Finite temperature total energy differences

By breaking time down into small intervals dt, the equations of motion can then be solved directly using finite difference algorithms [48]. In the simplest form of MD the total energy of the molecular system is a conserved quantity. However, it is equally possible to carry out MD at constant temperature by employing one of a number of available thermostat algorithms [51]. When... 

Subsequently, Frantz calculated the total energy as a function of temperature for the different clusters, which resulted in the curves of Figure 11. The curves are seen to posses a change in the slope over a more or less narrow temperature interval, that becomes more narrow when the clusters are larger. These changes in the slope signal phase transitions (see Section 3.6). Ultimately, for the infinite system the slope of the energy will become discontinuous at the temperature of a phase transition, but for the smaller, finite systems, this transition is obviously smeared out. 

We have so far only set up the tools that will be needed in the zero-temperature analysis of the total energy. To span the set of possible structural outcomes for a range of different temperatures requires a knowledge of the entropy of the competing phases. In order to effect such a calculation, we must first revisit the finite-temperature properties of crystals discussed in the previous chapter with an eye to how the entropy of the competing phases (including the liquid) may be determined. 

For all three systems, they performed simulations in which the systems were first equilibrated in the high temperature, athermal limit. A jump was then made to finite temperature at which the system was still disordered, and then gradually cooled down. At each temperature, they performed 2 x 10 Monte Carlo steps. For the 7-16-7 system, they also studied other thermal treatments instead of slow cooling, they quenched to each required temperature from the athermal state, to a total of 39 different temperatures. As well, starting from the lowest temperature probed, they slowly reheated the system. In all cases, they monitored the polymer mean-squared end-to-end distances / rms, the specific heat Cv, and the energy per lattice site. Systems were always cooled into a microphase separated state. 

As mentioned in the introduction, nanoparticles have sizes below the thermodynamic limit, implying that their properties do not scale with their size. Obviously, it is therefore of interest to determine how their properties then scale. Eryiirek and Giiven have recently presented one theoretical study devoted to this aspect. They studied the thermodynamic properties of finite clusters with N = 39-55 atoms for which the interatomic interactions were modeled with a Lennard-Jones potential, eqn (2). To this end they performed molecular-dynamics simulations within the microcanonical ensemble for the different clusters, meaning that the total energy and the number of particles were kept fixed. From trajectories of 2 x 10 steps they determined various averaged values from which they subsequently determined, e.g., the temperature of the cluster,... 

After the pressure field is obtained at each vertex node, the finite difference formulation for the energy equation is represented in the gapwise direction at the centroid of each element. The location of temperature nodes is chosen to be the centroid of each element where the velocity, viscosity, and shear rate are most accurate and less averaging is required for the temperature dependent properties. The total enthalpy h is spht up into latent heat and sensible heat as... 

---