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Impact charging experiments with single particles of hundred micrometre size
Powder Technology 135 – 136 (2003) 14 – 22 www.elsevier.com/locate/powtec
Impact charging experiments with single particles of hundred micrometre size Tatsushi Matsuyama a,*, Masaomi Ogu a, Hideo Yamamoto a, Jan C.M. Marijnissen b, Brian Scarlett c a Soka University, Faculty of Engineering, Tangi-cho 1-236, Hachioji, Tokyo 192-8577, Japan Delft University of Technology, Faculty of Applied Science, Julianalaan 136, 2628BL Delft, The Netherlands c University of Florida, Particle Engineering Research Center, P.O. Box 116135, Gainesville, FL 32611, USA
b
Received 30 October 2002; received in revised form 9 April 2003; accepted 14 April 2003
Abstract This paper reports the results of an ongoing program, which investigates the electrostatic charge generated when a particle impacts and rebounds from a metal plate. Previous studies were conducted with particles, which were several millimetres in size. From those experiments, a theory of ‘charge relaxation’ was developed which predicts the equilibrium charge, which a particle will acquire after multiple impacts. In this paper, we report the development of a new apparatus, which is more sensitive by several decades. With this equipment, the experiments could be repeated with particles as small as 100 Am. The results show that the charge relaxation theory is still relevant in this range. However, it is necessary to modify this theory, both for the larger and the smaller particles, to account for the distribution of the charge on the surface of the particle. D 2003 Published by Elsevier B.V. Keywords: Online charge measurement; Faraday cage; Particle electrification; Charge relaxation model; Impact charging; Triboelectrification
1. Introduction When a particle collides with a wall, or when two particles collide, there may be a net transfer of electrostatic charge. An understanding of this phenomenon, which is called tribocharging, has practical applications, for example in the design of photo printing and electrostatic coating machines. The charge may also be sufficient to constitute a hazard in powder handling processes. As with all particle processes, it is necessary first to understand the fundamental mechanism when two particles contact or when a single particle collides with a target. We have previously reported experimental measurements of the charge acquired when a polymer particle is fired, from an air cannon, against a metal target. The velocity of the particle and its angle of impact were controlled variables in those experiments. Before striking the target, the particle passed through a Faraday cage, which recorded its initial charge. After rebounding, it was collected in a second * Corresponding author. Tel.: +81-426-91-8169; fax: +81-426-918169. E-mail address: [email protected] (T. Matsuyama). 0032-5910/$ - see front matter D 2003 Published by Elsevier B.V. doi:10.1016/S0032-5910(03)00154-2
Faraday cage or, alternatively the residual charge on the target was measured. Thus, the net transfer of charge could be deduced. Using this apparatus significant results were obtained [1 – 7], which demonstrated that the net charge acquired during an impact is dependent upon the initial charge, which the particle carries before the impact. During successive impacts, the particle acquires an equilibrium or maximum charge after which no further transfer occurs. The two principal authors of this paper have proposed a mechanism and model to explain this behavior, the ‘charge relaxation’ model [6,7]. The previous work was carried out with larger particles, several millimeters in diameter. This paper describes the results of an ongoing program to test the applicability of the model for smaller particles. This has required the construction of a new apparatus, which has a greater sensitivity to charge by several orders of magnitude. With this apparatus, we have been able to reduce the particle size to one hundred micrometers. The charges measured are still of the order of 104 electron charges. Nevertheless, this is sufficiently small that it is necessary to modify the charge relaxation model and to consider the manner in which this charge is distributed on the surface of the particle.
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
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2. Design of the new apparatus There are two basic problems to be addressed in devising an impact experiment which is suitable for smaller particles. As the particle becomes smaller, the charge that it carries also becomes correspondingly smaller. This requires an enhanced sensitivity of the charge measurement, particularly of the transferred charge. In order to achieve this, in the new equipment, the whole target is mounted in a Faraday cage whose total charge is monitored by a current integrating circuit with negative feedback and low input bias. The second problem is that a small particle has low momentum and decelerates significantly before striking the target. There is also the problem that a significant air flow to the target may, itself, transfer charge and produce misleading results. Thus, the use of an air cannon is precluded. In the new device the particle is accelerated in an aerosol beam generator [8], which focuses the trajectory of the particle and then projects it into a partial vacuum. In this case, a single orifice generator was sufficient to maintain a vacuum of 0.1 atm and to generate velocities as high as 50 m/s. With the use of multiple orifices, both of these limits can be considerably extended [8]. A schematic diagram of the new device is shown in Fig. 1. The particle is dropped by tweezers to the orifice one by one, and accelerated and focused by the aerosol beam generator. Immediately before striking the target, it passes through a double ring detector by which the velocity of the particle can be measured [9]. The particle then enters the Faraday cage and strikes the target. While the particle remains inside the cage, the net charge does not change and thus represents the initial charge carried by the particle. When the particle
Fig. 2. Schematic illustration of the process of charge measurement with the Faraday cage enclosing target and expected trace record (a) and (b) typical trace record. In the practical measurement, a reverse circuit of the amplifier records the signal in the opposite polarity.
