JOURNAL OF COLLOID AND INTERFACE SCIENCE 194, 239–248 (1997)ARTICLE NO. CS975107

Theory of Electrode Polarization in Dielectrophoresisand Electrorotation

2. Generalized Analytical Tools for the Study of Nonhomogeneous Fields in Ionic Media

Reginald Paul1 and Karan V. I. S. Kaler*

Department of Chemistry and *Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Received June 3, 1997; accepted July 24, 1997

field was neglected. This amounts to ignoring the influenceThe analysis begun in the previous paper on the role of electrode of the Stern layer.

polarization in experimental measurements using dielectro-With the development of micromachining and microelec-phoresis and electrorotation in a polar spherical coordinate frame

is extended to include other curvilinear coordinate systems. The tronic techniques it is now possible to design electrode sys-Green functions, as derived in the previous paper in terms of the tems that can simulate many different types of inhomoge-Dyson equation, is shown to obey a variational theorem that allows neous fields and, therefore, the restrictions from the firstthe inclusion of the equilibrium potential and the Stern layer effect assumption must be relaxed. It is also known that the Sternneglected thus far. q 1997 Academic Press layer and its dynamics play a very important role in the

Key Words: dielectrophoresis; electrorotation; electrode polar- dynamics and properties of the diffuse double layer. Theization; dielectrics; double layers.

method of the Green functions that was used is a very flexiblemethod that can be adapted to different coordinate systemsand in this paper we present this generalization. Moreover,I. INTRODUCTIONwe show that by expressing the Green functions in the formof Dyson’s equation a variational theorem can be provedIn the previous paper (1) we introduced the problems ofwhich allows the inclusion of the Stern layer effects in alow frequency anomalies that are encountered in the studiesvery natural manner.of biological cells by the methods of dielectrophoresis and

electrorotation. An account of the model system used andII. BASIC MODEL AND GOVERNING EQUATIONSsome of the background literature was also provided. This

introductory section was then followed by an extensive in- In order to present an analytical description we considervestigation of the role played by the phenomenon of elec- the problem to be defined in a space filled by a family oftrode polarization and charge screening by the double layer surfaces that are parallel to the surface of the nonplanaron the field experienced by a levitated cell from an hemi- electrode. This family of surfaces are defined by a functionspherical electrode. It was found that a frequency dependent

£(x , y , z) such thatcorrection or screening factor must be introduced in orderto calculate the true response of the cell. The application of

£(x , y , z) Å A1 . [1]this factor showed that, while the anomalies still persist,their magnitudes are smaller than expected when electrode Here A1 is a constant which for a specific value, A1 Å £el ,polarization is neglected. becomes the nonplanar electrode surface and (x , y , z) are

The results were, however, calculated after the following the coordinates of a point in a Cartesian frame of reference.assumptions had been made: From this definition it is seen that the region outside the

electrode is characterized by the range of values of £:( i ) The nonuniform field was assumed to originate froman hemispherical geometry. This allows the use of a polar

£el £ £ £ ` . [2]spherical coordinate system for the mathematical analysis.( ii ) The effects of the equilibrium potential that would

Specific points in space, P(x , y , z) , are obtained by thebe present at an electrode even in the absence of an appliedintersection of one of the above surfaces with two other

1 To whom correspondence should be addressed. surfaces:

239 0021-9797/97 $25.00Copyright q 1997 by Academic Press

All rights of reproduction in any form reserved.

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240 PAUL AND KALER

that the ions in the Stern layer are, indeed, tightly bound toj1(x , y , z) Å A2 j2(x , y , z) Å A3 . [3]the electrode surface then we may consider the electrodealong with these ions to constitute a rigid body. The surfaceWe thus obtain a curvilinear coordinate frame {£, j1 , j2} inof this rigid body, commonly referred to as the slippingwhich the governing equations must be written. The ultimatesurface, will be assumed to lie at £ Å £s , while the electrodechoice of this frame depends, obviously, on the shape of thesurface itself lies at £ Å £el . Beyond the Stern layer thereelectrode and upon such factors as the separability of thewill still be a predominance of counterions but these ionsgoverning partial differential equations.will not be rigidly bound to the electrode and constitute theIt is, in general, possible to consider completely arbitraryso-called diffuse double layer and at even greater distanceselectrode shapes but since we are primarily concerned withwhere the influence of the electrode has vanished the condi-the phenomenon of dielectrophoresis and electrorotationtion of charge neutrality will hold and this is the bulk regionsuch a level of generalization is neither necessary nor desir-of the electrolyte. At any vector position r the potential ofable. Most of the electrodes used allow a separation of thethis equilibrium situation is given by C 0(r) which is a timepartial differential equations into three ordinary differentialindependent potential that satisfies the equilibrium Poisson’sequations dependent upon the variables £, j1 , and j2 , respec-equation:tively. Furthermore, within reasonable assumptions the part

of the solutions that depend upon the variables j1 and j2

constitute a complete set of orthogonal functions En(j1 , j2) :Ç

2C 0(r) Å 0 4pem

∑N

jÅ1

zjen 0j (r) . [6]

