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Analysis of the electrostatic force on a dielectric particle with partial charge distribution
Journal of Electrostatics 67 (2009) 686–690
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Analysis of the electrostatic force on a dielectric particle with partial charge distribution Boonchai Techaumnat a, *, Tadasu Takuma b a b
Department of Electrical Engineering, Chulalongkorn University, Phyathai Road, Pathumwan, Bangkok, Thailand Tokyo Denki University, Japan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 January 2008 Received in revised form 22 January 2009 Accepted 7 March 2009 Available online 20 March 2009
This paper presents the analysis of the electrostatic force acting on a charged dielectric particle on a grounded plane. The force has been determined by a numerical field calculation method to make clear the effect of particle dielectric constant and charge distribution on the particle surface. The charge is treated to be distributed in three ways: (a) uniformly over entire surface, (b) partially on the upper, or (c) on the lower part of a particle. The calculation results show that, if a particle with dielectric constant 3p ¼ 3 is partially charged on the lower part by a zenith angle p/2, p/4 and p/8, the force shall be higher by 0.7, 4.3 and 20 times, respectively, than that for a uniform charging with the same charge amount. On the other hand, the force becomes weaker when charge is on the upper part. The effect of the particle dielectric constant is found to be dependent on the charge distribution. With charge uniform on the entire surface or on the upper part, the force always increases with the dielectric constant. However, when surface charge is restricted to a small area at the lower part of the particle (qq < p/4), the force may decrease with increasing the dielectric constant. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Charged particle Adhesive force Boundary element method Electric field Surface charge
1. Introduction Charged dielectric particles are used in many applications of electrostatics such as powder coating and electrophotography. Electrostatic force acting on the particles in such applications plays an important role in their performance. For example, the detachment of charged toner particles from an electrode is done by applying electric field which must supersede the adhesive force between the particles and the electrode. A simplified configuration of an isolated, charged spherical particle above a grounded plane is commonly applied in the analysis of the force on the particle. The attractive force Fa between the particle and the plane is often estimated simply by the method of images as
Fa ¼
Q2 16p3m 30 ðR þ dÞ2
(1)
where R is the radius of the particle, d the height of the particle from the grounded plane, Q total charge on the particle, 3m dielectric constant (relative permittivity) of the surrounding medium, and 30 the permittivity of free space. This equation is the
* Corresponding author. E-mail address: [email protected] (B. Techaumnat). 0304-3886/$ – see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2009.03.004
force between a charge Q at the particle center at a height h ¼ R þ d above the conducting plane and its image charge Q induced the plane. (Thus, the separation between the charges is equal to 2h). It gives the exact solution only when a uniform surface charge density rs ¼ Q =ð4pR2 Þ is on the sphere surface and the effect of the particle dielectric constant 3p, usually different from 3m, is neglected. An accurate analysis of the electric and electrostatic force on a dielectric particle for various values of d was done by Davis [1] using bispherical coordinates and by Fowlkes and Robinson using multipole images [2]. These analytical works show that the effect of the dielectric constant of the particle is negligible only when the particle is far away enough from the plane (about 10 times of particle diameter). For a particle closer to the plane, the force is greatly enhanced by increasing the dielectric constant 3p. For a constant 3p, the force increases with decreasing d, but converges to an upper limit as d approaches zero. A real-number parameter a has been incorporated into Eq. (1) to include such effect of polarization on the adhesion (increasing Fa) for a range of 3p, primarily in the case of uniform charging [3]. Measurement of the adhesive force on toners usually gives much larger values than that given by Eq. (1) [3]. A charge-patch model, which assumes nonuniform distribution of charge on the particle surface, has been proposed to explain the difference between the analytical and experimental results [3,4]. In this model, force is determined by simply assuming that the charge-patch and the
B. Techaumnat, T. Takuma / Journal of Electrostatics 67 (2009) 686–690
grounded plane form a parallel-plate capacitor [3]. However, this model provides only a rough approximation which takes into account neither curved surfaces nor the dielectric constant of the particle. A few numerical computations have been performed by the Galerkin-type Finite Element Method (FEM) for nonuniform charge distributions. One of them expresses the surface charge density as a linear sum of Legendre functions [5], which makes it rather difficult to apply the result to practical charge distributions. Another paper considers only the specific case of dumb-bell type charge distribution [6], in which a particle is charged in the angle range of 10 on the both sides of the particle. In this paper, we study the behavior of the adhesive electrostatic force between a charged dielectric particle and a ground plane when the particle is uniformly charged on a portion of its surface. We focus on more fundamental configurations, in which the surface charge is either on the lower part or on the upper part of the particle. Instead of using the simple capacitor model, we apply the boundary element method (BEM) to calculate the electric field on the particle surface. With the BEM, which is a numerical method, we can obtain the field solution for various conditions of charge distribution and particle dielectric constant. Furthermore, the BEM usually enables more accurate calculation than the FEM for such arrangements with very narrow regions between a particle and an electrode where electric field is often concentrated. The force is then determined from the Maxwell stress due to the electric field on the particle surface. The objective of this study is to clarify the effect of the partial charge distribution on the particle as well as the role of the particle dielectric constant on the attractive force. Note that we confine our interest here to the electrostatic force based on continuous representation of charge by the surface charge density. Other discussion on nonelectrostatic force and the other studies on toner adhesion may be referred to [7] and the references therein. The idea of proximity force has also been presented in several papers to explain the difference between the theoretical prediction and the experimental results [8].
