Equations Governing Single and Tandem DMA Configurations and a New Lognormal Approximation to the Transfer Function

Abstract

The basic equations governing the responses of single and tandem differential mobility analyzer (DMA) systems are summarized. Particle diffusion within the DMA resulting in broadening of the transfer function is included in this analysis. For tandem DMA (TDMA) work, a given particle diameter exiting the first DMA is modeled in the following conditioner as growing into a multi-modal lognormal distribution before entering the second DMA. Approximations and solution techniques for both single and tandem DMA systems are discussed. A new lognormal approximation to the DMA transfer function is introduced leading to a simple lognormal form for the TDMA response function. The maximum absolute error of the transfer function is 0.10 at 200 nm in the range plus or minus one and a half standard deviations of the lognormal fit. It is 0.035 at 3 nm in the range plus or minus two standard deviations of the lognormal fit. The maximum fractional error in the calculated TDMA response is no more than 0.08 at 200 nm and 3 nm in the range plus or minus one standard deviation of the lognormal fit.

DMA NOMENCLATURE

a*	=	= (− dln Z p /dln D p ) | D p *
b	=	radial classifier plate separation
D	=	particle diffusion coefficient
[Dtilde]	=	4π LD/(Q m + Q c ) dimensionless particle diffusivity
D p	=	particle diameter
D p *	=	particle diameter associated with Z p *
[Dtilde] p	=	= D p /D p * dimensionless particle diameter
[Dtilde] pg	=	geometric mean of lognormal Ω in [Dtilde] p space
erf(x)	=	error function
f c	=	aerosol charge fraction
f cond	=	aerosol conditioner function
f 0cond	=	fraction of conditioned aerosol in a mode
G cond	=	geometric mean diameter growth factor of conditioned aerosol mode
G DMA	=	dimensionless classifier geometry and flow factor
L	=	axial classifier inlet and outlet slit separation
M 0	=	zeroeth moment of transfer function
M Dg0	=	zeroeth moment of lognormal Ω in [Dtilde] p space
M Zg0	=	zeroeth moment of lognormal Ω in [Ztilde] p space
dN/dlog Dp	=	aerosol size distribution function
N 1	=	measured aerosol concentration downstream of DMA1
N 2	=	measured aerosol concentration downstream of DMA2
Q a	=	aerosol inlet flow rate
Q c	=	clean sheath air inlet flow rate
Q m	=	main excess air outlet flow rate
Q s	=	aerosol sampling outlet flow rate
R 1	=	outer radius of axial classifier center rod
R 2	=	inner radius of axial classifier housing
R in	=	aerosol inlet radius of radial classifier
R out	=	aerosol outlet radius of radial classifier
V	=	voltage applied to classifier
Z p	=	particle electric mobility
Z p *	=	transfer function centroid electric mobility
Δ Z p	=	transfer function base half width
[Ztilde] p	=	= Z p /Z p * dimensionless particle mobility
[Ztilde] pg	=	geometric mean of lognormal Ω in [Ztilde] p space
Greek Letters	=
β	=	= (Q s + Q a )/(Q m + Q c )
δ	=	= (Q s − Q a )/(Q s + Q a )
ϵ (x)	=	= x ·erf(x) +e − x 2 /√π
Δ φ	=	change in electric flux function from aerosol inlet to outlet
η CPC	=	CPC counting efficiency
σ	=	√G DMA [Dtilde] dimensionless classifier diffusion parameter
σ*	=	σ evaluated at centroid mobility Z p *
σ cond	=	geometric standard deviation of growth factor of conditioned aerosol mode
σ Dg	=	geometric standard deviation of lognormal Ω in [Dtilde] p space
σ Zg	=	geometric standard deviation of lognormal Ω in [Ztilde] p space
Ω	=	classifier transfer function
Ω d	=	diffusing transfer function
Ω nd	=	non-diffusing transfer function

