Dielectrophoretic force

J. theor. Biol. (1972) 37, l-13

Dielectrophomtic Force HJZXBERT A. POE&~ AND JOE S. CRANE$

Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74074, U.S. A. (Received 28 July 1971) Non-uniform electric fields produce motion (dielectrophoresis) of cells and other neutral bodies suspended in liquid media. The general force expression for this new and useful technique is derived. The equation is a general one, capable of including frequency-dependent conduction or polarization mechanisms as well as provision for various desired geometric distributions of dielectrics. It forms the basis for studying interpretive models of the dielectrophoretic response of suspended bodies such as cells and organelles. 1. Dielectrophoretic

Force

In the discussion we shall present a generalized theory of the dielectrophoretic force on a body suspended in a fluid. As we shall show from experimental experience with cellular dielectrophoresis, it is desirable that the theory be general enough to handle at least two basic mechanisms, one affecting conduction, the other affecting polarization. Having the generalized force expression then permits the later comparison with experiment of various specific model geometries and polarization mechanics. This should then permit more flexible,insight into the ways in which cells respond to electrical forces. In a non-uniform electric field (a.c. or d.c.) a neutral body such as a cell is usually constrained to move toward the region of greatest field intensity. Such motion induced by the action of a non-uniform electric field on a polarizable object is termed dielectrophoresis (Pohl, 1951, 1958). It is to be distinguished from electrophoresis which is motion caused by the action of an electric field (whether uniform or non-uniform) on a charged object. Die@ctrophoresis, acting in effect upon the dielectric constant rather than upon the charge, has proven to be useful in making selective separations of either living or inanimate systems, and in studying the nature of individual systems 7 NATO Senior Science Fellow on leave to Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge, England. $ Present address: Department of Physics, Cameron State College, Lawton. Oklahoma. 1 1 T.B.

2

H. A. POHL

AND

J. S. CRANE

(Pohl, 1960a,b, 1968; Pohl & Plymale, 1960; Pohl & Schwar, 1960; Pohl & Hawk, 1966; Crane & Pohl, 1968; Pohl & Pickard, 1969; Pohl & Crane, 1971). It has been shown that the dielectrophoretic response of living cells and organelles suspended in aqueous media varies greatly with the frequency of the applied field and with the conductivity of the liquid medium (Crane & Pohl, 1968; Wiley, 1970; Crane, 1970). A typical curve is shown sketched in Fig. 1. The amount of cells collected in a minute’s time at a fixed voltage, etc., is shown as a function of frequency. It can be seen that there are frequency ranges over which no collection occurs, i.e. in which the cells act as if they are less polarizable than the aqueous medium. In the curve shown there are indications of three frequency ranges in which cell collection at the high field region is strong, and other regions in which no collection occurs. The actual shape of the frequency vs. dielectrophoretic response curve varies with the type of organism and with its physiological state. Some cells show but a single peak, some show as many as three. We should like to be prepared to explain such response variations.

102

104 Log [frequency

FIG. 1. A typical (diagrammatic) suspension of living cells.

106

IO’

( Y 11

yield spectrum obtained with the dielectrophoresis

of a

Let us consider first a set of mechanisms by which frequency variant polarization could arise, then finally a generalized theory which can accommodate any such mechanism. For a perfectly insulating spherical particle of radius a and permittivity s2 suspended in an insulating fluid medium of permittivity .si the dielectrophoretic force can be shown to be: F = 2na 3 -4~ - MWz

5 +2s,

.Y

DIELBCTROPHORETIC

3

FORCE

where E is the r.m.s. value of the applied field strength. Starting with this equation and assuming the field to be that between concentric spherical electrodes, it is possible to show that the yield, y (expressed as the length of the chains of particles formed) is given by (Pohl, 1951, 1958) Y = ;;;;!;;)

r;;;;;;;;‘l

112,

(2)

