Accuracy of recovered moments for narrow mobility distributions obtained with commonly used inversion algorithms for mobility size spectrometers

ABSTRACT

Aerosol mobility size spectrometers are commonly used to measure size distributions of submicrometer aerosol particles. Commonly used data inversion algorithms for these instruments assume that the measured mobility distribution is broad relative to the DMA transfer function. This article theoretically examines errors that are incurred for input distributions of any width with an emphasis on those with mobility widths comparable to that of the DMA's transfer function. Our analysis is valid in the limit of slow scan rates, and is applicable to the interpretation of measurements such as those obtained with tandem differential mobility analyzers as well as broader distributions. The analysis leads to expressions that show the relationship between the inverted number concentration, mean size, and standard deviation and true values of those parameters. For narrow distributions (e.g., for a mobility distribution produced by a DMA with a 1:10 aerosol:sheath air flow ratio) under typical operating conditions, number concentrations and mean mobility obtained with inversion algorithms are accurate to within 0.5% and 1.0%, respectively. This corresponds to mean diameter retrieval errors of 1.0% for large particles and 0.5% for small (kinetic regime) particles. The widths (i.e., relative mobility variance) of the inverted distributions, however, significantly exceed the true values.

Copyright © 2018 American Association for Aerosol Research

EDITOR:

• Jingkun Jiang

1. Introduction

An aerosol mobility size spectrometer system consists of an aerosol charger followed by Differential Mobility Analyzer (DMA) and an aerosol detector, usually a Condensation Particle Counter (CPC). In the charger, particles acquire a known steady state charge distribution according to size. The resulting charged particles are size-selected in the DMA according to their electrical mobility and the DMA voltage setting and are then passed through to the detector. The distribution of particles passing through the DMA has a finite width of mobility. However, in the standard approaches to data analysis, they are generally treated as all of the centroid mobility of the classifier window (Knutson 1976). This “constant distribution” approximation results in a generally small but predictable error.¹ The following analysis derives the resulting biases in the moments–e.g., total number, mobility mean and standard deviation–of the recovered size distribution. The data are treated as continuous, not discretized, in this analysis.

There are many potential sources of error in the recovered size distribution–particle shape effects, limited data resolution, multiple charging and charge fraction uncertainties, uncertainties in transport losses or detection efficiencies, as well as issues specific to voltage scanning. Though many of these can have a far greater impact on the recovered distribution, the analysis here considers only the aforementioned inversion approximation. In particular, the scanning mobility particle sizer (SMPS; Wang and Flagan 1990) is widely used and can suffer from effects of detector arrival time smearing (Russell et al. 1995; Collins et al. 2002) and warping of the DMA transfer function (Collins et al. 2004; Dubey and Dhaniyala 2008; Mamakos et al. 2008) at fast scan rates. Our analysis assumes operation with steady classifier voltage and will only be accurate for SMPS data in the limit of slow scan rates, i.e., where the ratio of the mean gas residence time in the classifier to the time constant for the exponential voltage scan ramp is smaller than about 0.02. In this limit, distortions of the transfer function due to changes in classifying voltage as particles travel through the classifier are negligible relative to other sources of error (Collins 2002; Dubey and Dhaniyala 2008; Mamakos et al. 2008).

The narrow distribution results of the analysis in this work are particularly pertinent to tandem differential mobility analyzer (TDMA) measurements in which a mobility size spectrometer is used to measure the output distribution from another DMA, with or without an intervening aerosol conditioner. In most cases, relatively low aerosol concentrations mandate slow scan rates or stepped voltage operation. In such case, this analysis can provide accurate estimates of the biases resulting from use of the constant distribution approximation in the data analysis, as employed in the majority of SMPS systems. For example, the analysis derived here provides the mathematical basis for our companion paper (Stolzenburg and McMurry 2018). That Aerosol Research Letter describes a method to assess the performance of the mobility classifier and condensation particle counter in aerosol mobility size spectrometer systems.

