Molecular modelling Newtons laws

Molecular dynamics consists of the brute-force solution of Newton s equations of motion. It is necessary to encode in the program the potential energy and force law of interaction between molecules the equations of motion are solved numerically, by finite difference techniques. The system evolution corresponds closely to what happens in real life and allows us to calculate dynamical properties, as well as thennodynamic and structural fiinctions. For a range of molecular models, packaged routines are available, either connnercially or tlirough the academic conmuinity. 

In molecular dynamic models, polymer chains are modelled atom by atom. Forces on each atom exerted by its surrounding atoms are calculated according to the nature of the chemical bond and the distance between the atom and the surrounding atoms. Different mathematical relations are used for different type of bonds, with unique sets of parameters for each atomic species. The variables in the model are the locations of the atoms. At any instant of time, the accelerations of the atoms are calculated according to Newton s law and their velocities and positions are updated using a small timestep. Repeating the procedure provides the trajectories of all the atoms. Figure 1.1 shows an example of molecular models for a semi-crystalhne polymer. 

The molecular dynamics method is based on the time evolution of the path (p (t), for each particle to feel the attractions and repulsions from all other particles, following Newton s law of motion. The simplest case is a dilute gas following the hard sphere force field, where there is no interaction between molecules except during brief moments of collision. The particles move in straight lines at constant velocities, until collisions take place. For a more advanced model, the force fields between two particles may follow the Lennard-Jones 6-12 potential, or any other potential, which exerts forces between molecules even between collisions. 

According to this model, pressure is caused by the collisions that molecules make with the walls of its container. An analysis of this molecular motion using Newton s laws leads to an expression identical to the ideal-gas law. An important result of the theory is that the average kinetic energy of one mole of gas can be expressed in terms of the absolute temperature... 

It is therefore remarkable that 100 years or so before the laws of thermodynamics were formulated, Daniel Bernoulli developed a billiard ball model of a gas that gave a molecular interpretation to pressure and was later extended to give an understanding of temperature. This is truly a wonderful thing, because all it starts with is the assumption that the atoms or molecules of a gas can be treated as if they behave like perfectly elastic hard spheres—minute and perfect billiard balls. Then Newton s laws of motion are applied and all the gas laws follow, together with a molecular interpretation of temperature and absolute zero. You have no doubt... 

The basic law of viscosity was formulated before an understanding or acceptance of the atomic and molecular structure of matter although just like Hooke s law for the elastic properties of solids the basic equation can be derived from a simple model, where a flnid is assumed to consist of hypothetical spherical molecules. Also like Hooke s law, this theory predicts linear behavior at low rates of strain and deviations at high strain rates. But we digress. The concept of viscosity was first introduced by Newton, who considered what we now call laminar flow and the frictional forces exerted between layers within a fluid. If we have a fluid placed between a stationary wall and a moving wall and we assume there is no slip at the walls (believe it or not, a very good assumption), then the velocity profile illustrated in Figure... 

Newton tested his own ideas by rederiving the laws of Kepler, while Kepler had deduced his three laws from Tycho s observational data. So in fact, at the very foundation of modem Science we find a this very fruitful relationship between observation and theory. It is all too easy to forget that in the, not so distant past, the computers were humans [6]. To trace the pre-history behind the modem computers is yet another story [7]. In the case of Tycho Brahe, Johannes Kepler and Isaac Newton, using a modem vocabulary, it was Kepler who did the work of a computer , while Tycho Brahe provided the experimental evidence and Newton supplied the theoretical and mathematical models. Thanks to these pioneering scientists we perform our Molecular Dynamics simulations today [8-10]. 

Questions naturally arise as to the accuracy of predictions made by models. Quantum mechanical models of molecular processes are capable of fantastic accuracy. Models of planetary motion based on Newton s laws of motion are sufficiently accurate to put men on the moon and bring them back. Thermodynamics itself is a model of energy relationships, which Einstein once said is the only theory he was sure would never be overthrown. Unfortunately, hydrological and geochemical models deal with much more complex processes, and are less accurate. 

Newton had an incorrect theory of gases in which he assumed that all gas molecules repel one another and the walls of their container. Thus, the molecules of a gas are statically and uniformly distributed, trying to get as far apart as possible from one another and the vessel walls. This repulsion gives rise to pressure. Explain why Charles s law argues for the kinetic-molecular theory and against Newton s model... 

First of all, the use of molecular mechanics and dynamics pre-suppose a three-dimensional structure or a model built by sequence homology. Even if molecular mechanics involves large simplifications, it is an extremely powerful and reliable technique to study organic systems, except for reaction pathways. Molecular dynamics is a simple technique which is almost insensitive to temperature but extremely sensitive to the chosen time-step for the integration of the Newton s law. If the main interest of the user is to explore the conformational space of a system, then heating is the best choice (up to 1000 K or more). 

Classical molecular dynamics (MD) simulations rely on model potentials describing the interactions between the particles in the system, rather than a first-principles calculation of the system energy at each time step. Obviously, the quality of the outcome depends on the quality of the model potential. Model potentials are normally either fit to experimental results, or based on first-principles calculation. Since the majority of the model potentials is based on two-body interactions only, such potentials generally ignore the multibody character of real interactions. The time evolution of the system is computed by solving numerically Newton s laws of motion. Molecular dynamics or molecular simulation is a vast field with many fields of applications. For detailed discussions, see Refs. [9, 10]. 

Molecular dynamics studies the properties of matter or transport phenomena by constructing an atomic or molecular system with the initial microscopic state of a system specified in terms of the positions and momenta of the constituent atoms or molecules, which can be obtained either from theoretical consideration or from experimental results. Molecular dynamics also requires a model potential to simulate the interactions among atoms and molecules, in which way physics comes into play a role. The model interaction potential should obey the fundamental laws of physics and chemistiy and capture the important features of the intermolecular interactions that determine the property of interest. It needs to be remembered that the model is just an approximation to the interactions in the real world and the results from the molecular dynamics simulation have to be tested against proved theoretical or experimental findings, i.e., it should be able to reproduce some properties of matter like density distribution, transport coefficients, and so on. Starting from the initial setup of the system with given intermolecular potentials, Newton s equations of motion are integrated for each... 

The principles of molecular mechanics may be used in a molecular simulation calculation, which is a type of computational statistical me-chanics. The goal of molecular simulation is to analyze a theoretical model of molecular behavior in order to determine the macroscopic properties of a substance. In one approach, known as molecular dynamics (MD), Newton s laws of motion for individual particles and a set of potential energy terms describing the forces on the structures are applied to all of the atoms in the calculation. Integration of the resulting differential equations over a short time period leads to new locations and new velocities for the atoms. 

This repulsion gives rise to pressure. Explain why Charles s law argues for the kinetic-molecular theory and against Newton s model. 

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