A mechanical time-of-flight filter intended for measurement of velocities of nanoparticles exiting a gas aggregation source has been developed. Several configurations maximizing simplicity, throughput or resolution are suggested and investigated both theoretically and experimentally. It is shown that the data measured using such filters may be easily converted to the real velocity distribution with high precision. Furthermore, it is shown that properly designed filters allow for the monitoring of the velocity of nanoparticles even at the conditions with extremely low intensity of the nanoparticle beam. The popularity of this vacuum-based and fully solvent-free deposition technique developed in the 1990s5, which is based on the spontaneous nucleation of sputtered material accompanied by subsequent coalescence and coagulation of the growing particles, is primarily due to the possibility of fabricating high-purity nanoparticles with well-controlled size and morphology. As shown in numerous studies, mGAS systems are also very flexible concerning the materials from which the nanoparticles can be prepared that range from single-material metallic17, metal-oxide20 or plasma polymer NPs26 to multi-material ones37. Furthermore, the gas aggregation sources allow for the deposition of nanoparticles on virtually any substrate that can withstand the vacuum conditions. However, the way how the gas-phase synthesized NPs interact with a substrate and assemble on it is strongly dependent on many factors. Among them, one of the most important ones was found to be the velocity/energy of NPs upon their impact on a solid substrate. Related to this, two principal deposition regimes may be distinguished depending on the energy of incoming NPs39 that is commonly expressed as the kinetic energy per atom Eat. While the integrity and structure of both nanoparticle and substrate may dramatically change for Eat higher than the binding energy of the nanoparticle constituents, nanoparticle stays intact and preserves its chemical composition during its interaction with a substrate at lower kinetic energies40. In addition, it has been demonstrated that in the latter case, i.e. in the so-called “soft-landing” regime, the nanoparticles may be in some cases rebound from the substrate. This process is very sensitive not only to the physicochemical properties of both the substrate and incoming nanoparticle  but also to relatively small variations in the velocity of nanoparticles interacting with the substrate material42. For instance, as shown in41 for nanoparticles with a mean diameter between 10 and 100 nm, the bouncing may occur at velocities around several tens of m/s, i.e. in the velocity range common for magnetron-based gas aggregation sources with no additional nanoparticle acceleration. The reflection of nanoparticles and its proper control in turn allowed for a highly selective nanoparticle assembly on substrates, i.e. an important step towards the production of functional nanodevices43.Gas-phase synthesis of nanoparticles (NPs) by magnetron-based Gas Aggregation Sources (mGAS) has established itself as one of the most viable strategies of preparation of this class of nanomaterials in the last few decades45 and core/satellite47 nanoparticles or for the in-flight oxidation of metallic NPs48. Naturally, the critical parameter that determines the performance of this deposition strategy is the residence time of NPs in the zone, where they are modified. This is naturally given by their velocity and the length of the modification zone.Another field in which the velocity of nanoparticles has to be considered is their in-flight coating/modification. This technique utilizes the spatiotemporal separation of the core formation that takes place inside the aggregation chamber of mGAS and its subsequent modification/coating that occurs in an auxiliary zone located in between the output of mGAS and substrate. Such an approach was recently employed for the production of various types of core/shellBoth aforementioned examples highlight the necessity of the precise knowledge of nanoparticle velocity for the rational and effective design of the deposition procedure. Nevertheless, in contrast to carrier gas, whose speed may be easily calculated from the known geometry of the system and gas flow, the determination of the velocity of nanoparticles remains rather challenging using theoretical models as the size of the velocity slip effect is still generally unknown. Because of this, the precision of the calculations is, in many cases, questionable, and thus the direct measurement of the particle velocity is urgently needed.49 using a system based on an electrostatic deflection. However, this approach has intrinsic limitations as it requires the knowledge of the mass and charge