Fractionation of Aerosols by Particle Size and Material Composition Using a Classifying Aerodynamic Lens

: The fractionation of airborne particles based on multiple characteristics is becoming increasingly significant in various industrial and research sectors, including mining and recycling. Recent developments aim to characterize and fractionate particles based on multiple properties simultaneously. This study investigates the fractionation of a technical aerosol composed of a mixture of micron-sized copper and silicon particles by size and material composition using a classifying aerodynamic lens (CAL) setup. Particle size distribution and material composition are analyzed using scanning electron microscopy (SEM) and energy dispersive X-ray spectroscopy (EDX) for samples collected from the feed stream (upstream of CAL) and product stream (downstream of CAL) at varying operational pressures. The experimental findings generally agree with the predictions of an analytical fractionation model but also point to the importance of particle shape as a third fractionation property. Moreover, the results suggest that material-based fractionation is efficient at low operational pressures, even when the aerodynamic properties of the particle species are similar. This finding could have significant implications for industries where precise particle fractionation is crucial.


Introduction
The aerodynamic properties of submicron, gas-carried particles are relevant in processes where aerosols are formed.Knowledge of these properties is crucial not only in many technical processes, such as mining and recycling, but also in the health care sector [1][2][3].
Copper and silicon are materials commonly found in electronic devices.At the end of the product life cycle, one way to reobtain these materials from used product devices is grinding followed by material separation.Waste in the form of dust produced during these processes may still hold valuable particulate material.As mining's price and environmental impact increase, it is also becoming increasingly important that more knowledge about the aerodynamic behavior of aerosols is exploited so that recycling can be further enhanced via the fractionation of airborne particles.With this in mind, it is essential to consider the multidimensional characteristics of these particles.

Multidimensional Separation of Particles
The separation of particles, taking into account their multidimensional properties, has garnered significant attention in recent years.A plethora of measurement and separation methods have been developed, and a mathematical framework has been introduced to characterize multiple-property distributions of particle properties [4,5].As of now, more progress has been made in the separation of particles dispersed in the liquid phase than for those dispersed in the gas phase.Innovations have emerged that separate particles as a result of agglomeration based on attributes such as magnetic properties, size, and surface chemistry [6][7][8].Additionally, methods involving microchannels, microcolumns, and other flow field-based techniques have been developed [9,10].In these methods, often, a secondary force (e.g., an electric field) is applied perpendicular to the flow direction of the liquid containing the particles.In some instances, the separation forces are applied sequentially rather than spatially, enabling the selection of particles with specific properties.While these are just a few examples, the number of innovations can be expected to continue to grow in the near future.However, it should be noted that many of these new methods are tested on carefully selected model particle systems and may not be suited to industrial processes.The concept of applying two different forces is also prevalent in aerosol sciences.For example, Sandman et al. [11] demonstrated the separation of airborne particles using acoustic and electrical fields.Another recent trend is the integration of multiple existing classification methods into a single device, as exemplified by the centrifugal differential mobility analyzer (CDMA) presented by Rüther et al. [12].

Multidimensional Characterization of Aerosols
Despite the increasing importance of particle multidimensionality in scientific and industrial applications, offline analysis methods remain the most common practice for examining such properties.Recent years have seen some progress in the simultaneous, real-time characterization of multiple aerosol properties.The integration of different measurement methods enables the investigation of properties such as the effective density and fractal dimension of agglomerates present in aerosols [13].Yet, studies into bi-or multivariate property distributions are still relatively scarce.To obtain a bivariate histogram of aerosol particle properties, such as the particle mobility diameter and aerodynamic diameter, a three-step process is typically employed.First, the aerosol is classified by one property, e.g., mobility using a differential mobility analyzer (DMA), then classified again by a second property, e.g., aerodynamic diameter using an aerodynamic aerosol classifier (AAC), and finally counted, often using an electrometer or a condensation particle counter (CPC) [14].Each measurement populates an appropriate bin in the bivariate histogram with data from fractionated ensembles.For a full 2D distribution measurement, one classifier is held at a constant classification parameter (e.g., DMA voltage) while the other scans through its classification parameter; then, the first classifier setting is changed.Such a measurement can take several hours.

Aim of This Study
The objective of this study is to investigate an aerodynamic fractionation process that takes into account multiple properties of airborne particles.Unlike other laboratory studies, which often utilize model particle systems such as polystyrene spheres with narrowly distributed properties, this work studies a particle system which is significantly closer to those often found in industrial processes, as the particles examined here exhibit more complex shapes and broader size distributions.The diverse combinations of size and density in these particles result in a wide range of expected aerodynamic behaviors.The particular batches of copper and silicon particles were selected for this study because their aerodynamic diameters are similar enough to pose a substantial fractionation challenge.To attain a high degree of material fractionation and concurrently refine the size distribution of the microparticles, a classifying aerodynamic lens (CAL) setup is employed.Mixtures of copper and silicon particles with defined mass ratios are prepared, aerosolized, and subsequently fractionated using the CAL.Samples drawn from both the aerosol feed and product streams undergo analysis using offline methods such as scanning electron microscopy (SEM).The outcomes are then juxtaposed with predictions generated by an analytical model that employs the theoretical transfer function of the CAL.CALs have been used to fractionate nanoparticles by size before [13]; this work investigates the use of a CAL as a tool for material and size fractionation of microparticles.

