A short tutorial contribution to impedance and AC-electrokinetic characterization and manipulation of cells and media: Are electric methods more versatile than acoustic and laser methods?

While micro total analysis systems (μTAS) [Daridon et al. 2001, Kovarik et al. 2012] were originally developed for chemical analysis, lab-on-chip systems (LOCs) [Becker et al. 1995, Dürr et al. 2003, El-Ali et al. 2006, Fuhr et al. 1997, Goater et al. 1997, Wolf et al. 2006] were developed to monitor physiological cell parameters, cell manipulation and sorting, or for use as micro-bioreactors. Nowadays, a distinction between the two types of microsystems seems somewhat obsolete. With integrated sensors, the microsystems can be used for non-invasive, on-line monitoring of a cell’s physiological state and the effects of specific compounds [Bousse et al. 1994, Bousse and Parce 1994, Buehler et al. 2011] while neuro-chips can detect the action-potential patterns of electrically active neuronal networks [Gross et al. 1995, Haia and Spira 2012, Koester et al. 2010a]. Automated patch-clamp systems combine microfluidic structures to handle suspended single cells with patch electrodes for electrophysiological investigations [Koester et al. 2010b, Dunlop et al. 2008, Py et al. 2011]. The development of cell-based in vitro systems aims at reducing animal testing and costs in medical diagnostics or drug development. In these systems, microfluidic structures can be combined with integrated manipulation and sensor elements for cell or particle alignment, orientation, sorting, trapping, deformation or cell lysis and analysis [Barat et al. 2012, Becker et al. 1995, Dürr et al. 2003, Fiedler et al. 1998, Fuhr and Reichle 2000, Goater et al. 1997, Laurell et al. 2007, Nilsson et al. 2009, Sun and Morgan 2010, Schnelle et al. 2000].

Three different force fields are commonly used for contactless manipulation of biological particles or cells in microchips. All three are based on differences in the effective physical properties of the objects and their surrounding media. While optical traps (laser tweezers) exploit differences in the refractive indices (corresponding to permittivity differences at light frequencies for nonmagnetic objects) and are usually applied to trap single objects [Fuhr and Reichle 2000], acousto-fluidic cell manipulation methods have been elaborated for the complex handling of a multitude of objects including cells [Barat et al. 2012, Nilsson et al. 2009]. The technique is based on differences in the viscoelastic properties of the objects and the suspension medium. New setups allow for alignment, orientation, separation, sorting, and trapping [Guck et al. 2005, Laurell et al. 2007, Nilsson et al. 2009, Oberti et al. 2010]. Meanwhile, the performance of acoustofluidics [Glynne-Jones and Hill 2013] is comparable to that of earlier AC-electrokinetic techniques [Dürr et al. 2003, Müller et al. 1999, Schnelle et al. 2000].

In contrast to DC-electrokinetic techniques, such as electrophoresis and electro-osmosis, which are based on the interaction of permanent charges with the applied electric field, AC-electrokinetic techniques are based on the interaction of induced charges with the applied electric field. Accordingly, the induced force effects are based on differences in the effective electric polarizability of the objects and the suspension medium. Because AC-electrokinetics covers a larger frequency range than optical and acoustic methods and spans many frequency decades (<1 kHz to >1 GHz), AC-electrokinetics is more flexible in the choice of polarization effect to be exploited in the generation of the manipulation forces [Jones 1995, Hölzel 1997]. For a historical overview please see Gimsa et al. [1991] and Jones [1995].

Generally, the forces generated increase with the contrast between the objects and their surrounding media, i.e. the differences in the effective refractive indices, the viscoelasticities and the electric polarizabilities in optical, acoustic and AC-electrokinetic manipulation, respectively. For manipulation, an optimal gradient with respect to the object size and at an overall low energy absorption in and around the object is desirable. Different methods exist to increase the contrast for the three force fields, i.e. choosing an optimal field frequency/wavelength, temperature adjustment, staining, adjustment of the external medium properties, e.g. by changing viscosity and density in acoustic manipulation or in lowering the medium conductivity in AC electrokinetics.

Even though absorption will mainly result in temperature dissipation in all three techniques, the primary mechanisms of energy absorption are very different, ranging from direct acousto-mechanical vibrations to the absorption of single light quanta. Nevertheless, to the best of our knowledge the energy absorption in and around the objects, i.e. the actual load for the biological cells has not been compared for the three methods (for AC fields see Wachner et al. [2002], Retelj et al. [2013], Simeonova et al. [2002]).

Of course, it is advantageous if the property contrasts can be directly (label-free) exploited for the (physiological) characterization and/or handling of the object, such as is the case with the elastic properties of cancerous cells [Guck et al. 2005], the cell-membrane conductivity [Gimsa et al. 1989] or the abundance of membrane systems in the cells [Gimsa et al. 1991]. It would also be interesting to combine non-interfering force fields for certain manipulation purposes or to consider the cross talk of the induced effects [Barat et al. 2012, Fuhr and Reichle 2000, Schnelle et al. 2000].

Because AC-electrokinetics and electric impedance measurements use the same frequency range, both methods may (in principle) detect the same properties, e.g. cell proliferation in suspensions. Nevertheless, with adherent cells, only the impedance can be detected, e.g. from interdigitated electrode structures (IDES), preferably in the low kHz range [Ehret et al. 1997, Koester et al. 2010a].

