Measurement of Anisotropic Particle Interactions with Nonuniform ac

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Measurement of Anisotropic Particle Interactions with Non-uniform AC Electric Fields Bradley Rupp, Isaac Torres Diaz, Xiaoqing Hua, and Michael A. Bevan Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b04066 • Publication Date (Web): 22 Jan 2018 Downloaded from http://pubs.acs.org on January 29, 2018

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Measurement of Anisotropic Particle Interactions with Non-uniform AC Electric Fields Bradley Rupp,† Isaac Torres-Díaz,† Xiaoqing Hua, and Michael A. Bevan∗ Chemical & Biomolecular Engineering, Johns Hopkins University, Baltimore, MD 21218

Abstract Optical microscopy measurements are reported for single anisotropic polymer particles interacting with non-uniform AC electric fields. The present study is limited to conditions where gravity confines particles with their long-axis parallel to the substrate, such that particle can be treated using quasi-2D analyses. Field parameters are investigated that result in particles residing at either electric field maxima or minima and with long-axes oriented either parallel or perpendicular to the electric field direction. By non-intrusively observing thermally sampled positions and orientations at different field frequencies and amplitudes, a Boltzmann inversion of the time-averaged probability of states yields kT-scale energy landscapes (including dipole-field, particle-substrate, and gravitational potentials). The measured energy landscapes show agreement with theoretical potentials using particle conductivity as the sole adjustable material property. Understanding anisotropic particle-field energy landscapes vs. field parameters enables quantitative control of local forces and torques on single anisotropic particles to manipulate their position and orientation within non-uniform fields.

Introduction The ability to control an anisotropic colloidal particle’s position and orientation is a key enabling mechanism for micro-manipulation in nano- and micro- scale systems. Optical,1 magnetic,2 and electric3-4 “tweezers” are established methods to control particle positions. Such methods are based on an induced dipole interacting with a local field maximum or minimum to exert a net trapping force (based on suitable material property contrasts between particles and media). In the case of electric tweezers, contrast between frequency-dependent dielectric properties of particles and media determine where particles are located in non-uniform AC electric fields. Based on time-average induced dipole-field interactions,5-7 when particles are more polarizable than the medium, particles localize at electric field maxima, and when the medium is more polarizable than the particles, particles localize at electric field minima. Whereas transport of particles with induced dipoles in AC electric field gradients can control particle motion (i.e., dielectrophoresis8-9), particle position can be controlled via time-averaged static trapping of particles at electric field maxima or minima. Controlling anisotropic particle orientation with external fields typically involves aligning the particle’s long-axis with the field direction (often within concentrated ensembles1016 ). This is generally understood as the induced dipole being aligned with the particle long-axis, which has a minimum energy when aligned with the local field direction. However, it has been shown from microscopy observations that anisotropic shaped cells can be aligned in AC electric fields with either their long or short axis parallel to the field in uniform AC electric fields.17-18 Two orientations of synthetic anisotropic particles have been observed in AC electric field

∗ To whom correspondence should be addressed: [email protected] † These authors contributed equally to this work. Rupp, Torres-Díaz et al.