INTRODUCTION

The Differential Mobility Analyzer (DMA) was introduced more than three decades ago (Hewitt 1957; Knutson and Whitby 1975) and was soon used routinely as a sizing standard for submicron gasborne particles (Liu and Pui 1974). Soon thereafter, Knutson (1976) established the principles for using DMAs to measure submicron aerosol size distributions. Numerous refinements in instrumentation and data analysis methods have increased the time resolution and range of sizes that can be measured by electrical mobility classification, and this is now the most commonly used technique for measuring size distributions of 3–100 nm particles. Several commercial instrument systems complete with software are currently available for measuring aerosol mobility distributions. These instruments are referred to as Scanning Mobility Particle Sizers (SMPS; Wang and Flagan 1990). The data inversion is based on the approximation that the aerosol exiting the DMA is “monodisperse” relative to any other particle size-dependent parameter in the system. The end user of such a package no longer needs to know the actual workings of the DMA or the software. All that is required is to input the start and stop voltages (i.e., diameters), number of bins (i.e., resolution) and scan time. When the scan is complete the software outputs the size distribution as well as total number, surface, and volume. In most cases, the data represent the true distributions quite well.

The configuration of two DMAs in series, called the Tandem Differential Mobility Analyzer (TDMA) (Rader and McMurry 1986), was first exploited by Liu et al. (1978), McMurry et al. (1983), and Rader et al (1987) to measure particle growth/shrinkage due to hygroscopicity, gas phase reaction, and evaporation, respectively. Hygroscopicity TDMA (HTDMA) measurements of water uptake by atmospheric aerosols was recently reviewed by Swietlicki et al. (2008); observations from more than 100 articles on this topic were summarized in this article. TDMA systems can also be used to study the effects of other processes on mobility size including heating (Volatility TDMA; VTDMA) (Kuhn et al. 2005; Orsini et al. 1999; Philippin et al. 2004; Sakurai et al. 2003a; Sakurai et al. 2003b) and chemical reactions (Reactivity TDMA; RTDMA) (Gupta et al. 1995; Higgins et al. 2002; 2003; Liao et al. 2006; Liao and Roberts 2006; Nienow et al. 2004). TDMA systems can also be used to evaluate DMA Transfer Functions (Fissan et al. 1996; Hummes et al. 1996; Li et al. 2006; Stolzenburg and McMurry 1988; Stratmann et al. 1997).

In TDMA experiments, the raw data consists of number concentration downstream of the second DMA versus voltage on the second DMA. Growth factors can be determined fairly accurately from peak locations in these data. However, to make full use of the power of the TDMA, Rader and McMurry (1986) developed the exact equations governing the response of the system and solved them to obtain 1% precision in particle growth factors. An alternative approach for analyzing TDMA data was recently described by Cubison et al. (2005), which involved adapting the optimal estimation method of Rogers (1976, 1990). Some researchers have used SMPS software to reduce TDMA data. Again, this will yield fairly accurate peak locations and growth factors. However, the approximation made in SMPS analyses that the aerosol size distribution entering DMA2 is constant over the DMA2 mobility window is not valid for TDMA measurements. This approximation will lead to errors in values of the retrieved size distributions, which will be significantly wider than the true values.

This work provides a summary of the important equations governing the responses of the single DMA and TDMA systems including diffusional broadening of the DMA transfer function. The article includes a summary of our earlier work that has been widely used but not yet published (Stolzenburg 1988; Stolzenburg and McMurry 1988) as well as some significant new extensions to this work. For TDMA work, the output of the aerosol conditioner is modeled as a multi-modal lognormal distribution for a given input particle diameter. A new lognormal approximation to the DMA transfer function is introduced greatly simplifying the calculation of the diffusing transfer function. It also leads to a simple lognormal form for the calculated TDMA response function.

ANALYSIS FOR DIFFUSING AND NON-DIFFUSING DMA TRANSFER FUNCTIONS

The DMA transfer function, Ω, is defined as the probability of a particle of a given size successfully traversing the classifier (Knutson and Whitby 1975). Specifically, it is the probability of the particle, starting at the aerosol entrance of the classification zone, reaching the aerosol exit. Losses in the inlet and outlet zones outside the classification zone are not included in Ω.

A schematic of an axial flow DMA is shown in Figure 1a. R ₁ and R ₂ are the inner and outer radii of the annular classification zone. L is the axial length between the midpoints of the aerosol inlet and outlet slits. Q _a and Q _s are the aerosol inlet and outlet flows; and Q _c and Q _m are the entering particle-free and exiting particle-laden sheath flows. The voltage, V, applied to the center rod is used to select particle size.