2. Theory

This section summarizes the theoretical basis for extracting moments of mobility distributions measured using mobility size spectrometers. We begin with the general relationship between measurements and number distributions, and then discuss errors that accrue during data inversion when the measured mobility distribution is assumed constant within the DMA mobility window while “recovering” moments from measurements. This assumption is made with most commonly-used inversion software. Our analysis focuses on errors in the total number concentration, mean mobility, and standard deviation of the mobility distribution (i.e., the 0th, 1st, and 2nd moments). We obtain three sets of results for those errors. The first is a rapidly converging infinite series. With a diffusion-broadened DMA transfer function, the series eventually diverges, but appropriate truncation of the series still yields highly accurate results in many cases. The second set of results truncates this series to a single term, and is applicable if the DMA transfer function is “sufficiently” narrow. The third is applicable to narrow, near monodisperse, input distributions that are definitely not constant within the DMA mobility window.

2.1. Standard mobility spectrometer data analysis

The raw measured data from a mobility spectrometer system can be characterized for any input aerosol as[1]

The functionality of the DMA transfer function, , can be best described by introducing , the centroid mobility of the transfer window at a given voltage, . For non-diffusing aerosols, depends only on the dimensionless ratio , but an extra dependence (through particle diffusivity, ) is added for diffusing aerosols (Stolzenburg 1988; Stolzenburg and McMurry 2008). Neglecting multiple charging (), there is a unique relationship between (or ) and , and the rest of the factors within the integral of Equation (1[1] ) can be combined to give[2] where[3]

Once the mobility distribution function, , is found, it is relatively straightforward to calculate . Though the latter procedure may involve other approximations and sources of errors, none of that is addressed here. The focus here is on the recovery of , which is introduced for notational convenience in the following analysis. is essentially the DMA input mobility distribution function for singly-charged particles of the opposite polarity as .² However, it also includes post-DMA efficiency factors. From this point on, all references to “input distribution” refer to .

In the standard mobility spectrometer data analysis, ultimately based on the formulation of Knutson (1976) for steady-state operation, it is assumed that does not vary significantly within the relatively narrow non-zero width of such that it can be treated as a constant evaluated at and pulled out of the integral of Equation (2[2] ) to give[4]

The last equation can be rearranged to give[5] where is used to designate this standard “recovered mobility distribution” defined by the last equality in Equation (5[5] ). Combining the last equality with Equation (2[2] ), the recovered mobility distribution function can be written as[6]

Using this equation, the question now becomes how accurately does reflect . As many size distributions can be well-characterized by their total number concentration, mean size and standard deviation (i.e., the first three moments of the distribution), the accuracy of the recovered mobility distribution will be evaluated according to the accuracy of the recovered moments for both broad and narrow distributions. Though geometric mean diameter and standard deviation are most often used to characterize a size distribution, arithmetic mobility moments are used here to facilitate the mathematical manipulations.

The data inversion employs the single approximation over the width of the DMA transfer function. One may imagine an input distribution such that this is an exact relationship within a limited non-zero region of the distribution. Within this region the recovered mobility distribution will have no bias. However, in order to have well-defined moments, the distribution must go to zero at either end, necessitating the existence of regions where this condition is violated. In addition, the mobility width of the DMA transfer function increases at higher mobilities (lower voltages), even in the absence of diffusion, leading to an element of asymmetry in the voltage scan results. The combination of these two effects leads to a small bias in all the recovered moments, which to first order is independent of the characteristics of the input distribution. The following analysis will demonstrate the existence of and derive expression for these biases.

For the case of the non-diffusing transfer function, Knutson and Whitby (1975) described a very accurate method of recovering the mobility moments that eliminates these approximations and biases. Their analysis is in a number of respects very similar to what follows here, though restricted to the non-diffusing case. The analysis here, reduced to the non-diffusing case, is consistent with the Knutson and Whitby (1975) analysis.

