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Quantification of the dielectric constant of single non-spherical nanoparticles from polarization

forces: eccentricity effects

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2013 Nanotechnology 24 505713

(http://iopscience.iop.org/0957-4484/24/50/505713)

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IOP PUBLISHING NANOTECHNOLOGY

Nanotechnology 24 (2013) 505713 (8pp) doi:10.1088/0957-4484/24/50/505713

Quantification of the dielectric constant ofsingle non-spherical nanoparticles frompolarization forces: eccentricity effects

G Gomila, D Esteban-Ferrer and L Fumagalli

Institut de Bioenginyeria de Catalunya (IBEC) , C/Balidiri i Reixac 15-21, E-08028 Barcelona, SpainDepartament d’Electronica, Universitat de Barcelona, C/Martı i Franques 1, E-08028 Barcelona, Spain

E-mail: gabriel.gomila@ub.edu

Received 17 September 2013, in final form 29 October 2013Published 27 November 2013Online at stacks.iop.org/Nano/24/505713

AbstractWe analyze by means of finite-element numerical calculations the polarization force between a sharpconducting tip and a non-spherical uncharged dielectric nanoparticle with the objective of quantifying itsdielectric constant from electrostatic force microscopy (EFM) measurements. We show that for an oblatespheroid nanoparticle of given height the strength of the polarization force acting on the tip dependslinearly on the eccentricity, e, of the nanoparticle in the small eccentricity and low dielectric constantregimes (1 < e < 2 and 1 < εr < 10), while for higher eccentricities (e > 2) the dependence is sub-linearand finally becomes independent of e for very large eccentricities (e > 30). These results imply that aprecise account of the nanoparticle shape is required to quantify EFM data and obtain the dielectricconstants of non-spherical dielectric nanoparticles. Experimental results obtained on polystyrene, silicondioxide and aluminum oxide nanoparticles and on single viruses are used to illustrate the main findings.

(Some figures may appear in colour only in the online journal)

1. Introduction

Quantifying the dielectric constant, εr, of small 3Dnano-objects with a scanning probe microscope has beena long-standing challenge with potential applications inmany fields, from materials science and nano-electronicsto biology [1–14]. In particular, because the dielectricconstant is an intrinsic property of a material, quantifying εrallows identification of the material composition of nanoscaleobjects [14–16], which is crucial in materials science andbiomedicine but still remains a major issue [15–19]. However,measuring the dielectric constant of a small 3D object hasbeen hampered by difficulties in detecting its inherently weakdielectric signal and distinguishing it from the major sizeand shape effects of the object, as well as from non-localcontributions of the scanning probe.

Recently we have achieved it using quantitativeelectrostatic force microscopy (EFM) combined with precisefinite-element numerical calculations [14]. We demonstratedthat by measuring weak polarization forces with sub-piconewton resolution using a sharp, structurally stable

tip and by accurately taking into account geometricaleffects in the theoretical modeling, one can determine thedielectric response of single low-polarizable nanoparticles(down to ∼10 nm in radius) at low frequencies, namely,the quasi-static dielectric constant. The high accuracyachieved in the dielectric measurement enabled us to use thedielectric constant as the fingerprint to discriminate unlabeleddielectric nanoparticles of identical shape but differentcomposition, thus demonstrating quantitative and label-freematerial identification with a scanning force microscope. Thisapproach has shown unparallel sensitivity, selectivity andlateral resolution. It potentially enables one to distinguishdielectric nano-objects in a wide range of dielectric constantvalues (1 < εr < 1000), with a relative error smaller than10% (for εr < 10) and true nanometer resolution (∼2 nm tipradius).

In this paper we analyze the additional challenge imposedby a non-spherical shape of the nanoparticles. We show thatthe tip–nanoparticle polarization force is very sensitive to thenon-sphericity of the nanoparticle. Therefore, when analyzingelectrostatic forces in quantitative terms, the nanoparticle

10957-4484/13/505713+08$33.00 c© 2013 IOP Publishing Ltd Printed in the UK & the USA

Nanotechnology 24 (2013) 505713 G Gomila et al

Figure 1. (a) Schematic representation of the calculated system and boundary conditions used. (b) Calculated electric potential andequipotential lines (1 V applied) for an oblate spheroidal nanoparticle with eccentricity e = 2(D = 40 nm,W = 80 nm) and dielectricconstant εr = 4 polarized with a tip radius R = 5 nm, cone half-angle θ = 10◦ and cone height H = 1 µm located at z = 10 nm from thenanoparticle.

shape cannot be disregarded and simplified assumptions basedon parallel-plate or similar models do not appropriatelydescribe the electrostatic interaction. Instead, a preciseaccount of the nanoparticle shape is required to quantifyEFM data and obtain the dielectric constants of non-sphericalnanoparticles. From these results, we suggest that when thedielectric constant of the particle is known, EFM data could beused to estimate the non-sphericity of nanoparticles withoutthe need for complicated deconvolution procedures based ontopographic profile analysis.