This short review focuses on the various AC-electrokinetic effects that can be exploited in LOCs [Gimsa 2001, Jones 1995, Shih et al. 2013]. The frequency dependence of AC-electrokinetic effects such as electroorientation (EO), electrodeformation (ED), translation (dielectrophoresis; DP), aggregation or electrorotation (ER) can be used to characterize suspended single cells [Gimsa 1999, Gimsa et al. 1991, Griffin 1970]. These effects can also be exploited in the manipulation of cells for sorting, collecting and positioning in LOCs [Fuhr et al. 1997, Schnelle et al. 2000]. AC-electrokinetic effects arise from the interaction of the induced polarization of the objects with the inducing external field. Unlike impedance, AC-electrokinetic characterization requires a sufficiently high field strength (>1000 V/m) and the (microscopic) observability of the induced movements. Its advantage is a higher parameter resolution and the discrimination against electrode effects. Whereas impedance methods are integrative, AC-electrokinetic methods are differential, i.e. impedance determines the integral polarizability of suspension medium and the objects, while AC-electrokinetics is based on the difference in the effective polarizabilities of medium and object. Nevertheless, in both approaches, analytical descriptions are based on the same geometrical models, e.g. multi-shell ellipsoidal cell models.

While the observed effect depends on the field quality, the frequency dependence of the dipole moment induced in the object generates particular frequency spectra. These relations can be easily explained for DP and ER: while objects subject to DP are translated in a linear inhomogeneous field, they rotate in a circular homogeneous field in ER. The DP force and ER torque are proportional to the in-phase (mathematically: the real) and out-of-phase (mathematically: the imaginary) component of the induced dipole moment, respectively [Gimsa and Wachner 1998, Jones 1995]. Combining DP and ER spectra is ideal for the dielectric characterization of single objects, because the methods permit a method-based separate access to the real and imaginary components of the induced dipole moment. Nevertheless, the AC-electrokinetic effects are not independent of each another, and combined force spectra must be expected, e.g. broken DP and ER spectra with ranges separated by the reorientation of non-spherical objects. Such spectra can be observed in combined measurements of the EO, DP and ER spectra of ellipsoidal chicken red blood cells in isotonic media of different conductivities. An ellipsoidal single-shell model is the simplest possible approach for the explanation of such combined effects [Gimsa 2001, Yang and Lei 2007].

Interestingly, AC-electrokinetic forces can also be observed in the bulk of fluids when dielectric property gradients are introduced by local heating. In electrothermal traveling wave (TW) micropumps (ETTWμPs) [Fuhr et al. 1992 and 1994, Gimsa et al. 1997, Morgan et al. 2001, Perch-Nielsen et al. 2004, Liu et al. 2014] and electro-thermal micropumps (ETμPs) [Stubbe et al. 2013], the interaction of the external AC field with smeared polarization charges of the medium induces medium pumping.

Many of the cell manipulation or pumping elements can be produced as platinum or transparent indium-tin-oxide structures on glass chips, which may form the bottom plate of measuring chambers or self-sealing microfluidic systems.

Please note, that electric force effects on suspended objects or media can also be induced by the interaction of DC or AC fields with electric double layer (see: DC electrokinetics) or dipole charges (see: Debye effects). When an external AC field deforms the equilibrium distributions of such charges, objects or media are polarized. This polarization, as well as its frequency-dependent α-, β-, γ-, and δ-dispersions, may generate AC electrokinetic effects. Nevertheless, this paper focuses on AC field-induced polarizations at structural interfaces in inhomogeneous media (Maxwell-Wagner (i.e. β) polarizations and their dispersions). For a summary see e.g. Foster and Schwan [1996], Gimsa and Wachner [1998], Schwan et al. [1962].

The AC fields induce polarization charges at the structural interfaces inside inhomogeneous media (Maxwell-Wagner polarization). In suspensions or emulsions of micro- and nano-scale objects such as particles, oil droplets, air bubbles and biological cells, these structures are introduced by medium boundaries, e.g. biological membranes. Nevertheless, local heating may also introduce "smeared interfaces" into homogeneous bulk media, e.g. aqueous electrolytes. AC-electrokinetic effects are based on the forces generated by the interaction of these polarization charges with the inducing field. The observed effect depends on the temporal and spatial field properties. For objects, EO, ED, DP, ER, field-trapping/caging and travelling wave dielectrophoresis (TWDP) are well investigated. The frequency dependence of the dipole moment induced in the objects or volumes generates particular frequency spectra that provide information on the dielectric properties of the objects. Nevertheless, heated micro and nano volumes of a medium experience analogous forces, which can be used e.g. for pumping and mixing.

For biological cells, the general dielectric structure is introduced by two media of complementary electrical properties. While biological membranes have relative permittivities below 10 and specific conductivities in the range of 10^-6 S/m, aqueous solutions (cytoplasm, nucleoplasm and interstitial solution) have values of approximately 70 and 1 S/m, for their respective properties. Thus, from a physical point of view, the lipid and aqueous phases can qualitatively be represented by air (or vacuum) and water, respectively. The simple block-model in Figure 1 consists of air and water blocks confined by electrodes. The electrode charges will influence charges in water but not in air. These interfacial charges are carried by ions (according to conductivity differences) and dipoles (according to the permittivity differences).

Fig. 1

A geometrical transition into spherical objects, i.e. a water droplet and an air bubble immersed in air and water, respectively, will not alter the charge distributions qualitatively.

