Article pubs.acs.org/Langmuir
Direct Measurement of Field-Induced Polarization Forces between Particles in Air Ching-Wen Chiu† and William A Ducker* Department of Chemical Engineering, Virginia Tech, Blacksburg, Virginia 24061 S Supporting Information *
ABSTRACT: We have measured the effect of DC and AC electric fields (up to 15 kV/m) on the force between two 15 μm radius BaTiO3 glass spheres in air in the separation range 0−6 μm. These fields cause attractive forces that are much greater than van der Waals forces and therefore can be used to control and separate particles in many applications. The attractive force has a static and dynamic component. The static forces are about 9000 times greater than the prediction by multiple-moment methods but otherwise follow the expected trends for polarization forces. For example, the force scales with the square of the field, is constant over a range of field frequencies, and has the same force-separation profile predicted by multipole-moment methods. In contrast to the point dipole approximation, which depends inversely on the fourth power of the distance between the centers of the spheres, the measured static force approximately follows a power law, F ∝ −s−0.75, that depends on the separation, s, between the nearest points of the spheres. This power law is very similar to the prediction by multipole-moment method for separations less than 1/10th of the radius. The dynamic response force occurs at twice the frequency of the drive and has a similar amplitude to the static force. The electrical field also causes a large increase in adhesion.
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INTRODUCTION
these applications polarized particles interact directly with static charges, which is a related but distinct effect. A common example familiar to many atomic force microscope (AFM) users is the problem of very large attractive forces between the probe and the sample that can be removed by discharging the sample.19,20 Presumably this attractive force is caused by static charge on the sample that polarizes the AFM probe and leads to an attraction if the field is inhomogeneous. The motivation for this study is that there exists a very large attractive force between like dielectric particles in air that has not been extensively studied, yet in many cases it dominates the interparticle forces in air by orders of magnitude. Our objective is to characterize the force in terms of its dependence on field strength, frequency, and separation between the particles, and to compare to theoretical estimates. This is conveniently achieved by the colloidal probe method of atomic force microscopy, with the insertion of two electrodes in the AFM to produce a controlled electric field. The work described here is restricted to the case where the externally applied field (DC or AC) is parallel to the line joining the centers of two identical spheres in air, in which case the force is always attractive and parallel to the field. We also consider only a single example, two BaTiO3 glass spheres,
This paper describes measurements of the forces between two small particles that have been polarized by an external electrical field. Over the last 50 years considerable attention has been directed to the study of forces acting on particles, especially the van der Waals force, which has its origin in the thermally fluctuating dipoles in materials. The correlation in the direction and time of the dipoles is relatively weak but enough to produce an important force. Here we consider a related phenomenon: placing particles in a strong electric field induces dipoles in the particle that are correlated with each other. For this reason, the attraction between two (field-induced) polarized particles might be expected to be greater than the attraction caused by thermal fluctuations. Forces produced by external fields are also interesting because their strength can be manipulated through control of the electric field. In practice, there is a large external field-induced polarization force that is important in a number of applications such as xerography,1 electric field stabilized fluidized beds,2 electrostatic separation,3,4 electrorheological fluids,5−10 micro- and nanoelectromechanical systems (MEMS and NEMS),11 mineral purification,3,4 dielectrophoresis and/or electrophoresis for DNA separation,12,11 separation and alignment of carbon nanotubes (CNT),13−15 and directed self-assembly of colloidal particles.16−18 The force also is important in powder contamination in the semiconductor industry.1 In some of © XXXX American Chemical Society
Received: August 28, 2013 Revised: December 10, 2013
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because each particle strongly disturbs the field experienced by the other particle. Greater accuracy can be obtained by modeling the interaction with higher poles. Washizu and Jones26 (and later Cox et al.)31 have calculated the force between two polarizable particle where the force is considered as the sum of a set of linear multipoles (monopole, n = 0, dipole, n = 1, quadrupole, n = 2...) with sufficient n to obtain convergence. For example, for εm = 1000, their calculation included up to n = 200. They consider only a single dielectric response, which is suitable for our measurements in the range 1−100 kHz. For the multipolar force
which each have a diameter of approximately 30 μm. BaTiO3 has a very high static dielectric constant (typically about 200021,22) so is expected to produce large dipoles and large forces compared to low dielectric materials. Owing to the technological impact, there has been a sustained modeling effort in this area.5,23−28 In the point dipole approximation the time average force, Favg, between the two spheres is given by5 24πεmR6 2 βeff (ω)Erms 2 r4
Favg =
(1)
where εm is the permittivity of the medium, R is the radius of the sphere, r is the separation between the centers of the spheres, ω is the frequency in radians, Erms = E0/√2, where E0 is the amplitude of the external field, and βeff is the effective relative polarizability of the particles. Here we consider only a medium of air. βeff is a function of the frequency response of the materials: as the field frequency increases, the slower polarizations cannot respond and the value of βeff diminishes. Note that although the exponent of r in eq 1 is four compared to two in the Coulomb force, because the force depends on the distance between particle centers, the decay as a function of separation between the nearest points of the particle, s, (where the s = r − 2R) is weak compared to most surface forces. In a DC or a low-frequency AC field, the polarization is dominated by the migration of free charges, and therefore, is a function of the particle and medium conductivity, as described by the Maxwell−Wagner model.5,27 This migration is impeded at the interface between two unlike dielectrics and/or grain boundaries of the dielectric, causing an accumulation of free charges at interfaces.29 This is so-called “interfacial or space charge polarization” and is also referred to as “Maxwell− Wagner interfacial polarization”.30 In a higher-frequency AC field, the particle and the medium permittivities dominate the particle polarization because the free charges do not have enough time to respond to the alternation of the field. As the frequency of the field is increased, the permittivity of dielectrics decreases. This effect begins in the microwave region whereas our experiments only span 1−100 kHz, so this high frequency decrease in permittivity is not expected in our experiments. Parthasarathy and Klingenberg describe the effective permittivity as follows:5 2
2
βeff =
2
F = fz πR2εmE0 2
where fz is a distance- and εm-dependent value that is calculated analytically. Note that the force is still dependent on the square of the field strength. The inclusion of the higher order multipoles, with each higher order pole decaying with higher powers of r, causes faster decay of the force with separation. This decrease in range is not explicit in eq 6 but is contained in fz. For example, using the method of Washizu and Jones for εp = 5000, we find empirically that there is a simple power law scaling with the separation between the particles, s: s F ≈ −0.95nN(s /nm)−0.8 , for < 0.1 (7) R That is, the inclusion of the higher order multipoles means that the field-induced polarization force for highly polarizable spheres acts like a surface force, but with a lower power of s than, for example, van der Waals forces, making it a longer range force. At larger separations, the power law trends back to the r−4 dependence of eq 1. For a time varying electric field, E = E0 cos (2π f 0t), where f 0 is the frequency of the applied field and t is the time, the E2 dependence of eq 6 gives a response that is the sum of a time invariant (static) component and a component at twice the frequency of the electric field E2 =
F = fz πa 2ε0εmErms 2(1 + cos(4πf0 t ))
where βd =
εp + 2εm
, βc =
σp + 2σm
, tMW =
F=
εp − 2εm σp + 2σm
lim
βeff 2(ω) = βd 2
(4)
lim
βeff 2(ω) = βc 2
(5)
ωrMW →∞
24πε0εmR6 r4
βeff 2Erms 2(1 + cos 4πf0 t ).
