A measuring system for the time variation of size and charge of a single spherical particle and its applications

Chemical Engineering Science 61 (2006) 2212–2229www.elsevier.com/locate/ces

A measuring system for the time variation of size and charge of a singlespherical particle and its applications

Yoji Nakajima∗

Division of Materials Science and Engineering, Graduate School of Engineering, Hokkaido University, Sapporo, 060, Japan

Received 31 March 2004; received in revised form 21 September 2004; accepted 4 October 2004Available online 21 June 2005

Abstract

A single charged particle is trapped in a simple quadrupole electrode assembly and oscillated by means of a controlled electric field.A laser Doppler velocimeter (LDV) of the fringe mode is used for detecting the amplitude and phase lag of the particle oscillation withrespect to the driving ac field. A unique method for the LDV signal processing is presented that takes the advantage of the sinusoidaloscillation of the particle at a known frequency. Superimposed to the ac drive, a dc drive is added for highly accurate measurements ofparticle size and charge. This enables us also to discriminate the polarity of charge without Bragg cells.

In this paper, the basic principle of the method of the size and the charge measurements is explained and the accuracy of the measurementsis demonstrated experimentally. The errors in the size and charge measurements, respectively, are less than 1% and 3% with a confidencecoefficient of 99%. Since this apparatus repeats the measurements every 0.3 s for a single particle, the errors can be reduced to 0.1% and0.3% when the measured values over a period of 30 s, or over 100 data are averaged.

As some areas of its applications, experimental data are presented on the Rayleigh instability of evaporating charged droplets. It isshown that there are three types of Rayleigh fission. One of the types seemed to show occurrence of air breakdown around a micronsized spherical particle, which has not been reported so far. However experiments in highly insulating gas (SF6) revealed that it was notthe case but a type of Rayleigh fission. Nevertheless the experimental results gave important information on the charge limit of sphericalsolid particle due to electric breakdown in air at normal room conditions.

Some cares to ensure the advantages of the present method are presented, and possible improvements of the apparatus are also suggestedfor future use. 2005 Elsevier Ltd. All rights reserved.

Keywords: Charged droplet; Charged particle; Charge measurement; Size measurement; LDV; Electrodynamic trapping; Rayleigh fission

1. Introduction

Measurement of time variations in particle size and chargeproperties often yields novel information. For instance, evap-oration characteristics of small droplets (Taflin et al., 1988;Shulman et al., 1997; Widman et al., 1998), uptake dynamicsof HCl by a single sulfuric acid microdroplet (Schwell et al.,2000), the Rayleigh instability of charged droplets (Duft etal., 2002; Taflin et al., 1989), chemical reaction rate constantof droplets (Ward et al., 1987) have been studied from the

∗ Tel.: +81 1170 66591; fax: +81 11 7069593.E-mail address: [email protected].

0009-2509/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.05.004

measurement of the time variation in size of a single droplet.For these purposes, the electrodynamic balances (EDB) havebeen used to trap and to levitate a single aerosol particlefreely in a gaseous medium. In the experiments cited above,bihyperboloidal electrodes (Wuerker et al., 1959) with com-plex design and construction were used, so that the electricfield formed by the electrodes could be readily calculable.However, such bihyperboloidal electrodes are rather costlyto construct, and moreover, the electrode configuration re-stricts the scattered light collection angle for the optical re-ceiver within a rather narrow limit.

Instead, simpler forms of electrodes have been acceptedfor EDBs. Richardson et al. (1989) used a cylindrical andtwo hemispherical cap electrodes to measure the Rayleigh

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Y. Nakajima / Chemical Engineering Science 61 (2006) 2212–2229 2213

instability of charged droplets. Davis and Bridges (1994) ex-amined the cause of charged droplet fission prior to reachingthe Rayleigh instability limit by using two different sets ofdouble ring electrodes. Bhanti and Ray (1998) used a simi-lar type of EDB to Richardson’s for the measurement of in-stantaneous chemical reaction rate constant in a droplet. Zhuet al. (2002) measured mass transfer from an oscillatingsmall droplet using an octopole double ring electrode config-uration. In the present system, a more facile electrode con-figuration, namely two parallel rod electrodes located in thecentral plane of a pair of parallel plate electrodes, is adoptedfor forming a quasi-quadrupole ac electric field, which cantrap charged particles at the central portion of the electricfield. The parallel plate electrodes also provide a uniform acelectric field to oscillate the charged particle.

In the experiments cited above, the droplet size was mea-sured by using two elaborate optical methods based on theMie light scattering theory. One is now called the phase func-tion method (Bartholdi et al., 1980) and the other is called theoptical resonance method (Ashkin and Dziedzic, 1981). Boththe methods, in particular the latter, offer very accurate sizemeasurement if the refractive index of the droplet is knownprecisely. But both the methods require very complicatedand time-consuming computations. For several decades, hy-drodynamic approaches for measuring the sizes of clouddroplets and smoke particles have been developed. Wells andGerke (1919) utilized photographs of particle oscillation inan ac field for the first time, though they had to assume thecharge of the droplet. Using a laser Doppler velocimeter,Mazumder and Kirsch (1977) and Renninger et al. (1981)proposed a novel method, in which the phase lag of the mo-tion of particle relative to the oscillating driving force wasrelated to the relaxation time or aerodynamic size of the par-ticle. This system is now known as the E-SPART (electricalsingle particle aerodynamic relaxation time) analyzer.

In the present measuring system, a quasi-quadrupole elec-trode system mentioned above is incorporated in the mea-surement volume of a laser Doppler velocimeter of the fringemode. The basic idea of the system is similar to that of theE-SPART analyzer, but the equipment is extremely simpli-fied; it uses neither conventional LDV signal demodulatornor Bragg cells. Nevertheless the polarity of droplet chargecan be discriminated if necessary, and further, the accuracyof the size and charge measurements is comparable to thatattained by EDBs owing to the special method of LDV sig-nal analysis and also to the repeated measurements for asingle particle. In this paper, the unique features in the basicprinciple of the measuring system are elucidated and accu-racy of the size and charge measurements is demonstrated.

This measuring system has been developed for the ex-periment of droplet agglomeration enhanced by electrostaticforce and particle vibration (Nakajima and Sato, 2003). Toexemplify other applications, some experimental results onthe Rayleigh instability are presented, which show the com-plexity of the Rayleigh disruption of charged droplets. Anattempt to clarify the charge limit for a spherical solid parti-

cle sustainable in normal air follows. Then a semi-empiricalequation relating the particle diameter and the charge limitfor negatively charged particles is proposed on the basis ofTownsend theory. Although the theory of Townsend is notapplicable to positively charged particles, an experiment inargon gas showed that the difference due to the polarity wasnot very significant, and the prediction may be usable forboth positively and negatively charged particles.

2. Theoretical descriptions of the measuring system

2.1. Trapping electrodes

Wuerker et al. (1959) analyzed and examined the stabilityof charged particle motion in an exact bihyperboloidal elec-trode configuration to clarify the basis of electrodynamictrapping of charged particles. Here the simplicity of the trap-ping system is demonstrated because we might presume thata very sophisticated electrode configuration would be nec-essary for the trapping system.

In this experiment, a simplified electrode configuration asshown in Fig. 1 was used to trap and to levitate a chargedparticle at the center of the measurement volume of LDV. Itconsisted of a pair of parallel plates and two rods with twoinsulated spherical electrodes for each rod. An ac voltagewas applied to the rods to form a quasi-quadrupole ac elec-tric field, which electrodynamically trapped charged parti-cles as in the quadrupole mass spectrum analyzers. The plateelectrodes behaved like mirrors making the electric imagesof the rod electrodes. For analyzing the electric field distri-bution, the electric image method modified by the chargesimulation method was used. Fig. 2 shows the equipotentialand electric force lines thus obtained. As shown in the centerof Fig. 2, a saddle point in the x–z plane is formed along thecenterline of the electrode system. The electric field aroundthe saddle point is described below.

60

110

Plate electrodeSphericalelectrode

PTFE tube

Rod electrode(3mmφ)

18

16 75

700Hz1kV

1.75kVmax

0.4∼6.4kHz

unit:mm

+300V

+24V

z

y

x

Fig. 1. Quadrupole electrode assembly for trapping charged particle (fornegatively charged particles, polarity of diodes should be reversed).