2. Numerical calculations of the polarization forcebetween a sharp conducting tip and a non-sphericalnanoparticle

The dielectric polarization of a nanoscale object in an externalelectric field depends on the electric field distribution, theobject shape and size and its inherent dielectric properties,represented by the dielectric constant of the material. Inthe case of nanoparticles on metallic substrates polarized bythe conducting tip of an electrostatic force microscope, theelectric field is highly non-uniform and strongly dependent onthe nanoparticle itself, due to the sharp nature of the tip andto the small distance between the two polarizing electrodes(the tip and the substrate), which is comparable to the size ofthe nanoparticle. Under these conditions, and contrary to thecase of a nanoparticle in a uniform external electric field (seeappendix), the problem cannot be solved analytically and onehas to resort to a self-consistent resolution of the electrostaticpolarization problem by using numerical calculations.

To this aim, we have considered the tip–nanoparticlesystem shown schematically in figure 1(a). It consists of asharp metallic tip, modeled as a truncated cone of height Hand cone half-angle θ terminated with a tangent sphericalapex of radius R located in close proximity (at a distancez) to a dielectric uncharged nanoparticle with an oblatespheroidal shape, which lies on a conductive substrate. Theaxial symmetry axis of the tip is assumed to be aligned

with the center of the nanoparticle. The nanoparticle hasheight D, width W and relative dielectric constant εr. Theeccentricity of the nanoparticle is defined as e = W/D.Thus, for an oblate spheroid nanoparticle one has e >1. The surrounding medium is assumed to be dry airwith relative dielectric constant εr,air ∼ 1. A dc voltagebias VDC is applied between the conducting tip and theconducting substrate, which leads to an electric force actingon the tip which depends on the nanoparticle geometryand dielectric constant, as well as on the tip geometryand tip–nanoparticle distance. The electrostatic problem issolved by finite-element numerical methods using COMSOLMultiphysics (electrostatic module). Details on the boundaryconditions used are given in figure 1(a) (see also thesupplementary information of [14]). From the resolution ofthe problem, the electric potential distribution and hence theelectrostatic force acting on the tip, FDC(z, εr, e), is obtainedby using the built-in Maxwell stress tensor function.

Figure 1(b) gives an example of an electric potentialdistribution (1 V applied) calculated for a representativeexperimental situation of a tip with apex radius R = 5 nm,cone half-angle θ = 10◦ and an oblate spheroid nanoparticlewith height D = 40 nm, width W = 80 nm(e = 2) anddielectric constant εr = 4. The calculated electric force isconverted into a capacitance gradient via the static expressionC′(z, εr, e) = 2FDC(z, εr, e)/V2

DC. From this expression, weobtain the capacitance-gradient contrast, 1C′(z, εr, e) =C′(z, εr, e) − C′(z, 1, e), which gives the signal variationdetected by the tip when the nanoparticle is under the tipwith respect to the case when there is no nanoparticle (onlyair). Note that because the tip apex is located over thecenter of the nanoparticle, the capacitance-gradient contrastcalculated here corresponds precisely to the maximumcapacitance-gradient contrast measured over the center of thenanoparticle in a dielectric image taken at constant height inEFM experiments [14].

Calculations have been performed for oblate spheroidalnanoparticles with heights in the range D = 30–40 nm and

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Nanotechnology 24 (2013) 505713 G Gomila et al

eccentricities between e = 1 (spherical case) and e = 30. Inorder to mimic the uncertainty that may exist in an EFMexperiment concerning the nanoparticle width (difficult tomeasure accurately due to dilation and scanning effects), inthe calculations we varied the eccentricity of the nanoparticlewhile keeping its height fixed. This means that a change inthe eccentricity is equivalent to a change in the width of thenanoparticle. Note that under these conditions the volume ofthe nanoparticle increases quadratically with the eccentricity,V = (4π/3)(D/2)(W/2)2 = (π/6)D3e2. In all simulations thecone height was kept fixed to H = 1 µm, after verifyingthat a higher cone would not change the results, and thecone half-angle was fixed to θ = 10◦ (the nominal angle ofdoped silicon tip used in the experiments). The apex radius,instead, varied in the range R = 2–10 nm. Finally, we haveconsidered representative tip–sample distances in the range10–50 nm. Whenever possible we used values correspondingto our experimental results reported in [14].