where C is the particle suspension concentration, ri and rz are the respective radii of the inner and outer spherical electrodes, Vis the applied r.m.s. voltage, t is the elapsed time, and q is the viscosity of the suspending medium. Application of equation (2) to real systems (i.e. to real, not “perfect” dielectrics) shows good agreement with experiment in some respects. Linear response of the yield with cellular concentration, and with voltage, is usually observed, as is the inverse half-power dependence on viscosity, and upon the square root of the collection time. Equation (2), however, contains no provision for conduction or frequency effects. An adequate theory must provide explanations for the effects of the latter parameters. It is apparent that the permittivities, ez and si, of the particle and the medium should be allowed to be frequency and conductivity dependent. Attributing a frequency dependence to the apparent permittivity is a familiar idea. Most dielectric constant measurements, for example, consist of determining the effective parallel capacitance of the material and attributing it to an effective permittivity. As the effective capacitance changes with applied frequency, this permittivity must also change. The problem is then to devise a physical interpretation of that change. The same type of problem must be faced in explaining dielectrophoresis results. In this case, however, the physical interpretation must also explain the observed strong conductivity dependence. It is therefore likely that provision must be made in the theory for the simultaneous presence of two basic mechanisms, one affecting conduction, the other affecting polarization, Each basic mechanism may itself in turn be a composite of like mechanisms with separate relaxation times. There are a number of specific mechanisms that have been proposed as possible explanations for the variation of the dielectric constant of materials with frequency. They are: (i) dipole orientation of molecules or pendant groups of atoms (Smyth, 1955; Von Hippel, 1954); (ii) hyperelectronic polarization of the long conjugated molecular constituents (Rosen & Pohl, 1966; Hartman & Pohl, 1968; Wyhof & Pohl, 1970); (iii) “gating” or ionic flow bias mechanisms operating in cell membranes (Cole, 1940; Schwan, 1957; Schwan & Cole, 1960); (iv) interfacial polarization at various interfaces (Van Hippel, 1954; Maxwell, 1873); (v) relaxation of ionic double

4

H. A. POHL AND J. S. CRANE

layers such as in the surrounding ion atmosphere (Miles & Robertson, 1932; Schwarz, 1962; Schwan, 1962; Pollak, 1965). Before going on to a more detailed analysis of the suggested five mechanisms, let us consider a particularly simple general scheme to account for the observed frequency variations of dielectrophoretic yield. In considering an extension of equation (1) to include dielectric polarizations which are frequency dependent, we might imagine replacing the term &I(&2- 4 82 +2&r by the real part of its analogue where the terms &2 and ei are now complex quantities, a2 and B,, and g: is the complex conjugate of 8i, viz., by

We now focus on the difference term, (d2 -&) and pull out its absolute value. For a complicated particle such as a cell it is easy to imagine a set of polarization mechanisms simultaneously acting, each with a different relaxation time. The effective polarization, Ii?,/, for such a system might then typically appear (Smyth, 1955) as sketched in Fig. 2 (heavy curve). As the frequency increases,

102

104 Log [frequency

106 I YI I

108

FIG. 2. Diagrammatic sketch of the frequency dependence of the absolute values of the complex dielectric constants of a complicated “solid” dielectric (heavy line), and a conductive liquid such as slightly salty water (light line). The shaded portions are regions in which the “effective” dielectric constant of the solid exceeds that of the liquid medium.

the slower systems of polarization drop out successively. The polarization curve sketched for the liquid medium is that for a conductive medium in which, as is usual (Van Hippel, 1954), we consider the polarization complex

DIELECTROPHORETIC 4

= 63

5

FORCE

- i al/co)

(3)

and IhI = Je:+w42,

(4) where u1 is the conductivity and o is the angular frequency. The net “polarization” is shown in Fig. 3. The positive (shaded) portion of the curve then resembles the observed collectability or “yield” of cells (Fig. 1) such as is observed. We remark that the curve in Fig. 3 is the composite resultant of polarizations which individually only decrease with frequency increase. Since it is normal and usual in dielectric studies to lind the polarization to decrease

Id

74

6

Log

[frequency

IO6

( Y) ;”

FIG. 3. Diagrammatic sketch of the tkequency dependenceof the difference in effective dielectric constants of a solid and its surrounding liquid medium. Note resemblance of positive (shaded) regions to the experimental dielectrophoretic yield spectrum.

or “relax” with frequency (except, say, at absorption band edges) in just such a manner, the simple picture(Figs 2 and 3) just presented toexplain the observed frequency response of diekztrophoresis (e.g. Fig. 1) is an attractive one. Our present knowledge of the various polarization-frequency regimes is too fragmentary, however, to let one be certain of the validity of such a picture. Five specific mechanisms were mentioned as candidates for consideration in discussing the frequency and conductivity of effects on cellular dielectro-, phoresis. It is quite possible that all could be operative simultaneously, with the relative importance determined by the local experimental conditions such as frequency, conductivity, etc. At high frequencies the dipolar orientation (Smyth, 1955; Von Hippel, 1954) is likely to be the dominant effect, whereas at lower frequencies the gating mechanism, hyperelectronic polarization and ion layer mechanisms might be more important.

