Abstract

The force experienced by a neutral dielectric object in the presence of a spatially non-uniform electric field is referred to as dielectrophoresis (DEP). The proper quantification of DEP force in the single-cell level could be of great importance for the design of high-efficiency micro-fluidic systems for the separation of biological cells. In this report we show how optical tweezers can be properly utilized for proper quantification of DEP force experienced by a human RBC. By tuning the temporal frequency of the applied electric field and also performing control experiments and comparing our experimental results with that of theoretically calculated, we show that the measured force is a pure DEP force. Our results show that in the frequency range of 0.1-3 $MHz$ the DEP force acting on RBC is frequency independent.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  25. A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “AC electric-field-induced fluid flow in microelectrodes,” J. Colloid Interface Sci. 217(2), 420–422 (1999).
    [Crossref]
  26. H. Nili and N. G. Green, “Higher-order dielectrophoresis of nonspherical particles,” Phys. Rev. E 89(6), 063302 (2014).
    [Crossref]
  27. M. Castellarnau, A. Errachid, C. Madrid, A. Juarez, and J. Samitier, “Dielectrophoresis as a tool to characterize and differentiate isogenic mutants of Escherichia coli,” Biophys. J. 91(10), 3937–3945 (2006).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  32. N Lukas and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2012), Chap. 4.

2018 (1)

2017 (3)

Y. Qiang, J. Liu, and E. Du, “Dynamic fatigue measurement of human erythrocytes using dielectrophoresis,” Acta Biomater. 57, 352–362 (2017).
[Crossref]

S. M. Mousavi, S. N. S. Reihani, G. Anvari, M. Anvari, H. G. Alinezhad, and M. R. R. Tabar, “Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers,” Sci. Rep. 7(1), 4832 (2017).
[Crossref]

H. J. Jeon, H. Lee, D. S. Yoon, and B. M. Kim, “Dielectrophoretic force measurement of red blood cells exposed to oxidative stress using optical tweezers and a microfluidic chip,” Biomed. Eng. Lett. 7(4), 317–323 (2017).
[Crossref]

2016 (1)

M. A. Saucedo-Espinosa, M. M. Rauch, A. LaLonde, and B. H. Lapizco-Encinas, “Polarization behavior of polystyrene particles under direct current and low-frequency ($< 1kHz$<1kHz) electric fields in dielectrophoretic systems,” Electrophoresis 37(4), 635–644 (2016).
[Crossref]

2015 (2)

G. Pesce, G. Rusciano, G. Zito, and A. Sasso, “Simultaneous measurements of electrophoretic and dielectrophoretic forces using optical tweezers,” Opt. Express 23(7), 9363–9368 (2015).
[Crossref]

M. A. Saucedo-Espinosa and B. H. Lapizco-Encinas, “Design of insulator-based dielectrophoretic devices: Effect of insulator posts characteristics,” J. Chromatogr. A 1422, 325–333 (2015).
[Crossref]

2014 (8)

H. Nili and N. G. Green, “Higher-order dielectrophoresis of nonspherical particles,” Phys. Rev. E 89(6), 063302 (2014).
[Crossref]

I. S. Park, S. H. Park, S. W. Lee, D. S. Yoon, and B. M. Kim, “Quantitative characterization for dielectrophoretic behavior of biological cells using optical tweezers,” Appl. Phys. Lett. 104(5), 053701 (2014).
[Crossref]

I. S. Park, S. H. Park, D. S. Yoon, S. W. Lee, and B. M. Kim, “Direct measurement of the dielectrophoresis forces acting on micro-objects using optical tweezers and a simple microfluidic chip,” Appl. Phys. Lett. 105(10), 103701 (2014).
[Crossref]

M. Li, W. H. Li, J. Zhang, G. Alici, and W. Wen, “A review of microfabrication techniques and dielectrophoretic microdevices for particle manipulation and separation,” J. Phys. D: Appl. Phys. 47(6), 063001 (2014).
[Crossref]

T. Z. Jubery, S. K. Srivastava, and P. Dutta, “Dielectrophoretic separation of bioparticles in microdevices: A review,” Electrophoresis 35(5), 691–713 (2014).
[Crossref]

R. C. Gascoyne and S. Shim, “Isolation of circulating tumor cells by dielectrophoresis,” Cancers 6(1), 545–579 (2014).
[Crossref]

Y. J. Lo, Y. Y. Lin, U. Lei, M. S. Wu, and P. C. Yang, “Measurement of the Clausius-Mossotti factor of generalized dielectrophoresis,” Appl. Phys. Lett. 104(8), 083701 (2014).
[Crossref]

E. Du, M. Dao, and S. Suresh, “Quantitative biomechanics of healthy and diseased human red blood cells using dielectrophoresis in a microfluidic system,” Extreme Mech. Lett. 1, 35–41 (2014).
[Crossref]

2012 (1)

H. Park, M. T. Wei, and H. D. Ou-Yang, “Dielectrophoresis force spectroscopy for colloidal clusters,” Electrophoresis 33(16), 2491–2497 (2012).
[Crossref]

2010 (2)

2009 (1)

M. T. Wei, J. Junio, and H. D. Ou-Yang, “Direct measurements of the frequency-dependent dielectrophoresis force,” Biomicrofluidics 3(1), 012003 (2009).
[Crossref]

2006 (1)