2. Configuration of analysis and parameters We consider the configuration of a spherical, dielectric particle located on a grounded plane as shown in Fig. 1. The particle, having a radius R, is either (a) charged uniformly over its surface, (b) partially charged at the top or (c) partially at the bottom. These charging conditions are schematically shown in Fig. 2. For the case of the partially charged particle, qq denotes the boundary angle of the charged area. In all cases, the surface charge density is treated to be constant (¼rs) on the charged area. The configuration of charge on the lower part represents cases in which a large amount of charge accumulates on a small portion of the particle surface. By the electric force, this charged region tended to be attracted so as to
687
be in contact with the plane. On the other hand, the charging model in Fig. 2(b) is used to study the variation of force if the charged region is far from the contact point and the plane. The dielectric constant of the particle is denoted by 3p, and that of the surrounding medium is assumed to be unity (free space) for simplicity. On the surface of the particle, the boundary conditions are as follows: 1. Continuity of potential.
fext ¼ fint ;
(2)
where f is the potential and the superscripts ‘ext’ and ‘int’ are used to indicate that the value is on the exterior or the interior side of the surface, respectively. 2. Gauss’s law. With the normal component En of the electric field on the particle surface,
Enext 3p Enint ¼
rs on the charged area; and 30
¼ 0
(3)
on the uncharged area
int where rs is the surface charge density. In this equation, Eext n and En are considered to be positive in the direction from the interior to the exterior of the particle. When the particle partially charged on the upper part approaches the grounded plane, its position may become unstable and not exist in practice if the particle shape is perfectly spherical. This is because if the particle is slightly rotated in the q direction from the position in Fig. 2(b), the net electrostatic force will have both vertical and horizontal components and attract the charged portion to the conducting plane. However, it is still worth seeing the variation of the force with charge position. In the numerical analysis, we use the following values of qq and 3p:
qq ¼ p/2, p/3, p/4, p/6, and p/8, 3p ¼ 1, 3 and 5 (in some cases, also 2 or 4). This range of 3p includes the dielectric constant of toner particles, which is usually between 3 and 4.
3. Calculation method The boundary element method (BEM) for axisymmetrical (AS) arrangements is used for the calculation [9]. Owing to the axisymmerty of the configuration, the boundary of the particle can be represented by the contour q ¼ 0–p (r ¼ R), which is subdivided into line elements in the BEM. At any point x on each element, we express the potential f(x) as
fðxÞ ¼
X
fi Nif ðxÞ
(4)
i
and the outward normal component of the electric field En(x) as
En ðxÞ ¼
X
En;i NiE ðxÞ
(5)
i
Fig. 1. Configuration of a dielectric particle resting on a grounded plane.
where Nfi and NEi are dimensionless functions of the position x on the element, which are used to interpolating f and En from the values fi and En,i at node i, respectively. The values of fi and En,i are unknown and to be determined by a system of linear equations:
688
B. Techaumnat, T. Takuma / Journal of Electrostatics 67 (2009) 686–690
Fig. 2. Charging configurations of the dielectric particle: (a) uniformly charged, (b) partially charged on the upper part, and (c) partially charged on the lower part.
C int ðrÞfi ¼
Z
Enint ðxÞwðr; xÞdS þ
S
Z
vwðr; xÞ dS vn
fðxÞ
(6)
S
4. Results and discussion 4.1. Electric field
for the interior of the particle, and
C
ext
ðrÞfi ¼
Z
Enext ðxÞwðr; xÞdS
S
þ
Z
vwðr; xÞ fðxÞ dS vn
(7)
S
for the exterior of the particle. In the above equations, r is the position of the node i, w is a function that satisfies Laplace’s equation everywhere except at r, v=vn is the derivative in the normal direction outward from the region under consideration at r, and S is the particle surface including its image with respect to the grounded plane in Fig. 1. The constants Cint(r) ¼ Cext(r) ¼ 1/2 on smooth surface. The fundamental solution w is defined in axisymmetrical arrangements for r(r, z) and x(a, b) as
wðr; xÞ ¼
K
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nðm þ nÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi mþn
(8)
where K is the elliptic integral of the first kind, m ¼ r2 þ a2 þ ðz bÞ2 , and n ¼ 2ra. Eqs. (4)–(8) are the main equations for the BEM code utilized in this work. More details on the boundary element method can be found in Ref. [9]. On the surface of a charged particle, two sets of variables (fi and En,i) are used separately for the interior and exterior sides of the surface, respectively. The boundary conditions in Eqs. (2) and (3) are applied to relate the variables on these meshes. In the numerical field calculation, second-order curved elements are used to realize a high accuracy of the results. The potential and field on the elements are also interpolated by second-order polynomial functions. We subdivide the surface contour into 720 secondorder elements, i.e. 1441 nodes exist from the top to the bottom of the particle. (Nodes are at both ends and the middle point of the elements.) A small separation is inserted between the particle and the grounded plane to avoid a triple junction problem [10]. A preliminary calculation was performed to confirm that the separation ( ¼ 5 104 R) used here is sufficient for obtaining the converged value of force that acts on the particle when it is on the grounded plane. After obtaining the field solutions via the BEM, we can determine the force using an integral of the Maxwell stress as [11]
