ALBERT

Notes: The phenomenon of aerodynamic enrichment of heavy molecules seeded in supersonic free jets has been known since 1955. But its systematic exploitation in the generation of intensely focused molecular beams has been prevented by the lack of a quantitative and realistic explanation of the observed facts. Here, the aerodynamic focusing of CBr4, W(CO)6, and C2Cl6 molecules seeded in jets of He or H2 is studied experimentally, and found to be most singular under conditions similar to those known to produce sharply focused beams of microscopic spheres suspended in air jets. The gas mixture expands through thin-plate orifice into a vacuum chamber, forming a supersonic free jet. The spacial distribution of the heavy molecules in the jet is measured at varying distances L to the nozzle by scanning a thermocouple probe across a jet diameter. The probe is sufficiently small to interfere negligibly with the flow. The increment DV in the thermocouple voltage resulting from seeding the heavy gas on a given flow of He or H2 is seen to be a sensitive indicator of the local concentration of seed molecules in the jet. The following behavior is observed in terms of the same Stokes number or inertia parameter S that governs the simpler and better understood phenomenon of aerosol focusing. Below S=0.4 for H2 and S=0.2 for He, heavy molecule and aerosol beam widths are practically identical, and the boundary of the jet of heavy molecules is rather sharp. At higher values of S, aerosol beams show further reductionsin cross section, down to less than 10% of a nozzle throat diameter dn. In contrast, the measured heavy species minimal beam widths or waists at a distance L∼dn from the throat are around 0.5dn and 0.35 dn for jets of He and H2, respectively. In units of dn, these widths are several times larger than expected from elementary considerations on the defocusing effects due to Brownian motion (of the order of the square root of the molecular mass ratio between light and heavy molecules). Nonetheless, the thin-plate orifice nozzle yields considerably more concentrated jets of heavy gases than previously seen, with far-field enrichment factors for the seed species close to 50 in thecase of H2 jets. This technique, thus, appears to provide a greatly improved source for intense molecular beams. Aerodynamically focused beams have a sharp distribution of kinetic energies, being ideally suited for cross beam and beam surface studies. But they are not quite so optimal for spectroscopic studies because they require moderate source Reynolds numbers (of order 100), at which the heavy gas undergoes very little translational, rotational, or vibrational cooling.

Notes: Approximate scaling laws for the charge and size of the drops ejected from the apex of Taylor cones run in the cone-jet mode (electrospray) are now available for highly conducting electrolytes (10−4 S/m〈K〈1 S/m) electrosprayed at atmospheric pressure. In order to confirm that such laws do also apply to Taylor cones in vacuo, the current versus liquid flow rate curves I(Q) characteristic of a given liquid are investigated both in vacuum and in atmospheric pressure air. Although the sprays of drops differ profoundly in both cases, the two corresponding I(Q) curves are nearly identical for relatively involatile liquids such as tributyl phosphate. A discussion on the possible relation between the behavior of Taylor cones of electrolytes of organic liquids and liquid metal ion sources (K∼106 S/m) is attempted, yielding insights on the role of space charge. However, the electrical conductivity variable which dominates the behavior of liquid cones of electrolytes appears to be irrelevant in liquid metals. © 1998 American Institute of Physics.

Notes: The Boltzmann equations for a binary mixture of gases are considered in the asymptotic limit when their molecular weight ratio and the light gas Knudsen number are small quantities. A first mass-ratio expansion reduces the cross-collision operator of the light gas Boltzmann equation to a Lorentz form, uncoupling its kinetic behavior from that of the heavy gas. The light gas distribution function is then determined to first order in the Knudsen number, independently of the degree of nonequilibrium characterizing the heavy gas, whose influence is felt only through its hydrodynamic quantities. All transport coefficients arising are determined variationally for arbitrary interaction potentials using Sonine polynomial expansions as trial functions. A remarkable feature of this analysis is that it yields binary transport information (i.e., diffusion and thermal diffusion coefficients) from considering only the Boltzmann equation for the light gas. A second mass expansion reduces the cross-collision operator of the heavy gas equation to a Fokker–Planck form. The corresponding coefficients involve integrals over the light gas distribution function determined previously and are evaluated explicitly in terms of the hydrodynamic quantities and transport coefficients of the light gas. The heavy gas distribution function can be determined by solving a Fokker–Planck equation at dilutions large enough to make heavy–heavy collisions negligible, or by a new Knudsen number expansion when the molar fraction of the heavy gas is of order 1. In this latter case, the heavy gas kinetic behavior is independent of the light gas, being characterized by the same transport coefficients of the pure heavy gas. The problem is then reduced to a set of two-fluid hydrodynamic equations.

