Traceable methods for calibrating condensation particle counters at concentrations down to 1 cm−3

The concentration of nanometer-sized particles is frequently measured in terms of particle number concentration using well-established measuring instruments, such as condensation particle counters (CPCs). Traceability for these measurements can be achieved by means of calibration against a reference aerosol electrometer starting at concentrations >1000 cm−3. Here, two independent methods for extending traceability down to 1 cm−3 are described. The first method relies on a custom-made, reference optical particle counter while the second method combines electrometer measurements with a series of dilution steps. An inter-comparison of the two methods was carried out using polystyrene spheres with a nominal diameter of 100 nm in the concentration range 1 cm−3–100 cm−3. A CPC Model 3752 (TSI Inc, USA) was used as transfer standard. The obtained results showed a deviation of 1%–4% between the two methods, which was in agreement with the stated uncertainties.


Introduction
Condensation particle counters (CPCs) are indispensable in most areas of aerosol science and technology, with applications ranging from pure aerosol research to industrial practice.
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In many occasions, CPCs are used in the laboratory as reference instruments for particle number concentration 3 , e.g. for calibrating optical particle counters (OPCs) for clean-room facilities [1].
In the concentration range from ∼100 cm −3 to ∼20 000 cm −3 (the exact range depends on the CPC model), the relationship between the actual concentration and the concentration reported by a CPC is expected to be linear.This means that the CPC counting efficiency (CE) is constant.As the concentration decreases, the concentration reported by the CPC may start to scatter and at the lowest concentration limit, a CPC may report a concentration that is greater than the actual concentration (false count).This can result from small leaks in the CPC or contamination of the optics, especially if the CPC has been previously employed in measuring engine exhaust or other aerosols at high concentrations.In the higher concentration range, on the other hand, the reported concentration underestimates the actual concentration due to coincidence count losses, if a built-in coincidence correction factor is not available or if this correction factor is not accurate [2,3].
CPCs are typically calibrated against Faraday-cup aerosol electrometers (FCAE) with traceability to the International System of Units (SI) according to the ISO 27891 standard [4].However, due to the limitations of the FCAE, accurate measurements of aerosol concentration can only be performed above 1000 cm −3 [3,5].Assuming a proportional fit and extrapolating to zero is not a scientifically justified solution for accredited calibration laboratories or national metrology institutes providing calibrations of CPCs at concentrations below 10 3 cm −3 .
To overcome this issue, Owen et al [6] used a fixed diluter to verify the proportionality of their CPC from 1 cm −3 to 10 4 cm −3 .Traceability to the SI was achieved by comparing the CPC to an FCAE at high concentrations, fixing the constant of proportionality.
In this study, we compare two primary standards developed in Switzerland and Japan, respectively, for particle number concentration.The Swiss primary standard is based on a custom-made reference OPC whereas the Japanese standard relies on a reference FCAE and aerosol diluters.The goal of this exercise was to (i) contribute to the further development of the ISO 27891 standard by describing standardized methods for CPC (and OPC) calibration down to 1 cm −3 and (ii) provide evidence of equivalence to support future claims for internationally recognized calibration and measurement capabilities (CMCs) [7].This is the first inter-comparison of CPC calibration at such low concentration levels.

Swiss primary standard
The Swiss primary standard at the Federal Institute of Metrology (METAS) and its validation has been described in detail in previous publications [8][9][10].The experimental setup consists of three sections (figure 1): the aerosol generation system, a turbulent flow tube (homogenizer) and the particle detection system.Polystyrene (PS) aerosols in the size range 100 nm-15 µm are generated based on wet or dry dispersion using commercially available generators (in this study, only 100 nm PS spheres were generated).Depending on the specific application, the PS test aerosols can be sizeselected by an aerodynamic size classifier (AAC) or a differential mobility analyzer (DMA) to remove residue particles (here, an AAC and DMA were used in series).The homogenizer is a 4 m-long custom-made stainless-steel tube, oriented vertically, with an inner diameter of 16.4 cm.Dry filtered air (flow rate of 120 l min −1 , particle concentration less than 0.001 cm −3 ) enters the homogenizer from the very top and sweeps the PS particles down the tube, where they are further mixed by three turbulent air-jets.The sampling zone, located 3 m downstream of the aerosol inlets, accommodates two isokinetic sample probes, one for the reference OPC and one for the CPC under test (the test CPC).The aerosol spatial homogeneity lies within 2% for particle sizes up to 10 µm.
The detection system consists of a custom-made OPC placed right at the outlet of the homogenizer.The tube of sampling probe is straight to prevent any particle losses due to impaction.The sampled aerosol (60 ml min −1 ) enters the detection chamber through the 0.2 mm orifice of the inlet nozzle and is surrounded by a sheath-air flow (3.5 l min −1 ).The sheath air is filtered and recirculated.The laser beam is generated by a continuous-wave laser (Model Verdi V-5, Coherent Corp., USA) at a wavelength of 532 nm, and focused at the point of intersection with the aerosol stream with the use of a cylindrical lens.The width of the laser beam at the point of intersection with the aerosol stream is 0.7 mm as measured with a beam profiling camera (WinCamD, DataRay Inc., USA).Particles crossing the laser beam scatter light, which is detected by a photomultiplier tube placed at a 90 • angle.
The facility has found applications in the calibration of different particle counters, such as OPCs for clean rooms [8], optical particle size spectrometers for air quality monitoring [10], bioaerosol monitors [11,12] and aerodynamic particle sizers [13].
The aerosol flow measurement through the OPC is traceable to the national standard of flow and the particle counter has been calibrated against the primary frequency standard at METAS.The OPC has been validated against a reference CPC (i.e.calibrated with an FCAE according to ISO 27891) at particle size and concentration of 200 nm and 800 cm −3 , respectively (figure S1 in [10]).Inter-comparisons with the inkjet aerosol generator (IAG, a primary standard of Japan) [14,15] in the particle size range 500 nm-10 µm have revealed a good agreement between the two standards [9,10].
The facility depicted in figure 1 was used as primary standard in the concentration range <1000 cm −3 and the measurement was performed according to the ISO 21501-4 standard for OPC calibration [1].For the measurements above 1000 cm −3 , the test CPC was calibrated against an FCAE with size-selected PS particles according to the ISO 27891 standard [4].In both cases, the CE, which is equivalent to the 'detection efficiency (η)' in ISO 27891, was calculated as the ratio of the particle concentration reported by the test CPC to that measured by the reference instrument (OPC or FCAE).Note that the temperature in the laboratory was set to 21 • C during this study.