6

H.

A.

POHL

AND

J. S. CRANE

A survey of the literature indicates that the recent trend is to explain low frequency polarization results in terms of the surrounding ion atmosphere (Miles & Robertson, 1932; Schwarz, 1962; Schwan, 1962; Pollak, 1965) and its pliant relation to the relatively fixed surface charge. This position has been taken by Miles & Robertson (1932), O’Konski (1960), Schwan (1962) and Schwarz (1962). It is also the approach of Dintzis, Oncley & Fuoss (1954) in their explanation of the dispersion for polyelectrolytes such as those of molecular weight about 2 x 106, and which are comparable in size to DNA fragments. Although one notes this trend to explain low frequency polarization behavior in terms of ion atmospheres, the other processes cannot be ruled out completely. Their effects should be considered in any general treatment of the problem. Hyperelectronic polarization (a sort of Maxwell-Wagner polarization on a molecular scale) (Hartman & Pohl, 1968), biased ion passage across membranes (gating) (Cole, 1940), and interfacial polarization (Maxwell, 1873) can all be considered together with ion double layer and dipole orientation effects in terms of appropriate conductivities and polarizations which depend on frequency. Some are best considered from the viewpoint of polarizations which are frequency dependent, some as conduction processes with frequency dependence. We shall wish to be prepared to handle either type in a general expression for the dielectrophoretic force. 2. General Expression for Dielectrophoretic

Force

In the preceding discussion, a number of polarization mechanisms were considered which could be responsible for the observed dependence of cellular dielectrophoresis on field frequency and medium conductivity. To isolate the effective mechanism it is first necessary to have a theoretical expression for the yield or amount of dielectrophoretic collection. Accordingly we shall here obtain a general expression for the dielectrophoretic force and in a later paper present a model incorporating several specific mechanisms. With these specifications and the general force expression, the theoretical calculation of the yield-frequency-conductivity relations will then be made. We now turn to the derivation of the matter of the general expression for dielectrophoretic force on a body which sits in a fluid medium, both of which may be polarizable and conductive. The development prepares for the possibility that both the polarizing and the conduction mechanisms are basically separate mechanisms in their frequency responses. As was noted earlier, the consideration of materials as perfect insulators leads to a force expression (equation (1)) that is not compatible with the experimental results on cells. We must therefore determine a more general

DIELECTROPHORETIC

FORCE

7

expression which is applicable to real materials. These non-perfect dielectrics will in general exhibit both dielectric constants and conductivities. If the applied field is periodic there will also be a time lag of response. This time lag is most easily expressed as a phase difference between the applied field and the resulting current and polarization (Van Hippel, 1954). For a sinusoidal field impressed across a simple (linear) medium, the current, charge density, polarization, and other descriptive parameters will also be sinusoidal, but not necessarily in phase with the field. That is for E = EO e@’ we have J,

=

J,,

ej(Q+eJ),

D

=

Do

ef(~-b),

p

=

p.

ej(mt-eP),

p = p. &W-W

H = Ho ej(Q-h;, B : B, ej(C+b), M = MO ei(Ch%d f (5) where J, is the conduction current, D is the dielectric displacement, p is the volume density of charge, P is the electric polarization, B is the magnetic induction, H is the magnetic field intensity, M is the magnetization, t is the time, o is the angular frequency (o = 2x0, where u is the cyclic frequency), j is J-1, O1is the phase angle and i,, is the amplitude for the instantaneous value of the quantity i. For simple media (i.e. saturation effects absent), the relations D = EE and J, = aE (6) also hold. It follows that e, the dielectric constant or permittivity, conductivity, must be given by E = (Do/E,) e-jeD and u = (Jo/E,) e-jeJ,

and u, the (7) (8)

and that they are therefore complex. An alternate way of expressing these complex quantities is to break them up into real and imaginary parts such as 6 3 e’-j$ (9) and 0 = b’-j#. (10) The ratio of the imaginary part to the real part determines the amount by which the related quantities are out of phase. That is e”l~’ = tan Ob

(11)

and d/u’

= tan 0,.