Background and Theory
In aerosol dynamics, the particle relaxation time τ P plays a crucial role, representing the time for a particle's velocity to adjust to flow changes.For gas-suspended particles, τ P is determined by the volume equivalent particle diameter d P , particle mass density ρ P , dynamic gas viscosity η, and Cunningham correction C c , as given by Equation (1): The Cunningham correction accounts for particle slipping behavior [15], given by where λ g denotes the mean free path length of gas molecules as a function of pressure p.The values for A 1−3 are given by Rader (1990) as A 1 = 1.207,A 2 = 0.440, and A 3 = 0.78 [16].
The Stokes number Stk is a dimensionless parameter assessing the particle's travel distance within the relaxation time relative to a characteristic length ℓ.It is defined by where u denotes the mean gas velocity, which is assumed to be at equilibrium with the particles' velocity.Understanding Stk is pivotal in devices like impactors, where efficiency is linked to the particle's Stokes number.A specific threshold exists, ensuring particles with Stk equal to or greater than this value are captured with 100% efficiency.This work investigates the fractionation of aerosol particles on the basis of their relaxation time using a classifying aerodynamic lens (CAL) [17].This is a special form of an aerodynamic lens which applies a sheath gas on the central axis of a flow through an orifice [13].The CAL confines particles upstream to specific radial positions.Consequently, the passage through the orifice causes them to transition between streamlines.Thus, the downstream position becomes a function of the upstream radial position as well as the aerodynamic properties of the particles, leading to a narrow transfer curve with distinctive characteristics.The lens relaxation time τ L , which is the specific relaxation time of a particle optimally focused by a CAL, depends on CAL geometry and operational parameters, given in Equation (4): Here, Stk o , d L , A L , and Qv,L denote the optimum Stokes number (defined later), lens orifice diameter, cross-sectional area of the lens orifice, and volume flow rate through the lens, respectively.For convenience, the volume flow rate into the lens, which is at the inlet at ambient pressure p 0 , denoted as Qv,CAL , is often referred to; the latter, however, needs to be adjusted to the gas density at the (subatmospheric) operation pressure of the lens.This is achieved by adjusting the volume flow at the inlet Qv,CAL , with the pressure at the inlet p 0 (at normal conditions) compared to the operation pressure p just upstream of the lens orifice.

Aerosol Flow Conditions and Geometric Considerations
The predictability of material-and size-based fractionation in a CAL fundamentally relies on maintaining laminar flow.As detailed in our previous work [17], certain restrictions are necessary to ensure the efficiency of the fractionation process.The focusing of small particles is only possible when specific limits for Reynolds (Re), Mach (Ma), and Knudsen (Kn) numbers are not exceeded.These critical limits are denoted as Re c , Ma c , and Kn c , respectively.Furthermore, the available pumping capacity is another limiting factor in the CAL's performance.The equations for calculating Re, Ma, Kn, and their respective critical values, as given by Wang et al. [18], are provided below.The Reynolds number is calculated via the mass flow rate of the gas Qm by Note that Qm , in this case, is the total mass flow rate through the lens orifice, which is the sum of the aerosol and the sheath flow rate.The Mach number, given as compares the gas velocity u to the speed of sound in the gas, represented by c g .The Knudsen number of the gas, which indicates whether a fluid behaves as a continuum or as the motion of individual molecules, is given by It is worth noting that changes in λ g with operational pressure p enable adjustments to the lens's relaxation time τ L .
The optimal Stokes number Stk o is typically determined through numerical fluid simulations.However, it can also be estimated based on the Mach number (Ma) and Reynolds number (Re) of the flow, as demonstrated by Wang et al. [19,20].In this study, the optimal Stokes number is obtained by interpolating between several results to match determined key factors for the CAL, such as the Reynolds, Mach, and Knudsen numbers.The optimal Stokes number is also validated using computational fluid dynamics (CFD), as described in Section 3.1.

A Model for the Transfer Behavior of the CAL
A particle is optimally focused by the CAL if both Equations (1) and (4) are fulfilled.This concept, referred to as the iso-τ-principle, is expressed in Equation (8), where τ is denoted as the normalized relaxation time of the particle: However, it should be noted that the sampling orifice also captures particles near the central axis, leading to a broadening of the transfer behavior.Simulations and experiments indicate that the differential transfer function Ω of the CAL can be described in a manner analogous to the behavior of a differential mobility analyzer (DMA) [13,21].Therefore, the following form of the transfer function, described by Stolzenburg et al., is used [22]: Here, β and δ are parameters that define the shape of the transfer function.In the case of the DMA, the values of β and δ are a function of the DMA's shape and the sheath-toaerosol ratio.However, for the CAL, no simple function is available.Therefore, numerical particle and fluid simulations are applied.
A convolution of the iso-τ-principle with the width of the transfer function Ω results in a two-dimensional transfer characteristic of the CAL, accounting for both particle size d p and density ρ p .A typical CAL 2D transfer function (also called transfer probability) is illustrated in Figure 1 below.