of nanoparticles, does not allow in-situ monitoring of nanoparticle velocity and is applicable solely for the charged nanoparticles. Another method, a time-gated quadrupole mass filtering51 allows in-situ monitoring, but again only of the charged NPs. In addition, such systems allow filtering of nanoparticles only in a rather narrow range of their size. The velocity of charged NPs may also be in principle measured using a Wien filter52 or electrostatic time-of-flight filter53. However, the exclusion of neutral and opposite charged NPs is too severe limitation because aside from not being able to detect all nanoparticles, the charging of NPs is size-dependent and may distort the results and complicate the interpretation of the results. A theoretical possibility to measure the velocity of NPs including neutral ones would be based on the measurement of beam impuls54. Aside from other technical issues connected with this strategy, the sensitivity of such measuring systems is still not sufficient even for high-intensity beam mGASes. To overcome these principal drawbacks, a novel approach is investigated in this study that utilizes a mechanical slotted disk type filter originally developed for the measurement of the velocity of molecular beams (e.g.58) and used previously as a filter for mass selection of gas-phase synthesized NPs59. To our best knowledge, it has never been used for the measurement of velocities of nanoparticles, and thus the main aim of this study is (i) to test the applicability of such velocity filter for the evaluation of velocities and velocity distribution function of nanoparticles generated by mGAS and (ii) to show, how the selectivity and high throughput may be optimized by a proper design of the filter.The first attempt to measure the nanoparticle velocity was presented in60. It was equipped with a magnetron 81 mm in diameter set with copper target 3 mm thick. The orifice was 23.5 mm long, 3 mm in diameter. The aggregation length, i.e. distance from the target surface to the end of the orifice, was 175 mm. The source was connected to a high vacuum main chamber pumped by diffusion and rotary pumps. The working gas was argon , which was introduced via a flow controller . A capacitance gauge was used to measure pressure both in the aggregation chamber  and in the main deposition chamber . The magnetron was powered with a DC magnetron source .To test the performance of constructed filters, a gas aggregation source shown schematically in Fig. To investigate the filter performance, the NPs have been deposited at magnetron current 200 mA at two different aggregation chamber pressures 20 and 100 Pa corresponding to flowrates of the buffer gas 1 and 12.7 sccm. Under these conditions, Cu nanoparticles with mean diameters of 18 ± 3 nm and 26 ± 7 nm were produced as witnessed by scanning electron microscopy using JEOL (JSM-7200F) electron microscope operated in the secondary electron mode and 15 kV acceleration voltage . The thickness of the disks was 1 mm each . The third key parameter is the relative angle between the slits of the first and second disks φ. This should be higher than the angular width of the slit to avoid nanoparticle transmission at zero rotation. In addition, the higher the value of φ, the lower velocities are transmitted. In our experiments, the φ was fixed at 9°. The described setup with the specified geometry parameters allows measurement of nanoparticle velocity from 7 to 215 m/s.The basic configuration of the velocity filter, which is depicted in Fig. The second and more complex variant of the rotor was based on series of layers of very thin disks set one on the top of another, each rotated by an incremental angle relative to the previous see Fig. b. In thiExcept for the central shaft, motor connector, the disk holders, ball bearings and screws, all the remaining parts of both slotted disk and helical rotor devices were 3D printed on Prusa i3 MK3S printer from PETG polymer using a 0.4 mm nozzle. The printing resolution (layer thickness) was 0.2 mm.The transmitted nanoparticles were collected on a Quartz Crystal Microbalance sensor  using 0.5-in. crystals and Ø12 mm sensor opening. The QCM frequency (5 MHz) was read using a universal frequency counter  with 3 GHz reference, 1 mHz resolution and 0.1 Hz resolution repeatability, gate time 1 s and refresh rate 500 ms. All the data were logged in the computer and synchronized to other quantities, such as the actual rotation frequency and other operating conditions. To suppress noise in the case of low deposition rate measurements, the deposition rates were optionally smoothed over 5 or 20 measurements corresponding to 2.5 and 10 s measurement time. The smoothing allows noiseless detection of very low deposition rates, but the rate of rotation frequency