Aerosol Science 37 (2006) 1340–1346www.elsevier.com/locate/jaerosci

Conceptual possibilities for extending the particle size rangeof a differential mobility analyzer using longitudinal

and transversal electrodesManuel Alonso

National Center for Metallurgical Research (CSIC), Avenida Gregorio del Amo, 8, 28040 Madrid, Spain

Received 17 November 2005; received in revised form 25 January 2006; accepted 25 January 2006

Abstract

A new type of differential mobility analyzer (DMA) is proposed which is, in theory, applicable to the measurement of particle sizedistributions of aerosols over the full range of particle size. The basic concept of the full-range DMA can be best realized using aparallel-plate geometry. It contains two arrays of detectors, e.g. thin metal strips connected to electrometers. One array of detectorsis placed longitudinally along one of the plates, and the other one is placed transversally downstream of the aerosol entrance. Withthis arrangement, too low mobility particles which escape undetected in a conventional DMA settle onto the transversal detectorsand can thus be measured. For small particles collected along the longitudinal array of detectors, the resolution of the instrument isthe same as for a conventional multi-channel DMA. For the largest particles, which are collected on the transversal detectors, theresolution deteriorates as the particle size increases.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Aerosol instrumentation; Differential mobility analyzer; Wide range DMA; Multi-channel DMA

1. Introduction

Differential mobility analyzers (DMA) have proven very useful for the measurement of aerosol particle size dis-tributions in the submicrometer range. The long history of mobility analyzers can be traced back to the early worksof Zeleny and Langevin using condensers to measure the electric mobility of air ions. These works, as well as allthe later developments that followed, have been extensively described in an excellent review article of Flagan (1998).The first commercial instrument was based on the coaxial condenser mobility analyzer developed at the University ofMinnesota (Knutson & Whitby, 1975; Whitby & Clark, 1966). This coaxial cylindrical DMA with an outlet port toextract a fraction of particles within a specific range of mobility became the standard prototype of what may be termed‘single-channel’ analyzers. Further developments on single-channel analyzers have been mainly aimed at improvingthe measurement resolution for nanometer-sized aerosol particles and ions (Chen, Pui, Mulholland, & Fernandez, 1999;De Juan & Fernandez de la Mora, 1998; Winklmayr, Reischl, Lidner, & Berner, 1991; Zhang, Akutsu, Russell, Flagan,& Seinfeld, 1995).

E-mail address: malonso@cenim.csic.es.

0021-8502/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jaerosci.2006.01.012

M. Alonso / Aerosol Science 37 (2006) 1340 –1346 1341

A different approach was followed by Tammet and co-workers at the University of Tartu. In the early 1970s theydeveloped a coaxial DMA in which the outer cylinder was equipped with a series of insulated electrodes, each connectedto its own electrometer. In this manner, the number concentration of different-sized particles could be measuredsimultaneously, thus reducing considerably the time required for the measurement of particle size distributions. This‘multi-channel’ analyzer is especially appropriate for the measurement of rapidly fluctuating aerosols. A recent articlecontaining a description of this instrument and earlier references has been published by Tammet, Mirme, and Tamm(2002).

Regardless of whether the single- or the multi-channel design is adopted, the DMA effective length is chosen so as togive an optimum resolution for particles within a specific size range. Thus, short DMAs are intended for the measurementof aerosols in the lower end of the size spectrum, while long columns are generally used for the measurement of largerparticles.

In some practical applications one is confronted with the task of measuring aerosols whose particle size spans severalorders of magnitude (e.g. atmospheric aerosols). The measurement of the full particle size distribution requires the useof different instruments or, at least, different designs of a given apparatus (i.e. DMAs with different lengths). Clearly, forthese applications it would be desirable to have a single equipment which could perform measurements with relativelygood resolution within a wide range of particle size. The aim of the present paper is to propose the basic concept uponwhich such an instrument can be built.

2. Illustration of the working principle of the full-range DMA

The basic idea, illustrated in Fig. 1, is simple. Fig. 1a shows a conventional single-channel DMA with an outletport to withdraw particles of the desired size, but the idea is also applicable to multi-channel analyzers. For any givenapplied voltage, particles larger than those which are classified leave the DMA along with the excess flow. Since theapplicable voltage must be below that which causes electrical breakdown, there is, for a given electrode length, alimitation on the size of the particles which can be classified. By increasing the DMA length, one can increase themaximum ‘detectable’ particle size, at the expense of deteriorating the measurement resolution for small particles (e.g.diffusion broadening of the transfer function as a result of increasing aerosol residence time in the apparatus). Notethat in the case of a multi-channel instrument with electrometers fixed along the collector electrode, the resolution forsmaller particles is independent of the electrode length although, as for the single-channel design, longer electrodesare still required to extend the detectable particle size range. Of course, larger particles can also be measured witha conventional DMA not by increasing its length, but by reducing the flow rates. For too large particles, however,one would need too low, unstable, difficult-to-control flow rates which would give rise to unacceptable measurementerrors.