2.2. Arithmetic moments of the recovered mobility distribution

The moments to consider in this analysis are given by[7] for the input distribution and[8] for the recovered distribution. Note that is the k^th moment of the mobility distribution. The objective of this analysis is to assess errors in values of , , and , the total number concentration, mean mobility, and standard deviation. Substituting Equation (6[6] ) into Equation (8[8] )[9] and interchanging the order of integration gives is fixed in the inner integral of the last equation, allowing the transformationto give[10]

The inner integral of the right side of Equation (10[10] ) is defined as[11] with a nominal range of integration of . However, for typical operation of the DMA with flow ratio and limited diffusional broadening of the transfer function, the range in which is significantly greater than zero is well within . Within that more restricted range the following Taylor series converges[12] where is the binomial coefficient given by . Substituting Equation (12[12] ) into Equation (11[11] ) gives[13] where[14]

and are integrals over the DMA transfer function with various mobility weightings.³ We define them for mathematical concision, and do not focus on their physical significance.

2.2.1. Evaluation of the M_i* and P_k* integrals

In the evaluation of the integrals, the diffusing form of the transfer function (Stolzenburg and McMurry 2008; Stolzenburg 2018; and Equation (S2) of the online supplementary information [SI]) is used, dependent on , fixed flow ratios and , and the dimensionless diffusion parameter . The lengthy derivation of these moments is set forth in the SI and results in[15] for the even moments while the odd moments are all zero.

Equation (15[15] ) can now be substituted back into Equation (13[13] ) to obtain expressions for . Noting again that the odd moments are zero, and substituting , Equation (13[13] ) becomesBoth the inner and outer summations go to infinity, however, they are not independent as both derive from the same original infinite limit of . This dependence is indicated as[16] In the last equation, is proportional to particle diffusivity and thus to . This must be taken into account in the following evaluation of the moments. To make this dependence explicit, a constant of proportionality, , is defined such that[17] where is a constant independent of both and Equation (16[16] ) can then be rewritten as[18]

2.2.2. Infinite series expression for the I_r,k moments

To obtain the moments, substitute Equation (18[18] ) for the inner integral in Equation (10[10] ) to get[19] It is seen that, due to the dependence of the diffusion parameter on , the k^th moment of the recovered distribution, , depends on not only the k^th moment of the input distribution, , but on all higher moments of the input distribution as well. To relate more directly to alone and facilitate calculation of relative error, factors can be rearranged and, following Equation (17[17] ), the notation is introduced to indicate the diffusion parameter evaluated at the arithmetic mean of the input distribution. The result is⁴ [20] The main dependence of the moments on the total number concentration and mean size of the input distribution can be characterized as such that the main dependence on both factors, and , cancels out of the ratio of the final factor in parentheses in Equation (20[20] ). The primary dependence of this factor is thus on the relative width or standard deviation of the input distribution.

Note that, as the width of the transfer function in the form of and goes to zero, the only remaining non-zero term in the double summation above is the first () such that , indicating perfect recovery of the moments and the entire size distribution as expected.

2.2.3. Range of validity and series truncation: An approximate expression for I_r,k

The convergence and validity of the infinite series' of Equation (20[20] ) depends upon the convergence of Equation (13[13] ) or Equation (16[16] ), in which the zero odd moments have been eliminated. Minimally, this requires that the moments approach zero as increases. This condition depends directly on the width of the DMA transfer function.

