Interactive dipole model for two-sphere system

Journal of

ELECTROSTATICS ELSEVIER

Journal of Electrostatics 33 (1994) 385-392

Interactive dipole model for two-sphere system R i c h a r d D. S t o y Department of Electrical Engineering, Widener University, Chester, PA 19013, USA Received 8 November 1993; accepted after revision 1 June 1994

Abstract

Approximate "dipole" equations are derived for the interaction of two uncharged dielectric spheres immersed in a uniform electric field. Each sphere is treated as a point polarizable particle in a locally uniform field whose strength is that which would be present at the sphere's center if the sphere were removed. The response of each sphere is a dipole moment. In this way each sphere sees the uniform external field and the dipole field of the other sphere. These interactive dipole moments and forces are calculated for the two orientations of the spheres relative to the external field. Data are presented which show the range of validity of the results: the exclusion of all higher order multipoles is acceptable only as long as the spheres are separated by at least one radius.

1. Introduction In the last 40 years have witnessed several investigations into the forces exerted on one or two spherical particles immersed in a uniform external field or located near a planar boundary. Lebedev and Skarskaya [1] obtained closed-form equations for the charge and the force on a single conducting sphere which is touching one plate of a parallel-plate capacitor. Levine and McQuarrie [2] considered metallic spheres in bispherical coordinates to model the permittivity of simple gases. Davis [3] found the field and force on a dielectric sphere which is near a grounded conducting plane. In a long review article Goel and Spencer [4] discussed the electric forces on charged spheres and spheroids with conchoidal surfaces. O'Meara and Saville [5] found the force on two touching metal spheres which are located in a uniform field. In a pioneering work Goyette and Navon [6] found the potential and dipole moments in bispherical coordinates for two identical uncharged dielectric spheres which are immersed in a uniform field, provided that the spheres are far apart. Fowlkes and Robinson [7] found the force on a dielectric sphere which is subject to an externally applied electric field which is normal to the plane of the conducting substrate which is touching the sphere. Electrostatic problems involving two arbitrary uncharged dielectric spheres in a uniform field [8, 9] can be solved exactly in bispherical coordinates, 0304-3886/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 3 8 8 6 ( 9 4 ) 0 0 0 1 7-Q

R.D. Stov,'Journal o/" Electrostatics 33 (1994) 385 392

386

provided that the spheres do not touch. Unfortunately the starting point, the solution to Laplace's equation, always involves infinite series whose coefficients cannot be expressed in closed form [8, 9]. When the spheres are relatively close together we must use exact solution procedures to get accurate results for multipole moments, fields and forces. When the spheres are not so close together approximate, closed-form equations are very useful. Here we present a model for two interacting dielectric spheres in a uniform external field. The essence of which is: (1) each sphere is treated as a point polarizable particle in a locally uniform field whose strength is that which would be present at the sphere's center if the sphere were removed; (2) the response of each sphere is a dipole moment, thus each sphere sees the uniform external field and the dipole field of the other sphere. This model is equivalent to neglecting all multipole moments higher than dipole in solving the problem.

2. Theory Here we are using the rationalized MKSA system of units. Sphere 1 of radius R~ and permittivity el and sphere 2 of radius R2 and permittivity e2 are embedded in an infinite medium of permittivity e3. The distance between the centers of the spheres is designated D and the uniform external field is directed along the positive z axis with strength E0. The non-interactive dipole moments [10] of spheres 1 and 2 are designated Pl, and P2a, respectively:

~3 )R~Eo = az4rte3K1Eo, Pla = az47Ze3 \ e elI +--_2e3

(1)

P2~ = a z 4 r t e 3 ( ~ + ~ e 3 ) R 3 E o

(2)

= az4r~e3K2Eo.

Here az is the unit vector in the z direction. Let us momentarily place the origin of a spherical coordinate system at the location of the center of sphere 1 with the uniform field directed in the positive z direction. Then the electric field 1-10] due to and outside of this sphere is calculated in spherical coordinates to be E=Eo

a, cos0

1 + r3 ] + a o s i n 0

-1 +

(3)

where a, and ae are the radial and colatitudinal unit vectors, respectively, of the spherical coordinate system. At any point on the z-axis (0 = 0 or 0 = n and r > R1 ) we find

(

E = azEo 1 q-

r3

;.

(4)

R.D. Stoy/Journal of Electrostatics 33 (1994) 385-392

387

At any point in the plane z = 0 (0 = ~/2 and r > R~) we find

(5) The relevant geometries are shown in Figs. 1 and 2. We now locate sphere 2 at a distance D > R~ + R E from sphere 1 where sphere 2 sees the above field and produces the dipole moment P2b. P2b = az47~e3K2Eo

( ~ 1 +

D3 j ,

(6)

0 = 0,

(7) Sphere 1 now sees sphere 2 and produces the dipole moment Pxc.

pl¢ = a~4us3KiEo Plc = a~4ue3KxEo

[ I + --D~-~ 2K~/~+ 2Kx'd D3 j ], l-D3\

- ~

,

0=0,

(8)

0 = ~/2.