Fig. 1b illustrates how the range of applicability of a DMA can be extended without increasing its length or reducingthe flow rates. An additional array of transversal detectors would detect all the different-sized particles. With this

conventional DMA

(a)

longitudinal detectors

transversal detectors

full-range DMA

(b)

Fig. 1. The basic concept of a full-range DMA.

1342 M. Alonso / Aerosol Science 37 (2006) 1340 –1346

arrangement, no one particle, no matter how large, can avoid detection. Intuitively, one can see that the trajectoriesof different-sized very large particles remain quite close to each other near the bottom electrode: the resolution forvery large particles must, by necessity, be relatively poor. It can also be expected that widening the gap between thetwo electrodes would result in improved resolutions for the upper end of the particle size spectrum. Nevertheless, andleaving aside the resolution, since this instrument is capable of differential detection of particles of any size, one canrefer to it as a full-range DMA (FRDMA). Note finally that the FRDMA need not be especially long.

In summary, placement of a series of detectors (metal strips connected to electrometers) in the direction perpendicularto the aerosol flow gives us the possibility of having a single, compact instrument for the measurement of aerosols overa wide range of particle sizes.

3. A full-range DMA prototype and working equations

A possible realization of the FRDMA is schematically shown in Fig. 2. It consists of a rectangular channel withtwo parallel-plate electrodes, and having a small rectangular channel, with the same width as the large one, throughwhich the aerosol is fed. One of the electrodes is equipped with a series of thin metal strips attached to its inner surfaceand connected to appropriate electrometers. At a certain distance downstream of the aerosol entrance, a transversal,additional array of thin metal strips connected to electrometers is placed. When a certain fix voltage is applied betweenthe plate electrodes, the measured currents give the number concentration of particles (of a given polarity) for as manydiscrete size intervals as detectors are used. The unmeasured gaps left between those discrete particle size bins arefilled by successive application of varying voltages.

The transversal metal foils should be made as thin as practicable in order to minimize fluid flow distortion. Fur-thermore, they should be maintained at an appropriate electric potential to ensure that the electric field in the gapsbetween them be the same as the field E applied between the parallel-plate electrodes. Thus, the jth transversal detector,separated a distance Hj from the bottom electrode (see Fig. 3), should be kept at the potential Vj = EHj .

Fig. 3 shows the relevant geometry parameters needed for the determination of the transfer function and the mobilitybandwidth (resolution) for each detector. Diffusion effects will be intentionally neglected, for two reasons. First,these effects are only important for the smallest particles, i.e. those collected on the longitudinal detectors placedalong the top electrode, a situation which has been profusely addressed during the last two decades (Alonso, 2002;Kousaka, Okuyama, Adachi, & Mimura, 1986; Rossell-Llompart, Loscertales, Bingham, & Fernández de la Mora,1996; Stolzenburg, 1988). Second, the aim of the proposed instrument is not to improve the measurement resolution,but to expand the measurable particle size range of DMAs.

The main channel has a height H and a width W (not shown in the figure). The aerosol feeding channel has a heightHA and a width W equal to that of the large channel. The total distance between the aerosol entrance plane x = 0 andthe plane where the center lines of the transversal detectors lie is L. The metal strips connected to electrometers haveall the same width �.

topelectrode

bottom electrode detectors

excess

aerosol

sheath

Fig. 2. Sketch of the parallel-plate full-range DMA.

M. Alonso / Aerosol Science 37 (2006) 1340 –1346 1343

z

y

x

L

H

Hj

�

�

Li

HA

aerosol

sheath

(a)

H

Hj

HA

XL2

XL1

Xj1

Xj2

(b)

Fig. 3. (a) Definition of geometric parameters. (b) The boundary trajectories for determination of the transfer function.

The equation of motion for a particle of electric mobility Z is

ZE dx = u(y, z) dy, (1)

where E is the electric field between the plates, and u is the flow velocity which, in principle, depends on the twocoordinates normal to the flow direction.

