The theory of electrostatic separations: A review part II. Particle charging

Minerals Ealgl~ering, Vol. 2, No. 2, pp. 193-205, 1989

0892-6875/89 $3.00 + 0.00 © 1989 Pergamon Press plc

Printed in Great Britain

THE THEORY OF ELECTROSTATIC SEPARATIONS: PART II. PARTICLE CHARGING

A REVIEW

E.G. KELLY % and D.J. Spottiswood § % Dept of Chemical & Materials Engineering, University of Auckland, Auckland, New Zealand Metallurgical Engineering Dept, Colorado School of Mines Golden, CO, U.S.A.

(Received 1 September 1988)

ABSTRACT This paper, the second in a series on the theory of electrostatic separations, b r i e f l y r e v i e w s the methods and equipment available for the measurement of electric fields, electric charges, contact-charge accumulation, and conductivity. P a r t i c l e c h a r g i n g by the three major processes relevant to commercial separators (corona, or ion b o m b a r d m e n t , induction, and tribocharging) is then discussed. I. INTRODUCTION In any separator, the separation is brought about by suspending the particles in a m e d i u m and s u b j e c t i n g t h e m to a s e p a r a t i n g force that acts on some particle property. In the case of e l e c t r o s t a t i c s e p a r a t i o n s , the p r i m a r y separating force is given by F

-- Q . E

(I)

where F is the vectorial sum of all the forces, Q is the total charge, and E is the e l e c t r i c f i e l d i n t e n s i t y at a point P in space. While in reality secondary forces must also be considered, it follows that information a b o u t the two p a r a m e t e r s E (electric field strength) and Q (electric charge) are central to an understanding of electrostatic separations. However, whether or not a p a r t i c l e has a c h a r g e as it e n t e r s an e l e c t r i c field w i l l d e p e n d markedly on its conductivity, and thus knowledge of the relative conductivity of the particles is also important. The following is a brief review of the techniques available for electrostatic measurements. More d e t a i l e d r e v i e w s have been given by Secker and Chubb [I], and Cross [2]. II. PROPERTY MEASUREMENTS (I) Measurement of Electric Fields. E l e c t r i c f i e l d s at a surface or in a space can be measured by appropriate electrometers, of which there are two main types; capacitive probes [3,4] and field mills [5,6]. These instruments are based on the principle that when a grounded, electrically conductive, "sensing" surface is exposed to an electric field, an electric field is induced on it. The magnitude of the charge is proportional to the local field intensity and to the total area exposed to the field. A number of commercial instruments are based on the electric field mill. In these, the electric field falling on the sensing surface is interrupted by a rotating shaped sector. The periodic voltage generated is amplified by an AC a m p l i f i e r , the r e c t i f i e d o u t p u t of w h i c h is d i s p l a y e d on a meter. The advantage of this system is that the sensitivity can be quite high, and the charge sign can be determined. The main disadvantage is that the p h y s i c a l s i z e of t h e p r o b e l i m i t s t h e r e s o l u t i o n o b t a i n a b l e , and the r e s p o n s e characteristics mean that it is best suited to static or r e l a t i v e l y s l o w l y varying fields. There are a number of versions of capacitive probes. While differing in their treatment of the signal collected, they all o p e r a t e on s i m i l a r p r i n c i p l e s 193