M. Castellarnau, A. Errachid, C. Madrid, A. Juarez, and J. Samitier, “Dielectrophoresis as a tool to characterize and differentiate isogenic mutants of Escherichia coli,” Biophys. J. 91(10), 3937–3945 (2006).
[Crossref]

2004 (1)

I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

2002 (4)

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

N. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation,” Phys. Rev. E 66(2), 026305 (2002).
[Crossref]

P. R. C. Gascoyne and J. Vykoukal, “Particle separation by dielectrophoresis,” Electrophoresis 23(13), 1973–1983 (2002).
[Crossref]

H. Li and R. Bashir, “Dielectrophoretic separation and manipulation of live and heat-treated cells of Listeria on microfabricated devices with interdigitated electrodes,” Sens. Actuators, B 86(2-3), 215–221 (2002).
[Crossref]

2000 (1)

N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. I. Experimental measurements,” Phys. Rev. E 61(4), 4011–4018 (2000).
[Crossref]

1999 (2)

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “The role of electrohydrodynamic forces in the dielectrophoretic manipulation and separation of particles,” J. Electrost. 47(1-2), 71–81 (1999).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “AC electric-field-induced fluid flow in microelectrodes,” J. Colloid Interface Sci. 217(2), 420–422 (1999).
[Crossref]

1995 (1)

F. F. Becker, X. B. Wang, Y. Huang, R. Pethig, J. Vykoukal, and P. R. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proc. Natl. Acad. Sci. 92(3), 860–864 (1995).
[Crossref]

1986 (1)

1951 (1)

H. A. Pohl, “The motion and precipitation of suspensoids in divergent electric fields,” J. Appl. Phys. 22(7), 869–871 (1951).
[Crossref]

Alici, G.

M. Li, W. H. Li, J. Zhang, G. Alici, and W. Wen, “A review of microfabrication techniques and dielectrophoretic microdevices for particle manipulation and separation,” J. Phys. D: Appl. Phys. 47(6), 063001 (2014).
[Crossref]

Alinezhad, H. G.

S. M. Mousavi, S. N. S. Reihani, G. Anvari, M. Anvari, H. G. Alinezhad, and M. R. R. Tabar, “Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers,” Sci. Rep. 7(1), 4832 (2017).
[Crossref]

Anvari, G.

S. M. Mousavi, S. N. S. Reihani, G. Anvari, M. Anvari, H. G. Alinezhad, and M. R. R. Tabar, “Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers,” Sci. Rep. 7(1), 4832 (2017).
[Crossref]

Anvari, M.

S. M. Mousavi, S. N. S. Reihani, G. Anvari, M. Anvari, H. G. Alinezhad, and M. R. R. Tabar, “Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers,” Sci. Rep. 7(1), 4832 (2017).
[Crossref]

Ashkin, A.

Azadbakht, A.

Babaei, M.

Baek, S. H.

Ballerini, R.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

Bashir, R.

H. Li and R. Bashir, “Dielectrophoretic separation and manipulation of live and heat-treated cells of Listeria on microfabricated devices with interdigitated electrodes,” Sens. Actuators, B 86(2-3), 215–221 (2002).
[Crossref]

Becker, F. F.

F. F. Becker, X. B. Wang, Y. Huang, R. Pethig, J. Vykoukal, and P. R. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proc. Natl. Acad. Sci. 92(3), 860–864 (1995).
[Crossref]

Berg-Sørensen, K.

I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

Bjorkholm, J. E.

Capitanio, M.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

Castellanos, A.

N. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation,” Phys. Rev. E 66(2), 026305 (2002).
[Crossref]

N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. I. Experimental measurements,” Phys. Rev. E 61(4), 4011–4018 (2000).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “The role of electrohydrodynamic forces in the dielectrophoretic manipulation and separation of particles,” J. Electrost. 47(1-2), 71–81 (1999).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “AC electric-field-induced fluid flow in microelectrodes,” J. Colloid Interface Sci. 217(2), 420–422 (1999).
[Crossref]

Castellarnau, M.

M. Castellarnau, A. Errachid, C. Madrid, A. Juarez, and J. Samitier, “Dielectrophoresis as a tool to characterize and differentiate isogenic mutants of Escherichia coli,” Biophys. J. 91(10), 3937–3945 (2006).
[Crossref]

Chu, S.

Dao, M.

E. Du, M. Dao, and S. Suresh, “Quantitative biomechanics of healthy and diseased human red blood cells using dielectrophoresis in a microfluidic system,” Extreme Mech. Lett. 1, 35–41 (2014).
[Crossref]

Du, E.

Y. Qiang, J. Liu, and E. Du, “Dynamic fatigue measurement of human erythrocytes using dielectrophoresis,” Acta Biomater. 57, 352–362 (2017).
[Crossref]

E. Du, M. Dao, and S. Suresh, “Quantitative biomechanics of healthy and diseased human red blood cells using dielectrophoresis in a microfluidic system,” Extreme Mech. Lett. 1, 35–41 (2014).
[Crossref]

Dunlap, D.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

Dutta, P.

T. Z. Jubery, S. K. Srivastava, and P. Dutta, “Dielectrophoretic separation of bioparticles in microdevices: A review,” Electrophoresis 35(5), 691–713 (2014).
[Crossref]

Dziedzic, J. M.

Errachid, A.