Figure 2 shows the two complementary cases of the effective polarization of spherical objects that are less or more polarizable than their surrounding medium. Please note that their size allows for an electroquasistatic description of the polarization of micro-objects such as cells up to above 1 GHz. Figure 3 presents the single-shell model of a biological cell. For low external conductivity, biological cells may qualitatively behave like air bubbles (below membrane dispersion) or water droplets (above membrane dispersion) depending on the field frequency. While the cytoplasm is almost field-free at low frequencies, the membrane experiences extremely high field around its poles oriented in field direction (in a qualitative view the voltage dropping above the cell drops above its membrane).

Fig. 2

Fig. 3

Generally, AC-electrokinetic effects are modelled using the induced dipole moment of the objects. The dipole moment is proportional to the difference of the integrals over the effective local fields in the volume of the object and the undisturbed external field in the same volume in the absence of the object. The second integral is frequency-independent for constant external field strength. In the biological cell (Figure 3), the volume integral is dominated by the membrane field at low frequencies. At high frequencies, the membrane is capacitively bridged, the membrane-field contributions fade and the volume integral is dominated by the cytoplasmic field. Please note that the orientations of the dipole moments are antiparallel in the two cases.

According to physical principles, forces are generated by potential differences, e.g. the difference between the potentials at the site of the pole in the presence and absence of the object (Figure 4). Please note that the effective field in confocally shelled objects polarized in a homogeneous external field is constant and corresponds to that of Maxwell's equivalent body.

Fig. 4

Our influential-radius concept allows for the separation of the geometric and electric problems and a simple description of the induced dipole moment, in addition to the transmembrane potential [Gimsa and Wachner 2001]. The influential radius describes the dependence of the maximum pole potential to the shape of an ellipsoidal object. Generally, the induced complex dipole moment along semiaxis a for an object of volume V = 4π abc / 3 is:

with εeε0,fCMa*=fCMaℜ+fCMaJandEa${{\varepsilon }_{e}}{{\varepsilon }_{0}},f_{CM}^{a*}=f_{CM}^{a\Re }+f_{CM}^{a\mathfrak{J}}\,\text{and}\,{{E}_{a}}$being the permittivity of the external medium, the complex Clausius-Mossotti factor consisting of its real and imaginary part and the external field component along semiaxis a. Complex terms are marked by asterisks.

fCMa*$f_{CM}^{a*}$is given by the normalized potential difference at the site of pole ^a in the presence and absence of the ellipsoidal object (Figure 4):

with n_a being the depolarizing factor along semiaxis a [Gimsa 2001]. Please note that n_a + n_b + n_c =1 for ellipsoidal objects, i.e. n_a = n_b = n_c =1/ 3 for spherical objects. The distances from the object’s respective symmetry planes to those equipotential planes that are just touching the respective poles of a vacuum object with the shape of the actual object are along each semiaxis of an ellipsoidal object a_inf , b_inf , c_inf . Along semiaxis a the influential radius is:

While the three influential radii provide the solution for the geometric problem, the electric problem for a single shell ellipsoid can be solved, assuming a series circuit of three resistor-capacitor (RC) pairs for the internal, membrane and external media along each semiaxis. The length of the three external elements (e.g. Zea*$Z_{e}^{a*}$along semiaxis a) must be such that it ensures the maximal possible potentials at the poles.

There are at least two arguments for the correctness of a description by three stacked "finite" elements (RC pairs) along each semiaxis:

Fig. 5

The impedances Z^* of the three RC pairs along a semiaxis can be calculated from the geometries of prismatic elements given by their lengths l , i.e. length of the semiaxis ( l = a), membrane thickness ( l = d), and ( l = a_inf − a), as well as a constant cross-sectional area A:

with j⁼ (⁻1)^0.5and ω = 2π f , where σ , ε , ε₀, and f stand for the specific conductivity and relative permittivity of the considered medium, the permittivity of vacuum and the field frequency. From the voltage-divider chains, the potentials at the semiaxes poles can be obtained. Along semiaxis a we obtain:

with Ψ0aa=E0a$\frac{\Psi _{0}^{a}}{a}=E_{0}^{a}$being the external field or field component in the direction of semiaxis a. The maximum possible potential at the pole is ainfE0a,${{a}_{\inf }}E_{0}^{a},$which is reached at low enough frequencies for an infinitely high membrane impedance Zma*$Z_{m}^{a*}$. Analogously, the transmembrane potential ΔΨma*$\Delta \Psi _{m}^{a*}$at the pole of semiaxis a is:

The RC models obtained for the induced dipole moment and the transmembrane potential can easily be simplified for the limiting DC and very high-frequency cases, and for certain frequency bands or dispersion processes [Gimsa and Wachner 1999]. For ns pulses see Retelj et al. [2013] or Schoenbach et al. [2004].

The behavior of the cell model in the dispersion range of membrane polarization is of special interest. For a qualitative consideration, the permittivities of the internal and external bulk media, as well as the membrane conductivity, can be neglected. For simplicity, a spherical cell of radius a = r and a_inf = r_inf = 3r /2 is considered. The three finite elements described by equation (4) are reduced to:

with d , ε _m, and Cmspec=Cm/A$C_{m}^{spec}={{{C}_{m}}}/{A}\;$being the thickness, relative permittivity and area specific capacitance of the membrane. In the following, the induced membrane potential is considered as an example to demonstrate the simplicity of the influential radius ansatz. Using equations (7) and expanding the fraction by 1/Zm*${1}/{Z_{m}^{*}}\;$equation (6) reads:

The characteristic frequency of membrane polarization is obtained from the RC-time constant of the membrane:

Using equation (9), the absolute value of the induced transmembrane potential is obtained from equations (8) leading to the Schwan equation [Marszalek et al. 1990, Schwan 1983]:

In spherical objects, the magnitude of the induced transmembrane potential depends on the cosine of the angle between a line from the considered membrane site through the center of the sphere and the semiaxis which is oriented in field direction. At its pole, the membrane field is at a maximum of (r_inf /d)E₀ , i.e. the external field is amplified by a factor of r_inf /d corresponding to approximately 1000 for a standard cell geometry.

