# High resolution characterization of a sheathed axisymmetric variable supersaturation condensation particle sizer

Michel Attoui, Luis J. Perez-Lorenzo, Charles Brock, Juan Fernandez de la Mora

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Michel Attoui, Luis J. Perez-Lorenzo, Charles Brock, Juan Fernandez de la Mora. High resolution characterization of a sheathed axisymmetric variable supersaturation condensation particle sizer. Journal of Aerosol Science, 2023, 169, pp.106112. 10.1016/j.jaerosci.2022.106112. hal-04379540

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# High resolution characterization of a sheathed axisymmetric Variable Supersaturation Condensation Particle Sizer

Prepared for submission to J. Aerosol Sci.: 14/September/2022
Revised 21/November/2022

Michel Attoui, Luis J. Perez-Lorenzo<fnref n="2" />, Charles A. Brock<fnref n="3" />, Juan Fernandez de la Mora<fnref n="2" />

## ABSTRACT

Diffusive condensation particle counters (CPCs) injecting the aerosol near the symmetry axis within a larger flow of a vapor-containing sheath gas have previously exhibited a size resolving power considerably lower than theoretically expected. We hypothesize that, reexamining these instruments with particles of improved uniformity in composition, size and shape, should reveal a much better performance, well suited for high resolution particle sizing and basic heterogeneous nucleation studies. Accordingly, we investigate the performance of the Variable Supersaturation Condensation Particle Sizer (VSCPS) of Gallar et al. (2006) with an aerosol of uniform singly charged polyethylene glycol particles (3-9 nm diameter), produced by a bipolar electrospray, and size-selected by a differential mobility analyzer of high resolution. The condensation size resolution obtained with n-butanol vapor almost triples that previously demonstrated for this instrument, exceeds that established for any other CPC, and can apparently be further increased. Flow and temperature field calculations of the maximal supersaturation in the VSCPS would enable basic heterogeneous nucleation studies, as well as high resolution sizing of neutral or charged particles.

<footnote n="1">¹ LISA, UMR7583, Université de Paris, Institut Pierre Simon Laplace, Créteil, France</footnote>

<footnote n="2">² Yale University, Department of Mechanical Engineering and Materials Science, 9 Hillhouse Avenue, New Haven, CT, 06511-8286, USA</footnote>

<footnote n="3">³ NOAA Chemical Sciences Laboratory, 325 Broadway R/CSD2, Boulder, CO 80305</footnote>
# 1. Introduction

The ideal of determining size distributions of ultrafine particles without depending on their relatively inefficient charging followed by mobility classification has stimulated a diversity of studies on vapor condensation with variable supersaturation. Some have employed several condensation particle counters (CPCs) in parallel, each having a different cut diameter (Brock et al. 2000; Kulmala et al 2007; Stolzenburg et al. 2017; Williamson et al. 2018). Others have scanned continuously over the saturation ratio $S$, either relying on moderately fast variation of the proportion of the cold and hot streams combined in turbulent mixing CPCs (Seto et al. 1997; Gamero-Castaño et al. 2000, 2002; Sgro et al. 2004; Vanhanen et al. 2011), or on a slower temperature scanning in thermo-diffusive CPCs (Wiedensohler et al. 1994; Kuldinow et al. 2022). Given the greater ease of calculating the saturation ratio field in laminar CPCs, the study of Gallar et al. (2006) is of particular interest, as they achieved relatively quick scans of $S$ by varying the proportion of a dry and a vapor laden stream via fast-response flow controllers, which they mixed in a laminar thermodiffusive CPC. Their approach is ideally suited, not only for high resolution sizing, but also for basic heterogeneous nucleation studies.

Here we are particularly concerned with the size resolving power that can be achieved via variable-$S$ sizing, first because it is the key to enable basic research on heterogeneous nucleation, and also because there are a number of potential applications for a high-resolution nano-particle sizer in the fields of new particle formation and growth and nano-particle technology. Another application we are pursuing is to increase the resolution with which different viral particles can be distinguished from each other by determining two independent sizes: the mobility size and the condensation or Kelvin size. For non-spherical particles, the details of viral components sticking out of the main body would reflect differently in these two measures of size. It has become apparent in a few recent studies relying on particles of high uniformity (in size, shape, and composition), that the size resolution achievable by condensation may be potentially quite high. This seems to hold not only in the few relatively modern CPCs already investigated with improved test particles (Kuldinow et al., 2022; Fernandez de la Mora et al., 2022), but perhaps even more in previously existing CPCs that had been originally characterized with less homogeneous test particles. For instance, a recent evaluation (Brilke et al. 2020) of two different commercial versions of the widely used ultrafine CPC (UCPC) of Stolzenburg and McMurry (1991), has found that one of them has excellent resolving power. We are also much indebted to Dr. Mark Stolzenburg for sharing his calculations showing very steep theoretical activation curves in the UCPC, and even steeper in its shorter variant developed by McDermot et al. (1991). These two promising instruments share with that of Gallar et al. (2006) and their common ancestor of Wilson et al. (1983a,b) the special feature that the aerosol is injected in the center of a larger flow of vapor rich particle-free air. Consequently, most particles are located very near the axis of symmetry within the condenser, and thus proceed along the streamline where the maximum in the saturation ratio occurs, and where it is most homogeneous in the radial direction. Here we evaluate the size resolving power that might be expected of this family of instruments by slightly modifying and testing the sheathed-flow Variable Supersaturation Condensation Particle Sizer (VSCPS) of Gallar et al. (2006).

# 2. Experimental

**The VSCPS.** The instrument was originally referred to as the Variable Saturation Condensation Particle Sizer (VSCPS) by Gallar et al. (2006). Here it will be more briefly called the CPS, though noting that the full name is important to distinguish it from alternative condensation sizers relying on pulse height analysis at fixed saturation ratio (Weber et al. 1998). The saturator is composed of
a single, heated block including two identical rectangular cuboid volumes, intended to be at the same temperature. One of these volumes contains liquid butanol, while the other is dry (Fig. 1b). They will be referred to as the *wet* and the *dry* volumes. Fast-response mass flow controllers direct air flow through the two volumes, with flow rates $Q_w$ and $Q_d$, respectively. Air passing through the volume containing butanol yields an air/butanol vapor mixture. Ideally the butanol vapor would be at equilibrium ($S_w=1$) at the temperature of the saturator block ($T_s$); we show that this ideal was not reached in the original instrument.

<figure><img src="image_2.png" /><figcaption>Figure 1: Top Scheme (a): Global diagram of the full setup surrounding the CPS, indicating various flow rates. Bottom scheme: Detailed schematic of the CPS with the vertically oriented wicking material in the wet volume (above) and the aluminum wool in the dry volume (below), to increase saturation ratio and thermal homogeneity, respectively. The hot saturator components are represented in gray.</figcaption></figure>

Within the heated saturator block, the flows $Q_w$ and $Q_d$ are mixed well, yielding a combined stream with a saturation ratio $S_m$

$$S_m = w S_w \qquad (1)$$

$$w = Q_w/(Q_w+Q_d). \qquad (2)$$

The quantity $w$ will be referred to as the *wet fraction*. It may vary from 0 to 1, playing the vital role of the spectral variable in this size spectrometer. Mixing is achieved by joining the dry and
wet streams in a tee. To ensure thorough mixing, the tee is followed by a line about 1 cm long
densely packed with aluminum wool (wavy region to the right of Figure 1b).

