c Chapter 1 Introduction :

a) Monte-Carlo method considers only configuration space eliminating the momentum part of the phase space. No dynamics -> we can only study systems in equilibrium.

b)MD studies is usually done on liquid since solids and gases have well developed theoretical foundations

Chapter 2 Basic molecular dynamics :

a) Studies based on spherical particles that interact (interactions between pairs of atoms are responsible for providing the two principal features of interatomic forces)

b)Forces : 1-resistance to compression (repulsion at close range )

2-binding the atoms together meaning an attraction (this happens at closer ranges after a certain cutoff separation)

*** (these potentials are of great importance in formulating your model: example Lenard Jones potential)

c) Lenard Jones potential: this model is somehow like colliding balls that are both soft and smooth in this model we ignored the attractive tail that represents the Van Der Waals interaction which leads to computational simplicity ( but surely stays limited yet helpful for other models)

d)Boundary conditions:1-There is a difference between an infinite and a finite system and what system can be considered small (according to the size of the sample and the percentage of particle interacting with the wall we decide , for MD number of atoms is small so the walls are best eliminated)2-Periodic boundaries????

e) Initial state: the results of a simulation of certain duration should be insensitive to initial conditions. A simple choice is positions chosen on square lattice to give desired density. The velocities will depend on temperature and the center of mass should remain at rest to avoid any overflow.

f) Methodology :

1- Integration: The leap frog method model is first order but it has excellent energy conservation properties and it is a 2 step procedure.

g) Programming ( which I didn’t go through in details)h)Conservation laws: 2 things the computation must

pass the momentum which is intrinsic to the algorithm and periodic boundary conditions. The energy conservation which is sensitive to the integration method and the time step. Sometimes programming errors can be detected by violation of a conservation law.

i) Equilibrium :Before making measurements we should be sure that that different quantities relaxed to their equilibrium averages( by averaging over series of time steps) , but the problem is that they relax at different rates

j) Trajectories: MD studies the trajectories followed by single atoms. For solids: small vibrations over lattice site, for gases: long distance trajectories, liquids: small steps and no long range positional order. The difference on the trajectories shows from the diffusion coefficient which is the mean square of the atomic displacement.

Chapter 3 Simulating simple systems:

Methods for computing the interactions and integrating the equations of motion to generate the atomic trajectories, for simplicity we will be working with monatomic fluids based on LJ potential.a) Equations of motion: simply applying the

Lagrange equation for a Lagrange based on Newton’s second law.

b)Potential functions: MD uses a classical point of view by considering the particles point masses interacting through forces that depend on the separation of these particles. So mainly neglecting quantum effects due to the complexity they arise, or replacing these effects by some classical models to reduce errors.Example potentials: LJ interaction that is characterized by its strongly repulsive core and weakly attractive tail. But the continuity at rc

affects the energy conservation and the actual atomic motion. This can be handled by slightly fixing the potential but it has a lot of effects.

c) Interaction computations: All pairs method ( this method is inefficient when rc is small compared to the linear size of the simulation region) we have two techniques:1) All pairs method: simplest method but inefficient when interaction range is small compared to the linear size of the simulation region but it only works for small values of Nm.

2)Cell subdivision: we divide the simulation region into a lattice of small cells where the cell edges is more than rc length. So now interactions will only happen between atoms in the same cell or neighboring cells thus reducing the problem. For this method we use the linked list (associating a pointer with every data entry) not the sequential list because it’s extremely wasteful the storage for linked lists is known in advance.

3)Neighbor list method: we use this method to allow the list of pairs lying in interaction range to be useful over several successive time steps so it guarantees that no interacting pair is ever missed

d) Integration methods:

1)low order methods: leap frog(two step form p:61) , verlet method

Algebraically equivalent to the leapfrog method Advantages • Conservation of energy, • Higher stability than for predictor-corrector methods, • Lower memory requirements.

2) Multiple-value methods : use information from one or several earlier time Steps. Adams approach: use the acceleration at a series of previous time steps

Higher order than the Verlet method – require extra computations and storage. We consider multistep methods – disadvantage: step size h cannot be

Changed easily. The method implies two steps: • A predictor step providing an initial approximation to the propagated Solution • A corrector step yielding a refined approximation.

