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- Abstract
- 1. Introduction
- 2. Theory
- 3. Relative contributions of DEP and EP effects
- 4. Discussion
- 5. Conclusion
- References

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Nanotechnology. Author manuscript; available in PMC 2012 June 17.

Published in final edited form as:

Published online 2011 April 21. doi: 10.1088/0957-4484/22/24/245103

PMCID: PMC3254113

NIHMSID: NIHMS308385

Mark A. Reed: ude.elay@deer.kram

The publisher's final edited version of this article is available at Nanotechnology

See other articles in PMC that cite the published article.

We present here a study on overlooked aspects of alternating current (AC) electrokinetics-AC electrophoretic (ACEP) phenomena. The dynamics of a particle with both polarizability and net charges in a *non-uniform* AC electric trapping field is investigated. It is found that either electrophoretic (EP) or dielectrophoretic (DEP) effects can dominate the trapping dynamics, depending on experimental conditions. A dimensionless parameter *γ* is developed to predict the relative strength of EP and DEP effect in a quadrupole AC field. An ACEP trap is feasible for charged particles in ‘salt-free’ or low salt concentration solutions. In contrast to DEP traps, an ACEP trap favors the down scaling of the particle size.

Dielectrophoretic (DEP) traps have been successfully implemented for many applications [1, 2, 3]. Such traps operate through the interaction of induced polarization charges with non-uniform electric fields. According to the classic DEP theory [4], the sign associated with the real part of Clausius-Mossotti (CM) factor (K(*ω*), figure 1(a)) dictates the behavior of the particle. For Re[K(*ω*)]>0, particles will be directed toward a local electric field maxima (positive DEP, pDEP), while for Re[K(*ω*)]<0, particle is attracted to a local electric field minima (negative DEP, nDEP). Negative DEP traps offer distinct advantages for many applications since the particles are trapped away from electrodes, mostly with a quadrupole geometry [5, 6, 7, 8, 9]. The frequency that produces a change from pDEP to nDEP is referred to as the crossover frequency *f _{co}* and is given by (1/2

(a) Real part of CM factor (Re[K(*ω*)]) as a function of the frequency. Shaded area denotes the anomalous center trapping in the theoretical pDEP region. (b) Plot showing the range of the medium conductivity and frequency over which a homogeneous **...**

It is well known that most polarizable particles and molecules suspended in aqueous solutions will develop surface charges by either dissociation of surface chemical groups or adsorption of ions from the solution [11]. Therefore, when placed in a liquid with a spatially non-uniform electric field, colloidal particles experience not only a dielectrophoretic (DEP) force but also an electrophoretic (EP) force [12, 13, 14]. It is a widely-held notion that the EP effect in aqueous solution will ‘vanish’ upon high frequency AC fields due to the linearity of EP with electric field [13, 14, 15]. As a result, EP contributions are not taken into account in most of the high frequency electrokinetic experiments [6, 7, 8, 9]. However, the assumption that EP effects will vanish is only true for a spatially homogeneous AC field. A charged particle exposed to an oscillating *inhomogeneous* electric field can experience a cycle-averaged force. This so-called ponderomotive force [16] plays a significant role in a variety of physical systems such as Paul traps [17] and laser-based particle acceleration [18].

Here we show both experimentally and theoretically the non-vanishing ponderomotive EP effect in *high frequency* electric field and its application for trapping charged particles in aqueous solutions. In contrast to DEP traps [5, 6, 7, 8, 9], an AC electrophoretic (ACEP) would favor the down scaling of the particle size.

We start our analysis with the one-dimensional EP motion of a homogeneous spherical particle with mass *m*, charge *Q* and radius *a* in a high frequency AC electric field. The damping force due to the viscosity of the liquid is of the form −*ξ*, where the Stokes drag coefficient *ξ* can be approximated by *ξ* = 6*πηa*, and *η* is the dynamic viscosity. Here we assume that the hydrodynamic memory effect [19] can be neglected for the dragging force (i.e. the friction force is only dependent on the current velocity). Without loss of generality, we assume the particle moves in an electric potential consisting of two parts: a static part *U* (*x*) and a harmonically oscillating part *V* (*x*) cos*ωt*, where *ω* is the angular frequency. The magnitude of *V* (*x*) is not assumed small in comparison with *U* (*x*). Note that both potentials have a spatial dependence. This potential gives a static force
$F\left(x\right)=-Q\frac{\partial}{\partial x}U\left(x\right)$ and an oscillating force *f*(*x, t*) = *QE*_{0}(*x*) cos*ωt*, where
${E}_{0}\left(x\right)=-\frac{\partial}{\partial x}V\left(x\right)$.

