Forces and torques on conducting particle chains

Journal of Electrostatics, 21 {1988) 121-134

121

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

F O R C E S A N D T O R Q U E S ON C O N D U C T I N G P A R T I C L E C H A I N S

T.B. JONES

Department of Electrical Engineering, University of Rochester, Rochester, N Y 14627 (U.S.A.) and BRUCE RUBIN

Copy Products Research and Development, Eastman Kodak Company, Rochester, N Y 14650 (U.S.A.) (Received November 2, 1987; accepted in revised form April 14, 1988)

Summary An expression for the effective dipole moment of a chain of two unequal, conducting spheres with arbitrary orientation to a uniform applied electric field is determined using the method of electrical inversion. This result is used to obtain an expression for the dielectrophoretic force and to calculate the electrical alignment torque. The effective moment of non -contacting and unequal, conducting spheres, aligned parallel to the electric field, is also calculated using the method of images. The results are applicable to the dielectrophoretic collection and alignment of chains of biological cells in electrofusion procedures.

1. Introduction

Calculation of the dielectrophoretic (DEP) force and the alignment torque on chains or agglomerates of particles is readily achieved if an expression for the effective m o m e n t is available. In previous work, effective dipole moments for chains of two, three, and four conducting spheres were calculated using the method of images [1,2]. The calculation was done under the restriction that the spheres were of equal size and were aligned parallel to the applied electric field vector E. In the present work, the more general situation of two-sphere chains which are not aligned with the field is considered. Furthermore, the restriction to spheres of equal size is removed and both the cases of contacting and non-contacting spheres are worked. For touching spheres, the method of geometric inversion is used to solve the electrostatics problem, while for spheres not in contact, the method of images is employed. In all cases, we restrict the net charge of the chains to zero. The principal objective for this work is prediction of the forces and torques exerted on chains of biological cells in aqueous suspension. Recently, there has been considerable interest in the application of non-uniform electric fields and forces to cells and cell chains. The forces and torques exerted on single cells 0304-3886/88/$03.50

© 1988 Elsevier Science Publishers B.V.

122

have been studied extensively. Sauer has examined the basic formulation of the dielectrophoretic force on isolated cells and has used a rigorously derived force expression to study the interaction and mutual attraction of cell pairs [3]. He has also calculated the mutual torques of interacting cells [4]. However, the case of rigidly linked particle chains considered in the present work has not received attention. 2. Touching spheres The situation of touching spheres is readily attacked using the method of geometric inversion because, by proper choice of the sphere of inversion, the two spherical conductors invert to become plane parallel surfaces. Cases where the electric field is parallel (bk) or perpendicular ( 3_ ) to the line-of-centers of the particles are considered separately, and expressions are obtained for the vector components of the effective moments p ILand p ~. Then, superposition is employed to represent any general situation where the electric field is oblique to the line-of-centers of the chain. The method of solution is illustrated by first considering equal-sized spheres, for which case the results may be expressed in terms of well-known series.

2.1 Spheres of equal size Refer to Fig. la, which shows two conducting spheres in contact and aligned parallel to the electric field created by the charges + Q and - Q located at distances D >> R from the point of contact between the spheres. The sphere of

""

1- ~ *"

~)__

- -

"", ~ \

<-/

D

SPHERE OF INVERSION

r

G

P

Q

-Q \

/

(o)

4R

;--2.--I-

o,

4.

"1

;, (b)

Fig. 1. (a) Chain of two equal spheres parallel to the electric field E due to source charges + Q and - Q. (b) Inverse problem, showing image dipoles required to evaluate V' ( r' ) at inverse point P ' .

