Instant acquisition of high resolution mobility spectra in a differential mobility analyzer with 100 independent ion collectors: Instrument calibration

Abstract

A parallel plate differential mobility analyzer (DMA) having 100 independent current collectors is calibrated to relate the axial distances L_n between the inlet slit and the detector position to the particle mobility Z at given voltage difference V and sheath gas flow rate Q. Calibrating species are tetraheptylammonium bromide clusters (THABr) and polyethylene glycol (PEG35k, 5 nm in diameter), generated by a bipolar electrospray source, and purified in a cylindrical DMA. Gaussian fitting of the raw discrete mobility spectra in the form of ion current I_n versus collector position L_n, I_n(L_n), yield the mean value L_o of the collector position maximizing the signal for a given ion. The many (Z,V,L_o) triads obtained at given Q from many different DMA voltages and standard mobilities collapse into a single 1/(Z_iV_j) vs L_o curve when slight adjustments are made to the Z_i. For different flow rates, Q/(Z_iV_j) vs. L_o curves collapse also, as long as the peaks are moderately narrow. However, for sufficiently small Q/Z, the THABr cluster peaks become broad, and the curves Q/(Z_iV_j) vs. L_o cease to collapse precisely. In contrast, the data for PEG show that this behavior is not a low-Q (Reynolds number) effect from the growth of the two lateral boundary layers, but is rather due to the broad and non-Gaussian peak shapes obtained at low Q or high Z. The calibration is accordingly unaffected by the Reynolds number. This simplicity was unexpected, given the three-dimensional flow in this DMA with growing lateral boundary layers.

Copyright © 2020 American Association for Aerosol Research

EDITOR:

• Kihong Park

1. Introduction

We have recently reported the development of a parallel plate differential mobility analyzer (DMA) having 100 independent collectors on the second electrode, each connected to an operational amplifier that reads continuously the collected current (Perez-Lorenzo, Amo-Gonzalez, and Fernandez de la Mora 2019). This multi-channel DMA (M-DMA) hence produces full high resolution spectra in real time. The M-DMA was based on a commercial instrument (SEADM’s parallel plate DMA P5) whose lower electrode was made completely flat, and could be substituted by an electronic circuit board containing the 100 collector strips separated from each other by insulating strips. The metallic collector strips are 35 µm high laying over an insulating layer, which produces 201 insulator-conductor steps along the lower M-DMA electrode. Given that the M-DMA operates at Reynolds numbers (Re) often in excess of 100,000, a risk existed that the flow would become turbulent. Fortunately no indications of flow instability were detected even at the largest flow rate achieved (>1000 L/min). This favorable feature is likely connected to the small height of these steps (having a modest local Re), and is further associated with the stabilizing effect of the slightly convergent geometry of the DMA channel. Parallel plate DMAs have reached an outstanding performance level, with resolving powers in excess of 100 (Amo-González and Pérez 2018), and high transmissions, even of high mobility ions (Rus et al. 2010, Javaheri et al. 2008). They are also easily connected in tandem with minimal losses (Amo-González and Fernandez de la Mora 2017; Zamora et al. 2018; Amo-González et al. 2018; Amo-González et al. 2019). However, the flow is neither axisymmetric nor strictly planar, so the relation between the mobility Z, the DMA length L, the voltage difference V and the flow rate of sheath gas Q may not conform to the familiar Knutson-Whitby (1975) form (1) $$\mathit{ZVL}/Q=\mathrm{ }K=\text{constant}.$$(1)

One potential reason for the lack of two-dimensionality is that the side walls have growing boundary layers, which would normally result in a downstream acceleration of the flow in the center region of the channel, equivalent to an axial evolution of the flow rate per unit channel lateral width W. This effect has a dependence on Re, hence on Q, whence the geometric constant K might perhaps have a Reynolds number dependence, even in a DMA of constant cross section A. The situation is further complicated in the M-DMA prototype used here, because its slowly converging section implies an axially shrinking area A(x). Accordingly, one could expect that the geometric constant K would depend slightly not only on L, but perhaps also on Q: ZVL/Q = K(Q,L). For a conventional parallel plate DMA with a single outlet slit, these circumstances have posed no serious challenge, since K would at most depend on the single variable Q. K can accordingly be readily determined by a single calibration at given Q by use of an ion standard of known mobility. In our multidetector instrument the need to calibrate 100 different detectors at each flow rate becomes less trivial. Practical use of this instrument in precise mobility studies at high resolution therefore calls for a detailed calibration study such the one undertaken here. The dependence Z(L) we report is also of interest in planar DMAs with a conventional outlet slit. Furthermore, the effect of growing lateral boundary layers has never been previously studied in planar DMAs because they are relatively irrelevant at the high Reynolds numbers at which they are operated. Since such viscous effects gain importance at decreasing Q, our findings are also of interest to planar DMAs having an outlet slit.

The instrument described here is by no means the first DMA of its kind. DMAs having multiple detectors have been developed commercially, including the Fast Aerosol Size Spectrometers from Cambustion (DMS500 model) and TSI (3091 model). Both are widely used to capture particle size distributions produced in relatively fast transients, for instance in automobile exhaust emissions. Both instruments operate at modest flow rates, with no published data available on conditions under which turbulent transition might arise. They are accordingly not ideally fitted for high resolution nanoparticle studies. None of these prior instruments takes advantage of the relatively inexpensive electronic fabrication techniques used here, specifically developed for planar geometries. While planar fabrication methods would readily produce M-DMAs with 500 collectors, their development by conventional methods in the more common cylindrical geometry would be quite challenging.