(12)

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H. A. POHL

AND

I. S. CRANE

If these quantities are pictured in the complex plane, they can be considered as vectors which can be broken up into two components, one real and one imaginary. The amounts of phase lag depend on the components of e and 0 respectively, since these are the quantities relating D and J, to E. 3. Maxwell’s Equations For sinusoidal fields of the form given in equations (5), Maxwell’s equations for simple media become (King, 1945) e(V*E) VxE V-B VxH

= = = =

pr -jwB 0 (a+ jme)E.

(13) (14) (15) (16)

The boundary conditions at an interface between materials 1 and 2 are el(nl*W+e2(n2*EJ n, xE,+n,xE, n,xH,+n,xH, n,*B,+n,*B, The continuity

= = = =

-frlf--t12f9 0, -I,f-12f, 0.

(17) (18)

(1% (20)

equations for conservation of charge are V*(aE) +jwp, = 0

(21

and (22) V,(I,f+12f)+jo(?,f+r12f)-nl.(alEl)-n2.(a2E2) = 0. In the above equations, the usual meanings are given to the familiar symbols of E, B, H, e, w, 6. The less familiar symbols are defined as follows: n, and n2 are unit outward normal vectors, Pf is the volume density of free charge, nf is the surface density of free charge, and I, is the surface density of free current (current/width), which is different from zero only for perfect conductors. Eliminating Pf from equations (13) and (21) gives

V*[(e -ja/o)E] = 0 (23) everywhere except at the boundaries where of course V(a+j~e) # 0. Combining equations (17) and (22) to eliminate (n, f + qZf) and restricting ourselves to non-perfect conductors gives the conditions at the boundaries as (cl +joe,>(n,~E,)+(a,+joe,)(n,~E,) It is convenient at this point to define the expression K s a+jm

= 0.

(24)

(25)

DIELECTROPHORBTIC

FORCE

9

as the complex conduction factor and the quantity 5 = K/jo = s--j@0 (26) as the complex dielectric factor. Dividing equation (24) by jw then gives t,(n, *Ed + Mz -E,) = 0. (27) This gives the boundary condition for real media and sinusoidal fields as identical in form with that for ideal dielectric in static fields if the dielectric constant is replaced by the complex dielectric factor <. 4. General Energy and Force Equations In order to develop a generalized force equation it is necessary to first determine a general expression for the electric energy of a system. The force can then be obtained by taking the negative gradient of this energy. For sinusoidal fields it is shown by Ring (1945) that the time averaged electric energy in a volume z is given by U = i j e,E2 dr, (2% t where E is the amplitude of the varying E field and not the r.m.s. value. Recalling from equation (26) that 8, = Re (0 = Re ((9, (29) where Re(<) = E’- co/o - E,, the effective permittivity, and where {* is the complex conjugate of 5 and Re (<) is the real part oft, allows the energy to be expressed as U = + j Re (E*{*E9 dr. (30) z Detining the quantity D as the “total displacement vector” by (see equations (6) and (26)) D = eE = D-jJ,/o (31) gives the energy as 17 = #j Re(E*D*)dr. (32) To determine the energy of a body in an external field requires the integral to be evaluated o=r all space. This is usually impossible to do directly, so that a simpler integration is required. It is possible to develop an expression for the energy in terms of an integral over the body alone. This development is similar to the procedure developed by Schwarz (1963) for the non-conducting case. In that case, the conduction current is zero and D is equal to D. Consider a field E, established by charged conductors of finite extent in a linear isotropic medium of complex dielectric factor {.+ with a resulting total

10

H. A. POHL

AND

J. S. CRANE

displacement of De. Now insert a body of complex dielectric factor 5 and denote the resultant field vectors by E and D. The change in energy will be given according to equation (32) as Ai7=;t j Re (E*D*-E,,.Dg) dr. (33) all space The integrand can be modified using the identity Re (E-D*-E,+D;) = Re {[E+(g,/
CE+(~,/5t)E,l*.(D-D,) = - CV~+(54/rX)V~,l*.(D-D,) = - (V+)*.(D-De) = - V($*)*(D-D,),

(35)

where (36)

* = 4 +(54/t4*)40. But -V$**(D-D,)

= -V@*(D-D,)]+$*[V(D-De)].

