Gas Rarefaction Research Articles

Rarefaction windows in a high-power impulse magnetron sputtering plasma

The velocity distribution function of the sputtered particles in the direction parallel to the planar magnetron cathode is studied by spatially- and time-resolved laser-induced fluorescence spectroscopy in a short-duration (20 μs) high-power impulse magnetron sputtering discharge. The experimental evidence for the neutral and ionized sputtered particles to have a constant (saturated) velocity at the end of the plasma on-time is demonstrated. The velocity component parallel to the target surface reaches the values of about 5 km/s for Ti atoms and ions, which is higher that the values typically measured in the direct current sputtering discharges before. The results point out on the presence of a strong gas rarefaction significantly reducing the sputtered particles energy dissipation during a certain time interval at the end of the plasma pulse, referred to as “rarefaction window” in this work. The obtained results agree with and essentially clarify the dynamics of HiPIMS discharge studied during the plasma off-time previously in the work: N. Britun, Appl. Phys. Lett. 99, 131504 (2011).

Modified high power impulse magnetron sputtering process for increased deposition rate of titanium

A modified version of high power impulse magnetron sputtering (HiPIMS) has been used to deposit titanium films at higher deposition rates than for conventional HiPIMS while maintaining similar pulse voltages and peak currents. In the present study, additional control parameters are explored through the chopping of the HiPIMS pulse into a pulse sequence. Experiments show that the use of sequences allows for an increase of the deposition rate of more than 45% compared to conventional HiPIMS. The increase in deposition rate is ascribed to a combination of reduced gas rarefaction effects, prevention of sustained self-sputtering, and a relaxation of ion trapping.

Simulation of the transient heat transfer between two coaxial cylinders

The transient heat flux between two coaxial cylinders is studied on the basis of the numerical solution of the nonlinear unsteady S-model kinetic equation. A large range of the gas rarefaction and two temperature ratios between the cylinders' walls are considered. The time evolutions of the averaged over the distance between the cylinders heat flux, pressure, and bulk velocity are analyzed. The steady state time, as a time of steady state flow establishment, is introduced. This time depends on the characteristic time of the system and on the gas rarefaction. It is found that this steady state time for the averaged heat flux varies approximately 18 times from the near free molecular to the hydrodynamic flow regime. In the slip flow regime, the analytical expression for the steady state temperature distribution between two cylinders is obtained. It is found that the transient heat flux evolution derived from the energy balance allows to estimate the time of steady state flow establishment in the slip flow regime.

General approach to transient flows of rarefied gases through long capillaries

An approach to model a transient flow of rarefied gas through a long capillary based on numerical data for flow rate previously obtained from the linearized and stationary kinetic equation was proposed. As an example, non-steady flow through a long circular tube connecting two reservoirs is considered. Pressures in the reservoirs and flow rates in the inlet and outlet are obtained as a function of time. It is shown that the behaviors of these quantities vary by changing the gas rarefaction from the free-molecular regime to hydrodynamic one. The typical time to reach the equilibrium state is calculated.

Unsteady rarefied gas flow through a slit

A transient gas flow through a rectangular slit is simulated on the basis of unsteady S-model kinetic equation. The flow into vacuum and at two pressure ratios between the reservoirs are considered for a wide range of the rarefaction parameter lying from the free molecular to the near continuum regime. The time of steady flow establishment is obtained and it is found that this time depends on the pressure ratio between the reservoirs and on the gas rarefaction. The time evolution of the macroscopic flow parameters is analyzed in detail. The present results are compared with the results on the orifice flow previously obtained by other authors.

Time-dependent experimental analysis of a thermal transpiration rarefied gas flow

Thermal transpiration is the macroscopic movement induced in a rarefied gas by a temperature gradient. The gas moves from the lower to the higher temperature zone. An original method is proposed here to measure the stationary mass flow rate of gas created by thermal transpiration in a micro-tube heated at its outlet. In addition, by means of a time-dependent study, parameters such as the pressure variation, the pressure variation speed, and the characteristic time of the system are analyzed. The experimental system is composed of a glass tube of circular cross section and two reservoirs positioned one at the inlet and one at the outlet of the capillary. The reservoirs are connected to two fast response time capacitance diaphragm gauges. By monitoring the pressure variation with time inside both reservoirs, it is possible to measure the macroscopic movement of the gas along the tube. Three gases, nitrogen, argon, and helium, are studied and three temperature differences ΔT = 37, 53.5, and 71 K are applied to the tube. The analyzed gas rarefaction conditions vary from near free molecular to slip regime. Finally, Poiseuille counter flows consistent with the experimental zero flow conditions of the thermal transpiration process are proved to be possible.

