of stable, independent existence. Not all substances are composed of molecules. Some substances are composed of
More informationShear Properties and Wrinkling Behaviors of Finite Sized Graphene
Shear Properties and Wrinkling Behaviors of Finite Sized Graphene Kyoungmin Min, Namjung Kim and Ravi Bhadauria May 10, 2010 Abstract In this project, we investigate the shear properties of finite sized
More informationFluid Dynamics. Massimo Ricotti. University of Maryland. Fluid Dynamics p.1/14
Fluid Dynamics p.1/14 Fluid Dynamics Massimo Ricotti ricotti@astro.umd.edu University of Maryland Fluid Dynamics p.2/14 The equations of fluid dynamics are coupled PDEs that form an IVP (hyperbolic). Use
More informationCourse: TDEC202 (Energy II) dflwww.ece.drexel.edu/tdec
Course: TDEC202 (Energy II) Thermodynamics: An Engineering Approach Course Director/Lecturer: Dr. Michael Carchidi Course Website URL dflwww.ece.drexel.edu/tdec 1 Course Textbook Cengel, Yunus A. and Michael
More informationPhysics Nov Cooling by Expansion
Physics 301 19-Nov-2004 25-1 Cooling by Expansion Now we re going to change the subject and consider the techniques used to get really cold temperatures. Of course, the best way to learn about these techniques
More informationThermodynamics of Fluid Phase Equilibria Dr. Jayant K. Singh Department of Chemical Engineering Indian Institute of Technology, Kanpur
Thermodynamics of Fluid Phase Equilibria Dr. Jayant K. Singh Department of Chemical Engineering Indian Institute of Technology, Kanpur Lecture - 01 Review of basic concepts of thermodynamics Welcome to
More informationVarious lecture notes for
Various lecture notes for 18311. R. R. Rosales (MIT, Math. Dept., 2-337) April 12, 2013 Abstract Notes, both complete and/or incomplete, for MIT s 18.311 (Principles of Applied Mathematics). These notes
More informationChapter 2: The Physical Properties of Pure Compounds
Chapter 2: The Physical Properties of Pure Compounds 2-10. The boiler is an important unit operation in the Rankine cycle. This problem further explores the phenomenon of boiling. A. When you are heating
More informationUnified Quiz: Thermodynamics
Fall 004 Unified Quiz: Thermodynamics November 1, 004 Calculators allowed. No books allowed. A list of equations is provided. Put your name on each page of the exam. Read all questions carefully. Do all
More informationIdeal Gases. 247 minutes. 205 marks. theonlinephysicstutor.com. facebook.com/theonlinephysicstutor. Name: Class: Date: Time: Marks: Comments:
Ideal Gases Name: Class: Date: Time: 247 minutes Marks: 205 marks Comments: Page 1 of 48 1 Which one of the graphs below shows the relationship between the internal energy of an ideal gas (y-axis) and
More informationIntroduction. Statistical physics: microscopic foundation of thermodynamics degrees of freedom 2 3 state variables!
