## Nanoparticle size analysis with relaxation of induced grating by dielectrophoresis

Optics Express, Vol. 14, Issue 12, pp. 5755-5764 (2006)

http://dx.doi.org/10.1364/OE.14.005755

Acrobat PDF (615 KB)

### Abstract

We propose an alternative approach to the use of dynamic light scattering (DLS) for the analysis of particle sizes ranging from 5 nm to 100 nm. This approach employs a combination of 1) diffusion, 2) density grating, and 3) dielectrophoresis (DEP), and measures the diffusion coefficient from the decay rate of the diffracted light intensity in the relaxation process of particle density modulation generated by DEP. Both the experiments and the theoretical analysis confirm the reliable determination of particle size independently of the refractive index. The new method records a decay signal directly without an autocorrelator and is expected to have a less extreme sensitivity dependence on particle size than DLS.

© 2006 Optical Society of America

## 1. Introduction

*D*, which is then converted to the diameter. However, there is an important drawback to DLS: the sensitivity of detection changes drastically depending on the particle size. This is because the efficiency of light scattering is proportional to the diameter to the sixth power (per particle) [2], or proportional to the diameter to the third power (per unit volume of particles).

*D*as in DLS; however, it includes an activation procedure by dielectrophoresis (DEP). It first generates a periodic density modulation of the particles, which are then released to diffuse until they reach a steady state, as shown in the following description. Although many reports on DEP [8–11

11. H. Watarai, T. Sakamoto, and S. Tsukahara, “In situ measurement of dielectrophoretic mobility of single polystyrene microparticles,” Langmuir **13**, 2417–2420 (1997). [CrossRef]

12. M. Terajima, “Translational diffusion of intermediate species in solutions,” Res. Chem. Intermed. **23**, 853–901 (1997). [CrossRef]

## 2. Description of the method

*a*) is 20 µm and the open space between adjacent teeth is 10 µm. When a radio frequency (r-f) voltage is introduced between the R and L sides, it causes an intense electric field between the teeth of R and L. Hence, the particles in the cuvette migrate by DEP force toward the space between R and L, because the DEP exerts forces on the particles with induced dipole moment toward the more intense electric field [8–11

11. H. Watarai, T. Sakamoto, and S. Tsukahara, “In situ measurement of dielectrophoretic mobility of single polystyrene microparticles,” Langmuir **13**, 2417–2420 (1997). [CrossRef]

*Λ*) is twice as large as that of the teeth of the electrodes; i.e.,

*a*=20µm but

*Λ*=40µm.

*I*starts to decrease according to

*I*

_{0}is the initial intensity at the instance when DEP is turned off,

*q*is defined using

*Λ*(the pitch of the density grating) as

*D*is the diffusion coefficient given by the Einstein-Stokes relation

*k*is the Boltzmann’s constant,

_{B}*T*is the temperature in Kelvin,

*η*is the viscosity of the dispersion medium, and

*d*is the diameter of the particles. From the exponential factor of the decay curve,

*D*is obtained using Eq. (1), and then is converted to

*d*by Eq. (3) based on the known

*η*and

*T*.

## 3. Derivation of the basic decay equation

*x-y*plane with the period

*Λ*formed by the DEP. The

*x-y-z*coordinates are defined as follows:

*x*, the distance normal to the grooves;

*y*, the distance along the grooves; and

*z*, the distance normal to the grating surface. The complex transmission amplitude can be defined as

*T*(

_{amp}*x*)

*·e*, where

^{iφ(x)}*T*(

_{amp}*x*) and

*φ(x)*are the transmittance and the phase due to the periodic density of the particles, respectively. Then, defining absorption

*µ*(

*x*) by

*T*(

_{amp}*x*)=

*e-*, we have

^{µ(x)}*e-*as the complex transmission amplitude of the grating.

^{µ(x)}·e^{iφ(x)}*u(x)*is an integrated value over the

*z*direction because the diffusion toward the

*z*direction does not contribute to the change in the complex amplitude. The particle density

*u(x)*is a function of the period

*Λ*, and may thus be expressed by the Fourier cosine series as

*x*is properly shifted to make

*u(x)*symmetric, and

*m*is a positive integer. Next, we introduce a term for the relaxation due to the diffusion of particles. The density expression including time

*u(x,t)*should satisfy the one-dimensional diffusion equation

*m*). Therefore, Eq. (4) becomes

^{2}q^{2}Dt*m*≥2 decreases very rapidly; we therefore conclude that the dominant term is only from the

*m*=1 component.