Fa ¼ 3E 30
Z
1 EEn E2 n dS; 2
Fig. 3 shows the values of 3pEint n versus angle on the interior side of the particle surface for the cases of a uniformly charged particle, rs/30 ¼ 1 V/m and 3p ¼ 1, 3, and 5. The normalizing factor is E0 ¼ Q/(4p30R2). It can be seen from the figure that electric field is concentrated near the bottom pole, which is closest to the grounded plane, as expected. The peak electric field at the bottom pole is significantly intensified by the dielectric constant of the particle, as also indicated by Davis [1]. As rs/30 ¼ 1 V/m, we have from Eq. (3) that Eext n on the exterior side is positive for all q. Hence, the field direction is outward from the surface charge. On the interior side, the direction is outward from the surface charge for q (measured from the bottom pole) larger than an angle q0, which is approximately the same (about 52 ) for the range of 3p used here. For smaller q, the field is inward to the surface charge on the interior side. Fig. 4 shows an example of the normalized electric field on a particle that is partially charged on the bottom part for qq ¼ p/4, 3p ¼ 3 and rs / 30 ¼ 1 V/m. As the field is highly nonuniform, the field vectors are normalized to have the same length, and the field strength is indicated by the grey scale in each figure. With charge only on a partial portion of the particle surface, there exists an abrupt change in electric field at q ¼ qq, where charge density changes from rs to zero. Fig. 4 shows that the magnitude of the field is comparatively small on the uncharged portion. Similarly to the
(9)
S
where S is the particle surface, n is the unit normal vector, and 3E is the dielectric constant of the surrounding medium, which is equal to unity for air in our analysis.
Fig. 3. Normalized electric field (multiplied by dielectric constant) on the interior side of the particle surface having uniform charge distribution, rs ¼ 30.
B. Techaumnat, T. Takuma / Journal of Electrostatics 67 (2009) 686–690
689
Fig. 4. Electric field on a dielectric particle charged on the lower part for qq ¼ p/4, 3p ¼ 3 and rs/30 ¼ 1 V/m: (a) exterior and (b) interior.
case of a uniformly charged particle, the electric field in the interior is outward from the surface charge for q > q0, and inward near the bottom pole. As the angle q0 here is much smaller than that in Fig. 3, electric field is highly concentrated at the pole. It should also be noted that the field is given in Figs. 3 and 4 for the same charge density rs. If the same total charge is taken, the electric field shall be significantly stronger for the case of a partially charged particle in Fig. 4 than that in Fig. 3. 4.2. Force As already mentioned, we are interested in the adhesive force that attracts the particle to the grounded plane for three configurations of charge: (a) uniform over the entire surface, (b) only on the upper part and (c) only on the lower part of the particle. Our numerical calculation results for the case of uniform charge agree well with that by the analytical method of Ref. [1]. The following formula is used to express the attractive force when a uniformly charged particle rests on a grounded plane [3],
Q2 Fa ¼ a ; 4p3m 30 ð2RÞ2
(10)
where Q is the total charge amount and a is a constant representing the effect of polarization. From our calculation results, the factor a ¼ 1.0, 1.29, 1.61, and 1.95 for 3p ¼ 1, 2, 3, and 4, respectively. The result for 3p ¼ 4 agrees well with a ¼ 1.9 given in Ref. [3]. In Fig. 5(a) and (b), the attractive force Fa is shown for the case of charge on the upper and lower parts, respectively, for various values of qq. In both figures, the case of uniform charge is given as qq ¼ p for comparison. Fa is given per the square of V0 ¼ Q/(4p30R), the potential on the particle surface in the absence of the grounded plane. (That is, the force is displayed on the graph for the particle having the same total charge.) It is clear from Fig. 5 that the force magnitude depends highly on the spatial distribution of charge. The force is stronger when charge is located on the lower half, but weaker when charge is on the upper half. With charge on the upper part of the particle [Fig. 5(a)], the force decreases with decreasing qq since charge amount is positioned farther from the grounded plane. Fa converges to the solution of the corresponding case of a point charge at the top pole of the particle, in which Fa ¼ Q 2 =ð64p30 R2 Þ if 3p ¼ 1. The convergence can be observed from Fig. 5(a) at qq about p/8. With charge on the upper part, the figure shows that the increase of 3p consistently increases
Fig. 5. Force on the sphere with charge (a) on the upper part and (b) on the lower part of the particle.
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B. Techaumnat, T. Takuma / Journal of Electrostatics 67 (2009) 686–690 Table 1 Attractive force on a typical toner particle charged by 10 mC/g on the lower part.
qq p p/2 p/4 p/8
Attractive force, Fa (nN)
3p ¼ 1
3p ¼ 3
3p ¼ 5
2.46 5.31 20.7 90.6
3.96 6.92 21.0 81.2
5.66 8.83 22.6 78.6
The particle radius is 5-mm, mass density is 1000-kg/m3.
Fig. 6. Variation of vertical force with the boundary angle qq for the case of charge on the lower part of the sphere.