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mediated deposition of concentrated particles on electrodes,19 which provides further evidence for controlling anisotropic particle orientation in electric fields beyond simple alignment of the long axis with the field. Such observations of different anisotropic particle alignments in AC electric fields are supported by theoretical state diagrams based on models for ellipsoids without20 and with21 coatings. Practically, such models indicate that ellipsoids with three different axes can be oriented in up to six different orientations on a surface,22 which suggests additional understanding of such behavior could be practically useful. Direct measurements of the time-averaged interaction of non-uniform AC electric fields with anisotropic colloidal particles vs. frequency is essential to validate existing theories. Theoretical understanding how field shape and frequency control particle position and orientation for given material properties could provide the capability to design, control, and optimize materials and processes involving anisotropic particles. However, the authors are not aware of any direct measurements of such interactions. Although transport of anisotropic particles in AC electric field gradients (i.e., dielectrophoresis)8-9 is often used to measure an electric force balanced by hydrodynamic drag, this interaction is not identical with the one that traps a particle in a particular orientation at an electric field. During transport processes, minimization of hydrodynamic drag often aligns particles along their long-axis relative to the direction of motion. One approach to measuring the time-average interaction of colloidal particles with nonuniform AC electric fields in the absence of flow has previously been demonstrated for spherical particles.5-7 By measuring thermal sampling of a single particle about its equilibrium position in an AC electric field, a time-averaged probability distribution can be constructed. The measured probability distribution can be inverted via a Boltzmann relationship to obtain the local energy landscape about the potential energy minimum position.23 Measuring energy landscapes (i.e., energy vs. position) vs. frequency was used to show how spherical particles change their equilibrium position and thermal sampling between electric field maxima and minima, which was captured by theoretical models. By performing measurements between co-planar electrodes on a microscope slide with a well-defined spatially varying electric field,24 it was possible to model the potential energy landscapes with no adjustable parameters. This yielded a theoretical potential that could be employed for arbitrary electric field configurations, which are potentially easier to design and control than optical frequency electromagnetic fields for optical trapping.2526 The aforementioned analysis has been used to measure other field mediated interactions,5-7, 2729 and is routinely used to measure particle-substrate interactions,30-32 but has not been adapted to investigate anisotropic particles interacting with non-uniform AC electric fields. In this work, uni-axial superellipsoidal particles are measured as they thermally sample positions and orientations at equilibrium for different field frequencies and amplitudes. For the relatively heavy micron scale SU-8 particles investigated in this work (Fig. 1a), particles adopt quasi-2D states (with their long axis parallel to the substrate) that are positioned at either electric field maxima or minima and oriented with their long-axis parallel or perpendicular to the field direction. By generating measured probability distributions of thermally sampled states consisting of position and orientation, measured energy landscapes are obtained by a new analysis for anisotropic particles based on a Boltzmann inversion.23 Measured energy landscapes are compared with theoretical models including three-dimensional particle-substrate33 and particle-field22 interactions to interpret results and to generalize potentials for a variety of particle shapes, materials, and field conditions in a broad range of applications.

Rupp, Torres-Díaz et al.


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Fig. 1. (a) SU-8 photoresist particles with superellipsoidal shape (with inset dashed line showing n = r = 4 in Eqs. (2) and (5)) for non-uniform AC electric field experiments. (b) Schematic of anisotropic particle (uni-axial superellipsoid) with arbitrary orientation between parallel coplanar electrodes as parameterized by Euler angles (ϕ, θ, ψ). Principal particle axes depicted in color (primed axes) show particle coordinates. Black axes (unprimed axes) show lab coordinates. (c) Projected coordinate system observed in optical microscope (top view).

Theory Anisotropic particle potential energy The general coordinate system for a tri-axial superellipsoid with semi-axes (rx, ry, rz) is defined by the laboratory coordinates (x, y, z) and the particle coordinates (x’, y’, z’) (Fig. 1b,c). The particle orientation in laboratory coordinates is specified by Euler angles (θ, φ, ψ). The net interaction energy of a tri-axial superellipsoid, which depends on position and orientation, includes the particle-wall (upw) and particle-field interactions (upf), is given by u net ( x, y, z,θ , φ,ψ ) = u pw + u pf ,

(1)

where for a superellipsoid particle surface given by,

(

r

x rx + y ry

)

r nr

n

+ z rz = 1,

(2)

the particle-wall interaction energy for an arbitrary orientation of the particle has been quantified using Derjaguin approximation,33 u pw = uepw =

Z Γ

exp ( −κ h ) ,

2

 zv eψ p   kT   zv eψ w  Z = 64πε m  tanh tanh    4k T  ,  zv e   4k BT  B

(3)

where Γ is the Gaussian curvature of the particle evaluated at the minimum separation distance h between surfaces, κ–1 is the Debye length, εm is the medium dielectric constant, k is Boltzmann’s constant, T is the absolute temperature, e is the electron charge, zv is the valence of ions in the bulk, and ψp and ψw are the particle and wall surface potentials. The gravitational potential of a superellipsoidal particle is u gpf = v p ( ρ p − ρ m ) gz ,

(4)

where z is the particle center elevation, ρp is particle density, ρm is medium density, g is acceleration due to gravity, and vp is the superellipsoid volume,34 which for n = r = 4 (Fig. 1a) is



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vp =

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rx ry rz Γ (1 4 ) Γ (1 4 ) Γ ( 5 4 ) Γ (1 2 ) = 6.482rx ry rz , Γ (1 2 ) Γ (9 4) 2