Japanese primary standard
2.2.1.Overview.The calibration method with the Japanese primary standard at the National Institute of Advanced Industrial Science and Technology (AIST) uses a reference FCAE and diluters and determines the counting efficiencies step-by-step from high to lower concentrations.The calibration procedure begins with a calibration at a high concentration, 40 000 cm −3 in this study, to determine the CE of the test CPC, denoted by η 40k (Step 1).In the next measurement (Step 2), the ratio of the CE at a lower concentration, 2000 cm −3 in this study (η 2k ), to that at the concentration in Step 1 (i.e.40 000 cm −3 ), is determined.The product of this ratio, η 2k ⁄η 40k , and η 40k that is obtained in Step 1 gives the CE η 2k .Further measurements for Steps 3, 4, and 5 are performed to obtain the CE ratios η 100 ⁄η 2k , η 10 ⁄η 100 , and η 1 ⁄η 10 , and determine the counting efficiencies at lower concentrations, i.e. η 100 , η 10 , and η 1 at concentrations of 100 cm −3 , 10 cm −3 , and 1 cm −3 , respectively (figure 2).

Measurement setups.
The measurement of Step 1 is based on the calibration method described in ISO 27891 [4] with a reference FCAE, which is a simple parallel comparison between the test CPC and reference FCAE with equal sampling flow rates (0.3 l min −1 in this study).A flow splitter is used to split the aerosol in two equal parts.Figure 3 shows the measurement setup.An electrospray aerosol generator (EAG, Model 3480, TSI Inc., USA) is used to generate primary aerosols.More details are given in appendix A.
The measurement of Steps 2-5 follows a procedure developed by AIST with the setup shown in figure 4. The difference from Step 1 is the addition of another set of diluters in the tubing between the flow splitter and test CPC, which is denoted as '2nd diluter set' while the 'Diluter set' in figure 3 for Step 1 remains between the EAG and DMA and is denoted as '1st diluter set'.More details are given in appendix A.
The reference FCAE in figures 3 and 4 is the one used as the Japanese national primary standard for airborne particle number and charge concentrations and consists of a Faraday cup (in-house production), an electrometer (Model 6430 by Keithley Instruments, Inc., USA), a flow meter (Model DF-240 BA by Cosmo Instruments Co. Ltd, Japan), and a flow controller (Model MC-4000 by Lintec Co. Ltd, Japan).Detailed information on the reference FCAE is given in [16].
Throughout Steps 1-5, the EAG was operated with a diluted PS particle suspension, dry air without carbon dioxide gas and a capillary of 40 µm inner diameter.The PS particle suspension was prepared by mixing 300 µl raw PS particle suspension (SC-0100-D, JSR Life Sciences Corp., Japan) with 100 µl 20 mM of ammonium acetate aqueous solution and 600 µl of ultrapure water.Since the nominal particle concentration of the raw PS particle suspension (1% solid content) was 1.8 × 10 13 ml −1 according to the manufacturer, the concentration of the diluted suspension was estimated to be 5.4 × 10 12 ml −1 .
The background particle concentration was maintained low by using very clean particle-free air for dilution and make-up flows.For example, in the measurement at 1 cm −3 , the background concentration was less than 0.002 cm −3 and negligibly low.
The temperature in the laboratory was 22 • C during this study.