Predicting the Material Separation Efficiency Using an Analytical Model
The prediction of fractionation efficiency on the basis of the two-dimensional transfer characteristic of the CAL becomes feasible either when the individual properties of the involved particles are known or when assumptions about the correlation within an ensemble distribution of these properties are made [5].In the context of this study, the distributions of particle size and density are based on individual particle data.
If more than one of the attributes selected for fractionation is distributed on a spectrum, knowledge of the multidimensional form of the transfer curve is essential.In this specific case, the transfer behavior is simplified by considering only the bulk density of the particle materials.Thus, the multidimensional transfer behavior is represented by two independently acting transfer curves, one for each material.This approach enables prediction of the number distribution of products from any given feed material distribution, provided that the bulk densities and the individual feed number distributions are known.Scanning electron microscopy (SEM) allows for determining the number distribution in feed and product samples.However, these number distributions alone are insufficient as the model input and for direct comparison because the total number of particles in the sample is more influenced by the number of particles counted than by the material composition.The material composition is determined using energy dispersive X-ray spectroscopy (EDX).Nevertheless, as the transfer curve is number-based, it is essential to convert the mass-based material composition into a number distribution.
The SEM image analysis gives histogram data of particle counts C i within defined size ranges called bins.By normalizing with the total number of counted particles, the number fraction x i is defined for copper as and the definition for silicon follows analogously.During the separation process, the particle mixture is aerosolized.The total particle concentration of the aerosol is the sum of the concentration of the involved species N tot = N Cu + N Si .Assuming the overall shape of the particle size distribution for one species remains stable when sampled from the aerosol, the aerosol mass concentration corresponding to each bin i of the normalized histogram M i can be derived for copper as follows, while M Si i is calculated analogously: where N i is the number concentration fraction in bin i and m i is the individual particle mass for bin i derived from the mean diameter d i of that bin, assuming the particles to be spheres of uniform density: The mass fraction of copper in a sample ω Cu is measured using EDX, and is calculated as follows: Inserting Equations ( 11) and (12) into Equation ( 13) allows one to solve for the total aerosol number concentration of one species, given here for silicon: With this, an arbitrary total aerosol concentration N tot is chosen.The number concentrations N Cu and N Si are then derived from the mass fraction ω Cu .In turn, this allows one to build a histogram from the number concentration fractions N i in the feed sample for each material, as shown in Equation (11).
Subsequently, the transfer function Ω is applied to each bin of the corrected histogram individually, resulting in the following equation for the adjusted concentration in the product sample N Cu i,Product : Finally, the product mass fraction ω Cu Product is calculated from inserting N Cu i,Product into Equation (11) and then applying Equation (13).

Model Assumptions and Restrictions
The derivation above shows how to predict a product mass fraction ω Cu Product from the feed mass fraction ω Cu Feed and histogram counts C i .For the implementation in MATLAB (The MathWorks, Inc., Natick, MA, USA), however, we opted to work on the basis of the moments of the counted particle size distributions, namely, the geometric mean diameter d g and the geometric standard deviation σ g .This allows for greater flexibility and higher resolution, because it allows one to choose narrower bin sizes.However, it requires a few additional steps, which were left out in the section above to keep the derivation as simple as possible.Thus, it is important to note that this model, due to its limited inputs, comes with certain assumptions and restrictions:

•
The model is based on the convolution of distribution and transfer functions.

•
Ensemble data are considered, not individual particle properties (d g,Stk , σ g , ρ p,bulk ).• The distribution of particle size is considered to follow a log-normal distribution type.• Particle-particle interactions (e.g., hetero-and homo-agglomeration) are not considered.

•
Particle shape is assumed to be spherical, with other shapes not currently accounted for.This means that concepts such as effective particle density and fractal dimension are not considered.

•
Particle losses (e.g., diffusion, wall interactions) are included in the CAL transfer function.
Although referred to as an analytical fractionation model, it discretizes the distribution function to calculate the total mass of the number distribution.The number of bins used in this process regulates both the model's resolution and the runtime for a single virtual experiment.

Materials and Methods
The CAL was designed based on model calculations and constraints using CAD Software (SOLIDWORKS, Dassault Systems, France) and later realized in stainless steel.
The key parameters for this design are listed in Table A1.