On sheath energization and Ohmic heating in sputtering magnetrons

In most models of sputtering magnetrons, the mechanism for energizing the electrons in the discharge is assumed to be sheath energization. In this process, secondary emitted electrons from the cathode surface are accelerated across the cathode sheath into the plasma, where they either ionize directly or transfer energy to the local lower energy electron population that subsequently ionizes the gas. In this work, we present new modeling results in support of an alternative electron energization mechanism. A model is experimentally constrained, by a fitting procedure, to match a set of experimental data taken over a large range in discharge powers in a high-power impulse magnetron sputtering (HiPIMS) device. When the model is matched to real data in this way, one finding is that the discharge can run with high power and large gas rarefaction without involving the mechanism of secondary electron emission by twice-ionized sputtered metal. The reason for this is that direct Ohmic heating of the plasma electrons is found to dominate over sheath energization by typically an order of magnitude. This holds from low power densities, as typical for dc magnetrons, to so high powers that the discharge is close to self-sputtering, i.e. dominated by the ionized vapor of the sputtered gas. The location of Ohmic heating is identified as an extended presheath with a potential drop of typically 100–150 V. Such a feature, here indirectly derived from modeling, is in agreement with probe measurements of the potential profiles in other HiPIMS experiments, as well as in conventional dc magnetrons.

Gaseous mixtures in vacuum systems and microfluidics

In vacuum technology, one deals with gaseous mixtures more frequently than with a single gas, but the information about transport phenomena in mixtures published in the open literature is very poor. Moreover, methods to modelmixture flows are more complicated than those for single gas. The aim of this work is to review general approaches to modeling mass, heat, and momentum transfer through gaseous mixtures over the whole range of gas rarefaction. This review is written in an easy, accessible manner avoiding hard mathematical derivations, though an extensive list of references is provided for readers wishing to find more details about the field. Results for some classical problems such as velocity slip and temperature jump coefficients, Poiseuille flow,Couette flow, and heat transfer for gaseous mixtures are presented in graphical form. A comparison of these results with those corresponding to a single gas is presented, which shows the peculiarities of the transport phenomena in mixtures and gives us an idea how to model mixture flows in vacuum systems and microfluidics.

Flows of rarefied gaseous mixtures in networks of long channels

A method is presented to calculate the steady flows of rarefied gaseous mixtures in networks of long channels. The approach is based on the kinetic level. First, the McCormack linearized kinetic model is solved to obtain the local flow properties in the channels in a wide range of gaseous rarefaction and mole fraction. Second, the global flow properties including the flow rates and the distribution of the pressure and the mole fraction are deduced. An integral equation is introduced in order to determine the flow rates as functions of the differences of the partial pressures between the two ends of each channel. The conservation of mass at the nodes of the network results into a system of linear algebraic equations. The overall mathematical problem is solved iteratively. Pressure driven flows of He/Xe and He/Ar through an example network of circular tubes are calculated at intermediate values of the gaseous rarefaction. The results of the flow rates and the pressures and the mole fractions at the nodes in the whole system and the representative distributions of the pressure and the mole fraction along the channels are presented and commented on.

Acceleration Profile of a Flat Flyer Driven by Detonation Isentrope

AbstractIn the design of explosive devices, understanding of the behavior of explosively propelled matter is one of the important steps to optimize the performance of the device. In a typical flat, metallic flyer and explosives charge system, the flyer reaches its maximum velocity after a certain degree of expansion of the detonation gas. During this expansion, the flyer is deformed in an arced‐shape by the incoming rarefaction from nearby surfaces. In this work, an acceleration/deformation profile of an explosively propelled flat, metallic plate was studied based on the isentropic expansion of detonation gas and subsequent rarefaction intrusion to the center of the flyer. In order to properly describe the arced deformation of the flyer, a rather simplified new term of the pressure release ratio behind the flyer η is introduced based on the expansion of the detonation isentrope behind the flyer. A theoretical model was built to predict the behavior of an explosively driven flyer and the rarefaction intrusion into the center of the explosives charge. The results are compared to a hydrocode simulation and exhibit favorable agreement in a limited application.