Measurement procedures and derivation of counting efficiencies.
Step 1.As explained earlier, the measurement in Step 1 follows the procedure described in ISO 27891 [4] and more details are given in [17].The CE of the test CPC, η CPC , is calculated with the following equation (1) which is based on equation (1) of the ISO 27891 standard [4]: C CPC is the particle concentration indicated by the test CPC, β the flow splitter bias correction factor, p the number of net charges on a particle, and ϕ p the fraction of particles with p charges.The summation gives the multiple-charge correction factor for the calibration aerosol for up to three charges, and is unity when all particles are singly charged and greater than unity when multiply charged particles are present.η FCAE is the detection efficiency of the reference FCAE, which is essentially unity, and accounts for corrections for current and flow rate measurements (η I and η Q , respectively), and the non-ideal characteristics of the Faraday cup (η FC ) due to particle losses in the inlet tube and particle penetration through the inner filter [16], i.e. η FCAE = η I • η Q • η FC .The particle losses in the inlet tube are estimated with the equation for diffusive deposition from a laminar flow in a straight tube (3.5 cm in length in this study) of a circular cross section [18].C FCAE is the particle concentration by the reference FCAE calculated with the following equation (2) based on the assumption that all particles in the calibration aerosol are singly charged: I FCAE and Q FCAE are the current and volume flow rate recorded by the reference FCAE, respectively, and e the elementary charge.The current and volume flow rates are calculated based on measurements with an electrometer, flow meter, thermometer, and pressure transmitters that are calibrated with traceability to the SI.The thermometer and pressure transmitters are used to derive the actual volume flow rate at the inlet of the Faraday cup.The results of the calibrations are reflected in the above η I and η Q and their associated uncertainties.
In the parallel measurement of calibration aerosols with the reference FCAE and test CPC, the DMA voltage is toggled repeatedly between on and off at 1 min interval while C FCAE and C CPC are recorded [4,16,17].This voltage cycling is repeated for at least ten times.The value of η CPC is calculated for each voltage-on period.
Step 2. Two measurements are performed to determine the ratio of η 2k /η 40k .The first measurement, which is denoted as Measurement (a), is performed with the dilution ratio of the first diluter set (DR 1,a ) equal to 1 (i.e.no dilution).With this setting, the particle concentration at the inlet of the reference FCAE (C inlet FCAE,a ) is as high as the generation of the calibration aerosol allows, which was 80 000 cm −3 with the 100 nm PS particles used in this study.The dilution ratio of the second diluter set (DR 2,a ) is adjusted so that the inlet concentration of the test CPC (C inlet CPC,a ) is equal to the target concentration.In this study, DR 2,a was adjusted to about 1/2 because the target concentration was 40 000 cm −3 .The adjustment of DR 2,a was done by adjusting the reading of the test CPC, with its CE at 40 000 cm After Measurements (a) and (b), the ratio of the ratios r a and r b is calculated as R ab = r b /r a , which is eventually equal to η 2k /η 40k as explained in appendix B. Finally, η 2k is derived by multiplying η 40k obtained in Step 1 by R ab = η 2k /η 40k , i.e.
In the actual procedure, the recording of C FCAE,a and C CPC,a in Measurement (a) is repeated ten times by 1 min voltage on/off cycling to derive the average of the r a ratios, i.e.
Measurements (e) and (f) of Step 4 for η 10 /η 100 and (g) and (h) of Step 5 for η 1 /η 10 are similarly performed to obtain η 10 and η 1 , respectively.The nominal inlet concentrations and dilution ratios in Steps 2-5 are summarized in table 1.
The recording of C FCAE and C CPC is repeated ten times with periodic voltage on/off cycling, and the calculation of R is done with average ratios, such as R cd = r d /r c , similar to Step 2. The duration of voltage-on periods is set to be either 1 min, 2 min, or 5 min, while that of voltage-off periods is always 1 min.The longer voltage-on durations are for suppressing the fluctuation of the concentration indicated by the test CPC by taking longer time-averaging at low concentration levels in Measurements (e)-(h).The voltage-on durations set in this study are given in table 1.

Remarks on the stepwise dilution method.
The first remark to note is that the stepwise dilution method does not require knowledge of precisely quantified dilution ratios of the first and second diluter sets since the dilution ratios cancel out in the derivation of the counting efficiencies (see appendix B).The requirements for the diluters are the capability of fine adjustment of the dilution ratio in the desired range and the stability of the dilution ratio within a typical duration of measurement, e.g. a few hours, especially for the second diluter set.During method development, we found that the diurnal variability (u diu ) in appendix C differed depending on the stability of the diluters used in the second diluter set.The diurnal variability increased because the DR 2 terms did not completely cancel out, for example, between Measurements (a) and (b) in equation B.7 for R ab , due to a drift of DR 2 .The diluters used in AIST's setup were those that had the smallest diurnal variability among those tested.This method relies on the good linearity of concentration measurement by the reference FCAE, instead of a fully characterized performance of diluters.Note that the linearity of concentration measurement by the reference FCAE corresponds to the variability of the slope of current calibration in [16], and that good linearity means the slope remains unchanged with varied current levels.
Second, the unequal flow rates at the flow splitter and the use of conductive silicone tubes between the flow splitter and second diluter set in Steps 2-5, which might cause deviation of β from unity, do not compromise the measurement accuracy since β cancels out in the derivation of the counting efficiencies (see appendix B) as far as it remains constant.To keep β constant during measurement in each step, the bending shape of the silicone tubes was maintained unchanged.
Third, this method is applicable to other particle sizes and materials without significantly increasing the calibration uncertainty.An important factor that affects the calibration uncertainty is the particle concentration at the reference FCAE when the first diluter set is turned on in Measurements (b), (d), (f), and (h).If this concentration is too low to be measured by the reference FCAE, the calibration uncertainty increases.To achieve the level of uncertainty in this study with other particle sizes or materials, the undiluted concentration (C inlet FCAE,a ) must be comparable to or greater than 80 000 cm −3 .For example, this method is applied at AIST to 30 nm sucrose and 50 nm poly-alpha-olefin particles generated with the EAG to achieve similar uncertainties.
Fourth, this method is also applicable to CPCs of inlet flow rates other than 0.3 l min −1 .Indeed, this method is applied at AIST to CPCs having an inlet flow rate of 1 l min −1 and 1.5 l min −1 .For the CPCs with 1 l min −1 and 1.5 l min −1 inlet flow rate, the particle concentration in Step 1 and Measurement (a) is usually set at 10 000 cm −3 instead of 40 000 cm −3 .The increase from 10 000 cm −3 to 40 000 cm −3 in this study is because the uncertainty of η CPC obtained in Step 1 was unacceptably large, since the current was too low (ca.8 fA) at 10 000 cm −3 for the reference FCAE with the flow rate set equal to that of the test CPC (0.3 l min −1 ).The uncertainties are about the same between calibrations of CPCs with 0.3 l min −1 inlet flow rate starting at 40 000 cm −3 and those of CPCs with 1 l min −1 or 1.5 l min −1 inlet flow rate starting at 10 000 cm −3 .The concentration of 2000 cm −3  in Measurements (b) and (c) in this study was chosen as the logarithmic mid-point between 40 000 cm −3 in Step 1 and Measurement (a) and 100 cm −3 in Measurement (d) at which a comparison would be performed with METAS.