194

E.G. KELLYand D. J. SPOTTISWOOD

w h i c h involve d e t e r m i n a t i o n of a number of c a p a c i t a n c e s and v o l t a g e s in the system; p r i m a r i l y those across the grounded sample, the sample to probe a i r gap, and those of the instrument itself. N e g a t i v e - f e e d b a c k e l e c t r o m e t e r s use n e g a t i v e feedback to greatly increase the input resistance of the m e a s u r i n g circuit amplifier. This v i r t u a l l y puts the probe at ground potential, and allows the accurate d e f i n i t i o n of the area that the probe actually sees. The s y s t e m is a l s o s i m p l e and c o m p a c t , w i t h no m o v i n g p a r t s . Its p r i n c i p l e d i s a d v a n t a g e is t h a t a c c i d e n t a l c h a r g i n g of the p r o b e m a y l e a d to f a l s e readings. S i n g h and H e a r n h a v e d e s c r i b e d an e l e c t r o s t a t i c probe that is c a p a b l e of d e t e c t i n g the c h a r g e on i n d i v i d u a l p a r t i c l e s as s m a l l as 50 microns. It can also detect bipolar charging of particles [7]. The p o s i t i v e - f e e d b a c k e l e c t r o m e t e r has a v i b r a t i n g device in the p r o b e h e a d between the sample and the sensing surface. The instrument can be made very sensitive with a fine spatial resolution. In the p r e s e n c e of space charges such as corona, more s o p h i s t i c a t e d biased probes must be used [8,9], a l t h o u g h under s o m e c o n d i t i o n s , the f i e l d m i l l p r i n c i p l e can still be used, in w h i c h case it is used in a form known as the e l e c t r o s t a t i c flux meter [10,11]. (2) M e a s u r e m e n t of E l e c t r i c Charges. While in principle the above capacitive probes can be used to measure electric charges at surfaces as well as electric fields, they are not suitable for the i r r e g u l a r g e o m e t r y of r e a l p a r t i c l e s . In s u c h a s i t u a t i o n , p e r h a p s the simpliest and the most accurate method for m e a s u r i n g electric charge is with a F a r a d a y pail. Its o p e r a t i o n is based on Gauss' Law, w h i c h states that Q = $ w h e r e ~ is the electric flux

(2) (Coulomb).

Thus, when a charged p a r t i c l e is placed in the pail, the flux from the charge on the p a r t i c l e produces an equal charge on the outside of the pail that can be m e a s u r e d by an electrometer. The outside of the pail must be shielded to e n s u r e t h a t no stray charges are picked up, and methods of c a l i b r a t i n g the pail have been d e s c r i b e d [12]. T h e "separation tower" has been suggested as a suitable method of m e a s u r i n g the c h a r g e - t o - m a s s r a t i o of s m a l l p a r t i c l e s [13]. The method involves t r i b o c h a r g i n g of the p a r t i c l e s in a fluidized bed, from w h i c h they d i s c h a r g e into a tower where they fall freely between two electrodes. The particles are c o l l e c t e d at the b o t t o m in i n d i v i d u a l l y shielded Faraday trays, and their p l a c e m e n t a l l o w s the c h a r g e - t o - m a s s r a t i o to be d e t e r m i n e d . The m a j o r l i m i t a t i o n of the method is that it m e a s u r e s tribocharging, and thus gives little information about the charging that will occur by other mechanisms. (3) M e a s u r e m e n t of C o n t a c t - C h a r g e Accumulation. In an effort to q u a n t i f y tribocharging, a n u m b e r of s t u d i e s h a v e m e a s u r e d contact-charge a c c u m u l a t i o n [14,15]. Many of these studies appear to give c o n f l i c t i n g results, but this can be a t t r i b u t e d to inadequate c o n s i d e r a t i o n of the d i f f e r i n g e x p e r i m e n t a l conditions ( p a r t i c u l a r l y t h e i r d e v i a t i o n from equilibrium), and the d i f f i c u l t y in d e t e r m i n i n g the truly relevant properties of t h e m a t e r i a l b e i n g studied. One of the more successful studies used a s p h e r i c a l m e t a l c o n t a c t o r t h a t c o u l d be t o u c h e d , in a c o n t r o l l e d and r e p r o d u c a b l e way, to an i n s u l a t o r s p e c i m e n . C h a r g i n g a n d b a c k f l o w time constants were determined, and it was shown that c o n t a c t - c h a r g e a c c u m u l a t i o n on insulators is due to slight e l e c t r i c a l conductivity [16]. (4) M e a s u r e m e n t of Conductivity. Meaningful measurement of c o n d u c t i v i t y is very d i f f i c u l t because over and above the problems of dealing with an irregular particle, the significance of inherent and induced "contamination" clearly indicate that most reported data on p a r t i c l e s will be very specific to the given experiments. Cohen's review on the c o n d u c t i v i t y of alumina is a good i l l u s t r a t i o n of these p r o b l e m s [17]. L a w v e r and W r i g h t [18] d e s c r i b e d a c o n c e n t r i c cell for measuring the c o n d u c t i v i t y of a bulk of g r a n u l a r material. They considered this type of cell a p p r o p r i a t e for the d e t e r m i n a t i o n of c o n d u c t i v i t y r e l e v a n t to h i g h tension s e p a r a t i o n b e c a u s e p a r t i c l e / p a r t i c l e contact w o u l d be significant in

