Connection between thermophoresis and thermodiffusion in a liquid binary mixture

In a liquid suspension, thermophoresis is the motion of a suspended particle under a temperature gradient. In a liquid binary mixture, thermodiffusion is the generation of a composition gradient upon application of a temperature gradient. A quantitative connection is established between the two phenomena without making assumptions about their mechanisms. It is shown that Galilean invariance and the choice of a Galilean reference frame play a key role in that connection. The results are not restricted to very dilute suspensions or mixtures.     

Turbulent particle transport in magnetized plasmas

Particle transport in magnetized plasmas is investigated with a fluid model of drift wave turbulence. An analytical calculation shows that magnetic field curvature and thermodiffusion drive an anomalous pinch. The curvature driven pinch velocity is consistent with the prediction of turbulence equipartition theory. The thermodiffusion flux is found to be directed inward for a small ratio of electron to ion pressure gradient, and it reverses its sign when increasing this ratio. Numerical simulations confirm that a turbulent particle pinch exists. It is mainly driven by curvature for equal ion and electron heat sources. The sign and relative weights of the curvature and thermodiffusion pinches are consistent with the analytical calculation.     

Driving forces and polymer hydrodynamics in the Soret effect

A temperature gradient induces different driving forces on the components of a mixture which translates into their segregation. We show that these driving forces constitute the physical picture behind the thermodiffusion effect, and provide an alternative expression of the Soret coefficient which can be applied to both colloidal suspensions and molecular mixtures. To verify the validity of the formalism, we quantify the related forces in an Eulerian reference frame by non-equilibrium molecular simulations. Furthermore, we present an analytical argument to show that the hydrodynamic interactions need to be accounted for to obtain the proper scaling of the thermophoretic force. This result combined with the presented expression satisfactorily explains the experimentally known size dependence of the thermodiffusion coefficient in dilute polymer solutions.     

Transport of volume in a binary liquid

In this paper, the transport of volume in a binary liquid mixture is theoretically investigated in three steps, with strong implications for the measurement of mutual diffusivities in non-dilute mixtures. In a first step, the velocity of volume transport is determined from the transport velocities of the two components and the thermodynamic relation of state of the liquid mixture in equilibrium. The role played by Galilean invariance and the choice of a rigid frame of reference for reckoning current densities is highlighted. The divergence of the volume-transport velocity field is found to involve the isothermal compressibility and the thermal expansivity of the liquid together with the spatiotemporal variations of pressure and temperature. In a second step, a linear-response relation is introduced between the interdiffusion current density and the gradient of composition; this relation phenomenologically defines the mutual diffusivity of the binary liquid in a manifestly Galilean-invariant way. In a third step, it is examined whether the practical measurement of that diffusivity in a constant-volume container entails a vanishing mass-transport or volume-transport velocity. From a singular-perturbation analysis of the hydrodynamic equations, it is shown that the mass-transport velocity vanishes in the limit of a diffusion of composition that is much slower than the diffusion of momentum. As a consequence, the volume-transport velocity does not vanish during interdiffusion even though the law of additive volumes of the components holds. The physical meaning of the non-vanishing volume velocity is interpreted by means of the thermodynamic results obtained in the second step. Some of the conclusions carry over to multicomponent liquid mixtures.     

Second law of thermodynamics in volume diffusion hydrodynamics in multicomponent gas mixtures

We presented the thermodynamic structure of a new continuum flow model for multicomponent gas mixtures. The continuum model is based on a volume diffusion concept involving specific species. It is independent of the observer?s reference frame and enables a straightforward tracking of a selected species within a mixture composed of a large number of constituents. A method to derive the second law and constitutive equations accompanying the model is presented. Using the configuration of a rotating fluid we illustrated an example of non-classical flow physics predicted by new contributions in the entropy and constitutive equations.     