2. Experimental

A sketch of the experimental apparatus is given in Figure 1. We used a tandem DMA system configured to transmit positive particles from a bipolar electrospray (ES) source capable of yielding primarily singly charged ions of both polarities (Fernandez de la Mora and Barrios 2017). The inner diameters (ID) of the positive and negative silica capillary emitters used were 150 µm and 100 µm, respectively. Both were pulled under a flame producing emitting tip diameters down to 20–30 µm. All efforts to maintain a consistent experimental environment were made. The recirculation pumps for both DMAs were kept at constant revolutions throughout each experiment. The recirculating flow in both closed circuits included a cooler upstream the pump, which maintained fixed temperatures close to room temperature. The M-DMA had a cooler placed downstream of the pump and set at room temperature. DMA₂ had its cooler set below room temperature just before the pump, such that the gas exiting the pump would be close to room temperature. The temperature was monitored just before entering both DMAs. The sample flow rate into each DMA was kept constant throughout each experiment. The positive and negative electrospray currents were kept constant. The DMAs were given ample time to reach an equilibrium temperature before starting an experiment. Temperature was not directly controlled, but was monitored for the room and the DMA exterior/interior. Temperature changes that took place exceptionally in some experiments will be explicitly mentioned when necessary. Samples were changed daily to avoid electrolytic degradation.

Figure 1. Left: Sketch of the experimental apparatus. Right: Detail of the channel geometry.

Samples: Tetraheptylammonium bromide (THABr; Sigma Aldrich) was sprayed in a methanol solution (Baker, 20 mM), with the same solution used in the positive and the negative spray. Poly(ethylene glycol) 35,000 (PEG35k, from Fluka) was sprayed in methanol in the positive mode at 10 µM concentration, with 50 mM of ammonium acetate. Ammonium acetate came from a 5 M aqueous solution (Sigma Aldrich) and was added to reduce the initial size of the sprayed drops. When generating PEG35k particles, the negative spray used the same buffer as the positive spray, though without addition of PEG35k. This then resulted mainly in small solvated acetate anions that effectively reduced the charge of the originally multiply charged PEG cations, without appreciably modifying their size distribution.

DMA circuits: The cylindrical DMA (referred to as DMA₂, although placed upstream in the tandem DMA system) is a half-mini DMA including a circularizer ring to make the entering sample flow highly axisymmetric (Fernandez de la Mora 2017). It was run in closed loop with fixed sample flow rate q₂ in and out between 5.5 and 7.5 SL/min (Table 1), and sheath gas flow rate Q₂. Its voltage V₂ was selected to pass a single species into the parallel plate M-DMA. The voltages V₂ selected in each case are collected in the V₂ column in Table 1. The M-DMA takes a sample flow q₁ reported in Table 1. This flow enters into a small chamber immediately upstream the inlet slit to the M-DMA through a tube with 1/8” inner diameter. This small choice of diameter was a considerable blunder, which resulted in a jet dynamic pressure 16 times greater than if the tube ID had been twice as large. The penetration of this jet through the slit of the M-DMA led to an asymmetric and broadened peak shape, especially at modest values of the sheath gas flow rate. Unfortunately this detail escaped our attention until the study was completed. In order to moderate the loss of resolving power associated with the non-uniform penetration of this jet through the central region of the inlet slit, q₁ is generally considerably smaller than q₂. In some cases q₁ was null (the ions being drawn into the M-DMA by its internal electric field penetrating through the slit), though we cannot preclude some periodic flow ingestion due to a possible unsteadiness of the flow at the inlet slit interface between the incoming jet and the sheath gas flow. The balance of flows q₂-q₁ is removed from the closed circuit through an open port in the small chamber preceding the inlet slit of the M-DMA. In future work, besides using a wider aerosol jet, there are alternative strategies to decrease the sample flow rate through DMA₂ while maintaining sufficient transmission for tandem DMA measurements. One consideration is that the losses are not primarily in the connection between both DMAs. They are more serious in the region upstream of the first classification, where space charge and recombination of oppositely charged particles play a larger role. It is therefore possible to use the split flow strategy of Chen and Pui (1997) where the aerosol flow passing through the inlet slit of DMA₂ is less than the aerosol flowing through the bipolar ES chamber, the balance going through the annular chamber preceding the inlet slit and leaving the DMA as waste sample.

Table 1. Characterization of the various experimental series collected.