The theory of electrostatic separations

195

such a separator. Mugeraya and Prabhakar investigated the effect of applied voltage, compaction, moisture, and temperature on the conductivity of b e a c h sand minerals [19]. Other workers have concentrated on measuring the conductivity across a single grain. In one instance, a number of mineral grains were rested between two p a r a l l e l c o n d u c t i n g rods [20], w h i l e the U.S. B u r e a u of M i n e s used one particle at a time [21]. A l t h o u g h there is now a better understanding of the effect of semiconductor characteristics in electrostatic separations, most phenomena have so far been studied by empirical methods. It is possible that more fundamental studies using the techniques developed for semiconductors [22] will be of b e n e f i t . C a r t a et al [23] h a v e d e s c r i b e d how the w o r k f u n c t i o n , F e r m i level VF, forbidden g a p Vg, a n d c h a r g e c a r r i e r c o n c e n t r a t i o n s (n and p) can be determined.

III. MECHANISMS OF PARTICLE CHARGING While there are a number of ways of charging particles [24], only three are serious contenders f o r c h a r g i n g p a r t i c l e s in c o m m e r c i a l e l e c t r o s t a t i c separations; corona or ion b o m b a r d m e n t c h a r g i n g , i n d u c t i o n c h a r g i n g , and tribocharging. These are illustrated in Figure I, and will be discussed in this section. Other mechanisms, such as photoelectric [25] and pyroelectric [26] c h a r g i n g h a v e b e e n s h o w n to be to e x p l o i t a b l e , but have had little commercial success. (I) Corona Charging. The highest particle charge levels in electrostatic separation are achieved by ion bombardment (corona charging), the basic concepts of which are illustrated in Figure la. Essentially, the method involves the charging of particles as they pass between two electrodes. Due to the h i g h v o l t a g e used, the gas b e t w e e n the e l e c t r o d e s is ionized, and these ions charge the particles by bombardment. (a) Coronas. Under normal conditions gases are non-conductors. However, if the p o t e n t i a l b e t w e e n two e l e c t r o d e s is raised to a sufficient level, the ionization and conductivity of the gas increases greatly as a corona discharge occurs. Further increase of the voltage eventually leads to an uncontrollable current flow due to spark-over or arcing (the actual spectrum of processes is somewhat more complex [27]). Figure 2 shows the typical form of the corona voltage-current relationship. Practical systems involve asymmetrical electric fields because the electrodes are of significantly different size and shape; for example, a w i r e and a cylinder (drum). In these cases the corona between the electrodes consists of two parts; a relatively narrow "glow" region at the small electrode (because it has the steepest field gradient), and a "dark" region over the remaining space to the large electrode. The nature of the corona differs, depending on whether the wire electrode is negative or positive. A positive corona has a g e n t l e g l o w - l i k e color, is relatively steady and uniform near the wire electrode, and can be produced in any gaseous medium. A negative corona concentrates as tufts of glowing gas spaced at intervals along the wire, but is possible only with gases such as oxygen that provide electron attachment. Both positive and negative c o r o n a d i s c h a r g e s e a c h h a v e t h e i r a p p l i c a t i o n s , a l t h o u g h the negative corona is preferred b e c a u s e it has a h i g h e r s p a r k - o v e r v o l t a g e that a l l o w s a m o r e intense corona to be produced in air. The glow region is where gas ionization occurs, while the dark region contains neutral m o l e c u l e s plus a small f r a c t i o n of a n i o n s and e l e c t r o n s moving, respectively, to the negative and positive electrodes. Typically, particles to be separated are charged in the dark region: the glow region is normally so narrow that its only function is to generate electrons. While higher charge d e n s i t i e s on particles can be a c h i e v e d in the g l o w r e g i o n b e c a u s e of the higher field strength there, charging is erratic and thus less efficient [28]. In practice, it is desirable to maximize the corona current, without suffering spark breakdown. C a l c u l a t i o n of the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s is

E.G. KELLY and D.J.SPWTISW~~D

196

complicated for all but the simplest of geometries 1291, although techniques for more complex geometries have been described 130,311.