To this mixed stream of fixed temperature T_s, fixed saturator total flow Q= Q_w+Q_d, and controllable
saturation ratio wS_w, an aerosol flow q is added, injected in the middle of the flow through a
capillary. The injection of the aerosol stream into the center of a sheathing, vapor-laden flow is the
key characteristic of high-resolution, laminar flow CPCs designed for nano-particle detection
(Wilson et al. 1983a,b; Stolzenburg and McMurry, 1991). Prior to this injection, the aerosol flow
traverses the whole length of the heated saturator without contacting either volume, to achieve the
same saturator temperature T_s as the sheath gas, but it contains no working vapor (nor water vapor
either in this work). Therefore, the distribution of vapor immediately after aerosol/sheath injection
is non uniform. Nevertheless, when the ratio q/Q is small, this nonuniformity is quickly dissipated
by vapor diffusion into the dry core. Stolzenburg and McMurry (1991) have therefore argued that
the dilution effect from the dry aerosol affects only modestly the maximal saturation ratio. Here
we will determine this effect quantitatively. The initial gas velocity is also non-uniform, even if
the ratio q/Q is chosen for maximal flow stability, such that the core jet enters the sheath gas stream
at the same mean velocity as the neighboring streamlines. Indeed, the initial axial velocity profile
in the mixing region is rather complex, since both the core and the peripheral flows are confined
by walls that bring the velocity locally to zero. The initial velocity profile will then tend to have
two maxima, one at the center of the aerosol jet, and another somewhere within the sheathing flow.
Furthermore, the finite thickness of the capillary wall results initially in a finite wake of low speed
fluid separating the other two roughly parabolic velocity profiles.

A unique characteristic of the VSCPS is that the aerosol injection region and the condenser are
vertically oriented downward below the saturator, unlike in conventional CPCs. The reason for
this design is that condensed fluid dripping down the walls of the condenser and vertical portion
of the saturator (after mixing Q<sub>w</sub> and Q<sub>d</sub>) would provide wetted walls that would saturate the
combined saturator flow Q<sub>sat</sub>, resulting in an inability to control S in the condenser by varying Q<sub>w</sub>
and Q<sub>d</sub>. Instead, the condensed fluid on the walls of the condenser flows by gravity toward the
optical detector. This fluid is collected by a small reservoir along the wall of the condenser prior
to the focusing of the condenser exit flow into the optical detector, so it does not foul the detector.
This fluid was removed in this work by pumping it out manually with a syringe after taking a few
spectra. A practical benefit of this approach is that co-condensed water cannot contaminate the
butanol reservoir in the wet volume of the saturator, which is a problem for conventional CPCs for
ambient sampling in humid environments.

For sizing purposes, it is not necessary to achieve S<sub>w</sub>=100%. But in order to study heterogeneous nucleation, one needs to know accurately S<sub>w</sub> as a function of Q<sub>w</sub>. For this purpose, it is simplest to be sure that the residence time in the wet volume is long enough to achieve complete saturation at the largest Q<sub>w</sub> to be used. The original CPS instrument had its wet volume filled with aluminum wool to increase the air/liquid contact area. It also used a perfluorinated organic compound as the working fluid, in contrast to the n-butanol used in the current work. To test the level of saturation achieved by the aluminum wool, we partially filled the wet volume with water, and used a hygrometer to record the saturation ratio S attained at various values of Q<sub>w</sub>. While we intended to attain saturator flow rates in the range of 1 l/min, we found that S<sub>w</sub> was only 20% at Q<sub>w</sub>=0.1 l/min, and would have been considerably smaller with the less diffusive vapors of n-butanol to be used here. We therefore placed the aluminum wool in the dry volume (to increase heat transfer to the gas), and installed 9 vertical porous surfaces (~1 mm thick) parallel to the general direction of the flow (Figure 1). The wetted surfaces made contact with the top and bottom of the volume, forcing
the air flow to divide into 10 narrow parallel channels. No porous material was used to cover any
of the 6 surfaces of the aluminum volume. Under these conditions, we found that the flow was
saturated with water vapor at all flow rates up to Q<sub>w</sub>= 1.5 l/min. At Q<sub>w</sub> = 2 l/min the relative
humidity was 86.6%. These tests were carried out at room temperature, but a similar level of
saturation is expected independently of temperature. Since water vapor is about 3 times more
diffusive than butanol, we anticipate that butanol will saturate the air stream up to about Q<sub>w</sub> = 0.5
l/min. In the present study we will explore conditions at flow rates above 0.8 l/min, at which S<sub>w</sub> <
1. For instance, for a flow rate of 0.8 l/min with n-butanol in the wet volume, we estimate S<sub>w</sub> ~0.8.
This limitation is repairable in future work, as there is sufficient space for increasing the number
of parallel channels.

**Sheathing.** The instrument includes aerosol injection into a sheathing, vapor-laden flow, a key
design element originally introduced by Wilson et al. (1983a), and perfected by Stolzenburg and
McMurry (1991) to minimize particle losses and allow activation and detection of particles down
to 3 nm or less. The aerosol flow *q* is injected into the center of an axisymmetric stream by a
centered capillary (2.2 mm ID, 3.2 mm OD), where it is surrounded by vapor laden sheathing flow
(*Q*) confined by a circular conduit (6.4 mm ID). The greatest relevance of this sheathing for our
present purposes follows from its effect on sizing resolution, as it exposes most of the particles to
the nearly radially uniform thermodynamic state prevailing in the center of the stream (i.e., the
axial symmetry of the temperature and vapor concentration forces their radial gradients to vanish
at the center streamline, although both evolve substantially in the axial direction along the length
of the condenser). Any fraction of the aerosol travelling sufficiently far from the center would
experience a different maximum value of S, thus widening the structure of the activation efficiency
curve. For that very reason, it is important that the capillary be accurately placed in the center of
the stream. To approximate this ideal we developed a centering routine based on a limited freedom
to move the outer cylinder with respect to the capillary, and on a removable brass piece centering
the other two during mounting.

Another important issue is the stability of the aerosol jet surrounded by the sheathing flow. Such
mixing flow geometries are notoriously unstable even at modest Reynolds numbers. If the core
aerosol jet becomes oscillating, the saturation ratio experienced by the particles will vary in time,
limiting the resolution. This delicate mixing region has been investigated in the UCPC of
Stolzenburg and McMurry (1991), and over a wider range of flow rates in the shorter variant of
McDermot at al. (1991). Both studies report no visual evidence of turbulent transition. Here the
possibility to measure relatively high resolving powers will provide a sensitive indicator not only
of the onset of flow instability, but also of the flow conditions maximizing resolution.