3. Results

3.1. Eccentricity effects on the tip–nanoparticle electrostaticforce

Figure 2(a) shows the variation of the capacitance-gradientcontrast, 1C′(z, εr, e), as a function of tip–nanoparticledistance, z, for the case of a solid oblate spheroid of dielectricconstant εr = 6.3, height D = 39 nm and eccentricities e = 1,2, 5, 10 and 30 (for e > 30 the curves overlap). The tip radiusis R = 5 nm. For all eccentricities considered the contrastdecays with increasing tip–nanoparticle distance, as expected.In the spherical nanoparticle limit (bottom curve, e = 1)the contrast shows a strong decay with distance, reachingpractically zero at a few tens of nanometers (∼50 nm)from the nanoparticle. Instead, for larger eccentricities thecontrast decays more slowly and it still gives a measurablevalue at distances larger than 50 nm. We note that forlarge eccentricities (e > 30) the contrast approaches the onecorresponding to an infinitely large dielectric film of thesame thickness (and dielectric constant) as the nanoparticle(dashed line in figure 2(a)). We then conclude that for smalleccentricities (1 < e < 2) the contrast decays with distancein a similar way as it decays for a spherical nanoparticle (apower-law decay, see below), while for large eccentricities thedecay is more similar to the one found for a thin dielectricfilm (a power law plus a logarithmic decay due to the conecontribution, see [20]).

The dependence of the capacitance-gradient contrast onthe eccentricity can be better understood by plotting it asa function of the eccentricity at a fixed tip–nanoparticledistance. In figure 2(b) we show it for the sametip–nanoparticle system given in figure 2(a) and for differenttip–nanoparticle distances, z= 10, 16, 22 and 30 nm (from topto bottom). We see that for fixed tip geometry, particle heightand measuring distance, the dielectric contrast is an increasingfunction of the lateral diameter of the nanoparticle. For smallnon-sphericities (1 < e < 2) the increase is roughly linear.This fact can be better appreciated in the inset of figure 2(b),

Figure 2. (a) Calculated capacitance-gradient contrast at the centerof the nanoparticle as a function of the tip–nanoparticle distance fora nanoparticle of fixed dielectric constant εr = 6.3 and heightD = 39 nm, and different eccentricities e = 1, 2, 5, 10 and 30.Symbols are the calculated points, while continuous lines areinterpolated curves. The dashed line corresponds to the case of alarge dielectric film of the same thickness and dielectric constant asthe nanoparticle. (b) Idem, but as a function of the eccentricity ofthe nanoparticle for different tip–nanoparticle distances, z = 10, 16,22, 30 nm. Inset: zoom of (b) in the small eccentricity range(1 < e < 2) normalized by the calculated value for e = 1. (c) Idem,as a function of the dielectric constant of the nanoparticle at fixedtip–nanoparticle distance z = 9.5 nm and different eccentricities inthe range 1 < e < 2 (from bottom to top). The horizontal line is thevalue measured on a single viral particle of spheroidal shape withexperimental parameters equal to the ones used in the calculations(1C′virus = 7.15 zF nm−1, see [14]). Tip geometry data: R = 5 nmand θ = 10◦.

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Nanotechnology 24 (2013) 505713 G Gomila et al

where we plot the capacitance-gradient contrast at a givendistance as a function of the eccentricity normalized by thecorresponding value obtained for a spherical nanoparticle ofdiameter equal to the height of the nanoparticle. The curvesare linear, showing a slope b = 1–1.4 for the tip–particledistances considered. For larger eccentricities (2 < e < 30)the dielectric contrast increases sub-linearly and saturates to aconstant value for very large eccentricities (e > 30).

Finally, in figure 1(c) we show the dependence of thecapacitance-gradient contrast on the dielectric constant of thenanoparticle at fixed tip–particle distance (z = 9.5 nm) anddifferent eccentricities in the small eccentricity regime (1 <e < 2, from bottom to top). The dependence on the dielectricconstant is qualitatively similar to the one displayed by aspherical nanoparticle (e = 1, bottom line) but it shows largervariations with increasing eccentricity of the nanoparticle. Itcan be shown that with respect to a spherical particle thecontrast increases roughly proportionally to the eccentricityof the nanoparticle for 1 < εr < 10.

