Supersonic Beams at High Particle Densities: Model Description beyond the Ideal Gas Approximation † Wolfgang Christen,* ,† Klaus Rademann, † and Uzi Even ‡ Institut fu ¨r Chemie, Humboldt-UniVersita ¨t zu Berlin, 12489 Berlin, Germany, and Sackler School of Chemistry, Tel AViV UniVersity, 69978 Tel AViV, Israel ReceiVed: March 30, 2010; ReVised Manuscript ReceiVed: July 8, 2010 Supersonic molecular beams constitute a very powerful technique in modern chemical physics. They offer several unique features such as a directed, collision-free flow of particles, very high luminosity, and an unsurpassed strong adiabatic cooling during the jet expansion. While it is generally recognized that their maximum flow velocity depends on the molecular weight and the temperature of the working fluid in the stagnation reservoir, not a lot is known on the effects of elevated particle densities. Frequently, the characteristics of supersonic beams are treated in diverse approximations of an ideal gas expansion. In these simplified model descriptions, the real gas character of fluid systems is ignored, although particle associations are responsible for fundamental processes such as the formation of clusters, both in the reservoir at increased densities and during the jet expansion. In this contribution, the various assumptions of ideal gas treatments of supersonic beams and their shortcomings are reviewed. It is shown in detail that a straightforward thermodynamic approach considering the initial and final enthalpy is capable of characterizing the terminal mean beam velocity, even at the liquid-vapor phase boundary and the critical point. Fluid properties are obtained using the most accurate equations of state available at present. This procedure provides the opportunity to naturally include the dramatic effects of nonideal gas behavior for a large variety of fluid systems. Besides the prediction of the terminal flow velocity, thermodynamic models of isentropic jet expansions permit an estimate of the upper limit of the beam temperature and the amount of condensation in the beam. These descriptions can even be extended to include spinodal decomposition processes, thus providing a generally applicable tool for investigating the two-phase region of high supersaturations not easily accessible otherwise. Introduction On the basis of the experimental and theoretical foundations of molecular flow and effusion laid out by Knudsen a hundred years ago, 1,2 studies of gases at low pressure have led to the first observation of collision-free streams of molecules by Dunoyer in 1911. 3 The advancement and further development of these atomic and molecular beams to an extremely powerful experimental technique are primarily due to the dedication of Stern, 4-7 who was awarded the Noble Prize in physics in 1943 in part for his seminal contributions to this molecular ray method. Other milestones in this context were the theoretical design study of a supersonic beam source by Kantrowitz and Grey, 8 the experimental realization of supersonic nozzle jet expansions by Becker and Bier, 9 and not least the invention of pulsed jet expansions by Hagena. 10 Quite a substantial number of disciplines in chemistry, physics, and engineering could emerge and progress only due the availability of an intense, collimated flow of collision-free particles with a narrow velocity distribution. Until today supersonic molecular beams continue to constitute an extremely versatile and rather popular experi- mental tool in science and technology. 11-14 They are of prime importance in both basic and applied research fields such as analytical chemistry, cluster science, heterogeneous catalysis, optical spectroscopy, quantum physics, surface science, and thin film growth. Focusing on the applications, more fundamental aspects of the jet expansion of compressible fluids have not been investigated in great detail. Although the dynamic evolution of a supersonic flow is determined by the conservation of energy, mass, and momentum, these conservation laws need to be supplemented with an appropriate characterization of the material properties of the working fluid via a suitable equation of state (EOS). In the past, several attempts have been made to include real fluid properties in a description of the nozzle flow, e.g., by a consideration of the heat of condensation or by employing a van der Waals or Redlich-Kwong EOS. 15-26 Still, in almost all publications of supersonic beams merely the isentropic expansion of an ideal gas into vacuum is considered. While the advantage of this approximation is simplicity, the ideal gas model description fails at elevated particle densities, i.e., in both low temperature and high pressure applications. This failure has been impressively confirmed in several intrigu- ing observations of velocity distributions of supersonic jets of helium 27-30 and hydrogen 31 at cryogenic conditions, and of carbon dioxide at supercritical conditions. 32,33 But even for rare gases at room temperature and modest stagnation pressures of 400-600 kPa, the mean flow velocity deviates from the ideal gas prediction, as has been demonstrated recently utilizing highly accurate time-of-flight measurements of supersonic beams of tagged metastable argon atoms. 34 Likewise, predictions and limitations of the classical nucleation theory for cluster formation have been addressed in numerous calculations and experiments. † Part of the “Klaus Mu ¨ller-Dethlefs Festschrift”. * To whom correspondence should be addressed, christen@wolfgang- christen.net. † Humboldt-Universita ¨t zu Berlin. ‡ Tel Aviv University. J. Phys. Chem. A 2010, 114, 11189–11201 11189 10.1021/jp102855m  2010 American Chemical Society Published on Web 07/26/2010