Consider first the transfer function �i (Z) for the ith longitudinal detector, placed at a distance Li along the x-axis.�i (Z) is the probability that a particle of mobility Z hits the ith detector. As depicted in Fig. 3b, the particles of mobilityZ have trajectories which are bounded by two limiting trajectories, one starting at (x = 0, y = HA) and ending at(x = xL1, y = H), and the other one starting at (x = 0, y = 0) and ending at (x = xL2, y = H). The two limitingx-coordinates, which are the same for all the longitudinal detectors, can be easily found from integration of (1) overthe appropriate limits, resulting in

xL1(Z) = QC

WEZ, (2)

xL2(Z) = QA + QC

WEZ, (3)

where

QA =∫ W

0dz

∫ HA

0u(y, z) dy (4)

and

QC =∫ W

0dz

∫ H

HA

u(y, z) dy (5)

are, respectively, the flow rates of aerosol and clean sheath air. The transfer function for the ith longitudinal detector isthen given by the fraction of the segment of length xL2 − xL1 which intersects the detector:

�i (Z) =⎧⎨⎩

min[Li2, xL2(Z)] − max[Li1, xL1(Z)]xL2(Z) − xL1(Z)

if xL1 �Li2 and xL2 �Li1,

0 if xL1 > Li2 or xL2 < Li1

(6)

where

Li1 = Li − (�/2), (7)

Li2 = Li + (�/2). (8)

1344 M. Alonso / Aerosol Science 37 (2006) 1340 –1346

To evaluate the transfer function for the jth transversal detector, the limiting x-coordinates for the boundary trajectorieshave to be evaluated at the plane y=Hj . In this case, the values of the limiting coordinates are specific for each transversaldetector. Integrating the equation of motion results in

xj1(Z) = hjQC

WEZ, (9)

xj2(Z) = QA + hjQC

WEZ, (10)

where

hj = Hj − HA

H − HA. (11)

As before, the probability that a particle of mobility Z hits the jth transversal detector is given by the fraction of thesegment of length xj2(Z) − xj1(Z) which overlaps the detector of width �:

�j (Z) =⎧⎨⎩

min[L2, xj2(Z)] − max[L1, xj1(Z)]xj2(Z) − xj1(Z)

if xj1 �L2 and xj2 �L1,

0 if xj1 > L2 or xj2 < L1.

(12)

In the last expression,

L1 = L − (�/2), (13)

L2 = L + (�/2). (14)

We turn now into the calculation of the instrument resolution in ideal conditions, that is, leaving aside diffusioneffects. The particle of minimum mobility which is collected in the ith longitudinal detector is that whose trajectorystarts at the plane (x = 0, y =HA) and ends at (x =Li + �/2, y =H). The particle of maximum mobility which settleson the ith longitudinal collector has a trajectory starting at (x = 0, y = 0) and ending at (x = Li − �/2, y = H). Thecentroid mobility is that of a particle which travels from (x = 0, y = HA/2) to (x = Li, y = H). Integration of (1) overthe corresponding limits leads to

Zi,min = QC

WLi2E, (15)

Zi,max = QA + QC

WLi1E, (16)

Zi,c = (1/2)QA + QC

WLiE(17)

with Li1 and Li2 being given by expressions (7) and (8).The relative bandwidth (resolution) for the ith longitudinal electrometer thus becomes

�Zi

Zi,c= 2

(QA + QC)/(1 − (�/2Li)) − QC/(1 + (�/2Li))

QA + 2QC. (18)

If the collector strips are very thin, in the limit Li?�/2 the resolution attains a constant value which depends only onthe flow rates of aerosol and clean sheath air:

�Zi

Zi,c→ 2QA

QA + 2QCfor Li?�/2. (19)

The approximation (19) becomes better for detectors placed farther from the entrance, where relatively large particlesare collected.

M. Alonso / Aerosol Science 37 (2006) 1340 –1346 1345

10-3 10-2 10-1 100

0.1

1E = 10 V/m

tran

sfer

func

tion,

ψ (-

)

electric mobility, Z (cm2/Vs)(a)

0.1

1

10-310-410-510-6

E = 104 V/cm

tran

sfer

func

tion,

ψ (-

)

electric mobility, Z (cm2/Vs)(b)

Fig. 4. Example of transfer functions. FRDMA geometry and operating conditions: length, L = 200 mm; main channel height, H = 10 mm; channelwidth, W = 10 mm; aerosol feeding channel height, HA = 0.5 mm; detector width (�): longitudinal = 0.2 mm; transversal = 10 mm; aerosol flowrate, QA = 0.5 l min−1; clean air flow rate, QC = 10 l min−1. Applied electric field, E: (a) 10 V cm−1; (b) 104 V cm−1. In each plot, the dashedcurves represent the transfer functions for the transversal detectors; the full lines, for the longitudinal detectors.