In summary, we have seen that for a nanoparticle offixed height a variation in the eccentricity of the nanoparticle(increasing the width) strongly affects the polarization forcebetween the tip and the nanoparticle. We have shown that forsmall eccentricities (1 < e < 2) and low dielectric constants(1 < εr < 10), the variation of the capacitance-gradient signalwith respect to the one obtained for a spherical nanoparticleincreases roughly proportionally to the eccentricity of thenanoparticle. For higher eccentricities, the dependence issub-linear and finally saturates to a value independent ofthe eccentricity for very large widths. An explanation forthis characteristic behavior is presented in the discussion(section 4).

3.2. Eccentricity effects on the quantification of the dielectricconstant of non-spherical nanoparticles by EFM

We can now analyze the impact of these results on theextraction of the dielectric constant of a single non-sphericalnanoparticle from polarization force measurements obtainedwith an electrostatic force microscope. As we mentionedbefore, the dielectric constant can be experimentally obtainedby detecting the maximum capacitance-gradient contrast overthe center of the nanoparticle in an EFM constant-heightdielectric image [14]. This value is then fitted to the theoreticalprediction of finite-element numerical calculations in whichthe geometry and size of the tip and of the nanoparticle areaccurately taken into account and using the dielectric constantof the nanoparticle as the only fitting parameter. We note thatwhile the effective tip geometry can be precisely determinedby a specific tip calibration procedure that proved to beextremely reliable even for sharp tips [14], determining theexact geometry of a nanoparticle, and in particular, of its widthcan be an issue due to tip dilation and scan effects [21, 22]common to all atomic force microscopy (AFM) techniques.These artifacts have no impact when the nanoparticles areknown to be spherical because the diameter can be safelydeduced from the height, accurately measured by AFM.However, when the nanoparticle is not spherical, determining

Figure 3. Extracted dielectric constants as a function of theeccentricity of four nanoparticles measured in [14], namely, a singlevirus (magenta), and single aluminum oxide (red), silicon dioxide(green) and polystyrene (blue) nanoparticles. Symbols arenumerically calculated points and solid lines are the correspondinginterpolating function. The dashed lines represent the approximationof equation (3) with b = 1 and 1.4. Experimental parameters: (i) PSnanoparticle: D = 30 nm, z = 12 nm, R = 5 nm and1C′exp,PS = 1.7 zF nm−1; (ii) SiO2 nanoparticle: D = 33 nm,

z = 9 nm, R = 7 nm and 1C′exp,SiO2= 4.9 zF nm−1; (iii) Al3O2

nanoparticle: D = 28 nm, z = 14 nm, R = 5.5 nm and1C′exp,Al2O3

= 2.67 zF nm−1; (iv) virus: D = 39 nm, z = 9.5 nm,

R = 5 nm and 1C′exp,virus = 7.15 zF nm−1. Measured eccentricities:ePS = eSiO2 = eAl2O3 = 1 and evirus = 1.3, which enabled theextraction of the dielectric constants εPS = 2.69, εSiO2 = 4.47,εAl2O3 = 8.75 and εvirus = 6.3, respectively.

the exact width and hence the eccentricity of the object canbe difficult. Thus, it is important to analyze the effect ofnanoparticle eccentricity on the extracted dielectric constantsto evaluate its impact on the values obtained.

Figure 2(c) shows the dielectric signal of 1C′exp,virus =

7.15 zF nm−1 (red horizontal line) that we measured on anon-spherical nanoparticle—a single virus shown in figure4 of [14]. The theoretical curves given in figure 2(c) arecalculated for the geometrical parameters correspondingto the calibrated probe geometry and to the topographicmeasurement taken on the virus. The intersections of themeasured value with the theoretical curves give the extracteddielectric constant. As can be seen, the value extractedstrongly depends on the eccentricity of the nanoparticle. Thedielectric constants obtained in the present case are plotted infigure 3 (magenta) as a function of the eccentricity, showingthat the dielectric constant decreases from εr ∼ 12, whenassuming a spherical shape (e = 1), down to εr ∼ 3.5, whenassuming a width of the nanoparticle double than its height(e = 2). A similar analysis can be performed for nanoparticlesof different materials. In figure 3 we also show the resultsthat we obtained for polystyrene (PS), silicon dioxide (SiO2)and aluminum oxide (Al2O3) nanoparticles, correspondingto those that we previously measured and reported in figure1 of [14]. In particular, the measurement parameters ofthe three nanoparticles are the following ones: (i) a PSnanoparticle with height D = 30 nm measured at z = 12 nmwith tip apex radius R = 5 nm, giving dielectric signal of