(10)
In summary, from theory we expect to observe a force that has both a static and dynamic component, that scales with the square of the applied field, and that has a strong frequency dependence near DC and at microwave frequencies. At small separations and high permittivity, the force should approximate a power law. Important previous experimental work on field-induced forces includes that of Wang et al.,32,33 who measured the force between 6 mm SrTiO3 particles for separations between 0.01 and 0.8 mm using a weighing balance. Their experimental results confirm that the field-induced force is proportional to the square of electric field, as predicted by the point-dipole approximation but the measured force at small separations was
(3)
Applying two frequency limits to eq 2: ωrMW →∞
(9)
Likewise, a time varying force at twice the frequency of the drive is also expected for the dipolar force.
(2)
σp − σm
(8)
Substitution into eq 6 for the multipolar force gives
βc + βd (ωtmw )
εp − εm
1 2 E0 [1 + cos(2π (2f0 )t )] 2
= Erms 2[1 + cos(2π (2f0 )t )]
2
1 + (ωtmw )2
(6)
This describes the effect that, at low frequency, the response is dominated by the conductivity and, at high frequency, the response is dominated by the permittivity. Equation 1 considers only dipolar interactions, which is not accurate when the separation between the particles is very small B
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much greater than predicted by the point-dipole approximation. The field-induced force has also be indirectly estimated from the yield stress of a set of BaTiO3 particles subject to an electric field using rhoemetry,34 but rheometry measures the collective behavior of microparticles in fluids, not the behavior of individual particles. Mittal et al. have measured the AC electric field-induced dipole−dipole force between two 3.00 μm diameter polystyrene latex in KCl solutions with different concentration by modified optical tweezers.35 Their results show that the dipolar interactions become weaker with the increase of salt concentration, as the difference between particle and medium conductivities becomes smaller. Their experimental results also confirm that the field-induced force is proportional to the square of electric field, as predicted by the point-dipole approximation. In contrast to that work, one of the motivations of this work is to examine the applicability of eq 1 for forces in high dielectric constant materials, such as BaTiO3 and small separations where we expect the point-dipole approximation is inapplicable. Agreement between experiment and calculation would help to validate the use of eqs 6 and 7 as an approximation in particle systems. The experiments described here utilize the AFM colloidal probe method, where the force on particles is measured by attaching a BaTiO3 glass particle to an AFM cantilever. The particles are placed in an electric field and the force between them is measured as a function of separation, field strength and frequency (DC and 1−100 kHz) at very low humidity, to minimize surface conduction effects. The measured forces are compared to eqs 9 and 10.
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Figure 1. Schematic of apparatus. The sphere attached to the cantilever is referred to as the “top sphere”, whereas the sphere attached to the AFM scanner is referred to as the “bottom sphere”. The bottom plate was grounded, and the voltage of the top plate was varied. cantilever on a NSC12 chip of tipless, uncoated silicon cantilevers (MikroMasch). To control the humidity, the AFM was enclosed in a glovebag (GLOVE BAG Inflatable Glove Chamber, Glas-Col LLC) that was continuously supplied with dry air. A humidity of fc (the colloid probe is driven above resonance) the amplitude of response diminishes. Note that the responses at 3f 0 and 4f 0 are observed even in the thermal noise spectrum of a cantilever but in this case might also have contributions due to fields produced by the spheres. For example, when the colloid probe oscillates at 2f 0, it sets up a time varying field of the same frequency, which polarizes the bottom particle at 2f 0, which interacts with the f 0 dipole on the colloid probe, causing a force with a frequency of 3f 0, etc. We will now focus on the static and dynamic forces as a function of field strength, frequency, and separation. Force Is Independent of the Frequency in the Range 1−100 kHz. We have examined the dependence of the force on the external field frequency, f 0. The purpose is to examine whether there is a frequency-dependent polarization of the BaTiO3 glass particles in this range. The data in Figure 6 clearly
Figure 4. Effect of 60 V (8.6 kV/m) and 5 kHz field on the force acting on the cantilever. The horizontal axis is the distance between the cantilever and the top of the bottom plate, minus one particle diameter, so that the cantilever is in the same position relative to the electrodes in all three experiments. The force on the cantilever-only is very small, about −6 nN (force noise is about 1 nN), and independent of separation. The spring constants are 7.7 N/m for the cantilever-only and cantilever-bottom sphere experiment and 4.2 N/m for the two sphere experiment.
relatively small and independent of distance in this range (because the separation exceeds 30 μm). To reduce the influence of the cantilever, the remainder of the paper focuses on forces at separations less than about 3 μm where the cantilever force is a smaller fraction of the total force. In our quantitative comparison to theory (Figure 8) we account for the effect of the cantilever. AC Fields Produce Both a Static and a Dynamic Force. By experiment, we find that the AC field-induced force has both a time invariant component, which we call the static response, and an oscillatory component, which we call the dynamic response. Figure 5 shows a discrete Fourier transform of the deflection response to a 8.6 kV/m, 10 kHz electric field when the particles are separated by1.85−2.0 μm. The largest responses occur at DC and at twice the field frequency, as predicted by the simple polarization model described in eq 9 or 10, so the measured forces are consistent with an induced
Figure 6. Static component of the AC field-induced force as a function of field frequency and strength for a constant particle separation of 2.5 μm and a constant plate separation of 7 mm.
