|Figure 1. A schematic of a polarisable particle suspended within a point-plane electrode system. When the particle polarises, the interaction between the dipolar charges with the local electric field produces a force. Due to the inhomogeneous nature of the electric field, the force is greater in the side facing the point than that on the side facing the plane, and there is net motion towards the point electrode. This effect is called positive dielectrophoresis. If the particle is less polarisable than the surrounding medium, the dipole will align counter to the field and the particle will be repelled from the high field regions, called negative dielectrophoresis.
This effect was first termed Dielectrophoresis by Pohl . The dielectrophoretic force, , acting on a homogeneous, isotropic dielectric sphere, is given by:
where r is the particle radius, is the permittivity of the suspending medium, is the Del vector operator, E is the rms. electric field and the real part of the Clausius-Mossotti factor, given by:
where and are the complex permittivities of the medium and particle respectively, and with the conductivity, the permittivity and the angular frequency of the applied electric field.
The frequency-dependence of indicates that the force on the particle varies with the frequency. The magnitude of also varies depending on whether the particle is more or less polarisable than the medium. If is positive, then particles move to regions of highest field strength (positive dielectrophoresis); the converse is negative dielectrophoresis where particles are repelled from these regions. By careful construction of the electrode geometry which creates the electric field, it is possible to create electric field morphologies so that potential energy minima are bounded by regions of increasing electric field strengths. In such electrodes, particles experiencing positive dielectrophoresis are attracted to the regions of highest electric field (typically the electrode edges, particularly where adjacent electrodes are close), whilst particles experiencing negative dielectrophoresis are trapped in isolated field minima.
If a polarisable particle is suspended in a rotating electric field, the induced dipole will form across the particle and rotate in synchrony with the field. However, if the angular velocity of the field is sufficiently large, the time taken for the dipole to form (the relaxation time of the dipole) becomes significant and the dipole will lag behind the field. This results in non-zero angle between field and dipole, which induces a torque in the body and causes it to rotate asynchronously with the field; the rotation can be with or against the direction of rotation of the field, depending of whether the lag is less or more than 180°. This is shown schematically in figure 2.
|Figure 2. A schematic of a polarisable particle suspended in a rotating electric field generated by four electrodes with 90° advancing phase. If the electric field E rotates sufficiently quickly, the induced dipole M will lag behind the electric field by an angle related to the time taken for the dipole to form (the relaxation time). The interaction between the electric field and the lagging dipole induces a torque G in the particle, causing the particle to rotate. This effect is known as electrorotation.
The general equation for time-averaged torque G experienced by a spherical polarisable particle of radius r suspended in a rotating electric field E is given by:
where Im[K(w)] represents the imaginary component of the Clausius-Mossotti factor. The minus sign indicates that the dipole moment lags the electric field. When viscous drag is accounted for, the rotation rate R(w) of the particle is given by [Arnold and Zimmerman 1988]:
Note that there are two significant differences between the expression for torque as shown in equation (3) and the force in equation (1). Firstly, the relationship with the electric field is as a function of the square of the electric field rather than of the gradient of the square of the electric field. Secondly, the torque depends on the imaginary rather than the real part of the Clausius-Mossotti factor. This is significant in that a particle may experience both dielectrophoresis and electrorotation simultaneously, and the magnitudes and directions of both are related to the interaction between the dielectric properties of particle and medium. As an example of this, a plot of the frequency-dependence of a solid particle exhibiting a single dielectric dispersion is shown in figure 3.
|Figure 3. A plot of the real (solid line) and imaginary (dotted line) parts of the Clausius-Mossotti factor. The magnitude and signs of the real and imaginary parts govern the magnitude and direction of the dielectrophoretic force and electrorotational torque respectively.|
Travelling-wave dielectrophoresis is a linear analogue of electrorotation; instead of electrodes being arranged in a circle, they are laid out as "tracks", but the relationship of phases (each electrode being 90o phase advanced from the last) remains. This produces an electric field wave which "travels" along the electrodes [Hughes et al. 1996]. When this wave interacts with a polarisable particle, a dipole is induced which moved with the electric field peak. If the travelling wave is moving sufficiently quickly, the dipole will lag behind the field in a similar manner to that observed in electrorotation. However, as the wave is travelling in a linear (rather than rotational) manner, the result is the induction of a force rather than a torque, with particles moving along the electrodes like a train travelling along tracks. This is demonstrated schematically in figure 4.