* dj1dj2w(j1 , j2)E*m (j1 , j2)En(j1 , j2) Å Cndmn . [4]Here the various parameters appearing are defined as fol-

lows:Here w(j1 , j2) is a weighting function, Cn is a normalizationconstant, and dmn is the Kronecker delta. Equation [4] allows (i) em is the permittivity of the medium in which thean arbitrary functions of r to be expressed as a linear combi- electrolyte is present.nation: ( ii ) zje is the charge carried by the j type ion ( j Å 1, 2,

rrrN) in the solution.F(r) Å F(£, j1 , j2) Å ∑

n

cn fn(£)En(j1 , j2) . [5] ( iii ) n 0j (r) is the equilibrium number density of the j type

ion at the vector position r .

The boundary conditions that must be obeyed by the po-Here, cn are constant coefficients of expansion and the settential C 0(r) areof functions fn are dependent upon the variable £ only. The

precise forms of the functions En(j1 , j2) will depend uponthe shape of the electrode chosen and thus upon the coordi- LimÉ

£ r £sC 0(r) Å z , LimÉ

£ r `C0(r) Å 0. [7]

nate frame. It is important to bear in mind the fact that theexperimental device being used will determine the specific Here z is the well known ‘‘zeta potential’’ of electrophoresis.terms in the potential expansion that must be used. In most When a time dependent field is applied to the system theof the currently employed instruments it is the dipolar term above equilibrium situation is perturbed and we assume thatof the expansion that plays the dominant role. Therefore at the slipping surface a uniform potential V ( t) Å Ve0ivt

although expansions of the type given by Eq. [5] are very develops. The diffuse double layer also responds to thisgeneral in nature in practice one or two terms may play a potential and we assume that the diffuse double layer poten-significant role in the experiment. tial, CD(r , t) , is linearly related to V ( t) in the manner

Since the contents of this paper are of a general naturespecific choices for the functional forms of the three defining

CD(r , t) Å c(r)Ve0ivt , [8]surfaces £(x , y , z) , j1(x , y , z) , and j2(x , y , z) will not bemade. The choice where £ Å r , j1 Å u, and j2 Å f was

where the function c(r) is hitherto an unknown functionmade in the previous paper (1) .and it is one of the aims of this paper to derive an expressionWe first consider the equilibrium situation in which nofor it.external field has been applied but an electrolyte solution

Further away from the diffuse double layer lies the bulkhas been placed next to the electrode. As is discussed inregion in which the charge density is zero and the potentialmost text books on electrochemistry (see, for example, Old-Ce0ivt must obey Laplace’s equation:ham and Myland (2)) , this arrangement results in the devel-

opment of a charge on the electrode surface with a concomi-tant production of a tightly bound Stern layer. If we assume Ç

2CB Å 0. [9]

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241GENERAL ELECTRODE POLARIZATION IN BIOELECTRICS, 2

The function CB(r) may be expanded in the form given by Proceeding in precisely the same manner as in the previouspaper with the variable £ replacing the radial coordinate rEq. [5]:and retaining all the other symbols for the various fields weobtain two linear partial differential equations for c and fj ,CB(r) Å ∑

n

qnxn(£)En(j1 , j2) . [10]respectively:

In the absence of the double layer the coefficients qn can be (i) Ç2c / a(r)c Å f1(r) , [17]calculated by invoking boundary conditions on the electrodesurface that require the continuity of CB with the applied wherepotential V . In the presence of the double layer, however, thebulk is screened from the electrode in a, hitherto, unknown

a(r) å 0 4pe 2

emkT∑N

jÅ1

z 2j n 0

j (r) [18]manner so that although we may formally write

CB(r) Å ∑n

BnCBn(£)En(j1 , j2) , [11]f1(r) å 4pe 2

emkT∑N

jÅ1

z 2j n 0

j (r)fj(r) å ∑N

jÅ1

bj(r)fj(r)

the constants Bn can no longer be calculated from a simple bj(r) Å 4pe 2

emkTz 2

j n 0j (r) . [19]

boundary condition. Furthermore, since the double layer hasa frequency dependence these coefficients must also be fre-quency dependent quantities. A particle that has been levi- The boundary conditions obeyed by c(r) aretated will most likely be present in the bulk region and thusexperience a field due to CB rather than V ( t) . The frequency LimÉ

£ r £sc(r)

dependent parameters Bn will, therefore, color the frequencydependent electrical properties measured by experimenta- Å 1 0 1

VLimÉ

£ r £scB(r) , LimÉ

£ r `c(r) Å 0 [20]tion. It is imperative that the calculation provide an algorithmfor the computation of this additional frequency dependence.