Under the conditions considered in this paper, the dynamics of the particle can be described by a secular motion *S*(*t*) on a time scale typically longer than one oscillating cycle *τ* = 2*π/ω*, which is decoupled from the rapidly oscillating micromotion *R*(*S, t*) (Appendix B). Therefore, the particle motion can be written in form, *X*(*t*) = *S*(*t*)+*R*(*S, t*), with the constraint *R*(*S, t*) = 0 (bracket denotes time averaging). If the amplitude of the rapidly oscillating motion is much smaller than the characteristic length of the non-uniform electric field
${E}_{0}/\frac{\partial {E}_{0}}{\partial x}(\text{i.e.}2Q\frac{\partial {E}_{0}}{\partial x}\ll m{\omega}^{2})$, it is reasonable to assume |S| |R| (where the dependence on *t* and *S* are henceforth dropped for brevity). Therefore *R* can be considered a small perturbation to *S*, and thus the equation of motion, *mẌ* = −*ξẊ* + *F*(*X*) + *f*(*X, t*) can be expanded to the first order in *R*,

$$m(\ddot{S}+\ddot{R})=-\xi (\dot{S}+\dot{R})+F(S)+f(S,t)+R\frac{\partial}{\partial x}(F\left(x\right)+f(x,t)){|}_{x=S}$$

(1)

The rapidly oscillating terms on each side of equation (1) must be approximately equal, *m* ≈ −*ξ + f*(*S,t*). The oscillating term
$R\frac{\partial F\left(x\right)}{\partial x}$ term is neglected by assuming
$\xi \omega \gg \frac{\partial F\left(x\right)}{\partial x}$, which is reasonable for highly damped environments such as water. By integration, we obtain the rapid micromotion component as,

$$R\left(S,t\right)\approx -\frac{{f}_{o}\left(S\right)}{m{\omega}^{2}\left[1+{(\xi /m\omega )}^{2}\right]}(cos\omega t-\frac{\xi}{m\omega}\mathit{\text{sin}}\omega t)$$

(2)

where *f*_{0}(*x*) = *QE*_{0}(*x*). The rapid micromotion *R*(*S, t*) is thus an oscillation at the same frequency as *f*(*x, t*). The oscillating amplitude depends on the position of the secular motion through *f*_{0}(*S*), the driving frequency, the damping coefficient, and the particle mass.

The secular motion *S*(*t*) can therefore be found by averaging equation (1) over one period of the rapid micromotion, and by replacing *f*_{0}(*x*) with *QE*_{0}(*x*),

$$m\ddot{S}=-\xi \dot{S}+F\left(S\right)-\frac{{Q}^{2}{E}_{0}\left(x\right)\frac{\partial {E}_{0}(x)}{\partial x}{|}_{x=S}}{2m{\omega}^{2}\left[1+{(\xi /m\omega )}^{2}\right]}$$

(3)

It should be noted that the above analysis is valid for *any viscosity of the medium* when
$\omega \gg \sqrt{2Q\frac{\partial {E}_{0}}{\partial x}/m}$ (Appendix B). This inequality is valid for most experiments [6, 7, 8, 9] performed in an aqueous environment around MHz range.

Let us consider three related cases based on equation (3).

Under this circumstance, the averaged secular motion is described by *m* = −ξ. By integration, we obtain *S* = *A*_{1}*e*^{−}^{ξt}^{/}* ^{m}* +

The third term on the right hand side of equation (3) provides a ponderomotive EP force for the secular motion due to the *non-uniformity* of the electric field. The particle is directed towards a point with weaker electric field. Moreover, due to the squared dependence on charge *Q* in equation (3), the repelling of the particle from regions of high electric field intensity holds true for both positive and negative charges. We need to emphasize that even though the time average over one period of both *f*(*S, t*) and *R*(*S, t*) is zero at a fixed point, *averaging over the micromotion* in the *non-uniform* electric field is the essential mechanism that causes the movement of charged particles.