(5)

where Γ is the complete gamma function (not to be confused with Gaussian curvature in Eq. (3)). An explicit description of the time-average potential energy of a particle having a uniform induced polarization within an applied electric field is limited to few particle geometries. In particular, it is only possible when the Laplace equation can be solved via separation of variables35 and for uniform polarization fields inside particle geometries with a second-degree surface.36 Because superellipsoids do not satisfy these constraints, superellipsoid particle polarization is approximated by an equivalent ellipsoidal particle polarization (i.e., same volume and axis ratio), which is given by the sum of the potential along the particle three principal axes as20, 22

udepf = −

3ε mv p

4

(f

CM ,x

)

Ex′ 2 + fCM ,y E y′ 2 + fCM ,z Ez′ 2 ,

(6)

where E’ is electric field in the specified particle-coordinate, and each variable denoted as fCM is a Clausius-Mossotti factor, defined along the particle j–axis as (fCM,max is largest fCM,j as the frequency tends to zero)

(7)

where rj is the length of the particle’s j-axis, the subscripts p and m denote particle and medium properties, and the complex permittivities are given as, (8) where ε is permittivity, ε0 is free space permittivity, σ is conductivity, and ω is field frequency. Aj is a geometric factor that arises from the solution of the Laplace equation in ellipsoidal coordinates. For a sphere (rx = ry = rz), Aj = 1/3, which recovers the expected expression.8 Coplanar Electrode Field The electric field, E, components in Eq. (6) parallel thin-film coplanar electrodes of separation, d, is24

  sin xˆ   cos xˆ   − tan −1  , Ex = E0  tan −1    sinh zˆ   sinh zˆ    E   cosh zˆ + cos xˆ cosh zˆ + sin xˆ   Ez = 0  ln   , 2   cosh zˆ − cos xˆ cosh zˆ − sin xˆ  

(9)

where xˆ = π ( x + d ) / 2d + π / 4 , zˆ = π z / 2d , and E0=Vpp/πdg, where Vpp is the peak to peak voltage, dg is the electrode gap. Rupp, Torres-Díaz et al.


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Position & Orientation Transitions Using the model in Eqs. (6) and (7), the value of fCM indicates frequencies when the particle is more polarizable than the medium (fCM,j > 0) and when the medium is more polarizable than the particle (fCM,j < 0). The sign of fCM determines whether a particle has a potential energy minimum at either the electric field maximum (when fCM,j > 0, Eq. (6) is minimized by obtaining most negative value at field maximum) or the electric field minimum (when fCM,j < 0, Eq. (6) is minimized by obtaining smallest positive value at field minimum). The frequency at which the particle and medium are equally polarizable (fCM,j = 0) is the crossover frequency that determines when the particle changes position between the field maximum and field minimum, which for the j–axis is given as

ωc , j

( (

)  ) 

 (σ p − σ m ) A jσ p + (1 − A j ) σ m = −  ( ε p − ε m ) A j ε p + (1 − A j ) ε m 

0.5

(10)

,

A different crossover frequency determines when particles change orientation. The relative values of fCM,j for each axis determine how orientation of a particular particle axis along the primary field direction minimizes energy in Eq. (6). When fCM,j > 0, the largest |fCM| value determines which axis aligns with the field. In contrast, when fCM,j < 0, the smallest |fCM| value determines which axis aligns with the field. The frequency at which two axes are equally polarizable (fCM,j = fCM,k) determines the crossover when particles change orientation. The transition between the j–axis aligned with the field to the k–axis aligned with the field is  (

ω j ,k =  − c2 ± ( c22 − 4c0 c4 )

0.5

) 2c 

0.5

(11)

4

c0 = (σ p , j − σ m ) s j sk2 − (σ p ,k − σ m ) sk s 2j

c2 = ( ε p − ε m ) ( e j sk2 − ek s 2j ) + (σ p , j − σ m ) s j ek2 − (σ p ,k − σ m ) sk e 2j