M. Castellarnau, A. Errachid, C. Madrid, A. Juarez, and J. Samitier, “Dielectrophoresis as a tool to characterize and differentiate isogenic mutants of Escherichia coli,” Biophys. J. 91(10), 3937–3945 (2006).
[Crossref]

Finzi, L.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

Flyvbjerg, H.

I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

Gascoyne, P. R.

F. F. Becker, X. B. Wang, Y. Huang, R. Pethig, J. Vykoukal, and P. R. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proc. Natl. Acad. Sci. 92(3), 860–864 (1995).
[Crossref]

Gascoyne, P. R. C.

P. R. C. Gascoyne and J. Vykoukal, “Particle separation by dielectrophoresis,” Electrophoresis 23(13), 1973–1983 (2002).
[Crossref]

Gascoyne, R. C.

R. C. Gascoyne and S. Shim, “Isolation of circulating tumor cells by dielectrophoresis,” Cancers 6(1), 545–579 (2014).
[Crossref]

Giuntini, M.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

González, A.

N. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation,” Phys. Rev. E 66(2), 026305 (2002).
[Crossref]

N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. I. Experimental measurements,” Phys. Rev. E 61(4), 4011–4018 (2000).
[Crossref]

Green, N.

N. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation,” Phys. Rev. E 66(2), 026305 (2002).
[Crossref]

Green, N. G.

H. Nili and N. G. Green, “Higher-order dielectrophoresis of nonspherical particles,” Phys. Rev. E 89(6), 063302 (2014).
[Crossref]

N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. I. Experimental measurements,” Phys. Rev. E 61(4), 4011–4018 (2000).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “The role of electrohydrodynamic forces in the dielectrophoretic manipulation and separation of particles,” J. Electrost. 47(1-2), 71–81 (1999).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “AC electric-field-induced fluid flow in microelectrodes,” J. Colloid Interface Sci. 217(2), 420–422 (1999).
[Crossref]

Hecht, B.

N Lukas and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2012), Chap. 4.

Hong, Y.

Huang, Y.

F. F. Becker, X. B. Wang, Y. Huang, R. Pethig, J. Vykoukal, and P. R. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proc. Natl. Acad. Sci. 92(3), 860–864 (1995).
[Crossref]

Jeon, H. J.

H. J. Jeon, H. Lee, D. S. Yoon, and B. M. Kim, “Dielectrophoretic force measurement of red blood cells exposed to oxidative stress using optical tweezers and a microfluidic chip,” Biomed. Eng. Lett. 7(4), 317–323 (2017).
[Crossref]

Juarez, A.

M. Castellarnau, A. Errachid, C. Madrid, A. Juarez, and J. Samitier, “Dielectrophoresis as a tool to characterize and differentiate isogenic mutants of Escherichia coli,” Biophys. J. 91(10), 3937–3945 (2006).
[Crossref]

Jubery, T. Z.

T. Z. Jubery, S. K. Srivastava, and P. Dutta, “Dielectrophoretic separation of bioparticles in microdevices: A review,” Electrophoresis 35(5), 691–713 (2014).
[Crossref]

Junio, J.

M. T. Wei, J. Junio, and H. D. Ou-Yang, “Direct measurements of the frequency-dependent dielectrophoresis force,” Biomicrofluidics 3(1), 012003 (2009).
[Crossref]

Kim, B. M.

H. J. Jeon, H. Lee, D. S. Yoon, and B. M. Kim, “Dielectrophoretic force measurement of red blood cells exposed to oxidative stress using optical tweezers and a microfluidic chip,” Biomed. Eng. Lett. 7(4), 317–323 (2017).
[Crossref]

I. S. Park, S. H. Park, S. W. Lee, D. S. Yoon, and B. M. Kim, “Quantitative characterization for dielectrophoretic behavior of biological cells using optical tweezers,” Appl. Phys. Lett. 104(5), 053701 (2014).
[Crossref]

I. S. Park, S. H. Park, D. S. Yoon, S. W. Lee, and B. M. Kim, “Direct measurement of the dielectrophoresis forces acting on micro-objects using optical tweezers and a simple microfluidic chip,” Appl. Phys. Lett. 105(10), 103701 (2014).
[Crossref]

Y. Hong, J. W. Pyo, S. H. Baek, S. W. Lee, D. S. Yoon, K. No, and B. M. Kim, “Quantitative measurements of absolute dielectrophoretic forces using optical tweezers,” Opt. Lett. 35(14), 2493–2495 (2010).
[Crossref]

LaLonde, A.

M. A. Saucedo-Espinosa, M. M. Rauch, A. LaLonde, and B. H. Lapizco-Encinas, “Polarization behavior of polystyrene particles under direct current and low-frequency ($< 1kHz$<1kHz) electric fields in dielectrophoretic systems,” Electrophoresis 37(4), 635–644 (2016).
[Crossref]

Lapizco-Encinas, B. H.

M. A. Saucedo-Espinosa, M. M. Rauch, A. LaLonde, and B. H. Lapizco-Encinas, “Polarization behavior of polystyrene particles under direct current and low-frequency ($< 1kHz$<1kHz) electric fields in dielectrophoretic systems,” Electrophoresis 37(4), 635–644 (2016).
[Crossref]

M. A. Saucedo-Espinosa and B. H. Lapizco-Encinas, “Design of insulator-based dielectrophoretic devices: Effect of insulator posts characteristics,” J. Chromatogr. A 1422, 325–333 (2015).
[Crossref]

Lee, H.