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Nanotechnology 24 (2013) 505713 G Gomila et al

1C′exp,PS = 1.7 zF nm−1; (ii) a SiO2 nanoparticle with D =

33 nm, z = 9 nm, R = 7 nm and 1C′exp,SiO2= 4.9 zF nm−1;

(iii) a Al3O2 nanoparticle with D = 28 nm, z = 14 nm, R =5.5 nm and 1C′exp,Al2O3

= 2.67 zF nm−1. In the three cases,the extracted dielectric constants again show a remarkabledependence on the eccentricity, which is more pronounced onincreasing the dielectric constant of the material.

The reason for the dependence of the extracted dielectricconstant on the eccentricity of the nanoparticle can beunderstood by analyzing the dependence of the capacitance-gradient signal on these two parameters, εr and e. Wehave previously shown that for spherical low-polarizablenanoparticles with 1 < εr < 10 the capacitance-gradientcontrast as a function of tip radius, nanoparticle diameter,tip–sample distance and dielectric constant follows thephenomenological expression [14]

1C′spher(z, εr) =

(c0

(R+ r0)(D− d0)

(z− D)2 + d1(D− d2)

)log(εr) (1)

where c0 ∼ 2.25 zF nm−1 and r0 ∼ 11.8 nm are constants,while d0 = d0(R) ∼ 8–10 nm, d1 = d1(R, εr) ∼ 2–3 nm, d2 =

d2(R, εr) ∼ 12–0 nm are smooth functions (see the fullequation and its interval of validity in the supplementaryinformation of [14]). We can generalize this expression tothe case of non-spherical nanoparticles of relatively smalleccentricities 1 < e < 2 by noting that in this range of valuesthe capacitance-gradient contrast increases roughly linearlywith the eccentricity (see inset of figure 2(b)). Accordingly,the phenomenological expression can be generalized tonon-spherical nanoparticles as

1C′non-spher(z, εr, e) ≈

(c0

(R+ r0)(D− d0)

(z− D)2 + d1(D− d2)

)× (1+ b(e− 1)) log(εr) (2)

where b ∼ 1–1.4 for 10 nm < z < 30 nm and 1 < εr <

10. By comparing equations (1) and (2) one can concludethat for a given measured capacitance-gradient contrast, therelationship between the extracted dielectric constant whenthe nanoparticle is assumed spherical, εr,sph, and when it isassumed non-spherical with a given eccentricity e, εr,non-sph, is

εr,non-sph = (εr,sph)1

1+b(e−1) . (3)

We verified the goodness of this remarkably simple relation infigure 3, where the dashed lines represents equation (3) withb = 1 and 1.4. Clearly, the exact computed values reasonablyapproach the curves delimited by equation (3), thus fullyvalidating it.

These results clearly show that to reliably determinethe dielectric constant of a nanoparticle, its lateral width, inaddition to its height, needs to be accurately determined. Inthe case of the measured virus, a refined topographic analysis(see [14]) provided an eccentricity evirus = 1.33, which ledto a relative dielectric constant of εr,virus ∼ 6.3. Instead, thenanoparticles were known to be approximately spherical, sothat by assuming e = 1 we obtained the dielectric constantsεPS = 2.69, εSiO2 = 4.47 and εAl2O3 = 8.75, in excellentagreement with the dielectric constants of the corresponding

Figure 4. Extracted dielectric constants as a function of theeccentricity of the single virus calculated for different tip radii,namely, (i) R = 5 nm and θ = 10◦ (black, corresponding to themeasurement of [14]), (ii) R = 25 nm and θ = 15◦ (orange) and(iii) R = 100 nm and θ = 25◦ (violet). Symbols are numericallycalculated points for a realistic EFM setup, based on the measuredparameters of the virus D = 39 nm, z = 9.5 nm, εvirus = 6.3. Solidlines are the corresponding estimation for a uniform electric fieldgiven by equation (A.4) (see appendix).

bulk materials, thus further supporting the assumption on thespherical nature of the nanoparticles.