change must be slower than the averaging time to avoid shifting or smudging of the logged data.A semi-analytical model of the device suitable for the numerical calculation has been developed in order to fit the measured data as well as to predict the behavior of the measurement system. The basic idea of the model is to use a number of test particles travelling parallel to each other along normal to the disks planes across the entrance slit of the rotor Fig. a. Each oα, the relative angle of the disks φ and the radius of the slit (the radius where the beam is passing) r0, it is possible to define the length of the arc of the slit aαaφFor the purpose of the model, let’s look at the device from the surface of the cylinder, the radius of which is the distance between the rotor axis and the NPs beam, being unfold into a plane, see Fig. f represents the frequency of the disks rotation. The incident trajectory  is defined asn + 1 is the number of input trajectories and i is the index of selected trajectory going from 0 to n.The tangential velocity of the disks is given bydi and rotation frequency f are given by the two dashed blue lines in Fig. l) is labelled δ, see Fig. vmax of the nanoparticle to pass through the system is then given by the lower line, while the minimum velocity vmin by the upper line. The angle δ may be expressed either in terms of velocity as a ratio between tangential velocity vrot of the slits and the incident NPs velocity vin asbo is the thickness of the disks and l is the distance of the centers of the disks. Note that for δmax the limit beam passes the outer corner of the disk 2, while for δmin it passes the inner corner. The velocity of the transmitted NPs maybe then expressed asThe limits of the incident velocity of the transmitted NPs for given trajectory After substitution of all variables into Eqs.  and 9)  9) and adcrit of the incident NPs trajectory from the upper slit border that allows NPs to pass may be expressed asThe Eqs.  and 11)11) preseAfter substitution and adjustments the first transmitted trajectory isf, NPs velocity v and trajectory i to determine whether the nanoparticle will pass or not through the system. The signal s at the given rotation frequency is then obtained by the sum of the blocked/passed counts c = {0,1} over the trajectory index and the possible NPs velocities multiplied by the probability of the specific NPs velocity pv.With Eqs. , 11) an an11) anThe mean velocity transmitted through the filter assuming the finite thickness of the rotor disks may be assumed in the formTo obtain the simulated signal, the above-described calculations have to be executed for a frequency range and the peak amplitude has to be adjusted to the measured data. A program to simulate the behavior of the system at various configurations and to fit measured data has been written in Delphi XE2.f, which is also the output of the model. By shifting from frequency to velocity and sending the number of trajectories n to infinity, the relation  on the rotation frequency relation  gives thr number . In our α. In these figures, the horizontal cut of the map corresponds to one virtual measurement of the dependence of the number of detected particles on the rotational frequency of the filter with the velocity of NPs specified by the vertical position of the cut. On the other hand, the vertical cut shows the transmissivity of NPs velocities at the specific rotation frequency. Comparing the numerical results presented in Fig. α provide a narrower window of velocities at which nanoparticles at the given rotation frequency may pass the filter, i.e. better velocity resolution. For instance, while the range of nanoparticle velocities that contribute to the overall detected signal for the rotational frequency of 100 Hz and α equal to 5° is from 51 to 161 m/s, this range drops down to 72 to 81 m/s for the same rotational frequency and angular width of the slit 1°. However, the higher velocity resolution of the slotted disks system for the lower values of α is counterbalanced by the substantial decrease of the number of the NPs that pass the velocity filter, hence, to the lower signal. To give some example, the signal intensity, i.e. the integral of the number of NPs that went through the time-of-flight filter operated at the frequency of 100 Hz, decreased by a factor of 18 as the angular width of the slits changed from 5° down to 1°. The transmissivity of the 5° disks is 9.5%. The possible way how to reduce this unwanted effect is to increase the number of slits. Increasing the number of slits in case of the slotted disks rotor is, however, possible only to a certain degree. For the higher number of the slits, the nanoparticles (in the available frequency range) would start to pass through more than one slit in the