Fa. By contrast, if we consider that the charge and the plane form a system of capacitor, in which the dielectric constant is equal to 3p, the attractive force Facap per unit area is a function of 3p, voltage difference Vcap and separation D as cap
Fa
¼
1 Vcap 2 23p D
(11)
Hence, the force will decrease with increasing 3p. The reason for this disparity is that Fa shown here is not only the force that acts directly to the charge but also includes that on the particle because of polarization. In Fig. 5(b) for the case of charge on the lower part of the particle, the behavior of the force is more complicated than that of Fig. 5(a). At a fixed value of 3p, a decrease in qq increases the force. This is obviously due to the presence of larger amount of charge near the grounded plane. For example, when qq ¼ p/8, the force is approximately 35 or 13 times higher than that on a uniformly charged particle for 3p ¼ 1 or 5, respectively. The variation of force with 3p depends on the boundary angle qq. For large qq (greater than about p/ 4), the force increases with increasing 3p, which is the same as that in the case of charge on the upper part. However, Fa decreases with increasing 3p for qq smaller than about p/6. The results exhibit the effect of the particle dielectric constant, which cannot be predicted by a simple charge-patch model. In addition, it should be noted that if the distribution of surface charge is more complicated (for example, the dumb-bell distribution assumed in Ref. [6]), Fig. 5 indicates that the charge portion at the bottom pole will have a predominant role in determining the attractive force behavior. Fig. 6 presents the force when the charge is on the lower part of the particle with qq as the abscissa. As Fa is shown on logarithmic scales, the linear increase with decreasing qq means that the force becomes singular as qq decreases to zero, independent of 3p. Needless to say, qq ¼ 0 corresponds to the configuration of a point charge at an infinitesimal distance from a grounded plane. From the aforementioned calculation results, we may estimate the attractive force in a practical case of toner particles in laser printing. Consider a typical toner particle of 5-mm radius and 1000kg/m3 mass density. The toner is assumed to be charged to 10 mC/g. If the charge distribution is uniform and the effect of 3p is neglected, the estimation by Eq. (1) gives Fa ¼ 2.46 nN, which is about one order of magnitude smaller than experimental values [3]. Table 1 summarizes the force on this toner particle for various values of 3p and qq on the lower part (except the case qq ¼ p, corresponding to
a uniform charge distribution). With the charge distribution still being uniform but the effect of the dielectric constant 3p ¼ 3 taken into account, Fa becomes 60% higher, i.e. 3.96 nN. If both the dielectric constant and the partial distribution of charge are assumed, the force will be about as high as 7.0, 21 and 81 nN, respectively, for qq ¼ p/2, p/4 and p/8. Therefore, the difference between the measurement results and the analysis [4,12] is possibly due to the spatial distribution of charge on the toner particles. 5. Conclusion From the numerical results, we have shown that the force attracting a charged dielectric particle to a grounded plane can vary greatly with the distribution of the charge on particle surface. That is, the force is stronger when the charge exists on the lower part of the particle, and it is weaker when the charge exists on the upper part of the particle. The increase in the dielectric constant of the particle results in stronger force in most cases. However, when charge is concentrated on a small area near the bottom pole of the particle, force may decrease with increasing the dielectric constant. The results presented here could explain the discrepancy between the measured force and the analytical one in applications such as toners in electrophotography. Acknowledgments The authors would like to thank Dr. Masami Kadonaga at Ricoh Company for his discussion on the practical data of toners. References [1] M.H. Davis, Electrostatic field and force on a dielectric sphere near a conducting plane – a note on the application of electrostatic theory to water droplets, Am. J. Phys. 37 (1969) 26–29. [2] Wm.Y. Fowlkes, K.S. Robinson, The electrostatic force on a dielectric sphere resting on a conducting substrate, in: K.L. Mittal (Ed.), Particles on Surfaces 1: Detection, Adhesion and Removal, Plenum Press, New York, 1988, pp. 143–155. [3] D.A. Hays, Toner adhesion, J. Adhes. 51 (1995) 41–48. [4] D.A. Hays, Electric field detachment of charged particles, in: K.L. Mittal (Ed.), Particles on Surfaces 1: Detection, Adhesion and Removal, Plenum Press, New York, 1988, pp. 351–360. [5] J.Q. Feng, E.A. Eklund, D.A. Hays, Electric field detachment of a nonuniformity charged dielectric sphere on a dielectric coated electrode, J. Electrostat. 40 & 41 (1997) 289–294. [6] J.Q. Feng, D.A. Hays, Theory of electric field detachment of charged toner particles in electrophotography, J. Imaging Sci. Technol. 44 (2000) 19–25. [7] M.C. Dejesus, D.S. Rimai, E. Stelter, T.N. Tombs, D.S. Weiss, Adhesion of silicacoated particles to bisphenol-A polycarbonate films, J. Imaging Sci. Technol. 52 (2008) 010503. [8] L.B. Schein, W.S. Czarnecki, Proximity theory of toner adhesion, J. Imaging Sci. Technol. 48 (2004) 412–416. [9] C.A. Brebbia, S. Walker, The Boundary Element Technique in Engineering, Butterworths Publishing, 1980. [10] T. Takuma, Field behavior at a triple junction in composite dielectric arrangements, IEEE Trans. Elect. Insul. 26 (1991) 500–509. [11] D.J. Griffiths, Introduction to Electrodynamics, Prentice Hall, NJ, 1989. [12] M. Takeuchi, Adhesion forces of charged particles, Chem. Eng. Sci. 61 (2006) 2279–2289.
