## 1 Introduction

DEP is an important electrokinetic technique for micromanipulating and transporting micro- and nanoscale biological particles suspended in aqueous media [1-3]. DEP is the translational movement of an electrically polarizable body by the action of a nonuniform electric field. It is often implemented by applying radio frequency electrical potentials to microfabricated electrodes immersed in liquid (typically of low conductivity). Biological particles amenable to DEP manipulation include: cells, viruses, nucleic acids (DNA and RNA), proteins, etc.

One of the important applications of DEP is determining dielectric properties of small volume samples. Crossover measurements have been a standard method for characterizing cells, viruses and colloidal bioparticles [4-8]. This method often involves preparation of suspension media for a range of controlled conductivities in order to infer a dielectric property, e.g. surface conductivity. Typically, after the sample has been suspended in a suitable DEP chamber, the applied signal is varied until the samples shows a transition from positive to negative DEP (pDEP to nDEP). The signal frequency at transition (i.e. the zero force point, where the polarizability of the particle equals that of the suspending medium) is the recorded crossover frequency for that particular medium conductivity. Unfortunately, the DEP crossover technique requires (i) the existence of both pDEP and nDEP, (ii) substantial time to prepare a range of medium conductivities and perform experiments, (iii) substantial number of specimen samples needed for each crossover experiment, and (iv) the need for considerable operator skill and avoidance of error.

An alternative method for determining the dielectric properties of particles is the initial collection rate. The rate of accumulation of samples into specified collection volume is approximately proportional to the DEP force. The rate, therefore, enables the dielectric properties of the particle to be inferred, e.g. by fitting the rates to the frequency-dependent polarizability, which can be modeled by the real part of the Clausius–Mossotti (CM) function. The collection rate technique, coupled with the need to circumvent problems associated with crossover measurements and the popular use of programmable automated switching software, e.g. LabVIEW™, motivates using collection rates as a means of determining dielectric properties of samples.

A useful method for measuring particle collection rates and inferring dielectric properties is pulsed DEP; measurements have been reported for cells and their constituents, e.g. DNA, RNA, viruses [4, 9-17]. Figure 1 shows the typical experimental setup for collection rate measurements using planar microelectrodes. Pulsed DEP is the application of DEP for short time durations, typically varying from milliseconds to seconds and the pulse shapes vary from rectangular, ramped, triangular, and so forth. Typically, under the influence of pDEP nanoparticles collect at the edges of the microelectrodes when the signal is switched “on,” and are subsequently released during the “off” phase. Continuously pulsed DEP is also called amplitude modulated (AM) DEP because the pulse physically consists of a sinusoidal waveform with period much less than the pulse duration, and with an amplitude that depends on the shape of the pulse [18].

One of the problems facing the determination of collection rates is that the number of nanoparticles accumulating in the capture region is dependent on the localized concentration at, or slightly above, the planar electrode array—practically this is seen to vary even if the bulk concentration is nominally the same between samples. In previous work using fluorescently labeled particles [12-14], this problem was addressed by normalizing (i.e. taking the ratio) of fluorescence with respect to the initial fluorescence. However, ratio normalization itself can be problematic, particularly when the value of the fluorescence is low. In the regime of low-level fluorescence, it is preferable to adopt alternative methods.

In this paper, we present a novel approach for measuring collection rates and determining the dielectric properties of fluorescent nanoparticles. Our recent research investigating the properties of AM DEP [19] is significantly advanced to allow changing of the carrier frequencies so that between the DEP being switched on, the frequency is switched, or “hops” to a selected frequency suitable for estimating the dielectric properties. The system process is thus titled, dual-cycle DEP. The issue arising from variable nanoparticle concentration localized near the planar array is addressed by using dual frequencies, one as a “control” and the other as a “probe.” The ratio of the “probe” and “control” collection rates is used to infer the dielectric parameters. In addition, frequency-dependent collection rates are often semiquantitatively compared to the real part of the CM function [4, 8], fitted by ad hoc multivariate methods [17] and crossover frequencies [16]. For the first time, we derive first-principle bivariate statistical algorithms for systematic fitting the real part of the CM using the method of least squares that minimizes residual error.

Clearly, there is considerable motivation for using DEP collection rates for determining dielectric properties of nanoparticles using dual-cycle DEP, yet there is no systematic, quantitative analysis of dual-cycle DEP, or indeed a theoretical framework to utilize this electrokinetic process. This work attempts to remedy this deficiency in the literature by developing mathematical and statistical models for dual-cycle DEP. These models are used to infer the dielectric properties of polymer nanospheres from experimental data.