show that the static component of the AC field induced force is not a function of frequency in this range (see also the Supporting Information). From eqs 9 and 10, it is clear that the 2f 0 dynamic force has the same origin as the static response, so the force should also have the same frequency response, i.e., be frequency independent. However, our sensor has its own mechanical response, with a resonance at 53 kHz, so we expect to see a resonance when the applied field is at half this frequency, and indeed this is observed (see the Supporting Information). We observe no resonance other than the mechanical resonance of the cantilever, so we conclude that there is no resonance of dipolar response in the frequency range 1−100 kHz. The mechanical resonance does show the interesting property that the quality factor (Q) of the cantilever−sphere is much lower for the field-induced force than for the thermal force (5 vs 300), which indicates that the polarization force
Figure 5. Discrete Fourier transform of the cantilever deflection for an applied voltage of 60 V (8.6 kV/m) when the spheres are separated by 1.85−2.0 μm. The thermal noise produces a peak at the resonant frequency of the colloid probe, fc =53 kHz. Note that the amplitude is plotted on a log scale. E
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field strength (data not shown) as expected for a Coulomb force rather than a polarization force. Force Is Much Greater in Magnitude than the Predicted Force. Figure 8 compares the static component
provides strong damping compared to the viscous damping on the colloidal probe. Force Scales with the Square of the Field Amplitude. Equation 9 predicts that both the static and dynamic response of the field-induced polarization force should scale with the square of the field strength. Figure 7a is a plot of the amplitude
Figure 8. Net force at 8.6 kV/m and 5 kHz compared to the functional form of the static components of eqs 10 (point-dipole approximation) and 9 (multipole-moment method). The net force is defined as the measured force acting on the colloid probe minus the force without the sphere on the end of the cantilever. Each of the theoretical curves has been multiplied by 9000 to facilitate comparison to the experimental results. The dynamic component of the measured force varies too rapidly to be seen on this separation scale.
of the measured force to two theoretical predictions of the force, from eqs 10 (point-dipole approximation) and 9 (multipole-moment method), each multiplied by 9000. This data is for a 5 kHz signal, but recall that the force is independent of frequency in the range 1−100 kHz, so the comparison applies to this range of frequencies. The data in Figure 8 is the net force between the spheres, that is the difference between the measured force between the colloid probe and the bottom sphere and the force between the cantilever and the bottom sphere. This is an attempt to account for the force acting between the bottom sphere and the cantilever, which should not be included in the sphere−sphere interaction. The magnitude of the correction can be seen in Figure 4. This correction is an approximation: the local field experienced by all molecules in the field will be altered by the addition of another object in the field. The multipole-moment calculation shown in Figure 8 is for ε = 5000 and 500 poles. In the range 200−500 poles, the calculated force changed by less than 1% at the separations shown. In the range ε = 1000−5000, the force changed by less than 6% for distances greater than 30 nm, so the calculation is for the maximum dielectric constant. The important point is that the measured force in an AC field is much greater in magnitude than the predicted force; the measured force is about 9000 times the calculated force. Static Component of Force Has the Same Functional Form at the Multipole-Moment Calculation. Although the magnitude of the static force is much greater than expected, the functional form is very well predicted by the multipole-moment calculation of Washizu and Jones applied for an AC field (eq 9)
Figure 7. Effect of electric field strength on the force at a fixed separation. (a) The response at 2f 0 at fixed separation, 1.85−2.0 μm, and fixed field frequency. The lines show least-squares fits. (b) The static response at a variety of fixed separations and a single field strength of 100 kHz.
of response at a fixed separation of about 1.85−2.0 μm as a function of the square of the field strength. Figure 7a shows the response at 2f 0 for two different drive frequencies, one below the colloid probe mechanical resonance (f 0 = 1 kHz) and one above (f 0 = 100 kHz). At both frequencies, the response is proportional to the square of the field strength, again consistent with eqs 9 and 10. Figure 7b shows how the static response to a 100 kHz field varies with the square of the applied field strength. In common with the dynamic response, the static response is also proportional to the square of the field. One final note on the effect of field strength is that the amplitude of the f 0 response appears to be approximately proportional to the F
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for a particle with dielectric constant in the range ε = 1000− 5000, which is applicable for BaTiO3. This is clearly shown in Figure 8 on both a linear and a log scale for the separation range 0−5700 nm. The best fit of a power law to the measured data is F = −6600nN(s/nm)−0.75
(11)
which has a similar exponent to the best fit to the multipolemoment method (−0.81), which is also the same as for calculations for a conducting sphere, described by Davis.40 There is only a slight difference in fit quality between an exponent of −0.81 and −0.75. Note that the force acts like a power law depending on the separation between the nearest points on the sphere and not the centers of the sphere: it is acting like a surface force. We rationalize this effect as being due to the fact that the field that each part of each particle experiences is increasingly distorted by the proximity of the other particle. Force As a Measure of Particle Dielectric Constant. The huge discrepancy between the magnitudes of the measured and theoretical multipole force currently prevents the force from being used as a measure of particle dielectric constant. However, a path toward a method of measuring the dielectric constant is suggested by the fact that Washizu and Jones found that the dielectric constant had a large effect on the functional form of the force. At low dielectric constant (ε = 2), the multipole force is approximately equal to the point-dipole force and they steadily diverge up to a saturation at about ε = 1000. If future measurements of the forces between particles with a range of dielectric constants show that the functional form is dependent on the dielectric constant, then the functional form could be used as an empirical measure of the dielectric constant. Of course, this putative method would be much more satisfactory if the gulf in magnitudes between theory and experiment could also be explained. Dynamic Component of the Force Also Depends on the Separation. Equation 9 predicts that the 2f 0 component of the force has the same functional form as the static force. To test this prediction, we took our deflection−time data where the separation was varied and made distance bins that had enough time to obtain a good Fourier transform. The time bins corresponded to separation bins, so the values are averages over a range of separations. The resulting data (Figure 9) shows that the amplitude did increase at smaller distances. The best fit to a power law has an exponent of −0.59, but because of the scatter in the data, the fit for an exponent of −0.59 is not much better than for the −0.75 that was found for the static component. Thus the static and dynamic have qualitatively similar amplitudes and distance dependence, as expected from eq 9. Force of Adhesion between Particles Also Scales with E02. The adhesion force, or force required to separate two particles from contact, is important because it determines the force or work required to separate systems of particles that are coagulated, and the yield stress of a particle slurry. It can in principle be calculated from knowledge of the force−separation law, but in practice this calculation is difficult because surface roughness makes it difficult to estimate the “separation” in contact. Whereas the adhesion is usually considered to be a given for a particular material, coating, and medium, fieldinduced adhesion offers the opportunity of controlling the adhesion. Measurements of the force required to separate two BaTiO3 glass particles in air are shown in Figure 10. It is clear that the adhesion depends on the strength of the field and,
Figure 9. Measured dynamic component of the AC field-induced force as a function of separation for two BaTiO3 glass spheres. The lines represent the best fits to power laws. For the 2f 0 compenent, the best fit has an exponent of −0.59.