Predictor step: an extrapolation to time t +∆t of values from earlier time steps

Advantages: suited for more complex problems were as rigid bodies and constrained dynamics, where greater accuracy at each timestep is required Disadvantage: more storage time and work.

e) Initial state :1)Initial coordinate2)Initial velocities 3)Temperature adjustment ( when there is a

gradual energy drift due to numerical integration error further velocity adjustments will be required over the course of the run.

f) Performance measurements:1)Accuracy ( for every integration method while

considering that to a limited extent accuracy can be sacrificed in the cause of speed )

2)Reproducibility : when we reach equilibrium the initial state memory is lost , and for simulations that vary only in the initial random velocities the kinetic energy varies with time

3)Efficiency 4)Trajectory sensitivity : trajectories are extremely

sensitive to the smallest perturbation so they will differ from a simulation to another but since we only take the average of these trajectories this problem is solved

Chapter 4 Equilibrium properties of simple fluids:A great deal of averaging is needed to have only the useful data; Statistical mechanics relates such MD averages to their thermodynamic counterparts

a) Thermodynamic measurement

1) Relation to statistical mechanics: number of atoms and energy (energy drift assumed suppressed) are fixed in MD simulations so the ensemble for discussing the equilibrium is microcanonical ensemble

2) Error analysis: not like experiment using MD simulation there is no guarantee that successive estimates are sufficiently unrelated.Systemic errors (finite size effects, interaction cutoff, integration)Inadequate sampling errorStatistical error (due to random fluctuations in the measurement) , to solve the problem of correlation we can average over blocks and what is needed is a criterion for choosing the minimal necessary block length

3) Energy4) Equation of state

b) Structure

1)Radial distribution function : the fluid state is characterized by the absence of any permanent structure , but there is structural correlations , g(r) RDF a function that describes the spherically averaged local organization around any given atom ,the potential energy and pressure can be found as integrals involving g (r) For example for a simple monatomic fluid g(r) shows how on average the neighborhood seen by an atom consists of concentric shells of atoms when density increases these shells become distorted showing on the RDF as additional peaks

2)Long range order: RDF addresses the local structure; long range order corresponds to the presence of a lattice structure. Note: when studying solidification in a finite system the best result is if the shape and size of the region allows integral number of unit cells otherwise there will be imperfections in the ordered state leading to a reduction in the apparent long range order.

c) Packing studies

1)Local structure: to describe the spatial organization and the local atomic organization RDF is not enough we use the voronoi method which is a geometric simulation method describing the shape and surrounding of an atom

2)Voronoi subdivision 3)Algorithm: the method for determining which

atoms can contribute to a particular polyhedron assumes that the region has been divided into cells the atoms required are obtained by first scanning a range of cells around the one containing the atoms under examination then sorting the atoms found into ascending distance order.

d) Cluster analysis

1)Cluster algorithm: there is 2 ways to see if an atom belongs to a cluster, one option is to consider the energy that binds an atom to another atom that is already in the cluster or more simply by interatomic distances with having a certain threshold distance. Spatial properties of the cluster include the radius of the gyration and moments of the mass distribution.

Chapter 5:

a) Transport coefficient

1) Background: there is some simple transport coefficient for simple systems and other more complicated; it is the constant factor relating the response of a system to an imposed driving force

2) Diffusion: D the diffusion coefficient is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles, after a long time t the asymptotic behavior will reach a plateau and D will begin to drop to zero, the main source of uncertainty in the trajectories is the strongly repulsive potential.

3) Shear viscosity: The shear viscosity of a system measures is resistance to flow. A simple flow field can be established in a system by placing it between two plates and then pulling the plates apart in opposite directions. Such a force is called a shear force, and the rate at which the plates are pulled apart is the shear rate for formulas

(http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_21/node6.html)

4) Thermal conductivity: In physics, thermal conductivity, k, is the property of a material that indicates its ability to conduct heat. It appears primarily in Fourier's Law for heat conduction.

b)Measuring transport coefficients:

1) Direct evaluation of diffusion: the computation will include a framework for measurements extending over a series of timesteps, with calculation including initialization, actual process of making and accumulating measurements , and a final summary , an important feature is that samples are overlapped to provide extra results

2) Diffusion from the velocity autocorrelation function : this is based on the appropriate autocorrelation function

c) Space time correlation functions:

1)Definitions: spectroscopic techniques measure the spectra of microscopic dynamical quantities, MD provides similar information directly from the trajectories ( important formulas and limits p: 134-135) Note: structure factor: is a mathematical description of how a material scatters incident radiation. Sound attenuation: The combined effect of scattering and absorption.

2)Computational methods3)Program details: we start the calculation by

evaluating the Fourier sums for the density and the three current components one longitudinal and two transverse along each of the three k directions.