In this case, it is convenient to express *F*(*x*) and *E*_{0}(*x*) in equation (3) by potential energy derivatives and the motion equation is obtained as,
$m\ddot{S}=-\xi \dot{S}-Q\frac{\partial}{\partial x}(U(x)+{\mathrm{\Phi}}_{sp}(x)){|}_{x=S}$, where Φ* _{sp}*(

$${\mathrm{\Phi}}_{sp}\left(x\right)=\frac{Q{E}_{0}^{2}\left(x\right)}{4m{\omega}^{2}\left[1+{\left(\xi /m\omega \right)}^{2}\right]}$$

(4)

As a consequence, the particles will move towards a point where
$\frac{\partial}{\partial x}\left(U\left(x\right)+{\mathrm{\Phi}}_{sp}\left(x\right)\right)=0$ and oscillate there, which is described by equation (2). In other words, under this situation the charged particle will oscillate at the bottom of the effective pseudopotential. Since *f*_{0}(*S*) is a complex function of time, the motion toward and around the bottom of the effective pseudopotential depends on the detailed form of *E*_{0}(*x*), and other parameters of the system (frequency *ω*, viscosity *ξ*, etc.).

To this end, we have investigated in detail the EP behavior of a charged particle in a high frequency
$(\omega \gg \sqrt{2Q\frac{\partial {E}_{0}}{\partial x}/m)}$ AC electric field. A translational motion is unarguably possible for *charged* particles in an *non-uniform* AC field. As a result, simply ignoring the EP effect in AC trapping field may not be correct.

Taking the EP effect into consideration for the quadrupole trapping field shown in the inset figure 1(b), the total instantaneous force on the particle will be *m* − *ξ* + *F _{ep}* +

The ratio of the EP versus the DEP ponderomotive force will determine the dominant mechanism. For arbitrarily complex trap geometries or potentials, this requires a case by case numerical calculation. We here choose a symmetric geometry for illustrative purposes, which is a good approximation to many experimentally realized traps [5, 6, 7, 8, 9]. Assuming a quadrupole AC electric potential
$\phi (x,y,t)=Vcos\left(\omega t\right)\left({x}^{2}-{y}^{2}\right)/2{r}_{0}^{2}$ (note that we have considered the one-dimensional motion without loss of generality. Namely, in the case of a quadrupole trap the motions of a particle are independent in each dimension), the ponderomotive force for both DEP and EP can be written as *F _{dep}* = −

$${k}_{\mathit{\text{dep}}}=-\mathit{\text{Re}}\left[K\left(\omega \right)\right]\frac{2\pi {\varepsilon}_{m}{a}^{3}{V}^{2}}{{r}_{0}^{4}}$$

(5)

$${k}_{ep}=\frac{{Q}^{2}{V}^{2}}{2m{\omega}^{2}{r}_{0}^{4}\left[1+{\left(\xi /m\omega \right)}^{2}\right]}$$

(6)

In order to hold the particle in the center of the trap, the ponderomotive force for both EP and DEP should be a restoring force (i.e. *k _{ep}* and

From the above analysis we see that both EP and DEP effects are able to trap the particle in the center of the device. Since the trap stiffness can be experimentally estimated through the equipartition theorem as *k* = *κ _{B}T*/

Trap stiffness as a function of the applied AC voltage amplitude squared (*V*^{2}) at 3 MHz. The experiment was performed on *a* = 240 *nm* polystyrene bead in a device with *r*_{0} = 4 *μm*. The squares are the experimental trap stiffness extracted from the **...**

To compare the relative strengths of the EP and DEP in trapping behavior, a dimensionless parameter *γ* is defined as,

$$\gamma \equiv \left|\frac{{k}_{ep}}{{k}_{\mathit{\text{dep}}}}\right|=\frac{{Q}^{2}}{4\left|\mathit{\text{Re}}\left[K\left(\omega \right)\right]\right|\pi {\varepsilon}_{m}{a}^{3}m{\omega}^{2}\left[1+{\left(\xi /m\omega \right)}^{2}\right]}$$

(7)

In order for EP effect to dominate in trapping dynamics, it requires *γ* 1. This will happen at a large net charge *Q*, a small particle size a, or a low working frequency *ω*. figure 3 (a) plots the *γ* value as a function of *Q* and a at a fixed frequency (1 MHz), while figure 3 (b) shows the *γ* dependence on *Q* and *ω* at a fixed particle size (0.5 *μm*). It should be emphasized that even though the *γ* parameter derived here is based on a quadrupole electric field, the general conclusion holds true for other geometries: the EP effect can dominate the trapping dynamics in the case of a sufficiently high charge in a *non-uniform* electric field.