**Linear control of the wet fraction w.** As shown schematically in Figure 1, the two sheath gas flows $Q_w$ and $Q_d$ determining the wet fraction w are governed by a pair of Gas Flow Controllers (GFC: Aalborg GFC17, 1000ml/min Air). The computer uses a DAQ card (National Instruments, USB6001) to supply an analog voltage (0-5V) to each of the controllers. This voltage sequentially sets them to the respective flow rates $Q_w$ and $Q_d$, keeping their sum constant, and sweeping directly over the variable w with preestablished initial value, final value and stepping increment. Both Aalborg GFCs have 1% full scale accuracy (±10 ml/min). Flowrate repeatability is specified at 0.5% full scale accuracy (±5ml/min). Flowrate uncertainty negatively affects the precision of the mixing fraction, and ultimately, the resolving power of the instrument. To improve the quantification of the two sheath flow channels, each was independently measured with a pair of Microbridge Mass Airflow Sensors (MMAS: Honeywell AWM3100V, F.S. 200sml/min) run in parallel, enabling a maximum total flow $Q_w + Q_d$ of 0.8 L/min. This sensing array is located
between the GFC and the heating volumes. The MMAS feature an excellent repeatability (0.5% of the measured value), though without an independent calibration their reading tolerance is no better than the Aalborg GFC, except at low flow values. The DAQ card recorded the analog flowrate output of both GFCs and all four MMASs. In the present study we operated predominantly at large enough flow rates for the GFC control signal to provide an accurate measure of the actual flow rates. Furthermore, because the 4 additional sensors have not yet been accurately calibrated, no use was made of their signals in the data inversion. For a number of experiments, including those using a total sheath flow above 0.8 l/min, the four additional sensors were not even included in the circuit. Since each of the GFCs provides at most 1 l/min, in the experiments carried out at $Q_w+Q_d=1.2$ l/min, *w* was limited to the range $0.166<w<0.833$.

**Aerosol generator.** We relied on a bipolar electrospray source (Fernandez de la Mora and Barrios, 2017) and a half-Mini differential mobility analyzer (DMA) of relatively high resolving power (Fernández de la Mora and Kozlowski, 2013; Fernández de la Mora 2017). The DMA sheath gas was run in recirculating flow, such that the flow of polydisperse aerosol entering the instrument was identical to the monodisperse aerosol flow selected.

**Control of the aerosol flow q.** The monomobile aerosol classified by the DMA was split in a tee between an electrometer (drawing typically 3 l/min) and the CPS (drawing up to 0.072 l/min). In initial tests the aerosol from the bipolar ES source was pushed into the DMA from a compressed gas line. However, the slight variations in the compressor pressure produced considerable variations in the small flow of aerosol *q* taken by the CPS. To stabilize *q*, the gas going into the bipolar electrospray source and the DMA was taken from the atmosphere, and passed through a diffusion drier and a HEPA filter (Figure 1a). Upon exiting the DMA, the monodisperse aerosol was accordingly at a pressure $p_i$ moderately negative with respect to the lab atmosphere. In order to control *q*, the exit line of the CPS going to the optical detector had accordingly to be at a pressure $p_o$ slightly more negative than $p_i$. This line was pumped by an optical particle counter (OPC) detector, which initially drew part of its critical flow (2.8 l/min) from the CPS, the balance coming through a tee from the same dried and filtered ambient air feeding the aerosol generator. This tee arrangement was, however, unsatisfactory, as some of the grown drops from the CPS did impact on the walls in the tee. For this reason, we suppressed the tee and operated the OPC at the substandard flow rate delivered by the CPS (Figure 1a). In order to control *q* we placed a needle valve in the line pumping the CPS, drawing the sum of flows $Q_w+Q_d+q$. This resulted in an excellent control and an acceptable stability of *q*, which was measured via the pressure difference $\Delta p$ through a thin metal capillary placed upstream of the aerosol flow inlet to the CPS. This flowmeter was calibrated with TSI's flowmeter model 4143. To minimize calibration errors at the low flow rate used in this work, the calibration used substantially larger *q* values, while checking that the response was well within the flowmeter's linear response region.

**Optical detector.** The droplets generated by the CPS were detected and counted using one of two particle sensors. The first was an RNet-301 aerosol particle sensor (Particle Measuring Systems, Boulder, CO, USA). As already noted, this OPC was operated at the substandard flow rate $Q+q$, by placing a valve between the OPC exit and the pump. This unorthodox modus operandi affects the counting efficiency by an unknown factor that may depend on $Q+q$. Nevertheless, the activation curves are determined at constant value of this flow rate, so this uncontrolled factor is fixed for a particular *w* sweep. This ambiguity affects our reported activation curves only through the coincidence error. When the OPC was run in this fashion, flooding was almost completely eliminated. This OPC claims a 5% coincidence error at a concentration of $10^6$ particles/ft$^3$. At the nominal flow rate of 2.8 l/min this corresponds to 1648 counts/s. According to Collins et al. (2013)
the true count rate $\lambda$ is related to the measured (uncorrected) value $\lambda_m$ as $\lambda_m=\lambda e^{-\lambda\tau}$, where $\tau$ is the dead time. From this given {error, count rate} pair it follows that $\tau_{2.8}(s)=-\ln(0.95)/1648$, or $\tau_{2.8}=31.1$ µs. The subscript 2.8 refers to the standard flow rate of the OPC, at which this coincidence error was inferred by the manufacturer. At the non-standard sample flow rate of 0.5 l/min used in our most relevant experiments, $\tau$ would be augmented by the flow rate ratio 2.8/0.5 to become $\tau_{0.5}=172$ µs. With this value the maximum possible $\lambda_m$ is $1/(e\tau)=2138$ s⁻¹. This estimate of $\tau$ must be grossly off, as in many tests we have measured uncorrected count rates as high as 6185.63 s⁻¹. This implies that $\tau$ is at most $1/(e6185.63 s⁻¹)=59.47$ µs. If our count rate had achieved its maximum value at 6185.63 s⁻¹, the activation curves measured would have presented a maximum rather than being monotonically increasing. For this upper limit for the dead time, when the maximal count rate is 4000 s⁻¹, the steepness of the corrected activation curve is 98.9% of the uncorrected one, and the shift of $w$ at 50% activation is 0.36%. Figure 2a illustrates the effect of the coincidence error by representing the artificial normalized activation curve $1/2\{Erf[50\sqrt{\pi}(w-0.5)]+1\}$ (left curve) and its coincidence corrections for $\tau=59.47$ ms, with maximal count rates of 4000 s⁻¹ (middle) and 6000 s⁻¹ (right). The values $w_o$ at which $P=1/2$ are $w_o=0.5$; 0.5011; 0.503, respectively, representing rather small relative shifts of at most 0.6%. Figure 2b shows the difference between corrected (light color) and uncorrected (black) activation curves for our most extreme data set, when the maximal count rate was about 6000 s⁻¹. The diameter of the seed particles was varied most extensively in this experimental series. The figure puts into perspective the smallness of the coincidence correction (both in $w_o$ and in the steepness of the rise), especially at the scale of the variation in $w$ over the full data set. The figure contains duplicated measurements for most particle diameters, showing that the experimental variation is comparable to the difference between corrected and uncorrected curves. Accordingly, in view of the fact that the value used for $\tau$ is just an estimate, coincidence corrections will be implemented only for activation curves whose maximal uncorrected count rate approaches 6000 s⁻¹. For further verification that the coincidence correction does indeed result in the small effects noted, we have made additional measurements under substantially reduced maximal uncorrected counts (Figure SI-1). A final remark on the inversion is worthy of note. The relation between corrected and uncorrected counts goes through a maximum, and therefore includes two branches. In section 2 of the Supplemental Information we present the reasons why only the first branch is relevant to our data, even though some of the raw $w$ sweeps actually go through a maximum followed by a region of slight decay.