2214 Y. Nakajima / Chemical Engineering Science 61 (2006) 2212–2229

Parallel plate electrodes

Rod electrode

x

z

Fig. 2. Equipotential and electric force lines in the quadrupole electrodeassembly.

Fig. 3. Absolute value of electric field vector in the center of the quadrupoleelectrodes.

The absolute value (or norm) of the electric field vector inthe area enclosed by the dotted square in Fig. 2 is illustratedin Fig. 3. The square (6×6 mm) covers enough wide area incomparison with the diameter of the measurement volume,which was approximately 2 mm in diameter. As seen fromFig. 3, the absolute value forms a cone. It follows that the

components of the electric field can be approximated by

Ex ≈ Axx sin t, Ez ≈ Azz sin t , (1)

where x and z are the horizontal and the vertical coordinatesas shown in Fig. 2. The angular frequency of the ac voltageapplied to the rod electrodes is denoted by , and Ax andAz are the proportionality constants. In Eq. (1) Az shouldbe −Ax because the electric field must satisfy the Laplaceequation.

The position of a charged particle, (x, z), in this field maybe written as follows, provided that the particle is so smallthat we can presume a linear relationship between the dragforce and particle velocity:

md2x

dt2 + Rdx

dt= qEx, m

d2z

dt2 + Rdz

dt= qEz − mg, (2)

where m is the particle mass, R the drag coefficient, q theparticle charge, and g the acceleration constant due to grav-ity. Here we notice from Eq. (1) that Ex does not depend onz, and Ez is independent of x. There are various structures ofelectrodes that can form such an electric field although validonly locally, where every component of the electric field isnearly proportional to the corresponding coordinate centeredat the saddle point. Thus the equations can be solved for xand z independently.

In case the charged particle oscillates around z0 with asmall amplitude, we may put Ez ≈ Azz0 sin(t) in Eq. (2).Then we have an approximate solution for z,

z = −qAzz0(m sin t + R cos t)

(R2 + m22)+ z0. (3)

Hence the driving force in z-direction becomes

qAzz sin(t) − mg

= − q2A2zmz0

2(R2 + m22)− mg + qAzz0 sin(t)

+ q2A2zz0m

2(R2 + m22)cos(2t)

− q2A2zRz0

2(R2 + m22)sin(2t). (4)

The first term on the right-hand side of Eq. (4) is a steadyforce yielded by the electrodynamic effect. This term is pos-itive when z0 < 0 and gives the charged particle a levitationforce. At the equilibrium state, it balances with the gravita-tional force, mg. As for the x-direction, the same calculationshows that the electrodynamic effect exerts a steady forcedirected to x = 0 on the charged particle. In this manner,charged particles are trapped near the centerline of the twoparallel rod electrodes.

As shown in Fig. 1, a bias dc voltage on the ac trap-ping voltage is applied to the spherical electrodes so thatthe charged particles drift along the centerline (i.e., y-axis)towards the center of the electrodes by electrostatic repul-sion. Another dc bias applied to the parallel plate electrodes


may help dislocated particles to turn back to the trappingzone. The polarity of the dc biases in Fig. 1 is for positivelycharged particles, and all the Zenner diodes should be re-versed in polarity for negatively charged particles.

2.2. Amplitude and phase measurement by LDV

As is well known, the phase lag of the oscillation of aparticle driven by an acoustic or ac electric field reflectsthe size of the particle. If we use an ac electric field, theamplitude of the particle oscillation gives information onthe charge of the particle as well. Fig. 4 is an example ofthe LDV light signal (beat signal) scattered from a sphericalparticle oscillating in the interference fringes formed by twocoherent laser beams. The sinusoidal curve in the figurerefers to the driving ac voltage for particle oscillation.

In this example, the amplitude, A, is several times largerthan the fringe spacing, , and so we can readily notice in thebeat signal the points at which the particle motion reverses.Hence the phase lag of particle displacement, , with respectto the driving ac voltage can be determined as depicted in thefigure. Further, by counting the number of peaks between thereversing points and using a suitable interpolation method,we can estimate the amplitude of the particle oscillation. Thismethod is straightforward but adequate only for amplitudegreater than the fringe spacing (Sato and Nakajima, 1990),and it is not necessarily suitable for the present measurement.

As the hydrodynamic formula relating the phase lag to theparticle size is appropriate only for small amplitude of theparticle oscillation, the amplitude should be as small as pos-sible for precise measurement. When the amplitude becomessmall enough for the precise measurement, the waveform

0 200 400 600 800 1000

73µm9.5µm

Time, µs

Inte

nsity

of

scat

tere

d lig

ht

drive signal beat signal

Fig. 4. Example of Doppler signal from a single particle with an amplitude,A, larger than the fringe spacing, (A ≈ 7.7=73 m). The phase lag corresponds to the time interval between the reversal points of the drivesignal and the particle oscillation.

0 5 10 15Time, ms

Inte

nsity

of

scat

tere

d lig

ht

Fig. 5. Example of beat signal for an amplitude smaller than the fringesize.

of the beat signal looks like a sinusoidal wave as shown inFig. 5. To extract adequate information for the phase andamplitude measurements from the beat signal, we aim toexpand the signal in a Fourier series as below.

As will be explained later, a superposition of a suitable dcbias on the ac electric field for driving the charged particle isadvantageous for the present method. (The dc bias circuit isnot shown in Fig. 1.) Then the location, x, of the particle inthe measurement volume of the LDV is assumed here to be

x = A sin(pt − ) + ut + x0, (5)

where A is the displacement amplitude, p the angular fre-quency of particle oscillation, the phase lag of particledisplacement with respect to the driving ac voltage, u thesteady velocity, t the time, and x0 the initial location of theparticle. Provided that the diameter of the measurement vol-ume is much larger than the displacement of the movingparticle, the light intensity across the fringe planes may beassumed to vary in a cosine wave. Hence the beat signal,namely the scattered light intensity from the moving particlein the fringes, may be written as

ip = B cos(kx) = B cosk(ut + x0) coskA sin(pt − )− B sink(ut + x0) sinkA sin(pt − ), (6)

where B is the constant that depends mainly on the opti-cal system and the particle size. The constant k is the wavenumber of the fringes, i.e., 2 divided by the fringe spacing. In reality, a very low frequency (or even dc) componenttermed the pedestal is accompanied. More precise explana-tion of the beat signal including the pedestal for the fringemode LDV may be found elsewhere (Roberds, 1977), butEq. (6) is practical enough because the pedestal componentin beat signal is not used in the present method.

To expand Eq. (6) in a Fourier series, use is made of thefollowing series expansions (Pipes and Harvill, 1970) i.e.,

cosa sin(b) = J0(a) + 2∞∑

n=1

J2n(a) cos(2nb),


sina sin(b) = 2∞∑

n=1

J2n−1(a) sin(2n − 1)b, (7)

where Jn(a) is the Bessel function of the first kind and oforder n. Then we have

ip = BJ 0(kA) cos(kut + kx0) + B

∞∑n=1

J2n−1(kA)

× [cos(2n − 1)pt + kut − (2n − 1) + kx0− cos(2n − 1)pt − kut − (2n − 1) − kx0]

+ B

∞∑n=1

J2n(kA)[cos2npt + kut − 2n + kx0

+ cos2npt − kut − 2n − kx0]. (8)

This shows that the spectrum of the beat signal consistsof infinite number of spectral lines located at the angularfrequencies of ku, p±ku, 2p ± ku, 3p±ku, and so on.The intensity of each line is proportional to Jn(kA), andhence the amplitude, A, determines the spectral distribution.

To elucidate the way for calculating the amplitude and thephase lag of the particle oscillation from the beat signal, thedata shown in Fig. 5 is processed as an example. The abscissaof the figure is the time. The sampling rate was (1/15)s−1.Although the data was truncated at 1000 (=15 ms) in thefigure, the actual record consisted of 5000 data (=75 ms),which included about 85 cycles of the particle oscillation.The frequency of the particle oscillation was 1131 Hz. Whenthe beat signal was being transmitted to a computer forrecording, an automatic control system intermitted the trap-ping ac voltage and superimposed a suitable dc componenton the driving ac voltage so as to make ku/2 ≈ 1131/4.This resulted in an equally spaced spectrum lines as can beseen from Eq. (8). As the measurement was carried out for asingle particle repeatedly, such a control was possible by re-ferring to the preceding measurement. Further explanationsof the automatic control system are omitted here.