Based on this *γ* parameter, we are then able to make a consistent explanation for our experiment and other DEP trapping experiments with a quadrupole electric field [6, 7, 8, 9]. The key parameter involved is the amount of net charge. Notice that the charge we used in the derivations above is the *effective* net charge rather than the bare charge [24]. A charged surface in contact with a highly conductive liquid creates an induced electric double layer (EDL). A significant fraction of the particle's charge is neutralized by the strongly bounded counterions in the Stern layer. The charged particle plus the thin Stern layer is further screened by diffusive counterions within a characteristic Debye length *λ _{D}*. To determine the effective charge

$${\mu}_{E}=\frac{2}{3}\frac{\varepsilon \zeta}{\eta}f(\alpha )$$

(8)

where *α* = *a/λ _{D}* is the ratio of particle radius to the Debye length of the electrolyte solution,

$$\zeta =\frac{{Q}_{\mathit{\text{bare}}}}{4\pi \varepsilon a\left(1+a/{\lambda}_{D}\right)}$$

(9)

where *Q _{bare}* is the bare charge of particle. The electrophoretic mobility

$${\mu}_{E}=\frac{{Q}_{\mathit{\text{bare}}}}{6\pi \eta a\left(1+a/{\lambda}_{D}\right)}f\left(\frac{a}{{\lambda}_{D}}\right)$$

(10)

The effective charge of particle can thus be estimated as,

$${Q}_{\mathit{\text{eff}}}=\frac{f\left(a/{\lambda}_{D}\right)}{\left(1+a/{\lambda}_{D}\right)}{Q}_{\mathit{\text{bare}}}$$

(11)

At a high ionic concentration *c*, the Debye length (*λ _{D}* ~ c

In contrast, it is easy to see from equation (11) that *Q _{eff}* →

Finally, we briefly comment on the scaling performance of both DEP and EP trap stiffness in the quadrupole trapping device. As shown in equation (5) and (6), reducing the device size *r*_{0} helps to enhance the trapping strength for both EP and DEP in a same fashion
$\left(\sim 1/{\mathit{\text{r}}}_{0}^{4}\right)$. Secondly, increasing the applied voltage has the same impact on the trap stiffness (~ *V*^{2}) and the maximum voltage that can apply is limited by the breakdown field and other electrokinetic effects (e.g. electro-thermal flow [28]). Most importantly, the DEP trap strength decreases with the volume of the particle. Conversely, EP traps prefer smaller particles since the trap stiffness increases with decreasing the mass of the charged particle, which makes the ACEP effect very attractive for single molecule trapping.

In summary, we have elucidated the importance of EP effects in a *non-uniform* AC electric field. The relative contributions of both DEP and EP effects in a quadrupole trapping field are studied and an important dimensionless parameter *γ* is obtained, which presents a consistent explanation for both the anomalous trapping behavior in ‘salt-free’ deionized water and most other DEP trapping experiments with salt solution. EP traps prefer smaller particles, as long as the particles are sufficiently charged. Therefore, it might be feasible to trap single molecules by the EP effect.

This research is supported by the U.S. National Human Genome Research Institute of the National Institutes of Health under grant No. 1R21HG004764-01.

Our planar quadrupole trapping device was fabricated on a SiO_{2}/Si wafer. The insulating SiO_{2} layer has a thickness of 3 *μm*. Four Au/Cr (~ 400/50 nm) electrodes were formed on top of this insulating substrate by UV-lithography and a double layer liftoff process. The tip to tip distance (2*r*_{0}) for each electrode pair ranges from 2 to 8 *μm*. The microfluidic channel was formed by poly(dimethylsiloxane) (PDMS) using SU-8 as a molding master [29]. Oxygen plasma treatment was used to permanently bond the PDMS to the device surface and form an anti-evaporation microfluidic channel. An inlet and an outlet were punched through before assembling. Once the device was assembled, it could be repeatedly used for a long time.