FIG. 1 Schematics of (a) axial flow and (b) radial flow Differential Mobility Analyzers.

A schematic of a radial flow DMA (Zhang et al. 1995) is shown in Figure 1b. R _in is the midpoint of the aerosol inlet slit, R _out = 0 ¹ is the radius associated with the outlet flow and b is the distance between the plates. Q _a , Q _s , Q _c , and Q _m are the same as for the axial flow DMA and V is the voltage applied to the top plate.

Non-Diffusing Transfer Function

Non-diffusing particles follow particle streamlines (analogous to fluid streamlines, Knutson and Whitby [1975]). When plotted against electric mobility, Z _p , the transfer function, Ω _nd , is triangular (or trapezoidal for unequal aerosol flows) with centroid mobility, Z _p *, equal to

and base half width

For the axial flow DMA, the change in the electrical flux from the aerosol inlet to the outlet is:

and for the radial flow DMA this quantity is (Zhang et al. 1995)

The rest of the analysis applies to both geometries of the DMA.

Particle diameter, D _p , is related to Z _p via

where e is the elementary charge, n _p is the number of elementary charges on the particle, C _s is the Cunningham slip correction and μ is the gas viscosity.

The non-diffusing transfer function can now be written as (Knutson and Whitby 1975)

This can be converted to non-dimensional form as

where the dimensionless mobility is defined as

along with dimensionless flow parameters

and

The dimensionless form of the non-diffusing transfer function, Ω _nd , is plotted in Figure 2.

FIG. 2 Non-diffusing transfer function, Ω _nd , showing probability of transiting the DMA for a particle with electrical mobility, Z _p = [Ztilde] _p · Z _p * where Z _p * is the centroid of the transfer function.

Diffusing Transfer Function

To a good approximation diffusing particles spread in an ever-expanding Gaussian cross-stream profile about the corresponding non-diffusing particle streamline (Stolzenburg 1988), as shown in Figure 3. At the aerosol exit the final standard deviation, σ, of this profile in non-dimensional form is given by

where G _DMA is a non-dimensional geometry factor (Appendix A),

and D is the particle diffusion coefficient.

FIG. 3 Schematic of axial flow DMA showing non-diffusing particle streamline (a) and Gaussian cross-stream probability distributions (b) of diffusing particles.

The diffusing transfer function, Ω _d , plotted in Figure 4 can now be given in non-dimensional form as (Stolzenburg 1988; Eq. 2.69)

where ϵ (x) = x ·erf(x) +e ^{− x 2} /√π and erf(x) is the error function. As σ goes to zero, this reduces to Ω _nd ([Ztilde] _p , β, δ) as given in Equation (7). (Stolzenburg 1988; Appendix D). The transfer functions shown in Figure 4 correspond to an aerosol-to-sheath flow ratio of β = 1.5/15 with equal aerosol flows (δ = 0) at DMA centroid diameter D _p * = 3 nm. Diffusing transfer functions, Ω _d , for the TSI 3085 nano DMA (σ* = 0.058) and the TSI 3081 long DMA (σ* = 0.165) are compared to the non-diffusing transfer function, Ω _nd (σ* = 0), as a function of dimensionless mobility, [Ztilde] _p .

FIG. 4 Transfer functions, Ω, as a function of dimensionless particle mobility, [Ztilde] _p , for a flow ratio of β = 1.5/15 (δ = 0) at DMA centroid diameter D _p * = 3 nm. Diffusing transfer functions, Ω _d , for TSI 3085 nano DMA (σ* = 0.058) and TSI 3081 long DMA (σ * = 0.165).

Moments of the Transfer Function

For both diffusing and non-diffusing particles. (Stolzenburg 1988; Eqs. 2.88–2.91) the zeroeth moment of Ω is

The mean, E, and standard deviation, S, of Ω about the centroid, [Ztilde] _p = 1, are given by

and

Integrated DMA Response Function

The typical instrument configuration involving a single DMA is shown in Figure 5. This system measures an unknown size distribution, dN/dln Dp ₁, by passing the aerosol through a charge conditioner, a DMA and then into a detector such as a condensation particle counter (CPC). The DMA is characterized by its transfer function, Ω₁ (V ₁, [Ztilde] _p1), and the CPC by its detection efficiency, η _CPC (D _p1). The charge conditioner is characterized by the charge fraction f _c (Z _p1,n) = f _c (D _p1,n , n) where n is the number of charges on the particle, Z _p1 is the particle electric mobility and D _p1,n is the particle diameter associated with mobility Z _p1 and n charges.