Figure 10. Relationship between the electric field-induced adhesion force and the square of the applied field. The plot shows the difference in adhesion with and without the electric field.
more precisely, scales approximately with the square of the field. We also find that the adhesion is independent of the frequency of the applied field, so the adhesion has the same characteristics as the force-at-a-distance and therefore is dominated by the polarization force. It is important to note that the measured adhesion was also a function of the number of times that the particles were contacted: the adhesion gradually increased with each contact, as described previously.19,20,41 This effect was ascribed to the transfer of charge in contact, which produces an electrostatic force between the particles. For the data shown in Figure 10, we measured the adhesion force in the presence and the absence of the field, and subtracted the adhesion in the absence of the field, to remove this contact effect from the data. G
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CONCLUSIONS Our experiments show that an electric field produces an attractive force between two BaTiO3 glass particles in air that is much greater than a van der Waals force and approximately follows a power law that depends on the separation between the particles. The adhesion force is also much greater in the presence of the field. The force is induced by either a DC field or an AC field. The DC field-induced force is greater in magnitude, because it includes contributions from the conductivity of the particles, and increases with time, which we attribute to migration of charge. The AC field-induced force has three principal components: static, f 0, and 2f 0. The static and 2f 0 components are expected for a polarization force, and the f 0 component we attribute to charging. The magnitude of the static force is about 9000 times greater than predicted by the multipole-moment model. The dramatic mismatch between theory and experiment is a necessity for many of the applications of field-induced forces described in the introduction, because the theoretically predicted forces are very weak. This suggests an opportunity for future modeling. With the glaring exception of the magnitude, the static force agrees well the expectations of a polarization force. The measured force (a) has a very similar separation dependence to the multipolemoment calculation of Washizu and Jones for particles with dielectric constant over 1000, (b) is proportional to the square of the field, and (c) is independent of the field frequency in the range 1−100 kHz. The field also produces a force at twice the field frequency, which has similar properties to the static response.
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ASSOCIATED CONTENT
S Supporting Information *
Additional material as discussed in the text. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Present Address †
Taiwan Semiconductor Manufacturing Company, Limited (TSMC). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was funded by the American Chemical Society Petroleum Research Fund, under Award No. PRF 49114-ND5. The authors thank Professor Masao Washizu for providing the FORTRAN code that was used in the multipole moment calculations.
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(30) Hughes, M. P. Nanoelectromechanics in Engineering and Biology; CRC Press: Boca Raton, FL, 2003. (31) Cox, B.; Thamwattana, N.; Hill, J. Electric field-induced force between two identical uncharged spheres. Appl. Phys. Lett. 2006, 88, 152903. (32) Wang, Z. Y.; Peng, Z.; Lu, K. Q.; Wen, W. J. Experimental investigation for field-induced interaction force of two spheres. Appl. Phys. Lett. 2003, 82, 1796−1798. (33) Wang, Z. Y.; Shen, R.; Niu, X. J.; Lu, K. Q.; Wen, W. J. Frequency dependence of a field-induced force between two high dielectric spheres in various fluid media. J. Appl. Phys. 2003, 94, 7832− 7834. (34) Rankin, P. J.; Klingenberg, D. J. The electrorheology of barium titanate suspensions. J. Rheol. 1998, 42, 639−656. (35) Mittal, M.; Lele, P. P.; Kaler, E. W.; Furst, E. M. Polarization and interactions of colloidal particles in ac electric fields J. Chem. Phys. 2008, 129. (36) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Measurement of Forces in Liquids Using a Force Microscope. Langmuir 1992, 8, 1831− 1836. (37) Hutter, J. L.; Bechhoefer, J. Calibration of Atomic-Force Microscope Tips. Rev. Sci. Instrum. 1993, 64, 1868−1873. (38) Meyer, G.; Amer, N. M. Novel Optical Approach to Atomic Force Microscopy. Appl. Phys. Lett. 1988, 53, 1045−1047. (39) CRC Handbook of Chemistry and Physics: CRC Press: Boca Raton, FL, 2013. (40) Davis, M. H. Two Charged Spherical Conductors in a Uniform Electric Field: Forces and Field Strength, United States Air Force Project Rand RM-3860-PR 1964. http://www.rand.org/content/dam/rand/ pubs/research_memoranda/2008/RM3860.pdf. (41) Bunker, M. J.; Davies, M. C.; James, M. B.; Roberts, C. J. Direct observation of single particle electrostatic charging by atomic force microscopy. Pharm. Res. 2007, 24, 1165−1169.
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dx.doi.org/10.1021/la403318g | Langmuir XXXX, XXX, XXX−XXX
Direct Measurement of Field-Induced Polarization Forces between Particles in Air Ching-Wen Chiu† and William A Ducker* Department of Chemical Engineering, Virginia Tech, Blacksburg, Virginia 24061 S Supporting Information *
ABSTRACT: We have measured the effect of DC and AC electric fields (up to 15 kV/m) on the force between two 15 μm radius BaTiO3 glass spheres in air in the separation range 0−6 μm. These fields cause attractive forces that are much greater than van der Waals forces and therefore can be used to control and separate particles in many applications. The attractive force has a static and dynamic component. The static forces are about 9000 times greater than the prediction by multiple-moment methods but otherwise follow the expected trends for polarization forces. For example, the force scales with the square of the field, is constant over a range of field frequencies, and has the same force-separation profile predicted by multipole-moment methods. In contrast to the point dipole approximation, which depends inversely on the fourth power of the distance between the centers of the spheres, the measured static force approximately follows a power law, F ∝ −s−0.75, that depends on the separation, s, between the nearest points of the spheres. This power law is very similar to the prediction by multipole-moment method for separations less than 1/10th of the radius. The dynamic response force occurs at twice the frequency of the drive and has a similar amplitude to the static force. The electrical field also causes a large increase in adhesion.