The particles used in the experiments are polystyrene beads (*Polysciences, Warrington, PA*) of two diameters (0.481±0.004 *μm* and 0.982±0.013 *μm*). The surface of these particles is functionalized with carboxylate group (-COOH). The parking area for each group is around 320 *Å*^{2}/group. Table A1 summarizes the properties for these two kinds of beads. These COOH surface groups are the origin of the negative charges (COO^{−}). Scanning electron microscopy (SEM) revealed that all the particles had a pronounced spherical shape.

The solution used in the experiment was prepared by the following steps: (i)The beads were firstly diluted to a density of ~ 10^{6} particle/mL by deionized (DI) water (milli-Q grade, resistivity 18 *M*Ω · *cm*), in order to eliminate the particle-particle interactions during the experiment. (ii)In order to thoroughly remove the residual ions from the stock solution, the beads prepared in step 1 were washed five times in DI water by centrifuging the beads in a 10 mL tube at 13500 G for 10 min, re-suspending in DI water each time. Salt solutions with 0.1X, 1X and 10X phosphate buffered saline (PBS) (*Sigma, St. Louis, MO*) were also prepared, following the same steps (i) and (ii). The final conductivity of ‘*salt-free’* solution was measured as 0.1 *μS/cm* (EC 215 Multirange Conductivity Meter, Hanna Instruments) and this slowly goes up to maximum 2.0 *μS/cm* during the course of an experiment (mostly due to the gas absorption).

The particle conductivity is estimated as *σ _{p}* =

Diameters (μm) | Parking area (Å^{2}/group) | Number of -COOH groups^{a} | Net charges Q (e)^{b} |
---|---|---|---|

0.481±0.004 | 158.2 | 4.59 × 10^{5} | 8.4 × 10^{4} |

0.982±0.013 | 7.9 | 3.83 × 10^{7} | 1.8 × 10^{5} |

The crossover frequency for salt solutions (PBS solutions) obeys the classic DEP theory very well. In contrast, the crossover frequency for ‘*salt-free’* solution is abnormally low (down to around 20 kHz). This abnormal trapping behavior for ‘*salt-free’* solution was repeated for at least five separate experiments with more than one single particle trapping observations per experiment.

The device was wire-bonded and mounted onto a printed circuit board (PCB). Potentials in the form of *U* – *V* cos*ωt*, produced by a function generator (*Tektronix AFG3252*) together with a voltage amplifier (*Tabor Electronics, Model 9250*), were delivered to the device through 50 Ω BNC cables and monitored by an oscilloscope (*Tektronix DPO 4104*).

The motion of the charged particle was monitored by an optical microscope (*Olympus BX51*) and the video was taken by a high-sensitivity digital CCD camera (*Olympus DP70*) with the highest shutter speed as fast as 1/44000 s. We use a particle tracking algorithm which has been described in detail elsewhere [32] to extract the motion fluctuations. The videos were decomposed into frame sequences using the software VirtualDub (http://www.virtualdub.org). The particle tracking was then carried out using the NIH ImageJ platform (http://rsbweb.nih.gov/ij/) with a particle tracking algorithm (https://weeman.inf.ethz.ch/ParticleTracker).

It should be noted that the video based position extraction method does not measure the instantaneous particle position and has a problem of ‘motion blur’, which results from time-averaging a signal over a finite integration time (shuttle time or acquisition time) [33]. This will lead to the underestimation of the real variance and overestimation of the trap stiffness for each data point in figure 2 (main text). However, for a fixed optical setup, the relative relations between each measured variance (the slope) remain unchanged.

Considering a particle of charge *Q* and mass *m* and in a quadrupole linear (2D) trap of characteristic radius *r*_{0} and electric potential,
$\phi \left(x,y,t\right)=\frac{U-Vcos\omega t}{2{r}_{0}^{2}}\left({x}^{2}-{y}^{2}\right)$, which by definition provides a spatially non-uniform electric field, with DC and AC components in *x*-dimension

$$\mathrm{\Phi}\left(x,t\right)=-Q\frac{\partial}{\partial x}\phi \left(x,y,t\right)=-F\left(x\right)+f\left(x,t\right)$$

(B.1)

where

$$F\left(x\right)=-m{\omega}^{2}\frac{a}{4}x$$

(B.2)

$$f\left(x,t\right)=m{\omega}^{2}\frac{q}{2}xcos\omega t$$

(B.3)

and the trap dimensionless parameters are defined as

$$\begin{array}{cc}a=\frac{4QU}{m{r}_{0}^{2}{\omega}^{2}}& q=\frac{2QV}{m{r}_{0}^{2}{\omega}^{2}}\end{array}$$