123

inversion has radius 2R and its center is at the contact point. It is important to note that, for equal spheres located midway between the two source charges, the zero net charge condition is met automatically and the potential of the spheres is zero. The objective is to determine an expression for the dipole component of the potential V(r) at point P due to the charge separation induced in the chain by the electric field E = Q/2~eD 2. The spherical surfaces of the particles invert to the parallel planes shown in Fig. lb [5 ]. The source charges + Q and - Q invert to a "source" dipole p' located at the origin, and the regularly spaced image dipoles are equivalent to the charges induced in the particle chain. The inverse potential V' (r') is calculated by summing the contributions to the potential from all the image dipoles. The transformations required to do the problem are p' =32~eR3E

(1)

r' = 4R 2/r

(2)

V = (2R/r) V'

(3)

The potential due to the image dipoles is p'

p'

V ' = - , ~ I~ 47~e(nnR-r') 2 ~- ~.=147~e(4nR+r') 2

(4)

If we assume that the source charges go to infinity (D >> R) while E stays finite, then r' << R and eqn. (4) becomes V' -

-p'r' 1 -7~e(4R) - - - - - ~ ,=1 ~ ---5 n

(5)

If eqns. (1)-(3) are used, the result is an expression for V in the primary variables. 4R3E 1 L ~ V= - ~ r .=1

(6)

which may be compared to the general expression for the potential due to a dipole on the axis at 0= O. Set V(r)=

Pelf cos 0 e=o 4~er 2

(7)

and the expression for the effective moment PI, is 1 Pll = 16~eR3E ~, .-5 = 19.23~eR3E n=ln

(8)

which agrees with the results previously obtained using direct application of

124 the method of images. Note that this derivation is more straightforward than a prior analysis [6]. A similar approach is used to determine the effective m o m e n t of a chain of two spheres with its line-of-centers aligned perpendicular to the electric field. Refer to Fig. 2a, which illustrates the arrangement of charges and the location of the point P. Note that the sphere of inversion is unchanged from Fig. la. The source dipole p' and its images, which alternate in sign, are shown in Fig. 2b. It is convenient to define the angles A0~, where rr

tan ~0~ ~ O n ~

(9)

4nR

Now, the expression for V' is Y'

--

2he P' ~

s(n/2--A0") n=l (-- 1 ) ' c °(4nR) 2

(10)

where 3 0 , << n/2. Employing the same procedure used to obtain eqn. (8), the effective dipole moment p~ may be identified as

SPHEREOF" rI P INVERSION~..~~/t ~"--F~""x\ ~

-Q ® (a)

I"

4R

pl

/

4R

2R-:

_~l t

"1

tpl

(b) Fig. 2. (a) Chain of two equal spheres perpendicular to the electric field E due to source charges + Q and -Q. (b) Inverse problem, showing image dipoles required to evaluate V' (r') at inverse point P'.

125

oo ( _ 1 ) . - 1 p . =8~eRaE ~ n 3 -7.2123~eRaE

(II)

n=l

Comparison of eqns. (8) and (11) reveals that the effective induced dipole m o m e n t is a strong function of the alignment of the chain and the electric field. When the chain is parallel to the field, the m o m e n t is more than twice the value of two non-interacting spheres. When the spheres are perpendicular to the field vector, the dipole m o m e n t is approximately 10% lower than this value. Equations (8) and ( 11 ) are in agreement with the results of Jeffery and Onishi [7].

2.2 Unequal spheres parallel to the electric field Consider two electrically conducting spheres of radii RA and RB which are in contact and aligned parallel to the electric field vector (Fig. 3a). Without loss of generality, we assume that fl=RB/RA >t 1. The sphere of inversion has

-

VW~L ~r'

SPHEREOF INVERSION I

~

I~

/

~'

(o)

2B

"I L

pl

(DIPOLES)

o

_q~

/

p' qm

t

o

-q~

(MONOPOLES) (b) Fig. 3. (a) Chain of two touching unequal spheres of radii R A and RB parallel to the electric field E. (b) Inverse problem, showing image dipole and monopole contributions to V' (r') at inverse point P'.