( ii ) Ç2fj / SO j(r)fj Å f2j(r) , [21]The potential C(r , t) at any given point will be a linearcombination of all the above potentials and it must satisfy

wherea Poisson’s equation analogous to Eq. [6]:

C(r , t) Å C 0(r) / dC(r , t) [12] SO j(r) å 0 zje

kTÅC 0

rÅ / iS 2j [22]

dC(r , t) Å (c(r)V / CB(r))e0ivt [13]

f2j(r) å 0 zje

kTÅC 0

rÅCB 0 iS 2j c

Ç2C(r , t) Å 0 4p

em∑N

jÅ1

zjenj(r , t) . [14]

S 2j Å

vlj

kT. [23]

In order to calculate the total time dependent potentialC(r , t) it is essential that we know the particle density nj(r ,

In order to solve Eq. [21] boundary conditions must alsot) . This quantity is determined by the dynamical equationsbe provided. To obtain an explicit form for one of the bound-governing the motion of the ions. The ionic velocity and inary conditions the expression for the ionic velocity given byconsequence the ionic motion is dependent upon the systemEq. [15] has to be written in some detail. The form of theof forces that are acting on it.vector operator Ç in generalized curvilinear coordinates isUnder steady state conditions with the ionic velocity writ-well known (see, for example, Margenau and Murphy (3))ten as the gradient of a velocity potentialand thus the ionic velocity vector may be written as:

vj Åzje

lj

ÇFH j [15]vj Å

zje

ljF e

£

Q£

ÌFH j

Ì£/

ej1

Qj1

ÌFH j

Ìj1

/ej2

Qj2

ÌFH j

Ìj2G . [24]

an expression for the density nj(r , t) can be obtained andit is the familiar Boltzmann distribution: Here, e

£, ej1

, and ej2are unit vectors along the curvilinear

coordinate axes and the quantities Q£, Qj1

, and Qj2are de-

fined as follows:nj(r , t) Å n`j e0z je /kT [C(r , t )/FH j(r , t ) ] . [16]

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242 PAUL AND KALER

III. TRANSFORMATIONS OF THE PARTIALQ 2

qÅ S Ìx

ÌqD2

/ S Ìy

ÌqD2

/ S Ìz

ÌqD2

; qÅ £, j1 , j2 . [25] DIFFERENTIAL EQUATIONS INTO INTEGRALEQUATIONS INCORPORATING THE BOUNDARY

CONDITIONSEach of the terms in Eq. [24] have very definite physicalsignificance. From the manner in which we have introduced The two basic partial differential Eqs. [17] and [21] arethe curvilinear coordinate frame it will be recalled that mo- similar to each other in overall structure so that it is sufficienttion along the axis £ corresponds to a motion in which the to consider a generic equation whose treatment can be suit-ion moves from one surface parallel to the electrode surface ably adapted to each of the specific cases. With this strategyto the next in the complete family of surfaces that fill the in view, for the present section, we consider the followingspace. Thus the first term in Eq. [24] is the component partial differential equation:of the velocity with which this transport occurs. Since theelectrode is, in most cases, composed of material that cannot

Ç2U(r) / sP (r)U(r) å LO (r)U(r) Å F(r) . [29]be penetrated by the ions a reasonable boundary condition

for this velocity component will beHere, L̂(r) is a shorthand for the operator Ç2 / sP (r) . Inorder to transform this into an integral equation we considera Green function G(r , r *) for the operator L̂(r) defined asLimÉ

£ r £s

ÌFH j

̣Š0. [26]

LO (r)G(r , r*) Å 4pd(r 0 r *) , [30]Using Eq. [11] in this result we obtain

where d(r 0 r *) is the well known Dirac delta function.There are two boundary conditions connected with each of

LimÉ£ r £s

ÌÌ£

fj Å1VLimÉ

£ r £s

ÌÌ£

CB the differential equations discussed in the previous section:( i) A specification of the value of the solution or its

derivative normal to the slipping surface. This boundaryÅ 1

V∑n

BnC *Bn(£s)En(j1 , j2) , [27] condition we denote by the symbol B1(U) and indicate its

dependence upon the specified information in the followingway:

where a prime implies a derivative of a function with respectto its argument. It is important to realize that Eq. [27] is