(B.4)

The equation of motion in the *x* direction is

$$m\ddot{x}=-\xi \dot{x}+F\left(x\right)+f\left(x,t\right)$$

(B.5)

Introducing a dimensionless viscosity *b* and a dimensionless time *τ*

$$\begin{array}{cc}b=\frac{2\xi}{m\omega}& \tau =\omega t\end{array}$$

(B.6)

and replacing

$$x=exp(-b\tau /4)P$$

(B.7)

then equation B.5 takes the form

$$\ddot{P}=hP+\frac{1}{2}qP\phantom{\rule{0.2em}{0ex}}cos\tau $$

(B.8)

where

$$h=\frac{{b}^{2}}{16}-\frac{a}{4}$$

(B.9)

Without loss of generality, the solution of equation B.8 can be written in the form of the Floquet expansion

$$P\left(t\right)=exp(-i{\int}_{0}^{\tau}\sigma (\tau )d\tau )\sum _{n=-\infty}^{\infty}{P}_{n}\left(\tau \right)exp\left(in\tau \right)$$

(B.10)

where *P _{n}*(

$$-{n}^{2}{P}_{n}-{\sigma}^{2}{P}_{n}+2n\sigma {P}_{n}=h{P}_{n}+\frac{q}{4}({P}_{n+1}+{P}_{n-1})$$

(B.11)

where the dependence on *τ* is henceforth dropped for brevity, and *n* takes all integers. To secure a nontrivial solution, one equates the infinite tridiagonal determinant of the system with zero. This yields the equation for the infinite number of Floquet-Lyapunov (FL) exponents *σ*, which define an infinite number of solutions for the system. We will seek the solution for the case of particular experimental interest, *q* 1, under arbitrary dragging parameter *b* and DC parameter *a*. Looking for the non-oscillating (*P*_{0}) and oscillating terms to the lowest non-vanishing powers in *q* (*P*_{±1}), equation B.11 simplifies into

$$-{\sigma}^{2}{P}_{0}=h{P}_{0}+\frac{q}{4}\left({P}_{1}+{P}_{-1}\right)$$

(B.12)

$$-{P}_{\pm 1}-{\sigma}^{2}{P}_{\pm 1}\pm 2\sigma {P}_{\pm 1}=h{P}_{\pm 1}+\frac{q}{4}{P}_{0}$$

(B.13)

reducing the infinite homogenepus system into the finite one of order 3. Thus

$${P}_{1}+{P}_{-1}=-A\frac{q}{2}{P}_{0}+O\left({q}^{2}\right)$$

(B.14)

which, when replaced in equation B.12, gives the FL exponents

$${\sigma}_{1,2}=\pm i\sqrt{h-\frac{A}{8}{q}^{2}}=\pm i{\sigma}_{0}$$

(B.15)

where

$$A\approx \frac{1}{1+4h}+O\left({q}^{2}\right)$$

(B.16)

Note that the above analysis is valid for an *arbitrary* drag parameter *b* and arbitrary DC parameter *a*. The equation B.13 yields

$$\begin{array}{cc}{P}_{1}=-\frac{q/4}{1\mp 2i\sqrt{h}}{P}_{0}& {P}_{-1}=-\frac{q/4}{1\pm 2i\sqrt{h}}{P}_{0}\end{array}$$

(B.17)

Notice that it is easy to show that *P*_{±2} are proportional to *q*^{2}, *P*_{±3} to *q*^{3}, therefore *P*_{±1} are the leading coefficients in expansion of small *q* of the rapid oscillating part of the Floquet expansion in equation B.10, oscillating with the driving frequency *ω*. Since in *P*_{0}(*t*) ~ exp(± ∫ *σ*_{0}*dτ*), *σ*_{0} can be either real or imaginary, the latter case producing secular oscillations of the frequency *ωσ*_{0} which is in the limit *b =* 0, *a =* 0 equal to
$\omega q/2\sqrt{2}$, obviously much smaller than *ω* when *q* 1. However, when *b* > 1 (typical for aqueous environments), *ωσ*_{0} is real, and when combined with the exponent in equation B.7 yields − (*b*/4 *σ*_{0}) < 0 when *q* 1. This means that *x*(*t*) is proportional to an exponentially decreasing function of time, exp [− (*b*/4 *σ*_{0})*ωt*]. It is interesting to note that the two exponent factors are quite different in size. Thus,
$\left(b/4-{\sigma}_{0}\right)\approx \frac{{q}^{2}/16}{\sqrt{h}\left(1+4h\right)}\ll 1$, while
$\left(b/4+{\sigma}_{0}\right)\approx b/4+\sqrt{h}\underset{a\to 0}{\to}b/2\sim 1$. This means only the solution corresponding to *σ*_{1} will define long time motion in water, the other being a short-time transient.