Notes: The phenomenon of ion evaporation from charged liquid surfaces is at the basis of electrospray ionization, a source of a stunning variety of gas phase ions. It is studied here by producing a monodisperse cloud of charged droplets and measuring the charge q and diameter dr of the residue particles left after complete evaporation of the solvent. When the droplets contain small monovalent dissolved ions, the electric field E on the surface of their solid residues is found to be independent of dr. One can thus argue that the source of small ions in electrospray ionization is field-emission, and not other proposed mechanisms such as Dole's charged residue model. A consequence of the observed independence of E on dr is that the rate of ion ejection is simply related to the rate of solvent evaporation, estimated here as that for a clean surface of pure solvent.The reduction G(E) brought about by the electric field E in the activation energy for ion evaporation has thus been inferred as a function of the measured field E in the range 1.5〈E(V/nm)〈3.25. It agrees surprisingly well with the so-called Schottky hump from the image potential model (IPM), GIPM=(e3E/4πε0)1/2. This remarkably simple result is paradoxical in view of two major objections raised earlier against the use of the IPM for ion evaporation from liquids. However, the correct mechanism (first introduced by Iribarne and Thomson) leading to an attractive interaction between the liquid surface and the escaping ion is not the creation of an image charge, but the polarization of the dielectric liquid by the ion. In the limit of a large dielectric constant ε(very-much-greater-than)1, the image force and the polarization force coincide numerically, though the later sets in much faster and is apparently free from the paradox raised by Röllgen. Also, the dielectric nature of the liquid and its strong screening of the net charges near its surface resolves another paradox raised by Fenn regarding the discrete distribution of charges. This screening also introduces a correction in the model proposed by Iribarne and Thomson for G(E), making its predictions virtually indistinguishable from those of GIPM(E). In conclusion, small ions observed in electrospray ionization are produced by field-emission. Measured ionization rates are well represented by results from a "polarization potential model'' which appears to be physically sound. These predictions coincide with those from the IPM in the limit ε(very-much-greater-than)1, the only case studied so far. © 1995 American Institute of Physics.

Notes: When concentrated solutions of NaI in formamide with electrical conductivities K larger than 1.1 S/m are electrosprayed from a Taylor cone-jet in a vacuum, ions are evaporated at substantial rates from the surface of the meniscus and the drops. This constitutes a new source of ions and nanoparticles, where the relative importance of these two contributions is adjustable. The currents of ions are measured independently from those associated with drops by a combination of stopping voltage analysis and preferential scattering in a gas background. The magnitude E of the electric field at the surface of the drops and at the apex of the cone-jet is controlled through the electrical conductivity K of the liquid and its flow rate Q through the jet. E is related through available scaling laws for Taylor cone-jets to the ratios K/Q or I/Q, where I is the current of drops emitted by the jet. Ion currents are very small or null at typical K/Q values used in the past. A relatively small initial ion current is attributed to a few particularly sharp features present, perhaps associated with small satellite drops. At still higher K/Q this first ionization source saturates, and ion evaporation from the main drops begins to dominate (E∼1 V/nm). E can then be determined with little ambiguity, and the associated ion current is also measured over a broad enough range of electric fields to determine the ionization kinetics. At still higher K/Q the ion current from the drops approaches saturation, and ion evaporation directly from the meniscus becomes dominant. The total spray current then presents the anomaly of increasing rapidly at decreasing liquid flow rate. The ion current from the meniscus can also be measured in this regime over a broad range of K/Q, with qualitative agreement with the ionization measurements from the drops. But the relation established between K/Q and E becomes suspect because ion and drop currents are now comparable. A third approach to infer the ionization rate is based on the related disappearance of Coulomb explosions of the drops above a critical K/Q. These results are congruent with the model of Iribarne and Thomson, with an activation barrier for ion evaporation equal to 1.7 eV−(e3E/4πε0)1/2. © 2000 American Institute of Physics.