|Figure 4. A schematic showing a polarisable particle suspended in a travelling electric field generated by electrodes on which the applied potential is 90° phase-advanced with respect to the electrode to its left. If the electric field E moves sufficiently quickly, the induced dipole M will lag behind the electric field , inducing a force F in the particle. This causes the particle to move along the electrodes, a phenomenon known as travelling-wave dielectrophoresis.|
The value of the force FTWDis given by equation (5) [Huang et al. 1992]:
where l is the wavelength of the travelling wave. This phenomenon was first observed by Batchelder . However, it remained largely unexplored until work by Masuda et al.  on using synchronously induced motion in low frequency travelling fields that travelling wave dielectrophoresis became significant. This work led to Fuhr et al.  effectively rediscovering asynchronous travelling wave dielectrophoresis. Since then, a large corpus of study has been conducted on this technique, including theoretical studies (Huang et al. ; Hughes et al. ) and demonstrations of practical devices for electrostatic pumping (e.g. Fuhr et al. 1994) and biotechnological applications (e.g. Morgan et al. )
In practice, dielectrophoretic manipulation of particles is performed on planar electrode arrays. Such electrode arrays employ patterned electrodes fabricated of gold or a similar conductor deposited on glass or silicon. A common electrode configuration used in AC Electrokinetics research is a quadrupole arrangement where four electrodes point towards a central enclosed region [Huang and Pethig 1991], as shown in the schematic in figure 5.
|Figure 5. A schematic of typical quadrupole electrode microstructures used in dielectrophoresis experiments. The electrodes are typically fabricated of gold on a glass microscope slide. The gap between opposing electrodes in the centre of the array is typically of the order 10-50 µm across, but can be as small as 500nm or as large as 1mm. To induce dielectrophoretic motion in particles suspended near the electrode array, electrodes would be energised such that a and c are of the same phase, and b and d are in antiphase to them. To cause electrorotation in particles within the central gap, electrodes b, c and d would be phase-shifted by 90°, 180° and 270° with respect to electrode a.
These electrodes may be of a variety of designs, according to the degree of field non-uniformity required [Hughes 1998]. The advantage with this arrangement is twofold. Firstly, electrodes can easily be used for dielectrophoresis, electrorotation or both. By energising the electrodes such that the phase angle of applied AC field between adjacent electrodes is 180°, particles will experience only dielectrophoresis. If the phase angle is reduced such that adjacent electrodes differ by 90°, the particle will experience both electrorotation and dielectrophoresis. The second advantage of the quadrupole design is that at the centre of the electrode array, there exists a well-defined electric field minimum, which is surrounded by well-defined field maxima at the electrode edges. This definition is advantageous when attempting to manipulate nanometre-sized particles, where Brownian motion is significant and may be sufficient to enable particles to "escape" from a weak forcefield "trap". This is illustrated in figure 6, which shows a simulation of the electric field in a plane 5 µm above the plane of the electrodes shown in figure 5. The dark circular region at the centre of the electrode array is the field minimum in which particles become trapped by the surrounding ring of high field gradient. The field maximum extends along the inter-electrode gaps, where particles will accumulate by positive dielectrophoresis.
|Figure 6. A simulation of the electric field in the plane 5 µm above the electrode array shown in figure 5. The simulation was performed using a Method of Moments software suite developed by the author at the University of Wales at Bangor. As can be seen, there is an electric field minimum (dark region) at the centre of the electrode array, surrounded by a ring of high electric field gradient. Particles experiencing negative dielectrophoresis are repelled into this minimum and become trapped. The electric field strength is high (white regions) along the electrode edges, where particles collect by positive dielectrophoresis.|
Advances in fabrication technology such as the use of electron beam lithography have allowed the manufacture of electrodes with feature sizes of the order 100 nm, covering many square cm. By reducing the size of inter-electrode gaps, electric fields of 107 Vm-1 can be generated by applying 10V across a 1 µm inter-electrode gap. More advanced dielectrophoretic traps have been developed [Schnelle et al. 1993] in which two sets of electrodes were stacked so as to form an octopole field "cage". This type of geometry has allowed the controlled manipulation of populations of particles in all 3 dimensions.
Since the force experienced by a particle undergoing dielectrophoresis scales as a function of the particle volume, it was believed for many years that a lower threshold of particle size existed, below which the dielectrophoretic force would be overcome by Brownian motion. It was held that to increase the force would require electric fields of such magnitude that local medium heating would increase local fluid flow, again acting to prevent dielectrophoretic manipulation. As electrode fabrication techniques were relatively crude, generating electric fields of sufficient magnitude required large potentials to be applied across large inter-electrode volumes, and consequently particles with diameters less than about 1 µm could not be trapped. Indeed, Pohl  speculated that particles smaller 500nm would require excessively large electric fields to trap against the action of Brownian motion.