126 radius 2RB with its center at the point of contact. The new set of transformations used to relate the various quantities in the original and inverted geometries are eqns. (1)-(3 ) with RB substituted everywhere for R. It is convenient to define A = 2fiRs and B = 2RB. Figure 3b shows the placement of the image dipoles for the inverted problem. If it is constrained to zero net charge, then due to its asymmetry (fie 1) the particle chain cannot be at zero potential. This finite potential Vo is accounted for by introduction of a point charge q* (monopole) at the center of inversion. The value of q* is established by setting the potential of the parallel electrodes to zero [5], that is 2RB Vwall --

'

q* Vo-~ - -

rA

4~zer'A--0

(12)

Equation ( 12 ) merely imposes the condition of a finite potential on the chain of spheres. The zero net charge condition must be met by an additional constraint. The image monopoles are also shown in Fig. 3b. If all image monopole and dipole contributions to V' are added, the result, correct to first order in r', is

V'

:__1....1 q*21

22

47re{[ 2 - ~ a -~ 16f12R~ p' ]

P 23 q ,

+[

16f12R~ q*Y~2'

32--fl-~,]r } '

(13)

where the summations are defined as

oo { 2fl_n(fl+ l )2 } 21=n=, ~ 2n(fl+l)[(n2-n)(fl+l)2+fl] { (_2n- I) (f12-1) l 22=,=1 ~ [(n2-n)(fl+l)2+fl]2J 2a=n=lZ ÷ [n(fl+l)l _ 1] a ~ [n(fl+l)-fi] ~ { [n(O+1)13 2p a l }

(14a) (14b) (14c)

It is evident from the analysis of section 2.1 above that the V' potential terms linearly dependent upon r' invert back to the desired dipole potential, i.e. V~ r - 2. Therefore, the zero-order term in eqn. (13) is set to zero by specifying the charge q*: •2P' q* = --SflRA ~1

(15)

It may be shown that eqn. (15) constrains the net charge of the chain to zero. The potential V' then becomes

47re(32flaRaA) 2 a - - ~ - ~

(16)

127

Equation (16) for the potential may be transformed back into an expression for V(r) and the effective m o m e n t evaluated. A simple basis for comparison of the results for all fl>~ 1 is established by constraining the total volume of the particle chain to be constant and equal to the volume of two identical spheres of radii R. Then, R3= ~RA ' 3 (1 +/~3), and the effective moment expression becomes: PH(fl)-

16~eflaRa l+f13

~3

- 4~-,1 2 E=K~(fl)E

(17)

2.3 Unequal spheres perpendicular to the electric field The situation of two unequal spheres with line-of-centers perpendicular to the applied electric field is illustrated in Fig. 4a. We again impose the restriction that the particle chain has zero net charge, but symmetry about the midplane between the two source charges + Q and - Q maintains the chain at zero potential so that q* = 0. Therefore, the potential contributions to V' arise only from the image dipoles shown in Fig. 4b. It is instructive to compare these

I SPHERE OF INVERSION

(a)

-Qe

I

Ip' °

°BIB° (b)

Fig. 4. (a) Chain of two touching unequal spheres of radii R Aand RBperpendicular to the electric field E. (b) Inverse problem, showing image dipole contributions to V' (r') at inverse point P'.

128

dipole images and their placement to those in Fig. 2b. The angles AOAn and AOBn are defined in Fig. 4b.

r'

r'

AOAn-2[nA+(n_I)B]

AOBn-2[nB+(n_I)A]

and

(18)

Following the same methods as in section 2.2 and imposing the constraint of constant total volume again, the new effective moment expression is

87~eRaf13E~ { Pi(fl)--

1+fl 3

2

1

[(fl+l)n] 3

n=l

[(fl+l)n-1]

--K. (fl)E

3

1 } [(fl+l)n-fl] 3 (19)

2.4 Discussionof results Calculated results for PH (fl) and P i (fl) are shown for fl>~ 1 in Fig. 5. Note that the values for these moments at fl= 1 correspond to those determined in section 2.1, eqns. (8) and (11). As fl-~ oo, the normalized parallel and perpendicular moments both asymptotically approach a value of 8, which is expected when one sphere is much larger than the other. The constant volume constraint means that p II (fl) = P II (fl - 1 ) and p ± (fl) = p i (fl - 1). Equations ( 17 ) and (19) indeed satisfy these conditions. I

I

I

I

I

[

z

t

I

o cv;

ALIGNMENT (~

6~ Od

COEFF.