Optimization of a Classifying Aerodynamic Lens for Material Fractionation
Selecting the initial parameters is a complex process.In addition to iterative CFD simulations, a stochastic optimization method was applied.The modeling process and its results are detailed in a previously published study [17].Figure 2 displays exemplary particle trajectories over typical velocity contours in a two-dimensional representation of a CAL operating at 400 Pa with a total incoming flow of 2 slm nitrogen, obtained using CFD (FLUENT 6.2, Fluent Inc., Now owned by Ansys Inc., Canonsburg, PA, USA).A Lagrangian method was chosen as the solver for this two-dimensional, axially symmetric case.It is important to note that this abstraction does not encompass the complete complex geometry of the sampling orifice or the critical orifice at the CAL inlet.The chosen abstraction allows for simulating the flow from the beginning of the mixture of aerosol and sheath gas to the start of the sampling orifice, where laminar flow is prevalent.
velocity magnitude (m/s)  Particle tracking analysis was used to determine the relaxation time-dependent transfer function, as shown in Figure 3.The transfer probability Ω is expressed as a function of the normalized particle Stokes number, where Stk p /Stk L is obtained by calculating the ratio between the injected number of particles N in with Stokes number Stk p and the number of particles collected at the sampling orifice N out , as shown in Equation (16).
By fitting Equation ( 9) to the CFD data points, values for the parameters β = 0.251 and δ = 0.2406 were obtained and subsequently used in the analytical fractionation model of the CAL process.The adjusted R 2 value for this fit shows, with 0.9813, a less-than-perfect fit because the Stolzenburg equation is a symmetric function.However, the width of the transfer curve is still well fitted.This general shape persists even as the operational pressure varies.Copper was selected as the particle material for particle tracking within the CFD simulation.For each investigated particle size, an equal number of particles was injected from the aerosol inlet position at a velocity equal to the gas velocity at that position.Near the theoretically expected optimally focused particle size for this CAL setup (the expected most-penetrating particle diameter was ≈350 nm for a CAL operated at 400 Pa, 2 slm using a 20 mm diameter lens), the stepping size between investigated particle sizes was reduced to achieve higher resolution.A stick boundary condition was applied for particles reaching the walls, and no external forces such as gravity were considered.To calculate the drag force, a user-defined function was applied to calculate the Cunningham correction according to the values provided by Rader.Additionally, the compressibility of the nitrogen carrier gas was taken into account.The particle size was converted into a particle Stokes number Stk p so that the transfer probability could be derived using Equation (16).The transfer probability was then normalized using the Stokes number of the lens Stk L .The optimal Stokes number Stk o , which equals Stk L , was found to be 0.5429 for this constellation.It is imperative that the pipes maintain a circular shape and concentricity throughout the entire length of the device to ensure predictable focusing effects.To achieve this, seamless steel pipes are chosen for the construction, onto which additional parts are welded externally, followed by rotational material removal.Additionally, vacuum tightness is rigorously established by selecting appropriate fittings and tested with the help of leakage tests.Key parameters of the final CAL design are summarized in Table A1.

Preparation of Particle Mixtures
All aerosols in this study are generated from mixtures of copper (NG04EO1003, Nanografi, Çankaya, Ankara, Turkey) and silicon particles (NG01EM3006, Nanografi, Çankaya, Ankara, Turkey), with manufacturer information provided in Table 1.The particles are weighed to achieve a total mixture mass of 10 g.The smaller copper particles are layered on top of the silicon particles, and the batch is thoroughly mixed for 2 min using a laboratory shaker (Vortex-Genie™ 2, Scientific Industries SI™, Bohemia, New York, NY, USA; setting: vortex level 8).This method allows for the production of batches with varying mass percentages. 1 not provided by manufacturer.

Setup for Fractionation
To determine the fractionation efficiency of the CAL, the experimental setup shown in Figure 5 is employed.It consists of the following parts: 1.
Aerosol production unit.

2.
Reference filtration sample of feed stream.

4.
Filtration sample of product stream.

Aerosol Production Unit
The powder mixtures are aerosolized using a dust generator (SAG 410/L, TOPAS, Germany).This dry powder dispersion device is equipped with a rotating rake, a conveyor belt, and a Venturi nozzle (d nozzle,TOPAS = 0.7 mm).Nitrogen (purity > 99.995%) serves as the carrier gas, and the dust generator operates at an admission pressure of 3.5 bar.Both the rake rotation rate (preparation rate) and conveyor belt speed (feed rate) are set to 50%.The aerosol production unit leads the aerosols to a distribution chamber, from which the CAL sampling line and the reference samples are fed.The dimensions of this chamber are given in Table A3 found in the Appendix A.

Reference Filtration Samples of Feed Stream
Samples from the feed aerosol stream are deposited onto an ePTFE-membrane filter fixed onto an ISO K 63 meshed O-ring using a 40 mm diameter O-ring.The ISO K 63 meshed O-ring is positioned between two adapter parts (KF 40 to ISO K 63), ensuring that all incoming aerosol passes through the filter medium.The filter material is circular and cut from the ePTFE-membrane (AS ePTFE-membrane 200-116-R00980, R+B Filter, Langenbrettach-Langenbeutingen, Baden-Württemberg, Germany) with a diameter of approximately 40 mm.The flow rate to the feed aerosol sample filter, denoted as QF , is set at 1.35 slm (1 slm = 10 3 sccm at standard conditions), defined by a critical orifice located after the filter and before a vacuum pump (MD 8C, Vacuubrand, Wertheim, Baden-Württemberg, Germany).