The spectrum was calculated by using DFT (discreteFourier transformation). Before the DFT analysis, the Han-ning window (Blackman and Tukey, 1958) was appliedto the raw data to avoid the unfavorable effect of limitedlength of data and noises. The result is shown in Fig. 6.We find spectrum lines at frequencies of 291, 840, 1423,1972, 2555 Hz, and so on. These conform very well tothe frequencies predicted by Eq. (8) if we take ku/2 tobe 291 Hz. Eq. (8) implies that the intensity of each linespectrum should be proportional to Jn(kA), where n isthe ordinal number of the harmonics corresponding to thefrequencies. Except for n = 0, each harmonic should havetwo frequency components (spectrum lines) with the sameintensity. Fig. 6 also reflects this fact. As kA determines theintensity distribution of the spectrum, we can find the am-plitude of the particle oscillation, A, from the spectrum. Todo this, the measured intensity of spectrum line for the nthharmonic was divided by the sum of seven intensities for

0 2000 4000 6000 8000

0th

1st

2nd 3rd

4th

5th

6th

Inte

nsity

, arb

itrar

y un

it

Frequency, Hz

with hanning window

Fig. 6. Amplitude spectrum of beat signal shown in Fig. 5.

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1

0.5

1

1.5

2

reduced amplitude kA(=2πA/δ )

0th

1st

2nd

3rd4th

|Jn(

kA)|

/Σ |

J m(k

A)|

m=

0

6

Nor

mal

izat

ion

cons

t, Σ

|Jm

(kA

)| m

=0

6

Σ|Jm(kA)|m=0

6

Fig. 7. Relative intensity of each harmonic component in amplitudespectrum.

harmonics from 0th to 6th to yield the normalized intensity:

h(n) = |Jn(kA)|/

6∑m=0

|Jm(kA)| . (9)

Fig. 7 shows the theoretical dependence of the intensitieson kA, though it includes only harmonics up to 4th to avoidgraphical complexity. Since the higher order components dieout for small kA, the intensity of 0th order plays an importantrole. If the steady component had not been superimposed onthe oscillating particle motion, the two frequency compo-nents in each harmonic order would degenerate into a singlecomponent. As a result, the 0th order component in Fig. 6would have been submerged in the pedestal component. Forthis reason the superposition of a suitable dc component onthe ac driving voltage is preferable for the measurement ofsmall amplitude.

Now let us seek the value of kA that fits the spectrumdistribution in Fig. 6. We see that the second harmonic hasthe highest intensity, the 0th and the 3rd come after it withalmost the same intensity, the 4th follows them and the 1stis the lowest among them. This can be seen in Fig. 7 atkA = 2A/ = 3.5 as indicated by the broken vertical line.The fringe spacing, , was 9.5 m in this measurement andso the amplitude was evaluated to be 5.29 m. In the actual


Table 1Possible solutions for phase lag, , and initial phase, kx0, for several harmonic modes (∗ and kx∗

0 are directly determined by DFT for and kx0, butthey may be folded in the range between − and )

Mode ∗ (a) (b) (c) (d)

0th Undefined : undefinedkx0 = kx∗

0

Fundam. −/2 ∼ 0 = ∗kx0 = kx∗

0

= ∗ + kx0 = kx∗

0 ±

/2 ∼ = ∗kx0 = kx∗

0

= ∗ − kx0 = kx∗

0 ±

Second −/2 ∼ 0 = ∗kx0 = kx∗

0

= ∗ + kx0 = kx∗

0

0 ∼ /2 = ∗ − /2kx0 = kx∗

0 ± = ∗ + /2kx0 = kx∗

0 ±

Third −/3 ∼ −/6 = ∗kx0 = kx∗

0

= ∗ + kx0 = kx∗

0 ±

−/6 ∼ 0 = ∗ − /3kx0 = kx∗

0 ± = ∗kx0 = kx∗

0

= ∗ + 2/3kx0 = kx∗

0

= ∗ + kx0 = kx∗

0 ±

0 ∼ /6 = ∗ − /3kx0 = kx∗

0 ± = ∗ + 2/3kx0 = kx∗

0

/6 ∼ /3 = ∗ − 2/3kx0 = kx∗

0

= ∗ − /3kx0 = kx∗

0 ± = ∗ + /3kx0 = kx∗

0 ± = ∗ + 2/3kx0 = kx∗

0

Fourth −/4 ∼ 0 = ∗ − /4kx0 = kx∗

0 ± = ∗kx0 = kx∗

0

= ∗ + 3/4kx0 = kx∗

0 ± = ∗ + kx0 = kx∗

0

0 ∼ /4 = ∗ − /2kx0 = kx∗

0

= ∗ − /4kx0 = kx∗

0 ± = ∗ + /2kx0 = kx∗

0

= ∗ + 3/4kx0 = kx∗

0 ±

system, the theoretical intensity distributions h(n) in Eq. (9)for n from 0 to 6 were tabulated in a computer for various kA,and the observed intensity distribution was compared withthem to find the value of kA that gave the best-fit distributionby using the least mean square technique. This method waspractical for the amplitude range 0.1kA20.

The phase measurement is straightforward. As is seenfrom Eq. (8), we have two frequency components for the firstorder harmonic, whose phase lags are +ku and −ku. Wecan calculate them from the results of DFT for correspondingfrequencies to give 2 by addition. The use of Hanningwindow turned out to be very effective for precise phasemeasurement. In case |J1(kA)| < |J2(kA)|, it is preferableto use the second harmonic for the phase measurement.

In practice, however, the value of varies between /2and or between 0 and −/2 according to the polarityof the particle charge, and value of kx0, between − and. Therefore the ranges of the phase arguments in Eq. (8),(2n − 1) ± kx0 and/or 2n ± kx0, exceed the range be-tween − and , over which the four-quadrant arctangentcan cover as a single valued function. As a result, angles

greater than or less than − are folded in the range be-tween − and . Then we will have several possible valuesfor when we consider the effect of folding. Table 1 liststhese possible values for a few harmonics, where ∗ andkx∗

0 are the four quadrant angles directly obtained from DFTanalysis. (See Appendix A.) Note that ∗ and kx∗

0 are deter-mined for each harmonic and are not necessarily commonto different harmonic modes. This table indicates that when∗ is in the range between −/2 and 0 for the fundamentalmode we will have two sets of solutions, i.e., (∗, kx∗

0)

and (∗ + , kx∗0 ± ). The double sign, ±, should be so

selected that kx∗0 ± lies in the range from − to . In this

way we have two different values for , which differ fromeach other by , and we cannot discriminate the polarity ofcharge. But if we use two harmonics, say the fundamentaland the second, we will have four sets of solutions for and kx0. Then we can find a unique set of and kx0, whichis common to both the harmonics. As Jn(kA) has zeros, ev-ery harmonic vanishes at some values of kA specific to theorder of the Bessel function, and therefore, the table is pre-pared for several harmonics. Here we have to note that two


harmonics should be selected such that one is the odd modeand the other even; otherwise we cannot find a unique solu-tion. Although this method was not used in the experimentbecause the polarity of droplet was known from the charg-ing condition, a computer simulation confirmed the method.A detailed explanation of the table will be given inAppendix A.

2.3. Size and charge measurement

As is well known, we can determine the particle radiusfrom the phase lag. But we know also that the Stokes lawfor the drag of a sphere in steady motion is not necessarilyapplicable to the drag of an oscillating sphere even when theReynolds number is infinitesimally small. Then a simplifiedBasset type solution for the drag of an oscillating sphere(Hinze, 1959) is used here. The drag force, Z, exerted on asphere of radius rp in a stationary oscillation at a velocityof Ap sin(pt) can be written as

Z = 6f rpAp

(1 + √

2

)sin(pt)

+(

√2

+ 2

9

)cos(pt)

, = rp

√p/, (10)

where is the kinematic viscosity of surrounding fluid, f

the density of fluid, and the size parameter defined above.This can be derived from the Navier–Stokes equations forincompressible flow by neglecting the non-linear convectionterm, and therefore, the amplitude should be very small. Itreduces to Stokes law when approaches zero.