Using the influential radius notion to separate the geometric and electric problems, analogous derivations are possible for spheroidal or ellipsoidal objects with conductive membranes [Gimsa and Wachner 1999, Gimsa 2001].

In AC electrokinetics, the observed effect depends on the field quality, i.e. the time-dependent distribution, orientation or inhomogeneity of the field. The frequency dependence of the induced dipole moment generates particular frequency spectra for the forces or torques. The sketches in Figures 6-8 summarize the effects observed in homogeneous (Figures 6, 7a), inhomogeneous (Figures 7b, c), rotating (Figure 8a), and TW fields (Figure 8b).

Fig. 6

Fig. 7

Fig. 8

Equations describing the observed effects can be derived from the different kinds of interaction of the induced dipole moment and the inducing field. For simplicity, it is assumed that small objects are polarized in a homogeneous field, i.e. the characteristic distances of the field inhomogeneity are large in comparison to the object size. At field frequencies beyond the hydrodynamic time constants, the objects experience time-averaged effective forces and torques. Simplified equations can be derived within an orthonormal coordinate system of unit vectors i , j , k.

The ED force along the oriented semiaxis a (Figure 6a) is:

the EO torque in linear polarized fields (Figure 6b) is:

(for EO in circular polarized fields, please see: [Gimsa 2001]). Evenly polarizable objects attract each other with:

over distances larger than the object size, where |fCMa|=|fCMaℜ|2+|fCMaℑ|2$\left| f_{CM}^{a} \right|=\sqrt{{{\left| f_{CM}^{a\Re } \right|}^{2}}+{{\left| f_{CM}^{a\Im } \right|}^{2}}}$. For DP (Figure 7b) and AC-field trapping (Figure 7c) or the vertical attraction and lifting in TWDP (Figure 8b) the force is:

with γ standing for the site-dependent degree of field inhomogeneity.

Objects trapped by positive DP are consequently attracted to the areas of highest field strength, i.e. to electrode sites of high curvature. Even though electrode contact may be acceptable for the trapping of multitudes of small objects with passivated electrodes or at frequencies high enough to avoid electrode processes, negative DP (Figure 7c) is safer [Fuhr and Reichle 2000, Gimsa 1999, Grom et al. 2006, Schnelle et al. 2000]. Please note that optical traps exploit 'positive DP', i.e. a polarizability (permittivity/refractive index) of the (nonmagnetic, biological) objects that is higher than that of the surrounding medium at light frequencies. This is possible because field maxima can be generated at a distance from the optical 'antennas' (i.e. lenses, etc.) due to the different wave length - antenna size ratios at radio and optical frequencies.

Usually, 4-electrode chambers driven by 90°-phase-shifted signals are used to generate rotating electric fields (Figure 8a). The induced dipole moment rotates with the frequency of the inducing field. Nevertheless, in case of dispersion of the induced dipole moment at a given frequency, the dipole moment possesses a phase-shift with respect to the external field vector. Their interaction generates a permanent torque N _c around semiaxis c:

The torque is highest when the frequency of the exciting field corresponds to the relaxation time of a polarization process. The reason is that only the out-of-phase component of the induced dipole moment contributes to torque generation (cross-product of field and induced dipole moment). Accordingly, dispersion processes are a prerequisite of ER. To the best of our knowledge no attempts were published to detect an equivalent torque generated by circularly polarized light in dispersion ranges of the refractive index.

TWDP (Figure 8b) can be considered as a linearization of ER [Hagedorn et al. 1992, Hughes et al. 1996]. The traveling effect generates a field gradient in the traveling direction of the field. For an object oriented with semiaxis a in the traveling direction of the field and an object-related field distortion that is small in comparison to the electrode-period length (a_inf << n_el x), the TWDP force is:

with x and n_el standing for the electrode distance and the number of electrodes for a full field period, e.g. four electrodes driven by 90°-progressively phase-shifted signals.

Please note that in the considerations above, the field magnitudes and time-dependent orientations in equations (7) to (12) depend on how the external field is defined. Because the effects are not independent of each other, combined force spectra are observed, e.g. the reorientation of non-spherical objects in linear (inhomogeneous) DP, rotating ER, and TW fields, respectively (for details see: Gimsa [2001], Goater et al. [1997], and Becker et al. [1995]).

The influential-radius concept can easily be extended to describe the impedance of suspensions of three-axial single-shell cells or homogeneous particles using the mixing equations of Maxwell and Wagner [Pauly and Schwan 1959, Asami et al. 1980]. For a random orientation, each of the three different Clausius-Mossotti factors contributes by one third (this is equivalent to the notion that each of the three semiaxes of all particles evenly contributes to the impedance of a population of even ellipsoidal particles):

where σ ^* and p refer to complex specific conductivities and the volume portion of the suspended objects, whereas the indices S and e refer to the whole suspension as well as the external medium. Please note that the factor 3 in the left-hand term stems from the definition of the Clausius-Mossotti factor [Asami et al. 1980]. Accordingly, for objects oriented with semiaxis a in the external field direction we obtain:

This expression is generally identical to that given by Foster and Schwan [1996] (equation 44a on page 44).