End corrections for rarefied gas flows through capillaries of finite length

The influence of end effects on rarefied gas flows through moderately long capillaries is investigated by implementing the concept of effective capillary length, representing a sum of the real length of capillary and its increment. To calculate the length increment, a flow field near the inlet cross section of capillary is analyzed on the basis of the linearized kinetic equation. Far from the inlet inside of capillary, the solution is matched at the level of the distribution function with the one-dimensional solution corresponding to the flow in an infinite capillary. Far from the inlet outside of capillary, the gas is assumed to be in equilibrium at a specific pressure and temperature. The capillary length increment has been calculated as a function of the gas rarefaction. Using these results it is possible to estimate a flow rate through a moderately long capillary without hard calculations for the complete geometry.

Time-resolved measurement of film growth during high-power pulsed magnetron sputtering (HIPIMS) of titanium

The growth rate during high-power pulsed magnetron sputtering (HIPIMS) of titanium is measured with a temporal resolution of up to 25 µs using a rotating shutter concept. According to that concept a 200 µm slit is rotated in front of the substrate synchronous with the HIPIMS pulses. Thereby, the growth flux is laterally distributed over the substrate. By measuring the resulting deposition profile with profilometry, the temporal variation of the growth flux per pulse is deduced. The time-resolved growth rates are measured for 0.25, 0.5 and 1 Pa with pulse lengths of 50, 200 and 400 µs for an average power of 100 W. We can clearly identify, the individual phases of a HIPIMS pulse consisting of ignition, current rise, gas rarefaction, plateau/self-sputtering, and afterglow as described in the literature. In addition, the maximum film growth is only reached after gas rarefaction, indicating a dynamic change in local transport properties. After the end of the HIPIMS pulse, the growth rate decays following two time constants of 100 µs and of ∼ms, respectively. The first is consistent with the decay of the ion flux in the afterglow; the second with a decay of reactive neutrals. The absolute comparison of growth rates indicates that a reduction of the efficiency to 30% for very short pulses is typical for a true HIPIMS plasma.

Benchmark problems for mixtures of rarefied gases. I. Couette flow

The planar Couette flow for gaseous mixture He–Ar is calculated by the direct simulation Monte Carlo method based on ab initio potential over the whole range of the gas rarefaction for several values of the mole fraction and for two values of the wall speed. The smaller value of the speed corresponds to the limit when the nonlinear terms are negligible, while the larger value describes a nonlinear flow. The shear stress, velocity gradient, temperature, and mole fraction profiles are presented. The reported results can be used as benchmark data to test model kinetic equations for gaseous mixtures. To study the influence of the intermolecular potential, the same simulations are carried out for the hard sphere molecular model. A relative deviation of the results based on this model from those based on the ab initio potential are analyzed. It is pointed out that the difference between the shear stresses of the two potentials for the linearized solution is within 1%, while it reaches 6% for the nonlinear cases.

Accelerated discrete velocity method for axial-symmetric gaseous flows

An accelerated discrete velocity method is presented for the simulation of axial-symmetric single-component gas flows in the whole range of gas rarefaction. The method relies on the Bhatnagar–Gross–Krook (BGK) kinetic equation. A diffusion-type moment equation is derived on the basis of the cylindrical kinetic one. Both equations are solved by discretizing the spatial and molecular velocity spaces in a joint iteration. The relationship between the cylindrical and the two-dimensional moment equations is shown. A stability analysis is developed to show the improved iteration performance of the accelerated method. Pressure-driven flow in a tube is simulated to study the method. The accelerated scheme is superior compared to the non-accelerated standard one at intermediate and high values of the rarefaction parameter in terms of the number of iterations required and the computational time. The flow rates and the velocity profiles are compared to the analytical discrete ordinate solution of the problems. There is a very good agreement between the results provided by the two methods.

Transient heat transfer flow through a binary gaseous mixture confined between coaxial cylinders

The unsteady transient behaviour of both single and binary mixture rarefied gas confined between two coaxial cylinders at rest caused by a sudden change of the temperature of one of the surfaces is studied numerically by using Direct Simulation Monte Carlo. For single gas flow, the results are compared to the numerical solution of the Navier–Stokes–Fourier equations. The study covers the response of the system from the initial moments, when the change in the temperature is introduced, up to the point where the system reaches the steady state for different degrees of gas rarefaction within the entire range of Knudsen number, various values of the temperature difference, from linear to strongly non-linear conditions, and several radius ratios. Numerical results are reported for two binary mixtures (Ne–Ar and He–Xe) with various mole fractions.