We incorporate Eq. (10) into the equation of motion, inwhich we assume that the phase of electric field is advancedby with respect to particle velocity, Ap sin(pt). isof course related to by = − /2.

4r3pp

3

dAp sin(pt)dt

+Z=qE0 sin(pt+), (11)

where p is the particle density, q the electrostatic charge ofthe particle, and E0 the field strength for particle oscillation.From this equation, we have

rp = 3f

√

2(2p + f )√

2p

3(tan − 1)

+√

9 + 9 tan + 16(p/f ) − 10 tan

, (12)

q = 6f rpAp

E0 cos

1 + rp

√p

2

. (13)

For the measurement in normal air, the discrepancy be-tween rp thus calculated and the one calculated from con-ventional Stokes low is within ±5% when 10 80 (or−100 − 170) for p = 1 g/cm3. As for the charge,it ranges from −3% to 19%. These discrepancies decreasewith increasing p such that they are inversely proportional

to the square root of p. Usually such a small differencemay be ignored, but the systematic dependency of the dis-crepancies on may bring about a false trend and can giverise to substantial error when we deal with, e.g., the timevariations in size and/or charge of an evaporating droplet.

3. Experimental setup and measured results

Fig. 8 is the schematic diagram of the measuring system.A 30 mW He–Ne laser was used to form stationary inter-ference fringes at the center of the quadrupole electrodes tomeasure the particle motion in the direction perpendicular tothe parallel plate electrodes. The spacing of the fringe planeswas 9.5 m, and the diameter of the measurement volumewas adjustable with a collimator but usually 2 mm. Dropletsto be measured were generated by a vibrating orifice aerosolgenerator (TSI model 3050). To charge the droplet by in-duction charging, a small ring electrode applied a dc elec-tric field to the tip of the liquid jet as it comes out from themicro orifice of the generator. The voltage to the electrodecould be adjusted within ±6 kV in response to the requireddroplet charge and its polarity determined the polarity of thecharge. The liquids for droplets were alcohol solutions oflow- or non-volatile liquids to be tested. Alcohol was com-pletely evaporated in an electrostatic curtain, which con-sisted of 16 metal rods forming a cylindrical cage. A high acvoltage was applied to every second rod and the rest of therods were grounded. The ac electric field between the rodsrepulsed the charged droplet inside the cage by the electro-dynamic effect. A shutter placed on the top of the measure-ment cell was opened for a short moment to lead the chargeddroplets into the upper section of the cell. The trapping acfield prevented the charged particles from flowing down intothe measuring section, because the ac electric field formedby the upper rod electrode of the quadrupole electrode as-sembly gave rise to electrodynamic repulsion on the chargedparticles from the rod electrode. A very short intermissionof the trapping ac voltage allowed some charged particles toenter into the measuring section. This process was repeateduntil a single charged particle was incidentally trapped inthe measuring section. When a single particle was success-fully trapped, the LDV signal took the form of an FM wavewithout amplitude modulation like the one shown in Fig. 4.Once a single droplet was trapped, the measurement couldbe repeated every 0.3 s. Further explanation on the opera-tion of the apparatus may be found elsewhere (Nakajimaand Sato, 2003).

Another thing that should be pointed out is the use of aspatial filter located at the focal plane of the light-collectinglens of the optical receiver. As is well known, the visibil-ity of the beat signal deteriorates when the particle size be-comes larger than the fringe spacing (Farmer, 1972, 1974).Roberds (1977) proposed a formula for the signal visibil-ity, and showed that the visibility can be improved with theuse of a small light collection aperture at the expense of


Signalprocessor

(computer)

Air pump

Syringe pumpOscillator

DCsupply

Shutter

Oscil-lator2

Oscil-lator1

Attenu-ator

Ch.0

Ch.1

Ch.1

Ch.0

CRT MO Drive

+

600Ω Amp

Amp

Trigger

Optical receiver(with spatial filter)

Vibrating orificeaerosol generator(with induction charger)

Measurementcell

Digitizer

Laserbeams

Digitizer

Relays

Electrostaticcurtain

_

Storageoscillo

Acrylic pipe(enclosure)

510Ω

510Ω

Spatial filter

laser beamfocus points

hyperbolic slit

Fig. 8. Schematic diagram of the measuring system.

sensitivity. In this experimental apparatus, a hyperbolic slitshown in Fig. 8 was used as a spatial filter (or laser beamstop) at the focal plane of the optical receiver to improve thevisibility. The spatial filter made the visibility greater than0.5 for particles smaller than 30 m in diameter for a fringespacing of 9.5 m at the minimum sacrifice of sensitivity.

3.1. Accuracy of the size and charge measurement

To examine the accuracy of the measuring system, thetime variations in the size and charge of DnOP (Di-n-octylphthalete) droplets were measured. An example is shownin Fig. 9. The upper and the lower bands of data points inthe figure, respectively, denote the droplet diameter and thecharge. These were recorded every 2 s for 10 h. As DnOP isa non-volatile liquid in normal room conditions, the size ofdroplet is expected to remain constant. In fact, the diameterwas fairly stable and it was calculated to be 6.873±0.030 m(or less than 0.5% error in size) with a confidence coef-ficient of 99%. As the major source of the error in size

6.90

6.85Dia

met

er, µ

m

0 2 4 6 8 10Elapsed time, h

52

53

54

55

Cha

rge,

fC

Diamter : 6.873 ± 0.030µm

Confidence Coeff.= 99%

Charge : 52.89 ± 0.46fC

Di-n-Octyl Phthalate(DOP)

Fig. 9. Example of accuracy check.

measurement is the error in phase measurement, Eq. (12) isdifferentiated with respect to to obtain the sensitivity of


0 10 20 30 40 50 60 70 80 900

1

2

3

4

5

Sens

itiv

ity

of s

ize

to p

hase

err

or

rela

tive

to

opti

mum

con

diti

on

Phase lag in particle velocity ψ, deg.

Fig. 10. Sensitivity of size to phase error relative to optimum condition.

the size to the phase. We found the sensitivity to be the low-est or optimum at = 45. Fig. 10 shows the sensitivityrelative to the optimum. The data in Fig. 9 was collected atthe optimum (=45). Since we may expect that the errorin phase measurement by the Fourier analysis is indepen-dent of in principle, we can see from Fig. 10 that the errorfor the range of between 15 and 75 would be less thantwice that at the optimum. Therefore we can use a fixed fre-quency for a fairly wide range of size. The size ratio mea-surable with a fixed frequency will be 4 if we admit 1% er-ror in size with a 99% confidence coefficient for the presentapparatus.

As for the droplet charge, it decreased graduallydue to recombination with ions from surrounding gasmedium, local electron avalanches, and some other pos-sible causes such as evaporation of ions. In Fig. 9, wefound a relatively stable interval shown by the whitearrow, and the average was taken over the interval togive 52.89 ± 0.46 fC with a confidence coefficient of99%. Although sensitivity analysis was not performedfor charge measurement, we may expect an error ofless than 3% if the size was measured within 1%error.

Since this apparatus can repeat the measurement ev-ery 0.3 s for a single particle, the error will be reducedby a factor of 0.1 by taking the average of measuredvalues over a period of 30 s, or over 100 data: as iswell known, the error or the standard deviation of anaverage is inversely proportional to the square root ofthe number of data averaged. Therefore the accuracyof measurement can be improved to a higher level ifthe time variations are not very rapid in comparisonwith the averaging period. As the digitizer (AD con-verter) used in this apparatus was not a state-of-the-artequipment, and it caused considerable phase error; theuse of high precision digitizer is expected to improve theaccuracy.

3.2. Rayleigh instability of charged droplets

As a charged droplet evaporates, the charge density onthe droplet surface becomes higher and higher. When theCoulombic repulsion overcomes the cohesion due to sur-face tension, the droplet disrupts. The critical charge for theCoulombic fission is called the Rayleigh limit. For invis-cid conductive liquids, the critical charge was theoreticallypredicted to be (Rayleigh, 1882; Hendricks and Schneider,1963):

|QRL| = 2√

2D3p, (14)

where is the permittivity of the surrounding gas medium(≈ 10−9/36 F/m for air), the surface tension and Dp thedroplet diameter. This formula has been widely used in vari-ous scientific fields, even in nuclear fission (e.g., Chandezonet al., 2001). A number of experiments have been carried outto study the mechanism. But the mechanism of the Rayleighdisruption has not been well understood.