Following an analogue procedure for the transversal detectors, which will again be labelled by a subscript j, thefollowing expressions are arrived at:

Zj,min = hjQC

WL2E, (20)

Zj,max = QA + hjQC

WL1E, (21)

Zj,c = (1/2)QA + hjQC

WLE, (22)

where hj is given by (11) and L1 and L2 by (13) and (14), respectively. The relative mobility bandwidth for the jthtransversal detector becomes

�Zj

Zj,c= 2

(QA + hjQC)/(1 − (�/2L)) − hjQC/(1 + (�/2L))

QA + 2hjQC. (23)

For a sufficiently long FRDMA, L?�/2, and the last expression simplifies to

�Zj

Zj,c≈ 2QA

QA + 2hjQCfor L?�/2. (24)

This last equation shows how the resolution worsens for transversal detectors closer to the bottom electrode, that is,for the largest particles.

All the above derived equations are independent of the specific form of the flow velocity profile, because in theintegration of the equation of motion the velocity field u(y, z) always appear in integrals over y and z of forms(4) or (5).

It must be noted that for all the detectors, longitudinal and transversal, the transfer function is not symmetricalabout the centroid Zi,c or Zj,c, i.e. Zi,c �= (Zi,min + Zi,max)/2 and Zj,c �= (Zj,min + Zj,max)/2. This is a well-knownfact (see, for instance, Tammet, 1970). The transfer functions have, in general, a trapezoidal form, as can be seenin Fig. 4, where an illustrative example of the possibilities of the proposed FRDMA is shown. In this example, the

1346 M. Alonso / Aerosol Science 37 (2006) 1340 –1346

multi-channel analyzer contains eight longitudinal detectors and four transversal ones. The separation distance betweenadjacent detectors is constant in a log scale. In case (a) the applied field is small, 10 V cm−1, and permits measuringparticles with mobility up to about 1 cm2 V−1 s−1, typical of air ions. In the other extreme, case (b), the large fieldapplied (104 V cm−1) allows measurement of particles with diameter slightly above 10 �m, in spite that this particularFRDMA is relatively short (20 cm).

The width � of the transversal detectors, as well as the spacing between them, should be carefully selected. If thespacing-to-width ratio is too large, too many particles would escape undetected and one would be forced to applysequentially several different values of the electric field E in order to measure the entire size distribution. This wouldresult in a longer measuring time. In the other extreme, if the spacing-to-width ratio is too small there would occur asort of screening effect, in which each detector would be partially hidden by its lower neighbor, so that a fraction ofparticles would precipitate onto the ‘wrong’ detector.

4. Concluding remarks

The practical implementation of the concept of the FRDMA proposed in the present article presents, in principle, nospecial technical difficulties. As a matter of fact, TSI has recently commercialized a fast mobility spectrometer, basedon the multi-channel analyzer developed by Tammet and co-workers, consisting of an array of longitudinal detectorsplaced along the external electrode of a concentric-cylindrical DMA. The proposed FRDMA contains an additionalarray of detectors placed transversally to the flow. The FRDMA is best realized using a parallel-plate geometry, becausethe transversal detectors can be directly attached to the lateral walls of the rectangular channel.

The additional transversal detectors allow measurement of the largest particles, no matter how large, of any aerosolpopulation. Besides, since it is a multi-channel type analyzer, the measurement of the entire particle size distributioncan be carried out very rapidly, thus allowing sizing of fluctuating aerosols. However, the resolution worsens withparticle size. An unavoidable problem, common to all mobility analyzers, is that of multiple charging, which makesthe mobility-size inversion algorithm quite involved for too large, heavily charged, particles. However, as noted byTammet et al. (2002), it is possible to carry out the mobility-size conversion unambiguously if an appropriate chargingtechnique, simultaneously combining diffusion and field charging, is used.

Despite these (not too serious) two drawbacks, the author believes that the FRDMA can be quite useful for sizingaerosols having particle sizes spanning several orders of magnitude because, at present, there is no single instrumentavailable which can perform such task.

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