(12)

c4 = ( ε p − ε m ) ( e j ek2 − ek e 2j )

s j = σ m + (σ p , j − σ m ) A j

(13)

e j = ε m + (ε p − ε m ) Aj Quai-2D analysis

The net potential energy of an anisotropic particle is a function of up to six variables including three translational degrees of freedom (x, y, z) and three rotational degrees of freedom (ϕ, θ, ψ). Analysis of all dimensions as well as methods to analyze cases of reduced dimensionality for comparison with experiment have previously been reported.22 For noninteracting infinitely dilute particles at position x and orientation θ (see Fig. 1), the probability of sampling different positions and orientations is given by z f yf φf ψ f

p ( x,θ )


y,z,φ ,ψ

=

∫ ∫ φ∫ ψ∫ p ( x, y, z,φ,θ ,ψ ) dz dy dφ dψ zi yi

(z

f

i

i

)(

)(

)(

− zi y f − yi φ f − φi ψ f − ψ i


)

,

(14)

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where p(x,y,z,φ,θ,ψ) = exp[-unet(x,y,z,φ,θ,ψ)] is obtained from a Boltzmann relationship to the potential energy in Eq. (1). The probability in Eq. (14) projected onto a lower dimension can be related to a projected low dimension potential energy landscape by Boltzmann’s equation as

u net ( x,θ ) − u net ( xr ,θ r ) kBT

= ln  p ( xr ,θ r ) 

y,z,φ ,ψ

p ( x,θ )

y,z,φ ,ψ

. 

(15)

where xr and θr are the reference position and orientation (e.g., electrode center, aligned with field). Further reduction in the spatial and orientational dimensionality of probability distributions or energy landscapes can performed in a manner similar to the specific case in Eqs. (14) and (15).

Materials & Methods Particle Fabrication. Particles with a superellipsoidal shape were fabricated using photolithography. Omnicoat (MicroChem), was spin coated onto a silicon wafer to provide a sacrificial layer. On top of the Omnicoat, SU-8 2002 (MicroChem) was spin coated to the desired particle thickness. A photomask was used to pattern the particles at an exposure energy of 50 mJ/cm2. The particles were lifted off of the substrate with Remover PG (MicroChem), which dissolved the Omnicoat layer. The particles were then rinsed with isopropyl alcohol and dispersed in DI water. The resulting particle geometry, shown in Fig. 1a, corresponds to a superellipsoid (described by Eq. (2)) n=r=4 and rx=ry5 kT. Exponentially longer observation times are required to obtain sufficient statistics necessary to resolve energy landscapes for energies > 5 kT (energy landscapes obtained via Boltzmann inversion (Eq. (15)) of a measured probability distributions are sampled exponentially less at higher energies relative to the energy minimum23). Fig. 3(d) also shows the averaged projections of the 2D energy landscape onto 1D landscapes for position, 〈unet(x)〉θ, and orientation, 〈unet(θ)〉x. The results in Fig. 2 show the particle is thermally sampling in the vicinity of the electrode gap center (x=0) with the long-axis aligned perpendicular to the x-component of the field (θ=90o). This same analysis is applied for different frequencies and amplitudes for the results presented in the following sections. Rupp, Torres-Díaz et al.


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Fig. 3. Potential energy landscapes for four states of particle position and orientation, including: (a) parallel to the field at the electrode edge (0.15Vpp 0.1MHz, E0 = 0.50 kV/m), (b) perpendicular to the field at the electrode edge (0.2Vpp 0.3MHz, E0 = 0.67 kV/m), (c) perpendicular to the field at the middle of the electrode gap (1.5Vpp 0.7MHz, E0 = 5.0 kV/m), and (d) parallel to the field at the middle of the electrode gap (1.5Vpp 1.4MHz, E0 = 5.0 kV/m). Surfaces are two-dimensional theoretical energy landscapes, net 〈u (x,θ)〉, from Eq. (1). Measured landscapes are projected onto x and θ are shown by circles and compared to projected theoretical landscapes shown by lines (using Eq. (14)).