<figure><img src="image_3.png" /></figure>

Figure 2: Comparison of corrected and uncorrected activation curves. (a), fictitious activation curve $1/2\{Erf[50\sqrt{\pi}(w-0.5)]+1\}$ (left) with corresponding coincidence-corrected curves, calculated for a dead time $\tau=59.47$ µs, for maximal counts of 4000 s⁻¹ (middle) and 6000 s⁻¹ (right). A measure $\Delta w$ of the width of the activation curve is the length of the upper leg (segment on the horizontal line $P=1$) of the right triangle whose hypotenuse is a segment in the interval $0\le P\le1$, tangent to the activation curve at its inflexion point (dashed inclined line). (b) Experimental activation curves
with maximal signal ~6000 s⁻¹ corrected (light color) and uncorrected (black), showing a relatively small coincidence correction.

The second detector was TSI's 7610 CPC. It was connected to the CPS with a tee to provide the balance flow to its nominal 1.5 l/min flow. This detector uses butanol as its working fluid and relies on a heated optical head, so it was unaffected by the condensed butanol and its vapors carried by the inlet flow. Also, impaction losses were not noticeable at the lower flow rates involved. This second sensor is considerably slower than the first, and detects all particles larger than ~10 nm. It is thus sensitive to some particles not activated by the CPS when using a low value of w (i.e., when examining the activation of larger sizes in the CPS). As a result, the activation curves measured were a combination of the activation in the CPS and the CPC. Because the activation probability of TSI's CPC at given operating conditions of the CPS is independent of the scanning variable w, the response of the CPS could be determined as the difference between the response of the TSI detector minus its value at w=0. The need for this correction naturally limited the largest particles we could investigate: Because the useful signal was reduced by a factor equal to the detection probability of TSI's detector, it became essentially zero at about 10 nm.

**Test aerosol.** To measure sharp activation curves, it is not enough to simply select with a DMA of high resolution a narrow mobility range from an aerosol with a broad size distribution. First, the DMA may have a resolving power insufficient to map the actual sharpness of the activation curve. Second, most aerosol sources tend to produce not only particles of different sizes, but also of different composition and morphology, both of which could affect the measured activation curve. For instance, the work of P. Wagner and colleagues at the University of Vienna has shown that tungsten oxide aerosols generated by the oxidation of a tungsten wire in air yield activation curves much broader than purified clusters including a single salt ion or cluster (Fernandez de la Mora, 2011). This behavior must be due in part to chemical heterogeneity (a mixture of several tungsten oxides; Tolfo, 1975). The evaporation of noble metals can be arranged such as to produce dominantly pure metal particles, but even so, substantial heterogeneity in shape may arise (Fernandez de la Mora et al. 2003). To determine the real size resolution of the CPS, it is accordingly important to achieve uniformity not only in mobility but also in size and chemical composition. An increasing number of studies has achieved this goal with small mobility-selected electrosprayed salt clusters, both in unipolar sources with charge reduction, or, with improved purity, in bipolar electrospray sources. However, these cluster sources are not able to purify particles larger than about 4-5 nm (of interest here due to the limited small particle transmission of the CPS). These relatively large clusters typically include a substantial number *n* of individual salt molecules, which are not easy to mobility-isolate from their neighbors. Consequently, particles of very similar mobilities but potentially different shapes could be simultaneously present in the classified aerosol. One way to sidestep this risk is to use liquid particles. A comparable option to be pursued here is to use soft polymeric materials whose clusters tend to form globular liquid-like particles, with mobilities indistinguishable from spheres of the same mass. For instance, it is known from high resolution mobility/mass studies that the width of the mobility peaks of mass-selected ions of polyethylene glycol (PEG) is so narrow as to be unquantifiable (Trimpin et al. 2007; Larriba et al., 2014). This means that PEG ions collapse into nearly spherical shapes irrespective of their mass, so mobility selection results in size as well as shape homogeneity. Excellent chemical homogeneity can be obtained by electrospraying relatively pure solvents (de-ionized water) seeded with volatile salts of high purity and moderately high concentrations of
polymer. In a recent study (Przybilak et al. 2022), we have confirmed that the activation curves of PEG particles selected by mobility alone from a broad size distribution are as narrow as those selected from pure clusters of a PEG molecule with exactly 28 monomer molecules (PEG-28 with a molecular mass 1251 amu). Here we will use a slightly different material to be referred to as PEG-9k. It was synthesized at Rohm Haas over 20 years ago by joining together with a chemical linker a variable number *n* of members of a relatively narrow distribution of PEG. The mobility spectrum of Figure 3 shows that this product is reasonably well preserved after spending two decades in our refrigerator. The original mobility spectrum (electrosprayed from a PEG-9k solution of about 1 µM concentration in 100 mM aqueous ammonium acetate) showed recognizable cluster peaks up to the hexamer (nonamer in the mass spectrum), with a concentration of the pentamer of about 1/20 that of the monomer (Saucy et al. 2004). The mobility spectrum of Figure 3 corresponds to a sprayed sample substantially more concentrated, as seen by the abundance of clusters beyond *n*=12. This sample has the interest of providing numerous well defined peaks in the size range from 3 to 10 nm, which facilitates our tests. Saucy et al. (2004) quote a monomer mass of 8600 Da for this sample. However, they do not explain how this value was determined, and refer to it also as PEG-9k. For this reason, we have determined the mobility of these peaks. Based on two measurements analogous to that in Figure 3 taken at a flow rate of sheath gas of 148 l/min (±0.2%, based on a Sensirion Inc. SFM3000 Mass Flow Meter), we find a mobility Z₃=0.0959 cm²/V/s for the trimer. Converting the mobilities for clusters from *n*=1 to 12 into mobility diameters via the free molecule limit of Millikan's formula [*d*<sub>Z</sub>=$d_p$+*d*<sub>o</sub>= (2.02/*Z*)¹/² under ambient conditions, with diameters in nm and *Z* in cm²/V/s)], and plotting *d*<sub>Z</sub> versus *n*¹/³ gives a straight line with a displacement *d*<sub>o</sub>=0.356 nm. Alternatively, plotting these *d*<sub>Z</sub> data versus the mass diameter *d*<sub>m</sub>=(6*m*<sub>o</sub>*n*¹/³/ρπ)¹/³ using a bulk density of 1.125 g/cm³, and requiring that the slope of the resulting straight line be unity, yields a mean molecular weight of 8960 Da for the monomer. This falls within the ambiguity range of the previous report for this sample (8.6-9 kDa), and confirms that the original substance is well preserved. The corresponding monomer mass diameter is 2.93 nm. However, there are two sources of ambiguity in our size determination. First, the flow rate of sheath gas includes an error of 2%. Also, the density of 1.125 g/cm³ corresponds to PEG of low molecular weight, and is expected to be larger at larger masses. In view of this ambiguity, the size of the various clusters will be assigned as 3*n*¹/³ (nm). Notice that this is a mass diameter rather than a mobility diameter. For *n*≥11, when individual peaks are no longer clearly identifiable, the size assignment has been based on the DMA voltage selected. For example, for *n*=1-10 in the spectrum of Figure 3, the measured linear relation between peak voltage *V*<sub>n</sub>¹/² vs. *n*¹/³ gives almost exactly (inset to Figure 3):