3.3. Tip radius effects

Until here, we have discussed the eccentricity issue for asharp tip of 5 nm radius, representative of highly dopedsilicon tips normally used for AFM topographic imagingwhich can also be employed for electrostatic measurements tomeasure the dielectric constants of small scale objects, such asnanoparticles, with high spatial resolution. The term ‘sharp’ isused here in comparison with conducting coated tips typicallyused in EFM that have larger radii. In particular, metal-coatedtips have nominal radii in the range 20–30 nm that easilyincrease by a few or even tens of nanometers during dielectricexperiments owing to tip wear. Doped-diamond coated tips,on the other hand, are resistive to wear but have larger radii,typically in the 50–100 nm range. For dielectric measurementsof small objects, sharp silicon tips are then preferable tocoated tips for two reasons: (1) their sharp apex (R < 10 nm)allows detailed structural imaging of small objects and inparticular discriminating single nanoparticles from clusters ofnanoparticles; (2) they remain extremely structurally stable,which is the key to reliably extracting the dielectric constant.

In the case of using larger tips, however, similarconclusions concerning the effects of the nanoparticleeccentricity on the polarization force can be reached. We showthis in figure 4, where we compare the extracted dielectricconstant as a function of the eccentricity for the nanoparticleanalyzed in figure 3 for three different tips, namely, asharp silicon tip (R = 10 nm, θ = 10◦), a metal-coated tip(R = 25 nm and θ = 15◦) and a diamond-coated tip (R =100 nm and θ = 25◦). In all three cases, the extracteddielectric constants have a strong and similar dependenceon the eccentricity (symbols), showing a slightly stronger

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Nanotechnology 24 (2013) 505713 G Gomila et al

decay with increasing tip radius. We thus conclude that,independently of the tip used, geometric effects due tothe eccentricity of the nanoparticle can strongly impact theextracted dielectric constant and hence need to be properlytaken into account in the theoretical modeling to avoidsystematic errors. Finally, we note that, even in the case oflarge tip radius, simplified approximations based on uniformelectric field approximations (continuous lines in figure 4,see appendix) do not provide an accurate description of theeccentricity effects.

4. Discussion

We have seen that the polarization force between aconducting tip and a non-spherical nanoparticle of fixedheight and variable width (on a conductive substrate) dependssignificantly on the eccentricity of the nanoparticle. Thisdependence shows two remarkable facts, namely, (i) it isrelatively smooth (linear) for small eccentricities (1 < e < 2)and (ii) it saturates to a value independent of e for very largeeccentricities (e > 30).

This behavior cannot be quantitatively understood fromarguments based on simplified approximate models. Inthe appendix we have developed a model in which thepolarization force is calculated from the interaction of theinduced nanoparticle dipole and its image in the tip, byassuming as a first approximation that the electric fieldpolarizing the nanoparticle is uniform and independent of thenanoparticle itself (which are very crude approximations forthe present problem). In this case, the polarization force actingon the tip increases as a power law of the eccentricity withexponent between 3 and 4 for small eccentricities and doesnot show any saturation for large eccentricity values.

The reason for this different behavior can be understoodwith simple arguments. In the case of an EFM setup, thepresence of the nanoparticle in the gap below the tip severelymodifies the electric field polarizing the nanoparticle, whichin particular depends on the eccentricity of the nanoparticle.As a result the induced electric dipole gets reduced, as does itsdependence on the eccentricity, resulting in a much smootherdependence of the polarization force on this parameter.Furthermore, the polarizing electric field generated by anEFM setup is highly localized in the region below the tip apex.As a result, when the eccentricity of the nanoparticle increasesand the height of the nanoparticle remains constant (increaseof width), an increase in volume of the nanoparticle is not fullyreflected in the polarization response because the nanoparticleis mainly polarized locally and because the extremes of thenanoparticle are further from the measuring probe. Hencefor large eccentricities the electrostatic interaction saturatesbeyond a certain width of the nanoparticle because thepolarization response is no longer affected by an increase inthe lateral size of the nanoparticle (a similar phenomenon hasbeen described for the case of thin dielectric films of finitelateral size [20]). In contrast, in the case of a nanoparticle ina uniform electric field, the polarizing field and the measuringelectrode affects the whole volume of the nanoparticle andhence an increase in volume results in a proportional increase

of the induced electric dipole, which translates into a quadraticincrease of the electric force with the increased volume of thenanoparticle, thus giving rise to a non-saturating response.

Beyond the origin of the specific dependence ofthe polarization force on the nanoparticle eccentricity,the important result is that this dependence cannot beneglected as it can potentially affect the measurement ofits dielectric constant using EFM. As a consequence, weemphasize that accurate quantification of the lateral widthof the nanoparticle is essential to apply this techniqueto non-spherical nanoparticles. This is not straightforwardusing a scanning force microscope, because the width ofan object obtained from the topography image is inevitablyaffected by tip dilation effects and scan artifacts [21, 22].One can circumvent these issues by precisely characterizingthe topographic response of the microscope to sphericalnanoparticles in a large range of sizes, and use it to determinethe actual width of non-spherical objects.