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Analysis of the electrostatic force on a dielectric particle with partial charge distribution Boonchai Techaumnat a, *, Tadasu Takuma b a b
Department of Electrical Engineering, Chulalongkorn University, Phyathai Road, Pathumwan, Bangkok, Thailand Tokyo Denki University, Japan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 January 2008 Received in revised form 22 January 2009 Accepted 7 March 2009 Available online 20 March 2009
This paper presents the analysis of the electrostatic force acting on a charged dielectric particle on a grounded plane. The force has been determined by a numerical field calculation method to make clear the effect of particle dielectric constant and charge distribution on the particle surface. The charge is treated to be distributed in three ways: (a) uniformly over entire surface, (b) partially on the upper, or (c) on the lower part of a particle. The calculation results show that, if a particle with dielectric constant 3p ¼ 3 is partially charged on the lower part by a zenith angle p/2, p/4 and p/8, the force shall be higher by 0.7, 4.3 and 20 times, respectively, than that for a uniform charging with the same charge amount. On the other hand, the force becomes weaker when charge is on the upper part. The effect of the particle dielectric constant is found to be dependent on the charge distribution. With charge uniform on the entire surface or on the upper part, the force always increases with the dielectric constant. However, when surface charge is restricted to a small area at the lower part of the particle (qq < p/4), the force may decrease with increasing the dielectric constant. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Charged particle Adhesive force Boundary element method Electric field Surface charge
1. Introduction Charged dielectric particles are used in many applications of electrostatics such as powder coating and electrophotography. Electrostatic force acting on the particles in such applications plays an important role in their performance. For example, the detachment of charged toner particles from an electrode is done by applying electric field which must supersede the adhesive force between the particles and the electrode. A simplified configuration of an isolated, charged spherical particle above a grounded plane is commonly applied in the analysis of the force on the particle. The attractive force Fa between the particle and the plane is often estimated simply by the method of images as
Fa ¼
Q2 16p3m 30 ðR þ dÞ2
(1)
where R is the radius of the particle, d the height of the particle from the grounded plane, Q total charge on the particle, 3m dielectric constant (relative permittivity) of the surrounding medium, and 30 the permittivity of free space. This equation is the
* Corresponding author. E-mail address: [email protected] (B. Techaumnat). 0304-3886/$ – see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2009.03.004
force between a charge Q at the particle center at a height h ¼ R þ d above the conducting plane and its image charge Q induced the plane. (Thus, the separation between the charges is equal to 2h). It gives the exact solution only when a uniform surface charge density rs ¼ Q =ð4pR2 Þ is on the sphere surface and the effect of the particle dielectric constant 3p, usually different from 3m, is neglected. An accurate analysis of the electric and electrostatic force on a dielectric particle for various values of d was done by Davis [1] using bispherical coordinates and by Fowlkes and Robinson using multipole images [2]. These analytical works show that the effect of the dielectric constant of the particle is negligible only when the particle is far away enough from the plane (about 10 times of particle diameter). For a particle closer to the plane, the force is greatly enhanced by increasing the dielectric constant 3p. For a constant 3p, the force increases with decreasing d, but converges to an upper limit as d approaches zero. A real-number parameter a has been incorporated into Eq. (1) to include such effect of polarization on the adhesion (increasing Fa) for a range of 3p, primarily in the case of uniform charging [3]. Measurement of the adhesive force on toners usually gives much larger values than that given by Eq. (1) [3]. A charge-patch model, which assumes nonuniform distribution of charge on the particle surface, has been proposed to explain the difference between the analytical and experimental results [3,4]. In this model, force is determined by simply assuming that the charge-patch and the
B. Techaumnat, T. Takuma / Journal of Electrostatics 67 (2009) 686–690
grounded plane form a parallel-plate capacitor [3]. However, this model provides only a rough approximation which takes into account neither curved surfaces nor the dielectric constant of the particle. A few numerical computations have been performed by the Galerkin-type Finite Element Method (FEM) for nonuniform charge distributions. One of them expresses the surface charge density as a linear sum of Legendre functions [5], which makes it rather difficult to apply the result to practical charge distributions. Another paper considers only the specific case of dumb-bell type charge distribution [6], in which a particle is charged in the angle range of 10 on the both sides of the particle. In this paper, we study the behavior of the adhesive electrostatic force between a charged dielectric particle and a ground plane when the particle is uniformly charged on a portion of its surface. We focus on more fundamental configurations, in which the surface charge is either on the lower part or on the upper part of the particle. Instead of using the simple capacitor model, we apply the boundary element method (BEM) to calculate the electric field on the particle surface. With the BEM, which is a numerical method, we can obtain the field solution for various conditions of charge distribution and particle dielectric constant. Furthermore, the BEM usually enables more accurate calculation than the FEM for such arrangements with very narrow regions between a particle and an electrode where electric field is often concentrated. The force is then determined from the Maxwell stress due to the electric field on the particle surface. The objective of this study is to clarify the effect of the partial charge distribution on the particle as well as the role of the particle dielectric constant on the attractive force. Note that we confine our interest here to the electrostatic force based on continuous representation of charge by the surface charge density. Other discussion on nonelectrostatic force and the other studies on toner adhesion may be referred to [7] and the references therein. The idea of proximity force has also been presented in several papers to explain the difference between the theoretical prediction and the experimental results [8].