Results and discussion
The intercomparison was carried out using PS spheres with nominal diameter of 100 nm in the concentration range 1 cm −3-100 cm −3 .A CPC Model 3752 (TSI Inc., USA) with the inlet flow rate set at 0.3 l min −1 (low-flow mode) was used as transfer standard (i.e.test CPC).According to the manufacturer, this CPC counts single particles up to concentrations of 100 000 cm −3 .To extract more information about the CE profile of the CPC, the instrument was also calibrated in the range 2 000 cm −3 -60 000 cm −3 against an FCAE as shown in figure 5.These measurement points provide a useful insight into the performance characteristics of the test CPC.The results are summarized in table 2. A detailed uncertainty analysis for the measurements by METAS can be found in [8].More information on the evaluation of the counting efficiencies and their associated uncertainties by AIST is given in appendix C.
The CPC exhibited a counting efficiency CE ∼1 at a concentration of 2 000 cm −3 .At higher concentrations CE started slowly to decrease, reaching 0.91 at ∼60 000 cm −3 .The reason was most likely coincidence count losses and insufficient correction for it by the built-in correction algorithms.At concentrations ⩽130 cm −3 the CPC CE remained high (⩾0.97)without any indication of false counts.The measurements by METAS and AIST showed a deviation of 1%-4% which was consistent with the stated measurement uncertainties.It should be noted that the counting efficiencies of AIST were consistently greater than those of METAS by as much as 4% in the range 10 cm −3 -1000 cm −3 .Since the differences were within the stated uncertainties by the two laboratories, the reasons for the differences may have been already fully accounted for, but perhaps there may have been additional sources for the observed differences such as slight performance change of the test CPC during the intercomparison.For example, while the detection efficiency was calibrated at METAS both before and after shipping from Switzerland to Japan and then from Japan to Switzerland and found unchanged, there may have been a slight drift in the detection efficiency that was smaller than the calibration uncertainty of about 3% at 95% confidence level.To calculate a mean CPC CE in the range 1 cm −3 -130 cm −3 , the arithmetic average of the measurement points needs to be calculated.Since METAS provided more data points than AIST (i.e. 6 instead of 3) in the range 1 cm −3 -130 cm −3 , average CE METAS values were calculated in the range 15 cm −3 -56 cm −3 and 115 cm −3 -130 cm −3 .In the end, the average CE value, which consisted of the three METAS and three AIST CEs, was found to be 0.999 ± 0.015 depicted by a solid brown line and shaded orange area in figure 5.The uncertainty was expressed as standard deviation of the mean multiplied by a coverage factor k = 2 (95% confidence The CE slightly exceeded unity at concentrations ⩽2060 cm −3 and at 1.80 cm −3 in the measurements reported by and METAS, respectively.It is usually considered that the CE of CPCs cannot exceed unity even in the plateau region because there is some area in the condenser section in CPCs where the saturation ratio is less than unity and hence condensation growth does not occur.A possible reason for the fact that the measured CE is equal to or greater than unity is an error in the flow rate used in the calculation of concentration in the CPC, which can make the apparent CE greater than unity.However, the nominal flow rate of the test CPC (0.3 l min −1 ) agreed very well with the flow rate measured at AIST (0.3005 l min −1 ± 0.0016 l min −1 at 22 • C, with a coverage factor of 2), which does not explain the CE ⩾1.The reason why the CE of the test CPC was about unity seems to be due to embedded and undisclosed corrections in the test CPC.The magnitude of the internal correction seemed to be about +2%, which was derived from the fact that the concentration in the test CPC's output files was about 2% greater than the concentration calculated from the counts per second found in the same output files for the corresponding measurement periods assuming the flow rate of 0.3 l min −1 at concentrations where coincidence count loss is negligible.The application of internal correction was further confirmed by an experiment with an IAG with 155 nm particles, in which the number of particles generated per second by the IAG (30 s −1 , which corresponds to concentration of 6 cm −3 at 0.3 l min −1 ) was compared with the counts per second by the test CPC.The ratio (equivalent to number-based CE, instead of concentration-based) was 0.978.This means that about 2% of particles were not counted by the test CPC, probably due to less-than-100% condensation growth efficiency and/or losses in the test CPC.Therefore, the observed CE of about unity for the test CPC was not due to an error in the calibration but can be explained as a result of the +2% internal correction for the 2% lost counts.
The test CPC 3752 used in this study was new (the intercomparison took place within one year from purchase date) and was used at METAS as secondary standard for low concentration measurements (typically below 50 000 cm −3 ) to prevent contamination of the optics.It is therefore not surprising that the CPC response remains linear down to 1 cm −3 .CPC performance at low concentrations depends on the model and maintenance state and can therefore vary significantly with time.The CE profile shown in figure 5 applies to the specific CPC under test at the time of calibration and should not be assumed to universally apply to every CPC of the same series.
It is worth noting that the quantity 'particle number concentration' is calculated by dividing counts (dimensionless) by volume.According to the document The International System of Units (SI) published by Bureau International des Poids et Mesures, 'there are also some quantities that cannot be described in terms of the seven base quantities of the SI, but have the nature of a count.Counting quantities are also quantities with the associated unit one.The unit one is the neutral element of any system of units-necessary and present automatically.There is no requirement to introduce it formally by decision.Therefore, a formal traceability to the SI can be established through appropriate, validated measurement procedures', as the ones described in this study [19].
This inter-comparison provides a step towards formal traceability to the SI by validating two different measurement procedures for low particle concentrations.It also serves as an intersection between the ISO 21501-4 and ISO 27891 standards for OPCs and CPCs, respectively [1,4].In effect, the measurement procedure by METAS describes a procedure for calibrating CPCs based on a reference OPC, while the method developed by AIST can be applied to calibrate CPCs (and OPCs too) against a reference FCAE at concentrations as low as 1 cm −3 .