Numerical methods for instability mitigation in the modeling of laser wakefield accelerators in a Lorentz-boosted frame

Modeling of laser-plasma wakefield accelerators in an optimal frame of reference [1] has been shown to produce orders of magnitude speed-up of calculations from first principles. Obtaining these speedups required mitigation of a high-frequency instability that otherwise limits effectiveness. In this paper, methods are presented which mitigated the observed instability, including an electromagnetic solver with tunable coefficients, its extension to accommodate Perfectly Matched Layers and Friedman’s damping algorithms, as well as an efficient large bandwidth digital filter. It is observed that choosing the frame of the wake as the frame of reference allows for higher levels of filtering or damping than is possible in other frames for the same accuracy. Detailed testing also revealed the existence of a singular time step at which the instability level is minimized, independently of numerical dispersion. A combination of the techniques presented in this paper prove to be very efficient at controlling the instability, allowing for efficient direct modeling of 10 GeV class laser plasma accelerator stages. The methods developed in this paper may have broader application, to other Lorentz-boosted simulations and Particle-In-Cell simulations in general.     

Thermal conduction and thermal diffusion in dilute polyatomic gas mixtures

A novel theoretical basis for the evaluation of the thermal conductivity and the thermal diffusion ratios of dilute polyatomic gas mixtures is derived within the semi-classical isotropic kinetic theory. New expressions for species diffusion coefficients, thermal diffusion coefficients, and thermal diffusion ratios are also obtained by using an expansion vector based upon the total energy of the molecules. Finally, practical and accurate expressions for the thermal conductivity and the thermal diffusion ratios are derived by using the recent theory of iterative transport algorithms, as developed by the authors. The resulting expressions can be used in either theoretical calculations or computational models of multicomponent flows.     

Mechano-chemistry; diffusion in multicomponent compressible mixtures

In the present work we derive the volume continuity equation and demonstrate its use to define the volume frame of reference in the multicomponent, compressible systems. The volume velocity (material velocity) is a unique frame of reference for all internal forces and processes, e.g., the mass diffusion, heat flow, etc. No basic changes are required in the foundations of linear irreversible thermodynamics except recognizing the need to add volume to the usual list of extensive physical properties undergoing transport in every continuum. The volume fixed frame of reference allows the translation of the Newton’s discrete mass-point molecular mechanics into continuum mechanics and the use of the Cauchy linear momentum equation of fluid mechanics and Navier-Lamé equation of mechanics of solids.Our proposed modifications of Navier-Lamé and energy conservation equations are self-consistent with the literature for solid-phase continua dating back to the classical interdiffusion experiments of Kirkendall and their subsequent interpretation by Darken in terms of diffuse volume transport. We do show that the local diffusion processes do not change the centre of mass of the system and that the stress and viscosity depend only on the local volume velocity.     

Enskog-Landau kinetic equation for multicomponent mixture. Analytical calculation of transport coefficients

The Enskog-Landau kinetic equation is considered to describe non-equilibrium processes of a mixture of charged hard spheres. This equation has been obtained in our previous papers by means of the non-equilibrium statistical operator method. The normal solution of this kinetic equation found in the first approximation using the standard Chapman-Enskog method is given. On the basis of the found solution the flows and transport coefficients have been calculated. All transport coefficients for multicomponent mixture of spherical Coulomb particles are presented analytically for the first time. Numerical calculations of thermal conductivity and thermal diffusion coefficient are performed for some specific mixtures of noble gases of high density. We compare the calculations with those ones for point-like neutral and charged particles. Received 10 June 1999 and Received in final form 15 October 1999     

Thermal diffusion in disperse systems

This paper discusses a theory for a new effect, the migration of solid dispersed particles initiated by a nonuniform temperature field. The reason for the motion is the inhomogeneity of the properties of a thin protective layer around a particle. The example of ionic dispersion shows that the sign of the coefficient of thermodiffusion depends on the magnitude of the electrostatic potential at the particle surface and the thickness of the Debye layer and that the coefficientis larger than the values known for molecular systems by a factor of 100 to 10000. In contrast to molecular systems, in disperse systems thermodiffusion should play a much more important role. Zh. éksp. Teor. Fiz. 115, 1721–1726 (May 1999)     