The first group to break this threshold was that of Washizu and co-workers  who used positive dielectrophoresis to precipitate DNA and proteins as small as 25 kDa. This step downward in size was accelerated by improvements in technologies for electrode fabrication, principally the use of electron beam fabrication. This renewed interest in manipulation of sub-micron particles, and subsequent work by Fuhr and co-workers [Fuhr 1995; Müller et al. 1996] and Green et al. [1995, 1997] demonstrated that viruses of 100 nm diameter could be manipulated using negative dielectrophoresis. It was also demonstrated that latex spheres of 14 nm diameter could be trapped by either positive or negative dielectrophoresis, as demonstrated by Müller et al.  and reproduced here in figure 7. Subsequent work by Hughes and co-workers demonstrated that by varying the frequency of the applied electric field, Herpes viruses can be trapped using either positive or negative electric fields . Another study demonstrated that molecules of the 68 kDa-protein avidin can be concentrated from solution by both positive and negative dielectrophoresis [Bakewell et al. 1998].
|Figure 7. A photograph of 14 nm diameter, fluorescently labelled latex spheres precipitated from an aqueous solution (conductivity adjusted to 2.5 mSm-1 by addition of KCl) by dielectrophoresis. The electrodes were of similar construction to those shown in figure 5, with 500 nm between adjacent electrodes and 2 µm across the inter-electrode gap. In photograph (a), the spheres collect in the high-field regions between the electrode edges when a 8Vpp, 2MHz signal is applied. In photograph (b) the spheres collect in a ball at the centre of the trap, which is levitated a few µm above the electrode plane when the applied signal is changed to 10Vpp, 10MHz. Collection of particles is rapid and begins immediately after the field is applied. This experiment was performed at the Humboldt University of Berlin, reproduced with permission. Taken from Müller et al. |
The most obvious nanotechnological application of this technique is in the concentrating of "parts" of molecular machinery to one site. This could be either in the form of bringing together of two interlocking molecules in a highly dilute solution, or collecting "fuel" for a nanomachine from solution. Furthermore, by attracting microspheres and colloidal gold particles which have been functionalised using antibodies, Velev and Kaler  have used dielectrophoresis to construct microscopic biosensors by "stacking" particles of different types, which bind on contact.
In 1996, Müller et al. Demonstrated that a single 1 µm latex sphere could be trapped by negative dielectrophoresis, and subsequently repeated the experiment with a latex sphere of 650 nm diameter. Further work by Hughes and Morgan  demonstrated that single Herpes viruses, viral capsids and latex spheres as small as 93 nm diameter could be held in stable traps using negative dielectrophoresis, as shown in figure 8. This study also considered in some detail the mathematical conditions for a particle to be trapped by either positive or negative dielectrophoresis, concluding that for stable trapping, an approximate minimum particle radius r is given by the expression:
where k is Bolzmann's constant, T is the temperature, and D d is the thickness of the electric field barrier of magnitude , E2. Since the factors D d and E2 are related to the size of the electrode array, they place limits on the minimum size of particle an array can be used to manipulate. Studies by Hughes et al. indicated that for electrode structures with 5Vp-p applied to electrodes with 6 µm between opposing electrodes and 2 µm between adjacent electrodes, the minimum stable trapping radius for single particle is approximately 30 nm. This research indicated that the key variable governing D d is the size of the centre of the trap, which in that research was an area of approximately 5 µm radius centred on the middle of the electrodes. Therefore it should theoretically be possible to reduce the whole array to this size; this would benefit the use of this technique in nanotechnology since the manipulator would be of the order of a few microns across. Furthermore, since D d scales linearly with distance and E2 scales as the inverse cube of distance, equation (5) indicates that reducing the size of the trap by a factor x whilst maintaining the same activating potentials will increase the efficiency of the trap by reducing the minimum trappable particle radius by a factor x2/3.
|Figure 8. A pseudo-3D image of a single 93nm latex sphere suspended in a trap by negative dielectrophoresis, using an electrode structure similar to that shown in figure 5, with 6 µm between opposing electrode tips and 2 µm between adjacent electrodes. The medium conductivity was adjusted to 5mSm-1 by addition of EDTA, and the applied signal was 5Vpp, 15 MHz. The bead occupies the space at the point of the "spike" at the centre of the electrode structure. The vertical dimension was extracted from light emission data from fluorescence micrographs. This experiment was performed at the University of Glasgow, reproduced with permission. Taken from Hughes and Morgan |
Due to the fact that this trapping is contactless, with particles being repelled from the electrode edges, particles may move upwards and will eventually leave the trap. In order to prevent this happening closed field cages, such as those fabricated by Schnelle et al.  must be employed. An octopole cage of sufficiently small dimension could in theory allow full 3D manipulation of single particles of less than 30 nm diameter, and has been demonstrated to function for aggregates of 14 nm particles. The technology to produce such electrodes for single-molecule manipulation exists, but the principal problem to be overcome is the observation of the trapped particle.