B1(U) Å a LimÉ£ r £s

U / b Lim£ r £s

ÌỤŠc . [31]only a formal boundary condition since the parameters Bn

are still unknown. To obtain these parameters that constitutethe far field or bulk potential parameters a further set of

The parameters a and b are given the values 0 or 1 dependingequations are needed and these will be discussed in a subse-upon whether the value of the solution or its normal deriva-quent section. A second boundary condition that a reasonabletive on the slipping surface has been provided and c is thefunction fj must satisfy is at the far surface given byactual value of this data. For example in the case of thevelocity potential fj this information is provided by Eq.[27], accordingly we make the choice: U Å fj , a Å 0, b Å

LimÉ£ r `fj Å 0, LimÉ

£ r `

ÌÌ£

fj Å 0. [28]1, c Å 1/V LimÉ

£ r £sÌCB /Ì£. While what we have just

described is the solution boundary condition, for the Greenfunction, on the other hand, we impose the corresponding

Equations [17] and [21] are the basic governing equationshomogeneous boundary condition which involves setting

which along with the necessary boundary conditions dis-c Å 0:

cussed above provide a pair of coupled linear partial differ-ential equations that must be solved in order to compute the

B1(G) Å 0. [32]electrical and velocity potentials. This is most convenientlyachieved by converting these differential equations into inte-gral equations that incorporate the boundary conditions. This ( ii ) A specification of the value of the solution and /or

its derivative normal to a surface that has been placed at anapproach has the advantage of not only incorporating theboundary conditions but under suitable approximations re- infinite distance from the slipping surface. As in the first

boundary condition this is symbolically designated by B2(U)duces the problem to quadratures which may have to beevaluated numerically. and in precisely the same way we write

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243GENERAL ELECTRODE POLARIZATION IN BIOELECTRICS, 2

function and its derivative to vanish. The total solution toB2(U) Å a LimÉ

£ r `U / b LimÉ£ r `

ÌỤŠc *. [33] the partial differential equation is now given by

For the Green function, as before, we impose the correspond- U(r) Å 14p©S

da *FaG(r , r*)ÌU(r *)Ì£ *ing homogeneous boundary condition which involves setting

c * Å 0:

/ bU(r *)ÌG(r , r*)Ì£ *

GB2(G) Å 0. [34]

With the Green function described in the above manner/ 1

4p * dr *G(r , r*)F(r *) . [38]we seek the solution of Eq. [29]. This may be done in thefollowing steps:

( i) In Eq. [29] we change the variable from r to r*. For the two partial differential equations that we have in(ii ) Multiply the resulting equation on the left-hand side the present problem the application of the above method

by the right-hand side of Eq. [30] and vice versa. leads to the following two integral equations:( iii ) Integrate both sides over all values of r * to obtain

c(r) Å 0 14p©S

da *ÌG(r , r *)Ì£ *

c(r *)U(r) Å 1

4pC(r) / 1

4p * dr *G(r , r*)F(r *) , [35]

/ e 2

ekT∑N

jÅ1

z 2j * dr *G(r , r *)fj(r *) [39]

where, C(r) is a solution to the homogeneous partial differ-ential equation:

fj(r) Å 0 14p©S

da *ÌCB(r *)Ì£ *

Gj(r , r*)LO (r)C(r) Å 0. [36]

/ zje

4pkT * dr *Gj(r , r*)ÇÇ *C 0(r *)rÇÇ *CB(r *)It is easily verified that the second term in Eq. [35], byvirtue of the presence of the Green function, satisfies thehomogeneous boundary conditions c Å c * Å 0. Thus it is / ivlj

kT * dr *Gj(r , r*)c(r *) . [40]necessary to select the function C(r) in such a manner thatthe boundary conditions specified for U(r) are satisfied. This

Here G(r , r *) and Gj(r , r *) are the Green functions for Eqs.is readily accomplished by writing[17] and [21], respectively. These two integral equations arecoupled but can be easily separated by the substitution of

C(r) Å©S

da *Eq. [39] into Eq. [40] and vice versa after the appropriatechange of variables. The results can be written in the follow-ing compact forms1 FaG(r , r*)