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INTRODUCTION
these applications polarized particles interact directly with static charges, which is a related but distinct effect. A common example familiar to many atomic force microscope (AFM) users is the problem of very large attractive forces between the probe and the sample that can be removed by discharging the sample.19,20 Presumably this attractive force is caused by static charge on the sample that polarizes the AFM probe and leads to an attraction if the field is inhomogeneous. The motivation for this study is that there exists a very large attractive force between like dielectric particles in air that has not been extensively studied, yet in many cases it dominates the interparticle forces in air by orders of magnitude. Our objective is to characterize the force in terms of its dependence on field strength, frequency, and separation between the particles, and to compare to theoretical estimates. This is conveniently achieved by the colloidal probe method of atomic force microscopy, with the insertion of two electrodes in the AFM to produce a controlled electric field. The work described here is restricted to the case where the externally applied field (DC or AC) is parallel to the line joining the centers of two identical spheres in air, in which case the force is always attractive and parallel to the field. We also consider only a single example, two BaTiO3 glass spheres,
This paper describes measurements of the forces between two small particles that have been polarized by an external electrical field. Over the last 50 years considerable attention has been directed to the study of forces acting on particles, especially the van der Waals force, which has its origin in the thermally fluctuating dipoles in materials. The correlation in the direction and time of the dipoles is relatively weak but enough to produce an important force. Here we consider a related phenomenon: placing particles in a strong electric field induces dipoles in the particle that are correlated with each other. For this reason, the attraction between two (field-induced) polarized particles might be expected to be greater than the attraction caused by thermal fluctuations. Forces produced by external fields are also interesting because their strength can be manipulated through control of the electric field. In practice, there is a large external field-induced polarization force that is important in a number of applications such as xerography,1 electric field stabilized fluidized beds,2 electrostatic separation,3,4 electrorheological fluids,5−10 micro- and nanoelectromechanical systems (MEMS and NEMS),11 mineral purification,3,4 dielectrophoresis and/or electrophoresis for DNA separation,12,11 separation and alignment of carbon nanotubes (CNT),13−15 and directed self-assembly of colloidal particles.16−18 The force also is important in powder contamination in the semiconductor industry.1 In some of © XXXX American Chemical Society
Received: August 28, 2013 Revised: December 10, 2013
A
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because each particle strongly disturbs the field experienced by the other particle. Greater accuracy can be obtained by modeling the interaction with higher poles. Washizu and Jones26 (and later Cox et al.)31 have calculated the force between two polarizable particle where the force is considered as the sum of a set of linear multipoles (monopole, n = 0, dipole, n = 1, quadrupole, n = 2...) with sufficient n to obtain convergence. For example, for εm = 1000, their calculation included up to n = 200. They consider only a single dielectric response, which is suitable for our measurements in the range 1−100 kHz. For the multipolar force
which each have a diameter of approximately 30 μm. BaTiO3 has a very high static dielectric constant (typically about 200021,22) so is expected to produce large dipoles and large forces compared to low dielectric materials. Owing to the technological impact, there has been a sustained modeling effort in this area.5,23−28 In the point dipole approximation the time average force, Favg, between the two spheres is given by5 24πεmR6 2 βeff (ω)Erms 2 r4
Favg =
(1)
where εm is the permittivity of the medium, R is the radius of the sphere, r is the separation between the centers of the spheres, ω is the frequency in radians, Erms = E0/√2, where E0 is the amplitude of the external field, and βeff is the effective relative polarizability of the particles. Here we consider only a medium of air. βeff is a function of the frequency response of the materials: as the field frequency increases, the slower polarizations cannot respond and the value of βeff diminishes. Note that although the exponent of r in eq 1 is four compared to two in the Coulomb force, because the force depends on the distance between particle centers, the decay as a function of separation between the nearest points of the particle, s, (where the s = r − 2R) is weak compared to most surface forces. In a DC or a low-frequency AC field, the polarization is dominated by the migration of free charges, and therefore, is a function of the particle and medium conductivity, as described by the Maxwell−Wagner model.5,27 This migration is impeded at the interface between two unlike dielectrics and/or grain boundaries of the dielectric, causing an accumulation of free charges at interfaces.29 This is so-called “interfacial or space charge polarization” and is also referred to as “Maxwell− Wagner interfacial polarization”.30 In a higher-frequency AC field, the particle and the medium permittivities dominate the particle polarization because the free charges do not have enough time to respond to the alternation of the field. As the frequency of the field is increased, the permittivity of dielectrics decreases. This effect begins in the microwave region whereas our experiments only span 1−100 kHz, so this high frequency decrease in permittivity is not expected in our experiments. Parthasarathy and Klingenberg describe the effective permittivity as follows:5 2
2
βeff =
2
F = fz πR2εmE0 2
where fz is a distance- and εm-dependent value that is calculated analytically. Note that the force is still dependent on the square of the field strength. The inclusion of the higher order multipoles, with each higher order pole decaying with higher powers of r, causes faster decay of the force with separation. This decrease in range is not explicit in eq 6 but is contained in fz. For example, using the method of Washizu and Jones for εp = 5000, we find empirically that there is a simple power law scaling with the separation between the particles, s: s F ≈ −0.95nN(s /nm)−0.8 , for < 0.1 (7) R That is, the inclusion of the higher order multipoles means that the field-induced polarization force for highly polarizable spheres acts like a surface force, but with a lower power of s than, for example, van der Waals forces, making it a longer range force. At larger separations, the power law trends back to the r−4 dependence of eq 1. For a time varying electric field, E = E0 cos (2π f 0t), where f 0 is the frequency of the applied field and t is the time, the E2 dependence of eq 6 gives a response that is the sum of a time invariant (static) component and a component at twice the frequency of the electric field E2 =
F = fz πa 2ε0εmErms 2(1 + cos(4πf0 t ))
where βd =
εp + 2εm
, βc =
σp + 2σm
, tMW =
F=
εp − 2εm σp + 2σm
lim
βeff 2(ω) = βd 2
(4)
lim
βeff 2(ω) = βc 2
(5)
ωrMW →∞
24πε0εmR6 r4
βeff 2Erms 2(1 + cos 4πf0 t ).