Fig. 11 is an example of such measurements with thepresent apparatus for evaporating DBP (di-buthyl phthalate)droplets. The diameter of the DBP droplet reduced from 11to 4 m during the evaporation. But the reduction was notcontinuous and we found several occurrences of Rayleighdisruption. The values attached at the moments of discon-tinuous size reductions are the fractions of volume loss atthe disruption. The charge remained virtually constant untilthe Rayleigh disruption occurred. The percentages attachedto the charge variation are the critical charges relative tothe Rayleigh limit, i.e., |QRL| given in Eq. (14), and thecharge loss fractions at the disruption. We see from the fig-ure that the Rayleigh fission occurs at about 95% of theRayleigh limit for this case. In this experiment on DBP, theRayleigh fission was accompanied by 10.20% mass loss andabout 25% charge loss. This corresponds to so-called “rough

0 1000 2000 3000 4000 50002

4

6

8

10

12

50

100

150

200

Dro

plet

cha

rge,

fC

Dro

plet

dia

met

er, µ

m

Evaporation time, s

Diameter

Charge

Di-n-Butyl Phthaleteσ=34.1mN/m

94.8%

95.6%

94.9%

95.3%

95.5%

70

-15%

-9%

-3%

-15%

-18%

-25%

-23%

-10%

-27%

Fig. 11. Time variations in size and charge of evaporating DBP droplet.


0 100 200 300 400

4

6

8

10

Ddr

ople

t cha

rge,

fC 100

70

50

30

Diam.Charge

Evaporation time, sD

ropl

et d

iam

eter

, µm

Pentadecane

150

σ = 26.6 mN/m

volumeloss 1%

Fig. 12. Time variations in size and charge of evaporating Pentadecanedroplet.

fission” mode (Fernández de la Mora, J., 1996) that emitsa few but relatively large daughter droplets. This mode wasreported by Abbas and Latham (1967) for water, aniline andtoluene, and by Taflin et al. (1989) for DBP.

Fig. 12 shows another mode, namely, the “fine fission”mode, in which the mass loss is as small as 1% and the chargeloss is 10.20%. These data were collected for a dropletof pentadecane. The loss of charge was 7.13%, and theloss of mass was so small that the discontinuity in the sizevariation was rather difficult to notice without magnificationas shown in the figure. Numerically, it was found to be0.7.1%. The data reported by Taflin et al. (1989) suggestedthat hexadecane and heptadecane also belong to this mode.Using high-speed microscopic images, Duft et al. (2003)succeeded in taking detailed photos of Rayleigh fission ofa charged ethylene glycol droplet, which deformed to anellipsoid with two sharp tips on its poles and finally ejectedfine liquid jets from both tips. The charge loss was reportedto be 33%, and the mass loss, 0.3%. At the same time, theyreported that about 100 daughter droplets 1.5 m in diameterwere expelled from a 24 m parent droplet, which implied amass loss of 2.4%. The data in a previous paper by Duft etal. (2002) showed that the charge loss was on the order of20% on the average. These facts suggest that ethylene glycolpresumably disrupts in the fine fission mode. The disruptionobserved in the experiments mentioned above occurred at90.100% of the Rayleigh limit except for data by Taflinet al., which showed somewhat very low values for all theliquids they tested.

Richardson et al. (1989) reported an extreme case of the“fine fission” mode for a droplet of sulfuric acid. The chargeloss was about 50% but no mass loss was observed. Thedisruption occurred at 92% of the Rayleigh limit on an aver-age, though their measured data for charge were rather scat-

0 1000 2000 3000 4000 5000

4

6

8

10

12

14

16

Evaporation time (s)

Dro

plet

dia

met

er, µ

m

Dro

plet

cha

rge,

fC

DiameterCharge

Glycerolσ = 64 mN/m

84%

82%

80%

79%

78%

Magnified plot

400

300

200

100

60

-31%

-31%

-28%

-34%

-35%

9.7

9.8

9.9

Fig. 13. Time variations in size and charge of evaporating glycerol droplet.

tered. Fig. 13 shows another example of the extreme casemeasured by the present apparatus for glycerol droplets. Thecharge losses were comparatively large but no discontinu-ous reduction in size was seen. Experiments were repeatedmany times for glycerol droplets, but a discontinuous sizereduction was never observed even in the occasional caseswhen the charge loss exceeded 50%. Another observationmade was that the discharge occurred at around 80% of theRayleigh limit, which was appreciably lower in compari-son with other liquids. As the Rayleigh limit for glycerolis high due to its high surface tension, the field strength atthe droplet surface could exceed 60 MV/m. According to thetheory of Fernández de la Mora (1996), high electrical con-ductivity of liquids favors the “fine fission” mode, but un-like sulfuric acid, glycerol is not a high conductive liquid.For these reasons, it was quite reasonable to suppose thatthe result shown in Fig. 13 might indicate an occurrence ofthe electric breakdown, which brought the charge limit ofsmall spherical solid particles in normal air. Then the effectsof polarity and medium gas on the relationship between themaximum charge and droplet size were examined, becausethe breakdown depends on them while the Rayleigh fissiondoes not. A highly insulating gas (i.e., SF6 gas) was usedfor the medium gas and an appreciable increase in the max-imum charge was expected. The measurements were carriedout for both polarities.

An example of the comparison with the correspondingresults in normal air is illustrated in Fig. 14, in which thecharge is plotted against the size. As is seen in the figure, thecharge remained almost constant during the evaporation untilit hits the limit line, which is independent of the mediumgases and the polarities. Therefore, we had to conclude thatthe large losses in droplet charge shown in Fig. 13 were nota result of the air breakdown but the disruption of glyceroldroplets in the “extremely fine fission” mode. It has not


been clarified as to what happens on the droplet surface inthe “extremely fine fission” mode, but probably, the parentdroplet ejects a number of very fine daughter droplets, whosecharge to mass ratio can be very high within the Rayleighlimit. Note that the charge to mass ratio of a droplet at theRayleigh limit is inversely proportional to D1.5

p , i.e., we have

|QRL|/m = 12√

12/2pD3

p from Eq. (14).

3.3. Maximum charge sustainable for spherical particlesin normal air

It has been frequently assumed that electric breakdownin normal air occurs at a field strength of around 3 MV/m.But the smaller the space of high electric field becomes,the higher the critical field strength. In fact, the observeddata for glycerol droplets (e.g., Figs. 13 and 14) indi-cated that the field strengths at the surfaces of dropletsamounted to 50 MV/m for 16 m diameter, 62 MV/m for9.8 m and 90 MV/m for 4.4 m, which were much higherthan 3 MV/m. Therefore, small particles can carry a largeramount of charge than what is normally expected. Therehave been several theoretical predictions (Harper, 1967;Pedersen, 1967; Elghazaly and Castle, 1987; Pedersen,1989; Crowley, 1991; Nakajima and Sato, 1999a,b) for themaximum charge of spherical solid particles sustainable innormal air. Fig. 15 shows these with the Rayleigh fissiondata for glycerol droplets in normal air in a bold brokenline. The papers by Pedersen (1967, 1989) did not actuallyrefer to the breakdown around a charged isolated particle.The curves labeled “Pedersen” in Fig. 15 were calculated bythe author and his colleague (Nakajima and Sato, 1999a,b)after Pedersen’s novel work. Fig. 15 shows that the cal-

Droplet diameter, µm

Dro

plet

cha

rge,

fC

3 4 5 7 8

Rayleigh limit

Glycerol

in airpositive

in SF6negative

20

in SF6positive

in airnegative

30

40

50

70

100

Actual limit

6

Fig. 14. Maximum charge vs. droplet diameter for glycerol in air andSF6 (positive and negative).

1 5 10 500.01

0.05

0.1

0.5

1

γ = 2γ0

γ = γ0 /2

Particle diameter Dp , µm

Max

imum

cha

rge

q max

, p

C

Pedersen(−)

measured forglycerol droplets(±)(Rayleigh fission)

Nakajima & Sato(−)

3

Haper(−)

Elghazaly & Castle(+)

Crowley(±?)