MAXWELL-Theorie (in Medien) und Quantentheorie in einem rotierenden Bezugssystem

On the basis of our 3-dimensional conceptions of the electromagnetic quantities [1] the MAXWELL theory is represented in a rotating frame of reference. For such a frame the MAXWELL equations obtain additional terms depending on the angular velocity (analogous to the CORIOLIS term etc. in NEWTON ian mechanics). Using these results with the help of the DIRAC equation we investigate quantum mechanics in a rotating frame of reference. Also in this case we obtain interesting additional terms depending on the angular velocity. The need for such a theory is obvious for many reasons: 1. Our Earth is such a rotating laboratory. Later the rotation effects must not be neglected. 2. In magnetic resonance physics theoretical questions with respect to rotating systems are important. 3. Interesting statements result for a rotational JOSEPHSON effect of superconductors. — All our calculations are carried out up to the first order in the angular velocity.     

The use of a non-spherical reference potential in statistical mechanical perturbation theory

A statistical mechanical perturbation theory for the pair correlation function and thermodynamic properties of molecular fluids is presented in which the reference potential function is non-spherical. With this choice the short-range molecular repulsive forces can be properly taken into consideration and attractive forces, such as those resulting from electric moments, treated as the perturbation. Calculations are presented for the first-order perturbation term to the Helmholtz free energy due to quadrupolar forces in models of liquid nitrogen and chlorine, and due to dipolar forces in liquid hydrogen chloride. For these calculations the rigid diatomic model and its modification appropriate to heteronuclear molecules were used for the reference potentials. It is found that the lowest-order perturbation terms here are proportional to the second power of the dipole or quadrupole moments, and not the fourth power as had been found previously using a spherical reference potential function. This second-order dependence on the electric moment is especially important in the case of mixtures, where it leads to an explanation for the occurrence of negative azeotropes in binary mixtures of species with quadrupole moments of opposite sign.     

Precise determination of the Soret,thermodiffusion and isothermal diffusion coefficients of binary mixtures of dodecane,isobutylbenzene and 1,2,3,4-tetrahydronaphthalene (contribution of the University of Mons to the benchmark test)

Within the framework of an international benchmark test, the Soret coefficient S T , the thermodiffusion coefficient D T , and the isothermal mass diffusion coefficient D of the three binary systems formed with dodecane, isobutylbenzene and 1,2,3,4-tetrahydronaphthalene (with a mass fraction of 0.5 in each component at a temperature of 25C) have been measured. Convective coupling in thermogravitational columns has been used to determine D T based on the Furry-Jones-Onsager theory and the so-called open-ended capillary technique for D . The ratio of D T to D , obtained in these two independent experiments, provides the Soret coefficient. In one case, we were able to use laser Doppler velocimetry to measure the modifications of the velocity amplitude due to thermodiffusion, which is also a way to obtain the Soret coefficient. Our results for these three coefficients are in good agreement with those obtained by other benchmark tests.     

Linear time domain model of the acoustic potential field

A new time domain formulation of the acoustic wave is developed to avoid approximating assumptions of the linearized scalar wave equation that limit its validity to low Mach particle velocity modeling or to a smooth potential field in a stationary medium. The proposed model offers precision of the moving frame while retaining the form of the widely used linearized scalar wave equation although with respect to modified coordinates. It is applicable to field calculations involving transient waves with unlimited particle velocity, propagating in inhomogenous fluids or in those with time varying density. The model is based on the exact flux continuity equation and the equation of motion, both using the moving reference frame. The resulting closed-form free space scalar wave equation employing total derivatives is converted back to the partial differential form by using modified independent variables. The modified variables are related to the common coordinates of space and time following integral expressions involving transient particle velocity representing wave radiated by each point of a stationary source. Consequently, transient field produced by complex surface velocity sources can be calculated following existing surface integrals of the radiation theory although using modified coordinates. The use of the proposed model is presented in a numerical simulation of a transient velocity source vibrating at selected magnitudes, leading to the determination of the propagating pressure and velocity wave at any point.     

Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope

This paper discusses the influence of an endoscope on the peristaltic flow of a couple stress fluid in an annulus under a zero Reynolds number and long wavelength approximation. The inner tube is uniform, rigid, while the outer tube has a sinusoidal wave traveling down its wall. Analytical expressions for the axial velocity, stream function and axial pressure gradient are established. The flow is investigated in a wave frame of reference moving with the velocity of the wave. Numerical calculations are carried out for the pressure rise, frictional forces and trapping. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail.     

