Simple Hard Disk Collisions. (5min) (uakron.edu) Deriving the formula for pressure from the motions of molecules was easy when we were talking about ideal gas molecules and we even got a compact, exact result (aka. the ideal gas law). The problem gets more complicated when the molecules can collide with each other as well as with the walls. This complexity undermines our ability to get an exact solution, but we can obtain a numerical solution by integrating all the collisions with respect to time and computing the average pressure as a result. The process begins with computing collision times of molecules with walls. This computation is simple enough that you should be able to do it even if you can't write a molecular simulation program.

Comprehension questions:

1. Two molecules with MW=16 and 0.36nm diameter are traveling in 2D with velocities: (643, -133) and (133, -643) m/s. Their initial positions in a 5 nm box are (2,4) and (3,1). Estimate the time (ns) and nature of the first collision, whether with a wall or with another molecule. 2. Two molecules with MW=16 and 0.36nm diameter are traveling in 2D with velocities: (133, -266) and (-133, 266) m/s. Their initial positions in a 5 nm box are (2,4) and (3,1). Estimate the time (ns) and nature of the first collision, whether with a wall or with another molecule. 3. A molecule with MW=16 and 0.36nm diameter is traveling in 2D towards the east wall of a box. Its velocity is given by (vx, vy) = (123, 345) m/s. It starts off being 2nm from the east wall. How far must the molecule travel before it contacts the wall? 4. A molecule with MW=16 and 0.36nm diameter is traveling in 2D towards the east wall of a box. Its velocity is given by (vx, vy) = (123, 345) m/s. It starts off being 2nm from the east wall. How much time(ns) before it contacts the wall?

Molecular Nature of Energy, Temperature, and Pressure By Etomica.(uakron.edu, 17min) We can use a free website (Etomica.org, DMD Module) to visualize the ways that molecules interact, resulting in the average properties that we see at the macroscopic level. The oversimplified nature of the ideal gas model becomes really obvious and the improvement of the hard sphere model is easily understood. Including both attractive and repulsive forces, as in the square well potential model, leads to more surprising behavior. The two effects may cancel and make the Z factor (Z=PV/RT) look like an ideal gas even though it is not. Also, the adiabatic transformation between potential energy and kinetic energy leads to spikes in temperature as molecules enter each other's attractive wells. In certain cases, you might see molecules get stuck in each others' wells. This is effectively "bonding." This bonding is limited at very low density because it requires a third interaction to occur during the collision in order to stay bonded. This requirement lies at the fundamental basis of what is known as "unimolecular reaction," a fairly advanced concept that is easily understood by watching the video.

FYI: some computers have difficulty with Java security. A workaround is to: (1) Click the Modules link at etomica.org (2) Under Download, click the appropriate link for your computer then extract the zip file. (3) In the "bin" directory, click the "launcher". (4) Select "Piston-cylinder SWMD" or "Discontinuous MD 3D" depending on the module of interest. If you would prefer to enable Java and run the simulations online: (1) control panel (2) Search for "Java" (install Java if you don't see this icon) (3) Security (4) Edit site list, http://www.etomica.org, Add (5) Access the DMD module from etomica.org and click "Run Simulation" (6) Download/save the "WebStart" program (NOT the Applet) of interest from etomica to a folder of your choice (6) Double-click the program and accept all queries.

Comprehension Questions: 1. What is the average temperature (K) illustrated in the screencast? Is it higher or lower than the initial temperature? Explain. 2. What is the average pressure (bar) illustrated in the screencast? 3. Go to the etomica.org website and perform your own simulation with the piston-cylinder applet starting with 100 molecules and assuming the square well poential model. You can run the simulation in fast mode, but let the molecules collide for 2500 ps. Then report the average value of T,P,U,Z. (Hint: compute Z from its definition, and be careful with units.)

Etomica MD simulation in 3D (uakron.edu, 11min) To get a reasonably realistic equation of state for square well spheres, we should perform the simulation in three dimensions. Fortunately, a free molecular simulator is provided by etomica.org. This screencast shows how to access the simulation, vary the temperature and density, and read the output values for energy and pressure. With this output, you can compute the compressibility factor (Z) and internal energy departure (U-U^{ig})/RT over a range of T and ρ values. If you select the "repulsion only" (hard sphere) potential model, you can compute properties like those in Fig. 7.9. If you select the "repulsion and attraction" (SW sphere) potential model, you can ompute properties like those illustrated in Fig. 7.7. Molecular simulation provides an accurate numerical solution for the pressure and energy of an assumed intermolecular potential that can be used to validate EOS correlations at all conditions of T and ρ Once we have the EOS of the SW fluid characterized in terms of s and e, it becomes a straightforward exercise to find the best s and e that match experimental data for a given compound. At that point, we have characterized the nanoscopic forces between the molecules. Knowing these forces enables us to conceive and design nanostructural devices with a level of insight not previously available. (See above for hints about circumventing or enabling security for Java.)

Comprehension Questions: 1. Use the 3D simulator for hard spheres to compute the value of Z^{HS}at a packing fraction of 0.35. Simulate for 3000ps. Compare your value to the other values in Figure 7.9 by plotting it along with the others. 2. Use the 3D simulator for SW spheres to compute the value of Z^{SW}at a density of 1.25 g/cm3. Use the parameters for argon as developed in Example 7.9, and set your temperature to 10000K. Simulate for 3000ps. Compare your value to the other values in Figure 7.7 by plotting it along with the others. 3. Can you anticipate any experimental difficulties with performing experiments at 10000K? What advantages does this suggest about MD simulation vs. experimental measurements?