$$n^{1/3} = [(V_n/V_1)^{1/2} - 0.1214]/0.8816. \qquad (3)$$
<figure><img src="image_4.png" /><figcaption>Figure 3: Raw mobility spectrum of concentrated PolyPEG-9k using the electrometer as the detector. The number n of PEG-9k units associated with each peak is indicated above it. The inset shows the experimental relation between the normalized DMA voltage and the level n of clustering.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: OPC count rate as a function of w in the CPS in raw (a) and normalized form (right) obtained for the pentamer ion (PEG-45k; n=5) at fixed Q =0.8 l/min, with a decreasing series of q values indicated in cm³/min in the legend (a). The normalized right figure includes fits to error functions. The shift of the curves to the right with increasing q is due to vapor dilution by the dry aerosol. CPS conditions: Qsat=0.5 l/min; Ts=34 °C; Tc=10 °C. Values of w were scanned upward in steps of 0.0025 every 2 s. DMA blower run at 3500 rpm.</figcaption></figure>

### 3. Results

**Structure of the activation curves.** Figure 4a shows a typical series of raw CPS spectra represented as detector signal C(w) (in counts/s) versus w. The various scans are acquired with variable q (indicated in the left legend, most of them measured twice to check repeatability), with fixed Q =0.5 l/min, Tc = 10, Ts = 34 °C and particle size (n=5). All the activation curves start on the left, at low values of w, with a slight constant background signal. This background count rate Cmin is associated with a small contribution of doubly charged particles that are activated at smaller w than the dominant singly charged particles. Then, at a critical w that depends on CPS parameters, one encounters a relatively sharp step, rising to a constant maximum signal Cmax. The right figure converts the raw data into activation curves via the normalization

$$P(w)=[C(w)-C_{min}]/(C_{max}-C_{min}). \quad (4)$$

Note that this definition of P(w) is not based on the concentration of original (inactivated) particles inferred by the electrometer, but on the maximum OPC signal achieved, which we interpret as
100% activation. This interpretation ignores diffusive losses, which should nevertheless be accounted when the CPS is used for size distribution determination purposes. Some of the curves exhibit a slight decay past the maximum, which we attribute to impaction losses of grown drops prior to their entry in the optical detector. These losses are not always present because they depend on small changes in the orientation of the tube following the exit of the CPS, which was not consistent between all experiments. When this orientation is correctly fixed, the upper asymptote is completely flat, confirming the interpretation of the 100% activation.

**Delay correction.** When we perform a CPS scan by varying the ratio *w* of wet flow over total sheath flow, although the CPS flow rates are moderate and the volumes in the saturator are large, these do not cause any delay because the total CPS exit flow *Q*<sub>sat</sub> + *Q*<sub>aer</sub> is constant and (at *Q*<0.5 l/min) *S*<sub>w</sub> is also constant. Unsteadiness due to *w* sweeping arises downstream from the point where the wet and dry streams mix. The estimated volume from this point to the optical detector is about 5.6 cm³. At the smallest *Q* of 500 cc/min at which our most relevant results are reported, the delay time associated with the CPS is at most 0.67 s. This is a significant delay compared to the response time of the OPC, but is several times shorter than the delay time in TSI's CPC used as detector. To evaluate the effect of this delay, we have performed several *w* scans with TSI's CPC at various scan rates at a single particle size. The scan rate is defined as the ratio between the step change in *w* and the dwell time during which CPC counts are accumulated for each experimental datum. Figure 5 shows delay-uncorrected activation probabilities as light symbols for the dimer ion of polyPEG-9k. Black symbols are delay-corrected. The scanning step in mixing ratio was Δ*w*=0.003333 in this case, while the dwell time varied from 0.2s to 5 s, as indicated in the legend. Note the fine collapse of the data by a simple delay linear with scan rate, with a delay time of 3.196 s. Similar tests were carried out with the OPC detector without the tee, giving a delay time of 1.5 s at a total flow rate of 0.566 l/min. Unless expressly indicated, none of our reported measurements will include delay corrections. All will be carried out with a dwell time of 2 s, for which the shift in *w* for the uncorrected data of Figure 5 is of only 0.0053 for the TSI detector, and about half that value for the OPC detector. This shift is only slightly larger than the dispersion of the corrected data seen in Figure 5. For a typical full scan covering 100 values of *w*, the scan time would have been 200 s. A rough estimate of the scan time to take a size spectrum with delay correction would be 20 s with the flow rates selected. Considerably smaller scan times could be achieved with a CPS design optimized for faster response, but this has not been our focus here.

<figure><img src="image_6.png" /></figure>

Figure 5. Detection probability as a function of *w* for different scan rates for particles of mass diameter 3.78 nm (the dimer of polyPEG-9k) for the TSI detector at *Q*=0.4 l/min; *q*=0.067 l/min;
$T_s=34$ °C; $T_c=10$ °C. Light symbols are uncorrected activation probabilities. Black symbols are delay-corrected. The curves are simply displaced to the right proportionally to $dw/dt$, with little deformation.

**Effect of the proportion of sheath to sample gas.** Before a systematic determination of the activation probability curves for various particle sizes and CPS flow rates, we carried out a limited exploration of the boundaries in $(Q, q)$ space under which the mixing of the two streams would become unstable. Unlike its flow visualization precedents (Stolzenburg and McMurry, 1991; McDermott, et al. 1991) this study was based on a direct measurement of the steepness of the activation curves for either the dimer or the pentamer of PolyPEG-9k. In a first experimental series with the dimer, we fixed $q=0.072$ l/min and varied $Q$ from 0.2 to 0.8 l/min. The steepness monotonically increased with $Q$ in all cases, without any signs of an abrupt transition to turbulent mixing. In a second series of tests with the dimer cluster and fixed $Q=0.8$ l/min, we varied $q$ from 0.045 to 0.114 l/min, producing a monotonic increase in the sharpness of the curves upon decreasing $q$. The results of these two tests are collected in Figure 6 in terms of the parameter $FWHM_w$, defined below, inferred from the measured activation curves. All the data confirm the intuitively expected increased resolution with increasing proportion of sheath over aerosol flow $Q/q$. The circular data points in Figure 6 are the only ones in this study obtained with the TSI detector. The triangles and diamonds included in Figure 6 are for the pentamer ion PEG-45k at variable $q$, respectively at $Q$ of 0.8 and 0.5 l/min.