Similar considerations should apply to nano-objectssuch as nanotubes and nanowires [25, 26], for which theelectric polarization is also expected to be dependent onthe cross-section eccentricity, and for which an accuratedetermination of the width from a scanning force topographicimage finds similar limitations as for the case of nanoparticles.Furthermore, although calculations have been done here fornanoparticles with a spheroid shape, it is also expected thatsimilar conclusions remain valid for nanoparticles whoseshape departs from a spheroid (e.g. nano-crystals), althoughminor quantitative differences due to detailed shape effectsmay arise. Instead, the present results are expected to be littleaffected by the presence of atomic scale local structures inthe nanoparticles, since the nanoparticle electric polarizationdepends on the whole nanoparticle volume and not only on itssurface properties.

Finally, we note that the dependence of the polarizationforce acting on the tip on the actual eccentricity of thenanoparticle could be used to quantify the real shape of ananoscale object of known dielectric constant without theneed of topographic analysis. As an example, for the PS, SiO2and Al2O3 nanoparticles that we analyzed in figure 3, weobtained εr = 2.69, 4.47 and 8.75, respectively, by assumingthem perfectly spherical. Now if we assume that the actualdielectric constants of these nanoparticles should be exactlyequal to the dielectric constants of the corresponding bulkmaterials (2.6, 3.9 and 9.4, respectively), the eccentricity ofthe nanoparticles can be estimated from figure 3 from themeasured dielectric constants. Proceeding in this way, wewould obtain ePS = 1.06, eSiO2 = 1.1 and eAl2O3 = 0.97,respectively. These values fully support the assumption of aspherical nature for these nanoparticles.

5. Conclusions

In summary, we have shown that the non-sphericity ofdielectric nanoparticles significantly affects the polarizationforce acting on the tip of an EFM setup. In the smalleccentricity and dielectric constant ranges (1 < e < 2, 1 <εr < 10) the polarization force depends roughly linearly

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Nanotechnology 24 (2013) 505713 G Gomila et al

on the eccentricity (for a fixed nanoparticle height), whilefor larger eccentricities it shows a sub-linear dependenceuntil saturation for e > 30. Comparison of these resultswith those expected for a nanoparticle in a uniform externalelectric field show a smoother dependence, reflecting thelocalized nature of EFM measurements. We have discussedthe consequences of these results for the quantification ofthe dielectric constant of nano-objects from polarization forcemeasurements and we have shown that to obtain reliableresults, accurate measurement of the lateral width of the objectis mandatory. Finally, we have suggested that if the dielectricconstant of the nanoparticle is known, the measurementof its dielectric response could be used to determine thenon-sphericity of the nanoparticle. Similar conclusions shouldapply also to nano-objects of different shapes, such asnanowires or nanotubes. The present results, thus, offer asuitable framework for quantitative interpretation of EFMmeasurements on non-spherical nano-objects.

Acknowledgments

The authors are grateful for financial support by the SpanishMEC under grant TEC2010-16844. G Gramse and M AEdwards are acknowledged for useful discussions.

Appendix. Non-spherical dielectric nanoparticle in auniform external electric field

A crude approximation for the polarization force acting onthe EFM tip due a polarized nanoparticle consists in assumingthat it is equal to the force between the nanoparticle inducedelectric dipole and its image dipole in the tip [24], and furtherassuming that the induced electric dipole is generated byan uniform electric field independent of the presence of thenanoparticle. This assumption predicts an electrostatic forcewith a quadratic dependence on the induced electric dipoleF ∼ (peff)

2, where the effective induced electric dipole, for anoblate spheroidal nanoparticle, is given by [23]

peff(e, εr) =4π3

(D

2

)3 e2(εr − 1)1+ (εr − 1)Lz(e)

E0. (A.1)

Here, the electric field is assumed to be along the minor axisdirection and Lz(e) is the polarization factor in the directionof the applied field given by

Lz(e) =12

∫∞

0

dt

(t + 1)√(t + 1)( t

e2 + 1)2. (A.2)

In equation (A.1), E0 is the strength of the applied electricfield, e = W/D is the eccentricity of the spheroid, with Dand W being its height and width, respectively, and εr is thedielectric constant of the material. For the case of a sphericalnanoparticle one has e = 1 and Lz = 1/3, which gives

peff,sph =4π3

(D

2

)3 3(εr − 1)(εr + 2)