Classifying Aerodynamic Lens
For the dimensions of the CAL, please refer to Section 3.1.A critical orifice at the aerosol inlet of the CAL fixes the aerosol flow to QCAL,Ae = 1 slm.The necessary sheath flow is provided by a mass flow controller (MFC, D-6321, Bronkhorst, The Netherlands) set to an equal QCAL,Sh = 1 slm.Consequently, the total flow rate through the CAL orifice is Qv,CAL = QCAL,Ae + QCAL,Sh = 2 slm.The focusing pressure p CAL is measured using an absolute pressure gauge (THERMOVAC TTR 101N, Leybold, Germany) upstream of the CAL orifice (d CAL = 20 mm) and controlled through the manual vacuum valve v 2 (ESV-S04100, Pfeiffer, Aßlar, Hessen, Germany) located just before the vacuum pump (RU-VAC WS 1001+SOGEVAC SV 300, Leybold, Cologne, North Rhine-Westphalia Germany).An additional valve v 1 allows for the evacuation of the CAL and is used in tandem with the MFC for setting up the focusing pressure before the aerosol reaches the fractionation device.Figure A1, located in the Appendix A, provides a rendered image of the CAL's components.

Filtration Sample of Product Stream
The CAL is equipped with two flow exits: one for the CAL product sample stream and one for the CAL exhaust outlet.The product sampling orifice is designed to sample 10% of the mass flow through the CAL.It is connected via the product sampling filter and a junction to the control valve v 2 .The exhaust outlet is usually connected directly to the junction but can also be equipped with an additional filter if needed.Before each filtration experiment, all flow rates are verified using a flow meter (Mass Flow Meter 4140, TSI, Minneapolis, MN, USA).

Sample Preparation and Offline Evaluation
Samples are extracted from ePTFE-membrane filter cutouts.A carbon adhesive pad (EM-T6, Micro to Nano, Haarlem, ET, The Netherlands; d sample = 6 mm) is affixed to a cylindrical aluminum sample holder (d sample holder = 12.5 mm).This assembly is then stamped onto the filter cutout at five rough positions: the center, north, east, south, and west.Multiple positions are chosen to minimize potential variations in sampling location.This approach also ensures the collection of a sufficient amount of material, particularly in cases where the filter has a sparse coating.Subsequently, any particles that do not adhere to the carbon film are removed using pressurized air to safeguard the scanning electron microscope (SEM).For each sample, we again select five distinct starting positions.Near these positions, images are captured at three different resolutions.The SEM (JSM 7500F, Jeol, Akishima, Tokyo, Japan) is equipped with an energy dispersive X-ray (EDX) spectroscopy unit (Quantax, Bruker Corporation, Billerica, MA, USA).
SEM coupled with EDX enables the generation of false-color images at high resolutions.The operational parameters for SEM and EDX are detailed in Table 2. Subsequently, the differently colored particles in these images are designated as regions of interest (ROIs) using ImageJ (Wayne Rasband, NIH, Kensington, MD, USA).ImageJ also facilitates the extraction of particle data from these manually delineated ROIs, including measurements like Feret and projected area equivalent diameter.The projected area equivalent diameter is the diameter of a circle with an equal area to the projected area, which an investigated particle presents in an image.For reasons of simplicity, this equivalent diameter will be referred to as the area equivalent diameter d A from now on.Matlab is employed for calculating particle distribution data, which include the geometric mean diameter and geometric standard deviation, and for generating histograms of particle counts.Figure A3 in the Appendix A depicts the overall process of how the characteristics of feed and product material are obtained alongside a schematic of the prediction process.

Results
This study focuses on the fractionation of particles based on two primary properties: their material and their size.The results of both material and size analyses are presented below, starting with the size distribution of the feed material.

Particle Size Analysis
A representative image of the feed material is depicted in Figure 6a, while Figure 6b repeats this for a sample of the product stream taken at 400 Pa.Even without the false-color enhancement achieved by combining SEM with energy dispersive X-ray spectroscopy (EDX), one can readily distinguish between the particle species.Copper particles appear as spherical shapes, while silicon particles exhibit a characteristic polygonal morphology.SEM image analysis enables the measurement of various diameter types, such as Feret diameters of the longest or shortest axes, or area-based measurements, among others.
For spherically shaped particles, such as the Cu particles in this study, these diameters are often of nearly equal value, but for differently shaped particles, such as the polygonalshaped Si particles, this is not necessarily the case.The fractionation process is based on the relaxation time of the particles and thus is a function of Stokes diameter.It was found that from the diameters obtained by SEM image analysis, the area equivalent diameter approximated the selected Stokes diameter better than other diameters such as the Feret diameter.An exemplary result of the distribution of the area equivalent diameter is presented in Figure 7b, while a more detailed discussion about the particle morphology is found in Section 8.4.The results of the particle size analysis are also summarized in Table 3 below alongside the model predictions for the optimally focused Stokes diameter.The product particle size distribution is assessed by comparing samples from the feed material to those collected from the product stream downstream of the classifying aerodynamic lens (CAL).A histogram representing a product sample obtained at 400 Pa is presented in Figure 7b.