and both contribute to the width of the transfer function, with the former being dominant in the large particle non-diffusing regime and the latter dominating in the small particle diffusion-limited regime. As such, the relative width of the transfer function anywhere is of the order . Note then from Equation (15[15] ) that these moments are of the order . If the relative width is small () then the higher order moments decrease rapidly and the summation of Equation (16[16] ) approaches convergence quickly. For the non-diffusing transfer function (, ), the non-zero portion is confined to within which the approximation of Equation (12[12] ) is valid and the summation of Equation (16[16] ) always converges. For , the tails of extend beyond that range such that the higher order moments of Equation (15[15] ) eventually begin to increase and the infinite summation of Equation (16[16] ) no longer converges. However, a pseudo-convergence is obtained by truncating the summation at the moment of minimum value or fewer terms. If the final term in this truncated summation is small enough, this will still yield an accurate value of . As the transfer function gets broader, more terms are required in the summation to obtain an accurate result. At the same time, the order of the minimum decreases until there are no longer enough terms within the truncated series to obtain an accurate value of . Though not readily quantified, this represents a limitation on the applicable range of Equations (16[16] ) or (18[18] ) for as well as Equation (20[20] ) for the recovered moments.

For sufficiently small relative to 1, the higher order moments decrease fast enough that the series of Equation (16[16] ) or (18[18] ) may be truncated at without significant loss of accuracy. For and in the first equality of Equation (16[16] ), the moments required from Equation (15[15] ) can be simplified to[21] Truncating the summations for at in Equation (16[16] ) gives[22] Finally, employing in Equation (20[20] ) yields[23]

This relationship between the moments of the recovered distribution and those of the input distribution can be put in a more meaningful form by introducing the mean, , and standard deviation, , of the input distribution into the final factor of Equation (23[23] ):The moment relationship for involves the third moment, , of the input distribution. To evaluate this, note thatsuch thatThe first term in the last expression is just the third moment of the input distribution about its mean and can be designated asNote that in the case of an input distribution, , that is symmetric about its mean, , . From the definition of the standard deviation, , the integral in the second term is simply ; and from the definition of the mean, , the third term is zero. Finally the integral in the last term is just the total number or . Thus,

Using these expressions for , Equation (23[23] ) can be broken down into the individual moment relationships as follows:[24] The arithmetic mean and standard deviation of the recovered mobility distribution are given by[25]

Summarizing to this point, the above moment relationships are based on the assumption that the width of the spectrometer's DMA transfer function in the form of its relative standard deviation (as determined by and ) is small compared to 1. Up to this point, no assumptions about the width of the input distribution have been made.

2.2.4. Accuracy of the I_r,k moments for a narrow input distribution

For a narrow, i.e., near monodisperse, input distribution, the assumption that is nearly constant within the non-zero width of is no longer valid and the accuracy of the recovered size distribution may be compromised. If it is assumed that the input distribution is symmetric about its mean and that its relative standard deviation is much less than one, then to good approximation only terms to combined second order in width of size distribution or transfer function need be retained in the moment equations. In general, odd order terms are all zero. With these assumptions, Equation (24[24] ) can be simplified to[26] Again keeping only terms to combined second order, the mean⁵ , , and standard deviation, , of the recovered distribution, , can be obtained from Equations (25[25] ) and (26[26] ) as:[27] and the relative variance of the recovered distribution is given by[28]

The quantity is common to each of the last six equations (Equations (26[26] )–(28[28] )) and, in this context, can be seen as a measure of error, or bias, of each parameter (total number, mean, standard deviation) for the recovered distribution from the corresponding parameter for the input distribution. Bearing in mind the truncation of higher order terms here, and comparing this to transfer function characteristics given in Stolzenburg and McMurry (2008), it is seen that this quantity also represents the relative variance of the spectrometer's DMA transfer function averaged across the width of the input distribution,[29] Thus, as the transfer function width gets larger, so do the deviations and, as it gets smaller, the deviations go to zero.