0-

l

Electrode

C - Conductor N-Non-conductor

_+++ +++ ++ 6 QQ -D F6

+c +

_

_

_

+c+ - *-

N

_

_

_

_

+++

Iy+

_

-

T

-=_

+-

a++++++ @@

0

+

+

+

+

+

+

_

_

_

_

-

-

++ ++ + c+

0

o++++++

_=_-- -

+

+

+

+ 0++ _0 ++ +c+

C-

8 +

+ + + + + +

+

+

+v+

+

+

__--3ii-y 0+++ _ + + 7Y 0++ -F.

F.

0 a+++++

+

+

+

+

+

+F. +

++++t+

-F.

Chmging

-

-

Charged

Fig.1 Particle charging processes: (a) Corona charging. (b) & (c) Induction charging. (d) Particle/particle and (e) particle/surface tribocharging.

The theory of electrostatic separations

i

i

i

197

I I

i

Kilovolfs

Fig.2

Typical form of the corona voltage-current

relationship.

Lama and Gallo [27] have p r e s e n t e d an e m p i r i c a l e q u a t i o n prespark corona current for a needle-to-plane corona: Ima x = Vs(V s - Vt)S-2

for

the

maximum

(3)

where I is the maximum presparking current, V s is the sparking voltage, S is the ~ d l e to p l a n e e l e c t r o d e spacing, and V t is the corona threshold voltage. Thus, increasing the electrode spacing and the voltage increases the allowable current. With wire-to-plane coronas, the maximum current increases w i t h w i r e - t o - p l a n e d i s t a n c e , and is r e l a t i v e l y i n d e p e n d e n t of the w i r e diameter. Other empirical data has been reviewed by Fraas [32]. High concentrations of dust particles can, by t h e m s e l v e s h a v i n g a charge, lower the total space charge between electrodes. This lowers the field and thus the charging of the particles of interest [29]. The phenomena is more s i g n i f i c a n t in e l e c t r o s t a t i c p r e c i p i t a t o r s , but could be a p r o b l e m with electrostatic separators if there is a high proportion of dust in the air, or in the particle stream. H u m i d i t y is r e c o g n i z e d as having a significant affect on the corona. The effect is difficult to study. Not only does the presence of water affect the ionization processes in opposing ways [33], the number and density of water droplets in the air is dependent on the quantity, size and nature of "dust" n u c l e a t i o n sites [34]. Abdel-Salam showed that for positive wire-to-plane coronas the effects of humidity could be calculated under certain conditions. W i t h t h i n w i r e e l e c t r o d e s , the c o r o n a i n c e p t i o n v o l t a g e d e c r e a s e s w i t h increasing humidity, while at high voltages, the corona current decreases as the relative humidity increases [33]. (b) Particle Charging. The charge acquired by its dielectric constant, the field intensity, the gaseous medium. Charging in a corona is two main processes: ion b o m b a r d m e n t (or charging, the latter becoming i n s i g n i f i c a n t micron in diameter [35].

a particle depends on its size, and the concentration of ions in usually considered to be due to field) c h a r g i n g and d i f f u s i o n w i t h p a r t i c l e s g r e a t e r than I

Ion bombardment results from bombardment of the particles by ions moving under the influence of the applied electric field. If a spherical particle with a u n i f o r m l y distributed free surface charge Q is placed in a uniform electric field E in a gas, the free and induced charge on the particle distort the original°field, and impart to it a radial component. If an ion is attracted to the particle and approaches from an angle 8 for which the radial force is negative, it will be captured (Figure 3) and will add to the charge on the particle. T h e s e a d d i t i o n a l c h a r g e s c h a n g e the field around the particle (Figure 3b), eventually stopping charging altogether. The maximum free charge Qmax on the surface of the spherical particle is [29,36,37]: Qmax = ~eoKed2Eo

(4)

E.G. KELLY and D. J. SPOTrXSWOOD

198

where

K e = 2[ er,p er~p

(5)

+- 21 ] + I

and e is the dielectric constant of the particl~2 For a non-conducting particel~, K tends to unity, e for air is 8.85 x 10F/m, and E for air is a b o u t 3 x ~ 0 -b V/m, w i t h t~e result that the m a x i m u m c h a r g ~ d ~ n s i ~ o (Qmax/area) possible on a non-conducting particle is about 2.66 x 10 -v C/m .