rebounds and leaves, then the recorded signal represents the net charge left on the target. The difference represents the net charge, which has been transferred. The process is illustrated in Fig. 2(a) and a typical trace recorded by the storage oscilloscope is shown in Fig. 2(b). In this case, a particle of 200 Am diameter impacted at a velocity of 30 m/s. The initial charge was 563 fC and the net charge acquired was 84 fC. It can be seen from this trace that the minimum value of the acquired charge, which can be measured with this target, is about 50 fC.
3. The charge relaxation theory Fig. 1. Schematic illustration of the new apparatus. Important dimensions: target size = 10 14 mm; inner and outer box of the Faraday cage = 20 and 50 mm cubic; hole on top and side on the Faraday cage = f10 mm; orifice = f0.2 mm; diameter of the ring detector = f10 mm; distance of the rings = 6 mm.
When two electrostatically neutral materials are brought into contact, a double layer is formed due to the difference in their work function [10]. When the bodies are again
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T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
The breakdown potential, Vs, has been modified as a function of the product of the gas pressure, p, and the separation gap, d. The effect is considerable, as shown in Fig. 3 where the breakdown potential in air is plotted as a function of the gap at three different pressures, 760, 350 and 150 torr. In this same figure, the potential of an approaching particle is also plotted as a function of the separation of the particle and plate. In each case, the particle charge has been chosen to illustrate the critical value, which just contacts, tangentially, the Paschen curve. It is to this separation that the charge relaxation will occur and is the point which determines the value of the residual charge which will be carried on to the next collision. In Fig. 3, the breakdown potential, Vs, is a function of the separation, d. The potential generated, Vp, is a function of the separation as well as of the charge, q. The condition that they touch is that they should be both equal and tangential. Thus:
Fig. 3. Influence of pressure on Paschen’s curve.
separated an electrostatic charge, of opposite polarity, is left on each body. It remains to explain why the measurements of the charge transferred do not coincide with the difference in the work function of the two materials [5]. The senior authors of the paper have previously postulated two mechanisms, which modify this basic behavior [6,7]. In the first instance, if one of the bodies is not neutral but already carries a charge, then a field is induced in the second body, which modifies the net charge transferred. The second mechanism we have called ‘charge relaxation.’ When the bodies separate, the distance of separation is initially very small and thus the two equal charges, of opposite polarity, create a large field. If this field is sufficient to cause ionisation of the gas, then a portion of the charge is neutralised by conduction through the gas. If the initial charge carried by the particle before impact is insufficient to cause breakdown, then a net transfer of charge is acquired. If the total charge after that impact is sufficient to cause breakdown during separation, then a portion of the acquired charge will again be discharged and this is charge relaxation. Eventually, the initial charge on the particle is such that no net transfer occurs and this value is the equilibrium charge or a maximum limited by the breakdown. In order to estimate the equilibrium charge which will be acquired by a particle, it is necessary to know the breakdown potential of the surrounding gas and for this purpose we have assumed that Paschen’s law appertains [11]. This law predicts that there is a minimum potential at which gaseous breakdown will occur, dependent upon the gas pressure, p, and the contact gap, d. For air, at room temperature, the minimum potential difference, Vmin, is 330 V and the critical product of gas pressure and separation gap (pd)c is 5.67 mm torr [11]. When the pressure is changed, Paschen’s law may be modified as follows: Vs ¼ Vmin
ðpdÞ=ðpdÞc : 1 þ lnfðpdÞ=ðpdÞc g
Vs ðdÞ ¼ Vp ðd; qÞ;
ð2Þ
BVs BVp ¼ : Bd Bd
ð3Þ
Since there are analytical expressions to describe each curve, these simultaneous equations can be solved in order to translate the critical breakdown, Vs, into an equilibrium charge, q. In practice, the solution was made numerically and the results are shown, for the particles under consideration, in Figs. 4 and 5. In Fig. 4, the dependence of the equilibrium charge is plotted as a function of the particle size and in Fig. 5 as a function of the gas pressure. Since the equilibrium charge also depends on the relative permittivity of the particle, these calculations assume a value of unity, which represents an insulating material. The figures appertain to relatively high gas pressures when the mean free path of the gas is still small
ð1Þ Fig. 4. Maximum or equilibrium charge as a function of particle size.