In the special case of a tandem DMA experiment where the downstream DMA (DMA2) is part of an aerosol mobility size spectrometer system, the input distribution to the spectrometer is largely determined by the transfer function of the upstream DMA (DMA1) with mean equal to the centroid mobility, . Equation (28[28] ) then becomes[30]

In words, Equation (28[28] ) or (30[30] ) says that the relative variance of the recovered distribution is simply the sum of the relative variances of the input distribution and the mobility spectrometer's DMA transfer function. (Note the similarity to the analysis of propagation of independent errors.) Thus, the width of the recovered distribution will only be accurate if the width of the transfer function is small compared to that of the input distribution. This is not the case for the narrow input distributions considered in this section.

3. Results: Errors in recovered moments for triangular and lognormal input distributions

Three sets of equations–Equations (20[20] ), (24[24] ) and (26[26] )–have been developed to estimate the errors, or biases, in the moments of the recovered size distribution. Assuming a successful pseudo convergence of a truncated series, Equation (20[20] ) effectively gives the exact errors for any width transfer function and input mobility distribution while Equations (24[24] ) and (26[26] ) are restricted to narrow transfer functions and Equation (26[26] ) is further restricted to narrow input mobility distributions. The relative variance, , of the DMA transfer function is a key measure of the errors for narrow input distributions. For wider distributions, the error levels increase with additional significant terms in the summations depending on their shape, but the narrow distribution results still provide a ready estimate of the order of magnitude of errors in this case.

For typical operation of mobility spectrometers, the DMA inlet and outlet aerosol flows are equal () and, most often, one tenth of the sheath flow . In almost all instances, such that . Thus, in the case of negligible diffusional broadening of the transfer function, and the errors in the total number and mean mobility are on the order of 1–2% or less. This level of error is well within typical flow measurement uncertainties for the spectrometer's DMA and detector which also affect these two parameters.

For more precise results and to include the effect of diffusional broadening of the transfer function, the accuracy for recovery of the first three moments has been evaluated through comparison with the results obtained by numerical integration of Equation (9[9] ) for triangular and lognormal input mobility distributions. All simulations modeled a mobility spectrometer using a TSI nanoDMA (Model 3085) operated with 20 L/min sheath flow and 2 L/min aerosol flow (, ) at one atmosphere and 23°C.

3.1. Triangular mobility distributions

For triangular input mobility distributions, as if from an upstream DMA (with no diffusion effects), numerical results were obtained at centroid particle sizes of 30, 10, 5, and 3 nm. At the corresponding mean mobilities, , the total relative variance, , of the spectrometer's DMA transfer function ranged from 0.0017 to 0.04 (), the diffusion effect comprising just 1.5% of the value at 30 nm. The relative full width at half maximum of the triangles, , ranged from 0.01 to 0.9 while the corresponding relative variance, , ranged from 0.000017 to 0.14. For comparison, predicted moments were calculated from Equation (20[20] ) with and moments of the input distribution given by[31] For all runs, the numerically calculated mobility moments of the recovered distribution agreed with the predictions within the accuracy of the numerical integrations, i.e., to at least 8 significant figures.

Figure 1 shows plots of the relative errors in recovered total number concentration, , and mean mobility, , as well as the absolute error in the recovered relative variance, , calculated from Equations (20[20] ), (25[25] ) and (31[31] ) as[32] as a function of the relative width of the input distribution at the four tested sizes. All three errors increase with increased broadening of the DMA transfer function at smaller sizes as well as with broadening of the input distribution, though is virtually constant over variations in the width of the input distribution. Nevertheless, under these conditions, the relative errors in recovered total number concentration and mean mobility are less than 0.5% and 1%, respectively. Though the absolute error in the recovered relative variance of the input distribution increases for wider distributions, the relative error decreases to 1.5% to 4.8% for the widest distribution. However, for narrow distributions the absolute error asymptotically approaches a finite constant (relative error goes to infinity) slightly greater than the value of the relative variance of the DMA transfer function, , as indicated by the gray dot-dash lines.

Figure 1. Relative errors in recovered total number concentration and mean mobility and absolute error in the recovered relative variance as a function of centroid particle size and relative width of a triangular input distribution to the mobility spectrometer from Equations (20[20] , n = 5) and (32[32] ). For comparison, gray dot-dash lines indicate the value of the DMA transfer function relative variance at the centroid size (or twice that value on the right axis).