Synchronous reorientations of a population of suspended particles should be detectable as impedance transitions. For example, such reorientation can be induced by EO in DC, AC, optical, acoustic or hydrodynamic fields. The impedance of single, oriented cells can be detected in microfluidic devices [Sun and Morgan 2010].

Finally, the combination of equation (17) with equations (2) and (5) permits a simple theoretical description and the determination of cell parameters from experimental spectra. For this, the impedance terms in equation (5), each of them describing an RC pair according to equation 4, provide an easy ansatz because they can directly be simplified at certain frequency ranges [Gimsa and Wachner 1999]. This allows for extending Schwan's RC model for a suspension of spherical cells [Pauly and Schwan 1959, Foster and Schwan 1996] to suspensions of ellipsoidal cells (manuscript in preparation).

Using equations (2), (5) and (7), from equation (18) for a spherical object ( a = r ) we directly obtain:

Assuming equal absolute values for the real and imaginary parts of σS*$\sigma _{S}^{*}$after some calculations, a new expression for the characteristic frequency of suspensions of spherical cells f_c1_IMP in the membrane dispersion frequency range can be derived:

Please note that equation (20) converges to equation (9) for infinitely low cell concentrations (p=0). Because the expression slightly differs from that given by Foster and Schwan [1996] (equation 24) we checked that the limiting cases:

correspond.

In the following calculations, standard parameters of biological cells have been assumed in order to consider the induced transmembrane potential, the impedance of suspensions, as well as force and torque spectra, which are observed in AC electrokinetic experiments (Table 1).

The Clausius-Mossotti factor of a spherical cell of radius r with fCMrℑ=fCMaℑ=fCMbℑ=fCMcℑ$f_{CM}^{r\Im }=f_{CM}^{\mathfrak{a}\Im }=f_{CM}^{\mathfrak{b}\Im }=f_{CM}^{\mathfrak{c}\Im }$was calculated using equations (2) and (5) (Figure 9). Its real and imaginary parts are proportional to the spectra of the DP force (equation (14)) and ER torque (equation (15). The spectra are reduced to the membrane dispersion when the permittivities of the internal and external bulk media are neglected (dotted lines). Nevertheless, in experiments, a second dispersion is clearly observed in the DP and ER spectra. Its mechanism is based on the superseding of the distributions of field and polarization, which are dominated by the frequency-independent bulk conductivities by distributions according to the bulk permittivities which dominate at high frequencies. This effect is usually neglected for the induced transmembrane potential as obtained from equation (10). Its -3 dB frequency corresponds to the characteristic frequency of membrane dispersion, i.e. the first peak in the imaginary part of the Clausius-Mossotti factor.

Fig. 9

The specific impedance (compare to equation (4)) of a suspension of spherical cells was calculated by equation (18) for the parameters of Table 1 and p=0.1 (Figure 10, full lines).

Fig. 10

Even though bulk-media dispersions are clearly observed in experiments with cell suspensions especially at low external conductivities, equation (18) has been reduced to the membrane dispersion. For this, the permittivities of the internal and external bulk media were neglected in the classical Laplace ansatz as well as in equation (5). Even though the derivation was simpler when started from the second approach, the same equation (19) was obtained in both cases (Figure 10, dotted lines). Besides the ease of simplifying an RC model, it is the ad hoc introduction of area-specific parameters for the membrane (equations (7)) that simplifies the derivation. An expression similar to equation (19) was not provided by Foster and Schwan [1996].

From equation (19), an expression can be derived for the characteristic membrane-dispersion frequency fc1IMP$f_{c1}^{IMP}$(equation (20)). It was compared with the simplified equation 56b on page 46 in Foster and Schwan [1996] assuming zero membrane conductance:

The expressions are identical for cell-volume fractions of p=0, i.e. the characteristic frequency of a cell suspension fc1IMP$f_{c1}^{IMP}$(equation (20)) reduces to f_c1 of the single cell (equation (9)). Consequently, equation (9) corresponds to equations (21) – (23) for p = 0.

Figure 11 compares different cases for the volume-fraction dependence of fc1IMP$f_{c1}^{IMP}$. At first, we checked that the fc1IMP$f_{c1}^{IMP}$plots of equation (18) correspond to the full expression of Foster and Schwan [1996] (Figure 11, dotted lines). While fc1IMP$f_{c1}^{IMP}$slightly increases for increasing cell-volume fractions p in the simplified equation of Foster and Schwan [1996] (full lines), our simplified equation (20) shows an improved correspondence to the full expression, with fc1IMP$f_{c1}^{IMP}$slightly decreasing with p.

Fig. 11

In the literature, we did not find explicit experimental results on the cell-volume fraction dependence of fc1IMP.$f_{c1}^{IMP}.$Nevertheless, published data on blood cells reports an increase of the permittivity as well as a decrease of the conductivity for increasing volume fractions [Fricke 1953, Schwan et al. 1962]. The combination of these correlations suggests a decrease of fc1IMP$f_{c1}^{IMP}$for increasing volume fractions. Also, our own experiments showed a decrease of fc1IMP$f_{c1}^{IMP}$relative to f_c1 for increasing fractions of human red blood cells, especially for p>0.1 (results not shown).

Despite the increasing capacities in the computing power of new computers, simplified equations will be useful for the qualitative understanding of the relations of detected parameters with cell and media properties. The equations also allow for fast data interpretation in screening applications or the on-line detection of cell properties, e.g. in batch cultures or microfluidic systems [Sun and Morgan 2010].