2. Configuration of analysis and parameters We consider the configuration of a spherical, dielectric particle located on a grounded plane as shown in Fig. 1. The particle, having a radius R, is either (a) charged uniformly over its surface, (b) partially charged at the top or (c) partially at the bottom. These charging conditions are schematically shown in Fig. 2. For the case of the partially charged particle, qq denotes the boundary angle of the charged area. In all cases, the surface charge density is treated to be constant (¼rs) on the charged area. The configuration of charge on the lower part represents cases in which a large amount of charge accumulates on a small portion of the particle surface. By the electric force, this charged region tended to be attracted so as to
687
be in contact with the plane. On the other hand, the charging model in Fig. 2(b) is used to study the variation of force if the charged region is far from the contact point and the plane. The dielectric constant of the particle is denoted by 3p, and that of the surrounding medium is assumed to be unity (free space) for simplicity. On the surface of the particle, the boundary conditions are as follows: 1. Continuity of potential.
fext ¼ fint ;
(2)
where f is the potential and the superscripts ‘ext’ and ‘int’ are used to indicate that the value is on the exterior or the interior side of the surface, respectively. 2. Gauss’s law. With the normal component En of the electric field on the particle surface,
Enext 3p Enint ¼
rs on the charged area; and 30
¼ 0
(3)
on the uncharged area
int where rs is the surface charge density. In this equation, Eext n and En are considered to be positive in the direction from the interior to the exterior of the particle. When the particle partially charged on the upper part approaches the grounded plane, its position may become unstable and not exist in practice if the particle shape is perfectly spherical. This is because if the particle is slightly rotated in the q direction from the position in Fig. 2(b), the net electrostatic force will have both vertical and horizontal components and attract the charged portion to the conducting plane. However, it is still worth seeing the variation of the force with charge position. In the numerical analysis, we use the following values of qq and 3p:
qq ¼ p/2, p/3, p/4, p/6, and p/8, 3p ¼ 1, 3 and 5 (in some cases, also 2 or 4). This range of 3p includes the dielectric constant of toner particles, which is usually between 3 and 4.
3. Calculation method The boundary element method (BEM) for axisymmetrical (AS) arrangements is used for the calculation [9]. Owing to the axisymmerty of the configuration, the boundary of the particle can be represented by the contour q ¼ 0–p (r ¼ R), which is subdivided into line elements in the BEM. At any point x on each element, we express the potential f(x) as
fðxÞ ¼
X
fi Nif ðxÞ
(4)
i
and the outward normal component of the electric field En(x) as
En ðxÞ ¼
X
En;i NiE ðxÞ
(5)
i
Fig. 1. Configuration of a dielectric particle resting on a grounded plane.
where Nfi and NEi are dimensionless functions of the position x on the element, which are used to interpolating f and En from the values fi and En,i at node i, respectively. The values of fi and En,i are unknown and to be determined by a system of linear equations:
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Fig. 2. Charging configurations of the dielectric particle: (a) uniformly charged, (b) partially charged on the upper part, and (c) partially charged on the lower part.
C int ðrÞfi ¼
Z
Enint ðxÞwðr; xÞdS þ
S
Z
vwðr; xÞ dS vn
fðxÞ
(6)
S
4. Results and discussion 4.1. Electric field
for the interior of the particle, and
C
ext
ðrÞfi ¼
Z
Enext ðxÞwðr; xÞdS
S
þ
Z
vwðr; xÞ fðxÞ dS vn
(7)
S
for the exterior of the particle. In the above equations, r is the position of the node i, w is a function that satisfies Laplace’s equation everywhere except at r, v=vn is the derivative in the normal direction outward from the region under consideration at r, and S is the particle surface including its image with respect to the grounded plane in Fig. 1. The constants Cint(r) ¼ Cext(r) ¼ 1/2 on smooth surface. The fundamental solution w is defined in axisymmetrical arrangements for r(r, z) and x(a, b) as
wðr; xÞ ¼
K
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nðm þ nÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi mþn
(8)
where K is the elliptic integral of the first kind, m ¼ r2 þ a2 þ ðz bÞ2 , and n ¼ 2ra. Eqs. (4)–(8) are the main equations for the BEM code utilized in this work. More details on the boundary element method can be found in Ref. [9]. On the surface of a charged particle, two sets of variables (fi and En,i) are used separately for the interior and exterior sides of the surface, respectively. The boundary conditions in Eqs. (2) and (3) are applied to relate the variables on these meshes. In the numerical field calculation, second-order curved elements are used to realize a high accuracy of the results. The potential and field on the elements are also interpolated by second-order polynomial functions. We subdivide the surface contour into 720 secondorder elements, i.e. 1441 nodes exist from the top to the bottom of the particle. (Nodes are at both ends and the middle point of the elements.) A small separation is inserted between the particle and the grounded plane to avoid a triple junction problem [10]. A preliminary calculation was performed to confirm that the separation ( ¼ 5 104 R) used here is sufficient for obtaining the converged value of force that acts on the particle when it is on the grounded plane. After obtaining the field solutions via the BEM, we can determine the force using an integral of the Maxwell stress as [11]