The accuracy of the lower order estimates of error can be gauged as follows. Equation (24[24] ), corresponding to Equation (20[20] ) with , underpredicts by about 0.4 to 1.2% of the true error in going from 30 to 3 nm. The amount is relatively constant over variations in the width of the input distribution. This underprediction increases to double that for and six times that for . For in Equation (20[20] ) the underprediction becomes negligible. The lowest order predictions from Equations (26[26] )–(28[28] ) were obtained for narrow distributions and show none of the dependence of the higher order predictions on the width of the input distribution. These predictions are equivalent to the horizontal gray lines in Figure 1. They provide accurate results for narrow triangles with , with the same underprediction as Equation (24[24] ) for and roughly 1.7 and 3 times that for and , somewhat more accurate than Equation (24[24] ). These lowest order predictions are utilized by Stolzenburg and McMurry (2018) in developing a method to assess the performance of mobility spectrometers.

To relate arithmetic mobility moments to geometric diameter-based statistics often generated by data inversion software, two approximations are very useful. The first is a local power law relationship relating mobility and diameter as[33] is the mobility corresponding to , a reference mobility diameter associated with the input distribution to the mobility spectrometer. may be the mean, , or, in the case of a tandem DMA setup, the DMA1 centroid mobility diameter, , either of which could be approximated by the recovered geometric mean diameter, for a narrow distribution. If the data inversion software reports the geometric mean, and standard deviation, , in diameter space, the power law relationship allows these to be readily converted to mobility space using⁶ [34]

For a narrow input mobility distribution centered about , the following approximation is valid[35] Beginning from the definition of the geometric standard deviation[36] Applying these approximations to the recovered distribution, a value for can be obtained for use in Equation (28[28] ) or (30[30] ).

3.2. Lognormal distributions

For lognormal input mobility distributions, similar to many ambient distributions, the parameters of the mobility distribution were derived from corresponding diameter distribution parameters using Equation (34[34] ). The lognormal mobility distribution was then calculated directly from and . Numerical results were obtained for geometric mean mobilities corresponding to the same four particle sizes as for triangular input distributions. Geometric standard deviation in diameter ranged over 1.01, 1.02, 1.05, 1.1, 1.2, 1.3, … up to the limit of stability of the numerical integration which ranged from = 1.2 at 3 nm to 1.7 at 30 nm. Above this limit, the numerical integration simulated a plateau in the recovered distribution at high mobilities (small sizes) resulting in infinite moment integrals. The exact cause of this apparent anomaly was not readily deduced.

As above, predicted recovered moments were calculated from Equation (20[20] ) with and moments of this lognormal input distribution given by[37] The numerical results agreed with the predictions to more than 6 significant figures, with slightly reduced accuracy for wider distributions and higher moments. For slightly higher values of than the numerical integration stability limits, the summations of the predictions of Equation (20[20] ) also failed to converge, likely due to the violation at small sizes of the assumption prior to Equation (12[12] ) above of limited diffusional broadening of the DMA transfer function.

Though the error in arithmetic moments is accurately predicted by Equation (20[20] ), knowledge of the accuracy of recovery of the original geometric mean and standard deviation is of much greater interest in this case. There is no direct way to obtain these parameters from the arithmetic moments without knowledge or an assumption of the shape of the recovered size distribution. Since the input distribution is lognormal, the recovered distribution can be expected to be similarly shaped. However, the deviations from a true lognormal shape create large discrepancies in the values of the geometric parameters calculated from the first three arithmetic moments (Equation 37[37] applied to the recovered moments and inverted) compared to those obtained by direct integration. Thus, only numerical results for recovery of geometric parameters are presented here.