Conclusions
An inter-comparison between the National Metrology Institute of Japan (AIST) and Switzerland (METAS) was carried out using PS spheres with nominal diameter of 100 nm in the concentration range 1 cm −3 -100 cm −3 .A CPC Model 3752 (TSI Inc., USA), which according to the manufacturer counts single particles up to concentrations of 100 000 cm −3 , was used as transfer standard.The CPC was calibrated by the two laboratories against their primary national standards.To extract more information about the CE profile of the CPC, the instrument was also calibrated in the range 2 000 cm −3 -60 000 cm −3 against an aerosol electrometer.
The CPC exhibited a counting efficiency CE close to 1 in the concentration range 1 cm −3 -2 000 cm −3 .There was no indication of false counts at low concentrations below 100 cm −3 .At higher concentrations CE starts slowly to decrease, reaching 0.91 at about 60 000 cm −3 , most likely due to coincidence count losses.
The obtained results were in good agreement, with a deviation of 1%-4%, which was compatible with the stated uncertainties.The good agreement between the two participating laboratories, despite the fact that the two primary standards relied on different measurement principles, provide strong evidence for the proficiency of the participants in the field of CPC calibration, validating their individual measurement procedures.

A.1. Step 1
In the setup shown in figure 3, the EAG outputs primary aerosols to be conditioned for generation of size-classified and concentration-adjusted charged calibration aerosols.In this study, PS particles of 100 nm in diameter were aerosolized.For bipolar charge conditioning of the primary aerosol, an Am-241 ion source of 4 MBq is installed in the EAG.Since the 4 MBq Am-241 in the EAG is not sufficient to achieve a thoroughly 'equilibrated' charging state, another Am-241 charge conditioner is added after the EAG.The concentration of the calibration aerosol is adjusted by a set of two diluters in series ('Diluter set' in figure 3) between the EAG and the DMA: one is a 'loop-type' diluter with particle filters, a blower, a valve, and a flow meter, and the other is a diluter by addition of particle-free air through a mass flow controller (see [17] for more detail).After the diluters, there is a branch ('vent' in figure 3) with a particle filter to bleed aerosol flow that is in excess of the prescribed inlet flow rate of the DMA.The DMA classifies particles according to their electrical mobility diameter and consists of a column, either Model 3081 (Long) or Model 3085 (Nano) depending on classifying particle size, and a controller (Model 3080) by TSI Inc. (USA).For this study, a Long column was used for the 100 nm PS particles.The DMA was operated with a flow rate setting of 0.9 l min −1 aerosol and 7 l min −1 sheath air.The mass flow controller, which is used to add particle-free 'make-up' flow to the size-classified aerosol after the DMA, was off in this study since the aerosol flow rate of the DMA (0.9 l min −1 ) was equal to the sum of the flow rates of the reference FCAE, test CPC, and monitor CPC (0.3 l min −1 for each).The concentration of the output calibration aerosol from the DMA is continuously recorded with a monitor CPC (Model 3776, TSI Inc., USA).The calibration aerosol passes through a mixer (StaticMixer Model T8-15R by Noritake Co. Ltd, Japan) for complete mixing with the make-up air and then flows through a four-way flow splitter (Model 3708 by TSI Inc., USA).Two ports of the splitter are used to deliver the calibration aerosol to the reference FCAE and test CPC, while one of the remaining two ports is used as a tap for measurement of the line pressure at the splitter and the other is plugged.The connecting tubes from the flow splitter to the reference FCAE and test CPC are made identical with metal fitting parts by Swagelok Co. (USA) and geometrically symmetric with respect to the center axis of the flow splitter.The tubing upstream of the mixer is made of conductive silicone tubes.

A.2 Steps 2-5
The '2nd diluter set' in figure 4 that is added to the setup for Steps 2-5 consist of a rotating disc diluter (Model MD19-1i by Matter Engineering AG, Switzerland) and a bridge diluter produced in-house.Depending on the dilution ratio needed, only the rotating disc diluter, only the bridge diluter, or both the rotating disc and bridge diluters (connected in series) are used.The tubing from the flow splitter to the second diluter set and to the test CPC is replaced with conductive silicone tubes.The flow settings are different from Step 1.In this study, the flow rate of the reference FCAE was increased to 1.5 l min −1 to have higher particle current readings by the reference FCAE for a given particle concentration.The flow rate from the splitter to the second diluter set was either 0.5 l min −1 when the rotating disc diluter was used, or 0.3 l min −1 when the rotating disc diluter was not used.The make-up flow after the DMA was set at 0.8 l min −1 (or 0.5 l min −1 without the rotating disc diluter) so that the DMA's aerosol flow rate was regulated at 1.5 l min −1 while the sum of the flow rates of the reference FCAE, second diluter set, and monitor CPC was 2.3 l min −1 (or 2 l min −1 ).The sheath flow rate of the DMA was set at 10 l min −1 .Another difference in the measurement setup between Step 1 and Steps 2-5 in this study was that the second charge conditioner was omitted in Steps 2-5 since higher particle concentrations could be attained after the DMA by removing it.The influence of omitting the second charge conditioner to the determination of the counting efficiencies in Steps 2-5 is discussed in the uncertainty evaluation in appendix C.