Pedersen(+)

modified

2 20

0.02

0.2

2

Fig. 15. Predicted maximum charge of spherical particle under normalatmospheric conditions.

culated maximum charge depends not only on the size ofthe particle but also on the polarity of the charge, which isdenoted by the sign in parentheses. Here we have to notethat the critical values for the breakdown must be higherthan the observed data for glycerol droplets shown in thebold gray dotted line, because the data were limited by theRayleigh fission. In this sense, the predictions by Crowley(1991) and Harper (1967) would give too low values.

All predictions for negative charge, except for Crowley’s,were based on the theory of Townsend (e.g., Engel, 1965).The differences among them largely came from the differ-ences in the estimation of the coefficients for the ionizationof the medium gas and for the secondary electron emission ofthe cathode. Since very few data were available for very highelectric fields, these estimations would be plausible only forthe particles larger than 100 m. Nakajima and Sato (1999a)searched for the data usable in a very high field strength,and proposed a formula for the maximum charge, which isexpected to be valid for particles larger than 10 m in di-ameter. The contribution of this formula to their discussionsmay be found elsewhere (Nakajima and Sato, 1999a,b). Asthe papers were published in Japanese, a brief explanation isgiven in the Appendix B together with the Townsend theory.

4. Discussions

The present apparatus can measure the time variation inthe size and charge of a single droplet repeatedly by theuse of a simplified trapping electrode. The apparatus canbe constructed with commonly available instruments andwithout special skills. The advantage of the present methodis the high accuracy of the size and charge measurements.If the time variation is not too fast, we can improve theaccuracy by taking the moving averages of the measuredresults. But to ensure its accuracy, it is necessary to take into


consideration the following points:

(1) The amplitude of the particle oscillation should be assmall as possible. This is because the hydrodynamic re-lation between the size and the phase lag is appropri-ate only for an infinitesimally small amplitude. In thisexperiment, the amplitude was selected to be around5.2 m.

(2) The phase error of the digitizer should be compensatedas accurately as possible. In this experimental appara-tus, the phase error was compensated within ±0.05 atthe DFT processing stage for the frequency range of0.4–6.4 kHz. On the other hand, the sensitivity error andnon-linearity of the digitizer will not be vital factors,because the DFT analysis for the phase and amplitudemeasurements is not very sensitive to them. Further, thesampling rate of digitizer may not necessarily be veryhigh. A simulation showed that it is sufficient to selectthe sampling frequency of the digitizer to be 50 timesthe frequency of the driving ac voltage when the am-plitude of the particle is less than or the order of thefringe size. So a digitizer with 1 MHz maximum sam-pling frequency will suffice for usual purposes.

(3) The length of a data record for the DFT analysis shouldcover several tens of cycles of the oscillation of thedroplet. As a result, it will take at least several tens msfor a single measurement. In this experiment, one recordcovered 85 cycles and it took about 300 ms includingthe time for DFT analysis. The use of a faster computermay shorten the time to 100 ms. This will be preferablefor the moving average technique. At any rate, however,this method is not suitable for high-speed measurementin contrast to the E-SPART analyzer.

(4) The FFT algorithm is not recommended for use be-cause it analyzes the signal at predetermined frequen-cies while the present method requires Fourier coeffi-cients at arbitrarily given frequencies corresponding tonp ± ku (n= 0, 1, . . . , 6). Note that p and u are notgiven in advance but have to be determined accuratelyby the DFT analysis. Application of the Hannig win-dow (Blackman and Tukey, 1958) to raw data and lin-ear interpolation between the discrete data points wereshown to be very effective in improving the reliabilityof DFT analysis.

(5) The diameter of the laser beams should be selected tobe 2 mm or larger when droplets as large as 15 m in di-ameter have to be measured. This is because the dropletbegins to fall due to gravity when the trapping electricfield is intermitted during the data acquisition period.

(6) As the droplet evaporates, the intensity of scattered lightweakens considerably. So it is recommended to use a12-bit digitizer or an automatic sensitivity control sys-tem for the optical receiver. For this purpose, some ef-fective means for improving the signal visibility shouldbe carried out for droplets larger than the fringe spac-ing. The use of a small round aperture at the optical re-

ceiver (Roberds, 1977) may not be very advantageousin the sense of sensitivity. Instead, a hyperbolic slit asshown in Fig. 8 is recommended for use. The dropletsize actually measurable by this experimental apparatuswas from 18 to 3 m with 30 mW He–Ne laser. Whenthe laser beams were focused to 0.8 mm in diameter,the minimum measurable size was about 1 m but theaccuracy of the measurements had considerably deteri-orated because the optimum driving frequency for thissize became 10 times higher than that available for theapparatus.

(7) Unlike the E-SPART analyzers, the present method isnot usable when the droplet charge is too small becausethe particle cannot be trapped by electrodynamic means.Application of high voltage to the trapping electrodesis limited by the electric breakdown of the mediumgas, which discharges particle charge by recombinationwith ions generated by the breakdown. For the presentexperimental apparatus, the minimum charge is around5 fC for a particle 6 m in diameter. For larger particles,lowering the frequency of trapping electric ac field from700 to 100 Hz will be effective if necessary. This isbecause the trapping force can be strengthened by usinga lower frequency for large particles as can be seen fromthe first term on the right-hand side of Eq. (4).

In the present method, a suitable dc component is su-perimposed on the ac driving voltage to give the chargeddroplet a constant velocity component such that u=p/4k.This makes the measurements accurate, but the dc bias hasto be regulated in response to the charge and the size ofthe droplet. Therefore, we had to seek a suitable dc bias bytrials until we succeeded in starting the automatic controlsystem by giving incidentally an approximate value to thedesired bias. Once the system started, the dc bias was au-tomatically adjusted by referring to the preceding measure-ment. This initial procedure is time consuming and will bea severe defect when we wish to make measurements on avolatile liquid like water. To make use of the present appara-tus in the areas mentioned in the introduction of this paper,the following improvements may be recommended.

(1) If we use moving interference fringes instead of addinga constant velocity component to the droplet motion,the initial procedure can be skipped. Bragg cells, whichmodulate laser light waves with a radio frequency oftypically 40 MHz, have been widely used in LDV tech-nique to move the fringes at a constant velocity by shift-ing the frequency of one of two coherent laser beams.For the present method, however, the amount of fre-quency shift required is only a quarter the frequencyof driving ac voltage for droplet oscillation, which isextremely low for a Bragg cell frequency shifter. Thismay be resolved by using a combination of two fre-quency shifters, which can achieve a low amount ofshift, namely from dc to kHz. No revisions in the data


processing algorithm will be required for this improve-ment.

(2) Another improvement to shorten the time taken to ini-tiate the measurement is to use electrostatic guide andone-by-one droplet feed gate. As suggested before, trap-ping a single droplet in the measurement volume re-quired a rather tedious and time-consuming procedure.Masuda et al. (1980) proposed a quadrupole rail guideand electrostatic gate system to feed deuterium oxide(heavy water) pellet one-by-one into a fusion reactor inhigh vacuum. If such a system becomes available for thepresent apparatus in the future, we can carry out mea-surements with volatile matter more easily, and further-more, we will be able to perform experiments in reducedgas pressure, which are expected to yield essential andvaluable information for the theory of electric break-down and electron avalanche in strongly non-uniformelectric fields. The results will contribute valuable dataand a new approach to the micro discharging process,which is now becoming more and more important inthe industrial applications.

5. Conclusions

It was shown that a simple LDV system with its sens-ing volume located within a quadruple electrode assemblycan be used to measure the time variation of the size andcharge of a single evaporating droplet. A mechanism fortrapping charged particles in a quadrupole electric field wasdeveloped to simplify the use of an electrodynamic trap. Anew approach to a signal processing system for the LDVsignals for size and charge measurements was developedto improve the accuracy of the size and charge measure-ments.

The accuracy of the measurements was discussed usingactual experimental data: the errors in size and in charge,respectively, were approximately 0.1% and 0.3% when thedata were collected and averaged for a period of 30 s.