An alternative approach would be to use an AFM tip to generate field non-uniformity, and use positive dielectrophoresis to attract particles to it. These could be positioned by the AFM and then "released" by changing the frequency of the signal to the AFM tip so as to repel the particles from it. The principle behind this using positive dielectrophoresis to attract particles to a 3D, microscopic needle tip has already been demonstrated [Alimova et al. 1999] to coat silicon tips with nanodiamond.
In 1992 Gascoyne et al. demonstrated that positive and negative dielectrophoresis could be used to separate a heterogeneous population of normal and leukaemic murine erythrocytes into two separate populations. Since the direction of the induced dielectrophoretic force is determined by the dielectric properties of the particle, populations of particles having different dielectric properties can be induced to move in different directions by careful choice of applied field frequency and medium conductivity. Examples of this work in the submicrometre range have been demonstrated by Morgan et al.  in the separation of a mixture of Herpes Simplex and Tobacco Mosaic viruses into two distinct populations and the separation of latex spheres of different sizes, or of similar size but different surface treatments. Another example of this is the separation of 93nm diameter latex spheres according to small variations in the surface charge of the spheres, as demonstrated by Green and Morgan .
In a nanotechnological context, the dielectrophoretic separation of particles allows for the manufacture of a tuneable electrostatic filter. By altering the frequency of the applied electric field, particles for two populations can be either both attracted to the electrode array, both repelled from it, or one can be attracted whilst the other is repelled. Since the effect is volumetric repulsion acts for several microns into the solution, according to the magnitude of the electric field the result is to alter the local concentration of the two particle types dramatically.
Whilst dielectrophoresis is useful for attracting particles to specific points, there are applications which require the moving of particles between one position and another. This requirement may be to relocate particles for processing, for separation purposes, or to gather in particles from a larger volume than can be achieved by conventional dielectrophoresis alone. Two means of particle transport have been demonstrated in the literature. Firstly, conventional dielectrophoresis has been used to move 100 nm latex spheres in a thermal ratchet, where dielectrophoresis is used to rectify the motion of particles induced by Brownian motion [Rousselet et al. 1994]. The induced motion can be either towards or against the ratchet potential, according to whether particles experience positive or negative dielectrophoresis, or the time between activation of ratcheting pulses [Chauwin et al. 1995]. This technique has the advantage that since only two phases are required, the electrodes for inducing such motion are easy to fabricate.
The alternative, and more popular, means of transporting particles using AC electrokinetics is by the application of travelling-wave dielectrophoresis. Whilst the majority of work on travelling-wave dielectrophoresis has concerned micrometre-sized objects such as blood cells [Morgan et al. 1997], some work has been performed on the concentration of particles on a surface using so-called "meander" electrodes [Fuhr et al. 1994b]. These structures use four electrodes in a series of interlocking spirals to generate a travelling wave; at the centre of the spiral, the electrodes form a quadrupole-type electrode array. It has been demonstrated that by careful manipulation of the amplitudes of the potentials on these electrode structures, it is possible to "steer" the motion of particles across the array. Tools such as these could be used as the basis for "conveyor belts" for factories on a chip, wherein different chemical processes may be carried out of the same chip, with operations performed by electrostatic or chemical means and the resultant output transferred to a new process by AC Electrokinetic means [Ward 1997].