Material Composition Analysis
After the investigation in particle size of feed and product distributions, their material composition is analyzed using EDX and compared to a prediction made by the analytical fractionation model.

Feed Material Composition
The material composition of the feed material is summarized in Table 4.As anticipated, the mass percentage distribution of the dry feed material mixture closely approximated a 50% Cu and 50% Si composition.Remarkably, the mixture's composition remained relatively consistent even after aerosol dispersion and collection via filtration using the feed reference filtration sampling method (described in Section 5.2) operating at atmospheric pressures.A reference sample collected at 300 Pa following passage through a critical orifice exhibited a slightly greater difference in composition compared to the reference sample obtained at 1 atm.Measurements from different points on the samples yielded slightly varied mass percentage values, resulting in an observed variance within the sample.While Table 4 provides material compositions for multiple measurements on individual samples, reference feed samples were gathered and measured at each CAL operating pressure.An average of all these feed samples was employed as input for the analytical fractionation model, resulting in an average copper content of 45.19 wt%.

Product Material Composition
Figure 8 presents the product material composition prediction as a function of the operating pressure alongside the actual measured product composition for operation pressures up to 600 Pa.With increasing pressure, the Si fraction decreases, while the Cu fraction increases.The fractionation model also effectively predicts the mass fractions attained at operating pressures higher than 600 Pa.However, these results are not included here, because operating the CAL at pressures above 600 Pa would primarily focus on particles with aerodynamic diameters exceeding 7.5 µm (equivalent to Stokes diameters of 3.5 µm and 0.9 µm for silicon and copper, respectively).It is worth noting that the distribution of the feed materials indicates the scarcity of particles of these sizes.
The x axis in Figure 8 can be interpreted as the particle relaxation time using Equation ( 1), presenting a curve that closely resembles a typical Tromp curve, often used in fractionation method analysis [23].However, the predicted product mass fraction is strongly influenced by material properties, especially the width of the distributions.In this case, as the operating pressure increases, the predicted product mass fraction for copper approaches 100%, while the silicon fraction approaches 0%.
As previously mentioned, the results of the energy dispersive X-ray spectroscopy (EDX) material composition analysis are depicted in Figure 8.In general, the measured product mass fractions align with the trend indicated by the model prediction.However, the product mass fractions do not fully reach 0% or 100%.Furthermore, the change in product mass fraction commences earlier than anticipated by the fractionation model.
The difference between measurements at 400 and 500 Pa is relatively small, possibly attributed to the decreasing number of particles in the samples.

Discussion
A comparison between the histograms of feed Figure 7a and product Figure 7b samples shows that the overlap between the distributions of copper and silicon, visible through the mixed colors, is decreased in the product sample.The overall width of the single material distributions also decreases from feed to product.
While the fractionation model and experimental results exhibit alignment in their general trends, several significant discrepancies emerged.This section delves into potential explanations for the disparities between the theoretical model and real-world outcomes.

Potential Sources for Measurement Errors
Particle counting presents a formidable challenge due to its time-consuming nature and susceptibility to user bias introduced by both the microscope operator and the image annotation software.In an optimal scenario, the particles would be well separated on the images.However, this case involved particles with overlapping features, adherence to one another, and the presence of agglomerates.Consequently, only those particles of which the shape could be confidently discerned were counted.This led to a significantly lower count, resulting in an unreliable statistical basis.Robust statistics typically demand a minimum of well above 1000 counts, and some studies even suggest count numbers of over 60,000 [24].However, in some instances, less than 10% of that first threshold number could be counted.
Notably, the particle counting was performed on the same samples subjected to EDX analysis.In this context, the particle mass was more than sufficient to yield reliable results.The error output from the EDX software remained consistent across all measurements (silicon: 0.9313% points; copper: 2.486% points), with the standard deviation across various measuring points within a sample ranging from 1.36 to 6.58.The magnitude of deviation depended on the extent of sample coverage.Besides errors introduced through measurement methods, other sources contribute to deviations between the theoretical model predictions and the actual fractionation process.These factors include variations in process parameters and, to a lesser extent, discrepancies between the geometries employed in models and CFD simulations compared to the physical CAL.
Pressure monitoring during fractionation experiments revealed fluctuations, with deviations in the order of a few tenths of a millibar over time; while the sheath flow rate was controlled and monitored, the aerosol mass flow rate was regulated using a critical orifice, allowing measurements only before and after the experiments.
An additional concern could be the purity of the particles themselves.Cu and Si are prone to surface oxidization.However, only trace amounts of oxygen were observed in the EDX. Figure A2 in the Appendix A illustrates how the model prediction changes when the particle densities are altered to reflect the materials (ρ CuO = 6.315 g/cm 3 and ρ SiO2 = 2.196 g/cm 3 ).The prediction based on fully oxidized particles seems to agree better with the EDX data, especially for the samples taken at 400 Pa.However, this would contradict the EDX measurements of the pure materials, given in Table A4.Severe oxidization during the aerosolization process also seems unlikely due to room temperature being used and nitrogen being used as the carrier gas.