(9)

+z 0=0

£3

()

0 = ~/2

0=Tr -z Fig. 1. Geometry of the parallel orientation, 0 = 0.

388

R.D. Sto~/Journal

of Electrostatics

33 11994)

385-392

?=O

-++-e

e = K/2

%

T

8=X

EO

Fig. 2. Geometry of the perpendicular orientation, 0 = 42.

If we repeat this procedure infinitely many times we get the interactive moments p1 and pz in closed form.

dipole

p1 = a,4m3K,Eo

p2 = a,4m3 K2 E.

p1 = a,4m3K1Eo

p2 = a,4m3 K2 E.

(10) 1 - K21D3 1 - K1K2/D6 l-

’

K1/D3

1 - K1K2/D6 > ’

8 = n/2.

(11)

In this dipole approximation the effective local electric field seen by a sphere is E,, times the quantity in brackets to its right in the equations for the dipole moment. The dipole moments are now self-consistent in that the field produced by either sphere is exactly the value needed to induce the indicated dipole moment in the other sphere.

R.D. Stoy/Journal of Electrostatics 33 (1994) 385-392

389

When the spheres are identical (K1 = K2 = K), the dipole moments have the following forms:

Px=P2=p=az4~e3KEo

1+

D3 ,

0=~/2.

(13)

Equations can now be calculated for the force between the spheres. For the usual case of K > 0 we find that the force is attractive in the parallel orientation (0 = 0) and repulsive in the perpendicular orientation (0 = n/2). The force on a dipole depends on the spatial variation of the external field [101 in the region near the dipole: F = V(p. E).

(14)

If we imagine that sphere 1 is at the origin and sphere 2 is on the positive z axis then the field in the region of sphere 2 is

I 21(1+22jo3)]1

E2 = a, Eo 1 + ~

~_~-4~

The force evaluated at r = D, with Pl F2 =

_

6pip2

a z

47ze3D4 ,

6

= IPl[

,

0 =

(15)

O.

and P2

----[P21,

is

0 = 0.

(16)

The force on sphere 1 is clearly the negative of the force on sphere 2. With sphere 1 remaining at the origin we now place sphere 2 in the plane z = 0. The field in the vicinity of sphere 2 is then

[

K1 ( 1 _ K 2 / D 3 x ~

E2 = azEo 1 - 7

1 ~I(1-K2/D6 ] J '

0 = rt/2.

(17)

The force on sphere 2 is now 3pl "P2

F 2 = a, 47xeaD 4,

0 = rt/2.

(18)

The force on sphere 1 is the negative of the force on sphere 2.

3. Results The chief practical uses for this work are in the closed-form computation of dipole moments of the spheres and the forces between them. Arbitrary spheres present so many parameters which can be varied (D/RI, D/R2, et/e3 and e2/~3) that detailed comparison of the predictions of this work to the results from the exact solution will

R.D. Sto)'/Journal t~/'Electrostatics 33 (1994) 385 392

390

be confined to identical spheres. It is sufficient to state that Eqs. (10), (11), (16) and (18) are the asymptotic equations for dipole moments and forces as D/R1, D/R2 --, ~ . There is less practical interest in the perpendicular case (0 = rt/2) than there is in the parallel case (0 = 0) since an attractive force will move the spheres closer together. Consequently, extensive comparison of the present work to prior exact works [8, 11 ] is further restricted to the parallel orientation. For dipole moments of the spheres the quantities to be tabulated are the normalized dipole moments, i.e. the ratio of the actual dipole moment to the interactive moment as given by Eq. (12): Pactua~/P, and also the ratio of the actual dipole moment to the non-interactive moment from Eq. (1): Pactual/Pla. These data are presented in Table 1 in which the parameter d is the distance between the closest points on the two spheres (d = D - 2R1). As can be seen from the data the interactive model predicts the dipole moment with less than 1% error for d/R~ >~ 1.00. As the spheres approach more closely this interactive model yields dipole moments which contain less than 40% of the error that results from ignoring the interaction. The force between the spheres depends upon the interaction of all the multipoles in sphere 1 with all the multipoles in sphere 2. Only when the spheres are relatively far apart does the majority contribution to this force come from the dipole-dipole interaction. Since the exact solution for the potential [-8, 9-1 is available we can find the

Table 1 Normalized dipole m o m e n t s for two identical spheres

d/R1

el~e, a 0.010

0.100

10.00

100.0

1000.0

101°

1000.00

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.00(30

100.00

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

10.00

1.0000 0.9994

1.0000 0.9995

1.0000 1.0009

1.0000 1.0011

1.0000 1.0012

1.0000 1.0012

1.00

1.0005 0.9653

1.0004 0.9696

1.0011 1.0600

1.0018 1.0794

1.0019 1.0818

1.0019 1.082 I

O.10

1.0096 0.9126

1.0074 0.9221

1.0397 1.2407

1.0781 1.3640

1.0841 1.3816

1.0848 1.3836

0.01

1.0134 0.9037

1.0104 0.9139

1.0824 1.3276

1.2158 1.5977

1.2456 1.6510

1.2492 1.6574

in each pair of data the upper number is the ratio of the exact dipole m o m e n t to that calculated from the interactive model of this work; the lower number is the ratio of the exact dipole m o m e n t to that computed from non-interacting dipoles.