FIG. 5 Typical instrument configuration involving a single DMA showing entering size distribution, dN/dln D _p1, charge conditioner, DMA and condensation particle counter and their associated characterization functions.

Raw data from this system is of the form (Knutson 1976)

N ₁ (V ₁) is the measured total concentration (or, generally, the integrated response function) downstream of the DMA set at voltage V ₁. For simplicity, only the term for n = 1 is considered here:

In most cases, the variations in η _CPC (D _p1), f _c (D _p1,1) and dN/dln D _p1 are relatively small across the non-zero width of Ω₁ (Figure 6). Under these conditions the integrated response function can be approximated as (Knutson 1976)

where η _CPC , f _c and dN/dln D _p1 are evaluated at D _p1*, the particle diameter corresponding to Z _p1*.

FIG. 6 Typical shapes of functions in the integrand of the integrated response for a single DMA system showing that variations within the non-zero width of the transfer function, Ω₁, are small for all other functions.

To a good approximation over a sufficiently narrow range of sizes particle diameter and mobility can be related as

and

where

and

Noting that the variation in

is also small across Ω₁, it follows that Equation (19) can be rewritten as

Finally, applying Equation (14) results in

Solving for the unknown size distribution function gives

This is the standard accepted method for recovering the size distribution from a single DMA configuration. Equation (27) is an approximation that is only accurate when the relative variation of the product (dN/dln D _p1) · f _c (D _p1,1) · η _CPC (D _p1) is small across the non-zero width of the transfer function.

Integrated TDMA Response Function

The basic TDMA configuration is shown in Figure 7. It consists of a bipolar charge conditioner with charge fraction f _c (D _p1,1), a first DMA with transfer function Ω₁ (V ₁, [Ztilde] _p1), an aerosol conditioner, a second DMA with transfer function Ω₂ (V ₂, [Ztilde] _p2) and a CPC with detection efficiency η _CPC (D _p2). The aerosol conditioner is characterized by f _cond (D _p2,D _p1) which gives the normalized distribution of the final size, D _p2, given an initial size, D _p1. Appendix B presents the case in which f _cond (D _p2,D _p1) can be characterized by a multi-modal lognormal distribution.

FIG. 7 Typical instrument configuration involving tandem DMAs showing entering size distribution, dN/dln D _p1, bipolar charge conditioner, first DMA, aerosol conditioner, second DMA, and CPC and their associated characterization functions.

The integrated response, N ₂ (V ₂, V ₁), for the TDMA system is given by (Rader and McMurry 1986)

Typical shapes for the components of the integrand are shown in Figure 8. Making the same approximations as for the case of a single DMA, this becomes

The inappropriate application of single DMA software at this point has the effect of treating the variations in Ω₁ as small over the non-zero width of Ω₂. As seen in Figure 8, this is clearly not the case. If f _cond (D _p2,D _p1) is parameterized according to penetration, growth, and growth spread parameters, the application of such software leads to values accurate to within a few percent for the first two parameters but significantly overestimates the value of the third by ignoring the contributions due to the finite widths of the transfer functions. To achieve precision to less than one percent, the parameters can be found through a search routine (e.g., Rader and McMurry 1986).

FIG. 8 Typical shapes of functions in the integrand of the integrated response for TDMA system showing that variations within the non-zero width of the product of the transfer functions, Ω₁ · Ω₂, are small for all other functions.

Using Equation (21), Equation (29) can be rewritten in terms of dimensionless diameters as

This can then be put in a more useful form in which the unknown entering size distribution function is eliminated using Equation (27)

In most TDMA experiments, N ₁ and N ₂ are measured, Ω₁ and Ω₂ are known from theory and the objective is to determine the unknown aerosol conditioner characterization function, f _cond (D _p2,D _p1). This can be done using a search routine to find values for the adjustable parameters of f _cond (D _p2,D _p1) to optimize the fit of the calculated responses from Equation (31) to the measured responses.