Considerations of the structure-based, AC electrokinetic effects observed for objects are helpful for the intuitive explanation of analogous AC electrokinetic forces acting on the fluid media of ETTWμPs [Fuhr et al. 1992 and 1994, Perch-Nielsen et al. 2004] or ETμPs [Stubbe et al. 2013]. In electrothermal pumps, a temperature gradient alters the electric properties of the fluid media in certain regions of the pump medium. The actual temperature distribution in the pump media results from the interplay of internal and external (e.g. resistive or radiation) heating and heat dissipation across the channel walls and the metallic electrodes. Even though no defined structural interfaces are introduced by thermal gradients, forces evolve from the interaction of the field with the induced smeared charges which are qualitatively the same as at the interfaces of objects. The forces acting on the ion and dipole charges are transferred to water molecules, thereby inducing pumping. To induce a directed pump effect, thermally induced Prandtl convections should be avoided, in other words, thermal convections or the induced medium movement should not (significantly) change the temperature distribution [Stubbe et al. 2013]. For this, conductive heat dissipation should be high enough, i.e. the hydrodynamic cross-sections of the pump channels should be small and its surface-to-volume ratio high enough. For aqueous media around room temperature, these prerequisites are generally fulfilled at channel widths/heights below 400 μm. Please note that for plane electrode structures, broader (not higher) channel geometries are possible when the height allows for sufficient heat dissipation from the medium volume.

Generally, most of the AC electrokinetic effects observable with micro-objects can be used in media pumping or mixing. To induce a directed movement of the pump medium, it is suggestive to exploit the DP (Figure 7b) and TWDP (Figure 8b) effects. In relation to these force effects acting on objects, also pumping can be based on the interaction of induced charges, which are in-phase or out-of-phase with the pumping field.

Figures 12a and 12b show two pump geometries that exploit in-phase charge induction. To induce translation, an inhomogeneous field is required in DP, which generates imbalanced forces that act on the two hemi-ellipsoids of a symmetric object. Nevertheless, pump geometries are more flexible than electrode designs for the DP of particles. Generally, a break in symmetry is required which may be introduced by the channel geometry or the electric or temperature fields or some combination thereof.

Fig. 12

Pumping in Figure 12a is based on forces induced by a largely homogeneous electric field. The medium is asymmetrically heated with respect to the distribution of the electric field by Joule heating and a resistive heating element inside a channel of symmetric geometry. Heating forms a "smeared object" in the medium whose electric properties are characterized by a temperature-dependent increase in conductivity of approximately 2% and a permittivity decrease of approximately 0.4% per degree [Stubbe and Gimsa 2013]. As a result, the polarizability of the heated "smeared object" is increased below the Maxwell-Wagner frequency and decreased above this frequency when the ionic bulk conductivity is superseded by displacement currents. Below the Maxwell-Wagner frequency, the pump effect can intuitively be compared to the electrostatic pulling force acting on elder pith by the boundary-field effect of a plate capacitor. The "smeared object" is expelled from the field when the ionic bulk conductivity is superseded by displacement currents above the Maxwell-Wagner frequency resulting in a reversal of the pump direction [Stubbe et al. 2007]. In both cases, the system tends to replace the lower by the higher polarizable medium.

In Figure 12b, the "smeared object" is formed by Joule heating of the pumping field. Both the electric and the thermal fields are largely symmetrical with the same symmetry planes. A symmetry brake is introduced by the channel geometry. The pump effect can intuitively be explained by ED of the "object" in the homogenous field between the electrodes (compare to Figures 2, and 6a). The pump direction changes from "elongation" to "compression" around the Maxwell-Wagner frequency.

ETTWμPs (Figure 12c) exploit field-induced forces acting on charges which are induced in the bulk medium out-of-phase with respect to the field. Medium pumping (straight arrows) results from the superposition of all forces acting in the medium [Morgan et al. 2001, Ramos et al. 2005]. Charges and forces are induced most efficiently by field components which are oriented in the direction of the temperature gradient, i.e. perpendicular to the gray-step borders in Figure 12c. The white bent arrows sketch an instantaneous moment in the effective TW field distribution. For a clearer presentation, field components with opposite orientations are neglected. Close to the electrode plane, bent field lines bridge the gaps between neighbouring electrodes, i.e. AC components dominate the field. Travelling field components are more pronounced further away from the electrode plane (long bent arrows). Especially the interaction of these components with charges which are induced in the bulk out-of-phase with the field contributes to the overall pumping force. The situation is analogous to the field distribution in four-electrode ER chambers where strongly elliptical, i.e. largely linear AC fields are found in the vicinity of the electrodes, especially in between neighbouring electrodes. An ideally circular polarized field is found in the centre of the chamber [Maswiwat et al. 2006]. In analogy to the above considerations on ETμPs, the symmetry break in ETTWμPs arises from the directed TW-field movement.

While the bent arrows cross gray-steps that are almost oriented in pump direction in the central area of the pump channel, the gray-step borders are oriented in a more perpendicular way in the peripheral regions of the electrode array. Accordingly, pump-force contributions will be more effectively induced in the two peripheral areas.

Combined considerations of the electric and temperature field distributions suggest an increase in pump efficiency, e.g. due to external medium cooling in the center of the channel or alternatively, by a central gap in the electrode structure. Instead of one elongated "smeared object", two objects will be formed. Taken together, the two objects will exhibit more areas with force contributions that are aligned with the pump channel. The pumping efficiency can also be improved by externally induced longitudinal thermal gradients as already observed by Garcia-Sanchez et al. [2009].