Introduction Thermodynamics: phenomenological description of equilibrium bulk properties of matter in terms of only a few state variables and thermodynamical laws. Statistical physics: microscopic foundation
More informationA Coupling Tool for Parallel Molecular Dynamics Continuum Simulations
A Coupling Tool for Parallel Molecular Dynamics Continuum Simulations ISPDC 2012 Philipp Neumann and Nikola Tchipev 29.06.2012 ISPDC 2012, 29.06.2012 1 Contents Motivation The Macro Micro Coupling Tool
More informationSPH 302 THERMODYNAMICS
THERMODYNAMICS Nyongesa F. W., PhD. e-mail: fnyongesa@uonbi.ac.ke 1 Objectives Explain the Laws of thermodynamics & their significance Apply laws of thermodynamics to solve problems relating to energy
More informationMAHALAKSHMI ENGINEERING COLLEGE
MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI 621 213. Department: Mechanical Subject Code: ME2202 Semester: III Subject Name: ENGG. THERMODYNAMICS UNIT-I Basic Concept and First Law 1. What do you understand
More informationBig Idea 2: Chemical and physical properties of materials can be explained by the structure and the arrangement of atoms, ions, or molecules and the
Big Idea 2: Chemical and physical properties of materials can be explained by the structure and the arrangement of atoms, ions, or molecules and the forces between them. Enduring Understanding 2.A: Matter
More informationIV. Compressible flow of inviscid fluids
IV. Compressible flow of inviscid fluids Governing equations for n = 0, r const: + (u )=0 t u + ( u ) u= p t De e = + ( u ) e= p u+ ( k T ) Dt t p= p(, T ), e=e (,T ) Alternate forms of energy equation
More information23 The Born-Oppenheimer approximation, the Many Electron Hamiltonian and the molecular Schrödinger Equation M I
23 The Born-Oppenheimer approximation, the Many Electron Hamiltonian and the molecular Schrödinger Equation 1. Now we will write down the Hamiltonian for a molecular system comprising N nuclei and n electrons.
More informationSchool of Chemical & Biological Engineering, Konkuk University
School of Chemical & Biological Engineering, Konkuk University Chemistry is the science concerned with the composition, structure, and properties of matter, as well as the changes it undergoes during chemical
More informationCHAPTER 4. Basics of Fluid Dynamics
CHAPTER 4 Basics of Fluid Dynamics What is a fluid? A fluid is a substance that can flow, has no fixed shape, and offers little resistance to an external stress In a fluid the constituent particles (atoms,
More informationSupplementary Information for. Universal elastic-hardening-driven mechanical instability in α-quartz and quartz. homeotypes under pressure
Supplementary Information for Universal elastic-hardening-driven mechanical instability in α-quartz and quartz homeotypes under pressure Juncai Dong, Hailiang Zhu, and Dongliang Chen * Beijing Synchrotron
More informationRadiative & Magnetohydrodynamic Shocks
Chapter 4 Radiative & Magnetohydrodynamic Shocks I have been dealing, so far, with non-radiative shocks. Since, as we have seen, a shock raises the density and temperature of the gas, it is quite likely,
More informationShock Waves. = 0 (momentum conservation)
PH27: Aug-Dec 2003 Shock Waves A shock wave is a surface of discontinuity moving through a medium at a speed larger than the speed of sound upstream. The change in the fluid properties upon passing the
More informationIntroduction to Aerodynamics. Dr. Guven Aerospace Engineer (P.hD)
Introduction to Aerodynamics Dr. Guven Aerospace Engineer (P.hD) Aerodynamic Forces All aerodynamic forces are generated wither through pressure distribution or a shear stress distribution on a body. The
More informationGeneral Physical Chemistry I
General Physical Chemistry I Lecture 5 Aleksey Kocherzhenko February 10, 2015" Last time " Diffusion, effusion, and molecular collisions" Diffusion" Effusion" Graham s law: " " " 1 2 r / M " (@ fixed T
More informationTopic 3 &10 Review Thermodynamics
Name: Date: Topic 3 &10 Review Thermodynamics 1. The kelvin temperature of an object is a measure of A. the total energy of the molecules of the object. B. the total kinetic energy of the molecules of
More informationn v molecules will pass per unit time through the area from left to
3 iscosity and Heat Conduction in Gas Dynamics Equations of One-Dimensional Gas Flow The dissipative processes - viscosity (internal friction) and heat conduction - are connected with existence of molecular
More informationMore Thermodynamics. Specific Specific Heats of a Gas Equipartition of Energy Reversible and Irreversible Processes
More Thermodynamics Specific Specific Heats of a Gas Equipartition of Energy Reversible and Irreversible Processes Carnot Cycle Efficiency of Engines Entropy More Thermodynamics 1 Specific Heat of Gases
More informationCE 530 Molecular Simulation
1 CE 530 Molecular Simulation Lecture 1 David A. Kofke Department of Chemical Engineering SUNY Buffalo kofke@eng.buffalo.edu 2 Time/s Multi-Scale Modeling Based on SDSC Blue Horizon (SP3) 1.728 Tflops
More informationExtension of the Planar Noh Problem to Aluminum, Iron, Copper, and Tungsten
Extension of the Planar Noh Problem to Aluminum, Iron, Copper, and Tungsten Chloe E. Yorke, April D. Howard, Sarah C. Burnett, Kevin G. Honnell a), Scott D. Ramsey, and Robert L. Singleton, Jr. Computational
More informationLecture Notes Set 4c: Heat engines and the Carnot cycle
ecture Notes Set 4c: eat engines and the Carnot cycle Introduction to heat engines In the following sections the fundamental operating principles of the ideal heat engine, the Carnot engine, will be discussed.