H. J. Jeon, H. Lee, D. S. Yoon, and B. M. Kim, “Dielectrophoretic force measurement of red blood cells exposed to oxidative stress using optical tweezers and a microfluidic chip,” Biomed. Eng. Lett. 7(4), 317–323 (2017).
[Crossref]

Lee, S. W.

I. S. Park, S. H. Park, S. W. Lee, D. S. Yoon, and B. M. Kim, “Quantitative characterization for dielectrophoretic behavior of biological cells using optical tweezers,” Appl. Phys. Lett. 104(5), 053701 (2014).
[Crossref]

I. S. Park, S. H. Park, D. S. Yoon, S. W. Lee, and B. M. Kim, “Direct measurement of the dielectrophoresis forces acting on micro-objects using optical tweezers and a simple microfluidic chip,” Appl. Phys. Lett. 105(10), 103701 (2014).
[Crossref]

Y. Hong, J. W. Pyo, S. H. Baek, S. W. Lee, D. S. Yoon, K. No, and B. M. Kim, “Quantitative measurements of absolute dielectrophoretic forces using optical tweezers,” Opt. Lett. 35(14), 2493–2495 (2010).
[Crossref]

Lei, U.

Y. J. Lo, Y. Y. Lin, U. Lei, M. S. Wu, and P. C. Yang, “Measurement of the Clausius-Mossotti factor of generalized dielectrophoresis,” Appl. Phys. Lett. 104(8), 083701 (2014).
[Crossref]

Li, H.

H. Li and R. Bashir, “Dielectrophoretic separation and manipulation of live and heat-treated cells of Listeria on microfabricated devices with interdigitated electrodes,” Sens. Actuators, B 86(2-3), 215–221 (2002).
[Crossref]

Li, M.

M. Li, W. H. Li, J. Zhang, G. Alici, and W. Wen, “A review of microfabrication techniques and dielectrophoretic microdevices for particle manipulation and separation,” J. Phys. D: Appl. Phys. 47(6), 063001 (2014).
[Crossref]

Li, W. H.

M. Li, W. H. Li, J. Zhang, G. Alici, and W. Wen, “A review of microfabrication techniques and dielectrophoretic microdevices for particle manipulation and separation,” J. Phys. D: Appl. Phys. 47(6), 063001 (2014).
[Crossref]

Lin, Y. Y.

Y. J. Lo, Y. Y. Lin, U. Lei, M. S. Wu, and P. C. Yang, “Measurement of the Clausius-Mossotti factor of generalized dielectrophoresis,” Appl. Phys. Lett. 104(8), 083701 (2014).
[Crossref]

Liu, J.

Y. Qiang, J. Liu, and E. Du, “Dynamic fatigue measurement of human erythrocytes using dielectrophoresis,” Acta Biomater. 57, 352–362 (2017).
[Crossref]

Lo, Y. J.

Y. J. Lo, Y. Y. Lin, U. Lei, M. S. Wu, and P. C. Yang, “Measurement of the Clausius-Mossotti factor of generalized dielectrophoresis,” Appl. Phys. Lett. 104(8), 083701 (2014).
[Crossref]

Lukas, N

N Lukas and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2012), Chap. 4.

Madrid, C.

M. Castellarnau, A. Errachid, C. Madrid, A. Juarez, and J. Samitier, “Dielectrophoresis as a tool to characterize and differentiate isogenic mutants of Escherichia coli,” Biophys. J. 91(10), 3937–3945 (2006).
[Crossref]

Mashaghi, A.

Moosavi-Movahedi, A. A.

Morgan, H.

N. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation,” Phys. Rev. E 66(2), 026305 (2002).
[Crossref]

N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. I. Experimental measurements,” Phys. Rev. E 61(4), 4011–4018 (2000).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “The role of electrohydrodynamic forces in the dielectrophoretic manipulation and separation of particles,” J. Electrost. 47(1-2), 71–81 (1999).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “AC electric-field-induced fluid flow in microelectrodes,” J. Colloid Interface Sci. 217(2), 420–422 (1999).
[Crossref]

Mousavi, S. M.

S. M. Mousavi, S. N. S. Reihani, G. Anvari, M. Anvari, H. G. Alinezhad, and M. R. R. Tabar, “Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers,” Sci. Rep. 7(1), 4832 (2017).
[Crossref]

Nili, H.

H. Nili and N. G. Green, “Higher-order dielectrophoresis of nonspherical particles,” Phys. Rev. E 89(6), 063302 (2014).
[Crossref]

No, K.

Ou-Yang, H. D.

H. Park, M. T. Wei, and H. D. Ou-Yang, “Dielectrophoresis force spectroscopy for colloidal clusters,” Electrophoresis 33(16), 2491–2497 (2012).
[Crossref]

M. T. Wei, J. Junio, and H. D. Ou-Yang, “Direct measurements of the frequency-dependent dielectrophoresis force,” Biomicrofluidics 3(1), 012003 (2009).
[Crossref]

Park, H.

H. Park, M. T. Wei, and H. D. Ou-Yang, “Dielectrophoresis force spectroscopy for colloidal clusters,” Electrophoresis 33(16), 2491–2497 (2012).
[Crossref]

Park, I. S.

