The virial stress is the most commonly used definition of stress in discrete particle systems. This quantity includes two parts. The first part depends on the mass and velocity (or, in some versions, the fluctuation part of the velocity) of atomic particles, reflecting an assertion that mass transfer causes mechanical stress to be applied on stationary spatial surfaces external to an atomic‐particle system. The second part depends on interatomic forces and atomic positions, providing a continuum measure for the internal mechanical interactions between particles. Historic derivations of the virial stress include generalization from the virial theorem of Clausius (1870) for gas pressure and solution of the spatial equation of balance of momentum. The virial stress is stress‐like a measure for momentum change in space. This paper shows that, contrary to the generally accepted view, the virial stress is not a measure for mechanical force between material points and cannot be regarded as a measure for mechanical stress in any sense. The lack of physical significance is both at the individual atom level in a time‐resolved sense and at the system level in a statistical sense. It is demonstrated that the interatomic force term alone is a valid stress measure and can be identified with the Cauchy stress. The proof in this paper consists of two parts. First, for the simple conditions of rigid translation, uniform tension and tension with thermal oscillations, the virial stress yields clearly erroneous interpretations of stress. Second, the conceptual flaw in the generalization from the virial theorem for gas pressure to stress and the confusion over spatial and material equations of balance of momentum in theoretical derivations of the virial stress that led to its erroneous acceptance as the Cauchy stress are pointed out. Interpretation of the virial stress as a measure for mechanical force violates balance of momentum and is inconsistent with the basic definition of stress. The versions of the virial‐stress formula that involve total particle velocity and the thermal fluctuation part of the velocity are demonstrated to be measures of spatial momentum flow relative to, respectively, a fixed reference frame and a moving frame with a velocity equal to the part of particle velocity not included in the virial formula. To further illustrate the irrelevance of mass transfer to the evaluation of stress, an equivalent continuum (EC) for dynamically deforming atomistic particle systems is defined. The equivalence of the continuum to discrete atomic systems includes (i) preservation of linear and angular momenta, (ii) conservation of internal, external and inertial work rates, and (iii) conservation of mass. This equivalence allows fields of work‐ and momentum‐preserving Cauchy stress, surface traction, body force and deformation to be determined. The resulting stress field depends only on interatomic forces, providing an independent proof that as a measure for internal material interaction stress is independent of kinetic energy or mass transfer.

A new look at the atomic level virial stress: on continuum-molecular system equivalence

(1994). The Ontology of Complex Systems: Levels of Organization, Perspectives, and Causal Thickets. Canadian Journal of Philosophy: Vol. 24, Supplementary Volume 20: Biology & Society: Reflections on Methodology, pp. 207-274.

/pdf/the-ontology-of-complex-systems-levels-of-organization-2s6t23qq5v.pdf

The Ontology of Complex Systems: Levels of Organization, Perspectives, and Causal Thickets

A lattice Boltzmann model is proposed for solving low Mach number thermal flows with viscous dissipation and compression work in the double-distribution-function framework. A distribution function representing the total energy is defined based on a single velocity distribution function, and its evolution equation is derived from the continuous Boltzmann equation. A lattice Boltzmann equation model with clear physics and a simple structure is then obtained from a kinetic model for the decoupled hydrodynamic and energy equations. The model is tested by simulating a thermal Poiseuille flow and natural convection in a square cavity, and it is found that the numerical results agree well with the analytical solutions and/or the data reported in previous studies.

https://repository.ust.hk/ir/bitstream/1783.1-3549/1/PhysRevE.75.036704.pdf

Thermal lattice Boltzmann equation for low Mach number flows: decoupling model

In this paper, we quantify and discuss the physical and surface chemical processes leading to the formation, temporal evolution and sedimentation of dust grains in brown dwarf and giant gas planet atmospheres: nucleation, growth, evaporation and gravitational settling. Considering dust particles of arbitrary sizes in the different hydrodynamical regimes (free molecular flow, laminar flow, turbulent flow), we evaluate the equilibrium drift velocities (final fall speeds) and the growth rates of the particles due to accretion of molecules. We show that a depth-dependent maximum size of the order of amax ≈1 µm(upper regions) .. .100 µm(lower regions) exists, which depends on the condensate and the stellar parameters, be- yond which gravitational settling is faster than growth. Larger particles can probably not be formed and sustained in brown dwarf atmospheres. We furthermore argue that the acceleration towards equilibrium drift is always very fast and that the tem- perature increase of the grains due to the release of latent heat during the growth process is negligible. Based on these findings, we formulate the problem of dust formation coupled to the local element depletion/enrichment of the gas in brown dwarf at- mospheres by means of a system of partial differential equations. These equations state an extension of the moment method developed by Gail & Sedlmayr (1988) with an additional advective term to account for the effect of size-dependent drift ve- locities of the grains. A dimensionless analysis of the new equations reveals a hierarchy of nucleation → growth → drift → evaporation, which characterises the life cycle of dust grains in brown dwarf atmospheres. The developed moment equations can be included into hydrodynamics or classical stellar atmosphere models. Applications of this description will be presented in a forthcoming paper of this series.

/pdf/dust-in-brown-dwarfs-ii-the-coupled-problem-of-dust-4bfnzskxo6.pdf

Dust in brown dwarfs. II. The coupled problem of dust formation and sedimentation

Silicon carbide (SiC) as both the most important non-oxide ceramic and promising semiconductor material grows stoichiometric SiO 2 as its native oxide. During passive oxidation, a surface transformation of SiC into silica takes place causing bulk volume and bulk mass increase. This review summarizes state-of-the-art information about the structural aspects of silicon carbide, silica, and SiC–SiO 2 interfaces and discusses physicochemical properties and kinetics of the processes involved. A special section describes the electronic properties of carbide–oxide interfaces, which are inferior compared to Si–SiO 2 interfaces, limiting the use of SiC-based electronics. In the oxidation of SiC there is a variety of parameters (e.g., porosity, presence of sintering aids, impurities, crystallographic orientation, surface treatment, and atmospheric composition) influencing the process. Therefore, the kinetics can be complex and will be discussed in detail. Nonetheless, a general linear-parabolic time-law can be foun...

Silica on Silicon Carbide

IN 1904 there appeared a slender volume entitled “The Dynamical Theory of Gases”, in which the author, now Sir James Jeans, endeavoured to develop the theory “upon as exact a mathematical basis as possible”. It was an original and somewhat severe book, the reader being at once confronted with the difficulties of the law of distribution of molecular velocities and with statistical mechanics and the controversies and unresolved problems on the equipartition of energy. At that time the English student whose reading was confined to his own language had to choose between the somewhat arid treatises by Burbury and by Watson, or the translation of Meyer's attractive, almost chatty, but unmathematical book. Jeans' book, which was much more akin to Boltzmann's elegant “Vorlesungen uber Gastheorie”, found acceptance, despite its severity, and many of to-day's physicists, as well as mathematicians, were ’brought up on it’ so far as gastheory was concerned, and looked to it as to an oracle, sometimes difficult to fathom as is the way with oracles. An Introduction to the Kinetic Theory of Gases By Sir James Jeans. Pp. vi + 311. (Cambridge: At the University Press, 1940.) 15s. net.

S. Chapman

Papers

An Introduction to the Kinetic Theory of Gases