Experiment		V1 Range [kV]	V1 step [kV]	V2 [kV]	q1 [SL/min]^a	q2 [SL/min]^a
THAB150	Dimer Trimer Pentamer Hexamer Heptamer Octamer Nonamer Decamer	1.2–2.6 1.6–3.2 2.0–4.2 2.4–4.8 2.6–5.0 2.8–5.0 3.0–5.0 3.2–5.0	0.2	0.540 0.675 0.886 1.008 1.127 1.225 1.330 1.430	3.85	7 7 7 7 7 7 7 7
THAB200	Dimer Trimer Pentamer Hexamer	1.7–3.8 2.1–4.8 2.8–5.0 3.3–5.0	0.1	0.540 0.675 0.886 1.008	4.00	7 7 7 7
THAB250	Monomer Dimer Trimer	1.5–3.3 2.2–5.0 2.7–5.0	0.07	0.358 0.500 0.670	3.70–4.0	7.1 5.1 7
THAB425	Monomer Dimer	3.1–5.0 4.5–5.0	0.1	0.361 0.543	4.30	7 7
THAB150 wChoke	Dimer Trimer Pentamer Hexamer Heptamer Octamer Nonamer Decamer	1.50–1.50 1.50–1.85 1.50–2.55 1.50–2.90 1.85–3.25 1.85–3.25 2.20–3.95 2.20–3.95	0.3	0.520 0.652 0.863 0.975 1.090 1.190 1.280 1.375	3.85	7 7 7 7 7 7 7 7
PEG150mono	Monomer	2.8–5.0	0.2	4.40	0	5.9
PEG250xql	Monomer^+2Monomer^+3Monomer^+4	3.0–5.0 2.0–4.6 1.6–3.6	0.2	2.15 1.43 1.09	0	5.5 5.5 5.5
PEG250xqlow	Monomer^+2Monomer^+3Monomer^+4	2.0–4.6 1.4–3.0 1.0–2.2	0.2	2.12 1.42 1.07	0	5.5 5.5 5.5
PEG250xqlower	Monomer^+2Monomer^+3Monomer^+4	1.6–3.6 1.2–2.4 0.8–1.8	0.2	2.12 1.42 1.07	0	5.5 5.5 5.5
PEG250xqlowest	Monomer^+2Monomer^+3Monomer^+4	1.4–3.0 1.0–2.0 0.8–1.6	0.2	2.10 1.41 1.06	0	5.5 5.5 5.5

^a Standard L/min, corrected to 0 degrees C and atmospheric pressure (8% smaller than the room temperature flow rate).

In contrast to conventional DMA operation, the sample flow q₁ entering through the inlet slit of the M-DMA is not drawn into a detector through an outlet slit. Rather, an identical flow rate is steadily removed from the closed sheath gas circuit through a second opening made considerably downstream from the M-DMA. Therefore, unlike DMA₂, the sample flows in and out of the M-DMA are generally not matched, as the latter is always null, while the former was not null in the experiments with THABr clusters. This unconventional feature affects the calibration procedure because the DMA response does not depend only on Q₁, but on a combination between Q₁ and q₁ which for cylindrical and two-dimensional DMAs not drawing a mobility-selected sample is Q₁+q₁/2 (Knutson and Whitby 1975). In our situation the slit does not occupy the whole 17 mm width of the DMA at the axial position of the inlet slit, but only 7 mm, whence the flow rate relevant to the calibration is Q₁+17q₁/14. In the most unfavorable case (experiment THAB150wchoke) the relative correction is 17q₁/(14Q₁) = (17/14)3.85/(280*0.653) = 2.56%. This shift is typically smaller than the ambiguity involved in the determination of Q. Therefore, in what follows, we will ignore this small correction and speak only of the sheath flow rate, Q₁ or simply Q. This does not involve any unnecessary simplification of the real problem, as long as what we shall subsequently call Q is taken to mean Q₁+17q₁/14.

For given sheath gas flowrate Q₁ in the M-DMA, we determine the range of V₁ voltages (limited to 5 kV by our power supply) for which the species selected on DMA₂ falls within the sensing area of the M-DMA. We then collect mobility spectra at a series of equally spaced V₁ values within this range, as indicated in columns V₁ Range and V₁ step in Table 1. We capture 19 full spectra per second for a total acquisition time of 1 s (19 individual samples) at each V₁ value, using home-made software. The operation is repeated with several clusters, as also indicated in Table 1.

Detectors: The axial distances L of the centers of the 100 detectors from the center of the inlet slit are specified in the fabrication drawings as follows: (2a) $$L_1=15.885\text{mm},$$(2a) (2b) $$L_n=L_1+0.204\left(n-1\right)\mathrm{ }\text{mm},\mathrm{ }2\le n\le 23\mathrm{ }\left(L_{23}=20.373\text{mm}\right),$$(2b) (2c) $$L_n=20.{578\mathrm{ }\mathrm{e}}^{1.01(n-24)},\mathrm{ }24\le n\le 100;\mathrm{ }\left(L_{100}=44.001\text{mm}\right),$$(2c) with rounding errors of 1 µm. The real origin of the circuit may differ by ±50 µm from these values, primarily from imprecision in installing the collector on its seat on the M-DMA. This ambiguity affects all the detectors almost equally as a mounting translation. The initial step in L of 0.204 mm is fixed by the thicknesses of 0.153 mm and 0.051 mm for the insulator and the collector strips, respectively. The increasing step in L following the 24^th collector is chosen such that the L_n values change by 1% between consecutive collectors. For n ≥ 24 the insulator width remains fixed at 0.152 mm, while the width of the collecting metallic strip increases with n. The opening of the inlet slit is w = 0.6 mm, wider than the distance between any pair of adjacent collectors. However, the value of the electric field E within the DMA is considerably larger than the value E_o outside, which narrows down the particle beam width within the DMA by a factor (E_o/E)^1/2 (Vidal-de-Miguel et al. 2012), enabling resolving powers >100 in the DMA model having an outlet slit at L = 4 cm (Amo-González and Pérez 2018).

Voltage Calibration: In order to compare the actual voltage difference V₁ across the electrodes of the M-DMA with the voltage assigned to it by the computer control, a calibrated multimeter was used directly for V₁ < 500 Volt, and through a calibrated high voltage probe of 1000:1 at higher voltages. The overall error found was at most ±0.64%, and ±0.47% between 1.5 and 5 [kV] where the vast majority of experiments were performed. As a result, no voltage corrections were implemented in the data analysis.