second disk that gives rise to parasitic signal and thus complicates the analysis of the measured data. Therefore, if the input signal is too weak and the number of slits cannot be further increased without complications, the construction of the filter may be altered to helical channels, as suggested in62. The calculation for such a system shown in Fig. At first, a profile of signal obtained with different angular widths of the slits has been investigated by simulation. If a δ-function in the velocity of the incoming particles is assumed, the profile of the signal is a triangle, the width of which is determined by the slit angular width and the velocity of the nanoparticles. This is shown in Fig. At this point, it is worth noting that although the correction for finite disk thickness coming from Eq.  may seemFigure vpeak and different full width at half maximum (FWHM) are presented in Fig. v is assumed in the form:AG is a scale factor connected to the peak height of the distribution and wG is a so-called scale parameter, which is connected to the width of the distribution, while its FWHM is defined as:The aforementioned calculations supposed δ-function of the velocity of the incoming nanoparticles. However, this is non-realistic as certain velocity distribution has to be assumed. This leads to the widening of the signal peak width and the triangle edges are getting smoothed. To demonstrate this, examples of signals simulated using Eq.  for GausAL is a scale factor connected to the peak height of the distribution and wL is a real number parameter related to the width of the distribution. The FWHM of the distribution  and the distribution with the lowest RSS was taken as the result of the fit. The obtained results are presented in Fig. A typical example of a signal recorded by the QCM crystal, which is directly proportional to the number/mass of deposited NPs, in dependence on the rotation frequency is depicted in Fig. sing Eq. . Such fising Eq.  or log-nsing Eq.  with vpeFurther tests were performed with all four rotors described above for two different deposition conditions (20 and 100 Pa pressure in the aggregation chamber) to investigate to what degree it is possible to reconstruct the NPs velocity distribution at different experimental conditions. The results of the measurements and their fits are shown in Fig. 64) connected with the long-term operation of mGAS. On the other hand, the shift in the mean velocity of NPs towards higher values for higher pressure in the aggregation chamber is due to the higher velocity and density of the buffer gas that results in more intensive acceleration of the carried NPs. This is an important finding as it experimentally proves that the velocity of the nanoparticles that leave the mGAS may be easily controlled by the pressure/flow of the carrier gas.First, it is evident that whereas the position of the velocity peak in measured data is independent of the used rotor, it differs significantly for different pressures in the aggregation chamber of the mGAS system: in the case of pressure 20 Pa the mean velocity of NPs was found to be around 35 m/s, while for the pressure of 100 Pa the mean velocity was 93 m/s. Alongside with the constant values of mean velocities for fixed pressure in the aggregation chamber, also the FWHM of the velocity distribution function determined by the fitting of the experimental data stayed almost unchanged see Fig. . The cloThe second important finding that can be demonstrated in Fig. α of the slits was observed and the measured data approached for the lower values the real velocity distribution function. This effect, which is in agreement with the numerical model of the time-of-flight filter performance, has an important practical consequence as it clearly shows that in the case of low values of α the velocity distribution function with a sufficient level of precision may be obtained directly from the measured data using Eq. (Finally, gradual narrowing of the measured velocity peak with lowering the angular width sing Eq. , i.e. wiThe time-of-flight filters based either on slotted disks or on a helical rotor were found to be despite their relative simplicity a powerful tool for the precise measurement of velocities and velocity distributions of nanoparticles leaving the mGAS system. As shown in this study, such devices may be used in several different configurations maximizing simplicity, signal amplitude or resolution. Furthermore, it is demonstrated that irrespective of the particular configuration of the measuring setup and with it connected velocity resolution, the recorded experimental data may be easily recalculated to the real velocity distribution functions. This allows one to select a proper design for a particular application, including the possibility to perform the velocity measurements even at the conditions that provide deposition rate as low as 1 monolayer of NPs per several minutes. This is an important finding as it paves the way for the better characterization of beams of nanoparticles generated by gas aggregation sources, which is urgently needed in various applications in which the velocity/energy of nanoparticles plays an important role. In addition, such measured velocity distribution functions may be used as a benchmark for testing of the validity of the theoretical models.