Evaluating virial coefficients for multicomponent mixtures: hard sphere mixtures and flexible chains

A new algorithm to compute the virial coefficients of multicomponent mixtures is proposed. The number of graphs that must be evaluated increases dramatically in a multicomponent mixture so that it becomes difficult to enumerate and compute all possible graphs. However, once all of them are known and evaluated, the virial coefficient of the mixture can be evaluated for any composition. If one is interested in the virial coefficient of a mixture of a certain composition, then a simpler approach can be followed. Starting from the graphs of a pure fluid, we assign a random chemical identity to each of the molecules of the graph. The probability of assigning a given chemical identity is taken from the composition of the mixture. In this way composition is treated as a random variable within the Monte Carlo procedure which determines the virial coefficient. The algorithm is checked by comparison with the virial coefficients of binary hard spheres mixtures which are well known. Good agreement is found. The procedure is then extended to multicomponent mixtures of hard spheres. Finally the procedure is applied to the determination of the virial coefficients of a flexible molecule. For flexible molecules the possible configurations of the molecules are treated as different components of the mixture. In this way we present what appears to be the first determination of the third and fourth virial coefficients of polymers in the continuum.     

The thermodynamic characteristics of multicomponent reaction mixtures in the sub- and supercritical states

The paper presents mathematical models and calculation methods for solving particular research problems related to the thermodynamic characteristics of multicomponent and multiphase mixtures. The special features of chemical and phase equilibria in such mixtures are considered in the ideal gas approximation and taking nonideality into account. The conditions of equilibrium phase stability are studied for multiphase systems. The results of calculations of characteristic phase diagrams and binodal and spinodal are given for model systems with a fixed chemical composition, and a new interpretation of the mathematical model for localizing the critical point of a multicomponent mixture with a given composition is presented. A new interpretation of the well-known classic homotopy method is suggested for solving complex nonlinear systems of equations. Some anomalies of phase portraits and critical curves that are necessary to take into account in selecting (planning) experimental conditions and calculating chemical processes and reaction parameters are considered separately. The possibility of calculating thermodynamic and thermophysical properties (entropy, enthalpy, heat capacity, heat effects of reactions, and adiabatic heating) is demonstrated for the example of particular multicomponent nonideal mixtures. The conclusion is drawn that cubic equations of state can be used for predicting the deviations of these properties from the ideal gas state and their anomalies in the vicinity of the critical points of mixtures.     

Thermodiffusion and vaporization of metal from levitated droplet IV. relations for experimental data analysis and their application for evaporation of iron

In the present paper the mathematical relations for analysing experimental results obtained at levitated droplet vaporization were derived. The calculations resulted in a system of nonlinear equations where the unknown quantities characterize condensation rate, thermodiffusion and their temperature dependences. The numerical solution was performed for vaporization of iron in the flow of argon. The thermodiffusion and vapour condensation increase the vaporization rate 5.4-times at 2200 K. The thermodiffusion ratio is negative in order 10⁻³ and its temperature dependence is important. The temperature dependence of vapour condensation was determined with the accuracy of 2%, the rate constant of condensation was determined approximately as a consequence of experimental data inaccuracy. Theory. Numerical calculations.     

Discrete Velocity Models for Polyatomic Molecules Without Nonphysical Collision Invariants

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. Unlike for the Boltzmann equation, for DVMs there can appear extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and hence, without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species, but also for binary mixtures and recently extensively for multicomponent mixtures. In this paper, we address ways of constructing normal DVMs for polyatomic molecules (here represented by that each molecule has an internal energy, to account for non-translational energies, which can change during collisions), under the assumption that the set of allowed internal energies are finite. We present general algorithms for constructing such models, but we also give concrete examples of such constructions. This approach can also be combined with similar constructions of multicomponent mixtures to obtain multicomponent mixtures with polyatomic molecules, which is also briefly outlined. Then also, chemical reactions can be added.