Detector sensitivity. Figure 2 shows the set of spectra obtained for the dimer ion of THABr at a given Q₁, representing the current I_n (fA) collected in each detector versus the axial position L_n of the detector center (rather than the physically less relevant detector number n). The interpretation of the current I_n in terms of the ion concentration N is complicated by the discrete nature of the collectors. However, since all ions move normally to the collector at a uniform speed ZV/Δ, the collected current is (3) $$I_n=ze{A}_{n}NZV/\Delta$$(3) where Δ is the distance between the two DMA electrodes, e is the elementary charge, z the number of charges on an individual ion, N is the average ion concentration immediately above the detector, and A_n is the n-dependent area spanned by the detector plus one neighboring insulator. The axial dimensions of the detectors have been described following Equation (2)(2a) $$L_1=15.885\text{mm},$$(2a) . They occupy the full width W(x) of the channel (they even extend under the side insulator). From the I(L) data we determine the L_o values at each peak center via Gaussian fits. Strictly speaking the quantity that is theoretically represented by a Gaussian distribution is N ∼ I_n/A_n rather than I_n. However, the small (1%) variation in A_n between neighboring detectors results in minimal changes in the mean peak position whether the A_n correction is implemented or not. This analysis then results for a given flow rate in a set of (Z, V_, L_o) triads. In the spectra of Figure 2, each peak contains data points from 6 or more different collectors, which suffice to infer an accurate L_o via Gaussian fitting.

Figure 2. Raw mobility spectra obtained for the dimer ion for experiment THAB250 with a fixed value of Q/Q_o = 1.92 and V₁ values ranging from 2.2 kV to 5 kV, with voltage steps of 70 V. The continuous lines are Gaussian fits to the data, from which a mean value for L is obtained. The vertical coordinate is in fA.

The detector currents I_n are obtained from different amplifiers, whose gains are not identical. The main source of variation in sensitivity is the amplifier resistor R_n, for which the manufacturer gave 50 GΩ with a 30% error. In spectra taken at relatively low flow rate of sheath gas, mobility peaks are broad, spanning many detectors. Substantial spikes are seen in uncorrected spectra representing collector voltage versus collector position L. All the spectra presented accordingly represent the collected current based on the detector voltage times the measured amplification, where this spikiness is greatly reduced. The amplification was determined as follows: First we calibrate a single resistor (0.76 TΩ) with a high precision reference electrometer (Keysight B2987A), and employ it to calibrate all amplifiers. Using a known voltage source in series with the calibrated resistor, a known current (6570fA) is injected at each collector. The resulting change in amplifier voltage is then measured through the acquisition hardware and software used for the experiments. Laboratory temperature/humidity variations may affect the calibration as well as the feedback resistor (±1000 ppm/°C). The operational amplifier leakage current is also strongly temperature dependent (aside from the manufacturing tolerance). All in all, if the reference resistor would have a 0.5% change over the course of the tests, as well as another 0.5% for the feedback resistors (5 °C change), and if we take the operational amplifier leakage limits at 25 °C (±20fA LMP7721MA), then we obtain an ambiguity of about 1.5% in the amplification (0.995³).

Figure 3 provides additional representative examples of corrected mobility spectra obtained at all flow rates investigated for some THABr clusters and PEG35k ions.

Figure 3. Selected corrected I(L) mobility spectra each collecting data for multiple voltages at fixed ion and gas flow rate. The vertical scale is in fA. The continuous curves are Gaussian best fits to the data.

3. Results

3.1. DMA calibration with THABr cluster standards

Figure 4 shows the (Z, V, L_o) points collected for singly charged tetraheptylammonium bromide cluster cations in the form Q/(Q_oZV) versus mean collector position L_o. The sheath gas flow rates used in this study are made dimensionless with the value Q_o corresponding to experiment THAB150. The reason is that we do not have an accurate measurement method to determine Q, but can determine relative Q values with fair precision. There are as many of these cluster data series (fixed symbol in the figure) as voltages selected. The experimental voltages are included in Table 1 for each cluster series, by providing the first and last voltage (column 3) and the uniform voltage jump used (column 4). For instance, for experiment THAB150, the dimer data are taken at voltages {1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4, 2.6} (kV). The various data series for different clusters should collapse on top of each other for each Q₁, as long as the assigned Z values correspond precisely to the actual cluster mobilities. These mobilities were measured at ambient conditions in separate experiments with DMA₂. Given the small voltage calibration errors involved, any slight departures of some of the data from a unique curve must be due to errors in the assigned Z. Some slight systematic departures are indeed found for some of the clusters, and are removed for all the cluster series by a slight shift in the nominal mobility. The nominal and adjusted mobilities used in generating Figure 4 are reported in Table 2.

Figure 4. Representation of the group Q/(Q_oVZ) (s/cm²) for singly charged tetraheptylammonium bromide cluster cations. Zn in the legend denotes the mobility of the n^th cluster (Table 2). (a) Data for 8 clusters and multiple voltages at the reference flow rate Q = Q_o (THAB150 experiment), all falling precisely on a single curve. The dashed line is a quadratic curve fit discussed in the theory section. (b) Collapse of data for experiments THAB150, THAB200, THAB250 and THAB450 when selecting Q/Q_o ratios of {1, 1.49, 1.92 and 4.05} (colored symbols). Also included are the data for the smallest flow rate (Q/Q_o = 0.634), which exhibit a clearly higher downward curvature than the other series.

Table 2. Values of the adjusted mobilities Z (cm²/V/s) for the various THABr experiments compared to the nominal value of row 2. The last column reports the Q/Q_o ratio collapsing at small L the various Q_o/(QZV) vs. L curves.