(10)
In summary, from theory we expect to observe a force that has both a static and dynamic component, that scales with the square of the applied field, and that has a strong frequency dependence near DC and at microwave frequencies. At small separations and high permittivity, the force should approximate a power law. Important previous experimental work on field-induced forces includes that of Wang et al.,32,33 who measured the force between 6 mm SrTiO3 particles for separations between 0.01 and 0.8 mm using a weighing balance. Their experimental results confirm that the field-induced force is proportional to the square of electric field, as predicted by the point-dipole approximation but the measured force at small separations was
(3)
Applying two frequency limits to eq 2: ωrMW →∞
(9)
Likewise, a time varying force at twice the frequency of the drive is also expected for the dipolar force.
(2)
σp − σm
(8)
Substitution into eq 6 for the multipolar force gives
βc + βd (ωtmw )
εp − εm
1 2 E0 [1 + cos(2π (2f0 )t )] 2
= Erms 2[1 + cos(2π (2f0 )t )]
2
1 + (ωtmw )2
(6)
This describes the effect that, at low frequency, the response is dominated by the conductivity and, at high frequency, the response is dominated by the permittivity. Equation 1 considers only dipolar interactions, which is not accurate when the separation between the particles is very small B
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much greater than predicted by the point-dipole approximation. The field-induced force has also be indirectly estimated from the yield stress of a set of BaTiO3 particles subject to an electric field using rhoemetry,34 but rheometry measures the collective behavior of microparticles in fluids, not the behavior of individual particles. Mittal et al. have measured the AC electric field-induced dipole−dipole force between two 3.00 μm diameter polystyrene latex in KCl solutions with different concentration by modified optical tweezers.35 Their results show that the dipolar interactions become weaker with the increase of salt concentration, as the difference between particle and medium conductivities becomes smaller. Their experimental results also confirm that the field-induced force is proportional to the square of electric field, as predicted by the point-dipole approximation. In contrast to that work, one of the motivations of this work is to examine the applicability of eq 1 for forces in high dielectric constant materials, such as BaTiO3 and small separations where we expect the point-dipole approximation is inapplicable. Agreement between experiment and calculation would help to validate the use of eqs 6 and 7 as an approximation in particle systems. The experiments described here utilize the AFM colloidal probe method, where the force on particles is measured by attaching a BaTiO3 glass particle to an AFM cantilever. The particles are placed in an electric field and the force between them is measured as a function of separation, field strength and frequency (DC and 1−100 kHz) at very low humidity, to minimize surface conduction effects. The measured forces are compared to eqs 9 and 10.
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Figure 1. Schematic of apparatus. The sphere attached to the cantilever is referred to as the “top sphere”, whereas the sphere attached to the AFM scanner is referred to as the “bottom sphere”. The bottom plate was grounded, and the voltage of the top plate was varied. cantilever on a NSC12 chip of tipless, uncoated silicon cantilevers (MikroMasch). To control the humidity, the AFM was enclosed in a glovebag (GLOVE BAG Inflatable Glove Chamber, Glas-Col LLC) that was continuously supplied with dry air. A humidity of fc (the colloid probe is driven above resonance) the amplitude of response diminishes. Note that the responses at 3f 0 and 4f 0 are observed even in the thermal noise spectrum of a cantilever but in this case might also have contributions due to fields produced by the spheres. For example, when the colloid probe oscillates at 2f 0, it sets up a time varying field of the same frequency, which polarizes the bottom particle at 2f 0, which interacts with the f 0 dipole on the colloid probe, causing a force with a frequency of 3f 0, etc. We will now focus on the static and dynamic forces as a function of field strength, frequency, and separation. Force Is Independent of the Frequency in the Range 1−100 kHz. We have examined the dependence of the force on the external field frequency, f 0. The purpose is to examine whether there is a frequency-dependent polarization of the BaTiO3 glass particles in this range. The data in Figure 6 clearly
Figure 4. Effect of 60 V (8.6 kV/m) and 5 kHz field on the force acting on the cantilever. The horizontal axis is the distance between the cantilever and the top of the bottom plate, minus one particle diameter, so that the cantilever is in the same position relative to the electrodes in all three experiments. The force on the cantilever-only is very small, about −6 nN (force noise is about 1 nN), and independent of separation. The spring constants are 7.7 N/m for the cantilever-only and cantilever-bottom sphere experiment and 4.2 N/m for the two sphere experiment.
relatively small and independent of distance in this range (because the separation exceeds 30 μm). To reduce the influence of the cantilever, the remainder of the paper focuses on forces at separations less than about 3 μm where the cantilever force is a smaller fraction of the total force. In our quantitative comparison to theory (Figure 8) we account for the effect of the cantilever. AC Fields Produce Both a Static and a Dynamic Force. By experiment, we find that the AC field-induced force has both a time invariant component, which we call the static response, and an oscillatory component, which we call the dynamic response. Figure 5 shows a discrete Fourier transform of the deflection response to a 8.6 kV/m, 10 kHz electric field when the particles are separated by1.85−2.0 μm. The largest responses occur at DC and at twice the field frequency, as predicted by the simple polarization model described in eq 9 or 10, so the measured forces are consistent with an induced
Figure 6. Static component of the AC field-induced force as a function of field frequency and strength for a constant particle separation of 2.5 μm and a constant plate separation of 7 mm.
show that the static component of the AC field induced force is not a function of frequency in this range (see also the Supporting Information). From eqs 9 and 10, it is clear that the 2f 0 dynamic force has the same origin as the static response, so the force should also have the same frequency response, i.e., be frequency independent. However, our sensor has its own mechanical response, with a resonance at 53 kHz, so we expect to see a resonance when the applied field is at half this frequency, and indeed this is observed (see the Supporting Information). We observe no resonance other than the mechanical resonance of the cantilever, so we conclude that there is no resonance of dipolar response in the frequency range 1−100 kHz. The mechanical resonance does show the interesting property that the quality factor (Q) of the cantilever−sphere is much lower for the field-induced force than for the thermal force (5 vs 300), which indicates that the polarization force
Figure 5. Discrete Fourier transform of the cantilever deflection for an applied voltage of 60 V (8.6 kV/m) when the spheres are separated by 1.85−2.0 μm. The thermal noise produces a peak at the resonant frequency of the colloid probe, fc =53 kHz. Note that the amplitude is plotted on a log scale. E
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field strength (data not shown) as expected for a Coulomb force rather than a polarization force. Force Is Much Greater in Magnitude than the Predicted Force. Figure 8 compares the static component
provides strong damping compared to the viscous damping on the colloidal probe. Force Scales with the Square of the Field Amplitude. Equation 9 predicts that both the static and dynamic response of the field-induced polarization force should scale with the square of the field strength. Figure 7a is a plot of the amplitude
Figure 8. Net force at 8.6 kV/m and 5 kHz compared to the functional form of the static components of eqs 10 (point-dipole approximation) and 9 (multipole-moment method). The net force is defined as the measured force acting on the colloid probe minus the force without the sphere on the end of the cantilever. Each of the theoretical curves has been multiplied by 9000 to facilitate comparison to the experimental results. The dynamic component of the measured force varies too rapidly to be seen on this separation scale.