ANALYSIS FOR LOGNORMAL APPROXIMATION OF DMA TRANSFER FUNCTIONS

For an ideal DMA and non-diffusing particles the transfer function can be calculated exactly and is triangular in shape (Equation [7]; Knutson and Whitby 1975). Stolzenburg (1988) has derived an approximation to the transfer function for diffusing particles (Equation [13]). The result is roughly Gaussian in shape but the calculation requires evaluation of cumbersome error functions. In this section we present a lognormal approximation to the DMA transfer function that is able to closely approximate either form and is easy to calculate.

Lognormal Transfer Function

The DMA transfer function (both diffusing and non-diffusing) can be approximated quite well by a lognormal function. For values of [Ztilde] _p close to 1, the approximation that allows the DMA transfer function to be expressed as a lognormal distribution is simply

The lognormal approximation to the transfer function in dimensionless mobility space is then given by

where M _Zg0 is the zeroeth moment of Ω (V,[Ztilde] _p ) and [Ztilde] _pg and σ _Zg are the geometric mean and standard deviation. (See Appendix C for definition of the notation L [x, μ, σ] and basic properties of lognormal distributions.) Using Equations (14), (16), and (32) these become

Using Equations (20) and (C.4) the transfer function in dimensionless diameter space is derived from Equation (33) as

where M _Dg0, [Dtilde] _pg and σ _Dg are the zeroeth moment, geometric mean and standard deviation of Ω (V,[Dtilde] _p ) in dimensionless diameter space. Using Equation (34) these become

Applying Equation (C.1) gives

Figure 9 compares the lognormal approximation of the transfer function to the corresponding non-diffusing, triangular one (Knutson and Whitby 1975) and the diffusing one of Stolzenburg (1988) at 200 nm and 3 nm for the TSI 3081 long DMA and the TSI 3085 nano DMA, respectively. These conditions represent the extremes of the DMA measurement range. For 15 L/min sheath and 1.5 L/min aerosol flows the maximum absolute error for the lognormal approximation is 0.10 for the TSI 3081 at 200 nm over the entire range. It is 0.03 for the TSI 3085 at 3 nm. For 5 L/min sheath and 1.7 L/min aerosol flows the error is 0.16 for the TSI 3081 at 200 nm over the entire range and 0.10 for the TSI 3085 at 3 nm.

FIG. 9 Transfer functions for TSI 3081 and 3085 axial flow DMAs with different sheath and aerosol flows and voltage settings corresponding to 200 nm and 3 nm, respectively. Absolute error of lognormal transfer function is with respect to diffusing transfer function. The non-diffusing and diffusing transfer functions for the TSI 3081 at 200 nm are virtually indistinguishable.

Integrated DMA Response Function

Applying Equations (21) and (35) to the integrated single DMA response function in Equation (25) yields

Invoking the property in Equation (C.9) produces

Integrated TDMA Response Function

The general integrated TDMA response function is given in Equation (30). Assuming a unimodal lognormal conditioner function and applying Equations (B.2), (C.1), and (35).

where

Using Equation (C.10) on the inner integral this becomes

where

Applying Equations (C.2), (C.3), and (C.6) yields

Using Equation (C.10) again gives

where

Dividing Equation (45) by Equation (39) produces

Finally, applying Equations (C.2) and (C.1) the normalized integrated TDMA response can be written as

Using Equation (41) and noting that [Dtilde] _pg = D _pg /D _p * gives

Equation (48) can then be rewritten in dimensional form as

Figure 10 compares the lognormal approximation of the TDMA response with no conditioner to the corresponding non-diffusing one (Knutson and Whitby 1975) and the diffusing one of Stolzenburg (1988) at 200 nm and 3 nm for the TSI 3081 long DMA and the TSI 3085 nano DMA. The relative error is the difference of the lognormal response and the diffusing response divided by the peak diffusing response. For 15 L/min sheath and 1.5 L/min aerosol flows the maximum relative error for the lognormal approximation is 0.04 for the TSI 3081 at 200 nm over the entire range. At 3 nm it is 0.001 for the TSI 3085. For 5 L/min sheath and 1.7 L/min aerosol flows the error is 0.05 for the TSI 3081 at 200 nm over the entire range and 0.009 for the TSI 3085 at 3 nm.