(37) Using equation (23) and V * D = 0, we note that the last term in equation (37) is zero. We note in passing that the derivation given by Schwarz (1963) asserts div D = div Do. But div D = p if free charge is present, hence his result is not general. The use here of div D = 0 is not so restricted, and therefore gives a general result. The volume integral of the remaining first term can be transformed by the divergence theorem into a surface integral over the enclosing surfaces, giving V*(rl/*(D-D,)}dz - all ,s,,ce

= - So-m lim Jo il/*P-Do) + c

dS

j +*(D--Do)

k Sr

6

(38)

where the S, are the conductor surfaces and Se is a surface which increases to infinity. Since I+?*is established by the conductors, it is proportional to I/r at large distances. D is proportional to E and V$. This gives D proportional to l/r2 at large distances. Since the surface area is proportional to r’, the value of the Se integral is proportional to l/r and goes to zero as r + 0~). The S, integrals are associated with the work done, A W, at the conductors upon introducing the body and so are not associated with the energy of the body per se. Thus the first term in equation (34) gives the work done at the conductors and the second is non-zero only in the body. Equation (33) then becomes,

DIELECTROPHORETIC

11

FORCE

dr+AW. Au=* jbod, Re ~*D,*-(~X/~,)E,**D] (39) The time averaged mean energy of the body is therefore AB - A W = is, or W = 4 j Re [E-D,+-(tX/&JE,**D] dr = +7

Re {St[E.E,*-(r/e,)E,*.E]}

= t 1: Re{CC1-WWW)

dz

dr,

WI

and we have now reduced the integral to one over the body alone. This result reduces to the special case treated by Schwarz (1963) when 5 = E. It is now a simple matter to obtain the general force equation by taking the negative gradient of equation (40). That operation gives F = -aLY

Rekk:(l-$Eg*E]}dr.

(41)

Before this expression can be evaluated, one must know the original impressed field Ec, the resultant field E throughout the body, and the complex dielectric factor for all parts of the body. E, will be determined by the electrode geometry, r will depend upon the conduction and polarization mechanisms involved, and E will be specified in terms of the r’s by the boundary conditions of equation (24). If equation (41) is a general force equation, then it must contain provisions for each of the mechanisms discussed earlier. This is easily shown to be the case. The rotating dipole mechanism and the relaxation effects are provided for by allowing e to be complex. The gating mechanism and also any other conduction relaxation effects arise through the complex conductivity. The Maxwell-Wagner multi-layer membrane effects are direct results of the boundary conditions on E at the layer interfaces. And finally, the relaxation of an ionic atmosphere can be included simply by considering an extra outer layer on the body of interest to have a high and complex conductivity. In a subsequent paper (Crane & Pohl, 1972) we shall apply equation (41) to the case of living and dead yeast cells with satisfactory results. 5. summary The simple and rapid technique of dielectrophoresis as applied to suspensions of organelles or cells show that the gathering of these bodies at the region of strongest field readily reflects certain polarization mechanisms sensitive to their physiological state; and that these dielectrophoretic responses in a.c. fields are quite dependent upon the frequency of the applied electric field and upon the conductivity of the aqueous medium. In view of the frequency and conductivity effects, simple dielectric theory based on con-

