China
Africa
Explore chapters and articles related to this topic
Z-Partition Function
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
For a canonical ensemble in which the system is allowed to exchange heat with the environment at a fixed temperature, volume, and a number of particles, the partition function of the discrete spectrum can be express as: ()Z=∑n=0n=Ne−βEn,β=1kBT
Statistical Physics
Published in Rolando M.A. Roque-Malherbe, Adsorption and Diffusion in Nanoporous Materials, 2018
The canonical ensemble is a system in a heat bath at constant temperature and volume, and with a fixed number of particles N [8], that is, a system, which is in thermal equilibrium with a large bath. Since energy can flow to and from the bath, the system is, as was previously stated, described by the bath temperature T rather than by a fixed energy E [10]. Such a system and the statistical method based on it, are referred to as a canonical ensemble.
Simulation of Crystalline Nanoporous Materials and the Computation of Adsorption/Diffusion Properties
Published in T. Grant Glover, Bin Mu, Gas Adsorption in Metal-Organic Frameworks, 2018
The canonical ensemble occurs when a system with fixed number of particles N and volume V is held at constant temperature T (using an infinite heat bath). The canonical ensemble describes a system in contact with a heat bath. The classical canonical partition function Z of a gas of N identical classical particles is given by Z(N,V,T)=1Λ3NN!∫e−βU(rN)drN. Here, β is called the inverse temperature or thermodynamic beta and has units of inverse energy, the de Broglie thermal wave length Λ is just an arbitrary length scale to make the partition function dimensionless. The letter Z stands for the German word Zustandssumme, “sum over states.” The partition function encodes how the probabilities are partitioned among the different microstates, based on their individual energies, and the probabilities sum up to one. The probability Pm that the system occupies microstate m is Pm=1Z(N,V,T)e−βUm where the factor exp(−βUm) is known as the Boltzmann factor. Microstates with high energy are less probable, while microstates with low energy are much more probable.
Modeling of fracture behavior in polymer composites using concurrent multi-scale coupling approach
Published in Mechanics of Advanced Materials and Structures, 2018
Shibo Li, Samit Roy, Vinu Unnikrishnan
In molecular dynamics, the required force field is selected to describe the material being studied when the system is initialized. The atomic interaction behavior is completely defined by the choice of different interaction potentials. The molecular system will be allowed to evolve under an imposed pressure and temperature state. Periodic boundary conditions are often chosen in MD simulations to calculate bulk materials as it can approximate an infinite system using a relatively small system. In this article, LAMMPS, a molecular dynamics program developed by Sandia National Laboratories, is used as the MD solver. The OPLS force field has been chosen to simulate polymer atomic interactions in this work. The canonical ensemble, NVT ensemble, in which number of particles, volume, and temperature are conserved, is used in the molecular simulations. This ensemble represents a system in thermal equilibrium with a heat bath at the prescribed temperature (T). A suitable thermostat is used to add or remove energy from the MD system. Various types of thermostats like Nose–Hoover, Berendsen, Andersen, etc. are available in MD.
Nucleation on a sphere: the roles of curvature, confinement and ensemble
Published in Molecular Physics, 2018
Jack O. Law, Alex G. Wong, Halim Kusumaatmaja, Mark A. Miller
In order to develop a model that explains these features, we start with the result from classical nucleation theory (CNT). In the grand canonical ensemble, the thermodynamic potential (free energy) is the grand potential , where F is the Helmholtz free energy. The change in grand potential for the creation of a nucleus is where γ is the line tension, is the bulk free energy of the nucleating phase relative to the parent phase per unit area, P is the perimeter of the nucleus and is the area of the nucleus as a function of its perimeter. Figure 3 includes an attempted fit of Equation (4) to the simulated results, assuming the usual planar relation . It can be seen that Equation (4) not only fails to capture the increase in the free energy at large cluster sizes, it also cannot fit the shape of the initial barrier.