## Comments

Lira replied on Permalink

## Hard Disk-Wall Colisions (5min)

Simple Hard Disk Collisions. (5min) (uakron.edu) Deriving the formula for pressure from the motions of molecules was easy when we were talking about ideal gas molecules and we even got a compact, exact result (aka. the ideal gas law). The problem gets more complicated when the molecules can collide with each other as well as with the walls. This complexity undermines our ability to get an exact solution, but we can obtain a numerical solution by integrating all the collisions with respect to time and computing the average pressure as a result. The process begins with computing collision times of molecules with walls. This computation is simple enough that you should be able to do it even if you can't write a molecular simulation program.

Comprehension questions:

1. Two molecules with MW=16 and 0.36nm diameter are traveling in 2D with velocities: (643, -133) and (133, -643) m/s. Their initial positions in a 5 nm box are (2,4) and (3,1). Estimate the time (ns) and nature of the first collision, whether with a wall or with another molecule.

2. Two molecules with MW=16 and 0.36nm diameter are traveling in 2D with velocities: (133, -266) and (-133, 266) m/s. Their initial positions in a 5 nm box are (2,4) and (3,1). Estimate the time (ns) and nature of the first collision, whether with a wall or with another molecule.

3. A molecule with MW=16 and 0.36nm diameter is traveling in 2D towards the east wall of a box. Its velocity is given by (vx, vy) = (123, 345) m/s. It starts off being 2nm from the east wall. How far must the molecule travel before it contacts the wall?

4. A molecule with MW=16 and 0.36nm diameter is traveling in 2D towards the east wall of a box. Its velocity is given by (vx, vy) = (123, 345) m/s. It starts off being 2nm from the east wall. How much time(ns) before it contacts the wall?

Elliott replied on Permalink

## Etomica Molecular Simulator

Molecular Nature of Energy, Temperature, and Pressure By Etomica.(uakron.edu, 17min) We can use a free website (Etomica.org, DMD Module) to visualize the ways that molecules interact, resulting in the average properties that we see at the macroscopic level. The oversimplified nature of the ideal gas model becomes really obvious and the improvement of the hard sphere model is easily understood. Including both attractive and repulsive forces, as in the square well potential model, leads to more surprising behavior. The two effects may cancel and make the

Zfactor (Z=PV/RT) look like an ideal gas even though it is not. Also, the adiabatic transformation between potential energy and kinetic energy leads to spikes in temperature as molecules enter each other's attractive wells. In certain cases, you might see molecules get stuck in each others' wells. This is effectively "bonding." This bonding is limited at very low density because it requires a third interaction to occur during the collision in order to stay bonded. This requirement lies at the fundamental basis of what is known as "unimolecular reaction," a fairly advanced concept that is easily understood by watching the video.FYI: some computers have difficulty with Java security. A workaround is to: (1) Click the Modules link at etomica.org (2) Under Download, click the appropriate link for your computer then extract the zip file. (3) In the "bin" directory, click the "launcher". (4) Select "Piston-cylinder SWMD" or "Discontinuous MD 3D" depending on the module of interest. If you would prefer to enable Java and run the simulations online: (1) control panel (2) Search for "Java" (install Java if you don't see this icon) (3) Security (4) Edit site list, http://www.etomica.org, Add (5) Access the DMD module from etomica.org and click "Run Simulation" (6) Download/save the "WebStart" program (NOT the Applet) of interest from etomica to a folder of your choice (6) Double-click the program and accept all queries.

Comprehension Questions:

1. What is the average temperature (K) illustrated in the screencast? Is it higher or lower than the initial temperature? Explain.

2. What is the average pressure (bar) illustrated in the screencast?

3. Go to the etomica.org website and perform your own simulation with the piston-cylinder applet starting with 100 molecules and assuming the square well poential model. You can run the simulation in fast mode, but let the molecules collide for 2500 ps. Then report the average value of

T,P,U,Z. (Hint: computeZfrom its definition, and be careful with units.)Elliott replied on Permalink

## MD Simulation in 3D (11min)

Etomica MD simulation in 3D (uakron.edu, 11min) To get a reasonably realistic equation of state for square well spheres, we should perform the simulation in three dimensions. Fortunately, a free molecular simulator is provided by etomica.org. This screencast shows how to access the simulation, vary the temperature and density, and read the output values for energy and pressure. With this output, you can compute the compressibility factor (

Z) and internal energy departure (U-U)/^{ig}RTover a range ofTandρvalues. If you select the "repulsion only" (hard sphere) potential model, you can compute properties like those in Fig. 7.9. If you select the "repulsion and attraction" (SW sphere) potential model, you can ompute properties like those illustrated in Fig. 7.7. Molecular simulation provides an accurate numerical solution for the pressure and energy of an assumed intermolecular potential that can be used to validate EOS correlations at all conditions ofTandρOnce we have the EOS of the SW fluid characterized in terms of s and e, it becomes a straightforward exercise to find the best s and e that match experimental data for a given compound. At that point, we have characterized the nanoscopic forces between the molecules. Knowing these forces enables us to conceive and design nanostructural devices with a level of insight not previously available. (See above for hints about circumventing or enabling security for Java.)Comprehension Questions:

1. Use the 3D simulator for hard spheres to compute the value of

Zat a packing fraction of 0.35. Simulate for 3000ps. Compare your value to the other values in Figure 7.9 by plotting it along with the others.^{HS}2. Use the 3D simulator for SW spheres to compute the value of

Zat a density of 1.25 g/cm3. Use the parameters for argon as developed in Example 7.9, and set your temperature to 10000K. Simulate for 3000ps. Compare your value to the other values in Figure 7.7 by plotting it along with the others.^{SW}3. Can you anticipate any experimental difficulties with performing experiments at 10000K? What advantages does this suggest about MD simulation vs. experimental measurements?