4)Correlation analysis

d)Measurements : 1) Transport coefficients:We have three autocorrelation functions, velocity, pressure tensor, and heat current autocorrelation their integrals yield the transport coefficients D (diffusion coefficient), η (shear viscosity), λ (thermal conductivity)

Summary:

Dynamic properties describes the way collective behaviors cause macroscopic observables to redistribute or decay Evaluation of transport coefficients require non equilibrium conditionsTwo formulations to connect macroscopic to microscopic

Chapter 6:

a) Introduction : choosing the ensemble happens on the macroscopic level and it is very important, in MD we work with the microcanonical ensemble since the energy and volume is fixed. Modifying the dynamics allows MD to model the equilibrium behavior of such ensembles two methods: correcting deviations in the controlled parameter, the value fluctuates but these fluctuations can be regulated (feedback control), or by ensuring that the controlled parameter is strictly constant by mechanical constraints that fixes the equation of motion.

b)Feedback methods

1) Controlled temperature: temperature is proportional to the mean-square velocity so we

can vary the temperature by adjusting the rate at which time progresses, we introduce some virtual time t, and dynamic variable s, and insert its effect in the Hamiltonian and Lagrangian, this will not affect the final equilibrium results but it does influence their accuracy and reliability.

2)Controlled temperature and pressure: pressure can be adjusted by altering the container volume(here taken cubic) by uniform isotropic volume change, by rescaling the atomic coordinates, now V and s are treated as supplementary dynamic variables, and again we will have two additional Lagrange equations of motion.

3)Controlled pressure with variable region shape: this more flexible approach allows for the size and shape changes needed to accommodate lattice formation on freezing and for the study of structural phase transitions between different crystalline states ( some complicated calculations)

c) Constraint methods:

1)Constant temperature: enforcing constant temperature amounts to introducing a nonholonomic constraint (is a system whose state

depends on the path taken to achieve it. there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state), the equilibrium properties of this isothermal system is canonical ensemble.

2)Constant pressure and temperature

Chapter 7:

a) Introduction: two approaches to fluid transport: 1: Newtonian dynamics for spatially inhomogeneous systems were the boundaries play an important role and hard walls will be used to determine transport coefficients, 2: Combination of modified equations of motion and fully homogeneous systems

b)Homogeneous and inhomogeneous systems: homogeneous systems advantages: all atoms perceive a similar environment disadvantages: unphysical nature of the behavior

c) Direct measurement

1)Viscous flow: Couette flow: the fluid is confined between two parallel walls that slide relative to

one another at a constant rate. Poiseuille flow: fluid forced to flow between two fixed walls, the viscous nature of the fluid requires sustained work to maintain motion, Real walls is a complicated problem we require sufficiently rough to ensure nonslip flow , we can use a layer of fixed or tethered atoms that mimic the effect of a rough wall, stochastic walls: when atoms try to cross the walls its reflected back to the interior these effects of roughness and temperature are achieved by randomizing the direction of the reflected velocity and scaling its magnitude to match the wall temperature. Analysis of the flow requires the construction of cross-stream vx and T profiles based on a series of slices in the xy plane a general scheme for computing properties is the two dimensional grid subdivision of the simulation region. Some sources of error: small amount of slip at the walls and density variations across the flow, the fact that the transport coefficients depend on ρ and T and even the local shear rate.

2)Heat transport: MD is used to mimic the real experiment of heat flow between 2 parallel walls maintained at different temperatures if heat transform is only from conduction then λ can be found Fourier’s law, the simulation is similar to the

viscous flow but differs in that the external force is absent and the walls are kept at different temperatures. But we should be careful here there is a kind of thermal boundary layer exists close to each wall where the fluid temperature vary more rapidly than in the bulk so that the effective thermal gradient is overestimated.

d)Modified dynamics

1)Linear response theory2)Shear viscosity: for this case we use Couette flow ,

the shear viscosity study are based on the constant temperature dynamics, the temperature might change a little that is why we can use a thermostat in order to evaluate the temperature with respect to the local flow

3)Thermal conductivity: a fictious external field of an unusual kind is introduced it has the effect of driving atoms with a higher than average energy in the direction of the force while those of a lower energy are driven in the opposite direction, so the force generates heat flow at least for small values of the field produces the effect of an imposed temperature difference.

Chapter 7 Rigid Molecules:

a) Introduction: we will study molecules constructed form a rigidly linked atomic framework , its suitable for small relatively compact molecules

b)Dynamics:

1)Coordinates: rigid molecules can be linear and nonlinear , a rodlike linear molecule can be specified using 2 angular coordinates whereas the general case is 3 where we can use the Euler angles

2)Quaternion: set of equations first introduced by Hamilton (quaternion is the complex sum of a scalar and a vector) their advantage here is that no trigonometric functions are required in evaluating R

3)Equation of motion for nonlinear molecules: in terms of Euler angles we will have singularity in the matrix when sinѲ = 0 this is solved by using quaternion, the interactions between rigid molecules are usually expressed as sums of contributions from pairs of interaction sites on different molecules, it is enough to know the center of mass separation of the two molecules

and their orientations in order to be able to compute the interactions between the site pairs

4)Equation of motion for linear molecules: in this case we have only 2 degrees of freedom so we should treat it differently so the torque on a linear molecule can be written as a sum over interaction sites

5)Temperature control: temperature constraint must be based on the combined translational and rotational kinetic energy for each molecule we include a Lagrange multiplier term in the translational equation, and the temperature adjustment to correct numerical drift is applied separately to the translational and rotational motion.