Limitations of the Fractionation Model
The assumptions and restrictions going into the analytical fractionation model, listed in Section 2.3, impact the predicted results and their accuracy.
Notably, Figure 8 presents data for pressures up to 600 Pa.Although the CAL can operate above that pressure, these results are omitted here because both particle size distributions tailor off at this point, which significantly reduces the sample size.
Compared to alternative methods such as stochastic methods based on the law of large numbers (Monte Carlo method) [25,26], the presented method requires less computing power and thus does not need to be run on a graphics card.This comes at a loss of individual particle properties, however, as we employ ensemble descriptions instead of individual particles.At this stage, the conversion from particle number to particle mass distribution requires that all particles of one species share the same (bulk) density.Depending on several factors, individual particles are more or less likely to differ from bulk density in their effective density.In short, this model only considers the Stokes diameter of the particles.In our opinion, leaving out the effective density has a considerable impact on the accuracy of the prediction, as it also interacts with other factors such as the particle shape, agglomeration, and fractal dimension.

Particle Losses
The particles under investigation in this work exhibit diameters of several hundred nanometers or more, rendering diffusion-induced particle losses relatively inconsequential.However, losses stemming from impaction or interception are undeniably more than mere possibilities [27].Furthermore, filtration efficiency, representing a composite of all three loss mechanisms, becomes a pronounced function of particle size within the particle size range of approximately 1 µm.Notably, the analytical fractionation model does not simulate particle losses comprehensively; it solely incorporates some losses in the height of the CAL transfer function.Estimating losses occurring at the CAL entrance, specifically the critical orifice, proves to be challenging.Additionally, potential losses might occur beyond the lens orifice itself, primarily due to the presence of vertices, which, although the flow is predominantly laminar, could trap particles, especially smaller ones.

Influence of Particle Morphology
As seen in Table 3, there is no agreement between the model prediction of optimally focused Stokes diameter and the measured area equivalent diameter.One of the sources of this discrepancy is the limited number of particles counted at the lower and higher ends of the investigated pressure range, which is a result of the already narrow initial particle size distributions of the feed material.Another factor is the difference in particle shape, which neither Stokes nor area equivalent diameter account for.The shape of a particle significantly impacts its aerodynamic properties.However, translating particle shape into a simplified model is a complex endeavor.Primary challenges arise from the difficulty in observing particle shape.When particles are analyzed through optical techniques such as SEM or transmission electron microscopy (TEM), only twodimensional images are accessible.During SEM image analysis, particle circularity f C , in the form of the isoperimetric quotient, was assessed alongside their area and Feret diameter.The circularity is a function of the area and perimeter of the particle.The relationship between particle shape, material, and size is depicted in Figure 9.There is a clear correlation between material and circularity.However, converting the two-dimensional circularity information into a three-dimensional shape factor, directly linked to particle settling velocity, remains an intricate task, although some studies exist that relate sphericity to circularity for various general particle shapes [28,29].Reliable three-dimensional information on particles is obtainable through only a limited number of methods, such as X-ray tomography.However, these methods often exhibit limitations in observable particle sizes, causing a gap between particle sizes ranging from several hundred nanometers to a few micrometers [30,31].Additionally, particle shapes exhibit high individuality and noncontinuous distribution, unlike other properties such as size.Particle size itself is an abstraction from shape, akin to measures often used like circularity, concavity, and the like.The use of the Stokes equivalent diameter for the model calculation exposed the challenge in obtaining the distribution of that particle diameter, especially at the low sampling pressures, which the CAL fractionation process requires.Moreover, the diameters obtained from SEM analysis are area-based and necessitate conversion to the Stokes diameter for model application.The inadequacy of this conversion likely contributes significantly to the disparities between model predictions and measurements in this work.