Fa ¼ 3E 30
Z
1 EEn E2 n dS; 2
Fig. 3 shows the values of 3pEint n versus angle on the interior side of the particle surface for the cases of a uniformly charged particle, rs/30 ¼ 1 V/m and 3p ¼ 1, 3, and 5. The normalizing factor is E0 ¼ Q/(4p30R2). It can be seen from the figure that electric field is concentrated near the bottom pole, which is closest to the grounded plane, as expected. The peak electric field at the bottom pole is significantly intensified by the dielectric constant of the particle, as also indicated by Davis [1]. As rs/30 ¼ 1 V/m, we have from Eq. (3) that Eext n on the exterior side is positive for all q. Hence, the field direction is outward from the surface charge. On the interior side, the direction is outward from the surface charge for q (measured from the bottom pole) larger than an angle q0, which is approximately the same (about 52 ) for the range of 3p used here. For smaller q, the field is inward to the surface charge on the interior side. Fig. 4 shows an example of the normalized electric field on a particle that is partially charged on the bottom part for qq ¼ p/4, 3p ¼ 3 and rs / 30 ¼ 1 V/m. As the field is highly nonuniform, the field vectors are normalized to have the same length, and the field strength is indicated by the grey scale in each figure. With charge only on a partial portion of the particle surface, there exists an abrupt change in electric field at q ¼ qq, where charge density changes from rs to zero. Fig. 4 shows that the magnitude of the field is comparatively small on the uncharged portion. Similarly to the
(9)
S
where S is the particle surface, n is the unit normal vector, and 3E is the dielectric constant of the surrounding medium, which is equal to unity for air in our analysis.
Fig. 3. Normalized electric field (multiplied by dielectric constant) on the interior side of the particle surface having uniform charge distribution, rs ¼ 30.
B. Techaumnat, T. Takuma / Journal of Electrostatics 67 (2009) 686–690
689
Fig. 4. Electric field on a dielectric particle charged on the lower part for qq ¼ p/4, 3p ¼ 3 and rs/30 ¼ 1 V/m: (a) exterior and (b) interior.
case of a uniformly charged particle, the electric field in the interior is outward from the surface charge for q > q0, and inward near the bottom pole. As the angle q0 here is much smaller than that in Fig. 3, electric field is highly concentrated at the pole. It should also be noted that the field is given in Figs. 3 and 4 for the same charge density rs. If the same total charge is taken, the electric field shall be significantly stronger for the case of a partially charged particle in Fig. 4 than that in Fig. 3. 4.2. Force As already mentioned, we are interested in the adhesive force that attracts the particle to the grounded plane for three configurations of charge: (a) uniform over the entire surface, (b) only on the upper part and (c) only on the lower part of the particle. Our numerical calculation results for the case of uniform charge agree well with that by the analytical method of Ref. [1]. The following formula is used to express the attractive force when a uniformly charged particle rests on a grounded plane [3],
Q2 Fa ¼ a ; 4p3m 30 ð2RÞ2
(10)
where Q is the total charge amount and a is a constant representing the effect of polarization. From our calculation results, the factor a ¼ 1.0, 1.29, 1.61, and 1.95 for 3p ¼ 1, 2, 3, and 4, respectively. The result for 3p ¼ 4 agrees well with a ¼ 1.9 given in Ref. [3]. In Fig. 5(a) and (b), the attractive force Fa is shown for the case of charge on the upper and lower parts, respectively, for various values of qq. In both figures, the case of uniform charge is given as qq ¼ p for comparison. Fa is given per the square of V0 ¼ Q/(4p30R), the potential on the particle surface in the absence of the grounded plane. (That is, the force is displayed on the graph for the particle having the same total charge.) It is clear from Fig. 5 that the force magnitude depends highly on the spatial distribution of charge. The force is stronger when charge is located on the lower half, but weaker when charge is on the upper half. With charge on the upper part of the particle [Fig. 5(a)], the force decreases with decreasing qq since charge amount is positioned farther from the grounded plane. Fa converges to the solution of the corresponding case of a point charge at the top pole of the particle, in which Fa ¼ Q 2 =ð64p30 R2 Þ if 3p ¼ 1. The convergence can be observed from Fig. 5(a) at qq about p/8. With charge on the upper part, the figure shows that the increase of 3p consistently increases
Fig. 5. Force on the sphere with charge (a) on the upper part and (b) on the lower part of the particle.
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B. Techaumnat, T. Takuma / Journal of Electrostatics 67 (2009) 686–690 Table 1 Attractive force on a typical toner particle charged by 10 mC/g on the lower part.
qq p p/2 p/4 p/8
Attractive force, Fa (nN)
3p ¼ 1
3p ¼ 3
3p ¼ 5
2.46 5.31 20.7 90.6
3.96 6.92 21.0 81.2
5.66 8.83 22.6 78.6
The particle radius is 5-mm, mass density is 1000-kg/m3.
Fig. 6. Variation of vertical force with the boundary angle qq for the case of charge on the lower part of the sphere.