ÌU(r *)Ì£ *

/ bU(r *)ÌG(r , r*)Ì£ *

G . [37]

c(r) Å c0(r) / * dr 9K(r , r 9)c(r 9)In this equation the integration is carried out over a surfaceS enclosing the volume of interest which in the present case fj(r) Å f0j(r) / * dr 9Kj(r , r9)fj(r 9) . [41]consists of two of the parts discussed above; the first includesthe slipping surface, while the second part includes the sur-face at infinity. It is easy to see that this function satisfies the The precise forms of the kernels and the functions c0 and

f0j can be computed from the two Eqs. [39] and [40] above;homogeneous differential Eq. [36] by applying the operatorL̂(r) to both sides. Since the integration and the differentia- however, these quantities are fairly complicated and their

detailed structures will not be given here. The two integraltion are carried out with respect to the components of r *each of these will commute with the operator L̂(r) and equations in Eq. [41] can be solved by the standard iterative

method in which as a first approximation the functions cbecause of Eq. [30] either a delta function or its derivativewill result. Since one of the arguments, r *, is by definition and fj within the integral sign are approximated by c0 and

f0j , respectively, and then the method repeated to obtainon the surface, while the other, r , is within the enclosedvolume they cannot become equal, thus causing the delta improved approximations by replacing c and fj within the

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244 PAUL AND KALER

integral sign by the results of the previous step. In principle external field perturbations. In the methodology adopted inthis paper the continuity equation is replaced by Eq. [21]this process must be carried out an infinite number of times;

however, the analysis shows that c0 and f0j are directly along with the definitions given in the two subsequent equa-tions. Writing these results in the set of curvilinear coordi-proportional to the applied potential V through the parame-

ters Bn defined in Eq. [11] and since this work is restricted nates introduced earlier and separating the £ component ofthe operator Ç2 we getto a linear response to the applied field the iterative process

does not have to be carried out beyond the first order. Intro-ducing the boundary condition given by Eq. [20] and the Ì

Ì£ FQj1Qj2

Q£

Ìfj

Ì£ Gformal boundary condition of Eq. [27] the final versions ofthe two functions c and fj may be summarized in the follow-ing compact forms:

Å 0 ÌÌj1

FQ£Qj2

Qj1

Ìfj

Ìj1G 0 Ì

Ìj2FQ

£Qj1

Qj2

Ìfj

Ìj2G

c(r) Å C0(r) / ∑n ,m

BnCnm(r) [42]

fj(r) Å D0j(r) / ∑n ,m

BnDjnm(r) . [43] / zje

kTQ 2

j1Q 2

j2

ÌF 0

Ì£ÌÌ£

(fj 0 FB) 0 ivlj

kT

1 Q£Qj1

Qj2(fj / c) . [44]The details of the four functions, C0 , Cnm , D0j , and Djnm can

be worked out once the functions c0 and f0j and the twoIn writing this equation we have assumed that the unper-kernels K and Kj have been computed. These quantities canturbed potential C 0 depends upon the variable £ only. Inte-be obtained after some straightforward but tedious calcula-grating both sides of this equation over all £(£s £ £ £ `)tions.and applying the boundary conditions given by Eqs. [27]and [28] we obtainIV. CALCULATION OF THE BULK POTENTIAL

PARAMETERS

0 Ìfj

Ì£ Z£Å£s

Å 0 1V

ÌCB

Ì£ Z£Å£s

Å Q (£s)£

Q (£s)j1

Q (£s)j1

Dukhin and Shilov (4) have described a method wherebythe parameters in the harmonic (satisfying Laplace’s equa-tion) component of a generalized potential may be computedfor colloidal particles with spherical symmetry. This ap- 1 F0 Ì

Ìj1*

`

£s

d£Q

£Qj2

Qj1

Ìfj

Ìj1

0 ÌÌj2

*`

£sproach consists of taking an integral transform of the densityconservation law. As discussed in the previous paper, for ahemispherical electrode, an integral along the radial direction

1 d£Q

£Qj1

Qj2

Ìfj

Ìj2

/ zje

kT *`

£s

d£Q 2j1

Q 2j2extending from the surface of the electrode to a surface at

infinity results in a transform. This transform displays thefact that the difference between the radial components of 1 ÌF

0

Ì£ÌÌ£

(fj 0 FB) 0 ivlj

kT *`

£sthe total ionic fluxes at the two surfaces must be accountedfor in two ways:

1 d£Q£Qj1

Qj2(fj / c)G . [45](a) the total flux along the tangential directions at these

two surfaces,(b) the perturbation caused by the application of the ex-

ternal field. Here Q (£s)a , a Å £, j1 , j2 implies these quantities calculatedat the slipping surface of the electrode. Substituting in thisThe parameters in the potential must be such that these crite-result Eq. [11] for CB and Eqs. [42] and [43] for c and fj ,ria are satisfied.respectively, followed by some manipulation we get a resultThis approach can be generalized for the present purposesthat may be expressed in the following compact form:if the hemispherical electrode is replaced by the more general

geometry envisaged in this paper and it is realized that the∑n

C *Bn(£s)En(j1 , j2)Å Ia(j1 , j2)/∑

n

BnIbn(j1 , j2) . [46]bulk potential does indeed satisfy Laplace’s Eq. [9] , thusbeing a harmonic potential. The set of parameters Bn mustbe chosen in such a manner that the total amount of ionsarriving to the electrode surface along the coordinate £ is Multiplication of both sides by E*m (j1 , j2)w(j1 , j2) and

integration over j1 and j2 yieldsbalanced by the currents along the other coordinates and the

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245GENERAL ELECTRODE POLARIZATION IN BIOELECTRICS, 2

Bm Å Sm / ∑n

RmnBn LO 0G0(r , r*) Å 4pd(r 0 r *) . [52]

In order to obtain the Green’s function for the exact Eq.Sm å

1

C *Bm(£s)Cm

* dj1dj2w(j1j2)E*m (j1j2)Ia(j1j2) [30] we proceed in the following steps:

( i) Replace the right-hand side of Eq. [30] with the left-hand side of Eq. [52].Rmn å

1

C *Bm(£s)Cm

* dj1dj2w(j1j2)E*m (j1j2)Ibn(j1j2) .( ii ) Rearrange this result into the following form after a

change in variable, r r r1 :[47]

LO 0(r1)Z(r1 , r*) Å 0LO 1(r1)(r1)G(r1 , r*)This result can be written in a matrix form:

Z(r1 , r) å G(r1 , r*) 0 G0(r1 , r*) . [53]B Å S / RrB [48]

(iii ) In Eq. [52] change r* r r1 and use the right-handor side of the equation so obtained to multiply the left-hand

side of Eq. [53] and vice versa.B Å (1 0 R)01

rS. [49] (iv) Integrate both sides of this result over all r1 .(v) The final answer can be written, to within an arbitrary

solution of Laplace’s equation, asIn principle the matrices involved in Eq. [49] are of infiniteorder but in practice, as discussed in Section II, the particular

G(r , r *)experiment will dictate the number of terms in the potentialexpansion that must be retained, hence the order of the matri-ces may not exceed two or three and Eq. [49] is practically Å G0(r , r *) 0 1

4p * dr1G0(r , r1)LO 1(r1)G(r1 , r*) . [54]usable. Within the confines of linear response theory eachof the components of the vector B will be proportional to the

Equation [54] is an integral equation that connects theapplied potential V by frequency dependent proportionalityzero order Green function to the exact Green function andconstants whose form will be given once Eq. [49] has beenwas first developed in the realm of quantum field theory bysolved:Dyson (5) and is often referred to as the Dyson equation.Since Dyson’s equation is not an algebraic equation but isB Å a(v)V . [50]an integral equation it cannot be solved directly to yield anexpression for G(r , r *) in terms of G0(r , r *) . One of theThe components of the vector a(v) are the correction ormost direct means of obtaining an approximation to G(r , r *)screening factors, only a single one of which was discussedis to start with the zero order Green function and substitute itin the previous paper.for the exact function under the integral sign in Eq. [54],carry out an iterative procedure, and truncate the infinite

V. EXPLICIT CALCULATION OF THE GREEN’Sexpansion thus obtained at some convenient point:

FUNCTION

G(r , r *)While Eq. [30] is simpler than the original differentialEq. [29] since it is a homogeneous equation, except at thepoint r Å r *, it is still a nonautonomous equation for which Å G0(r , r*) 0 1

4p * dr1G0(r , r1)LO 1(r1)G0(r1 , r*)exact solutions are, in general, impossible to calculate. It ispossible, however, to obtain very good approximations start-

/ 1(4p)2 * dr1dr2G0(r , r1)LO 1(r1)G0(r1 , r2)ing from the solution of a simpler equation.

Let us suppose that the operator L̂(r) can be partitioned1 LO 1(r2)G0(r2 , r*) / rrr. [55]according to

LO Å LO 0 / LO 1 , [51] It is shown in standard text books on mathematical methods(see, for example, Margenau and Murphy (3)) that thisexpansion is convergent and is exact if all infinite terms arewhere the Green function G0(r , r *) for the operator L̂0

can be computed exactly in the curvilinear coordinate frame taken into consideration. However, in most real situationssuch a convergence is slow and there is no guarantee thatbeing considered in this paper:

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246 PAUL AND KALER

any finite order truncation will prove to be adequate unless respect to the exact Green function and the result evaluatedat G Å Gt . In this expansion we do not consider terms thatthe operator L̂1 contains a multiplicative parameter that is

very small and thus forces rapid convergence of the expan- are higher than the first order in the deviation of the trialGreen function from the exact Green function.sion. The presence of such a parameter is not at all assured.