Appendix B. Derivation of R ab = η 2k /η 40k
In Measurement (a), the indicated concentrations C FCAE,a and C CPC,a are expressed as: where η FCAE,a is the detection efficiency of the reference FCAE at the concentration level in Measurement (a) (i.e.80 000 cm −3 ), η CPC,a is the CE of the test CPC at the concentration level in Measurement (a) (i.e.40 000 cm −3 ) and is equal to η 40k , C * (DR 1,a ) is the concentration before the flow splitter when the dilution ratio of the first diluter set is DR 1,a .Note that the splitter bias correction (β) applies only to C FCAE,a while DR 2,a applies only to C CPC,a .The ratio r a is then expressed as: where η FCAE,b is the detection efficiency of the reference FCAE at the concentration level in Measurement (b) (i.e. 4 000 cm −3 ), η CPC,b is the CE of the test CPC at the concentration level in Measurement b) (i.e. 2 000 cm −3 ) and is equal to η 2k , C * (DR 1,b ) is the concentration before the flow splitter when the dilution ratio of the first diluter set is DR 1,b .Note that the splitter bias correction (β) is assumed to be independent of the concentration level.The ratio r b is then expressed as: With equations (B.3) and (B.6), R ab is derived as: Note that DR 2,a and DR 2,b cancel out because DR 2 is not changed between Measurements (a) and (b).η FCAE,a and η FCAE,b also cancel out because the detection efficiency of the reference FCAE is considered constant with respect to the concentration level.The uncertainty of η FCAE,a /η FCAE,b = 1 is discussed in appendix C. The following nine sources of uncertainty are considered for η CPC in Step 1.The first four uncertainty sources (i)-(iv) are for the variables in equations ( 1) and (2).Two of the nine sources (repeatability and day-to-day variability) are considered in (vii).
(i) Current measurement by the reference FCAE, u (I) The uncertainty of current measurement corresponds to the type B uncertainty of the correction factor η I for current measurement, which is part of η FCAE in equation (1).Evaluation of the random uncertainty component in measured current (I FCAE ), which becomes a major component at low particle concentrations, is included in the evaluation of the type A repeatability uncertainty in (vii).
The evaluation of the type B uncertainty referred to [16].
In [16], the standard uncertainty of η I , u (η I ), which is essentially equal to u (b) = 0.003 54 in equation ( 14) in [16], was derived by considering the uncertainty in the calculation of the slope b by linear regression analysis of the electrometer's calibration data at ±1 pA and ±0.5 pA and the uncertainty that considered possible local deviation of the slope from b, which is obtained by the above regression, at current scales smaller than picoampere (about 20 fA).In addition, the long-term variability of the slope evaluated from annual calibration data of the electrometer for seven years was evaluated as s (b) /b = 0.00211.By combining these, the standard uncertainty u (I) was evaluated as where η I and b were approximately equal to 1.
(ii) Volume flow rate measurement by the reference FCAE, u (Q) The uncertainty of flow rate measurement corresponds to the type B uncertainty of the correction factor η Q for flow rate measurement.Evaluation of the random uncertainty component in measured flow rate (Q FCAE ), which is essentially negligibly small, is included in the evaluation of the type A repeatability uncertainty in (vii).
(iii) Elementary charge, u (e) The uncertainty of the elementary charge is zero for the value of 1.602 176 634 × 10 −19 C defined in the SI [19].
(iv) Non-ideal characteristics of the Faraday cup, u (η FC ) The penetration efficiency of 100 nm particles through the inlet tube of the Faraday cup (3.5 cm long) at 0.3 l min −1 is estimated to be 0.9984 using equations ( 1) and ( 5) in [18].The filtration efficiency was experimentally determined to be 0.999 9 or greater for the filter in the reference FCAE.Since these efficiencies are essentially 100% (i.e.ideal), the uncertainty for non-ideal characteristics was considered zero.
(v) Flow splitter bias correction factor, u (β) The value of the flow splitter bias correction factor, β, and its uncertainty are evaluated based on the procedure given in Annex G of ISO 27891 [4].In this study, the value of β and its standard uncertainty, solely for repeatability, at flow rates of 0.3 l min −1 and particle size of 100 nm were 1.001 50 and 0.000 34, respectively.In addition, we combined a relative standard uncertainty of 0.002 3 for long-term variability, which we derived for our setup from a set of β values obtained over nine years for particle size range from 23 nm to 300 nm and flow rates of 1 l min −1 and 1.5 l min −1 , i.e.