Fa. By contrast, if we consider that the charge and the plane form a system of capacitor, in which the dielectric constant is equal to 3p, the attractive force Facap per unit area is a function of 3p, voltage difference Vcap and separation D as cap
Fa
¼
1 Vcap 2 23p D
(11)
Hence, the force will decrease with increasing 3p. The reason for this disparity is that Fa shown here is not only the force that acts directly to the charge but also includes that on the particle because of polarization. In Fig. 5(b) for the case of charge on the lower part of the particle, the behavior of the force is more complicated than that of Fig. 5(a). At a fixed value of 3p, a decrease in qq increases the force. This is obviously due to the presence of larger amount of charge near the grounded plane. For example, when qq ¼ p/8, the force is approximately 35 or 13 times higher than that on a uniformly charged particle for 3p ¼ 1 or 5, respectively. The variation of force with 3p depends on the boundary angle qq. For large qq (greater than about p/ 4), the force increases with increasing 3p, which is the same as that in the case of charge on the upper part. However, Fa decreases with increasing 3p for qq smaller than about p/6. The results exhibit the effect of the particle dielectric constant, which cannot be predicted by a simple charge-patch model. In addition, it should be noted that if the distribution of surface charge is more complicated (for example, the dumb-bell distribution assumed in Ref. [6]), Fig. 5 indicates that the charge portion at the bottom pole will have a predominant role in determining the attractive force behavior. Fig. 6 presents the force when the charge is on the lower part of the particle with qq as the abscissa. As Fa is shown on logarithmic scales, the linear increase with decreasing qq means that the force becomes singular as qq decreases to zero, independent of 3p. Needless to say, qq ¼ 0 corresponds to the configuration of a point charge at an infinitesimal distance from a grounded plane. From the aforementioned calculation results, we may estimate the attractive force in a practical case of toner particles in laser printing. Consider a typical toner particle of 5-mm radius and 1000kg/m3 mass density. The toner is assumed to be charged to 10 mC/g. If the charge distribution is uniform and the effect of 3p is neglected, the estimation by Eq. (1) gives Fa ¼ 2.46 nN, which is about one order of magnitude smaller than experimental values [3]. Table 1 summarizes the force on this toner particle for various values of 3p and qq on the lower part (except the case qq ¼ p, corresponding to
a uniform charge distribution). With the charge distribution still being uniform but the effect of the dielectric constant 3p ¼ 3 taken into account, Fa becomes 60% higher, i.e. 3.96 nN. If both the dielectric constant and the partial distribution of charge are assumed, the force will be about as high as 7.0, 21 and 81 nN, respectively, for qq ¼ p/2, p/4 and p/8. Therefore, the difference between the measurement results and the analysis [4,12] is possibly due to the spatial distribution of charge on the toner particles. 5. Conclusion From the numerical results, we have shown that the force attracting a charged dielectric particle to a grounded plane can vary greatly with the distribution of the charge on particle surface. That is, the force is stronger when the charge exists on the lower part of the particle, and it is weaker when the charge exists on the upper part of the particle. The increase in the dielectric constant of the particle results in stronger force in most cases. However, when charge is concentrated on a small area near the bottom pole of the particle, force may decrease with increasing the dielectric constant. The results presented here could explain the discrepancy between the measured force and the analytical one in applications such as toners in electrophotography. Acknowledgments The authors would like to thank Dr. Masami Kadonaga at Ricoh Company for his discussion on the practical data of toners. References [1] M.H. Davis, Electrostatic field and force on a dielectric sphere near a conducting plane – a note on the application of electrostatic theory to water droplets, Am. J. Phys. 37 (1969) 26–29. [2] Wm.Y. Fowlkes, K.S. Robinson, The electrostatic force on a dielectric sphere resting on a conducting substrate, in: K.L. Mittal (Ed.), Particles on Surfaces 1: Detection, Adhesion and Removal, Plenum Press, New York, 1988, pp. 143–155. [3] D.A. Hays, Toner adhesion, J. Adhes. 51 (1995) 41–48. [4] D.A. Hays, Electric field detachment of charged particles, in: K.L. Mittal (Ed.), Particles on Surfaces 1: Detection, Adhesion and Removal, Plenum Press, New York, 1988, pp. 351–360. [5] J.Q. Feng, E.A. Eklund, D.A. Hays, Electric field detachment of a nonuniformity charged dielectric sphere on a dielectric coated electrode, J. Electrostat. 40 & 41 (1997) 289–294. [6] J.Q. Feng, D.A. Hays, Theory of electric field detachment of charged toner particles in electrophotography, J. Imaging Sci. Technol. 44 (2000) 19–25. [7] M.C. Dejesus, D.S. Rimai, E. Stelter, T.N. Tombs, D.S. Weiss, Adhesion of silicacoated particles to bisphenol-A polycarbonate films, J. Imaging Sci. Technol. 52 (2008) 010503. [8] L.B. Schein, W.S. Czarnecki, Proximity theory of toner adhesion, J. Imaging Sci. Technol. 48 (2004) 412–416. [9] C.A. Brebbia, S. Walker, The Boundary Element Technique in Engineering, Butterworths Publishing, 1980. [10] T. Takuma, Field behavior at a triple junction in composite dielectric arrangements, IEEE Trans. Elect. Insul. 26 (1991) 500–509. [11] D.J. Griffiths, Introduction to Electrodynamics, Prentice Hall, NJ, 1989. [12] M. Takeuchi, Adhesion forces of charged particles, Chem. Eng. Sci. 61 (2006) 2279–2289.