~iiiii~iii~!~i~i~i~iiiiiiii ~ililiiiiiiiiiiiii:iiiiii

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::~ :::, :::,::::::

®

®

Fig.3 The electric field near an uncharged (a) and a partly charged (b) particle, showing how the region of charge capture (unshaded) decreases as the particle charges. The particle charge as a function of time t is usually expressed by

Q = Qmax

(6)

t + T

where T is the particle-charging 4 e

T

=

time constant,

given by:

o

(7)

ciqi6 where c~ is the ion con.~ntration 6 is the+ion mobility (m- /V.sec).

(m-3), qi is the ion charge

M a l a v e - L o p e z and Peleg [38] have suggested that electrostatic decay can be better analysed by using a linear relationship

(Coulomb),

and

charging

and

t = Qt - QO where Qe is constants. Typically, although it [39].

K I + K2t

(8)

the charge density at time t, QO is the initial charge,

KI, K 2 are

the c h a r g i n g time is of the order of a few milliseconds [36], may not immediately be d i s t r i b u t e d u n i f o r m l y over the surface

Barthelemy and Mora in the form Qmax = ~eoKsde2Eo

[40,41]

carried

this analysis

further and presented

Eq. 4

(9)

The theory of electrostatic separations

199

w h e r e d e is the diameter of an electrical equivalent ellipsoid, and K the charging shape f a c t o r (a f u n c t i o n of e l l i p t i c i t y z/x and the d i e l e @ t r i c constant e of the particle). Some values of K are: r,p s

Shape z/x

Charging shape factor K Conductor,

5 I 0

Ks = =

s

for:

Non-conductor,

36.2 3.01 0.666

Ks = 5

25.15 2.15 0.532

which shows that the effect of shape and conductivity are significant. For example, conductors will attain higher maximum charges than n o n - c o n d u c t o r s , e l o n g a t e d particles (z/x = 5) greater charges than spherical particles, and spherical particles greater charges than flakes (z/x = 0). Similar conclusions on the effect of shape were obtained by Vereshchagin et al. as a result of t h e i r t h e o r e t i c a l a n a l y s i s of the f o r c e s on a s e m i - e l l i p t i c a l particle [42]. Real particles may have sufficiently sharp surface points that they can build up enough charge to generate a corona that causes a loss of charge. Not only will this prevent that particle from attaining its maximum charge; it can also bombard other particles with opposing charges, so lowering their charge. The effect will lower the efficiency and is equivalent to charging by bi-ionizing electrodes [43]. Barthelemy and Mora [40] also considered the case of the charging of particles on a grounded surface. Under these conditions, the conductor leaks charge to ground, while the non-conductor's charge builds up. In practice, equilibrium charges % w i l l be e s t a b l i s h e d , w i t h the c o n d u c t o r having a small charge (because incomplete leakage), and the n o n - c o n d u c t o r h a v i n g a less t h a n maximum charge (because some leakage occurs), according to the relationship:

0max0e= 1.

{IK1

1. fn(E'/2K1-1}1/2

where fn(E) is a function of the electric field, for the particle.