More informationPhysics 53. Thermal Physics 1. Statistics are like a bikini. What they reveal is suggestive; what they conceal is vital.
Physics 53 Thermal Physics 1 Statistics are like a bikini. What they reveal is suggestive; what they conceal is vital. Arthur Koestler Overview In the following sections we will treat macroscopic systems
More informationAb Ini'o Molecular Dynamics (MD) Simula?ons
Ab Ini'o Molecular Dynamics (MD) Simula?ons Rick Remsing ICMS, CCDM, Temple University, Philadelphia, PA What are Molecular Dynamics (MD) Simulations? Technique to compute statistical and transport properties
More informationTable of Contents [ttc]
Table of Contents [ttc] 1. Equilibrium Thermodynamics I: Introduction Thermodynamics overview. [tln2] Preliminary list of state variables. [tln1] Physical constants. [tsl47] Equations of state. [tln78]
More informationSpeed Distribution at CONSTANT Temperature is given by the Maxwell Boltzmann Speed Distribution
Temperature ~ Average KE of each particle Particles have different speeds Gas Particles are in constant RANDOM motion Average KE of each particle is: 3/2 kt Pressure is due to momentum transfer Speed Distribution
More informationGASES (Chapter 5) Temperature and Pressure, that is, 273 K and 1.00 atm or 760 Torr ) will occupy
I. Ideal gases. A. Ideal gas law review. GASES (Chapter 5) 1. PV = nrt Ideal gases obey this equation under all conditions. It is a combination ofa. Boyle's Law: P 1/V at constant n and T b. Charles's
More informationSpring 2015, Math 111 Lab 2: Exploring the Limit
Spring 2015, Math 111 Lab 2: William and Mary February 3, 2015 Spring 2015, Math 111 Lab 2: Outline Limit Limit Existence Example Vertical Horizontal Spring 2015, Math 111 Lab 2: Limit Limit Existence
More informationLangevin Dynamics in Constant Pressure Extended Systems
Langevin Dynamics in Constant Pressure Extended Systems D. Quigley and M.I.J. Probert CMMP 2004 1 Talk Outline Molecular dynamics and ensembles. Existing methods for sampling at NPT. Langevin dynamics
More informationChapter 15 Thermal Properties of Matter
Chapter 15 Thermal Properties of Matter To understand the mole and Avogadro's number. To understand equations of state. To study the kinetic theory of ideal gas. To understand heat capacity. To learn and
More informationChapter 2 Experimental sources of intermolecular potentials
Chapter 2 Experimental sources of intermolecular potentials 2.1 Overview thermodynamical properties: heat of vaporization (Trouton s rule) crystal structures ionic crystals rare gas solids physico-chemical
More informationPV = nrt where R = universal gas constant
Ideal Gas Law Dd Deduced dfrom Combination of Gas Relationships: V 1/P, Boyle's Law V, Charles's Law V n, Avogadro'sLaw Therefore, V nt/p or PV nt PV = nrt where R = universal gas constant The empirical
More informationIntroduction to model potential Molecular Dynamics A3hourcourseatICTP
Introduction to model potential Molecular Dynamics A3hourcourseatICTP Alessandro Mattoni 1 1 Istituto Officina dei Materiali CNR-IOM Unità di Cagliari SLACS ICTP School on numerical methods for energy,
More informationHeat, Work, and the First Law of Thermodynamics. Chapter 18 of Essential University Physics, Richard Wolfson, 3 rd Edition