of the measured force to two theoretical predictions of the force, from eqs 10 (point-dipole approximation) and 9 (multipole-moment method), each multiplied by 9000. This data is for a 5 kHz signal, but recall that the force is independent of frequency in the range 1−100 kHz, so the comparison applies to this range of frequencies. The data in Figure 8 is the net force between the spheres, that is the difference between the measured force between the colloid probe and the bottom sphere and the force between the cantilever and the bottom sphere. This is an attempt to account for the force acting between the bottom sphere and the cantilever, which should not be included in the sphere−sphere interaction. The magnitude of the correction can be seen in Figure 4. This correction is an approximation: the local field experienced by all molecules in the field will be altered by the addition of another object in the field. The multipole-moment calculation shown in Figure 8 is for ε = 5000 and 500 poles. In the range 200−500 poles, the calculated force changed by less than 1% at the separations shown. In the range ε = 1000−5000, the force changed by less than 6% for distances greater than 30 nm, so the calculation is for the maximum dielectric constant. The important point is that the measured force in an AC field is much greater in magnitude than the predicted force; the measured force is about 9000 times the calculated force. Static Component of Force Has the Same Functional Form at the Multipole-Moment Calculation. Although the magnitude of the static force is much greater than expected, the functional form is very well predicted by the multipole-moment calculation of Washizu and Jones applied for an AC field (eq 9)
Figure 7. Effect of electric field strength on the force at a fixed separation. (a) The response at 2f 0 at fixed separation, 1.85−2.0 μm, and fixed field frequency. The lines show least-squares fits. (b) The static response at a variety of fixed separations and a single field strength of 100 kHz.
of response at a fixed separation of about 1.85−2.0 μm as a function of the square of the field strength. Figure 7a shows the response at 2f 0 for two different drive frequencies, one below the colloid probe mechanical resonance (f 0 = 1 kHz) and one above (f 0 = 100 kHz). At both frequencies, the response is proportional to the square of the field strength, again consistent with eqs 9 and 10. Figure 7b shows how the static response to a 100 kHz field varies with the square of the applied field strength. In common with the dynamic response, the static response is also proportional to the square of the field. One final note on the effect of field strength is that the amplitude of the f 0 response appears to be approximately proportional to the F
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for a particle with dielectric constant in the range ε = 1000− 5000, which is applicable for BaTiO3. This is clearly shown in Figure 8 on both a linear and a log scale for the separation range 0−5700 nm. The best fit of a power law to the measured data is F = −6600nN(s/nm)−0.75
(11)
which has a similar exponent to the best fit to the multipolemoment method (−0.81), which is also the same as for calculations for a conducting sphere, described by Davis.40 There is only a slight difference in fit quality between an exponent of −0.81 and −0.75. Note that the force acts like a power law depending on the separation between the nearest points on the sphere and not the centers of the sphere: it is acting like a surface force. We rationalize this effect as being due to the fact that the field that each part of each particle experiences is increasingly distorted by the proximity of the other particle. Force As a Measure of Particle Dielectric Constant. The huge discrepancy between the magnitudes of the measured and theoretical multipole force currently prevents the force from being used as a measure of particle dielectric constant. However, a path toward a method of measuring the dielectric constant is suggested by the fact that Washizu and Jones found that the dielectric constant had a large effect on the functional form of the force. At low dielectric constant (ε = 2), the multipole force is approximately equal to the point-dipole force and they steadily diverge up to a saturation at about ε = 1000. If future measurements of the forces between particles with a range of dielectric constants show that the functional form is dependent on the dielectric constant, then the functional form could be used as an empirical measure of the dielectric constant. Of course, this putative method would be much more satisfactory if the gulf in magnitudes between theory and experiment could also be explained. Dynamic Component of the Force Also Depends on the Separation. Equation 9 predicts that the 2f 0 component of the force has the same functional form as the static force. To test this prediction, we took our deflection−time data where the separation was varied and made distance bins that had enough time to obtain a good Fourier transform. The time bins corresponded to separation bins, so the values are averages over a range of separations. The resulting data (Figure 9) shows that the amplitude did increase at smaller distances. The best fit to a power law has an exponent of −0.59, but because of the scatter in the data, the fit for an exponent of −0.59 is not much better than for the −0.75 that was found for the static component. Thus the static and dynamic have qualitatively similar amplitudes and distance dependence, as expected from eq 9. Force of Adhesion between Particles Also Scales with E02. The adhesion force, or force required to separate two particles from contact, is important because it determines the force or work required to separate systems of particles that are coagulated, and the yield stress of a particle slurry. It can in principle be calculated from knowledge of the force−separation law, but in practice this calculation is difficult because surface roughness makes it difficult to estimate the “separation” in contact. Whereas the adhesion is usually considered to be a given for a particular material, coating, and medium, fieldinduced adhesion offers the opportunity of controlling the adhesion. Measurements of the force required to separate two BaTiO3 glass particles in air are shown in Figure 10. It is clear that the adhesion depends on the strength of the field and,
Figure 9. Measured dynamic component of the AC field-induced force as a function of separation for two BaTiO3 glass spheres. The lines represent the best fits to power laws. For the 2f 0 compenent, the best fit has an exponent of −0.59.