The measurement system can be used to study Rayleighinstability of charged droplets. Experimental data were pre-sented for low volatile liquids. Three types of Rayleigh fis-sions were observed. One is the “rough fissions” mode, inwhich the droplet mass loss at each fission is on the orderof 10% with a charge loss of 20 to 30%. The “fine fission”mode is classified into two types. In the first type, both themass and charge losses, in particular the former, are appre-ciably smaller than those in the “rough fission” mode. In thesecond type, the charge loss is very high (30.50%) whilethe mass loss is imperceptible, possibly caused by ejectionof very fine charged daughter droplets and not by air break-down around the droplet, because the experiments in highlyinsulating gas (SF6) showed the same discharging charac-teristics as in air.

The charge limit due to the electric breakdown of airaround a small solid spherical particle was considered.

Several existing predictions for the limit were comparedwith the data for glycerol droplets, for which it wasshown that the charge loss was caused by Rayleigh fis-sion. As no Rayleigh fission occurs for solid particles,the predicted limit for solid particles should be higherthan the observed limit for glycerol droplets. Never-theless, some predictions were below the glycerol datashowing that they may not be acceptable. For negativelycharged particles, a prediction based on the Townsendtheory was briefly explained (in Appendix B) to pro-pose a reasonable correlation for the charge limit due tothe electron avalanches in normal air. As for positivelycharged particles, the limit is still left unclarified, butthe correlation for negative charge may be applicable topositively charged particles because the experiment in ar-gon gas showed no significant difference between the twopolarities.

Notation

a, b dummy variable or constantsA displacement amplitude of particle

oscillation, mAx , Az Amplitudes of x and z components

of electric field in the quadrupoleelectrode assembly (Az = −Ax),V/m

B proportionality constant in Eq. (6),A

d gap size of a parallel plate electrodeappearing in Paschen curve, cm

Dp =2rp, particle diameter, mE electric field strength, V/mEx , Ez x and z components of electric field

in the quadrupole electrode assem-bly, V/m

E0 norm (or absolute value) of the elec-tric field for driving charged parti-cle, V/m

g acceleration constant due to gravity,kg m/s2

h(n) relative strength of line spectrum fornth harmonic, dimensionless

ip ac component of beat signal, AJn(z) Bessel function of the first kind of

order n, dimensionlessk =2/, wave number of laser inter-

ference fringes, 1/mm particle mass, kgn order of harmonics, dimensionlessne number of electrons bred by a single

electron starting from the cathode,dimensionless

p ambient pressure, Pa


p20 reduced pressure, i.e., the pressureat which the density of gas at 20 Cequals the density of actual gas sur-rounding the electrode, Pa

q particle charge, Coulombqmax maximum charge for a spherical

particle sustainable in normal air,Coulomb

QRL Rayleigh limit, Coulombr radius coordinate, mrp radius of charged particle, mr0 radius coordinate at which the field

strength becomes too weak for ion-ization collision to occur, m (Thecritical field strength is 24.5 kV/cmin normal air)

R drag coefficient, kg/st time, su steady component of particle mo-

tion,m/sVbk breakdown voltage for a parallel

plate electrode, Vx, y, z rectangular coordinates, mx0 initial position of the particle in the

fringes, see Eq. (5), mZ drag force, N

Greek letters

ionization coefficient, 1/cm secondary electron emission coeffi-

cient, dimensionless0 for brass in normal air estimated

from the Paschen curve, dimension-less

fringe spacing, m permittivity of medium gas (≈

10−9/36), farad/m =rp

√p/, size parameter, dimen-

sionless kinematic viscosity, m2/sf density of fluid, kg/m3

p density of particle, kg/m3

surface tension, N/m phase lag of particle displacement

with respect to applied ac electricfield, rad

= − /2, phase lag of particle ve-locity with respect to applied elec-tric field, rad

angular frequency of electric fieldfor trapping charged particle, rad/s

p angular frequency of particle oscil-lation, rad/s

Acknowledgements

This work has been performed under continuous stimula-tions from Prof. T. Sato of Hokkaido Institute of Technology.Some parts of this paper were results of collaboration withhim. The author would also like to express his deep gratitudeto the reviewers for their help in revising the manuscript;in particular, one reviewer kindly corrected the manuscriptwith helpful advice and thoughtful comments.

Appendix A. Explanation of Table 1 for polarity discrim-ination

From Eq. (8), for the leading terms of the beat signal wehave

ip = BJ 0(kA) cos(kut + kx0)

+ BJ 1(kA)[cos(p + ku)t − ( − kx0)− cos(p − ku)t − ( + kx0)]+ BJ 2(kA)[cos(2p + ku)t − (2 − kx0)+ cos(2p − ku)t − (2 + kx0)]+ BJ 3(kA)[cos(3p + ku)t − (3 − kx0)− cos(3p − ku)t − (3 + kx0)]+ BJ 4(kA)[cos(4p + ku)t − (4 − kx0)+ cos(4p − ku)t − (4 + kx0)]. (A.1)

The principal value of the arctangent is defined in the rangebetween −/2 and /2, but in DFT analysis we use the four-quadrant arctangent, which gives values from − to . Hencewe can determine a unique value of kx0 from the 0th mode,i.e., the first term on the right-hand side of Eq. (A.1). Notehere that we have to check the sign of the Bessel function,Jn(kA) in response to the value of kA because its negativesign makes the phase shift by . From the fundamental mode,i.e., the second term, we may obtain two phase components,A1 = + kx0 and S1 = − kx0, by DFT. As their valuesmay be folded in the range from − to , it is better toexpress them as A∗

1 =∗+kx∗0 and S∗

1 =∗−kx∗0, which are

obtainable directly from DFT complex coefficients. Thus wehave ∗ =(A∗

1 +S∗1 )/2 and kx∗

0 =(A∗1 −S∗

1 )/2, and then twosets of possible solutions for and kx0 can be found withthe aid of Table 1. As only one of the sets has the same valuefor kx0 as that calculated from the 0th mode, we can selectthe proper set and determine the correct value for , whichmakes it possible to discriminate the polarity of the charge.However, it should be noted here that J0(kA) has zeros atkA = 2.405, 5.520, 8.654 and so on, and also J1(kA) haszeros at kA=3.832, 7.016, 10.173 and so on. Therefore, it isnot always possible for us to determine the values by usingthe 0th and the fundamental modes. To avoid such difficulty,Table 1 includes harmonics up to 4thmode. In this appendix,the derivation of the equations listed in Table 1 is explainedto establish the method. The third mode is exemplified herebecause it appears as the most complicated case in the table.


Fig. 16. Map of angle corrections required for A∗3 and S∗

3 when and kx0 are given.

Fig. 16 shows the map of angle corrections required for A∗3

and S∗3 to obtain the correct values for A3(=3 + kx0) and

S3(=3− kx0) when the values of and kx0, respectively,are given by the abscissa and the ordinate of the figure. Thecalculated values for A3 and S3 by DFT are denoted by A∗

3and S∗

3 , which may be folded in the range between − and. We find that there are three cases when > 2/3 asshown on the right side of the figure. In the triangular sectiondenoted by ‘case 1’, the inequalities, −3 + 5 > kx0 > −3 + 3 and 3 − > kx0 > 3 − 3, hold. It follows that5 > A3 > 3 and 3 > S3 > , and therefore, A3 =A∗

3 +4and S3 = S∗

3 + 2. Then if and kx0 are in this triangle,=(A3+S3)/6=(A∗

3+S∗3 +6)/6=∗+ and kx0=(A3−

S3)/2= (A∗3 −S∗

3 +2)/2=kx∗0 +. As is supposed to be

in the range > 2/3 then 0 > ∗ > − /3. This resultis tabulated in Table 1 as case (b) for −/3 < ∗ < − /6and case (d) for −/6 < ∗ < 0.