THAB cluster	1	2	3	5	6	7	8	9	10
Z (cm^2/V/s)	0.971	0.654	0.528	0.402	0.355	0.317	0.293	0.269	0.250	Q/Qo
150wchoke		0.654	0.524	0.399	0.352	0.315	0.290	0.268	0.2486	0.653
150		0.654	0.521	0.400	0.351	0.314	0.289	0.2654	0.2465	1.00
200		0.654	0.522	0.4005	0.352					1.49
250	0.971	0.654	0.5115							1.92
450	0.978	0.654								4.055

Figure 4a is taken at the standard flow rate Q = Q_o, whence in reality it is a plot of 1/(VZ) vs L. We note once more that the quantities Q and Q_o referred to here for brevity as sheath gas flow rates are in reality the combination Q₁+17q₁/14 of entering sheath and aerosol flow rates. Operating at different flow rates of sheath gas yields curves 1/(ZV) vs. L similar to that shown in Figure 4a for the standard flow rate Q_o, except that the vertical scale varies with Q. However, choosing the ratio Q/Q_o (unknown here due to lack of accurate flow rate measurements) such that the Q/(Q_oZV) datum at a certain L coincides for the two flow rates, the two full curves (at all L_o values) fall precisely on top of each other as long as the flow rates are large enough for the peaks to be narrow. This is demonstrated in Figure 4b by comparing Q/(Q_oZV) data for experiments THAB150 with corresponding data for experiments THAB200, THAB250 and THAB450 (colored symbols) when selecting Q/Q_o ratios of {1, 1.49, 1.92 and 4.05}. Like the data at the standard flow rate, these other data sets use, at each Q, their own slightly shifted optimal mobilities (Table 2), such as to collapse each into a single curve. The mobility of the dimer was kept as a common standard at all flow rates. Except for a considerable variation in the case of the trimer (Z_max/Z_min = 1.024), the adjusted mobilities established at the various flow rates are almost identical to each other. The exceptionally low adjusted value Z₃ = 0.5115 cm²/V/s found in the trimer data for experiment THAB250 is attributable to the fact that the trimer spectra were recorded with a temperature of 25.7 °C in the M-DMA, while the corresponding dimer and monomer experiments were obtained at 22.7 °C.

Also included in Figure 4b are the data for the smallest flow rate (THAB150wchoke). With the choice Q/Q_o = 0.634, these lower flow rate data are tangent at small L to all the other data, but exhibit a clearly higher downward curvature at larger L. As argued in the theory section, part of the slight downward curvature of all these curves is due to the convergence of the channel. An additional downward curvature would be expected as a result of growing boundary layers near the lateral insulating walls, which would create a downstream widening dead region equivalent to an extra channel area reduction. Because the thickness of this boundary layer decreases with the square root of the Reynolds number (∼Q^−1/2), one would expect the negative curvature to increase at decreasing Q.

Another possible explanation for the anomaly of the THABr series at Q/Q_o=0.634 would be that the assumption of a Gaussian peak shape in the variable L is not exact, not even in the limit q/Q→0. In variable voltage mobility spectra (with a fixed sampling slit position) the current versus voltage curve for mobile ions is only approximately Gaussian, the approximation losing accuracy when Q/Z diminishes and the peaks broaden (see Figure 2 of Rosell et al. 1996). In contrast, the theoretical peak shape in a parallel plate DMA of constant cross section is exactly Gaussian in spectra taken at fixed voltage and variable L. This prediction is not met here for any sufficiently broad peak obtained, as all are clearly non-Gaussian, with a right tail (towards larger L) broader than the left tail. Suspecting that this asymmetry might be due to the divergence of our DMA geometry, we have analyzed in the Theory section the effect of channel divergence on diffusive broadening. We find that the peak shape is then not exactly Gaussian, with a wider tail on the right than on the left. However, for our experimental conditions, the predicted asymmetry is far less than observed, even for the relatively wide peaks associated in Figure 4b with the lowest L datum for Q/Q_o = 0.634 (FWHM_L = 13.8%). We are then forced to conclude that these tails are due to a flow nonideality.

Irrespective of the tail anomaly, the large peak width apparently associated with off-lying points in Figure 4b is primarily due to diffusion. In order to test if these anomalous data are due to boundary layer growth (at low Q) or to excessive peak width (irrespective of Q), we have carried out additional tests with the considerably less diffusive PEG35k particles.

3.2. DMA calibration with the PEG35k sample

As shown in the lighter regions of the two dimensional mobility spectrum of Figure 5, the PEG sample used was fairly monodisperse, producing relatively narrow peaks even without DMA selection. In spite of the inherent polydispersity of this sample, because of its much smaller diffusivity, it produces narrower peaks than the (strictly monomobile) cluster standards at comparable and even lower flow rates. By adjusting the positions and currents of the positive and negative ES emitters we could produce four different PEG35k ions with charge states z = {1, 2, 3, 4}. Their nominal mobilities are collected in the second row of Table 3. They were determined with DMA₂ by comparing under fixed flow conditions the PEG peak voltages V_PEG with the voltages V₂ and V₃ at which the dimer and trimer ions of THABr appeared: Z_PEGV_PEG = Z₂V₂ = Z₃V₃. The mobilities inferred from the dimer and trimer standards differed among themselves by 0.37%, due to the slight difference between the ratio of THABr trimer to dimer taken as standard and the ratio obtained in the calibration experiments (i.e., slight disagreement between Z₂V₂ and Z₃V₃ for the nominal Z₂ and Z₃ and the measured V₂ and V₃). The nominal mobilities of PEG 35 K reported in Table 3 are an average of all the measurements made with both cluster standards. While the nominal values given in Table 3 are for the center of the full polydisperse distribution of PEG35k, in the calibration experiments of the M-DMA, the PEG calibration standard was selected with DMA₂. If the selection voltage is not exactly the peak voltage for the distribution, the selected mobility may stray slightly away from the mean value recorded in the table. For this reason, we slightly shift the nominal mobilities to collapse all the data at given Q into a single curve. The procedure is very much as used before with the THABr cluster experiments: we only adjust Z for charge states 1, 3, 4, while keeping fixed the mobility of charge state 2. In all PEG experiments the maximal variation of adjusted mobilities across all the experiments was Z_max/Z_min = 1.0106 (triply charged PEG35k data).