Energy Landscapes for Different Particle States From Eqs. (6)-(8), it can be seen that the particle position and orientation within inhomogeneous fields depends on the particle properties and the field frequency. There are three factors that determine how a particle is polarized in an electric field: the particle and medium conductivity, the particle and medium dielectric constants, and the particle geometry. Theoretical predictions indicate that uni-axial ellipsoidal particles (rx = ry < rz), which are not too different from the superellipsoids in this study, can be positioned and oriented in the same number of quasi-2D positions and orientations at low field amplitudes. In particular, it is possible to position particles either at the electrode edges or at the electrode gap center and oriented either parallel or perpendicular to the electric field.22 In short, it is possible to obtain 4 different states for a uni-axial anisotropic particle with its long axis parallel to the substrate. The results in Fig. 3 show 4 states for uni-axial superellipsoidal particles in practice, which can be analyzed to understand the potential energy landscape for each condition (field amplitudes and frequencies reported in the Fig. 3 caption). The left column shows images and renderings for each state. The right row of Fig. 3 shows 2D and projected 1D potential energy



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Table 1. Parameters used to fit Eqs. (1) – (15) to two-dimensional potential energy landscapes in Fig. 3. a b c Values obtained from: optical microscopy, thin film analyzer, manufacturer values (MicroChem), d 41 e f 5, 30-31 handbook values, conductivity meter, and previous experiments with dielectric particles.

parameter rx, ry, rz (µm)a,b ρp (kg/m3)c ρm (kg/m3)d κ-1 (nm)e ψp = ψw (mV)f

value 0.9, 0.9, 4.9 1200 1000 100 -50

parameter dg (µm)a εp/ε0c εm/ε0d σm (µS/cm)e

value 95 3.2 78 1

Table 2. Particle conductivities from theoretical global fits to all eight 1D energy landscapes in Fig. 3 and from individual fits to each landscape for the particle long-axis, fCM,z (blue) and short-axis, fCM,x (red).

frequency 0.1 MHz 0.3 MHz 0.7 MHz 1.4 MHz

σx = σy (µS/cm) individual fit global fit 31 31 21 21 21

σz (µS/cm) individual fit global fit 55 53 53 53 53

landscapes for each measured field condition. Solid lines are global best fits of Eq. (1) (inserted in Eqs. (14), (15)) to the eight measured potential energy landscapes for 〈unet(x)〉θ in red and 〈unet(θ)〉x in blue. The global fit is based on the parameters in Table 1 and Table 2 with particle conductivity (σx=σy and σz) as the sole adjustable parameters. For the global fit, the weighted mean-squared-error was minimized between all eight measured and fit potentials. The resulting particle conductivities are consistent with prior measurements and models of dielectric particles with electrostatic double layers (such as the SU-8 particles treated with sulfuric acid37 in this work).5-6, 40 The systematic biases observed in some 1D projected landscapes result from constraints on the fitting procedure that require the correct qualitative behavior in terms of the correct position and orientation. As such, even though some lines do not go thought points with positive and negative deviations, the mean square error is still minimized with the added constrain that the correct qualitative behavior is observed at all frequencies. The 2D potential energy landscapes are plotted based on the fitted 〈unet(x)〉θ and 〈unet(θ)〉x for each experimental condition. Before further discussing the agreement between the measurements and models, the following section reports the frequency dependent fCM resulting from the fits in Fig. 3. At this point in the discussion, we make an additional note about the quasi-2D nature of these experiments and the multi-dimensional energy landscape given in Eq. (1). Although the projected theoretical energy landscapes in Fig. 3 provide a rigorous comparison with the experimental microscopy measurements that only track x and θ, lower dimensional cross sections of the complete energy landscape also provide reasonable approximations. In particular, good approximations of the projected landscapes (Eqs. (14), (15)) can be obtained from 2D crosssections of the full energy landscape (Eq. (1)) evaluated at: φ=90o, ψ=0o, zm (the most probable centroid elevation), and y=0 (no y-dependence in this geometry). Using this procedure, the resulting 2D cross-sections are close to the projected landscape in the vicinity of the energy minimum state; within 0.1kT for particles near electrode edges and 1kT for particles at the Rupp, Torres-Díaz et al.