In Figure 12c, the temperature gradient runs from top to bottom in the central area of the electrode structure. When the pumping effect is dominated by force contributions generated in this area, the pump direction can be reversed by reversing the orientation of the gradient, e.g. by external heating or cooling [Liu et al. 2014].

An intuitive explanation for this effect can be obtained from a comparison of the field distributions in the center of Figure 12c with Figure 2. In a thought experiment, the bent field lines are straightened in order to allow for an easier comparison to the relations presented in Figure 2. If straightened, the long field line would pass through a virtual, smeared object of reduced temperature, starting and ending in pumping medium with an elevated temperature near the electrode plane, given the temperature distribution of Figure 12c. Qualitatively, this corresponds to Figure 2a, i.e. the smeared object is less polarizable than the surrounding medium when the conductivity polarization dominates. Even though the polarizability of the smeared object decreases in the dispersion range, its polarizability increases relative to that of the surrounding medium. In ER, an increasing effective polarizability of the object results in counter-field rotation (Figures 8a and 9). Accordingly, pumping in Figure 12c will be induced against the direction of movement of the TW field.

A reversal of the temperature gradient, for example by external heating or cooling, will inverse the above relations, that is, the smeared object will become more polarizable than the surrounding medium (Figure 1b). The pumping direction will be reversed accordingly. When a temperature gradient is externally imposed, perpendicular to the channel, the pump effect can be improved by an additional electrode array at the top of the pumping channel, inducing a field that travels in a direction opposite to the field at the bottom [Liu et al. 2014].

Because TW pumping relies on out-of-phase polarization, dispersion processes are a general prerequisite for pumping. In the β-dispersion range, these may be dispersions of the polarization of "smeared structural interfaces" in the pumping medium, but also a thick dielectric layer between the electrode plane and the aqueous pump medium can form a dispersive structural interface. Such a layer may improve TW pumping performance by improving the TW field distribution inside the pumping medium and can also be used to adjust the dispersion frequency [Marczak and Diesinger 2009]. The layer excludes pumping medium from the low-force range close to the electrode plane where the spatial TW field effect is ineffective, because AC components of the electrostatic field dominate. Also medium volumes at distances far from the electrode plane will not experience a TW force because the parallax between the single electrodes will result in a virtually equipotential electrode-array area [Morgan et al. 2001]. For a high pumping force, the pump-channel geometry should confine the electrothermal (electrostatic) TW effect to a certain distance range from the electrodes.

Nevertheless, in the frequency range below approximately 10 kHz, out-of-phase components are also generated by electric dispersions of the double-layer polarization (electric α-dispersions, i.e. "concentration polarization") or dispersions of microfluidic flow patterns (electroosmosis, "hydrodynamic α-dispersion") (for a discussion see Georgieva et al. 1998, Gimsa and Wachner 1998, Ramos et al. 2005, Zimmerman et al. 2002). In principle, these effects can be seen as the AC versions of DC electrokinetic effects and their frequency-dependent dispersion. Typically above 200 MHz, dipole molecules may also exhibit out-of-phase components due to Debye dispersions. The exploitation of Debye effects is not well investigated for pumping and will not be considered in this manuscript. Also DC-induced electro-osmotic flows which are known for a long time, will not be considered.

Nevertheless, because the majority of papers on TW pumping (or AC field pumping in asymmetrical electrode arrays) deal with electro-osmotic effects, a short note should be added in relation to the underlying mechanisms [Ajdari 2000, García-Sánchez et al. 2006, Morgan et al. 2001, Ramos et al. 2005, Urbanski et al. 2006]. In a number of papers, the qualitative differences of the pumping mechanisms that are based on α- or β-dispersion processes are not made clear or are even confused. One reason is that the (usually lower) α- and (usually higher) β-dispersion frequencies are not always well separated, depending on channel size and electric properties. While β-dispersions depend on the (largely constant) permittivity and the specific conductivity of the aqueous medium, α-dispersions depend on the (pH dependent) surface charge and ionic strength, i.e. the Debye-Hückel double-layer length [Maier 1997, Neu et al. 2002, Zimmerman et al. 2002]. Because ion concentration determines conductivity and ionic strength, it is the most important adjustable parameter which determines the frequencies of the α- and β-dispersions for a given system. Nevertheless, in the α-dispersion range, the different mechanisms involved are more complex than in the β-dispersion range discussed above.

Before the boom of microfluidic TW pumps, these relationships between the different α- and β-dispersion effects had been studied in ER experiments on single particles [Georgieva et al. 1998, Hughes et al. 1996, Maier 1997, Neu et al. 2002, Zimmerman et al. 2002]. Three α-effects, one electric and two electrokinetic effects influence the behaviour of a particle. First, electric polarization shifts the ion concentrations inside its electric double layer ("concentration polarization" with a time constant determined by the ionic diffusion coefficient and particle radius); second, the charged particle experiences electrophoretic movement in the external field (time-constant determined by particle mass and Stokes's friction); and last, and most complex, the electro-osmotic flow which is generated inside the double layer propagates hydrodynamically into the surrounding medium [García-Sánchez et al. 2006, Ramos et al. 2005]. To the best of our knowledge, these known relations have not yet been taken into account for the qualitative interpretation of TW pumping, even though the relations are simpler in pumps, because the "particle", i.e. the charged pump-channel wall, does not undergo electrophoretic movement.