6) Initial state: molecular orientation randomly assigned, each angular velocity have a fixed magnitude based on the temperature, and a randomly chosen direction.

c) Molecular construction

1)General features: interactions between rigid molecules is introduced by specifying the locations of the sites in the molecule at which the forces act, so the total force between 2 different

molecules is the sum of the forces acting between all pairs of interacting sites, work is proportional to the square of the number of sites. Interaction sites may be related with the positions of the nuclei.

2)Model water: explanation of the rigid model TIP4P (with all details listed p:217)

3)Interaction calculation: we calculate the LJ potential energy and force contribution between the O sites, and the coulomb between all pairs of charges, we use all pair approach and assuming periodic boundaries. The distance between the centers of mass of molecules containing the sites specify the cutoff range and not the distance between the sites themselves, and ensures that there is no partially interacting molecules.

d)Measurements

1)Types of measurement: RDFs associated with distinct sites on the molecules that provide clues on local molecular arrangement, Rotational diffusion by looking at the rate at which molecules undergo orientational change , and the hydrogen bond network formed by the fluid (this doesn’t have an experimental counterpart)

2)Radial distribution functions: we consider 3 different site-site distribution functions that are accessible experimentally goo, goh, ghh

3)Rotational diffusion: is the measure of the rate at which the direction of the molecular dipole changes it is the mean-square rate of change in orientation.

4)Hydrogen bonds: the molecular structure of normal ice involves a diamond lattice the forces responsible for this loosely packed arrangement are attributed to hydrogen bonding, each pair whose energy lies below a certain threshold is regarded as linked by a hydrogen bond

e)Rotation matrix representation

1)Equations of motion: is an alternative method that considers the rotation matrices of the molecules directly, we can solve it using a leapfrog integration scheme.

2)Integration3)Interaction calculation: the rotation matrix

approach deals with a fluid of rigid molecules constructed from tetrahedral assemblies of soft spheres it differs from water in that the moment of inertia components is larger leading to similar

contributions to the sites velocity from the translational and angular velocities so that a larger timestep can be used. And since the interactions are short ranged the neighbor list calculations works with neighbor list , and also reflecting container walls are introduced alternative to the periodic boundaries

4)Wall interaction: when the molecular interaction site approach a container they are subjected to a repulsive force in a direction perpendicular to the wall, they act independently so any site near an edge or corner will experience two or three separate wall interactions, these wall forces ensure that the molecules remain within the container.

Chapter 9 Flexible Molecules

a)Introduction: two case studies: configurational properties of a single chain molecule in a solution, model of a surfactant in solution in which very short interacting chain molecules are just one of the components of a three component fluid.

b)Description of molecule

1)Polymer chains: chain properties is 2 categories equilibrium and dynamical, the dynamical one can only be studied by MD, the basic and simple model is a single chain in the vacuum studying the configurational properties of an isolated polymer , then chain in an inert soft sphere solvent. Chain density is an important parameter to determine the interaction with itself and other chains.

2)Chain structure: the monomers can be simple atoms modeled using a soft sphere potential while bonds with limited length variation can be produced by means of an attractive interaction between chain neighbors.

c) Implantation details

1)Interactions: it can be handled using the soft sphere functions, but excluding the bonded atom pairs which require an additional function for evaluating the forces between them.

2)Initial state: it is essential that the atoms of each chain be positioned so that the bond lengths are all within their permitted ranges, and no overlapping between atoms occur, this is easy with low densities , but it can get more complicated and the

packing becomes important , and adding the solvent.

d)Properties

1)Chain conformation: three main spatial properties of polymer chains, mean square end to end distance <R2> which help to find if the chain is an open or compact configuration, distribution of R2 to determine the importance of effects such as excluded volume, and the mean square radius of gyration <S2> that provides information on the entire mass distribution of the chain and plays a central role in interpreting light scattering and viscosity measurement.

2)Measurements

e)Modeling structure formation: we have three component fluids water, oil like liquid, amphibilic chain molecules (this cause water and oil to separate and the shape of the oil will depend on its density its either droplets or layers. This models assumes forces between the pairs of like atoms fww, foo involve LJ potential which is attractive except at close approach , while interaction between unlike pairs is soft sphere repulsion used to prevent overlap, this

case study focuses on micelle growth and how its size vary with time

f) Surfactant models

1)Interactions: for atom chains with two different species of atoms instead of one

LJ between pairs of like atoms and soft sphere repulsion between unlike atoms

Soft sphere repulsion between nonlinked monomers in the same chain irrespective of type

Bonding forces between linked monomers in the same chain

2) Initial state: only one of the chain monomers is placed on a lattice sites and the other monomers are then suitably spaced along the x axis, they are positioned in random locations chosen in a way that ensures that chains do not overlap, and the head tail direction is randomly chosen.