Figure 5. Two-dimensional spectrum of PEG35k, plotting ion current (color scale) versus the spectral variables of the two DMAs. The three lighter regions correspond from right to left to charge states z = 2, 3, 4. The light line clearly visible going through these three PEG35k peaks is the locus of ions having the same mobility in both DMAs, and could be used for DMA calibration.

Table 3. Electrical mobilities Z (cm²/V/s) used to invert the various PEG35k experiments at different Q/Q_o for variously charged PEG35k^+z ions (z = +1 to +4).

z	+1	+2	+3	+4	Q/Qo
Nominal^a	0.0786	0.158	0.236	0.312
250xql^b		0.158	0.238	0.3127	0.6517
250xqlow^b		0.158	0.2365	0.313	0.425
250xqlower^b		0.158	0.2355	0.313	0.337
250xqlowest^b		0.158	0.2355	0.313	0.291
150mono^b	0.0786				0.247

^a Determined by calibration with the dimer and trimer of THABr in DMA₂.

^b Slightly shifted mobilities leading to optimal collapse into one curve of all the data at given Q.

Figure 6a compares Q_o/(QZV) data for PEG in charge states z= {+2, +3, +4} (experiment PEG250xqlow, with Q/Q_o = 0.336) with those of Figure 4a for THABr. While Q in the PEG35k experiment is lower than in the anomalous data found in Figure 4b at Q/Q_o=0.634 THABr reference, these PEG35k data collapse quite well over the reference data representing the high Q asymptote. The PEG35k data obtained at Q/Q_o of 0.6517 and 0.425 are equally indistinguishable from the other two included in Figure 6a. Accordingly, there is really no low flow rate (Reynolds number) anomaly at Q/Q_o of either 0.634 or even 0.336. The anomaly seen at the lowest flow rate included in Figure 4b must then be due to the excessively wide non-Gaussian peaks forming under these conditions, which result in an inaccurate experimental value of L_o.

Figure 6. Comparison of the THABr reference data of Figure 4a (open symbols at Q/Q_o=1) with Q/(Q_oVZ) data for PEG35k. Z + z in the legend denotes the mobility of a PEG ion having z elementary charges. (a) PEG35k at Q/Q_o = 0.336 (black filled symbols) showing excellent collapse with the cluster data. (b) Comparison with reference data from Figure 4a of data for all PEG35k experiments when selecting Q/Q_o ratios of {0.651, 0.425, 0.336, 0.289 and 0.2473} (colored symbols). The data for Q/Q_o = 0.289 depart very slightly from the general trend, and those for Q/Q_o = 0.2473 (large filled gray circles) more clearly.

Figure 6b compares the reference data of Figure 6a to all available PEG35k data at decreasing flow rates: Q/Q_o ratios of {0.651, 0.425, 0.336, 0.289 and 0.2473} (colored symbols). The data for Q/Q_o = 0.289 depart very slightly from the general trend, and those for Q/Q_o = 0.2473 (large filled gray circles) more clearly. These very slightly anomalous points are also associated with wide peaks, and, therefore, do not imply a low Re effect associated to lateral boundary layer growth. We therefore conclude that, even at the smallest flow rates studied, the calibration curve Q/(ZV) for this DMA is for all practical purposes independent of Reynolds number. We shall subsequently obtain an approximate value for Q_o of 283.5 L/min, implying that the smallest and largest flow rates studied are 70.14 and 1150 L/min. This corresponds to channel Reynolds numbers of 4580 and 75,140.

4. Theoretical considerations

4.1. Converting the landing position L to a mobility Z

Let us consider a particle streamline starting at the plane of symmetry at the center of the entrance slit of a parallel plate DMA (Figure 1, right), where the flow velocity (U, V, W) is dominated by the x component U, and the field (0, E, 0) is almost exactly uniform, directed in the y direction. There is no z dependence for the velocity field at the symmetry plane, while the velocity in the z direction is null. Accordingly the mean particle trajectory ignoring diffusion obeys the differential equation (4) $$dx/U\left(x,y\right)=dy/\left[ZE+V\left(x,y\right)\right].$$(4)

Let’s make x and y dimensionless with the gap Δ between the two DMA electrodes, and U,V dimensionless with a characteristic velocity U_o to be later defined: (5a–d) $$\xi =x/\Delta ;\mathrm{ }\eta =y/\Delta ;u(\xi ,\eta )=U\left(x,y\right)/U_o;\mathrm{ }v(\xi ,\eta )=V\left(x,y\right)/U_o.$$(5a–d) Then (4)(4) $$dx/U\left(x,y\right)=dy/\left[ZE+V\left(x,y\right)\right].$$(4) becomes (6) $$d\xi /u(\xi ,\eta )=d\eta /[ZE/U_o+v(\xi ,\eta )].$$(6)