<figure><img src="image_7.png" /><figcaption>Figure 6: Width of the activation curves, expressed as $FWHM_w$, as a function of $Q/q$. Circles are obtained for the dimer ion PEG-18k, revealing a monotonic increase in resolution with increasing $Q/q$. Filled circles are from Figure 4, for $Q=0.8$ l/min with varying flow $q$. Open circles are for $q=0.072$ l/min with variable $Q$. Triangles and diamonds are for the pentamer ion PEG-45k at variable $q$, respectively at $Q$ of 0.8 and 0.5 l/min.</figcaption></figure>

Two closely related measures of activation curve width (or its inverse, quantifying resolution) will be used in this study. The first one is given by the difference between the two intercepts $w^-$, $w^+$ (length of the horizontal line placed at $P=1$ in Figure 2a) of the tangent line through its inflection point (dashed inclined line in Figure 2) with the lines $P=0$ and $P=1$:

$$ \Delta w/w_o = (w^+-w^-)/w_o, \qquad (5) $$

where $w_o$ is the wet fraction at the point with an activation probability of 50%. The experimental activation curves are generally well approximated by the expression

$$ P(w)=0.5+0.5 \text{Erf}[a(w-w_o)], \qquad (6) $$
whose maximal slope is $(dP/dw)_{max} = a/\pi^{1/2}$. By determining $a$ and $w_o$ via fitting the data to the functional form (6), the defined measure of width is obtained as

$$ \Delta w/w_o = \pi^{1/2}/(aw_o). \qquad (7) $$

Although the quantity $\Delta w/w_o$ has been previously used as a convenient measure of step width in cumulative spectra, it is of interest to relate it to the more conventional measure of peak width in differential spectra. Because the derivative of a step function is a Gaussian, the condition that the peak height be half the maximum requires that the slope of the step be half of the maximum slope:

$$ FWHM_w = 2[\ln(2)/\pi]^{1/2} \Delta w/w_o = 0.93947 \Delta w/w_o. \qquad (8) $$

The two measures of structure width defined differ only by 6%, being essentially interchangeable.

**The onset of unstable mixing.** The limited study just described has not identified any catastrophic loss of resolution under any flow condition at which the gas going through the wet volume is likely to be 100% saturated. Nevertheless, given the clear resolution advantage of increasing the $Q/q$ ratio, and given that further decrease in $q$ is presently precluded by limited signal, we have extended the study of $Q$ values above 0.8 l/min, even though the air coming from the wet volume will be incompletely saturated. Figure 7 shows activation curves obtained at $Q$ of 0.8, 1 and 1.2 l/min, at various values of $q$ (indicated in l/min in the caption in the form “$Q/q$”. The inset represents the corresponding widths $\Delta w/w_o$, showing a noticeable increase from 0.8 to 1 l/min, and a drastic increase from 1 to 1.2 l/min. There is accordingly a definite limit to how much one can increase the fraction of sheath gas without entering into a regime of unstable mixing. An even broader activation curve was obtained with $Q=1.2$ l/min, $q=0.084$ l/min, yielding $\Delta w/w=0.216$. This curve is not included in Figure 7 because the temperatures were 10 and 40 °C, making the corresponding data sets not directly comparable. At $Q=1.2$ l/min, the mean velocity of sheath gas is 62 cm/s, with associated Reynolds number of 414. The corresponding value at 1 l/min is $Re=345$, at which there is already an appreciable loss of resolution. These Reynolds numbers are too low to destabilize the main flow inside a cylindrical duct, but the stability is reduced here by the presence of the capillary wall, and by the shear layer associated with the aerosol jet. Another possible source of instability for the pure outer flow is that the converging region preceding the final cylindrical tube includes three sharp turns, at which unsteady vortices may potentially be shed and not decay before reaching the mixing region. The corresponding velocity and Reynolds number for the aerosol jet as it emerges from the capillary are 25.3 cm/s and 38.6.

<figure><img src="image_8.png" /><figcaption>Figure 7: Detection probability as a function of $w$ obtained at $Q \ge 0.8$ l/min. The inset shows the widths of the three activation curves versus $Q_{CP}$, almost tripling at 1.2 l/min.</figcaption></figure>
Improving the performance by increasing *Q* above 1 l/min is presently not possible, but may still be viable by changing the geometry. Figure 6 seems to suggest that decreasing the aerosol flow *q* always increases the resolving power. Nevertheless, perhaps another instability will set in at *q* values below those already tested. Furthermore, even in the absence of a small-*q* mixing flow instability, the advantage of a decreasing aerosol flow must eventually saturate, since an aerosol jet initially delivered right at the axis must broaden radially by diffusion. These issues will be addressed in the future by means of a modified instrument, since going below the smallest *q* already achieved is precluded by the smallest signal at which we can reliably determine *FWHM*<sub>w</sub>. This is in turn limited by the modest particle transmission of our instrument. Future transmission improvements to extract the aerosol flow from a larger transport flow, as demonstrated by Stolzenburg, P. H. McMurry, and used in TSI's family of UCPCs, should lead to substantial improvements in absolute counting efficiency. The potential broadening effect of diffusion, if encountered at smaller *q*, could be reduced by shortening our rather long insulator (McDermott et al. 1991).

Notwithstanding the monotonic dependence of the resolution with *q*/*Q* seen in Figure 6, the data show substantial variation in performance even at fixed *q*/*Q*. The widest structures seen in the open circles correspond to tests carried at relatively low values of *q*, down to 0.15 l/min.

<figure><img src="image_9.png" /></figure>

Figure 8: Coincidence-corrected detection probability as a function of *w* obtained for various Poly-PEG-9k clusters at *Q*=0.5 l/min, *q*=0.06 l/min, *T*<sub>c</sub>=10 °C; *T*<sub>s</sub>=34°C (data points), including best fit error functions (continuous lines). Most data sets are measured twice (Black and light color, respectively). The various curves from right to left are for *n* = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.29, 14.47, 17.85, 21.48, 25.33.

<figure><img src="image_10.png" /></figure>
Figure 9: Characteristics of the activation curves, comparing the data in Figure 8 (open circles) with analogous measurements carried at higher DMA resolution (filled circles). (a) Relative width of the activation curves versus *d*<sub>p</sub>, showing a drastic width reduction on going from the monomer to the trimer. (b) Particle diameter versus *w* at 50% detection probability.

<table><caption>Table 1: Size characteristics of 15 of the particles used in the measurement of the activation curves of Figure 8. Non-integer cluster numbers *n* are inferred from the selected voltage</caption><thead><tr><th>n</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11.29</th><th>14.47</th><th>17.85</th><th>21.48</th><th>25.33</th></tr></thead><tbody><tr><th>d<sub>p</sub><br>(nm)</th><td>3</td><td>3.78</td><td>4.33</td><td>4.76</td><td>5.13</td><td>5.45</td><td>5.74</td><td>6</td><td>6.24</td><td>6.46</td><td>6.73</td><td>7.31</td><td>7.84</td><td>8.34</td><td>8.81</td></tr></tbody></table>

**Size dependent response of the CPS at Q=0.5 l/min.** In view of the reduced performance observed at low and high values of *Q*, and also to guarantee that the wet stream is saturated, our study involving a broad range of particle sizes has operated at *Q*= 0.5 l/min with *q* between 0.063 and 0.066 l/min; *T*<sub>s</sub>=34 °C; *T*<sub>c</sub>=10 °C. The coincidence-corrected activation curves obtained under such conditions are shown in Figure 8. The Figure includes the selected cluster number *n*, also included in Table 1 together with the particle diameter. Most of the curves were acquired in duplicate, and show good repeatability. The continuous lines are best-fit error functions to the data. One can see a drastic decrease in structure width on going from the monomer to the trimer, and fairly sharp activation steps for *n*≥3, within a ~2% variation in *w*. The corresponding *FWHM*<sub>w</sub> data are represented versus particle diameter in Figure 9a. Figure 9b reports the measured relation between particle size and the value *w*<sub>o</sub> of the spectral variable *w* at 50% activation efficiency. Note that, with one exception, none of the *w* data have been corrected for delayed response. The exception is in Figure 9b, where the data for the three smallest particles were taken at a slightly faster scan speed than the other data, and exhibited a clear displacement of 0.011 in *w*<sub>o</sub>. These three (duplicated) points have accordingly been moved downwards by this amount to achieve a smooth curve. The whole mismatch found between these three data and the rest of them is not necessarily due to delayed response, but may be partly due to other cause already discussed, such as slight changes in sample flow rate and CPS temperatures.