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
Fig. 5. Maximum or equilibrium charge as a function of the pressure.
compared to the gap size. They illustrate that the equilibrium charge is strongly dependent upon the particle size within a relatively narrow range, 50 –300 Am. The charge is also significantly dependent upon the gas pressure, a lower pressure leads to a lower equilibrium charge.
4. Experimental results The predictions shown in Figs. 4 and 5 were tested in that regime of pressure and of particle size using the new experimental equipment. A collection of the data attained is shown in Fig. 6a– f. It is clear that there is a wide scatter on the data and that this is not only due to random error nor to the limited sensitivity. There is some extra mechanism, which causes the acquired charge to vary, even when it is carrying, ostensibly, the same initial charge. We postulate that this extra, significant mechanism is that the initial charge is not uniformly distributed over the surface of the particle, that it may well be carried in local spots. The substantial effect of the distribution of the initial charge can be substantiated in two ways, by experiment and by simulation. In order to demonstrate the effect experimentally, some tests were performed with polymer particles, which had been coated with a conducting surface. In previous work, it was demonstrated that when a metal particle strikes a metal target the initial charge is completely neutralized. Thus for a conducting particle, the net transfer measured has the same value as the initial charge but of opposite polarity. It is to be expected that a polymer particle with a conducting surface will exhibit the same behavior, that the charge can redistribute on the surface so quickly during the collision that it is completely relaxed. In order to test this hypothesis a number of polymer particles, varying in size from 150 to 250 Am were coated with nickel and were impacted onto a metal target. A SEM picture of a coated particle is shown in Fig. 7 and the results are shown in Fig. 8. They fall exactly on a line with a negative slope of 45j and show no scatter. These experiments
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add credence to the postulate that the large scatter in the data recorded when a non-conducting particle strikes a conducting target is caused by uneven distribution of the charge on the surface of the particle. A further demonstration of the mechanism of charge relaxation was made by varying the material of the metal target. Most of the experiments were made with a stainless steel target for which the work function is not exactly known. The work function for iron is 4.04 eV [11]. A series of experiments were made with targets of gold and platinum whose work functions are 4.24 and 6.27 eV, respectively [11]. In Fig. 9(a), the results for a polymer particle of 100 Am diameter impacting on the gold and stainless steel targets are compared and similarly for the platinum target in Fig. 9(b). There is no discernible difference in either case. These experiments add a convincing demonstration that the mechanism of charge relaxation is the operative mechanism in determining the net charge transfer from an impact.
5. Simulation of uneven charge distribution The effect of uneven charge distribution can also be illustrated using a model of a band of charge [4] distributed over a solid angle of 10j and with a varying inclination to the direction of impact. This model is illustrated in Fig. 10. The charge is axisymmetric and so the resulting field can be predicted analytically if a relative permittivity of unity is assumed. Consider a particle of radius a, which is carrying a ring of charge of density r on the portion of its surface, which is inclined at angle h to the forward direction of impact. The charge induces a potential at a point P, which is at distance d from the particle. This model is shown in Fig. 11. Treating the problem in spherical coordinates (r,h,/), the potential, p(u1,h2), induced at P is given by: pðh1 ; h2 Þ ¼
Z
h2
Z
h1
2p
0
ra2 sinhdhd/ ; 4pe0 R
ð4Þ
thus: pðh1 ; h2 Þ ¼
ra2 2e0
Z
h2
h1
sinhdh : R
We may substitute: R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ ða þ dÞ2 2aða þ dÞcosh:
Integration then yields: pðh1 ; h2 Þ ¼
npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra d 2 þ 2aða þ dÞð1 cosh2 Þ 2e0 ða þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio d 2 þ 2aða þ dÞð1 cosh1 Þ : ð5Þ
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Fig. 6. Results of the impact charging experiments with 100, 200 and 300 Am PS particles under pressure of 100 and 200 torr.
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
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Fig. 7. A SEM photograph of the nickel plated PE particle.
If the charge is spread uniformly over the surface, then h2=k, h1=0. Substituting these values yields the well-known expansion for a uniform sphere: pð0; pÞ ¼
ra2 : e0 ða þ dÞ
ð6Þ
Including the effect of the mirror image (d = 2d), the surface potential on the top of the particle (d = 0), Vp, is thus given as: Vp ðdÞ ¼ pðh1 ; h2 ÞAd¼0 pðh1 ; h2 ÞAd¼2d :
ð7Þ
The expression in Eq. (7) was used to calculate the relationship between the impact charge and an initial charge, which is carried as a band of charge distributed over a solid angle of 10j and with a varying inclination to the direction of impact. The properties of the particles
Fig. 9. Results of the impact charging experiment with gold and platinum target compared to the results with stainless steel target.