In the diamond shaped section denoted by ‘case 2’, adiscussion similar to the above yields A3=A∗

3 +2 and S3=S∗

3 +2. Then =(A3+S3)/6=(A∗3+S∗

3 +4)/6=∗+2/3and kx0 = (A3 − S3)/2 = (A∗

3 − S∗3 )/2 = kx∗

0. These resultsare reflected in the table as case (c) for −/6 < ∗ < 0, case(b) for 0 < ∗ < /6 and case (d) for /6 < ∗ < /3. Notehere that the section for /3 < < /2 in the diamond isexcluded because never lies in this range. As for ‘case 3’,we have = (A3 + S3)/6 = (A∗

3 + S∗3 + 6)/6 = ∗ +

and kx0 = (A3 −S3)/2 = (A∗3 −S∗

3 − 2)/2 = kx∗0 −. This

yields the same results as ‘case 1’ except for kx0 = kx∗0 −,

but kx0 = kx∗0 ± gives the same kx0 if kx0 is selected to

be an angle in the four quadrants.As suggested in the main text, we have to select two

harmonics such that one is an odd mode and the other is aneven mode. The reason for this is explained as below:

Suppose that and kx0 for case (a) in the second modefor −/2 < ∗ < 0 agree with those for case (a) in the fourthmode for −/4 < ∗ < 0, then case (b) for the second modeautomatically gives the same values of case (c) for the fourthmode. (Remember that ∗ and kx∗

0 are not always commonto different modes.) As a result, we have two values of ,which differ from each other by , and we cannot discrim-inate the polarity of the charge. Further inspection of Table1 shows that we have to select modes for the discriminationof the polarity in the manner mentioned above.

Appendix B. Semi-empirical equation for the maximumcharge

To begin with, the theory of Townsend (e.g., Engel, 1965)is outlined for the case of negatively charged particles. Anelectron, which can be generated by some causes such asthe gamma process described later and ionization by natu-ral radiation, is accelerated by the electric field formed bythe particle charge, and undergoes collisions with neutralgas molecules. If the electron gains enough energy from theelectric field before the collision, the electron sometimesliberates one electron from the neutral gas molecule leav-ing a cation. The number of electrons thus liberated per unitlength that a single electron goes through is called the ion-ization coefficient, . It depends on the gas properties and isa function of the electric field strength, E, and the ambientpressure, p (or more correctly, the number of gas moleculesper unit volume). Many forms of approximation for arepossible (Schumann, 1923), but usually it takes the formof = ap exp(−bp/E). However, we adopted the follow-ing form for better approximation in a wide range of field


strength. The following numerical correlation was obtainedfor dry air from the data collected by Dutton (1975):

p20= 132.1

1 +

(937.8

E/p20

)3

+(

473.3

E/p20

)6

× exp

− 3343

E/p20

, (B.1)

where p20 is the reduced pressure, which means the pressureat which the density of gas at 20 C is equal to the densityof actual gas surrounding the electrode. As Eq. (B.1) is anempirical equation, we have to use specified units, namely,kPa for pressure, V/cm for electric field and cm−1 for .The range of E/p20 in the original data was from 500 to7000V/(cm kPa), where Eq. (B.1) gives a very close approx-imation to the numerical data.

The total number of electrons bred by a single electronstarting from the surface of the negatively charged particle(or the cathode) of radius rp will be

ne = exp

(∫ r0

rp

dr

), (B.2)

where r0 is the radius coordinate at which the electric fieldbecomes too weak to enable ionizing collision. As a result,(ne −1) cations drift back to the cathode (i.e., the surface ofthe negatively charged particle). On impact with the cathode,each cation will release a secondary electron with a proba-bility, , the secondary electron emission coefficient. This iscalled the gamma process. Therefore the discharge will beself-sustaining if (ne − 1)1, and the following equalityholds at the critical point for breakdown:

exp

(∫ r0

rp

dr

)− 1

= 1. (B.3)

This is the Townsend criterion for self-sustaining discharge.Unfortunately, no suitable data for applicable to a wide

range of field strength were reported in the literatures. Butif we know , can be estimated from the Paschen curve,which relates the breakdown voltage, Vbk [V], for parallelplate electrodes to the gap size d [cm]. The extensive datafor dry air collected by Wind (1974) can be approximatedclosely by

Vbk =(

247.5p20d + 649√

p20d)

1 + 0.0512

p20d

+(

0.01572

p20d

)2

+(

0.02784

p20d

)4

. (B.4)

For an arbitrarily supposed value of p20d [kPa cm], we cancalculate Vbk/(p20d) or Ebk/p20 from this equation repre-senting the Paschen curve. Then Ebk/p20 thus obtained is

substituted into Eq. (B.1) to obtain /p20, which is multi-plied by p20d to give d. Since the integral in Eq. (B.3) canbe written as d for parallel plate electrodes, we have for from Eq. (B.3):

= 1/exp(d) − 1. (B.5)

This calculation was repeated for various values of p20d

to obtain a correlation between and Ebk/p20. The elec-tric field around a particle of radius, rp, with a charge, q,is E = q/4r2 at the radius coordinate, r, (r > rp). For agiven reduced pressure, we can calculate from Eq. (B.1)as a function of r to evaluate the integral in Eq. (B.3). Andfurther, to estimate for Ebk/p20 from the correlation men-tioned above, we may use the field strength at the particlesurface for Ebk , because the kinetic energy of a cation for thegamma process is largely determined by the field strengthnear the collision surface. In this way, we can calculate theleft-hand side of Eq. (B.3) and search for the critical chargethat satisfies Eq. (B.3) by trials. The numerical results forspherical particles in normal air were approximated by thefollowing equation for the diameter range between 1 m and1 cm:

qmax = 1.17 × 10−4D2p1 + 2.90 × 10−2D−0.54

p

+ 4.40 × 10−7D−1.4p . (B.6)

In this equation, the units are Coulombs for the maximumcharge qmax and meters for the diameter Dp.

The data for the Paschen curve represented by Eq. (B.4)were mostly collected for brass electrodes, on which thegamma process took place. It may be questionable if thegamma process really occurs on the surface of a negativelycharged particle, which is a non-metallic material. There areno free electrons on the surface, but there are lots of an-ions from which electrons may be readily liberated by cor-responding energy to electron affinity, which is less than thework function of metals. So it is not unlikely for the nega-tively charged particles to emit electrons when the surfaceis bombarded by cations.

As for positively charged particles, we cannot use theTownsend theory because a cathode is inevitably necessaryto emit secondary electrons by the gamma process. To clar-ify the corona discharge from a positive point electrode,Hartmann (1984) used Meek’s theory (1940), whose phys-ical picture is not as clear as that of the Townsend theory.Elghazaly and Castle (1987) extended the theory to spheri-cal particles to obtain the curve shown in Fig. 15 in the maintext. However, Hartman used an approximation for field in-tensity near the point electrode. As his approximation turnedout to be inappropriate for small spherical electrode, it wasmodified to an exact one for spheres (Nakajima and Sato,1999b). In Fig. 15, the result is shown by the curve labeled“modified”, which seems rather unreasonable.

We need some experimental results that support the abovetheoretical consideration on electric breakdown around asmall particle. As no evidence of breakdown was observed


5 6 7 8 9 1050

60

70

8090

100

200

300

12 14Droplet diameter, µm

150

Dro

plet

cha

rge,

fC

negative droplet

positive droplet

max charge in argon calc'd from Townsend theory

measured in air & SF6for positive & negative charge

Fig. 17. Electric breakdown observed in Argon gas.

in the experiments in air, an experiment was carried out inargon gas, in which the breakdown of gas is easier to occurthan in air. The result is shown in Fig. 17.

The maximum charge in argon gas for a given size largerthan 8 m looks different from that shown before. We cannotice also the polarity dependence. They both indicate thatthe breakdown in argon gas limited the charges. The resultof prediction by the Townsend theory for argon gas is alsoplotted in the solid line. It conforms quite well to the datafor negatively charged particles within 10% error. (It agreesexactly with the positive data, but this is merely accidental,because the prediction is valid only for negative charges.)From this, we may expect the calculation procedure for max-imum charge in air for negative polarity to be quite reason-able, and therefore, Eq. (B.6) is acceptable as a reasonableprediction. Although Eq. (B6) is for negatively charged par-ticles in air at normal room conditions, it may be usable alsofor positively charged particles because the difference dueto the polarity is not very appreciable as shown in Fig. 17.

When the droplets become smaller than 8 m, the break-down limit exceeds the Rayleigh limit, and then the maxi-mum charge in argon gas follows the limiting line in air andSF6 plotted in the bold broken line, which is shown also inFigs. 14 and 15.

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Documents

A measuring system for the time variation of size and charge of a single spherical particle and its applications