By neglecting high order terms in *q*, we can now write *P*(*τ*) in equation B.10 as,

$${P}^{\left(i\right)}\left(\tau \right)\approx {P}_{0}^{\left(i\right)}+({P}_{1}^{\left(i\right)}exp(i\tau )+{P}_{-1}^{\left(i\right)}exp(-i\tau ))$$

(B.18)

where *i*=1, 2 correspond to two eigen solutions in equation B.15,

$${P}^{\left(1,2\right)}=[1-\frac{q/2}{1+4h}(cos\omega t\mp 2\sqrt{h}sin\omega t]exp(\pm \omega {\int}_{0}^{t}{\sigma}_{0}dt)$$

(B.19)

The full solution for *x*(*t*) is then

$$x\left(t\right)=exp(-\frac{b}{4}\omega t)({A}_{1}{P}^{\left(1\right)}+{A}_{2}{P}^{\left(2\right)})$$

(B.20)

where *A*_{1} and *A*_{2} are the integration constants, depending on the initial conditions. If one assumes zero initial velocity at an initial position *x*_{0}, as well as adiabatic switching on of the external potential, then for *b* *q*, the solution takes the form

$$x\left(t\right)=exp[-(b/4-{\sigma}_{0})\omega t]{P}^{\left(1\right)}{x}_{0}$$

(B.21)

When only an AC field is applied (*a* = 0), it follows

$$x\left(t\right)=exp(-\frac{{q}^{2}/{b}^{2}}{1+{b}^{2}/4}\omega t)[1-\frac{q/2}{1+{b}^{2}/4}(cos\omega t-\frac{b}{2}sin\omega t)]{x}_{0}$$

(B.22)

However, when also b=0 (vacuum case)

$${P}^{\left(1,2\right)}=(1-\frac{q}{2}cos\omega t)exp(\pm \frac{\omega}{2\sqrt{2}}{\int}_{0}^{t}\mathit{\text{qdt}})$$

(B.23)

which yields

$$x\left(t\right)={x}_{0}(1-\frac{q}{2}cos\omega t)cos(\frac{\omega}{2\sqrt{2}}{\int}_{0}^{t}\mathit{\text{qdt}})$$

(B.24)

defining the secular angular frequency $\frac{q}{2\sqrt{2}}\omega $.

To make a connection of these results with the more general case of non-uniform electric field in the main text, it can be seen from equation B.3 and equation B.4 that
$q=2Q\frac{\partial {E}_{0}(x)}{\partial x}/(m{\omega}^{2})$, where in the case of quadrupole field,
${E}_{0}\left(x\right)=Vx/{r}_{0}^{2}$. By the same token, *f*_{0}(*S*)/(*mω*^{2}) in equation 2 in the main text takes the form *qS*/2 in case of the quadrupole field. equation 3 in the main text takes the form

$$m\ddot{S}=-m\omega \frac{b}{2}\dot{S}-m{\omega}^{2}\frac{a}{4}S-m{\omega}^{2}\frac{{q}^{2}/8}{1+4h}S$$

(B.25)

where the last term at the right-hand side of equation B.25 defines the ponderomotive potential for the case of a quadrupole AC potential, yielding the ‘simplest’ spatially inhomogeneous potential, linear in *x*.

The harmonic EP pseudopotential of equation 4 in the main text can be written as

$${\mathrm{\Phi}}_{sp}\left(x\right)\approx \frac{q/8}{1+4h}\frac{V}{{r}_{0}^{2}}{x}^{2}$$

(B.26)

Finally, the EP ‘trap stiffness’ for a linear quadrupole trap has the form

$${k}_{ep}=m{\omega}^{2}\frac{{q}^{2}/8}{1+4h}$$

(B.27)

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