Fig. 8. Result of the impact charging experiments with surface conductive particles.
used in the experiments were assumed and the result for a particle of 200 Am in diameter is shown in Fig. 12. The case of 0j corresponds to the charge being carried at the point of contact. In this case the effect is a maximum and the relationship has a slope of 45j. At the other extreme, the case of 180j corresponds to a spot remote from the contact. It is surprising that this charge has no effect and the particle behaves as though it is initially neutral. For comparison, the relationship for a uniformly charged particle is also plotted in Fig. 12. For all of the cases where the band of charge is in a forward direction of less than 35j, the particle looses some of its initial charge but does not fully attain the equilibrium charge. All of the lines pass, of course, through the point representing the neutral particle. It is interesting that the expected relationship between the acquired and the initial charge is
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T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
Fig. 10. A schematic illustration of a band of charge distributed.
always linear, regardless of how it is distributed. This implies that the relationship will be linear even for particles carrying multiple, randomly distributed charges, and that the effective transfer will lie between the two limits of the forward and backward spots. This is illustrated in Fig. 13, which illustrates schematically how the impact charge can be expected to deviate substantially from the case of the uniformly charged particle, depending upon the distribution of the initial charge on the surface of the particle. With this postulate in mind, these two limits were superimposed on the results of Fig. 6(a – f). It is seen that, indeed, almost all of the data is contained between these two limits and is distributed on both sides of the prediction for the uniformly charged particle. In order to illustrate this more clearly, the scattered data of Fig. 6(a – f) have been collated in the form of histograms, which show the distribution of impact charge in each case. Because the impact charge and the initial charge are always linearly related, the impact charge has been normalized by the maximum value possible, that due to the spot change at the point of contact. This normalisation is illustrated in Fig. 14 and enables a single histogram to be plotted for
Fig. 11. Coordinates for a surface integration for a partially charged particle.
Fig. 12. An example of discrete estimations of the effect of the localized charge.
each sequence of events. The results may be compared with those predicted by the band of charge model, which are shown in Fig. 15. It is premature to generalize from these few results but it does seem that the results for the smaller particles, 100 Am, fit better the sharp distribution, which is expected from this model and which might be expected from a single spot charge, which is randomly distributed. In this case, the chance of a localized charge in the forward direction is relatively large and all of the data is contained within the prescribed limits. As the particles become larger, the normalized charge is more randomly distributed and this might imply that the larger particles can carry more than one spot of charge. With the largest particles, 300 Am, there is also a significant fraction of occasions when the particle seems to gain more charge than would be gained by a neutral particle
Fig. 13. A schematic illustration of a concept of localization of initial charge and of two limiting situation.
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
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Fig. 14. Distribution of the normalized impact charging experiments with 100, 200 and 300 Am PS particles under pressure of 100 and 200 Torr, from the results in Fig. 7.
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T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
(pc)c
critical product of gas pressure and separation gap for the minimum breakdown potential (mm torr) p(h1,h2) surface potential of particle for a band of charge from h1 to h2 (V) q amount of charge (C) R distance (m) Vmin Paschen’s minimum breakdown potential (V) Vp surface potential on the top of the particle (V) Vs gas breakdown potential (V) e0 permittivity of vacuum (F/m) h angle (rad) r surface charge density (C/m2) (r,h,/) spherical coordinates
Acknowledgements Fig. 15. The histogram of the normalized impact charge predicted by the band of charge model.
which leads to apparently negative normalized values. This is a real result, which requires further investigation.
6. Conclusions The insight gained from those previous experiments strengthens the belief that multiple experiments on single particles give the maximum insight into the mechanisms, which dictate the particle properties. In this case, both the charge acquired during an impact and its distribution are relevant. The experiments reported support further the charge relaxation mechanism and confirm that it is applicable to particles as small as 100 Am. The distribution of the acquired charge demonstrates that it may remain localised on the surface of an insulating particle for a considerable time. This fact is very relevant to the design of processes, which are dependent upon the Coulomb attractive force. Nomenclature a radius of particle (m) d contact gap (m) p gas pressure (torr)
The authors are grateful to International Fine Particle Research Institute (IFPRI) for financial support of this project. We also appreciate the cooperation of Professor Nakajima, Hokkaido University, who kindly provided the nickel plated particles.