I. S. Park, S. H. Park, D. S. Yoon, S. W. Lee, and B. M. Kim, “Direct measurement of the dielectrophoresis forces acting on micro-objects using optical tweezers and a simple microfluidic chip,” Appl. Phys. Lett. 105(10), 103701 (2014).
[Crossref]

I. S. Park, S. H. Park, S. W. Lee, D. S. Yoon, and B. M. Kim, “Quantitative characterization for dielectrophoretic behavior of biological cells using optical tweezers,” Appl. Phys. Lett. 104(5), 053701 (2014).
[Crossref]

Park, S. H.

I. S. Park, S. H. Park, S. W. Lee, D. S. Yoon, and B. M. Kim, “Quantitative characterization for dielectrophoretic behavior of biological cells using optical tweezers,” Appl. Phys. Lett. 104(5), 053701 (2014).
[Crossref]

I. S. Park, S. H. Park, D. S. Yoon, S. W. Lee, and B. M. Kim, “Direct measurement of the dielectrophoresis forces acting on micro-objects using optical tweezers and a simple microfluidic chip,” Appl. Phys. Lett. 105(10), 103701 (2014).
[Crossref]

Pavone, F. S.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

Pazoki-Toroudi, H.

Pesce, G.

Pethig, R.

R. Pethig, “Review article dielectrophoresis: Status of the theory,” Biomicrofluidics 4(2), 022811 (2010).
[Crossref]

F. F. Becker, X. B. Wang, Y. Huang, R. Pethig, J. Vykoukal, and P. R. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proc. Natl. Acad. Sci. 92(3), 860–864 (1995).
[Crossref]

Pohl, H. A.

H. A. Pohl, “The motion and precipitation of suspensoids in divergent electric fields,” J. Appl. Phys. 22(7), 869–871 (1951).
[Crossref]

Pyo, J. W.

Qiang, Y.

Y. Qiang, J. Liu, and E. Du, “Dynamic fatigue measurement of human erythrocytes using dielectrophoresis,” Acta Biomater. 57, 352–362 (2017).
[Crossref]

Ramos, A.

N. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation,” Phys. Rev. E 66(2), 026305 (2002).
[Crossref]

N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. I. Experimental measurements,” Phys. Rev. E 61(4), 4011–4018 (2000).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “AC electric-field-induced fluid flow in microelectrodes,” J. Colloid Interface Sci. 217(2), 420–422 (1999).
[Crossref]

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “The role of electrohydrodynamic forces in the dielectrophoretic manipulation and separation of particles,” J. Electrost. 47(1-2), 71–81 (1999).
[Crossref]

Rauch, M. M.

M. A. Saucedo-Espinosa, M. M. Rauch, A. LaLonde, and B. H. Lapizco-Encinas, “Polarization behavior of polystyrene particles under direct current and low-frequency ($< 1kHz$<1kHz) electric fields in dielectrophoretic systems,” Electrophoresis 37(4), 635–644 (2016).
[Crossref]

Reihani, S. N. S.

V. Sheikh-Hasani, M. Babaei, A. Azadbakht, H. Pazoki-Toroudi, A. Mashaghi, A. A. Moosavi-Movahedi, and S. N. S. Reihani, “Atorvastatin treatment softens human red blood cells: an optical tweezers study,” Biomed. Opt. Express 9(3), 1256–1261 (2018).
[Crossref]

S. M. Mousavi, S. N. S. Reihani, G. Anvari, M. Anvari, H. G. Alinezhad, and M. R. R. Tabar, “Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers,” Sci. Rep. 7(1), 4832 (2017).
[Crossref]

Romano, G.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

Rusciano, G.

Samitier, J.

M. Castellarnau, A. Errachid, C. Madrid, A. Juarez, and J. Samitier, “Dielectrophoresis as a tool to characterize and differentiate isogenic mutants of Escherichia coli,” Biophys. J. 91(10), 3937–3945 (2006).
[Crossref]

Sasso, A.

Saucedo-Espinosa, M. A.

M. A. Saucedo-Espinosa, M. M. Rauch, A. LaLonde, and B. H. Lapizco-Encinas, “Polarization behavior of polystyrene particles under direct current and low-frequency ($< 1kHz$<1kHz) electric fields in dielectrophoretic systems,” Electrophoresis 37(4), 635–644 (2016).
[Crossref]

M. A. Saucedo-Espinosa and B. H. Lapizco-Encinas, “Design of insulator-based dielectrophoretic devices: Effect of insulator posts characteristics,” J. Chromatogr. A 1422, 325–333 (2015).
[Crossref]

Sheikh-Hasani, V.

Shim, S.

R. C. Gascoyne and S. Shim, “Isolation of circulating tumor cells by dielectrophoresis,” Cancers 6(1), 545–579 (2014).
[Crossref]

Srivastava, S. K.

T. Z. Jubery, S. K. Srivastava, and P. Dutta, “Dielectrophoretic separation of bioparticles in microdevices: A review,” Electrophoresis 35(5), 691–713 (2014).
[Crossref]

Suresh, S.

E. Du, M. Dao, and S. Suresh, “Quantitative biomechanics of healthy and diseased human red blood cells using dielectrophoresis in a microfluidic system,” Extreme Mech. Lett. 1, 35–41 (2014).
[Crossref]

Tabar, M. R. R.