3)Cluster properties: we analyze the micelle growth using the cluster analysis technique if two atoms are separated by less than the distance rClust they are regarded as belonging to the same cluster.

g) Surfactant behavior

1)Micelle growth: the system forms a number of compact O regions surrounded by chains that are typically with outward pointing W ends, at first the number of micelles will increase while their mean size will be small, then the number of micelles will start decreasing approximately inversely proportional to the mean size.

2)Structure formation: the best way to examine the behavior is by looking at the spatially organized structures as they form, in the system where there is equal concentrations of the O and the W species we have a state in which each of the species occupies one or more extended regions separated by surfactant chain layers, that are far from planar and whose form is affected by the periodic boundaries.

Chapter 10 Geometrically constrained molecules

a)Introduction: classical mechanics allows geometric relations between coordinates to be included as holonomic constraints.

b)Geometric constraints

1)Role of constraints: we assume that some bond lengths and angles are constant because at the prevailing temperature there is insufficient energy to excite the associated degrees of freedom out of their quantum ground states, or classically their vibration frequency is much greater than those of other modes and coupling with the rest of the system is weak

2)Problem formulation: for two neighboring atoms the constraint is the bond length that is fixed (interaction between pairs), and for the next nearest the angle is fixed (interaction between triplet), and other constraints like those used to maintain the planarity of the molecule. In Newton’s second law there will be an additional term g corresponding to the constraints, we have two ways to solve it the iterative relaxation procedure that modifies each pair of constrained coordinates in turn until all constraints are satisfied to the required accuracy (has no errors), or to solve the problem by first computing the Lagrange multipliers from the time differentiated constraint equations and using them to solve the equations of motion (it has some numerical integration methods

3)Atom and constraint indexing: the case of a simple chain subjected to bond length constraints and optionally to bond angle constraints, for ns monomers we have nc-1 constraints , and for the angles nc=ns -2 so the total constraints will be 2ns-3

c) Solving the constraint problem

1)Matrix method: solving the equations of motion together with the constraints is done by expressing the constraint equations in matrix form then solving the resulting linear algebra problem using standard numerical techniques.

2)Relaxation method: we begin by advancing the system over a single timestep while ignoring the constraints and then we start to adjust r to obtain corrected coordinates that satisfies the constraints, despite the numerical error experienced by the atomic trajectories the constrained bond lengths and angles always maintain their correct value.

d)Internal forces

1)Bond-torsion force: this motion provides the means for local changes in spatial arrangement of the polymer chain, figure of angles p: 279 with the complete derivation. The torque caused by a rotation about bond i produces forces on the four atoms, the sums of the forces and torques acting on the four atoms are zero, we can derive a relation between the torsional potential function and the dihedral angle Ѳ, and we find the minimum of this potential at Ѳ=0, and two other minima’s at (+-2∏/3) barriers at (+-∏/3) and a maxima at ∏.

2)Bond angle force: the same analysis holds for the bond angle variation where interactions ensure that these variations are limited, and the potential function will be expressed here in terms of α, the bond angle, and the potential has one minimum at 110o and to maxima at 90o and 130o.

3)Other interactions: pairs of atoms in each molecule that are neither directly linked by a constraint nor jointly involved in these three and four body forces interact with the usual LJ potential, or for faster computation by soft spheres.

e)Implantation details

1)Initial state and parameters: for simulations involving both multiple chains and solvent we must specify to independent densities for chains, and for solvent atoms. When choosing the initial velocities that correspond to a given temperature the effect of constraints must be taken into account. And the total number of degrees of freedom is reduced from 3 ns to 2ns + 1 for bond length constraints and to ns + 3 if bond angles are also considered.

2)Structural properties

f) Measurements 1)Constraint preservation2)Properties

Chapter 11 Internal coordinates:

a) Introduction: a way to solve the problem is by considering that only the internal coordinates of the molecule are those actually corresponding to the physical degrees of freedom, its complicated but it’s an effective solution to the problem.

b)Chain coordinates : apart from the first monomer which has six DOF’s each additional monomer contributes one DOF to the chain, each such DOF corresponds to torsional motion or twist around the

appropriate bond axis and is represented by a dihedral angle. A chain of nr + 2 sites has a total of nr

+ 6 DOF’s of which nr are internal.c) kinematic and dynamic relations : important

formulas and derivations p: 299d)Recursive description of dynamics

1)Spatial vector formulation: formulas and derivations2)Stacked operators: is a way to write the formulas in

a concise stacked form, using spatial vectors and spatial operators, but this approach is only practical for the shortest of chains.

3)Mass matrix inversion: a better way for solving longer chains.

4)Recursion relations: the recursions relations for propagating the velocity, force, and acceleration values along the chain.