FIG. 10 TDMA responses with no conditioner for pairs of TSI 3081 and 3085 axial flow DMAs with different sheath and aerosol flows and DMA1 voltage settings corresponding to 200 nm and 3 nm, respectively. The relative error is the difference of the lognormal response and the diffusing response divided by the peak diffusing response. The non-diffusing and diffusing responses for the TSI 3081 at 200 nm are virtually indistinguishable as are the diffusing and lognormal responses for the TSI 3085 at 3 nm.

CONCLUSIONS

The basic equations governing the responses of single and tandem differential mobility analyzer (DMA) systems have been summarized. In most cases, single DMA integrated response data can be inverted using the simple, closed-form “monodisperse” approximation. This same approximation can only be used to partially simplify the TDMA integrated response equation and does not result in a closed-form solution. Instead, a best-fit search procedure must be used to determine the values of adjustable parameters in the aerosol conditioner function. Use of SMPS software on TDMA data, while easy and accurate to within a few percent, is not valid and does not allow access to the full power of the TDMA configuration, that is, less than one percent precision and information on the spread of the growth factor distribution.

A new lognormal approximation to the DMA transfer function fits the diffusing transfer function quite well and greatly simplifies the calculations. When this is combined for tandem DMA work with a lognormal aerosol conditioner function it leads to a simple lognormal form for the TDMA response function which can be readily inverted. The maximum fractional error in the TDMA response calculated from the lognormal transfer function is no more than 0.08 at 200 nm and 3 nm. The lognormal approximation could be used in a TDMA data inversion procedure to properly account for diffusional broadening of the DMA transfer function.

APPENDIX A: EVALUATION OF GEOMETRIC FACTOR G _DMA

The geometric factor G _DMA is the non-dimensionalized integral of v ² r ² dt where v is particle velocity, r is radial position and t is time (Stolzenburg [1988]; Appendices B and C). The integral is along the non-diffusing particle streamline of the centroid mobility of the transfer function from the aerosol entrance plane to the exit plane. It can be expressed as:

where

and

ω _a and ω _s are defined by

where

is the total DMA flow and

is the flow fraction (assuming Poiseuille flow) between radial positions ω and ω = 1.

APPENDIX B: LOGNORMAL DISPERSION

A useful case to be considered here is when the aerosol conditioner function can be represented as a multi-modal lognormal distribution

where f _0cond,k is the fractional weight (∑ _k f _0cond,k = 1), μ _cond,k = G _cond,k · D _p1 is the geometric mean and σ _cond,k is the geometric standard deviation of the k ^th mode of the distribution. G _cond,k is the mean diameter growth (or shrinkage) factor for the k ^th mode.

One possible physical description for this situation is an aerosol for which there are several distinct types, each with a distribution of properties. Each particle type would correspond to one of the modes of the growth model with a lognormal distribution of properties. For the case in which a particle type has uniform properties, σ _cond = 1, and no size change, G _cond = 1, the corresponding term in Equation (B.1) reduces to f _0cond .

Typically, the parameters f _0cond , G _cond and σ _cond are not known, where for simplicity only the unimodal case is considered here. Best fit values for these parameters can be determined by fitting calculated integrated responses N ₂ to the corresponding measurements. The physically meaningful ranges for these fit parameters are

The unimodal aerosol conditioner function can be written in terms of dimensionless diameters as

where

APPENDIX C: LOGNORMAL PROPERTIES

To facilitate the derivations in the main text a shorthand notation for the lognormal distribution is introduced as follows

A number of properties of lognormal distributions are needed for the analysis. Simple ones include

and

Two more complex properties are

and (Hatch and Choate 1929)

where

The integral of the lognormal distribution is just unity

This combined with Equations (C.6) and (C.5) gives, respectively,

and

Acknowledgments

We gratefully acknowledge the assistance of Mr. Chongai Kuang, who proofread the manuscript and checked all equations. This research was supported, in part, by the Office of Science (BER), U.S. Department of Energy, grant DE-FG-02-05ER63997.

Notes

¹ As with the axial flow DMA and L, R _in and R _out are defined according to the change in electric flux function from inlet to outlet. The flux function at each of these is equal to the asymptotic value approached deep within the inlet or outlet. For the inlet R _in is half way between the walls of the inlet slit. For the outlet R _out is on the axis of the exit tube. Hence, R _out = 0.