S. M. Mousavi, S. N. S. Reihani, G. Anvari, M. Anvari, H. G. Alinezhad, and M. R. R. Tabar, “Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers,” Sci. Rep. 7(1), 4832 (2017).
[Crossref]

Tolic-Nørrelykke, I. M.

I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

Vykoukal, J.

P. R. C. Gascoyne and J. Vykoukal, “Particle separation by dielectrophoresis,” Electrophoresis 23(13), 1973–1983 (2002).
[Crossref]

F. F. Becker, X. B. Wang, Y. Huang, R. Pethig, J. Vykoukal, and P. R. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proc. Natl. Acad. Sci. 92(3), 860–864 (1995).
[Crossref]

Wang, X. B.

F. F. Becker, X. B. Wang, Y. Huang, R. Pethig, J. Vykoukal, and P. R. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proc. Natl. Acad. Sci. 92(3), 860–864 (1995).
[Crossref]

Wei, M. T.

H. Park, M. T. Wei, and H. D. Ou-Yang, “Dielectrophoresis force spectroscopy for colloidal clusters,” Electrophoresis 33(16), 2491–2497 (2012).
[Crossref]

M. T. Wei, J. Junio, and H. D. Ou-Yang, “Direct measurements of the frequency-dependent dielectrophoresis force,” Biomicrofluidics 3(1), 012003 (2009).
[Crossref]

Wen, W.

M. Li, W. H. Li, J. Zhang, G. Alici, and W. Wen, “A review of microfabrication techniques and dielectrophoretic microdevices for particle manipulation and separation,” J. Phys. D: Appl. Phys. 47(6), 063001 (2014).
[Crossref]

Wu, M. S.

Y. J. Lo, Y. Y. Lin, U. Lei, M. S. Wu, and P. C. Yang, “Measurement of the Clausius-Mossotti factor of generalized dielectrophoresis,” Appl. Phys. Lett. 104(8), 083701 (2014).
[Crossref]

Yang, P. C.

Y. J. Lo, Y. Y. Lin, U. Lei, M. S. Wu, and P. C. Yang, “Measurement of the Clausius-Mossotti factor of generalized dielectrophoresis,” Appl. Phys. Lett. 104(8), 083701 (2014).
[Crossref]

Yoon, D. S.

H. J. Jeon, H. Lee, D. S. Yoon, and B. M. Kim, “Dielectrophoretic force measurement of red blood cells exposed to oxidative stress using optical tweezers and a microfluidic chip,” Biomed. Eng. Lett. 7(4), 317–323 (2017).
[Crossref]

I. S. Park, S. H. Park, S. W. Lee, D. S. Yoon, and B. M. Kim, “Quantitative characterization for dielectrophoretic behavior of biological cells using optical tweezers,” Appl. Phys. Lett. 104(5), 053701 (2014).
[Crossref]

I. S. Park, S. H. Park, D. S. Yoon, S. W. Lee, and B. M. Kim, “Direct measurement of the dielectrophoresis forces acting on micro-objects using optical tweezers and a simple microfluidic chip,” Appl. Phys. Lett. 105(10), 103701 (2014).
[Crossref]

Y. Hong, J. W. Pyo, S. H. Baek, S. W. Lee, D. S. Yoon, K. No, and B. M. Kim, “Quantitative measurements of absolute dielectrophoretic forces using optical tweezers,” Opt. Lett. 35(14), 2493–2495 (2010).
[Crossref]

Zhang, J.

M. Li, W. H. Li, J. Zhang, G. Alici, and W. Wen, “A review of microfabrication techniques and dielectrophoretic microdevices for particle manipulation and separation,” J. Phys. D: Appl. Phys. 47(6), 063001 (2014).
[Crossref]

Zito, G.

Acta Biomater. (1)

Y. Qiang, J. Liu, and E. Du, “Dynamic fatigue measurement of human erythrocytes using dielectrophoresis,” Acta Biomater. 57, 352–362 (2017).
[Crossref]

Appl. Phys. Lett. (3)

I. S. Park, S. H. Park, D. S. Yoon, S. W. Lee, and B. M. Kim, “Direct measurement of the dielectrophoresis forces acting on micro-objects using optical tweezers and a simple microfluidic chip,” Appl. Phys. Lett. 105(10), 103701 (2014).
[Crossref]

I. S. Park, S. H. Park, S. W. Lee, D. S. Yoon, and B. M. Kim, “Quantitative characterization for dielectrophoretic behavior of biological cells using optical tweezers,” Appl. Phys. Lett. 104(5), 053701 (2014).
[Crossref]

Y. J. Lo, Y. Y. Lin, U. Lei, M. S. Wu, and P. C. Yang, “Measurement of the Clausius-Mossotti factor of generalized dielectrophoresis,” Appl. Phys. Lett. 104(8), 083701 (2014).
[Crossref]

Biomed. Eng. Lett. (1)

H. J. Jeon, H. Lee, D. S. Yoon, and B. M. Kim, “Dielectrophoretic force measurement of red blood cells exposed to oxidative stress using optical tweezers and a microfluidic chip,” Biomed. Eng. Lett. 7(4), 317–323 (2017).
[Crossref]

Biomed. Opt. Express (1)

Biomicrofluidics (2)

M. T. Wei, J. Junio, and H. D. Ou-Yang, “Direct measurements of the frequency-dependent dielectrophoresis force,” Biomicrofluidics 3(1), 012003 (2009).
[Crossref]