The *w*(*d*<sub>p</sub>) relation is highly relevant for size spectrometry, as well as for basic nucleation studies. Figure 9 includes also data obtained under the same CPS conditions, while running the DMA at higher resolution (3500 rather than 2500 rpm in the blower). The purpose of these tests was to figure out if the finite slope observed in the activation curves was due in part to imperfect aerosol monodispersity. Surprisingly, the activation curves are slightly but clearly less steep when the DMA is run at higher resolution, indicating that their widths are not controlled by the DMA. They cannot be determined at the CPS either, since it was run under identical conditions in both cases. It is clear from various repetitions of these measurements under similar conditions (Figure SI-2) that the quality of the electrospray has a large influence on the steepness of the activation curves. Note in Figure 8 a displacement in *w*<sub>o</sub> of about 0.018 between a first data series corresponding to *n* =1-4, and the next series covering the remaining values from *n*=4 to 25.33. Figure SI-1 in the Supplemental Information published in the web collects diverse results of *w*<sub>o</sub> vs. cluster number *n*, showing the level of variability observed. Part of this variation was due to slight changes in *q*, whose small value makes it difficult to control precisely. Slight instabilities on the saturator and condenser temperatures of about 0.2 °C also result in noticeable shifts in *w*<sub>o</sub>. For instance, Figure SI-1 includes one series taken with the condenser 0.9 °C cooler and the saturator 1°C hotter than in the other experiments (T<sub>c</sub>=10 °C, T<sub>s</sub>=34 °C), which resulted in a downward shift in *w*<sub>o</sub> of about 0.06.
<figure><img src="image_11.png" /><figcaption>Figure 10: Detection probability *P* as a function of particle diameter *d*<sub>p</sub>. Size activation curves inferred from the largest particles from Figure 8 at Q=0.5 1/min (dashed gray lines joining data points) are compared with the data from Figure 2b of Gallar et al. (black lines)</figcaption></figure>

**Size resolution.** We have a direct means to measure the width of the *P*(*w*) curves at constant particle size. However, for the purpose of determining the size resolution, the quantity of interest is the width of the *P*(*d*<sub>p</sub>) curves under fixed settings of the CPS. We have measured the two-dimensional dependence *P*(*d*<sub>p</sub>, *w*) essentially continuously over the variable *w*, but only at a few discrete particle sizes. These several continuous *P*(*w*) curves available at a moderate number of sizes may be cut vertically at several values *w*<sub>i</sub> of *w*, to provide a discrete number of *P*(*d*<sub>p</sub>) values at fixed *w*. But due to the steepness of the *P*(*w*) curves and the discrete number of particle sizes studied, most *w*<sub>i</sub> values yield only two or three *P*(*d*<sub>p</sub>) values different from 0 or 1, therefore defining rather roughly the desired *P*(*d*<sub>p</sub>) curve. The data points joined by gray dashed lines in Figure 10 illustrate this situation for the data of Figure 8. To construct this figure, we have used only the activation curves for the largest particles, where the spacing of the various experimental *d*<sub>p</sub> is less than at the smaller diameters. Accordingly, the quantity Δ*d*<sub>p</sub>/d<sub>p</sub> can only be inferred from our data with modest precision in this size range, and far less accurately for smaller particles. Within this ambiguity, we obtain Δ*dp*/d<sub>p</sub>=0.069 at 8.1 nm. Figure 10 includes as continuous lines the *P*(*d*<sub>p</sub>) data previously reported by Gallar et al. (2006), from which we obtain Δ*dp*/d<sub>p</sub>=0.234 at 8.6 nm. This is 2.8 times broader than the new data. Part of the visual difference in steepness between the black and the gray lines is that our inversion ignores diffusion losses by artificially assigning 100% conversion to the high *w* asymptote. Gallar et al. (2006) show a rising rather than a flat envelope to the various *P*(*d*<sub>p</sub>), because they account for diffusive losses as an essential piece of the data inversion process. Our *P*(*d*<sub>p</sub>) curves would evidently lose steepness if they were lowered by inclusion of transport losses. This effect has been corrected in the comparison just made of the steepness of the old and the new curves.

A bipolar electrospray source of clusters was recently used to obtain activation curve widths for KanomaxFMT's fast CPC, which also introduces sheath gas in a planar channel (Kuldinow et al. 2022). The smallest Δ*dp*/d*P* they report is 0.121, at 2.82 nm. Use of the same cluster source to examine the performance of the unsheathed Airmodus A10 turbulent mixing CPC has obtained Δ*dp*/d*P*~0.1 (Fernandez de la Mora et al. 2022). The resolving powers of these three CPCs are therefore quite comparable, though the CPS is better, and has substantial potential to improve further.
Notice that most of our activation curves look considerably steeper in their original $P(w)$ form (Figure 8) than in the derived $P(d_p)$ representation of Figure 10. Somewhat puzzled by this fact we have sought a more accurate method to infer resolution in diameter space from the measured resolution in $w$ space. For the fixed CPS flow rates $Q$, $q$, and temperatures $T_c$, $T_s$, at which $w$ scans are performed, $P$ is only a function of $d_p$ and $w$: $P=P(d_p, w)$. The cut condition that $P=1/2$ then establishes a relation between $d_p$ and $w_o$:

$$P(d_p, w_o)=1/2. \qquad (9)$$

This is the link enabling inferring size distributions from saturation scans. This $d_p(w_o)$ relation is known experimentally and is represented in Figure 9b. Differentiating (9) one sees that the slopes $(\partial P/\partial \ln d_p)_w$ and $(\partial P/\partial \ln w)_d$ defining the respective resolutions in $d_p$ and $w$ space are related via

$$(\partial P/\partial \ln d_p)_w = -(d_{lnw}/d_{ln d_p})(\partial P/\partial \ln w)_{d_p}. \qquad (10)$$

<figure><img src="image_12.png" /><figcaption>Figure 11: Absolute value of the logarithmic derivative of $d_p$ versus $w_o$ (the $w$ value at 50% activation efficiency), represented as a function of particle diameter (from the data of Figure 9b). This derivative provides the ratio between the respective resolving powers in $w$ and $d_p$ spaces.</figcaption></figure>

The two resolutions therefore differ through the factor $(d_{lnw}/d_{ln d_p})$, which may depart considerably from unity, making the activation curves steeper in one space than in the other. The important quantity $(d_{lnw}/d_{ln d_p})$ is calculable from the critical relation between $w_o$ and $d_p$ measured (Figure 9b) for the low DMA resolution data offering a wider size range. We have represented this experimental curve in terms of $\ln d_p$ versus $\ln w$, and seen that the first is well approximated by a cubic polynomial of the second. $(d_{ln d_p}/d_{lnw})$ is accordingly quadratic in $\ln w$. This derivative is shown in Figure 11 as a function of $d_p$. One can see that $(d_{ln d_p}/d_{lnw})$ increases substantially at large particle sizes, explaining why the width seen in $w$ space for the largest particles (left end of Figure 8) is so much narrower than that observed in $d_p$ space (Figure 10).