391

R.D. Stoy/Journal of Electrostatics 33 (1994) 385-392 Table 2 Normalized forces for two identical spheres

d/R1

el/e3 0.010

0.100

10.00

100.0

1000.0

101°

1000.00

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

100.00

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

10.00

1.0000 0.9988

1.0000 0.9990

1.0000 1.0018

1.0001 1.0023

1.0001 1.0024

1.0001 1.0024

1.00

0.9589 0.8926

0.9645 0.9061

1.0556 1.1834

1.0708 1.2431

1.0726 1.2504

1.0728 1.2513

0.10

0.7552 0.6169

0.7791 0.6527

1.9858 2.8276

2.6696 4.2733

2.7773 4.5105

2.7905 4.5394

0.01

0.7112 0.5657

0.7365 0.6026

3.8116 5.7344

11.7486 20.2893

14.4100 25.3165

14.7763 26.0108

In each pair of data the upper n u m b e r is the ratio of the exact force to that from (16); while the lower n u m b e r is the ratio of the exact force to that of non-interacting dipoles.

force by numerical integration of the z component of the stress tensor over the surface of a sphere using the following equation from Stratton [10]:

F= ~3(~ IE(E..) - ~]da. d Sphere L

(19)

Here n is the unit outward normal vector and the electric field E is computed just outside the surface of the sphere. The quantities to be tabulated are the normalized forces: the ratio of the actual force from (19) to that found from (16); and also the ratio of the actual force to that between two non-interesting dipoles as given by (1) and (2). These data are presented in pairs as in Table 2. Here the superiority of the interactive model over the non-interactive model is clear. However, neither model accurately predicts the forces when the spheres are separated by less than one radius. The author is firmly convinced that comparison of models should be based on their predictions of physically measurable quantities such as force and not on such inferred quantities as dipole moment.

4. Conclusion

Consideration of the dipole moment to the exclusion of all higher-order moments is acceptable only as long as the spheres are separated by at least one radius. This is

392

R.D. Stoy/Journal of Electrostatics 33 (1994) 385 392

a c o n c l u s i o n s i m i l a r to t h a t r e a c h e d by J o n e s et al. [12] e x c e p t t h a t t h e y d i d n o t h a v e the d a t a n e c e s s a r y to c o m p a r e t h e forces f r o m e a c h m o d e l .

References [1] N.N. Lebedev and I.P. skarskaya, Force acting on a conducting sphere in the field of a parallel plate condenser, Sov. Phys. Tech. Phys., (3) (1962) 268-270. [2] H.B. Levine and D.A. McQuarrie, Dielectric constant of simple gases, J. Chem. Phys., 49 (1968) 4181-4187. [3] 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. [4] N.S. Goel and P.R. Spencer, Toner particle-photoreceptor adhesion. In: L.-H. Lee (Ed.), Polymer Science and Technology, Vol. 9B: Adhesion Science and Technology, Plenum, New York, 1975. [5] D.J. O'Meara and D.A. Saville, The electrical forces on two touching spheres in a uniform field, Q.J. Mech. Appl. Math., 34 (1981) 9-26. [6] A. Goyette and A. Navon, Two dielectric spheres in an electric field, Phys. Rev. B., 13 (1976) 4320-4327. [7] W.Y. Fowlkes and K.S. Robinson, The electrostatic force on a dielectric sphere resting on a conducting substrate. In: K.L. Mittal (Ed.), Particles on Surfaces: Detection, Adhesion and Removal, Plenum, New York, 1988. [8] R.D. Stoy, Solution procedure for the Laplace equation in bispherical coordinates for two spheres in a uniform external field: parallel orientation, J. Appl. Phys., 65 (1989) 2611-2615. [9] R.D. Stoy, Solution procedure for the Laplace equation in bispherical coordinates for two spheres in a uniform external field: perpendicular orientation, J. Appl. Phys., 66 (1989) 5093-5095. [10] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941, pp. 152, 176, 205. [11] R.D. Stoy, Induced multipole strengths for two dielectric spheres in an external electric field, J. Appl, Phys., 69 (1991) 2800-2804. [12] T.B. Jones, R.D. Miller, K.S. Robinson and W.Y. Fowlkes, Multipolar interactions of dielectric spheres, J. Electrostatics, 22 (1989) 231-244.