(vi) Multiple charge correction factor ( ∑ ϕ p • p), u (MCC) The standard uncertainty of the multiple charge correction factor, MCC = ∑ ϕ p • p, which is determined by following the procedure in D.3.2 in ISO 27891 [4], is calculated as u (MCC) = (MCC − 1) / √ 3 assuming a rectangular probability distribution between 1 and 2 × MCC − 1, i.e. the contribution of multiply charged particles (MCC −1) may be as small as zero and may be twice as large as that was measured.In this study, the value of MCC was about 1. 001  In this study, measurements of Step 1 were repeated for 2 days, and 32 1 -min data were obtained for each day.The analysis of variance on this data set provided an average CE of 0.934 2 and a standard deviation for repeatability of 0.004 1, which resulted in the standard uncertainty u rep (η CPC ) = 0.004 1/ √ 32 = 0.000 7.In addition, day-to-day variability in the determination of η 40k was evaluated to be u d2d (η CPC ) = 0.002 03 by analyzing 3-day data sets of η 40k with 40 repetitions per day performed without the second charge conditioner.(The repeatability was evaluated with the second charge conditioner.)Note that the influence of the second charge conditioner was very minor and was considered to have no effect in the evaluation of the day-to-day variability u rep (η CPC ) /η CPC = 0.000 72/0.934 2 = 0.000 77 u d2d (η CPC ) /η CPC = 0.002 03/0.934 2 = 0.002 17.
(viii) Uncertainty due to particle size uncertainty, u d (η CPC ) Since the calibration in this study was performed at 100 nm in the 'plateau' region of the test CPC, where errors in particle size of the calibration aerosol have negligible influence on the determination of the CE, this uncertainty was considered zero.
Table C.1 summarizes these uncertainties and shows that the expanded uncertainty (coverage factor k = 2) of the CE (η CPC = 0.934 2, which is η 40k in Step 2) was 0.010 9 in this study.The expanded uncertainty was derived with the following equations (C.1) and (C.2) assuming that the uncertainty sources were independent of each other The result of the evaluation of R ab and η 2k in this study was R ab = 1.076 5 The following seven sources of uncertainty are considered for η 2k .The first one is the uncertainty of η 40k .The other six sources are for R ab .
(i) Uncertainty of η 40k , u (η 40k ) The value of the uncertainty of η 40k is taken from Step 1, that is where s (r a ) and s (r b ) are the experimental standard deviations of r a and r b , respectively, from ten-time repeated measurements for each.In this study, seven pairs of Measurements (a) and (b) were repeated in one day and the values of u ran (R ab ) /R ab ranged between 0.0014 and 0.0024.The largest value was taken as the representative uncertainty, that is u ran (R ab ) /R ab = 0.002 4.
(iii) Diurnal variability of R ab , u diu (R ab ) In this study, the average and standard deviation of R ab were obtained as 1.076 5 and 0.002 7, respectively, from the seven pairs of Measurements (a) and (b) in the above (ii).The average was taken as the result of the Step 2 measurement given at the beginning of this section.The standard deviation was used for the calculation of uncertainty due to diurnal variability as u diu (R ab ) /R ab = 0.002 7/1.076 5 = 0.002 5.
(iv) Day-to-day variability of R ab , u d2d (R ab ) The day-to-day variability was evaluated as the standard deviation of R ab values that were obtained for a CPC, which was different from the test CPC in this study, over five years at about 1.5 years interval with similar concentration levels (Measurements (a) and (b) performed at 10 k and 1 k), which was  assuming a uniform probability distribution with the center at 0.934 2 and the half width of 0.007 4.
(vii) Effect of the use of only one Am-241 charge conditioner, u chrg (R ab ) As discussed in section A.2, the second Am-241 charge conditioner is omitted in Steps 2-5 to increase particle concentration.The increase of particle concentration occurs because the charge state without the second charge conditioner is somewhat 'off the equilibrium' state and has a higher fraction of +1-charged particles due to insufficient ion concentration for bringing initially highly-positively charged particles to 'neutral'.The incomplete charge conditioning not only increases +1charged 100 nm particles but also increases multiply charged particles (+2-charged 151 nm particles, +3charged 196 nm particles) in the DMA-classified calibration aerosol.The magnitude of this effect was evaluated by measuring the CE at 40 000 cm −3 in Step 1 with and without the second charge conditioner, and not including the term for multiple charge correction (i.e. the summation term) in equation ( 1) for the calculation of the CE.It was found that the measured CE without the second charge conditioner (0.924 in this comparison) was lower than that with the second charge conditioner (0.933) by 0.009, which corresponds to an increase of about 1% in the fraction of +2-charged particles, assuming that the decrease of the CE occurred solely due to an increase of multiply charged particles in the calibration aerosol and that the multiply-charged particles were all doubly charged.Change in the fraction of multiply charged particles may affect the measurement of R ab if the CE of the test CPC is significantly different between the size of singly charged particles and the sizes of multiply charged particles.In this study, however, the CE at 100 nm and that at 151 nm are essentially the same because both sizes are in the plateau region.Therefore, the effect of the use of only one Am-241 charge conditioner is negligible, and the uncertainty due to this effect is considered zero, that is Table C.2 summarizes these uncertainties and shows that the expanded uncertainty (coverage factor k = 2) of the CE (η 2k 1= 1.006) was 0.018 in this study.The expanded uncertainty was derived with the following equations (C.4) and (C.5) assuming that the uncertainty sources were independent of each other.