One of the most remarkable properties of the Dyson equa- (iv) Inserting the function F(G) from Eq. [58] into[dF/dG]t and carrying out the differentiation we obtaintion is that it obeys a variational principle that enables us to

calculate approximations even when the convergence rate isvery slow. Since this property is not widely known but only FdF

dGGt

Å 0 14p

1LO 1G0 0 1 1 å 0 1 b [60]found in some papers and books (see, for example, Paul(6)) on field theory that are not readily accessible to elec-trochemists, we present it in some detail. This proof of thevariational theorem also serves to provide the algorithm for b Å 1

4pLO 1G0 / 1. [61]

practical calculations. We now present the proof in the fol-lowing steps:

In Appendix I we include some of the details needed in(i) By resumming the iterative expansion presented in

order to carry out this differentiation and to understand theEq. [55] it is easy to see that the Dyson equation can be

meaning of the various quantities involved. Inserting Eq.written in a completely equivalent form:

[60] into Eq. [59] we obtain

G(r , r*)F(G) Å F(Gt ) 0 (G 0 Gt )rb Å 0. [62]

Å G0(r , r*) 0 14p * dr1G(r , r1)LO 1(r1)G0(r1 , r*) . [56]

(v) In order to solve this equation for G we require b01 ,which is given in Appendix II and is of the form

It is convenient to write both Eqs. [54] and [56] in the formof matrix equations in which the summation over indicesare replaced by integrations over the continuous variables. b01 Å 1 0 1

4pLO 1G . [63]

It is also important to keep in mind that the operator L̂1

acts on the first index (variable) of the matrix immediatelyThis result contains the exact Green function G and we re-to its right. Thus Dyson’s equation can be written in twoplace it by another guessed approximate form Gt to obtainequivalent forms:an approximate inverse. Inserting this result in Eq. [62] andsolving for G an approximate form of the latter indicated by

G Å G0 01

4pG0rLO 1G Å G0 0

14p

GrLO 1G0 . [57] the symbol Ga is produced:

( ii ) Equation [57] may be expressed as a function F(G) Ga Å1

4pGtrLO 1Gt /

1(4p)2 GtrLO 1G0rLO 1Gt

for the exact Green function in the form

0 14p

GtrLO 1G0 01

4pG0rLO 1Gt / G0 . [64]

F(G) Å G0 01

4pGrLO 1G0 0 G Å 0. [58]

(vi) Let us suppose that we make an arbitrary choice of(iii ) If this matrix equation, F(G) Å 0, could be solved Gt and Gt and let the difference of these from the exact Green

then the exact Green function can be calculated. Unfortu- function benately, however, such an exact solution is not possible andwe seek suitable approximations. If we, therefore, consider

Gt Å G / dGt , Gt Å G / dGt , [65]a trial guessed solution Gt then a Taylor expansion of F(G)about this approximation would yield

where it has not been assumed that either dGt or dGt are inany way small. Upon the insertion of these trial functions

F(G) Å F(Gt ) / FdF

dGGt

r

r

(G 0 Gt ) / rrr Å 0. [59] into Eq. [64] we obtain an approximation Ga that differsfrom the exact Green function by dGa (G Å Ga / dGa ) . It caneasily be seen that with these substitution several terms canbe grouped together such that either forms of Eq. [57] canHere [dF/dG]t implies a partial derivative of Eq. [58] with

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247GENERAL ELECTRODE POLARIZATION IN BIOELECTRICS, 2

be applied to each group, causing them to vanish and the tional form of these functions up to the investigator and itis advantageous to incorporate any physical intuition avail-final result collected in the formable in the selection. Since in this paper we are only layingdown the general theoretical framework for the calculation

dGa Å1

4p FdGtrLO 1dGt /1

4pdGtrLO 1G0rLO 1dGtG . [66] of the effects of electrode polarization on dielectrophoresis

and electrorotation we will not make any specific choicesfor the trial functions for the present. It may be pointed out

This yields two equations: that the variational method outlined in this section will playa significant role when the dynamics of the double layer isinfluenced by the Stern layer and the Zeta potential.dGa

dGt

Å 14p

1LO 1S1 / 14pLO 1G0DrdGt [67]