12

H. A. POHL

AND

J. S. CRANE

sideration only of static dielectric constants as for perfect insulators proves to be inadequate. Preparatory to investigating experimentally which of a number of possible mechanisms are actually or probably involved in cellular dielectrophoresis it is necessary to provide a suitable theoretical framework to describe dielectrophoretic force. Since several mechanisms may well be simultaneously involved, and since some may be best looked upon as frequency-dependent polarizations while others are best regarded as frequency-dependent conduction processes, the general theory will need to provide accommodation for both types. The required general force expression which has these provisions is derived here (equation (41)). This is done after showing how the required complex dielectric factor, <, can be regarded in a dual light, either as a sum of two fundamental but complex (and frequency-dependent) quantities, 0 and E, the inherent conductivity and inherent dielectric constant; or as the complex sum of two real quantities, a, and E,, the effective (and hence experimentally apparent) conductivity and dielectric constant. The first pair, r~ = a(w) and E = E(W) may be regarded as the interpretive set; the latter pair, Q, and E,, as the observable set. Given a field distribution and a plausible set of assumptions regarding polarization and conduction mechanisms and their geometric distribution, equation (41) can then later be applied to test the assumed mechanisms by comparison with experiment. It is planned to present in a later paper the detailed application of such physical assumptions in a model of celhdar dielectrophoresis, and to compare the predictions with experiment. The authors acknowledge with sincere thanks the financial support received from the National Institutes of Health, Public Health Service, and from the Biomedical Grant support given by the National Institutes of Health through the Research Foundation of Oklahoma State University. One of us (H.A.P.) also wishes to expresshis appreciation to the Department of Physics of the University of California, Riverside for their considerable support and stimulus given him during his stay in that institution; and to Professor N. F. Mott for his cordiality during the author’s stay at the Cavendish Laboratory. REFERENCES COLE, K. S. (1940). Cold Spring Harb. quant. Symp. Biol. 8, 110. CRANE. J. S. (1970). Ph.D. Thesis. Oklahoma State University. CRANE; J. S. & PO&L, H. A. (1968). J. efectrochem. Sot. l15,v5S4. CRANE, J. S. & POHL, H. A. (1972). J. theor. Biol. 37, 15. DINTZIS, H. M., ONCLEY, J. L. & FUOSS,R. M. (1954). Proc. nom. Acad. Sci. U.S.A. 40,62. IIARTMAN, R. D. & POHL, H. A. (1968). J. Polym. Sci. A-I 6,113s. KING, R. W. P. (1945). Electromagnetic Engineering, Vol. 1. New York: McGraw-Hill. MAXWELL, J. C. (1873). A Treathe on Electricity and Magnetism. London: Oxford University Press.

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MILES, J. B. 8c ROBER~N, H. P. (1932). P&s. Rev. 40, 583. O’KONSKI, C. T. (1960). J. phys. Gem., Zthucu 64,605. POAL, H. A. (1951). /. a&. i=hys. 22,869. POHL, H. A. (1958). J. apjd.Z’l?ys. 29,1182. POHL, H. A. (196Oa). J. electrochem. Sot. 107, 386. POHL, H. A. (1960b). Scient. Am. 203,107. POHL. H. A. (1968). J. electrochem. tic. 115, 584. POHL; H. A. ist &NE, J. S. (1971). mop&: J. 11,711. POHL, H. A. I% HAWK, I. (1966). Science, N. Y. 152,647. POHL, H. A. & PACKARD, W. F. (1969). Dielectropboretic and Electrophoretic Deposition. New York: Elcctrcchemical Society Inc. Porn, H. A. & PLYMALE, C. E. (1960). J. electrochem. Sot. 107,390. POHL., H. A. & SCHWAR, J. P. (1960). J. electrochem. SOC. 107,383. POLLAK, M. (1965). J. them. Phys. 43,908. ROSEN, R. & POIU, H. A. (1966). J. Pofym. Sci. A 4,1135. SCHWAN, H. P. (1957). Adv. biol. med. Phys. 4,147. SCHWAN, H. P. & COLE, K. S. (1960). Med. Pbys. 3,52. SCHWAN, H. P., SCHWARTZ, G., MACZUK, J. & PAULY, H. (1962). J. phys. Cbem., Ztbacu 66,2626. SCHWARTZ, G. (1962). J. pbys. Gem.. Ithaca 66,2636. SCHWARTZ, G. (1963). J. cbem. Phys. 39,2387. SMYTH, C. P. (1955). Dielectric Behavior ud Structure. New York: McGraw-Hill. VON H~PEL, A. H. (1954). Dielectrics and Waves. New York: John Wiley. WYHOF, J. R. & POHL, H. A. (1970). J. Polym. Sci. A-2, 8. 1741. WILEY, K. (1970). M.S. Thesis. Oklahoma State University.