(10)

and K 1 is a leakage constant

The order of magnitude of fn(E)/K I is 10 -14 (equivalent total resistance of the particle), that is, it tends t~ zero for a non-conductor, and to infinity for a c o n d u c t o r . B e c a u s e the charging time is rapid, the attainment of a steady state charge is therefore as depicted in Figure 4. (2) Induction Charging. Induction charging is that process whereby initially uncharged particles in an electric field assume the f i e l d p o l a r i t y . If a c o n d u c t i n g p a r t i c l e then c o n t a c t s a c o n d u c t i n g surface, it c o n d u c t s charge of one polarity to the surface, leaving the particle with a net c h a r g e of the o p p o s i t e p o l a r i t y . Once charged the particle then tends to be repelled from the surface because it now has the same c h a r g e p o l a r i t y as the s u r f a c e (if the s u r f a c e is grounded, the other electrode at least has an opposite charge and attracts the particle). N o n - c o n d u c t i n g p a r t i c l e s , h a v i n g no net charge, are n e i t h e r attracted nor repelled by the field (Figure lb). Induction charging may also occur between conducting particles field by the manner illustrated in Figure Ic. The charging [40]

characteristics

Q = CpV[1-exp(-t/~pCp)]

will be described

in an electric

by an equation of the form of (11)

200

E . G . KELLY and D. J. SPOTTISWOOD

where C is the capacitance of the particle, V is the voltage differential, and ~ ~he particle's equivalent total resistance. P

(~

( ~ Conductor

Non-Conductor

Maximum Q

Maximum ¢ 0 I

Steady State

Time

Fig.4

Steady State

f Time

Steady state charge on (a) a conductor, and (b) a non-conductor [After Barthelemy and Mora (40)].

(3) Tribocharging. T r i b o - or frictional charging is that process whereby a charge exists on a material after the parting of a solid/solid contact (a similar phenomena also occurs with solid/fluid contacts). The process is illustrated in Figures Id & le, and the two solids can be any combination of conductor, semiconductor, or non-conductor (dielectric). The magnitude of the final charge will actually be the result of two processes; the charge transfer that occurs d u r i n g the contact, and the charge backflow that occurs as the materials are parted. Although the phenomena of tribocharging was first r e c o r d e d o v e r 2500 y e a r s ago, the p r o c e s s , particularly with regard to non-conductors, is still not fully u n d e r s t o o d , and t h e r e is still d e b a t e a b o u t the a c t u a l m e c h a n i s m s involved. Much of the confusion in the literature can be attributed to the experimental c o n d i t i o n s of d i f f e r e n t r e s e a r c h e r s not being equivalent. Problems immediately arise concerning the concept of "contact". Firstly, on the micro scale, there is the difficulty of knowing the true surface contact; even the best prepared surfaces can have relatively little true contact. Then, on the macro scale; some experiments have involved contact only, some sliding, some deformation of one component, and others a combination of these. Difficulties also arise from problems due to specimen contamination. However, p e r h a p s the m o s t s i g n i f i c a n t fact is that w h e n the c h a r g i n g i n v o l v e s a non-conductor, the process is necessarily one of nonequilibrium. H a r p e r [15] in his c o m p r e h e n s i v e review made it clear that the mechanisms depend on the given combination of m a t e r i a l s , and that metals and semiconductors behaved quite differently to non-conductors (of which there are two different types, those that charge freely, and those that are reluctant to charge). In r e v i e w i n g the l i t e r a t u r e , it is found that t h r e e g e n e r a l c l a s s e s of c h a r g e - t r a n s f e r m e c h a n i s m s h a v e b e e n p r o p o s e d ; e l e c t r o n or ion t r a n s f e r determined by bulk properties, electron or ion transfer determined by surface properties, and transfer related to mechanical dislodgement. (a) C o n t a c t E l e c t r i f i c a t i o n of M e t a l s (Conductors). When two metals are brought into contact, charge is transferred between t h e m u n t i l t h e i r F e r m i levels equalize. The charge Q retained is given by