References [1] H. Yamamoto, B. Scarlett, Part. Charact. 3 (1986) 117 – 121. [2] T. Matsuyama, H. Yamamoto, KONA (1989) 15 – 21. [3] T. Matsuyama, H. Yamamoto, IEEE Trans. Ind. Appl. 30 (1994) 602 – 607. [4] T. Matsuyama, H. Yamamoto, IEEE Trans. Ind. Appl. 31 (1995) 1441 – 1445. [5] T. Matsuyama, H. Yamamoto, Adv. Powder Technol. 6 (1995) 211 – 220. [6] T. Matsuyama, H. Yamamoto, J. Phys. D: Appl. Phys. 28 (1995) 2418 – 2423. [7] T. Matsuyama, H. Yamamoto, J. Phys. D: Appl. Phys. 30 (1997) 2170 – 2175. [8] O. Kievit, M. Veiss, P. Verheijen, J. Marijnissen, B. Scarlett, Chem. Eng. Commun. 151 (1996) 79 – 100. [9] P. Vercoulen, J.C.M. Marijnissen, B. Scarlett, J. Aerosol Sci. 26 (1995) S797 – S798. [10] W.R. Harper, Contact and Frictional Electrification, Oxford University Press, London, 1967. [11] Inst. Electros. Jap., Seidenki Handbook, Ohmusha, Tokyo, 1981.
Impact charging experiments with single particles of hundred micrometre size Tatsushi Matsuyama a,*, Masaomi Ogu a, Hideo Yamamoto a, Jan C.M. Marijnissen b, Brian Scarlett c a Soka University, Faculty of Engineering, Tangi-cho 1-236, Hachioji, Tokyo 192-8577, Japan Delft University of Technology, Faculty of Applied Science, Julianalaan 136, 2628BL Delft, The Netherlands c University of Florida, Particle Engineering Research Center, P.O. Box 116135, Gainesville, FL 32611, USA
b
Received 30 October 2002; received in revised form 9 April 2003; accepted 14 April 2003
Abstract This paper reports the results of an ongoing program, which investigates the electrostatic charge generated when a particle impacts and rebounds from a metal plate. Previous studies were conducted with particles, which were several millimetres in size. From those experiments, a theory of ‘charge relaxation’ was developed which predicts the equilibrium charge, which a particle will acquire after multiple impacts. In this paper, we report the development of a new apparatus, which is more sensitive by several decades. With this equipment, the experiments could be repeated with particles as small as 100 Am. The results show that the charge relaxation theory is still relevant in this range. However, it is necessary to modify this theory, both for the larger and the smaller particles, to account for the distribution of the charge on the surface of the particle. D 2003 Published by Elsevier B.V. Keywords: Online charge measurement; Faraday cage; Particle electrification; Charge relaxation model; Impact charging; Triboelectrification
1. Introduction When a particle collides with a wall, or when two particles collide, there may be a net transfer of electrostatic charge. An understanding of this phenomenon, which is called tribocharging, has practical applications, for example in the design of photo printing and electrostatic coating machines. The charge may also be sufficient to constitute a hazard in powder handling processes. As with all particle processes, it is necessary first to understand the fundamental mechanism when two particles contact or when a single particle collides with a target. We have previously reported experimental measurements of the charge acquired when a polymer particle is fired, from an air cannon, against a metal target. The velocity of the particle and its angle of impact were controlled variables in those experiments. Before striking the target, the particle passed through a Faraday cage, which recorded its initial charge. After rebounding, it was collected in a second * Corresponding author. Tel.: +81-426-91-8169; fax: +81-426-918169. E-mail address: [email protected] (T. Matsuyama). 0032-5910/$ - see front matter D 2003 Published by Elsevier B.V. doi:10.1016/S0032-5910(03)00154-2
Faraday cage or, alternatively the residual charge on the target was measured. Thus, the net transfer of charge could be deduced. Using this apparatus significant results were obtained [1 – 7], which demonstrated that the net charge acquired during an impact is dependent upon the initial charge, which the particle carries before the impact. During successive impacts, the particle acquires an equilibrium or maximum charge after which no further transfer occurs. The two principal authors of this paper have proposed a mechanism and model to explain this behavior, the ‘charge relaxation’ model [6,7]. The previous work was carried out with larger particles, several millimeters in diameter. This paper describes the results of an ongoing program to test the applicability of the model for smaller particles. This has required the construction of a new apparatus, which has a greater sensitivity to charge by several orders of magnitude. With this apparatus, we have been able to reduce the particle size to one hundred micrometers. The charges measured are still of the order of 104 electron charges. Nevertheless, this is sufficiently small that it is necessary to modify the charge relaxation model and to consider the manner in which this charge is distributed on the surface of the particle.