A modified expression of the electron entropy production in a plasma is deduced by means of the Kelly equations of state instead of the ideal gas equations of state. From the Debye–Hückel model which considers the interaction between the charges, such equations of state are derived for a plasma and the entropy is deduced. The technique to obtain the modified entropy production is based on usual developments but including the modified equations of state giving the regular result plus some extra terms. We derive an expression of the modified entropy production in terms of the tensorial Hermitian moments  The entropy production was studied since the middle of the XIX century by Clausius . It has ynomials . In ordeynomials  to compuynomials .D represents the Debye lengthV and the Boltzmann distribution function is considered for obtaining the average. Notice that the same temperature is considered for the ions and the electrons. This will be analyzed in On the other hand, by giving different equations of state, it is possible to not abandon the Coulombic interactions of charged plasma particles. The historic paper in Electrochemistry and Plasmas by Debye–Hückel  considers, Kelly  deduced ∑αNαVkT1−8ND,wherIt should also be noted that over time there have been many attempts to describe equations of state being some of them very sophisticated where the magnetic and electrical fields are included such as the work of Liboff and Lie  and recevelpiece  calculatvelpiece . Moreovevelpiece  successfFurthermore, Wergeland  was ableOn the other hand, Callen  proposedng Kelly  and Wergng Kelly . Then, wng Kelly  qα=Qα−5Pαquation  an an(36)σerrive atσe=−eneuequation . Accordingly, by using a technique based on Maxwell’s relations [Our purpose in this section consists of completing the set of equations of state for a plasma with two species: electrons and one type of ion. Since in many situations in Plasma Physics, it is just necessary to consider only the electrons, we analyze the Kelly equation for one species. We will obtain the electron entropy for a plasma composed of electrons where the effect of the ions are considered just in the equations of state. We obtain the entropy as a function of the volume elations , we obtaIn the case of considering only electrons and ions in a quasi-neutral plasma, the Kelly equation for the partial pressure of the electrons can be written asD is the Debye’s length and It is necessary to note that ergeland  and EquaIt is interesting that we can obtain the heat capacity It has to be highlighted that if we consider the expression of the usual entropy production, Equation , and undDue to the similarity with the relationship between energy and pressure in the case of photons, we can propose a correction for the entropy given by the following expressionEquation ), there If we calculate the Helmholtz free energy Equation  which paEquation  and 52)A=Aig+UcoSince the electron entropy and the internal energy cannot be written explicitly as fundamental equations  an an52) anEquation  has beenLet us begin by calculating as in Equation , we haveEquation  and usin∂Te∂t as∂Te∂t=2313/2,and∂se∂Te=32quations , we havequation (82)∇ne=1Let us consider the modified entropy production, Equations  or 93)  93) but If we compare the usual entropy production Equation , and theEquation , we can  Tokamak , can be Therefore, even if Furthermore, these last expressions show that the change in the expression of entropy production can be significant when equations of state very different from the ideal gas equations are used. Moreover, new products of fluxes and forces, rocesses .By using Equations  and 45)45) and rWe arrive atWe obtain an expression for the modified electron entropy production by means of the Kelly equations of state within the approximation described by Equations  and 41)41) but kThe new terms in the modified entropy production represent new products of fluxes and forces. Such products may not be so negligible in some circumstances which implies that some analysis can be done with respect the Linear Irreversible Thermodynamics . The modThe next step consists of obtaining the modified electron entropy production without using such approximations. Such expression will include the moment