The initial condition is (ξ,η) = (0, 0), and the impact position ξ = L/Δ of the ion trajectory at η = 1 must now be a function of ZE/U_o, as well as the Reynolds number, (7a) $$L/\Delta =F\left(ZE/U_o,\mathrm{ }Re\right),$$(7a) (7a)(7a) $$L/\Delta =F\left(ZE/U_o,\mathrm{ }Re\right),$$(7a) may be rewritten as (7b) $$ZE/U_o=G(L/\Delta ,Re).$$(7b)

We may define Re = U_oΔ/ν. However, since U_o is not an experimentally known quantity but the flow rate of sheath gas Q is in principle measurable, we now define U_o = Q/(ΔW_o), where W_o is a characteristic value of the DMA width in the z direction, say at the inlet slit. Therefore, defining the quantity Q’ such that Re = Q’/ν and the voltage difference V between the two DMA plates: (8a–c) $$Q’=Q/W_o;\mathrm{ }E=V/\Delta ;Re=Q’/v,$$(8a–c) (7b)(7b) $$ZE/U_o=G(L/\Delta ,Re).$$(7b) becomes: (9) $$ZV/Q’=K[L/\Delta ,\mathrm{ }Re].$$(9)

This result is the basis for the data analysis followed earlier.

The next level of possible simplification is to note that Re = Q’/ν will be typically fairly large (i.e., Re = 18,523 at Q = Q_o = 283.5 L/min). Accordingly, the expected Re dependence would be slight, approximately reducing (9)(9) $$ZV/Q’=K[L/\Delta ,\mathrm{ }Re].$$(9) to (10) $$ZV/Q’=K(L/\Delta )\mathrm{ }\mathit{for}\mathrm{ }Q’/v\gg 1.$$(10)

Interestingly, we have found that this high Re limit applies well even when Re is only a few thousands.

4.2. A proxy value for Q

Short of measuring Q directly, we can establish a quantity Q_a(Q) closely related to it. This proxy quantity we may introduce based on a model where the velocity field is strictly uniform in the y and z directions, while depending slightly on x, such that U(x)=Q_a/A(x). Note that this is not claimed to be true. It is simply a device to define the proxy quantity Q_a as follows. Making use of (1)(1) $$\mathit{ZVL}/Q=\mathrm{ }K=\text{constant}.$$(1) , A(x)dx/Q_a=dy/(ZE), and integrating from the inlet slit to y = Δ gives (11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11)

It is clear that Q and Q_a are not the same thing. In particular, for each of our experimental data sets Q is strictly constant for all (Z, V, L_o) triads, while Q_a may in principle depend on L_. Let us examine this dependence for our experiments.

For our DMA the channel height is constant, Δ = 1 cm, while its width W has a constant semi-angle of convergence α = 1.5° and takes the value W_o = 1.70 cm at the position of the inlet slit. W ceases to evolve exactly linearly past L = 37.4 mm, with a gradual curvature that eventually brings the convergence angle to zero. However, the insulator wall curvature is so small that a fit to the geometry in the region from L = 37.4 to the last detector is indistinguishable from a 1.5° angle. Therefore, over the whole length used for mobility analysis the area and its integral in (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) are: (12) $$\mathrm{A}\left(\mathrm{x}\right)=\mathrm{\Delta }({\mathrm{W}}_{\mathrm{o}}-2\mathrm{L}\mathrm{ }\text{tanα}),$$(12) (13) $$\underset{0}{\overset{L}{\int }}A\left(x\right)dx=\Delta ({\mathrm{W}}_{\mathrm{o}}\mathrm{L}-{\mathrm{L}}^2\hspace{0.17em}\text{tan}\hspace{0.17em}\alpha ).$$(13)

Plotting the right hand side of (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) together with the experimental group Q/(Q_oZV) in Figure 4a while using the true value of α leads to imperfect agreement no matter what value of Q/Q_o is chosen. However, the data agree qualitatively with (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) and (13)(13) $$\underset{0}{\overset{L}{\int }}A\left(x\right)dx=\Delta ({\mathrm{W}}_{\mathrm{o}}\mathrm{L}-{\mathrm{L}}^2\hspace{0.17em}\text{tan}\hspace{0.17em}\alpha ).$$(13) in being well fitted by a quadratic expression in L. The agreement is in fact excellent if we take α = 2.09°, Q_o = 283.5 L/min, as shown by the dashed line included in Figure 4a. The effective angle found differs appreciably from the instrument’s half angle of 1.5°, not too surprisingly given that the model used for the velocity field has no rigorous basis.

In summary, by modifying slightly the definition (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) of Q_a based on an artificial channel angle of 2.09° we obtain a proxy flow rate that is essentially independent of L under all the conditions tested for which the actual flow rate is constant. Unlike all other flow rates previously discussed, Q_a is not an unknown or ambiguous quantity. It is precisely defined by (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) and (13)(13) $$\underset{0}{\overset{L}{\int }}A\left(x\right)dx=\Delta ({\mathrm{W}}_{\mathrm{o}}\mathrm{L}-{\mathrm{L}}^2\hspace{0.17em}\text{tan}\hspace{0.17em}\alpha ).$$(13) with α = 2.09°. For all practical purposes of providing a relation between L and Z, we may now ignore the actual flow rate Q and use Q_a and (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) instead. One still needs to determine Q_a under given flow conditions to make use of (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) . One may simply introduce a mobility standard under these flow conditions, and measure the position L where it lands. The right hand side of (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) is then computable fixing Q_a. As already noted its value is 283.5 L/min for the data of Figure 5a.