In view of Figure 11, one wonders if the observed trend of increasing $(d_{ln d_p}/d_{lnw})$ will impose limitations on the size resolution attainable with large particles. Based on the prior data of Gallar et al. (2006), this does not appear to be the case below 20 or 30 nm, where the behavior is similar to that of 5-10 nm particles. For such large particles one may also make approximate predictions on the relation between $w_o$ and $d_p$ by assuming that it is given by the Kelvin diameter:

$$d_o/d_K = d_o/d_p = \ln S \quad \text{with} \quad d_o = 4\gamma w_o/(k_B T). \qquad (11a-c)$$

The characteristic length $d_o$ defined in (11c) is 3.62 nm for butanol at 22 °C. $\gamma$ is the surface tension of the condensed vapor, $v_o$ is the molecular volume of the vapor molecule, and $k_B T$ is the thermal
energy. Since nucleation naturally takes place principally at the point of maximal S, S in (11) should be interpreted as the maximal saturation ratio present in the condenser.

Together with (1), (13) provides the following simple relation between w and $d_p$:

$$ \frac{d_o}{d_p} = \ln S^* + \ln w, \quad (14) $$

It then follows that

$$ |d_{lnd_p}/d_{lnd_w}| = d_p/d_o. \quad (15) $$

Therefore, the derivative under discussion is indeed expected to increase monotonically with $d_p$ at large $d_p$.

The availability of $d_{lnd_p}/d_{lnd_w}$ information to infer activation curve widths in $d_p$ space from measured widths in w space enables estimates $FWHM_d$ more accurate than those previously based on the few data available in Figure 10. The smallest $FWHM_w$ shown in Figure 9 at 5 nm is slightly smaller than 2.5%, corresponding to a resolving power of 40 in w space. According to Figure 11, $d_{lnd_p}/d_{lnd_w}$ for this size is 1.8, resulting in $FWHM_d$ of 4.5% (diameter resolution of 22.2). This is comparable to the typical resolving power of commercial DMAs for particles larger than 5 nm (10 in mobility, 20 in diameter). The situation for the pentamer is even more favorable for the data of Figure 6, taken at smaller aerosol/sheath ratio than those in Figure 9. In this case $FWHM_w=2\%$ at 5 nm ($FWHM_d=3.6\%$, resolution 27.8). Figure 6 shows also $FWHM_w=2.5\%$ for the dimer ion, for which $d_{lnd_p}/d_{lnd_w}=1.44$ ($FWHM_d=3.6\%$). This width is less than half the value $\Delta d_p/d_p=0.069$ obtained directly from the diameter data of Figure 10, and almost one third the values obtained in prior cluster studies with other CPCs. Once some of the improvements to the instrument discussed in the conclusions are implemented, there are grounds to expect size resolving powers in the vicinity of 30, at least in the 3.8-5 nm size range.

In a previous study (Fernandez de la Mora, 2020), we have used symmetry considerations to argue that the resolving power of sheathed instruments should be substantially higher in parallel plate geometries (such as that used in KanomaxFMT's fast CPC) than in the axisymmetric geometry investigated here. Paradoxically, the best $\Delta d_p/d_p$ found for the planar KanomaxFMT instrument was 0.121, over three times broader than the best obtained here for an axisymmetric configuration.

It is apparent that achieving in practice a symmetric sheathing is harder in planar than in cylindrical geometries.

## 4. Conclusions

We have used relatively uniform PEG particles 3-9 nm in diameter to reexamine the performance of the Variable Supersaturation Condensation Particle Sizer (VSCPS) of Gallar et al. (2006). The instrument sizes particles by seeding them near the symmetry axis of a condenser, while quickly varying the supersaturation by changing the proportion of dry and vapor rich flows mixed upstream of the condenser.

We show that the size resolution $\Delta w/w \sim 2\%$ obtained is the highest yet observed in any CPC used for sizing purposes. It almost triples that previously reported for this instrument, and almost doubles that observed for other more modern CPCs. We also argue that improvements in the experimental technique and the CPS itself should further increase this performance. Therefore, the class of sheathed axisymmetric CPCs has potential to offer fast, high-resolution sizing of particles in the size range of a few to tens of nm, without the low efficiencies associated in this size range with electrical mobility techniques. While the present prototype could possibly take delay-corrected size spectra in 20-30 s, the scan times used here are about 10 times greater. Here we have not tested the instrument beyond 9 nm. However, Gallar et al. (2006) have demonstrated its use up
to 30 nm with no signs of loss of resolving power compared to their measurement at 9 nm. We are
not aware of any fundamental reason why even larger particles could not be analyzed.

Desirable future developments include:

(i) Refine the aerodynamic design to delay mixing instabilities in order to increase the sheath flow rate *Q* and achieve higher resolving power.

(ii) decrease small particle losses by extracting the aerosol flow from a larger transport flow (Stolzenburg and McMurry, 1991), and by shortening the inlet aerosol line. This will facilitate the study of smaller particles.

(iii) Extend the range of particles studied into larger sizes by use of recently developed DMAs having high resolution at particle diameters well above 10 nm (Perez-Lorenzo and Fernandez de la Mora, 2021).

(iv) limit the negative effect on size resolution due to diffusive broadening of the aerosol beam, presently due to the long insulator separating the condenser from the saturator.

(v) A much needed technical improvement is substituting our improvised OPC for one allowing much larger count rates, as commonly used in commercial CPCs.

(vi) The instrument will become a powerful tool for basic studies in heterogeneous nucleation as soon as calculations or other means to determine the velocity, temperature and saturation fields enable conversion of the experimental variable *w* into the corresponding thermodynamic state at the point of maximal rate of nucleation within the condenser. This development would allow unambiguous comparison of measurements with theory, in an instrument capable of generating nucleation data with unprecedented speed and control of all relevant variables.

(vii) Supplementary information included in the web reports a variety of additional data obtained under similar conditions, providing an indication of the variability in our measurements. An important message from these extended measurements is that the electrospray spray quality has a large influence on the maximal observable steepness of the activation curves.

## Acknowledgments

We thank Dr. Mark Stolzenburg for communicating privately his calculations on activation curves for various geometrical variants of the UCPC, and various *q/Q* ratios. This work was funded in part by AFOSR grants FA9550-18-1-0165 and FA9550-22-1-0097. We thank Dr. Daniel Saucy for providing the singular polyPEG sample used to generate all our seed particles. We are much indebted to Drs. Greg Lewis and Arantza Fernandez of Aerosol Dynamics, and to Dr. Derek Oberreit of Kanomax FMT for sharing their wealth of knowledge on CPC instruments. This study has benefitted greatly from a previous unpublished work in our laboratory (Przybilak et al., 2022).

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