Figure 1 .
Figure 1.Schematic illustration of the primary standard at METAS (drawing not in scale).MFC, DMA and AAC stand for mass flow controller, differential mobility analyzer and aerodynamic aerosol classifier, respectively.

Figure 2 .
Figure 2. Conceptual diagram of AIST's stepwise dilution method for calibration of a CPC at concentrations as low as 1 cm −3 .

Figure 3 .
Figure 3. AIST's measurement setup for Step 1.The numbers indicate flow rates in units of liter per minute.EAG stands for electrospray aerosol generator.

Figure 4 .
Figure 4. AIST's measurement setup for Steps 2-5.The numbers indicate flow rates in units of liter per minute.The numbers in the parenthesis are the flow rates when the rotating disc diluter is not used in the second diluter set.

− 3 (
Step 1) taken into account.With these diluter settings, the particle concentrations indicated by the reference FCAE and test CPC (C FCAE,a and C CPC,a , respectively) are recorded.Then the ratio of C CPC,a to C FCAE,a is calculated as r a = C CPC,a /C FCAE,a .The second measurement, which is denoted as Measurement (b), is performed with the setting of the second diluter set unchanged from Measurement (a), i.e.DR 2,b = DR 2,a , with the dilution ratio of the first diluter set (DR 1,b ) being adjusted so that the inlet concentration of the test CPC (C inlet CPC,b ) is equal to the next target concentration.In this study, DR 1,b was set to about 1/20 because the target concentration of Measurement (b) was 2 000 cm −3 .

r a = ∑ r a / 10 .
The recording of C FCAE,b and C CPC,b in Measurement (b) is similarly repeated ten times to derive the average, r b .Then R ab in equation (3) is calculated as R ab = r b /r a .Steps 3-5.In Step 3, in which the ratio of η 100 /η 2k is to be determined, Measurements (c) and (d) are performed.In Measurement (c), the dilution ratio of the first diluter set (DR 1,c ) is set equal to 1 (i.e.no dilution, similar to Measurement (a)) while the dilution ratio of the second diluter set (DR 2,c ) is adjusted so that the inlet concentration of the test CPC (C inlet CPC,c ) is equal to 2 000 cm −3 .Note that the reference FCAE indicates 80 000 cm −3 in Measurement (c), as high as the concentration in Measurement (a).With these diluter settings, the particle concentrations indicated by the reference FCAE and test CPC, C FCAE,c and C CPC,c , respectively, are recorded.Then the ratio of C CPC,c to C FCAE,c is calculated as r c = C CPC,c /C FCAE,c .In Measurement (d), the inlet concentration of the test CPC (C inlet CPC,d ) is reduced to 100 cm −3 by adjusting DR 1,d (i.e. about 1/20) while DR 2,d is unchanged from Measurement (c).Note that the reference FCAE indicates 4 000 cm −3 , as high as the concentration in Measurement (b).With these diluter settings, the particle concentrations indicated by the reference FCAE and test CPC, C FCAE,d and C CPC,d , respectively, are recorded.Then the ratio of C CPC,c to C FCAE,d is calculated as r d = C CPC,d /C FCAE,d .After Measurements (c) and (d), the ratio of the ratios r c and r d is calculated as

Figure 5 .
Figure 5. Counting efficiency of the test CPC (TSI 3752) as a function of the particle concentration.The measurement data by AIST and METAS are depicted as green diamonds and blue circles, respectively.Data at high concentrations which were determined using an FCAE as reference instrument without prior aerosol dilution are depicted as open symbols.The average counting efficiency (CEaverage) of the CPC at concentrations ⩽130 cm −3 is designated by a solid brown line.The shaded orange area designates the confidence interval (k = 2) of CEaverage.

3 )
In Measurement (b), the indicated concentrations C FCAE,b and C CPC,b are expressed as:

U 9 η 10 = 7 η 1 =
= k • u c (η 2k ) .(C.5) C.3.Steps 3-5The results of the evaluation of R cd and η 100 in Step 3, R ef and η 10 in Step 4, and R gh and η 1 in Step 5 in this study were:Step 3 R cd = 1.005 8η 100 = η 2k • R cd = 1.006 × 1.005 8 = 1.011.Step 4 R ef = 1.001 η 100 • R ef = 1.011 × 1.001 9 = 1.013.Step 5 R gh = 0.993 η 10 • R gh = 1.013 × 0.993 7 = 1.007.Five sources of uncertainty are considered in Steps 3-5, which correspond to the uncertainty sources (i)-(v) in Step 2. They were evaluated similarly to Step 2 and are summarized in table C.3.Note that the uncertainty source (v) contains a potential systematic bias (i.e.local deviation of slope in current measurement, which is 0.002 87 in relative standard uncertainty) that is first considered in Step 2 and then repeated in Steps 3-5.This causes correlation and was taken into consideration in the calculation of relative combined uncertainties in table C.3.The uncertainty sources (vi) and (vii) are omitted in Steps 3-5 since they need to be considered once only in Step 2.

Table 1 .
Nominal inlet concentrations, dilution ratios, and voltage-on durations in Steps 2-5 in this study.With these diluter settings (table 1), the particle concentrations indicated by the reference FCAE and test CPC (C FCAE,b and C CPC,b , respectively) are recorded.Subsequently, the ratio of

Table 2 .
Counting efficiency CE of the test CPC and associated expanded uncertainties U (coverage factor k = 2; 95% confidence interval) in the concentration range 1 cm −3 -67 000 cm −3 .Note that CE and U are dimensionless quantities.
Measurement repeatability and daily variability of the CE of the test CPC, u rep (η CPC ) and u d2d (η CPC ) 3)ndom effects in the determination of R ab , u ran (R ab ) As explained in section 2.2.3 the value of R ab is obtained as R ab = r b /r a from Measurements (a) and (b).The uncertainty due to the random effects in the determination of R ab is evaluated with the following equation (C.3)

Table C . 1 .
Uncertainty budget forStep 1 in this study.
a ) and s (r b ): Uncertainty of η FCAE,a /η FCAE,b , u FCAE (R ab ) The uncertainty of η FCAE,a /η FCAE,b accounts for a possible deviation of the value of η FCAE,a /η FCAE,b from unity.As explained in Step 1 in section 2.2.3, the detection efficiency of the reference FCAE (η FCAE ) is expressed as η FCAE = η I • η Q • η FC .Since η Q and η FC are independent of concentration, the value of η FCAE,a /η FCAE,b may deviate from unity if η I changes with concentration.The magnitude of a possible change of η [16] concentration in Measurement B from η I at concentration in Measurement A was estimated to be comparable with the local variability (u B(D) (b * ) = 0.002 87) and daily variability (u D (b * ) = 0.001 09) in equation (14) evaluated in[16].That is u FCAE (R ab ) /R ab =

Table C . 2 .
Uncertainty budget forStep 2 in this study.