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middle of the electrode gap. This agreement indicates that while the particle is not exactly 2D, it is reasonably approximated as quasi-2D. Because the 2D cross section is easier to evaluate (no integration or particle-wall interactions), studies of similarly heavy particles confined near a substrate could benefit from using this simpler result. Position & Orientation Transitions from fCM(ω) Values of fCM, vs. frequency obtained from the fits are shown in Fig. 4. Lines are fCM plotted from Eqs. (7) and (8) using the fit parameters obtained from the global fitting process (minimization of error across all potential energy landscapes using a single set of material properties, Table 1). Points are fCM obtained by fitting each experiment (minimization of error for each 1D potential energy landscape). Vertical dashed lines are crossover frequencies between different positions (Eq. (10)) and orientations (Eq. (11)) using the global fit parameters. As depicted in Fig. 4, the progression in position and orientation with increasing frequency is: (1) transitioning from the long-axis being parallel to perpendicular with respect to the field xcomponent while residing at the field maximum near the electrode edge; (2) transitioning from the electrode edge to the middle of the electrode gap while the long-axis remains perpendicular to the field x-component; and (3) transitioning from perpendicular to parallel orientation of the long-axis with respect to the field x-component of the field minimum at the middle of the electrode gap. The crossover frequencies between different states of position and orientation are accurately captured by the energy landscapes in Fig. 3 and frequency dependent fCM(ω) in Fig. 4. In particular, the minimum energy configuration is in agreement between the measured and modeled energy landscapes at each AC electric field frequency. At the lowest frequencies where all fCM,j > 0, the net sign of potential energy in Eq. (6) is negative. As such, particles reside at the electric field maximum and have particle axis with the largest fCM,j aligned with the largest electric field component to attain the lowest energy state (i.e., the most negative potential energy). The particle long-axis (particle frame z-axis) has the highest fCM,j value at low frequencies so that it is oriented with the largest field component (lab frame field x-component). The first transition (ω ≈ 0.2 MHz) corresponds to the particle changing orientation without changing position. At this condition, the particle short-axis (particle frame x-axis) has the highest fCM,j value so that it is oriented with the largest field component (lab frame field x-component). It is important to note that no transition occurs when fCM,z < 0 but fCM,x > 0 (ω ≈ 0.3 MHz). As long as one particle axis has a positive fCM,j value, it will correspond to the lowest energy state since a negative energy is always less than the positive energy associated with negative fCM,j values. When all fCM,j values become negative (ω ≈ 0.5 MHz) with further increasing AC field frequency, the particle transitions from the electrode edge at the field maximum to the electrode gap center at the electric field minimum. The potential energy in Eq. (6) becomes positive at these frequencies, so the lowest energy state corresponds to the smallest positive energy state. With the positional transition from the electrode edge to the electrode gap center at (ω ≈ 0.5 MHz), the lowest energy orientation is the particle short-axis (particle frame xaxis) aligned with the largest field component (lab frame field x-component). In this case the negative fCM,j with the smallest magnitude results in the lowest positive potential energy state. A final orientational transition (ω ≈ 0.7 MHz) occurs when the particle long-axis aligns with the largest field component as the result of the negative value of fCM,z having the smallest magnitude. At the highest frequencies, the plateau of the fCM values (Fig. 4) is related to the ratio of Rupp, Torres-Díaz et al.


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Fig. 4. Claussius-Mossotti factor, fCM, from theoretical global fits (Eq. (7)) to all eight 1D energy landscapes in Fig. 3 (lines) and from individual fits to each landscape (points) for the particle long-axis, fCM,z (blue) and short-axis, fCM,x (red). Table 2 reports conductivities used for global and local fits. Vertical dashed lines indicate crossover frequencies between positions (Eq. (10)) and orientations (Eq. (11)).

medium and particle dielectric constants, where at higher frequencies the contribution of conductivity to the particle polarization is minimized (Eq. (8)). Under these conditions, the particle polarization is more strongly a function of relative permittivity (Eq. (7)), meaning that fCM along the short- and long- axes have similar values with the difference determined by particle geometry via the parameter Aj in Eq.(7). Measurements vs. Model The measured and theoretical potential energy landscapes (Fig. 3) show agreement within the resolution of the measurements and the uncertainty of the model parameters including two adjustable quantities (i.e., conductivities along the particle long- and short- axes). In particular, all states observed at different frequencies are captured as global minima on the evolving potential energy landscapes given by Eq. (1). In addition, the transitions between different positions and orientations are captured by the cross-over frequency expressions in Eqs. (10) and (11) using the material properties in Eq. (8). Although the model considers a uniform polarized particle, characterized by fCM, it captures the observed states and transitions for a charged superellipsoid particle in a non-uniform AC electric field. This suggests that the first moment of the polarization along the particle axes determines its position and orientation, and higher moments may provide better accuracy in the shape of the energy landscapes, particularly in regions of high field gradients, as we further discuss below. There are some discrepancies between the measurements and the models, but these can be attributed to known limitations including measurement resolution and model assumptions. Optical microscopy spatial resolution limits42 introduce uncertainty in particle centroids less than the pixel size and/or optical diffraction limit. Spatial limitations also influence angles determined from the inverse tangent of particle endpoints relative to particle centroids. This introduces a spatial Gaussian uncertainty that effectively blurs energy landscapes in a manner that biases strong forces (large energy changes over short distances) to appear weaker than expected.23