Even though the electrokinetic and impedance properties of a system should be equivalent, this interrelation does not hold at the low frequencies in the α-dispersion range [Georgieva et al. 1998, Maier 1997, Neu et al. 2002, Zimmerman et al. 2002]. A qualitative explanation for this uncoupling is that the hydrodynamic time constants of the microsystems correspond to the α-dispersion-frequency range with frequencies typically below 1 kHz. In contrast, the field induced forces are time averaged over a field cycle for hydrodynamic reasons in the β-dispersion range, because the frequencies are too high to allow particle or medium movements to follow the alternations of the AC field.

Another reason for the more complex effects in the α-dispersion range is due to the fact that the forces originate from within the very thin electric double layers (typically <1 nm to <100 nm, depending on ionic strength). In a qualitative view, electro-osmotic TW pumping, e.g. by a square-topped four-phase field, is generated by the superposition of four electro-osmotic flow patterns [Morgan et al. 2001, Perch-Nielsen et al. 2004, Ramos et al. 2005]. The patterns, which are induced by the four consecutive electrode-drive patterns, sum up to the effective pumping effect. A similar mechanism induces electro-osmotic AC-field pumping in asymmetrical electrode arrays [Ajdari 2000, Urbanski et al. 2006].

While the actual forces are generated inside the very thin electric double layer in electro-osmotic pumps, forces are generated at the smeared structural interfaces inside the bulk medium of ETμPs or ETTWμPs (Figure 12) [Fuhr et al. 1992 and 1994, Morgan et al. 2001, Perch-Nielsen et al. 2004]. The two pump principles with their different localisations of force generation may overlap in an intermediate frequency range, causing a complex pumping behaviour [Garcia-Sanchez et al. 2009, Ramos et al. 2005]. For example, electrosomotic pump forces may induce pumping at the surface of the electrode array in one direction and the electrothermal TW effect may pump the bulk in the opposite direction.

For in-phase ETμPs, the pumping velocity v_inp (Figures 12a) was shown to be:

with FE$\left\langle {{F}_{E}} \right\rangle$being the volume-averaged pumping pressure given by the time-averaged volume force FE$\left\langle {{F}_{E}} \right\rangle $in the effective range of the pumping field ζ , i.e. in the pump channel volume [Stubbe et al. 2013]. ø , ε _e , η , and ΔT are a specific channel-geometry coefficient, the permittivity of the medium, the medium viscosity, and the average temperature difference of the medium across distance ζ , respectively. Please note that φ does not only refer to the channel in the range of the pump structure itself but to the whole geometry contributing to viscous friction. The numbers 1000 and 11 result from the simplified characteristic temperature-dependent decrease of the permittivity by approximately 0.4% and the conductivity increase by approximately 2% per degree [Stubbe et al. 2013]. 1/ε0εeσe−1$1/\left( {{\varepsilon }_{0}}{{\varepsilon }_{\text{e}}}\sigma _{\text{e}}^{-1} \right)$is the Maxwell-Wagner frequency, which is determined by the balance of the Ohmic and displacement currents (compare to equation (4)). For purposes of simplicity, it can assumed to be constant and temperature-independent. Equation 25 also applies to the structure of Figure 12b after φ and ΔT have been redefined.

While forces are generated by interaction with charges which are in-phase with the external field in DP and ETμPs, they are generated by the interaction with out-of-phase charges in ER and ETTWμP. The strong interrelations of the effects allows for the derivation of an expression for the pumping velocity v_outp of ETTWμP by analogy considerations:

(see also: Fuhr et al. [1992] and [1992], Gimsa et al. [1997]). Nevertheless, as considered above, the force generation within a given cross-section of the pump channel in the electrode-array area will exhibit a stronger deviation and be less efficient than in ETμPs (see discussion above). Also ΔT is not well defined in equation 26 (compare to Figure 12c). In the equation, the resulting lower efficiency of the ETTWμP can be considered by a redefinition of factor φ .

Figure 13 presents the theoretical time-averaged volume forces for ETμP and ETTWμP given by equations 25 and 26. As in Figures 9 and 10, the relations of the in-phase (ETμP) and out-of-phase (ETTWμP) effects are related through the Kramers-Kronig relation. While the dielectric parameters of water result in a magnitude of the positive (conductivity) force plateau of ETμPs which is roughly 11 times that of the negative (permittivity) plateau, the force-peak of ETTWμPs around the Maxwell-Wagner dispersion frequency corresponds to one half of the distance between the ETμP plateaux. Neglecting the prefactor terms in equations 25 and 26, the frequency-dependent terms provide the integer numbers eleven and minus one for the ETμP plateaux and six (11+1)/2 for the ETTWμPs peak. Obviously, the ETTWμP-peak force cannot reach the conductivity-plateau force of ETμPs.

Fig. 13

These considerations show that ETμPs outperform ETTWμP. ETTWμP need adjustment of the field frequency to the force peak, require more complex electrode structures and driving signals (see e.g.: Pethig et al. [2003], Liu et al. [2014], Marczak and Diesinger [2009]). The maximum performance of ETTWμP is lower than that of ETμPs for physical reasons (peak vs. plateau heights) and it can only be reached with technological tricks, such as even more complex electrode structures (e.g. two TW-electrode arrays), additional dielectric layers, externally imposed temperature gradients, etc. Alternatively, the lower performance can be compensated for by higher temperature gradients. This is problematic, especially in biotech applications.