For this test, the input distribution is a lognormal in mobility diameter space. The same ranges of input parameters as the previous test were used, though for this test the range of stability of the numerical integration was somewhat larger, from = 1.4 at 3 nm to 2.4 at 30 nm. Figure 2 shows plots of the relative error in recovered total number concentration, , and the absolute errors in the recovered log geometric mean diameter, , and the log geometric standard deviation squared, , as[38]

Figure 2. Numerical results for relative error in recovered total number concentration and absolute errors in the recovered log geometric mean diameter and log geometric standard deviation squared (Equation (38[38] )) as a function of geometric mean diameter and standard deviation of a lognormal input distribution to the mobility spectrometer.

Applying diameter distribution approximations similar to those of Equations (35[35] ) and (36[36] ) shows that the error parameters of Equation (38[38] ) are roughly equivalent to those of Equation (32[32] ) for the triangle, though here for the diameter distribution.

As before, all three error parameters increase with increased broadening of the DMA transfer function at smaller sizes as well as with wider input distributions. Also, as with the triangle, the errors within the range of calculations are relatively small except for the width measurement of narrow distributions. However, the sharp rise with increasing distribution width of the measures of both the mean and width suggest the potential of significant errors over an extended range. This seems counterintuitive to our basic understanding of mobility spectrometers.

It may be more appropriate to scale these errors by the relative width of the distribution as gauged by[39] where the gap between and is characteristic of the width. These relative errors can be calculated as[40] and are plotted in Figure 3 along with the original . It is now seen that the errors in recovered mean and width relative to the input width decrease rapidly as the input distribution gets broader. The finite absolute errors in both of these parameters for narrow distributions means that the relative errors go to infinity there. Similar behavior is seen for the corresponding width-relative parameters of the triangular input mobility distribution above.

Figure 3. Numerical results for relative error in recovered total number concentration (Equation (38[38] )), and errors in the recovered log geometric mean diameter and log geometric standard deviation relative to the input log geometric standard deviation (Equation (40[40] )). All are plotted as a function of geometric mean diameter and standard deviation of a lognormal input distribution to the mobility spectrometer.

4. Summary and conclusions

The finite width of the DMA transfer function introduces a bias into the measurements of a mobility spectrometer. The primary contribution to the bias in the recovered moments, mean and standard deviation of the input distribution can be characterized in terms of the relative mobility variance, , of the DMA transfer function. For narrow distributions and transfer functions the relative errors in the recovered total number concentration and mean mobility are simply and , respectively, while the recovered relative mobility variance is just the sum of the relative variances of the input distribution and the transfer function. For wider distributions and transfer functions, these errors increase beyond these simple relationships, with steeper increases for higher order moments. However, when scaled against the width of the input distribution, the relative errors in the recovered mean and width actually decrease for wider distributions. Consequently, for input distributions significantly wider than the DMA transfer function under typical operating conditions, the relative error in recovered total number concentration and the errors in recovered mean diameter and width relative to the width of the input distribution are on the order of 1–2% or less, thus validating the relative accuracy of the standard approach to data analysis. However, for narrow distributions of width on the order of or less than that of the DMA transfer function, only the total number concentration and mean are recovered accurately, generally to better than 0.5% for number and better than 1% for mobility mean relative to itself corresponding to 1% for diameter mean for large particles and 0.5% for small particles in the free molecular regime.