R. Pethig, “Review article dielectrophoresis: Status of the theory,” Biomicrofluidics 4(2), 022811 (2010).
[Crossref]

Biophys. J. (1)

M. Castellarnau, A. Errachid, C. Madrid, A. Juarez, and J. Samitier, “Dielectrophoresis as a tool to characterize and differentiate isogenic mutants of Escherichia coli,” Biophys. J. 91(10), 3937–3945 (2006).
[Crossref]

Cancers (1)

R. C. Gascoyne and S. Shim, “Isolation of circulating tumor cells by dielectrophoresis,” Cancers 6(1), 545–579 (2014).
[Crossref]

Comput. Phys. Commun. (1)

I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

Electrophoresis (4)

M. A. Saucedo-Espinosa, M. M. Rauch, A. LaLonde, and B. H. Lapizco-Encinas, “Polarization behavior of polystyrene particles under direct current and low-frequency ($< 1kHz$<1kHz) electric fields in dielectrophoretic systems,” Electrophoresis 37(4), 635–644 (2016).
[Crossref]

T. Z. Jubery, S. K. Srivastava, and P. Dutta, “Dielectrophoretic separation of bioparticles in microdevices: A review,” Electrophoresis 35(5), 691–713 (2014).
[Crossref]

P. R. C. Gascoyne and J. Vykoukal, “Particle separation by dielectrophoresis,” Electrophoresis 23(13), 1973–1983 (2002).
[Crossref]

H. Park, M. T. Wei, and H. D. Ou-Yang, “Dielectrophoresis force spectroscopy for colloidal clusters,” Electrophoresis 33(16), 2491–2497 (2012).
[Crossref]

Extreme Mech. Lett. (1)

E. Du, M. Dao, and S. Suresh, “Quantitative biomechanics of healthy and diseased human red blood cells using dielectrophoresis in a microfluidic system,” Extreme Mech. Lett. 1, 35–41 (2014).
[Crossref]

J. Appl. Phys. (1)

H. A. Pohl, “The motion and precipitation of suspensoids in divergent electric fields,” J. Appl. Phys. 22(7), 869–871 (1951).
[Crossref]

J. Chromatogr. A (1)

M. A. Saucedo-Espinosa and B. H. Lapizco-Encinas, “Design of insulator-based dielectrophoretic devices: Effect of insulator posts characteristics,” J. Chromatogr. A 1422, 325–333 (2015).
[Crossref]

J. Colloid Interface Sci. (1)

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “AC electric-field-induced fluid flow in microelectrodes,” J. Colloid Interface Sci. 217(2), 420–422 (1999).
[Crossref]

J. Electrost. (1)

A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, “The role of electrohydrodynamic forces in the dielectrophoretic manipulation and separation of particles,” J. Electrost. 47(1-2), 71–81 (1999).
[Crossref]

J. Phys. D: Appl. Phys. (1)

M. Li, W. H. Li, J. Zhang, G. Alici, and W. Wen, “A review of microfabrication techniques and dielectrophoretic microdevices for particle manipulation and separation,” J. Phys. D: Appl. Phys. 47(6), 063001 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. E (3)

H. Nili and N. G. Green, “Higher-order dielectrophoresis of nonspherical particles,” Phys. Rev. E 89(6), 063302 (2014).
[Crossref]

N. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation,” Phys. Rev. E 66(2), 026305 (2002).
[Crossref]

N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, “Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. I. Experimental measurements,” Phys. Rev. E 61(4), 4011–4018 (2000).
[Crossref]

Proc. Natl. Acad. Sci. (1)

F. F. Becker, X. B. Wang, Y. Huang, R. Pethig, J. Vykoukal, and P. R. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proc. Natl. Acad. Sci. 92(3), 860–864 (1995).
[Crossref]

Rev. Sci. Instrum. (1)

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73(4), 1687–1696 (2002).
[Crossref]

Sci. Rep. (1)

S. M. Mousavi, S. N. S. Reihani, G. Anvari, M. Anvari, H. G. Alinezhad, and M. R. R. Tabar, “Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers,” Sci. Rep. 7(1), 4832 (2017).
[Crossref]

Sens. Actuators, B (1)

H. Li and R. Bashir, “Dielectrophoretic separation and manipulation of live and heat-treated cells of Listeria on microfabricated devices with interdigitated electrodes,” Sens. Actuators, B 86(2-3), 215–221 (2002).
[Crossref]

Other (1)

N Lukas and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2012), Chap. 4.

Supplementary Material (7)

NameDescription
» Visualization 1       Visualization related to Fig.1(d)
» Visualization 2       Visualization related to Fig.2(a)
» Visualization 3       Visualization related to Fig.2(b)
» Visualization 4       Visualization related to Fig.2(c)
» Visualization 5       Visualization related to Fig.4(b)
» Visualization 6       Visualization related to Fig.4(b)
» Visualization 7       Visualization related to Fig.5(a)

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Figures (5)