5)Interactions: two kinds of interactions are used in this model, pair interaction used to prevent the overlap of the atoms located at the sites of the chain, simple soft sphere repulsion is enough because fixed bonds and angles will eliminate the effect of neighbor pairs. The second kind of interaction is the torsion potential associated whit each DOF. The value of the dihedral angle and the value of the fixed bond angle is chosen to ensure

that the ground stet has the correct amount of twist to produce a helix

6)Inertia tensor

e)Solving the recursion equation

1)Organizational matters: a spherical mass element with a finite moment of inertia about its own center is associated with each site. Each bond has a single mass attached to its far site except for the first bond which has masses attached at both ends.

2)Site coordinates and velocities: given the position and the orientation of the initial link the dihedral angles for all the subsequent links and the time derivatives of these quantities the site positions and velocities can be evaluated.

3)Link inertia and velocities: 4)link accelerations

f) Implementation details: there are no direct pair interactions between nearest, second or third nearest neighbor monomers, because they are taken into account by either the rigid structure or the torsional interactions. The initial state of the chain produces a coiled chain with a relatively large coil radius, and a local zigzag configuration for each

successive pair of bonds, if the bond angles and torsional interactions have been suitably chosen then at low temperature the chain should collapse into a helical ground state

g) Measurements

1)Equilibrium: if left unattended the total energy is subjected to a slight amount of drift

2)Chain collapse: to study the behavior of the chain as it gradually cools from a high temperature state , were the bond angles and the dihedral angles have been chosen so that the minimum ground state of the chain is a neatly coiled helix, and if the chain will collapse into its ordered configuration or not

Chapter 12 Many body interaction

a)Introduction: two models three body interaction , and the embedded atom method

b)Three body forces

1)the problem: the force between two atoms will now depend on all other atoms in the vicinity

2)formulation: model for liquid silicon treated simply we have two body interaction contribution and three body contribution that both depend on rc , the three body part is symmetric under permutation of the atom indices , and is invariant under translation and rotation, and the formula is presented in a way such that each atom prefers a considerably smaller number of immediate neighbors than close packing (to make things simpler)

3)implementation details: the neighbor list method should be used but after modification, and the list must include each atom pair twice ij and ji to identify all interacting atom triplets

c) Embedded atom approach

1)Interactions: the LJ potential works when the electron clouds responsible for the attractive and repulsive components of the interatomic interactions remain localized close to the individual atoms. In metals the valence electrons may be shared among atoms , to describe this interaction we need the embedded atom potential, that consists of two parts, a pair interaction between metal atoms (actually ions) that does not explicitly on the density, and a many

body term that depends on the local value of the density at the point where the atom is located

2)Implementation3)Structure measurements: for the RDF (only) there

is no difference except softening the peaks between the embedded atom fluid and the soft sphere fluid.

4)Collision modeling: collision between extended bodies where one body (the projectile) impacting at high speed at the surface of another larger body (the target) that causes ejection of material from the opposite surface, the bodies involved are usually metallic so we use with both of them the embedded atom approach.

Chapter 13 long range interactions

a)Introduction : problems involving electric charges and dipoles cannot be treated with short range forces. Two methods for solving the long range interactions, the Ewald ressumation technique by recognizing the interaction sums over periodic images, and two hierarchical methods: the tree technique in which subdivision of each cell is repeated until occupancy reaches unity, and another

method that employs a fixed number of cell subdivision.

b)Ewald method

1)Interaction resummation: it works with the ionic crystals, and in simulating charged and dipolar fluids with periodic boundaries, were it continues to work with the periodic replicas as well, this technique eliminate the discontinuity arising from truncated long range forces. The energy of the replicated system includes contributions from all replicas since no truncation is imposed, but the self interaction is prevented but atoms interact with their replica images. One can use spherical cutoff (rc < L/2) with boundary conditions, because the outermost replica shell is effectively surrounded by a conductive medium. For large system we can choose α in a way we don’t need this cutoff and this is the case for dipolar systems (because the sum over replica can remain conditionally convergent).

2)Dynamics: the Lagrange equations of motion for translation involve the usual soft sphere interaction and a contribution from the force produced from the dipolar interactions.

3)Properties: the fact that each molecule has a dipole moment means that the orientational order can be studied as well; liquid is characterized by short range structural order. Given the definition of some orientational quantities compute the average values over the atoms in a series of concentric shells centered on the atom of interest it’s like the extension of the RDF computation.

4)Measurements

c) Tree-code approach

1)Design considerations: subdivide the entire region into eight cells (if 3D) repeat the subdivision for each of the cells that contains more than single atom and continue this recursive process until no cells are multiply occupied, the interaction computation then considers each atom in turn and pairs it with all cells beginning with cells at the top of the hierarchy , the accuracy of the interaction calculation depends on the minimal range at which a cell is no longer required to undergo further subdivision. A compromise between accuracy and performance should be made.