~E I /

5~ Qg.

~-20 = .

. . . . .

19.25

41~

I

I-z

121 "~

I0

N .J

_J

z

----"~"-'K.21 2

rr" 0

I

Z

I I I)

3 4 5

7

I0

20

B = RB/RA Fig. 5. Normalized effective dipole moments for chains of unequal touching spheres parallel and perpendicular to the electric field, and alignment torque coefficient versus fl subject to fixed total volume constraint.

129 Superposition may be used to determine the vector moment of a chain of conducting spheres at any angle ~ with respect to the electric field (see the inset in Fig. 5 ). Define the parallel and perpendicular components of the electric field to be Ell and Ex. Then, the net moment is

peff =KII (fl)Eii +K j_ (fl)E±

(20)

The expression for the effective moment may be used to determine the DEP force: FDEp = (Pelf (fi)" F)E

(21)

Likewise, the torque of alignment for the chain can be determined. T=peff (fl) × E

(22)

This torque always tends to align the chain so that its line-of-centers is parallel to the electric field, and its magnitude is ITI = ½(Kil (fl) - K ~ (fl)) sin 2~

(23a)

See Fig. 5 for a plot of ]T(fl) I/sin 2~,versus ft. For the special case of two equal spheres (fl= 1 ), the result for the alignment torque is

IT(fi= 1) l =5neR3E2sin 2~ ~ ~=6.0103neRaE2sin 2~

(23b)

rt=l

3. Spheres not touching

Consider two spheres of radii RA and RB at a spacing J, aligned parallel and located midway between the charges + Q and - Q. For this situation, the method of images presents a straightforward means to compute the effective moment [1]. The spheres A and B are isolated and uncharged, so they are not at the same potential. Under these conditions, two types of charges are required: primary images induced by the source charges and neutralization charges, which are needed to maintain the zero net charge condition on each sphere.

3.1 Primary images The asymmetry of the chain when the particles have unequal radii requires that separate account be kept of all image charges. Refer to Fig. 6a, which shows the first several images in each sphere. Standard image theory is used to provide recursion formulae for the magnitudes and locations of all the image charges [8 ]. Some attention must be paid to the form of the equations, in order to avoid non-convergent series when the effective moment is calculated. For notational convenience in the analysis below, all lengths and charges defined in Fig. 6a are normalized to RA or Q, respectively. If 7= J/RA, A =D/RA, then

XA=l+7/2

and

Xs=fl+7/2

(24)

130

~I~ -dAy --dB,l ~

D

(

® -Q

(a)

(b) Fig. 6. (a) Chain of two non-contactingand unequal spheres of radii RA and R B parallel to the electric fieldE, showingfirst fourprimary images. (b) Neutralizationchargeplacementfor above.

The first four primary images and their locations (all normalized) are qA1 = [~J-- ( 1 + 7 / 2 ) ] -~

and

dA,=XA+qA1

(25a)

q~] = [zl-Jt- ( 1 + ~ / 2 ) ] -1

and

dhl=XA--q'B1

(25b)

qB~ =fl[zl-- (fl+ y/2) ] -~

and

(25C)

q~,l = f l [ A + ( f l + 7 / 2 ) ] - '

and

dm =XB +flqsl d'Al=XB--flq'A1

(25d)

The recursion formulae are different for the even and odd image terms. For n = odd,

qA,n ~-qA,n--1/ [dA,n-1 "~XA ]

and

dA,n----XA--1/[dA,,~_I+XA]

(26a)

q~,n=q'B,n-1/[dh,n-~ +XA]

and

d~,n-=XA--1/[d'B,n_I+XA]

(26b)

qB,n-~qB,n--1/[dB,n-1 + X B ]

and

ds,n=XB--fl2/[dB,n_l+XB]

(26C)

q'A,n =flqA,.-- , / [d~A,n-1 "~XB ] and d'A,n-=XB-fl2/[d'A,,~_~+XB] while for n = even,