Heat, Work, and the First Law of Thermodynamics Chapter 18 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 Different ways to increase the internal energy of system: 2 Joule s apparatus
More informationAb initio molecular dynamics. Simone Piccinin CNR-IOM DEMOCRITOS Trieste, Italy. Bangalore, 04 September 2014
Ab initio molecular dynamics Simone Piccinin CNR-IOM DEMOCRITOS Trieste, Italy Bangalore, 04 September 2014 What is MD? 1) Liquid 4) Dye/TiO2/electrolyte 2) Liquids 3) Solvated protein 5) Solid to liquid
More informationSpeed Distribution at CONSTANT Temperature is given by the Maxwell Boltzmann Speed Distribution
Temperature ~ Average KE of each particle Particles have different speeds Gas Particles are in constant RANDOM motion Average KE of each particle is: 3/2 kt Pressure is due to momentum transfer Speed Distribution
More informationStructural Bioinformatics (C3210) Molecular Mechanics
Structural Bioinformatics (C3210) Molecular Mechanics How to Calculate Energies Calculation of molecular energies is of key importance in protein folding, molecular modelling etc. There are two main computational
More informationENGR 292 Fluids and Thermodynamics
ENGR 292 Fluids and Thermodynamics Scott Li, Ph.D., P.Eng. Mechanical Engineering Technology Camosun College Timeline Last week, Reading Break Feb.21: Thermodynamics 1 Feb.24: Midterm Review (Fluid Statics
More informationME6301- ENGINEERING THERMODYNAMICS UNIT I BASIC CONCEPT AND FIRST LAW PART-A
ME6301- ENGINEERING THERMODYNAMICS UNIT I BASIC CONCEPT AND FIRST LAW PART-A 1. What is meant by thermodynamics system? (A/M 2006) Thermodynamics system is defined as any space or matter or group of matter
More informationNOTE: Only CHANGE in internal energy matters
The First Law of Thermodynamics The First Law of Thermodynamics is a special case of the Law of Conservation of Energy It takes into account changes in internal energy and energy transfers by heat and
More informationScientific Computing II
Scientific Computing II Molecular Dynamics Simulation Michael Bader SCCS Summer Term 2015 Molecular Dynamics Simulation, Summer Term 2015 1 Continuum Mechanics for Fluid Mechanics? Molecular Dynamics the
More informationInternational Physics Course Entrance Examination Questions
International Physics Course Entrance Examination Questions (May 2010) Please answer the four questions from Problem 1 to Problem 4. You can use as many answer sheets you need. Your name, question numbers
More informationTemperature Thermal Expansion Ideal Gas Law Kinetic Theory Heat Heat Transfer Phase Changes Specific Heat Calorimetry Heat Engines
Temperature Thermal Expansion Ideal Gas Law Kinetic Theory Heat Heat Transfer Phase Changes Specific Heat Calorimetry Heat Engines Zeroeth Law Two systems individually in thermal equilibrium with a third
More informationKINETIC THEORY OF GASES
LECTURE 8 KINETIC THEORY OF GASES Text Sections 0.4, 0.5, 0.6, 0.7 Sample Problems 0.4 Suggested Questions Suggested Problems Summary None 45P, 55P Molecular model for pressure Root mean square (RMS) speed
More informationLecture 03/30 Monday, March 30, :46 AM
Chem 110A Page 1 Lecture 03/30 Monday, March 30, 2009 9:46 AM Notes 0330 Audio recording started: 10:02 AM Monday, March 30, 2009 Chapter 1: Ideal Gases Macroscopic properties of gases Equation of state
More informationClassify each of these statements as always true, AT; sometimes true, ST; or never true, NT.
