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Properties of Participating Media
Published in John R. Howell, M. Pinar Mengüç, Kyle Daun, Robert Siegel, Thermal Radiation Heat Transfer, 2020
John R. Howell, M. Pinar Mengüç, Kyle Daun, Robert Siegel
The atoms or molecules of an absorbing or emitting gas are not stationary, but have a distribution of velocities associated with their thermal energy. If an atom or molecule is emitting at wavenumber ηij and at the same time is moving at velocity v toward an observer, the waves arrive at the observer at an increased η given by η = ηij[1 + (v/c)]. If the emitter is moving away from the observer, v is negative, and the observed wavenumber is less than ηij. In thermal equilibrium, the gas molecules have a Maxwell–Boltzmann distribution of velocities. This velocity distribution results in a spectral line shape with a Gaussian distribution: κη,ijSij=1γDln2πexp−η−ηij2ln2γD2.
Basic statistical concepts
Published in Nils O. Petersen, Foundations for Nanoscience and Nanotechnology, 2017
The Maxwell-Boltzmann distribution of speeds is a special case of the Boltzmann distribution which describes the fraction of molecules, f(v), in an ideal gas with a particular speed, v, which corresponds to a kinetic energy of Ekin=12mv2 $ E_{kin} = \frac{1}{2}mv^{2} $ where m is the mass of the molecule. This is a continuous distribution which has the form f(v)=(m2πkT)324πv2e-12mv2kT $$ f(v) = (\frac{m}{{2\pi kT}})^{{\frac{3}{2}}} 4\pi v^{2} e^{{ - \frac{{\frac{1}{2}mv^{2} }}{{kT}}}} $$
Structural properties of Fe–Ni/Cu/Fe–Ni trilayers on Si(100)
Published in Phase Transitions, 2021
Ananya Sahoo, Maheswari Mohanta, S. K. Parida, V. R. R. Medicherla
The diffusion of atomic species takes place when there is a spatial concentration gradient that is expected in as-deposited trilayers. The particle flux density is related to the concentration gradient by Fick's law, , where D is the diffusion coefficient and c the concentration of the atomic species. The diffusion coefficient D strongly depends on the temperature. If we assume Maxwell–Boltzmann distribution for atoms, then the probability for an atom to attain an energy E is given by , where A is a constant. If is the diffusion barrier energy then for , predominant diffusion occurs. In case of as-deposited Invar trilayer, cluster formation with slight variation in composition is anticipated. Each cluster is of either bcc or fcc structure. At room temperature, the diffusion is negligible so that clusters remain stable maintaining the mixed fcc and bcc phases. When the trilayer is annealed, the atoms in the cluster vibrate with higher amplitude and have more energy to cross the diffusion barrier. This leads to strong diffusion across the clusters of different composition and structure. The diffusion continues as long as there is a concentration gradient of atomic species. In case of Fe–Ni alloys, both Fe and Ni diffuse in opposite directions till the film attains a homogeneous composition.
Solid-to-super-critical phase change and resulting stress wave during internal laser ablation
Published in Journal of Thermal Stresses, 2018
Yan Li, Chong Li, Wenlong Yao, Xinwei Wang
In the previous discussion, the overall temperature is defined when the system achieves local thermal equilibrium, which can be verified by studying the velocity distribution at the location of interest and comparing it to the Maxwell–Boltzmann distribution. Figure 7 shows the Maxwell–Boltzmann distributions and the velocity distributions of the atoms in group 2 are shown in Figure 6a. The solid line is the Maxwellian distribution, and the red dots are the MD simulation results. The atomic velocity before the laser radiation is observed to closely follow the equilibrium Maxwell–Boltzmann distribution. During the 40 ps of laser heating, the thermal equilibrium cannot be fully established. After this point, however, the atomic velocity again follows the equilibrium Maxwell–Boltzmann distribution. The Maxwellian distribution is fitted using both the temperature and the velocity to subtract the effects of the macroscopic velocity.
Subsurface deformation studies of aluminium during wear and its theoretical understanding using molecular dynamics
Published in Philosophical Magazine, 2018
C. S. Tiwary, J. Prakash, S. Chakraborty, D. R. Mahapatra, K. Chattopadhyay
A perfect single crystal specimen is created by filling up the simulation box according to the crystal structure of pure Al which is followed by minimisation of total potential energy of the system by iteratively adjusting the atom positions. To achieve finite temperature, atom velocities are randomly assigned by Maxwell–Boltzmann distribution corresponding to 300 K temperature. Subsequently, a dynamic relaxation process is followed under NPT ensemble to achieve stress-free external surfaces while maintaining the system temperature at ∼300 K. Once dynamic equilibrium is achieved the system is loaded by displacement control loading for different loading case. At this point, it is worthwhile to mention that to resolve the vibrational motion of individual atom correctly we have to use incremental MD time step in the order of the femtosecond (10−15 second). This consideration directly limits the minimum strain rate that can be applied to achieve a physically meaningful total strain of ∼5–10% with the present available computational facility. For this reason, a strain rate of 109/s is used in the present study (higher compared to experimental value but the value is kept constant in all simulations in order to understand the effect of loading on pure aluminium). At this high strain rate stress wave is a major concern but the applied incremental displacement is distributed throughout the sample using affine transformation which will eliminate this problem of stress wave.