Rarefied gas flow through a cylindrical tube due to a small pressure difference

Flow of a rarefied gas through a cylindrical tube connecting two reservoirs maintained at a small pressure difference is considered using the axisymmetric version of the linearised BGK kinetic model equation subject to Maxwell diffuse–specular boundary conditions. This is a problem of five dimensions in phase space, solved in a fully deterministic manner using a parallelised discrete velocity algorithm. Results include flow rates as well as distributions of density and velocity perturbations, from the free molecular up to the slip regime and for length-over-radius (L/R) ratios ranging from zero (orifice flow) up to 20. The dependency of the results on gas rarefaction, wall accommodation and tube length is analysed and discussed. It is found that the Knudsen minimum appears only at L/R=20. Furthermore, in the case of L/R=0 it is confirmed that the results are practically independent of the accommodation coefficient. Comparing the present linear results with corresponding non-linear ones, it is seen that linearised analysis can capture the correct behaviour of the flow field not only for infinitesimally small but also for small but finite pressure differences and that its range of applicability is wider than expected. Also, the error introduced by the assumption of fully developed flow for channels of moderate length is estimated through a comparison with the present corresponding results.

Transient flow of rarefied gas through a short tube

A transient rarefied gas flow through a short tube is studied on the basis of the direct simulation Monte Carlo method. The mass flow rates through both inlet and outlet cross sections are calculated as a function of the time in the free-molecular, transitional and hydrodynamic regimes with respect to the gas rarefaction. Two values of the pressure ratio, i.e. 0.1 and 0.5, and two values of the aspect ratio, i.e. 1 and 5, are considered. A characteristic time equal to that needed to cross the tube radius with the most probable molecular speed is defined. A typical interval of time to establish the steady flow rates was calculated. It was found that this time is about ten and forty characteristic times for the shorter and longer tubes, respectively. The flow field for the longer tube is analyzed in details.

Macroscopic description of nonequilibrium effects in thermal transpiration flows in annular microchannels

Thermal transpiration flow of rarefied gases in annular channels is considered where the driving force for the flow is a temperature gradient applied in the channel walls. The influence of gas rarefaction, aspect ratio of the annulus, and surface accommodation coefficient on mass and heat transfer in the process are investigated. An analytical approach to the problem is conducted based on linearized Navier-Stokes-Fourier (NSF) and regularized 13-moment (R13) equations, and a closed-form expression for Knudsen boundary layers is obtained. The results are compared to available solutions of the Boltzmann equation to highlight the advantages of the R13 over the NSF equations in describing nonequilibrium effects in this particular thermally driven flow. Through comparisons with kinetic data, it is shown that R13 equations are valid for moderate Knudsen numbers, i.e., Kn<0.5 where NSF equations fail to describe the flow fields properly.

Slip flow and heat transfer in microbearings with fractal surface topographies

The effects of random surface roughness on slip flow and heat transfer in microbearings are investigated. A three-dimensional random surface roughness model characterized by fractal geometry is used to describe the multiscale self-affine roughness, which is represented by the modified two-variable Weierstrass–Mandelbrot (W–M) functions, at micro-scale. Based on this fractal characterization, the roles of rarefaction and roughness on the thermal and flow properties in microbearings are predicted and evaluated using numerical analyses and simulations. The results show that the boundary conditions of velocity slip and temperature jump depend not only on the Knudsen number but also on the surface roughness. It is found that the effects of the gas rarefaction and surface roughness on flow behavior and heat transfer in the microbearing are strongly coupled. The negative influence of roughness on heat transfer found to be the Nusselt number reduction. In addition, the effects of temperature difference and relative roughness on the heat transfer in the bearing are also analyzed and discussed.

Rarefied gas flow in a planar channel caused by arbitrary pressure and temperature drops

The paper presents a numerical analysis of the nonlinear rarefied gas flow through a long planar channel of finite length. The solution is constructed for the arbitrary pressure and temperature drops, including flow into vacuum. The obtained results for the flow rates are compared with the linearized solutions in the large range of degree of gas rarefaction. The boundaries of the validity of the linearized approximation are established.