Electrorotation was first examined scientifically in the early 1960s [e.g. Teixeira-Pinto et al. 1960] but was most fully explored by Arnold and Zimmermann [1982; 1988], who established the technique for direct measurement of the value of K(w) by measuring the time taken for a particle to rotate. This application has little bearing to nanotechnology, not least due to the difficulty of seeing such small particles rotate, but electrorotation may be useful as a means of inducing rotation in molecular machinery. Recent work by Berry and Berg  has involved using electrorotation to drive the molecular motor of E. Coli bacteria backwards at speeds of up to 2,000 Hz. Since the electrodes involved in that work were applying 10 V across inter-electrode gaps of 50 µm or more, considerably higher fields (and hence induced torques) could be applied with electrodes with inter-electrode gaps of the order of 1 µm. If one considers that electrorotation is an electrostatic equivalent of laser spanners [Simpson et al. 1997], then there exists a number of applications within current nanotechnological thought to which electrorotation might be applied. There are a number of advantages that favour electrorotation over its optical equivalent; most notable advantages are that electrorotation-induced torque is easily controlled by altering the frequency of the electric field, and that there does not need to be a direct optical path to the part to be manipulated. Electrorotation has been demonstrated for the induction of torque in micron-scale motors by Hagedorn et al. .
Aside from the engineering of electrodes small enough to generate large electric field non-uniformity, there are other factors that present problems with scaling AC Electrokinetic techniques from the micrometre to nanometre scale. These include Brownian motion induced by thermal noise in the surrounding medium and effects of conduction and convection of the medium induced by the high electric fields used for manipulating colloidal particles. For sufficiently high concentrations of particles, a diffusion force would also be significant; however, the majority of work on dielectrophoresis has used sufficiently low concentrations of particles for each particle to be regarded as single and isolated.
Brownian motion due to thermal noise is well documented [Einstein 1956] and a number of studies have considered the force required to trap particles against Brownian motion [Pohl 1978; Washizu et al. 1994; Hughes and Morgan 1998]. Experimental studies with proteins [Washizu et al. 1994; Bakewell et al. 1998] have demonstrated that Brownian motion does not prevent the trapping particles as small as 68 kDa (equivalent to a cube 6 nm along a face) by negative dielectrophoresis, or 25 kDa by positive dielectrophoresis. Recent studies (Ramos et al.  Green et al. ) on the forces involved in dielectrophoretic manipulation of sub-micrometre particles examined the balance of Brownian motion and dielectrophoretic forces by considering the amount of time required to determine, with 99.7% confidence (3 standard deviations) that an observed force is due to dielectrophoresis. Any force applied to a particle will ultimately result in what Ramos et al.  describes as "observable deterministic motion", but if the force is small the time taken to observe it may be of the order of years! However, the forces used in the experiments described in the literature are sufficiently large for the applied force to observably overcome Brownian motion to be of the order of a second or (often significantly) less. It may ultimately be possible to manipulate single small particles precisely using dielectrophoretic forces, but this will be more difficult to achieve for very small colloids and work needs to be done to study this.
Other disruptive effects are caused by the presence of very high electric fields covering small areas of the electrode array, creating electrohydrodynamic (EHD) motion in the fluid. An in-depth study of this subject has been conducted by Ramos et al. , and presented in summary form by Green et al. . In these works, the authors demonstrated that there are two major EHD forces which are significant in driving fluid flow around electrodes, those being electrothermal and electroosmotic. The former force is due to localised heating of the medium causing discontinuities of medium conductivity and permittivity, which was found to be insignificant due to the small volumes these discontinities occupy (as the electrodes are so small). The latter force is due to the interaction of the tangential electric field with the diffuse double layer above the electrode surfaces. This creates a fluid pumping action, the magnitude of which can be several orders of magnitude larger than the dielectrophoretic force. However, careful electrode design can allow the EHD forces to be used as an aid to particle manipulation [Green and Morgan 1998].
The techniques of AC Electrokinetics, principally dielectrophoresis and electrorotation, have much to offer the expanding science of nanotechnology. Whereas dielectrophoretic methods have been applied to the manipulation of objects in the micro-scale, recent work shows that the technique is applicable to the manipulation of nano-scale particles. This opens a wide range of potential applications for AC electrokinetics to the development of mainstream nanotechnology.
The fundamental challenge in the advancement of nanotechnology is the development of precision tools for the manipulation of macromolecules in solution phase. The techniques presented in this paper go some way to addressing this by providing tools for the trapping, manipulation and separation of biomolecules and small latex spheres using tools (electrodes) on the micrometre scale. There exist a number of avenues down which AC electrokinetics can be developed to meet the needs of nanotechnology more readily. These include the design of electrodes for trapping of smaller single particles, increasing the precision to which particles may be trapped, and examining the extent to which particles may be manoeuvred within the trap by alteration of the electric field.
The author would like to thank Dr. Hywel Morgan of the University of Glasgow and Dr Thomas Schnelle of the Humboldt University of Berlin for the experimental photographs presented in this paper, and Dr. Nicolas Green of the University of Seville for valuable discussions.
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