(26d)

131

qA,~ =flq'A,n-~/ [dA,n-l + X s l

and

dA,~=Xn--fl2/[dA,._~+Xs]

(27a)

r ! d ! qB,~=flqB,n-1/[ B,n-1 d-XB ]

and

dB,. , ' = X n --fl 2/[dB,n_ 1 +XB]

(27b)

qB,n =qB,n--1/[dB,n-I "[-XA ]

and

dn,n =XA--1/[dB,n_, + X A ]

(27C)

r t d v qA,n=qA,~-l/[ A,~--I+XA ]

and

' =XA - 1 /[dA,.-i ' +XA] dA,n

(27d)

3.2 Neutralization charges The condition of net neutrality for the two isolated particles must be imposed before the effective dipole m o m e n t can be calculated. To effect neutralization, charges QA and QB are placed at the centers of each sphere. These charges in turn have infinite sets of images which must be taken into account. Refer to Fig. 6b which shows the placement of the first few charges. The charges 2n,n and 2B,. are normalized to QA and QB, respectively, and all distances YA,n, ~)B,nare normalized to RA. 2A1=1

and

yAI=XA

(28a)

2B1=1

and

7m=XB

(28b)

The recursion formulae for n = odd are ,~.A,n=,~A,n_I/[YA,n_I'~-XA]

and

2B,n =fl2B,n--1/[YB,.--1 +XB ] and

7A,n=XA--11[YA.n_I T X A I

(29a)

~B,~=XB--fl2/[yB,~_I +XB]

(29b)

and for n = even

2g,n=fl2g,.--l/[7g,n--l +XB]

and

~)A,n=-XB--fl2/[~A,n_I+XB]

(30a)

2B,n=2n,.--l/[YB,n--l +Xg]

and

~B,n=XA--1/[~B,n_l d-XA]

(30b)

The neutralization charges in each sphere must be added separately and so we define A~dd, A5yen, ... to be summations of the odd or even terms of ~,A,nor ~'B,n, respectively.

3.3 Dipole moment computation The above equations may be used to calculate the dipole moment. It is easiest to evaluate the neutralization charges for each set of images as defined by eqns. (25)-(27 ). The neutralization conditions are then, for n= odd:

Q( --qA,n +q~,n ) + QA,nA°Add -QB,nA~ven =0

(31a)

Q(qB,n --q 'A,n ) --QA,nAeAven "a-r~"~B,n"'A°dd =0B

(31b)

and for n = even:

Q(--qB,n d- q ~A,n) d- QA,nA °Add- QB,nA~yen -~ 0

(32a)

132 20.0

, , , : - - - 19.25

L.~

I

I

I

,

,

,

Z

~

I

,

,

,

i

I

I

I

'

I

,

~U

aT 15.0

~ .... 13.77

Z W

~

B : Re/RA

o a bJ

_N _.1

~E 0Z

1 0 ' 0 ~ 4 . 0 8.0~-"1"

.

.

I

.

I

.

.

I

.

.

I

.

.

[

.

t __ .

I

I

[

I

I

I

I

0.5 1.0 NORMALIZED SPACING (AI R )

1.5

T

Fig. 7. Normalized effective dipole moment of non-contacting spheres aligned parallel to the electric field E versus spacing for various values offl subject to fixed total volume constraint.

, ) --QA,nA even Q( qA,~ --qs., A +QB,n'4Bodd = 0

(32b)

Equations (31) and (32) are solved for all QA,nand QB,n in terms of the source charge Q. The dipole m o m e n t is then computed using the following series.

Peff-~QRA ~ [qA,ndA,n+qB,ndB,n--q'A,~d'A,n--qB,nd~,n] n=l

- ~ RA[QA,n)'A,nYA,n--QB,~AB,nYB,n] (33) rt~l

Computed results for the normalized effective dipole moment (peff/neR3E) versus separation (A/R) are shown in Fig. 7 for several values of ft. Once again, the constant total volume constraint is used. Note that if A/R >> 1 or fl >> 1, Pelf/heR aE-~ 8, as expected.