The theory of electrostatic separations

Q = Co.V °

201

(12)

where V is t h e c o n t a c t potential difference, a n d C o is the c o n t a c t capacitaffce. C is readily m e a s u r e d and is related to the surface topography; it is e s s e n t i a l l y equal to the c a p a c i t a n c e of two p e r f e c t l y smooth bodies at a s e p a r a t i o n equal to the m e a n a s p e r i t y height [44]. A l t h o u g h earlier work [45] claimed that Q d e c r e a s e d as the bodies were parted (due to e l e c t r o n b a c k f l o w by tunnelling), the more recent w o r k indicates that this does not occur, and that Q is e n t i r e l y due to the e q u a l i z a t i o n of the Fermi levels on contact. (b) C o n t a c t C h a r g i n g B e t w e e n C o n d u c t o r and Non-conductor. It has for some time been recognized that the charge transferred between a conductor and a n o n - c o n d u c t o r tends to c o r r e l a t e with the work function d i f f e r e n c e b e t w e e n the two materials [46-49]. Such a correlation implies that electrons are transferred between conductor and non-conductor until thermodynamic e q u i l i b r i u m is established. Doubt about this m e c h a n i s m has arisen because it has o f t e n been assumed that for t h e r m o d y n a m i c e q u i l i b r i u m to be established, the charge must be d i s t r i b u t e d through the bulk of the m a t e r i a l ; s o m e t h i n g t h a t is c l e a r l y not possible given the low mobilities of c h a r g e s in n o n - c o n d u c t o r s. It is n o w r e c o g n i z e d t h a t t h e c h a r g i n g of n o n - c o n d u c t o r s o c c u r s at the surface, and that there are s u f f i c i e n t sites to account for the charges found in practice. Charge t r a n s f e r occurs fairly rapidly [50] (of the order of a few m i c r o s e c o n d s ) by e l e c t r o n s t u n n e l l i n g b e t w e e n the c o n d u c t o r and localized s u r f a c e s t a t e s in t h e n o n - c o n d u c t o r in the v i c i n i t y of the contact. The surface states act as "traps", p r o v i d i n g or a b s o r b i n g e l e c t r o n s [51-55]. Back tunneling (i.e., l o s s of c h a r g e as the b o d i e s s e p a r a t e ) is i n s i g n i f i c a n t unless the surface is very highly charged [54]. M u c h of the c o n f u s i o n in the e x p e r i m e n t a l r e s u l t s on contact c h a r g i n g is c l a r i f i e d by the w o r k of H o m e w o o d and R o s e - I n n e s [16] on charge a c c u m u l a t i o n from r e p e a t e d contacting. They showed that between each contact, as a result of a small e l e c t r i c a l c o n d u c t i v i t y in the non-conductor, charge r e d i s t r i b u t i o n away from the point of contact occurs, thus a l l o w i n g further c h a r g i n g to occur on recontact. Hence, while local e q u i l i b r i u m may be e s t a b l i s h e d on contact, the total charge on a particle depends on the contacting r a t e , the d i s t r i b u t i o n of the c o n t a c t s o v e r t h e s u r f a c e , the n a t u r e of the c o n t a c t (single "point", rolling, or sliding), and the c o n d u c t i v i t y of the material. It is p o s s i b l e that some charge transfer in frictional c o n t a c t i n g o c c u r s m a t e r i a l t r a n s f e r [56], but it is not c o n s i d e r e d a major m e c h a n i s m [57].

by

(c) C o n t a c t E l e c t r i f i c a t i o n Between Non-conductors. Duke and F a b i s h have d e s c r i b e d a model for p r e d i c t i n g the results of i n s u l a t o r / i n s u l a t o r c o n t a c t c h a r g i n g w h i c h t h e y a t t r i b u t e to the filling of intrinsic localized "bulk" m o l e c u l a r - i o n states in the o u t e r m o s t f e w m i c r o m e t e r s of t h e ( p o l y m e r i c ) materials i n v o l v e d [58]. S h i n b r o t p r o p o s e d an a l t e r n a t i v e m e c h a n i s m that i n v o l v e d the reversal of d o u b l e - l a y e r dipoles. Implications of this t h e o r y include an increase in c h a r g i n g w i t h surface deformation, and t h r e s h o l d s for contact p o t e n t i a l and surface r o u g h n e s s below w h i c h contact c h a r g i n g ought not occur [59]. (4) C h a r g i n g Rate. B a s e d on the d i s c u s s i o n above it can be seen that the c h a r g i n g rate of any p a r t i c l e can be d e s c r i b e d by an e q u a t i o n that considers the two processes that o c c u r ; t h e a d d i t i o n of n e w charge, and the loss of e x i s t i n g charge. Both p r o c e s s e s c a n be d e s c r i b e d by s i m i l a r e x p o n e n t i a l e x p r e s s i o n s , but with d i f f e r i n g time constants. For example [9] :

dQ = Qc[I

- exp(-tb/Tb)]