Figure 10. Relationship between the electric field-induced adhesion force and the square of the applied field. The plot shows the difference in adhesion with and without the electric field.
more precisely, scales approximately with the square of the field. We also find that the adhesion is independent of the frequency of the applied field, so the adhesion has the same characteristics as the force-at-a-distance and therefore is dominated by the polarization force. It is important to note that the measured adhesion was also a function of the number of times that the particles were contacted: the adhesion gradually increased with each contact, as described previously.19,20,41 This effect was ascribed to the transfer of charge in contact, which produces an electrostatic force between the particles. For the data shown in Figure 10, we measured the adhesion force in the presence and the absence of the field, and subtracted the adhesion in the absence of the field, to remove this contact effect from the data. G
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(6) Halsey, T. C. Electrorheological Fluids. Science 1992, 258, 761− 766. (7) Hao, T. Electrorheological suspensions. Adv. Colloid Interface Sci. 2002, 97, 1−35. (8) Klingenberg, D. J.; Zukoski, C. F. Studies On The Steady-Shear Behavior Of Electrorheological Suspensions. Langmuir 1990, 6, 15− 24. (9) Wen, W. J.; Huang, X. X.; Sheng, P. Electrorheological fluids: structures and mechanisms. Soft Matter 2008, 4, 200−210. (10) Gast, A. P.; Zukoski, C. F. Electrorheological Fluids As Colloidal Suspensions. Adv. Colloid Interface Sci. 1989, 30, 153−202. (11) Pethig, R. Review Article-Dielectrophoresis: Status of the theory, technology, and applications Biomicrofluidics 2010, 4. (12) Morgan, H.; Green, N. G. AC Electrokinetics: colloids and nanoparticles; Research Studies Press: Philadelphia, PA, 2003. (13) Krupke, R.; Hennrich, F.; von Lohneysen, H.; Kappes, M. M. Separation of metallic from semiconducting single-walled carbon nanotubes. Science 2003, 301, 344−347. (14) Zhang, Y. G.; Chang, A. L.; Cao, J.; Wang, Q.; Kim, W.; Li, Y. M.; Morris, N.; Yenilmez, E.; Kong, J.; Dai, H. J. Electric-field-directed growth of aligned single-walled carbon nanotubes. Appl. Phys. Lett. 2001, 79, 3155−3157. (15) Chen, X. Q.; Saito, T.; Yamada, H.; Matsushige, K. Aligning single-wall carbon nanotubes with an alternating-current electric field. Appl. Phys. Lett. 2001, 78, 3714−3716. (16) Zhang, K. Q.; Liu, X. Y. In situ observation of colloidal monolayer nucleation driven by an alternating electric field. Nature 2004, 429, 739−743. (17) Ryan, K. M.; Mastroianni, A.; Stancil, K. A.; Liu, H. T.; Alivisatos, A. P. Electric-field-assisted assembly of perpendicularly oriented nanorod superlattices. Nano Lett. 2006, 6, 1479−1482. (18) Velegol, D.; Anderson, J. L.; Garoff, S. Determining the forces between polystyrene latex spheres using differential electrophoresis. Langmuir 1996, 12, 4103−4110. (19) Gady, B.; Schleef, D.; Reifenberger, R.; Rimai, D.; DeMejo, L. P. Identification of electrostatic and van der Waals interaction forces between a micrometer-size sphere and a flat substrate. Phys. Rev. B 1996, 53, 8065−8070. (20) Gady, B.; Reifenberger, R.; Rimai, D. S.; DeMejo, L. P. Contact electrification and the interaction force between a micrometer-size polystyrene sphere and a graphite surface. Langmuir 1997, 13, 2533− 2537. (21) Petrovsky, V.; Petrovsky, T.; Kamlapurkar, S.; Dogan, F. Physical modeling and electrodynamic characterization of dielectric slurries by impedance sectroscopy (Part II). J. Am. Ceram. Soc. 2008, 91, 1817−1819. (22) Wada, S.; Hoshina, T.; Yasuno, H.; Nam, S. M.; Kakemoto, H.; Tsurumi, T.; Yashima, M. Size dependence of dielectric properties for nm-sized barium titanate crystallites and its origin. J. Korean Phys. Soc. 2005, 46, 303−307. (23) Chen, Y.; Sprecher, A. F.; Conrad, H. Electrostatic ParticleParticle Interactions In Electrorheological Fluids. J. Appl. Phys. 1991, 70, 6796−6803. (24) Colver, G. M. An interparticle force model for ac-dc electric fields in powders. Powder Technol. 2000, 112, 126−136. (25) Cox, B. J.; Thamwattana, N.; Hill, J. M. Electric field-induced force between two identical uncharged spheres Appl. Phys. Lett. 2006, 88. (26) Washizu, M.; Jones, T. B. Dielectrophoretic interaction of two spherical particles calculated by equivalent multipole-moment method. IEEE Trans. Ind. Appl. 1996, 32, 233−242. (27) Davis, L. C. Finite-Element Analysis Of Particle-Particle Forces In Electrorheological Fluids. Appl. Phys. Lett. 1992, 60, 319−321. (28) Davis, L. C. Polarization Forces And Conductivity Effects In Electrorheological Fluids. J. Appl. Phys. 1992, 72, 1334−1340. (29) Kao, K. C. Dielectric Phenomena In Solids: With Emphasis on Physical Concepts of Electronic Processes; Academic Press: Amsterdam, 2004.
CONCLUSIONS Our experiments show that an electric field produces an attractive force between two BaTiO3 glass particles in air that is much greater than a van der Waals force and approximately follows a power law that depends on the separation between the particles. The adhesion force is also much greater in the presence of the field. The force is induced by either a DC field or an AC field. The DC field-induced force is greater in magnitude, because it includes contributions from the conductivity of the particles, and increases with time, which we attribute to migration of charge. The AC field-induced force has three principal components: static, f 0, and 2f 0. The static and 2f 0 components are expected for a polarization force, and the f 0 component we attribute to charging. The magnitude of the static force is about 9000 times greater than predicted by the multipole-moment model. The dramatic mismatch between theory and experiment is a necessity for many of the applications of field-induced forces described in the introduction, because the theoretically predicted forces are very weak. This suggests an opportunity for future modeling. With the glaring exception of the magnitude, the static force agrees well the expectations of a polarization force. The measured force (a) has a very similar separation dependence to the multipolemoment calculation of Washizu and Jones for particles with dielectric constant over 1000, (b) is proportional to the square of the field, and (c) is independent of the field frequency in the range 1−100 kHz. The field also produces a force at twice the field frequency, which has similar properties to the static response.
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ASSOCIATED CONTENT
S Supporting Information *
Additional material as discussed in the text. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Present Address †
Taiwan Semiconductor Manufacturing Company, Limited (TSMC). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was funded by the American Chemical Society Petroleum Research Fund, under Award No. PRF 49114-ND5. The authors thank Professor Masao Washizu for providing the FORTRAN code that was used in the multipole moment calculations.
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REFERENCES
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