Nomenclature

	=	exponential dependence of mobility on diameter
	=	constant independent of both and
	=	particle number size distribution input to the mobility size spectrometer
	=	DMA input mobility distribution of singly-charged unipolar particles (including post-DMA efficiency factors)
	=	recovered mobility distribution
	=	particle diffusion coefficient
	=	particle mobility diameter
,	=	input, recovered geometric mean diameter
	=	probability that a particle of diameter carries charges
	=	dimensionless DMA geometry and flow factor (Stolzenburg 2018)
	=	kth moment of input distribution (Equation (7[7] ))
	=	3rd moment of input distribution about its mean (preceding Equation (24[24] ))
	=	kth moment of recovered distribution (Equation (8[8] ))
	=	Boltzmann's constant
	=	zeroeth moment, or area, of DMA transfer function
	=	ith moment of the DMA transfer function about (Equation (14[14] ))
	=	particle concentration measured by the mobility spectrometer's detector at DMA voltage
	=	number of elementary charges per particle
,	=	input, recovered total number concentration
	=	DMA migration Peclét number
	=	inverse moment of the DMA transfer function (Equation (11[11] ))
	=	particle charge
	=	DMA aerosol inlet flow rate
	=	DMA clean sheath air inlet flow rate
	=	DMA main excess air outlet flow rate
	=	DMA aerosol sampling outlet flow rate
	=	absolute temperature
	=	DMA voltage
	=	a rough measure of the transfer function relative width
	=	particle electrical mobility
	=	mobility full width at half maximum of a triangular input mobility distribution
	=	centroid mobility of DMA transfer function window at voltage
	=	dimensionless particle mobility
,	=	input, recovered arithmetic mean mobility
,	=	input, recovered geometric mean mobility

Greek symbols

	=	characteristic DMA flow ratio
	=	characteristic DMA flow imbalance ratio
	=	absolute error in parameter
	=	relative error in parameter
	=	combined sampling, transport and detection efficiency of the entire mobility spectrometer excluding
	=	dimensionless DMA diffusion parameter
	=	relative variance of the DMA transfer function
,	=	input, recovered standard deviation in diameter space
,	=	input, recovered geometric standard deviation in diameter space
,	=	input, recovered standard deviation in mobility space
,	=	input, recovered geometric standard deviation in mobility space
	=	DMA transfer function = probability that a particle of mobility successfully traverses the DMA at voltage

Additional information

Funding

This research was supported by the US Department of Energy's Atmospheric System Research, an Office of Science, Office of Biological and Environmental Research program, under grant number DE-SC0011780.

Notes

¹ The inversion algorithm described by Collins et al. (2002) eliminates this error by finely discretizing the mobility distribution function, thereby allowing variations within the DMA mobility window. The inversion technique described by Twomey (1975) is used to determine the size distribution; to our knowledge, this approach is not widely used by aerosol scientists.

² Equation (2[2] ) would be equally valid interpreting as simply the full mobility distribution entering the DMA. However, the following analysis as written relies on a unique relationship between and thus necessitating the restriction to singly-charged particles. A later note indicates how to incorporate multiple charge states.

³ and are very similar to and as defined in Stolzenburg (1988, Equations. I.23 and F.1, respectively) except that is held constant for the integrations in the earlier work while is held constant in the corresponding integrals here (Equations (11[11] ) and (14[14] ), respectively). as used in Stolzenburg and McMurry (2008) and above corresponds to the definition, keeping constant in the integration.

⁴ To account for multiple charge states, the corresponding diffusion parameters are related as . Use Equation (20[20] ) or any of the later derived forms to calculate the recovered moments for the mobility distribution of each charge state separately. Then, for each order moment, the recovered moment of the overall mobility distribution is just the sum of the recovered moments of the component charge states.

⁵ It is common practice to specify the location of a peak in a response curve for a system according to the location of the maximum response value, i.e., the mode. However, to track the mode through Equation (1[1] ) would require detailed information about all the terms within the integral including all the charging, transport and detection efficiency terms. Minimally, the curvature (second derivative) at the peak of both the transfer function and the input distribution as well as the slope (first derivative) of each of the efficiency terms would be required. Instead, it is assumed that the input distribution is essentially symmetric about the mode as it is for an aerosol selected by an upstream DMA with a minimally to moderately diffusion-broadened transfer function. In this case, the mean and the mode are essentially co-located.

⁶ Equations (33[33] )–(36[36] ) are applicable to narrow input as well as recovered distributions.