- Q[I

- exp(-tc/Tc)]

(13)

dN w h e r e Qc is the charge in the r e g i o n of contact d u r i n g c o n t a c t , t h is the time b e t w e e n contacts, t c is the time of contact, Th is the time c o n s t a n t for the initial decay of charge, Tc is the t i m e ~ o n s t a n t for b a c k f l o w to the contactor, and N is the number of contacts. S o l u t i o n of Eq. 13 leads to an e q u a t i o n of the form:

202

E, G, KELLYand D. J. SPornswooD

Q(N)

= K I - K 2 e x p ( - N K 3)

(14)

w h e r e KI, K 2 and K 3 are constants and N is the number of subsequent contacts. (5) Coehn's Rule. Coehn's rule is an attempt to predict the charges on two m a t e r i a l s after they have been in contact. Qualitatively, it states that "when two materials are c o n t a c t e d and s e p a r a t e d , the m a t e r i a l w i t h the higher d i e l e c t r i c constant becomes p o s i t i v e l y charged". The rule has been q u a n t i t a t i v e l y formulated as [60] Q/A = 15 x I0-6(er,i

- er, 2)

(15)

w h e r e Q / A is the s u r f a c e c h a r g e density d i e l e c t r i c constants of the two materials.

(C/m2),

and

er, I and

er, 2 are

the

Given the c o m p l e x i t y of t r i b o c h a r g i n g and its d e p e n d e n c e on trace components, it is u n r e a l i s t i c to expect that such a rule will be very reliable, and this h a s b e e n f o u n d to be the c a s e . H o w e v e r , as a b e t t e r u n d e r s t a n d i n g of t r i b o c h a r g i n g is developed, it is possible that some general r e l a t i o n s h i p may be found for p r e d i c t i n g tribocharges. Duke and F a b i s h [58] claim that their q u a n t i t a t i v e model of contact e l e c t r i f i c a t i o n is a step in this direction. IV. C O N C L U D I N G REMARKS The c o n c l u d i n g paper in this series will consider the p r a c t i c a l a s p e c t s of e l e c t r o s t a t i c separations, and will, in particular, c o n t r a s t t h e o r e t i c a l and e m p i r i c a l information. NOMENCLATURE C

= p a r t i c l e capacitance

(F),

(C/V)

P C

o

= contact c a p a c i t a n c e

c i = ion c o n c e n t r a t i o n

(F),

(C/V)

(m -3)

d = differential d e = d i a m e t e r of an electrical e q u i v a l e n t ellipsoid E E e

= electric field intensity o r

(V/m)

= u n i f o r m electric field intensity = relative p e r m i t t i v i t y

(m)

(V/m)

(dielectric constant), Eq. 2

er,p = d i e l e c t r i c constant of particle. F

= force

fn(E)

(N)

= a f u n c t i o n of the electric field

Ima x = m a x i m u m p r e s p a r k i n g current in a corona K

(A)

= constant

K e = given by Eq. 5. K

s

= c h a r g i n g shape factor

N = number Q

= total charge

(C)

Qe = e q u i l i b r i u m charge

(C)

Qmax = m a x i m u m free charge on the surface of a particle

(C)

The theory of' electrostatic separations

203

Qo " charge in the region of contact during contact

(C)

Qt " charge density at time t (C) QO = initial charge qi = ion charge

(C)

(C)

S = needle to plane electrode t = time

= time of contact

V = voltage V

o

(m)

is)

t b = time between contacts tc

spacing

is)

(s)

(V)

= contact potential difference

V s = corona sparking voltage V t = corona threshold voltage

iV)

iV) (V)

B 8 = angles 6 = ion or charge carrier mobility T = particle-charglng

(m2/V.s).

time constant

(s)

T b = time constant

for the initial decay of charge

T c = time constant

for backflow to the contactor

= electric P

(s)

is)

flux (C).

= particle's

equivalent

total resistance

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