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
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2. Design of the new apparatus There are two basic problems to be addressed in devising an impact experiment which is suitable for smaller particles. As the particle becomes smaller, the charge that it carries also becomes correspondingly smaller. This requires an enhanced sensitivity of the charge measurement, particularly of the transferred charge. In order to achieve this, in the new equipment, the whole target is mounted in a Faraday cage whose total charge is monitored by a current integrating circuit with negative feedback and low input bias. The second problem is that a small particle has low momentum and decelerates significantly before striking the target. There is also the problem that a significant air flow to the target may, itself, transfer charge and produce misleading results. Thus, the use of an air cannon is precluded. In the new device the particle is accelerated in an aerosol beam generator [8], which focuses the trajectory of the particle and then projects it into a partial vacuum. In this case, a single orifice generator was sufficient to maintain a vacuum of 0.1 atm and to generate velocities as high as 50 m/s. With the use of multiple orifices, both of these limits can be considerably extended [8]. A schematic diagram of the new device is shown in Fig. 1. The particle is dropped by tweezers to the orifice one by one, and accelerated and focused by the aerosol beam generator. Immediately before striking the target, it passes through a double ring detector by which the velocity of the particle can be measured [9]. The particle then enters the Faraday cage and strikes the target. While the particle remains inside the cage, the net charge does not change and thus represents the initial charge carried by the particle. When the particle
Fig. 2. Schematic illustration of the process of charge measurement with the Faraday cage enclosing target and expected trace record (a) and (b) typical trace record. In the practical measurement, a reverse circuit of the amplifier records the signal in the opposite polarity.
rebounds and leaves, then the recorded signal represents the net charge left on the target. The difference represents the net charge, which has been transferred. The process is illustrated in Fig. 2(a) and a typical trace recorded by the storage oscilloscope is shown in Fig. 2(b). In this case, a particle of 200 Am diameter impacted at a velocity of 30 m/s. The initial charge was 563 fC and the net charge acquired was 84 fC. It can be seen from this trace that the minimum value of the acquired charge, which can be measured with this target, is about 50 fC.
3. The charge relaxation theory Fig. 1. Schematic illustration of the new apparatus. Important dimensions: target size = 10 14 mm; inner and outer box of the Faraday cage = 20 and 50 mm cubic; hole on top and side on the Faraday cage = f10 mm; orifice = f0.2 mm; diameter of the ring detector = f10 mm; distance of the rings = 6 mm.
When two electrostatically neutral materials are brought into contact, a double layer is formed due to the difference in their work function [10]. When the bodies are again
16
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
The breakdown potential, Vs, has been modified as a function of the product of the gas pressure, p, and the separation gap, d. The effect is considerable, as shown in Fig. 3 where the breakdown potential in air is plotted as a function of the gap at three different pressures, 760, 350 and 150 torr. In this same figure, the potential of an approaching particle is also plotted as a function of the separation of the particle and plate. In each case, the particle charge has been chosen to illustrate the critical value, which just contacts, tangentially, the Paschen curve. It is to this separation that the charge relaxation will occur and is the point which determines the value of the residual charge which will be carried on to the next collision. In Fig. 3, the breakdown potential, Vs, is a function of the separation, d. The potential generated, Vp, is a function of the separation as well as of the charge, q. The condition that they touch is that they should be both equal and tangential. Thus:
Fig. 3. Influence of pressure on Paschen’s curve.
separated an electrostatic charge, of opposite polarity, is left on each body. It remains to explain why the measurements of the charge transferred do not coincide with the difference in the work function of the two materials [5]. The senior authors of the paper have previously postulated two mechanisms, which modify this basic behavior [6,7]. In the first instance, if one of the bodies is not neutral but already carries a charge, then a field is induced in the second body, which modifies the net charge transferred. The second mechanism we have called ‘charge relaxation.’ When the bodies separate, the distance of separation is initially very small and thus the two equal charges, of opposite polarity, create a large field. If this field is sufficient to cause ionisation of the gas, then a portion of the charge is neutralised by conduction through the gas. If the initial charge carried by the particle before impact is insufficient to cause breakdown, then a net transfer of charge is acquired. If the total charge after that impact is sufficient to cause breakdown during separation, then a portion of the acquired charge will again be discharged and this is charge relaxation. Eventually, the initial charge on the particle is such that no net transfer occurs and this value is the equilibrium charge or a maximum limited by the breakdown. In order to estimate the equilibrium charge which will be acquired by a particle, it is necessary to know the breakdown potential of the surrounding gas and for this purpose we have assumed that Paschen’s law appertains [11]. This law predicts that there is a minimum potential at which gaseous breakdown will occur, dependent upon the gas pressure, p, and the contact gap, d. For air, at room temperature, the minimum potential difference, Vmin, is 330 V and the critical product of gas pressure and separation gap (pd)c is 5.67 mm torr [11]. When the pressure is changed, Paschen’s law may be modified as follows: Vs ¼ Vmin
ðpdÞ=ðpdÞc : 1 þ lnfðpdÞ=ðpdÞc g
Vs ðdÞ ¼ Vp ðd; qÞ;
ð2Þ
BVs BVp ¼ : Bd Bd
ð3Þ
Since there are analytical expressions to describe each curve, these simultaneous equations can be solved in order to translate the critical breakdown, Vs, into an equilibrium charge, q. In practice, the solution was made numerically and the results are shown, for the particles under consideration, in Figs. 4 and 5. In Fig. 4, the dependence of the equilibrium charge is plotted as a function of the particle size and in Fig. 5 as a function of the gas pressure. Since the equilibrium charge also depends on the relative permittivity of the particle, these calculations assume a value of unity, which represents an insulating material. The figures appertain to relatively high gas pressures when the mean free path of the gas is still small
ð1Þ Fig. 4. Maximum or equilibrium charge as a function of particle size.