Fig. 1.
Fig. 1. (a) The schematic of OT setup including lenses (${\rm{L}}_{1}$-${\rm{L}}_{7}$), Mirror (${\rm{M}}_{1}$-${\rm{M}}_{3}$), dichroic mirrors (${\rm{DM}}_{1}$, and ${\rm{DM}}_{2}$), objective and condenser lenses, ${\rm{QPD}}_{1,2}$, amplifier and Camera. (b) Photograph of the sample chamber including the two parallel microelectrodes used for producing the ESNE. The electrodes had a width of $500\mu$m with a lengths of 15 mm and a inter-space of $100\mu$m. (c) A typical PSD plot of the recorded positional time series of a trapped $3\mu m$ PS bead immersed in DI water. The sampling frequency and laser power were 10 $kHz$ and 160$mW$, respectively. (d) Typical visualization of the trapped bead in the presence and absence of the electric field Visualization 1. $\Delta x_B$ denotes the displacement of trapped bead due to the exerted force.
Fig. 2.
Fig. 2. Fluorescent nano particle tracers (green color) show the fluid motion under the external electric field (${\rm{V}}_{pp}$=6 V) at different frequencies ($f$) of (a) 500 $Hz$, (b) 5 $kHz$, and (c) 50 $kHz$. The vortical movement of the fluid disappeared at a frequency 50 $kHz$ and above (data are not shown), which demonstrates the absence of EO effect (Visualization 2, Visualization 3, and Visualization 4). The scale bars in the panels a-c are 40$\mu m$.
Fig. 3.
Fig. 3. Measurement of DEP force exerted on a trapped 3$\mu$m PS bead. (a) The measured DEP force as a function of squared applied voltage. The solid red line shows the linear fit ($y=\alpha x$) to the experimental data point with a result of $\alpha$=0.0268$\pm$0.0006. (b) The measured DEP force as a function of lateral position of the bead from the electrodes (the middle point as reference). The red circled connected with a solid line with similar color represent results of simulation for a similar configuration. Each data point represents average over three measurement with sampling frequency of $1 kHz$. The standard error of the averaged data considered as error bar, which is not visible in the graphs as they are smaller than the representing symbols. The sampling duration time for (a) and (b), respectively was 2 and 1 seconds. The frequency of the external voltage source was 700 $kHz$ and the peak-to-peak voltage difference for (b) was ${\rm{V}}_{pp}$=10V.
Fig. 4.
Fig. 4. (a) The measured DEP force exerted on RBC-bead complex as a function of frequency with $V_{pp}=12V$. Each data point represents average over seven different measurements, which are individually presented in the inset. The solid red and blue lines, respectively, show theoretically calculated DEP force acting on the RBC-bead complex, and the bead alone. In this calculation $a_1$, $a_2$, and $a_3$ values considered to be 3.8$\mu m$, 1.2 $\mu m$, and 3.2$\mu m$, respectively. The thickness of the cell membrane was considered to be $d=$4.5 nm. The electrical parameters of the medium and polystyrene was considered as $\sigma _m$=1.1S/m, $\varepsilon _m$= 75$\varepsilon _0$, $\sigma _{ps}$=0.0025S/m, and $\varepsilon _{ps}$=2.55$\varepsilon _0$. (b) The visualization of the scene when a single bead (left column) and bead-RBC complex (right column) are subjected to a DEP force. $\Delta x_B$, and $\Delta x_{RB}$ denote the displacement of the single bead and RBC-bead complex respectively, when the voltage is applied to the electrodes (Visualization 5 and Visualization 6). (c) The net DEP force experienced by the RBC as a function of voltage squared at a frequency of 2 $MHz$. The red solid line represents fit to linear function of $y=\alpha x$ with the result of $\alpha$=0.0261$\pm$0.0008
Fig. 5.
Fig. 5. (a) The visual appearance of a bead and a RBC are kept by separate traps. The upper and lower images are taken when the switch is off and on, respectively (Visualization 7). (b) The measured net force acting on the bead at different conditions: The bead is attached to the RBC (blue triangles), the bead is close to the RBC but not attached (red circles), and the RBC is far away from the bead (black squares). The data points show average over two data sets with $V_{pp}=8V$.

Equations (5)

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v e o = μ e o E = ε m ζ e o η E ; F e o , d r a g = 6 π η r v e o
v e p = μ e p E = ε m ζ e p η E ; F e p , d r a g = 6 π η r v e p
F D E P = 2 π r 3 ε m R e [ K ( ω ) ] E 2
K i ( w ) = 1 3 ( ε m e m ε m ) + 3 X i ρ ( ε m e m + A 2 i ( ε m ε m e m ) ) ε m + A 2 i ( ε m e m ε m ) + 3 X i ρ A 2 i ( 1 A 2 i ) ( ε m e m ε m ) X i = 1 3 ( ε c y t o ε m e m ) ε m e m + A 1 i ( ε c y t o ε m e m ) A 1 i = a 1 a 2 a 3 2 0 d s ( s + a i 2 ) ( s + a 1 2 ) ( s + a 2 2 ) ( s + a 3 2 ) A 2 i = ( a 1 + d ) ( a 2 + d ) ( a 3 + d ) 2 × 0 d s ( s + ( a i + d ) 2 ) ( s + ( a 1 + d ) 2 ) ( s + ( a 2 + d ) 2 ) ( s + ( a 3 + d ) 2 ) ρ = a 1 a 2 a 3 ( a 1 + d ) ( a 2 + d ) ( a 3 + d ) F D E P , i = 2 π a 1 a 2 a 3 ε m d d x i j = 1 3 ( R e [ K j ( ω ) ] E j 2 )
P ( f ) = x ~ ( f ) 2 = k B T γ π 2 ( f c 2 + f 2 )