Chapter 11 THE NATURE OF GASES States of Matter Describe the motion of gas particles according to the kinetic theory Interpret gas pressure in terms of kinetic theory Key Terms: 1. kinetic energy 2. gas
More information(b) The measurement of pressure
(b) The measurement of pressure The pressure of the atmosphere is measured with a barometer. The original version of a barometer was invented by Torricelli, a student of Galileo. The barometer was an inverted
More informationEfficient viscosity estimation from molecular dynamics simulation via momentum impulse relaxation
JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 6 8 AUGUST 2000 Efficient viscosity estimation from molecular dynamics simulation via momentum impulse relaxation Gaurav Arya, Edward J. Maginn, a) and Hsueh-Chia
More informationEquations of State. Equations of State (EoS)
Equations of State (EoS) Equations of State From molecular considerations, identify which intermolecular interactions are significant (including estimating relative strengths of dipole moments, polarizability,
More informationLattice energy of ionic solids
1 Lattice energy of ionic solids Interatomic Forces Solids are aggregates of atoms, ions or molecules. The bonding between these particles may be understood in terms of forces that play between them. Attractive
More informationMD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky
MD Thermodynamics Lecture 1 3/6/18 1 Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order
More informationHeat, Work, Internal Energy, Enthalpy, and the First Law of Thermodynamics. Internal Energy and the First Law of Thermodynamics
CHAPTER 2 Heat, Work, Internal Energy, Enthalpy, and the First Law of Thermodynamics Internal Energy and the First Law of Thermodynamics Internal Energy (U) Translational energy of molecules Potential
More informationMolecular Clustering and Velocity Increase in Converging-Diverging Nozzle in MD Simulation
Molecular Clustering and Velocity Increase in Converging-Diverging Nozzle in MD Simulation Jeoungsu Na 1, Jaehawn Lee 2, Changil Hong 2, Suhee Kim 1 R&D Department of NaJen, Co. LTD, Korea 2 Dept. of Computer
More informationIntroduction to Thermodynamic States Gases
Chapter 1 Introduction to Thermodynamic States Gases We begin our study in thermodynamics with a survey of the properties of gases. Gases are one of the first things students study in general chemistry.
More informationChapter 5 Gases. Chapter 5: Phenomena. Properties of Gases. Properties of Gases. Pressure. Pressure
Chapter 5: Phenomena Phenomena: To determine the properties of gases scientists recorded various observations/measurements about different gases. Analyze the table below looking for patterns between the
More informationShock and Expansion Waves
Chapter For the solution of the Euler equations to represent adequately a given large-reynolds-number flow, we need to consider in general the existence of discontinuity surfaces, across which the fluid
More informationThe goal of thermodynamics is to understand how heat can be converted to work. Not all the heat energy can be converted to mechanical energy
Thermodynamics The goal of thermodynamics is to understand how heat can be converted to work Main lesson: Not all the heat energy can be converted to mechanical energy This is because heat energy comes
More informationAb initio molecular dynamics and nuclear quantum effects
Ab initio molecular dynamics and nuclear quantum effects Luca M. Ghiringhelli Fritz Haber Institute Hands on workshop density functional theory and beyond: First principles simulations of molecules and
More informationIntroduction to Simulation - Lectures 17, 18. Molecular Dynamics. Nicolas Hadjiconstantinou
Introduction to Simulation - Lectures 17, 18 Molecular Dynamics Nicolas Hadjiconstantinou Molecular Dynamics Molecular dynamics is a technique for computing the equilibrium and non-equilibrium properties
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 2 Course Objectives
correlated to the College Board AP Physics 2 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring Understanding 1.A:
More informationHONOUR SCHOOL OF NATURAL SCIENCE. Final Examination GENERAL PHYSICAL CHEMISTRY I. Answer FIVE out of nine questions
HONOUR SCHOOL OF NATURAL SCIENCE Final Examination GENERAL PHYSICAL CHEMISTRY I Monday, 12 th June 2000, 9.30 a.m. - 12.30 p.m. Answer FIVE out of nine questions The numbers in square brackets indicate
More information