Notes: As a result of the increasing inefficiency in the transfer of energy in collisions between species with a decreasing ratio of molecular masses, the Knudsen number range of validity of the Chapman–Enskog (CE) theory for binary gas mixtures decreases linearly with the molecular mass ratio. To remedy the situation, a two-fluid CE theory uniformly valid in the molecular mass ratio is constructed here. The analysis extends previous two-fluid theories to allow for arbitrary potentials of intermolecular interaction and arbitrary mass ratios. The treatment differs from the CE formulation in that the mean velocities and temperatures of the two gases are not required to be identical to lowest order. To first order, the streaming terms of the Boltzmann equation are thus computed in terms of the derivatives of the two-fluid hydrodynamic quantities, rather than the mean mixture properties as in the CE theory. As a result, associated with the nonconservation of momentum and energy for each species alone, two new "driving forces'' appear in the first-order integral equations. The amount of momentum and energy transferred per unit time between the species appear in the theory as free constants, which allow satisfying the constraint that all hydrodynamic information be contained within the lowest-order two-fluid Maxwellians. Simultaneously, this constraint fixes the rate of momentum and energy interchange in terms of the two-fluid hydrodynamic quantities and their gradients. The driving force d12 of the CE theory is directly related to the rate of interspecies momentum transfer, and the corresponding CE functions D1 and D2 appear here unmodified.But the physical interpretation of d12 is very different in the two pictures. On the CE side there is only one momentum equation, while d12 provides constitutive information fixing the diffusion flux (velocity differences) in the mass conservation equation. Here, the similar constitutive information associated to d12 is used to couple two different momentum equations. Although the CE theory captures some of the two-velocity aspects of the problem, no CE analog exists with the functions E1 and E2 associated here with temperature differences, which now require solving new integral equations. Finally, the presence of two velocities and two temperatures leads to four coefficients of viscosity and of thermal conductivity for the two stress tensors and heat flux vectors. Also, two thermal diffusion factors enter now into the expression for d12. Although all these new coefficients arise as portions of the overall CE transport coefficients, their independent optimal determination requires new developments. The corresponding variational formulation is presented here and used to first order to obtain explicit expressions for all two-fluid transport coefficients by means of Sonine polynomials as trial functions.

Notes: The collision integrals describing the rate of exchange of momentum and tensorial energy between the components in a binary mixture of neutral gases with very different atomic masses are determined for arbitrary values of their two temperatures and velocities, for realistic intermolecular potentials, and allowing for large departures of the heavy gas from equilibrium conditions. In the range of interest where the system is perturbed within times of the order of the slow relaxation time characterizing the transfer of energy between unlike molecules, the light gas distribution function is Maxwellian to lowest order, with corrections given asymptotically in powers of the small parameter m/mp formed with the ratio of the species molecular masses. Also, provided that the ratio Tp/T between the temperatures of the two gases remains much smaller than mp/m, the desired collision integrals may be evaluated asymptotically in powers of m/mp in all generality. The computation is carried out in detail for the case when the interaction between atoms is described by a Lennard–Jones potential. A combination of numerical computations with optimal matching of analytical expressions valid for large and small slip velocities leads to a set of compact formulas which hold for the limits of high and low temperatures and to a general approximate expression for all temperatures.

Notes: Abstract An eigenexpansion solution of the time-independent Brownian motion Fokker-Planck equation is given for a situation in which the external acceleration is a step function. The solution describes the heavy-species velocity distribution function in a binary mixture undergoing a shock wave, in the limit of high dilution of the heavy species and negligible width of the light-gas internal shock. The diffusion solution is part of the eigenexpansion. The coefficients of the series of eigenfunctions are obtained analytically with transcendentally small errors of order exp(−1/M), whereM ≪ 1 is the mass ratio. Comparison is made with results from a hypersonic approximation.

Notes: Summary Necessary improvements to a recent paper on the flow of a dusty gas by Datta and Mishra are presented: particularly dealing with the importance of particle phase compressibility, and the hyperbolic nature of the particle momentum conservation equation which prohibits downstream (wall) boundary conditions for the solid phase. Basic differences between particulate and ordinary flow boundary layers are discussed and the correct conservation equations are written.