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
Fig. 5. Maximum or equilibrium charge as a function of the pressure.
compared to the gap size. They illustrate that the equilibrium charge is strongly dependent upon the particle size within a relatively narrow range, 50 –300 Am. The charge is also significantly dependent upon the gas pressure, a lower pressure leads to a lower equilibrium charge.
4. Experimental results The predictions shown in Figs. 4 and 5 were tested in that regime of pressure and of particle size using the new experimental equipment. A collection of the data attained is shown in Fig. 6a– f. It is clear that there is a wide scatter on the data and that this is not only due to random error nor to the limited sensitivity. There is some extra mechanism, which causes the acquired charge to vary, even when it is carrying, ostensibly, the same initial charge. We postulate that this extra, significant mechanism is that the initial charge is not uniformly distributed over the surface of the particle, that it may well be carried in local spots. The substantial effect of the distribution of the initial charge can be substantiated in two ways, by experiment and by simulation. In order to demonstrate the effect experimentally, some tests were performed with polymer particles, which had been coated with a conducting surface. In previous work, it was demonstrated that when a metal particle strikes a metal target the initial charge is completely neutralized. Thus for a conducting particle, the net transfer measured has the same value as the initial charge but of opposite polarity. It is to be expected that a polymer particle with a conducting surface will exhibit the same behavior, that the charge can redistribute on the surface so quickly during the collision that it is completely relaxed. In order to test this hypothesis a number of polymer particles, varying in size from 150 to 250 Am were coated with nickel and were impacted onto a metal target. A SEM picture of a coated particle is shown in Fig. 7 and the results are shown in Fig. 8. They fall exactly on a line with a negative slope of 45j and show no scatter. These experiments
17
add credence to the postulate that the large scatter in the data recorded when a non-conducting particle strikes a conducting target is caused by uneven distribution of the charge on the surface of the particle. A further demonstration of the mechanism of charge relaxation was made by varying the material of the metal target. Most of the experiments were made with a stainless steel target for which the work function is not exactly known. The work function for iron is 4.04 eV [11]. A series of experiments were made with targets of gold and platinum whose work functions are 4.24 and 6.27 eV, respectively [11]. In Fig. 9(a), the results for a polymer particle of 100 Am diameter impacting on the gold and stainless steel targets are compared and similarly for the platinum target in Fig. 9(b). There is no discernible difference in either case. These experiments add a convincing demonstration that the mechanism of charge relaxation is the operative mechanism in determining the net charge transfer from an impact.
5. Simulation of uneven charge distribution The effect of uneven charge distribution can also be illustrated using a model of a band of charge [4] distributed over a solid angle of 10j and with a varying inclination to the direction of impact. This model is illustrated in Fig. 10. The charge is axisymmetric and so the resulting field can be predicted analytically if a relative permittivity of unity is assumed. Consider a particle of radius a, which is carrying a ring of charge of density r on the portion of its surface, which is inclined at angle h to the forward direction of impact. The charge induces a potential at a point P, which is at distance d from the particle. This model is shown in Fig. 11. Treating the problem in spherical coordinates (r,h,/), the potential, p(u1,h2), induced at P is given by: pðh1 ; h2 Þ ¼
Z
h2
Z
h1
2p
0
ra2 sinhdhd/ ; 4pe0 R
ð4Þ
thus: pðh1 ; h2 Þ ¼
ra2 2e0
Z
h2
h1
sinhdh : R
We may substitute: R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ ða þ dÞ2 2aða þ dÞcosh:
Integration then yields: pðh1 ; h2 Þ ¼
npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra d 2 þ 2aða þ dÞð1 cosh2 Þ 2e0 ða þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio d 2 þ 2aða þ dÞð1 cosh1 Þ : ð5Þ
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T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
Fig. 6. Results of the impact charging experiments with 100, 200 and 300 Am PS particles under pressure of 100 and 200 torr.
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
19
Fig. 7. A SEM photograph of the nickel plated PE particle.