E0. (A.3)

Figure A.1(a) shows (peff)2 normalized by the factor (p0)

2=

(4π/3(D/2)3E0)2 as a function of the eccentricity of the

Figure A.1. (a) Effective induced dielectric dipole square of ananoparticle in a uniform external electric field normalized by thefactor (p0)

2= (4π/3(D/2)3E0)

2 as a function of the eccentricity ofthe nanoparticle for a fixed height of the nanoparticle and for threedifferent dielectric constants εr = 2, 4 and 10. (b) Idem, but as afunction of the dielectric constant of the nanoparticle fornanoparticle eccentricities in the range 1 < e < 2. The horizontalline represents a hypothetical measurement of the induced effectivedipole, (peff,0/p0)

2= 4. (c) Dependence of the dielectric constant of

a non-spherical nanoparticle, εnon-sph, as a function of theeccentricity of the nanoparticle for an electric force associated withan induced electric dipole peff,0 = peff,sph(εr,sph) for εsph = 2, 4, 8,12 (equation (A.4)). The dashed line is the relationship betweenεnon-sph and εsph predicted from the numerical calculations for arealistic EFM setup (equation (3)).

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Nanotechnology 24 (2013) 505713 G Gomila et al

nanoparticle for a fixed nanoparticle height D. The quantity(peff)

2, which in this approximation is proportional to thepolarization force acting on the tip, strongly depends on theeccentricity of the nanoparticle (note the logarithmic scalein the y axis). In fact, for small eccentricities (peff)

2 scalesroughly as e3−4, thus showing a much stronger dependenceon this geometric parameter than in the case of a realisticEFM setup (tip–particle–substrate interaction), where for thesame range of eccentricities the force varies roughly linearlywith e (see inset figure 2(b)). Moreover, the response doesnot saturate when the eccentricity is increased to large values(e > 30), again in contrast with the behavior predicted for theEFM setup (see figure 2(b)). In this eccentricity range Lz(e) ∼1 and hence (peff)

2∼ e4 according to equation (A.1). In

figure A.1(b) we show the dependence of (peff)2 as a function

of the dielectric constant of the material for 1 < εr < 10 forsmall values of eccentricities 1 < e < 2. Again, a strong effectof the eccentricity on the electrostatic interaction is evidenced,much stronger than for the case of a nanoparticle in EFMsetups.

We can analyze the effect of the eccentricity on theextraction of the dielectric constant from EFM measurementsin this simple model. To this end, we assume a measuredelectrostatic force, F0 ∼ (peff,0)

2, from which the effectiveinduced electric dipole, peff,0 can be determined. Then, ifthe nanoparticle is assumed to be spherical, the extracteddielectric constant, εsph, would be obtained by solving peff,0 =

peff,sph(εsph) (equation (A.3)). If instead the nanoparticleis assumed non-spherical with a given eccentricity e, thedielectric constant would be obtained by solving peff,0 =

peff,non-sph (εnon-sph, e) (equation (A.1)). In this case therelationship between the extracted dielectric constants is

εr,non-sph =(3− 3Lz(e)− 2e2)− εr,sph(3− 3Lz(e)+ e2)

−(3Lz(e)+ 2e2)+ εr,sph(3Lz(e)− e2).

(A.4)

In the limit of a spherical nanoparticle (e = 1), it givesεnon-sph = εsph, as it should be. Note that equation (A.4)only gives physically relevant values in a small range ofeccentricities for which the ‘measured’ electric dipole canbe interpreted as produced by a spherical nanoparticle. Forlarger eccentricities, the ‘measured’ effective induced dipoleswould not be compatible with the assumption of a sphericalnanoparticle, so that εsph cannot be defined. It can be shownthat at least in the small eccentricity range, 1 < e < 2, andsmall dielectric constant range, 1 < εsph < 10, equation (A.4)holds. Figure A.1(c) (continuous lines) shows εnon-sph as afunction of the eccentricity for various values of εsph. It canbe seen that εnon-sph decreases very rapidly as the eccentricityincreases, a result qualitatively similar to the one obtained

for the EFM setup (see figure 2(c)). However, the similarityremains only qualitative since the values obtained from thenumerical calculations of a realistic EFM setup, which canbe approximated by equation (3) with b = 1–1.4 (dashedlines in figure A.1(c)), show a smoother dependence owingto the weaker dependence of the electrostatic force on theeccentricity of the nanoparticle.

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