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While this accounts for some discrepancies where the measured interaction indicates broader thermal sampling than the theory (e.g., θ-projection in Fig. 3d), discrepancies where the theory indicates broader thermal sampling than the measurement (e.g., x-projection in Figs. 3a,b) must arise for different reasons. Some discrepancies not accounted for by microscopy limitations are likely due to approximations, as the result of simplifying assumptions, rather than basic flaws due to incorrect physics. For example, some limitations in the predicted fCM (in Eq. (7), used as input in Eqs. (1) and (6)) could arise from uncertainties in physical properties, approximating particle shapes as ellipsoids, and a uniform polarization model (without considering electrostatic double layer in a core-shell model). The superellipsoid particle polarization in this work is based on ellipsoidal particle model, which is more likely to be a reasonable approximation when the particle dimensions are small compared to the field gradient’s characteristic length. While this condition is satisfied in the electrode gap center, significant gradients over small distances in the z-lab coordinate near the electrode edge might require different approaches (e.g., multipole moments) to accurately quantify non-uniformities in the polarization.43 In other words, our model quantifies the first polarization moment in the potential energy landscape, which is reflected in the good agreement with fit fCM values at different frequencies. However, difference between the measured potential energy landscape and our model are likely due to higher polarization moments (multipoles) generated by the particle geometry and high field gradients. In addition, polarization non-uniformities due to superellipsoid edge shape (not present on ellipsoids) can alter each axes’ polarization and the resulting fCM.44 Another issue is that anisotropic particles with a conductive surface layer (e.g., electrostatic double layer) cannot necessarily have their polarizability and fCM computed based on an effective bulk conductivity,45 which is often the approach for spheres.21 While each effect can be more rigorously considered with significantly more complex numerical models, the results in this work show that the measured AC field frequency dependent lowest energy states, transitions between position and orientation, and the majority of energy landscape features can be quantitatively captured with a minimal model using two adjustable fitting parameters. Future modeling work can explore how these first measurements of frequency dependent energy landscapes might be more rigorously interpreted and predicted.

Conclusions Measurements and models are reported for energy landscapes for single anisotropic particles in non-uniform AC electric fields. Applying relatively weak fields allows particles to thermally sample quasi-2D positions and orientations. Using a new non-intrusive approach to sensitively quantifying anisotropic particle-field interactions, we analyze measured particle coordinates to construct equilibrium distributions that are inverted via a Boltzmann relation to obtain energy landscapes. Energy landscapes vs. frequency show how particles have four unique states; particles can be positioned at either electric field maxima or minima and oriented with their long axis either parallel or perpendicular to the field direction. By fitting measured energy landscapes to a simplified particle-field potential based on ellipsoids and input dielectric spectra, particle position is shown to depend on the relative polarizability of the particle and medium whereas particle orientation is shown to depend on the relative polarizability of each particle axis. Modeled energy landscapes capture frequency dependent energy landscapes including



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minimum energy states and transitions between them, although some quantitative differences in thermal sampling near minima are likely due to imaging limitations as well as model approximations. The particle-field potentials measured and modeled in this work are general to different electric field shapes and enable electric field mediated micro-manipulation of single particles and possibly assembly of particle ensembles. Future extensions of the approaches developed in this work could also be applied to a range of different particle shapes (e.g., including systematic variations in superellipsoid particles with different aspect ratios and shape parameters).

Acknowledgments We acknowledge financial support by the National Science Foundation (CBET-1434993).

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For Table of Contents Use Only For Table of Contents Use Only (1.75 inches high x 3.25 inches wide)

Measurement of Anisotropic Particle Interactions with Non-uniform AC Electric Fields Bradley Rupp, Isaac Torres-Díaz, Xiaoqing Hua, and Michael A. Bevan Chemical & Biomolecular Engineering, Johns Hopkins University, Baltimore, MD 21218



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Measurement of Anisotropic Particle Interactions with Nonuniform ac

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