2)Tree construction: the tree is constructed at each timestep by successively subdividing occupied cells

one level of the hierarchy at a time until no multiple occupancies remain. The interaction computations involve a backtracking algorithm that traverses the oct-tree level by level starting from the top level each cell is accessed in turn and a decision is made whether it is sufficiently Far from the atom under consideration that its center of charge coordinates can be used to evaluate the forces or whether further subdivision is required, the system will have hard wall boundaries , so clearly a large region size is required if finite size effects are to be kept to a minimum.

3)Interaction computation: the function considers each atom in turn and pairs it with the averaged contents of all the cells at suitable levels in hierarchy, the level at which each cell is utilized depends on its distance from the atom, the total momentum is not strictly conserved.

d)Fast-multipole method

1)Background: the tree code approach regards all the charged particles in each cell as being positioned at a single point, for multipole expansions this amounts to ignoring all but the lowest order monopole term,

accuracy can be improved by extending the expansions to higher order.

2)Multipole expansion: the familiar multipole expansion of electrostatics

3)Recursion relations and derivatives: to evaluate the Legendre formulas

4)Hierarchical subdivision: we want to reduce the computational effort to O(Nm) this is accomplished by hierarchical subdivision of the region into cells with the cell size at each level being half that of the level above, interactions between atoms in neighboring cells at the lowest level are computed directly, but for more distant cells evaluating the interactions involves the use of multipole expansions to represent the cell occupants, this method assumes a fixed level of subdivision and it is a useful approach for problems where significant spatial inhomogeneity occurs since each level is eight times as many cells as the level above (in 3D) not too many levels are required

5)Operators on multipole expansions: this fast method relies on the fact that when the origin of the multipole expansion is changed the coefficients of the original and shifted expansions can be related, it is assumed that the distances involved are chosen to ensure the expansion converges, we can do

multipole to multipole translation, and also the multipole expansion can be converted into a local expansion about a distant origin. We note that L coefficients are multipole expansions and the M coefficients are local expansions. Each cell at the nth level associated with it a set of Lni of L coefficients describing a multipole expansion about its midpoint for all the atoms in the cell.

6)Traversing the cell hierarchy: after evaluating the Lni

for the lowest level we start ascending through the cell hierarchy and for each cell at level l summing the L coefficients of cells at the next lower level l + 1 after shifting each of the expansions to the midpoint of the current cell, these sets of cumulative L coefficients denoted by Lli can be used in evaluating multipole expansions associated with cells at the lth level in the hierarchy. And a descent algorithm to consider all non neighbors and far cells.

e) implementing the fast multipole method

f) Results

1)Accuracy and performance: to show the errors when considering all pairs of charges and how these errors change with the choice of some constants

2)Radial distribution function: it shows the strong preference for oppositely charged neighbors at close range; because of the overall charge neutrality this effect is limited to relatively short distances, at large distances the RDF drop below the expected 0.5 due to hard wall boundaries.

Chapter 14 Step potentials

a)Introduction: the limitations of continuous potentials are that they require the changes in interactions over each timestep to be small; otherwise uncontrolled numerical error can suddenly appear. Step potential method doesn’t face this problem because it advances the system by a series of discrete events, and we avoid the need for explicit numerical integration by employing impulsive collisions whenever atoms interact. The simple example of hard-sphere fluid subject to periodic boundary conditions will be discussed here.

b)Computational approach

1)Dynamics: 2 atoms will collide if and when their separation becomes equal to the atomic diameter. A negative solution correspond to trajectories that

intersected in the past, a bound pair can only escape from the well if the missing energy is provided by a third atom.

2)Cell subdivision: the simulation progresses by means of a time ordered sequence of collision events. Similar to other methods we need to reduce the computational effort so we introduce cells, but the problem here is that we need to keep track of which atoms belong to which cells. A good device is using local time when a collision occurs only atoms in the immediate neighborhood are of concern and there is no point in updating the coordinates of atoms much further away.

3)Event calendar: since assuming that we can predict all future events , the existence of an event calendar is important which must produce the next events and be modifiable , when two atoms collide their velocities are changed so that any information stored in the calendar regarding future events involving these atoms ceases to be valid, such events will have to be erased from the calendar and replaced by whatever new events are predicted

4)Properties: equilibrium and transport properties are the same between step potentials and the continuous case, the only difference is in the

quantities that depend directly on interactions such as pressure, for RDF the only difference is instead of the measurement being performed at fixed multiples of the timestep a new class of measurement event is required.

c) Event management

1)Calendar design: The calculation contains a list of future collisions and cell crossings as well as well as events corresponding to measurements of various kinds constructed at fixed time intervals, the calendar holds a lot of information so efficiency is required that focuses on execution time without neglecting space requirements. We use a binary tree each scheduled event is represented by a node in the tree, which contains information identifying the time at which the event is scheduled to occur and event details. The operators that are performed on the tree data are the following:

retrieve the earliest event add a new event delete an existing event initialize the tree contents

2)Theoretical performance3)Program details

d)Properties

1)Radial distribution function: the only difference between the soft sphere and the hard sphere is the sharpness in the peek.