VI. CONCLUSIONS

In this paper we have presented the general theory thatandis required for the computation of the effects of electrodepolarization on the parameters of the bulk potential CB . Itis this potential that determines the external field experienced

dGa

dGtÅ 1

4pdGtrS1 / 1

4pLO 1G0DrLO 11. [68]

by a cell that has been levitated during dielectrophoresis orelectrorotation. All electrical quantities that are calculatedbased on the assumption that the levitated cell experiencesThese equations show that the derivatives of the approxi-the external field due to the unscreened charge on the elec-mate Green function with respect to the trial functions aretrode have to be corrected by the screening vector a(v) .linear in the error that is incurred in the trial function and,The components of this vector represent the corrections byin fact, if the errors were zero these two functional deriva-which each point on the experimentally measured dielectro-tives would vanish. This is the statement of the variationalphoretic or electrorotation spectrum must be multiplied inprinciple that is introduced as a result of the relationshiporder to obtain the true response of the cell.between the zero order and exact Green functions displayed

In order to apply the theory to specific electrode geome-by Dyson’s equation. Mathematically this may be put suc-tries which have been obtained either by taking actual elec-cinctly as follows:trodes or by simulating them through micromachining tech-niques the above theory has to be specifically tailored to thesituation. This point is illustrated by the example of theFdGa

dGtGdG t Å 0,dGtÅ0

Å 0 [69]hemispherical electrode studied in the previous paper.

APPENDIX IFdGa

dGtGdG t Å 0,dGtÅ0

Å 0. [70]The notion of partial differentiation of a quantity com-

posed of a product of matrices with respect to one of thematrices is fairly straightforward when discrete indicies areEquations [69] and [70] provide us with a useful methodinvolved. For example the differentiation of the matrix prod-for ascertaining the ‘‘best’’ trial functions. If we select Gtuct C Å ArB may be carried out in the following manner:and Gt to be dependent upon sets of parameters: {a1 , a2 ,

rrran} and {b1 , b2 , rrrbm}, respectively, then the varia-Cij Å ∑

k

AikBk jtionally best approximation, Ga , to the exact Green functioncan be obtained by requiring the parameters to satisfy n /m simultaneous equations: dCij

dAmn

Å ∑k

dAik

dAmn

Bk j Å ∑k

dimdnkBk j

ÌGa

Ìai

Å 0 ( i Å 1, 2, rrrn) dC

dAÅ 1 1rB.

ÌGa

Ìbi

Å 0 ( i Å 1, 2, rrrm) . [71]In the case of continuous variables the same principles holdexcept that all unit matrices are now to be interpreted asDirac delta functions and a dot implies an integration. ThusThis, of course, still leaves the explicit choice of the func-

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248 PAUL AND KALER

the above example for continuous variables is interpreted asbrb01 Å 1 / 1

4pLO 1(G0 / G0rx / 4pLO 01

rx).follows:

We therefore requireC(r , r *) Å * dr1A(r , r1)B(r1 , r*)

(G0 / G0rx / 4pLO 01rx) Å 0.

dC(r , r *)dA(r2 , r3)

Å * dr1dA(r , r1)dA(r2 , r3)

B(r1 , r*) From Eq. [58] this will follow if and only if

Å * dr1d(r 0 r2)d(r3 0 r1)B(r1 0 r *) x Å 0 14pLO 1G .

Thus Eq. [63] follows from this result.dC

dAÅ 1 1rB.

REFERENCES

1. Paul, R., and Kaler, K. V. I. S., J. Colloid Interface Sci. 194, 225APPENDIX II (1997).

2. Oldham, K., and Myland, J. C., ‘‘Fundamentals of ElectrochemicalScience,’’ Chap. 9. Academic Press, Toronto, 1994.The matrix b is defined in Eq. [61] and its inverse may

3. Margenau, H., and Murphy, M. H., ‘‘The Mathematics of Physics Andbe written as Chemistry,’’ Chap. 5. Van Nostrand, Toronto, 1961.

4. Dukhin, S. S., and Shilov, V. N., ‘‘Dielectric Phenomena and the Dou-ble Layer in Disperse Systems and Polyelectrolytes,’’ Chap. III. Wiley,

b01 Å 1 / x, Toronto, 1974.5. Dyson, F. J., Phys. Rev. 75, 1736 (1949).6. Paul, R., ‘‘Field Theoretical Methods in Chemical Physics,’’ Chap. V.

Elsevier Scientific, Amsterdam, 1982.with x an unknown matrix. The product with b yields

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