Let us now return to the actual flow rate Q and its relation to Q_a. The two flows are in principle different. However, since there is a unique value of Q_a associated to each Q, the two flow rates must be univocally related. We therefore expect that the two Reynolds numbers defined based on the two flow rates would be uniquely dependent on each other: (14) $$Q/(W_{o}v)=F[Q_a/(W_{o}v)].$$(14)

However, we can go beyond statement (14) to show that the two flow rates must in fact be directly proportional to each other. This is clear from the fact that the Q_a/(ZV 283.5 L/min) and Q/(Q_oZV) as plotted in Figure 4a are exactly the same functions of L. Therefore, taking their ratio, (15) $$Q_a/(283.5\mathrm{L}/\text{min})\mathrm{ }=\mathrm{ }Q/Q_o.$$(15)

The flow rates Q_a and Q are accordingly strictly proportional, with a still unknown proportionality constant that could be established by a single experiment.

In summary, the calibration of a DMA with many detectors requires not just one geometrical constant Q/(VZ) = K, but a function K(L). This function we have carefully determined experimentally for our planar DMA, and is fitted with excellent approximation by Equations (11)(11) $$Q_a=\frac{ZV}{{\Delta }^2}{\int }_0^{L}A\left(x\right)dx.$$(11) and (13)(13) $$\underset{0}{\overset{L}{\int }}A\left(x\right)dx=\Delta ({\mathrm{W}}_{\mathrm{o}}\mathrm{L}-{\mathrm{L}}^2\hspace{0.17em}\text{tan}\hspace{0.17em}\alpha ).$$(13) . Given the function K(L), if the operating flow rate Q is not precisely known, the remaining task of relating Z to L at all L values is as simple as the mobility standard calibration often used with DMAs having only one detector (Ude and Fernández de la Mora 2005). This final task gives also the flow rate Q_a, and will give the actual flow rate Q once the constant factor in (15)(15) $$Q_a/(283.5\mathrm{L}/\text{min})\mathrm{ }=\mathrm{ }Q/Q_o.$$(15) is experimentally determined.

We should stress that the simplicity of the calibration procedure described for this DMA is a result of the finding that the function K(L) shows no Reynolds number dependence, even at the lowest flow rates used [Q_a,min (L/min)=283.5*0.2474 = 70.14]. This feature is expected in axisymmetric or strictly two-dimensional instruments, but not so in a three dimensional device having lateral walls. Perhaps this simplifying feature will not be reproduced in other parallel plate DMAs in the absence of two special features of the instrument studied here, both of which reduce the width of the boundary layers. First, the flow is slightly accelerated, second, the DMA length L_max/Δ = 4.4 is relatively short.

4.3. Diffusive broadening in slowly converging DMAs

The anomalous mobility tails observed would not be theoretically expected in a DMA with uniform cross sectional area A. In order to establish if these wide tails are due to the x-dependence of A(x), the online supplementary information Section analyzes the problem of diffusion broadening for slowly varying A(x). We conclude that a slightly asymmetric mobility distribution is expected at modest sheath gas flow rates, but its predicted magnitude is much smaller than that observed. The prominent tails found at moderate Q must accordingly be due to flow nonidealities.

5. Conclusions

Contrary to our expectations and to misleading initial symptoms, a single quadratic calibration curve Q_a/(ZV) versus L is seen to accurately represent all our data independently of Reynolds number, enabling relating the particle mobility to the position of each collector strip. Apparent shifts from this general trend are observed at low flow rates in situations yielding wide peaks (FWHM > 10%). These shifts, however, are not due to an additional flow rate (Reynolds number) dependence of the Q_a/(ZV) versus L curve, but to the excessive width of the transfer function for the small particles considered.

The present study suffers from two main limitations. One is the fact that an imperfect connection between the two DMAs has reduced the resolving power of the parallel plate DMA. Another shortcoming is that a single calibration constant required to connect the proxy flow rate Q_a with the actual flow rate Q has not yet been measured. Notwithstanding the still missing Q/Q_a factor, introducing a mobility standard at any unknown flow rate Q and determining its landing position L_o yields the needed Z(L) relation for this particular operational flow rate. Combined with the calibration information provided here (Q_a/(ZV) versus L curve) for all 100 collectors, this single calibration is applicable to all the detectors, yet is as straightforward as the calibration previously applied to conventional DMAs having fixed outlet slit positions.

It is of interest to remark that the present study is the first where SEADM’s high resolution parallel plate DMA has been used to study particles as large as our PEG sample (5 nm in mobility diameter). Figure 3g displays spectra at the largest flow rate used for PEG (Q/Q_o = 0.6517; Q∼120 L/min), demonstrating a resolution in excess of 25. The ion is quadruply charged, and lands at L = 4 cm (the position of the outlet slit in SEADM’s P5 DMA) when the applied voltage is 1.6 kV. At this flow rate, the singly charged ion would be classified at 6.4 kV.

Acknowledgments

Yale’s contributions to this study were funded by SEADM, directly as well as via the loan of a parallel plate DMA modification of their P5 model, suitable to interface with the multidetector collector electrode and electronics. We are thankful to Mr. Michele Genoni and Mr. Derek Kuldinow for their contributions to this study.

Conflict of interest

Following Yale rules Juan Fernandez de la Mora declares an interest in the companies SEADM and NanoEngineering Corporation commercializing the two DMAs used in this study.

Additional information

Funding

We gratefully acknowledge partial support from AFOSR grant FA9550-18-1-0165 and NIH SBIR Phase I Grant Number 1 R43 GM131542-01A1 (National Institute of General Medical Sciences).