If the charge is spread uniformly over the surface, then h2=k, h1=0. Substituting these values yields the well-known expansion for a uniform sphere: pð0; pÞ ¼
ra2 : e0 ða þ dÞ
ð6Þ
Including the effect of the mirror image (d = 2d), the surface potential on the top of the particle (d = 0), Vp, is thus given as: Vp ðdÞ ¼ pðh1 ; h2 ÞAd¼0 pðh1 ; h2 ÞAd¼2d :
ð7Þ
The expression in Eq. (7) was used to calculate the relationship between the impact charge and an initial charge, which is carried as a band of charge distributed over a solid angle of 10j and with a varying inclination to the direction of impact. The properties of the particles
Fig. 9. Results of the impact charging experiment with gold and platinum target compared to the results with stainless steel target.
Fig. 8. Result of the impact charging experiments with surface conductive particles.
used in the experiments were assumed and the result for a particle of 200 Am in diameter is shown in Fig. 12. The case of 0j corresponds to the charge being carried at the point of contact. In this case the effect is a maximum and the relationship has a slope of 45j. At the other extreme, the case of 180j corresponds to a spot remote from the contact. It is surprising that this charge has no effect and the particle behaves as though it is initially neutral. For comparison, the relationship for a uniformly charged particle is also plotted in Fig. 12. For all of the cases where the band of charge is in a forward direction of less than 35j, the particle looses some of its initial charge but does not fully attain the equilibrium charge. All of the lines pass, of course, through the point representing the neutral particle. It is interesting that the expected relationship between the acquired and the initial charge is
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T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
Fig. 10. A schematic illustration of a band of charge distributed.
always linear, regardless of how it is distributed. This implies that the relationship will be linear even for particles carrying multiple, randomly distributed charges, and that the effective transfer will lie between the two limits of the forward and backward spots. This is illustrated in Fig. 13, which illustrates schematically how the impact charge can be expected to deviate substantially from the case of the uniformly charged particle, depending upon the distribution of the initial charge on the surface of the particle. With this postulate in mind, these two limits were superimposed on the results of Fig. 6(a – f). It is seen that, indeed, almost all of the data is contained between these two limits and is distributed on both sides of the prediction for the uniformly charged particle. In order to illustrate this more clearly, the scattered data of Fig. 6(a – f) have been collated in the form of histograms, which show the distribution of impact charge in each case. Because the impact charge and the initial charge are always linearly related, the impact charge has been normalized by the maximum value possible, that due to the spot change at the point of contact. This normalisation is illustrated in Fig. 14 and enables a single histogram to be plotted for
Fig. 11. Coordinates for a surface integration for a partially charged particle.
Fig. 12. An example of discrete estimations of the effect of the localized charge.
each sequence of events. The results may be compared with those predicted by the band of charge model, which are shown in Fig. 15. It is premature to generalize from these few results but it does seem that the results for the smaller particles, 100 Am, fit better the sharp distribution, which is expected from this model and which might be expected from a single spot charge, which is randomly distributed. In this case, the chance of a localized charge in the forward direction is relatively large and all of the data is contained within the prescribed limits. As the particles become larger, the normalized charge is more randomly distributed and this might imply that the larger particles can carry more than one spot of charge. With the largest particles, 300 Am, there is also a significant fraction of occasions when the particle seems to gain more charge than would be gained by a neutral particle
Fig. 13. A schematic illustration of a concept of localization of initial charge and of two limiting situation.
T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
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Fig. 14. Distribution of the normalized impact charging experiments with 100, 200 and 300 Am PS particles under pressure of 100 and 200 Torr, from the results in Fig. 7.
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T. Matsuyama et al. / Powder Technology 135 – 136 (2003) 14–22
(pc)c
critical product of gas pressure and separation gap for the minimum breakdown potential (mm torr) p(h1,h2) surface potential of particle for a band of charge from h1 to h2 (V) q amount of charge (C) R distance (m) Vmin Paschen’s minimum breakdown potential (V) Vp surface potential on the top of the particle (V) Vs gas breakdown potential (V) e0 permittivity of vacuum (F/m) h angle (rad) r surface charge density (C/m2) (r,h,/) spherical coordinates
Acknowledgements Fig. 15. The histogram of the normalized impact charge predicted by the band of charge model.
which leads to apparently negative normalized values. This is a real result, which requires further investigation.
6. Conclusions The insight gained from those previous experiments strengthens the belief that multiple experiments on single particles give the maximum insight into the mechanisms, which dictate the particle properties. In this case, both the charge acquired during an impact and its distribution are relevant. The experiments reported support further the charge relaxation mechanism and confirm that it is applicable to particles as small as 100 Am. The distribution of the acquired charge demonstrates that it may remain localised on the surface of an insulating particle for a considerable time. This fact is very relevant to the design of processes, which are dependent upon the Coulomb attractive force. Nomenclature a radius of particle (m) d contact gap (m) p gas pressure (torr)
The authors are grateful to International Fine Particle Research Institute (IFPRI) for financial support of this project. We also appreciate the cooperation of Professor Nakajima, Hokkaido University, who kindly provided the nickel plated particles.
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