2)Free path distribution: (distribution of path lengths between collisions) this value can only be studied for hard spheres because soft spheres don’t have the collision theory.

3)Efficiency

e)Generalization

1)Outline: two useful extensions of the hard sphere approach: the construction of polymer chains, and the way in which both rotational motion and inelasticity can be added.

2)Hard-sphere polymer chains3)Rotation and inelasticity: when collisions occur not

only is there a change in translational motion but the spins of the spheres also change. There is also one more feature the inelasticity.

Chapter 15 time-dependent phenomena

a)Introduction: we extend the MD approach to a class of problem in which the behavior is not only time dependent but the properties themselves are also spatially dependent in ways that are not always predictable, we will focus on MD applications in fluid dynamics.

b)Open systems: usually MD deals with closed systems, but when we have an open system they are out of thermodynamic equilibrium and in many cases spatially inhomogeneous and time dependent usually the existence of atoms is irrelevant to the fluid picture but we want to learn what the molecular constituents of fluid do while the fluid is flowing across the obstacle. We have two examples: thermal convection a horizontal layer of fluid is heated form below and the resulting interplay between the upward flow produced by heating and the downward flow produced by gravity leads to the formation of structured flow patterns, all this will depend on the Rayleigh number Ra. For flow past a rigid obstacle the behavior is governed by Reynolds number Re, we need parameter combinations that produce correct values of the dimensionless numbers Ra and Re to be able to observe the behavior and also the duration must be long enough

c) Thermal convection

1)motion in a gravitational field: we can include the effect of uniform gravitational field by extending the hard sphere method, since all will have the same acceleration but they will be parabolic rather than linear, so the prediction of cell crossings that calls for special attention since parabolic motion means that it is possible for the atom to leave the cell through the face or edge of entry.

2)Hard wall boundaries: the boundaries used are rigid; the walls can be smooth giving energy conservation (specular collision) or rough walls where atoms lose all memory of its prior velocity. Walls are divided into a series of strips so collisions will be divided between these 2 kinds.

3)Flow analysis: each quantity is computed for all atoms in every cell at a given instant and the results are averaged over time, we use the grid method, the size of the grid is important we want the smallest spatial structures and the most rapid changes in the flow to be resolved, at the same time we have a problem that a smaller number of atoms participating in the average for a single cell the larger the fluctuations.

4)Results: in simulation of this kind here is normally an initial transient phase during which the system gradually evolves into a final state appropriate to the choice of parameters.ccc

d)Obstructed flow

1)Boundaries and driving force: two kinds of boundaries, the region boundaries which can be made periodic so that the flow recirculates providing that any memory of nonuniform flow is erased, and the obstacle boundary which also should be rough to ensure that the adjacent fluid layer is at rest

2)Results: relatively large systems are required in order to allow the flow patterns to develop properly.

Chapter 16 Granular dynamics

a)Introduction: Transport of granular materials can be described my methods analogous to MD modeling although the constituent particles are no longer the molecules and particles of MD. The grains themselves are irregularly shaped often covered with asperities and are normally poly disperse, grain

collision is highly inelastic and friction is important for forming heaps a lot of factors (electrostatic forces, moisture, adhesion…) can all affect the behavior.

b)Granular models

1)Background: a simplified representation of the salient features of the grain structure and interactions is a prerequisite for simulation. The widespread models are based on inelastically colliding hard or soft spheres, rotational motion may or may not be included, normal and tangential velocity-dependant damping forces are used to represent the inelasticity of the interactions, static friction is partially covered by a force that opposes any sliding motion while particles are in contact. In all cases energy is no longer a conserved quantity.

2)Particle interaction: what distinguishes the interactions used for granular media from those in MD studies of molecular systems is the presence of dissipative forces that act over the duration of each collision, a normal damping force, a frictional damping, and static friction. The translational and

rotational accelerations depend on the sum of the above forces for all interacting pairs.

c) Vibrating granular layer

1)Two-dimensional version: the model here represents a horizontal layer of grains in a container whose base oscillates sinusoidaly in the vertical direction. The container sides are assumed periodic while the top boundary which plays a very minor role in the calculation is elastically reflecting.

2)Three-dimensional version: only a minimum change, the granular particles become spheres instead of disks.

d)Wave pattern: the analysis of this category of problem relies heavily on the ability to directly visualize the system as the simulation progresses.