Conclusions
Classifying aerodynamic lenses demonstrates substantial potential for material fractionation, even when dealing with mixtures of particles possessing rather similar relaxation time distributions.Particularly, at lower operating pressures below 300 Pa, the CAL efficiently separated silicon from copper particles, as indicated by the results presented in Figure 8.The CAL's narrow and differential transfer curve facilitates exploration of diverse focused combinations of particle size and material, achievable by selecting a specific relaxation time.This is accomplished by controlling the operating pressure, which offers rapid adjustments but necessitates a certain level of pumping power.Although the low Reynolds number criterion imposes some limitations on the maximum flow rate through a CAL, techniques such as parallelization and alternative designs [32] hold promise for future improvements in this regard.
The primary obstacle impeding progress in low-pressure aerodynamic fractionation research at this time is the lack of online measurement methods applicable in the pressure range of 100 to 1000 Pa and for particle sizes ranging from 100 nm to 5 µm [13].This limitation is especially pronounced in the realm of particle counting.Optical detection methods often prove inadequate for particles below 500 nm, and techniques aimed at increasing particle size through methods akin to condensation particle counters (CPCs) are hindered by the requirement of achieving oversaturation, an unattainable feat at such low pressures.The second most commonly applied method for particle counting relies on electrometers, necessitating particle charging.However, particles exceeding 100 nm in size often carry multiple charges, rendering the counting process complex.
An area that, to the best of our knowledge, remains underexplored is the determination of mass load limits for aerosols entering a CAL (or regular aerodynamic lens, for that matter).Future research endeavors should encompass investigations into the fractionation of more complex material mixtures involving more than two components, as well as a dedicated exploration of particle shape as an individual particle property.
In summary, the exploration of multidimensional fractionation approaches, particularly in the context of low-pressure aerodynamic fractionation, holds promise for precise material fractionation.
Funding: This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in the priority program 2045 "Highly specific multidimensional fractionation of technical fine particle systems" under grant number 382122643.

Data Availability Statement:
The dataset and code are available on request from the authors.

Conflicts of Interest:
The authors declare no conflicts of interest.

Symbols
The following mathematical symbols are used in this manuscript: The results of an EDX analysis of pure powder samples are given in Table A4.Oxygen is only present in minor quantities.Tungsten was included to check for contamination through previous experiments.The error is given as the 1 − σ estimation by the EDX instrument.

Figure 1 .
Figure 1.Pseudocolor plot of two-dimensional transfer function of a CAL operated at 400 Pa focusing pressure and 4 × 10 −5 kg/s total mass flow rate as a function of particle size d p and density ρ p , resulting in a typical "banana" shape.

Figure 2 .
Figure2.Particle trajectories as a function of the Stokes particle diameter over contours of gas velocity magnitudes for a stream of 2 slm total through a CAL operated at 400 Pa.The y axis is scaled by a factor of 10 for better visibility.Stokes diameters of 100, 400, and 800 nm were chosen to illustrate the possible outcomes of particle focusing.

Figure 3 .
Figure 3. Transfer curve of CAL as a function of normalized Stokes number Stk p /Stk L , obtained by particle tracking within a CFD model of the CAL operated at 400 Pa and fitted with the transfer function from Equation (9).

Figure 4
Figure4depicts a schematic of the CAL, presented as cut through its longest axis, thus revealing inner details of the involved parts.

Figure 4 .
Figure 4. Schematic of the classifying aerodynamic lens.Values for characteristic parameters are found in Table A1 in the Appendix A. The schematic is not proportional to the actual dimensions.Pinking is used to indicate a break introduced for readability.

Figure 5 .
Figure 5. Setup for the fractionation of aerosolized Cu and Si powders, including a small schematic for the principle of fractionation by relaxation time.Refer to Table A2 for a definition of the symbols used in this figure.

Figure 6 .
Figure 6.Representative SEM images of (a) feed stream, sampled at 1 atm and (b) product stream, sampled at 400 Pa, gathered via filtration at 1 atm and 400 Pa, respectively.

Figure 7 .
Figure 7. Histograms of projected area equivalent diameter distribution of (a) feed stream, sampled at 1 atm, and (b) product stream, sampled at 400 Pa, gathered via filtration at 1 atm and 400 Pa, respectively; analyzed using SEM and ImageJ; normalized by the total counts for each material.

Figure 8 .
Figure 8.The measured product mass fractions are compared to the prediction by the model.Measurements are given as points and the model predictions are given as dashed lines for copper in pink and for silicon in purple.

Figure 9 .
Figure 9. Observed shape characteristics of the feed material, shown as bivariate histogram of area diameter and circularity.

FigureFigure A2 .
FigureA2below illustrates how the model prediction changes when the particle densities are adjusted to those of fully oxidized particles.

Table 1 .
Material properties as given by manufacturer.

Table 2 .
SEM and EDX operation parameters.

Table 3 .
Model prediction of optimally focused Stokes diameter in comparison to the results of the SEM image analysis.Samples taken at 1 atm are considered feed material samples.Samples taken at lower pressures are product samples.
* Prediction is given as center of transfer function.Diameters 25% smaller or greater than the given value fall into the range of finite transfer probability.

Table 4 .
Influence of sampling method on observed material composition of feed material.
Appendix A.4. Presence of Other Elements in the Cu and Si Powders * Adjustable parameters.

Table A4 .
Mass percentages as result of EDX analysis of sample of pure materials, including oxygen and some additional elements, present in the samples.