3.4 Comment At small particle spacings, the image method for chains of conducting, uncharged, and non-contacting spheres presents certain computational difficulties. It is found that when 0
133 It may seem surprising that the limiting value of the effective m o m e n t Pelf for two isolated spheres coincides with that for a chain of two similar electrically contacting and touching spheres with zero net charge when A/R---, O. That this is the case may be shown readily by examining the contribution to the net moment of the neutralizing charges as the spheres approach contact. This is to say that the second sum in eqn. (33) goes to zero in the A/R~O limit. 4. Conclusion

In this paper, calculation of the effective dipole moment for chains of conducting spheres has been extended to unequal spheres having arbitrary alignment with respect to the applied electric field. Using the general vector expression for the effective moment, eqn. (20), the dielectrophoretic force and the alignment torque may be determined. The superposition method used in this paper for chains of conducting spheres may be employed to determine the alignment torque on a dielectric particle chain. The first step is to determine the dipole moments for the chain parallel and perpendicular to the electric field. Then, superposition is used to obtain the vector expression for the moment; the DEP force and alignment torque follow from eqns. (21) and (22). Unfortunately, neither image methods nor electrical inversion are useful for dielectric spheres. Other means, such as multipolar expansions or boundary value solutions must be employed to evaluate Pll a n d p ± . Such torque calculations are of interest in several diverse situations where chains of particles are placed under the influence of electric fields. One example is in the alignment of cell chains prior to electric-field-initiated cell fusion. Whether the desired cell pairs are formed using an ac electric field [10] or using biochemical agents [ 11 ], for good yield of fused cells it is important that the chains be aligned with the electric field prior to application of the fusion pulse. Accurate models for chains of cells necessarily involve layered spheres with ohmic and (possibly) dielectric loss, and solutions for the effective moments Pl and p± for such chains do not exist. However, the expressions for parallel and perpendicular moments provided in the present work do establish limits for results yet to be obtained with realistic cell chain models. For example, the torque expressions developed in this work may be useful in prediction of cell chain rotational dynamics and identification of the optimal set of parameters for alignment in electrofusion applications. Acknowledgment

This work was supported principally by a grant from the Particulate and Multiphase Processes Program of the National Science Foundation. Additional financial assistance from the Copy Products R&D Group of Eastman Kodak Company is gratefully acknowledged.

134

References 1 T.B. Jones, J. Appl. Phys., 60 (1986) 1247. 2 T.B. Jones, J. Appl. Phys., 61 (1987) 2416. 3 F.A. Sauer, Interaction forces between microscopic particles in an external electromagnetic field, in: A. Chiabrera, C. Nicolini and H.P. Schwan (Eds.), Interactions between Electromagnetic Fields and Cells, Plenum, New York, NY, 1985, pp. 181-202. 4 F.A. Sauer and R.W. SchlSgl, Torques exerted on cylinders and spheres by external electromagnetic fields, in: A. Chiabrera, C. Nicolini and H.P. Schwan (Eds.), Interactions between Electromagnetic Fields and Cells, Plenum, New York, NY, 1985, pp. 203-251. 5 W.R. Smythe, Static and Dynamic Electricity, McGraw-Hill, New York, NY, 1968, Sec. 5.09. 6 T.B. Jones, J. Appl. Phys., 62 (1987) 362. 7 D.J. Jeffery and Y. Onishi, J. Phys. A: Math. Gen., 13 (1980) 2847. 8 E. Weber, Electromagnetic Theory, Dover Press, New York, NY, 1965, Sec. 21. 9 D.J. Jeffery, J. Inst. Math. Its Appl., 22 (1978) 337. 10 U. Zimmermann and J. Vienken, J. Membr. Biol., 67 (1982) 165. 11 M.M.S. Lo, T.Y. Tsong, M.K. Conrad, S.M. Strittmatter, L.D. Hester and S.H. Snyder, Nature, 310 (1984) 792.