*µ(x,t)*and

*φ(x,t)*are both proportional to

*u(x,t)*. Let us assume that the groove position coincides with the maximum [cos(

*qx*)=1] or the minimum [cos(

*qx*)=-1] of the periodic function. Then the electric field

*E*at an angle

*θ*and at infinite distance can be calculated as the sum of the contribution of

*N*grooves at the maxima and the other

*N*grooves at the minima, the latter being shifted by

*Λ*/2 from the maxima. Hence the electric field

*E*is written as

*n*is the refractive index of the dispersion medium and

*λ*

_{0}is the wavelength in a vacuum. The equation (7) can be explained as follows. The last factor, [sin(

*Nδ*/2)/sin(

*δ/2*)], is a familiar textbook formula [13] for a field by a linear array of

*N*grooves. The term exp(-

*µ*+

*iφ*) represents the contribution from

*N*grooves at the density maxima. The other term, exp(

*µ-iφ*)·exp(

*iδ*/2), is the contribution from the other

*N*grooves at the minima, which are displaced from the maxima by

*Λ/2*, corresponding to the phase factor exp(

*iδ/2*). A graphical representation of (7) with a typical set of parameters is shown in Fig. 4. Increases in intensity are found at

*δ/2*=

*π*(first diffraction order) and

*δ/2*=3

*π*(third diffraction order), in accordance with the increases in

*φ*, as described above for Fig. 1.

*δ*=2

*π*, so

*eiδ*

^{/2}=-1 and the last term of Eq. (7) becomes

*N*. Therefore, the intensity

*I*for the first diffraction order is simplified to

*µ*=

*µ*exp(

_{0}*-q*),

^{2}Dt*φ*=

*φ*exp(-

_{0}*q*), where

^{2}Dt*µ*

_{0}and

*φ*

_{0}are the initial values for the absorption and phase, respectively, we finally obtain

*I*derived in Eq. (10) shows the dominant decay of exp(-

*2q*). The value

^{2}Dt*F*in Eq. (11) is the factor of deviation from dominant decay, and has a dependence on time (time being multiplied by

*q*) as shown in Fig. 5. It is approximately unity if

^{2}D*µ*and

_{0}*φ*are small, and approaches unity exactly when

_{0}*q*≥2.

^{2}Dt## 4. Experiment and results

*t*=0. The results for seven similar silica particles of different diameters ranging from 5 nm to 80 nm are plotted. In the experiments four traces have been recorded for each sample to estimate the reproducibility of measurements.

*D*values are obtained. Then the diameters

*d*of the particles are calculated using Eq. (3) assuming

*T*=298 K and

*η*=0.89x10

^{-3}Pa·s.

## 5. Discussion

*D*. Here DLS is based on the so-called intensity auto-correlation function [5], which includes the factor

*τ*is the time difference and

*K*is a value called the modulus of scattering vector, which is given by

*n*is the refractive index of the dispersion medium,

*λ*

_{0}is the wavelength, and

*θ*is the angle of observation used in scattering experiments. The decay factor (12) appearing in DLS is formally similar to exp(-

_{s}*2q*) in the decay function Eq. (1) of the proposed method.

^{2}Dt*2q*and

^{2}D*2K*. For example, under typical conditions,

^{2}D*d*=10 nm,

*T*=298 K,

*η*=0.89x10

^{-3}Pa·s,

*Λ*=40 µm,

*λ*=0.6 µm (wavelength of DLS apparatus), and

*n*=1.33; using Eq. (2), (3) and (13), these values are calculated as

*K*

^{2}

*D*for DLS is ten thousand times larger than 2

*q*; therefore, DLS requires a fast data collection system with micro-second resolution, while an inexpensive detection system with millisecond resolution is sufficient for the new method.

^{2}D11. H. Watarai, T. Sakamoto, and S. Tsukahara, “In situ measurement of dielectrophoretic mobility of single polystyrene microparticles,” Langmuir **13**, 2417–2420 (1997). [CrossRef]

*u(x)*has a particle density that is opposite that in Fig. 3; i.e., the polarity of

*µ(x)*and

*φ(x)*are inverted. However, because Eq. (9) and Eq. (10) are unchanged even if

*µ*or

*φ*is replaced by (-

*µ*) or (-

*φ*), the signals of raw traces always increase as in Fig. 5 regardless of the type of DEP. The drawback of using DEP for particle analysis is the influence of the electric conductivity of the medium of dispersion. If the medium has significant conductivity, the strength of the electric field decreases and the DEP force that moves the particles also decreases. The resulting low modulation of the grating leads to low sensitivity. Development of a countermeasure against this conductivity effect would thus be an important subject for the future improvement of the system.

12. M. Terajima, “Translational diffusion of intermediate species in solutions,” Res. Chem. Intermed. **23**, 853–901 (1997). [CrossRef]

14. D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation Spectroscopy: The method of cumulants,” J. Chem. Phys. **57**, 4814–4820 (1972). [CrossRef]

17. E. R. Pike and B. McNally, “Theory and design of photon correlation and light-scattering experiments,” Appl. Opt. **36**, 7531–7538 (1997). [CrossRef]

*ρ*and the volume of a particle by

*V*, the particle volume fraction can be expressed by

*ρV*. Then the delay in the phase

*φ (=2πn/λ*is proportional to

_{0})*ρV*for particles sufficiently small compared to the wavelength [2]. The diffracted light intensity from a grating is, therefore, proportional to (

*ρV*)

^{2}from Eq. (9) for a non-absorptive solution. In the case of scattering measurements (DLS), on the other hand, the scattering intensity for a particle is known to be proportional to

*V*

^{2}for small particles. Hence, a solution including

*ρ*particles should exhibit a signal intensity proportional to

*ρV*

^{2}because the phases of light scattered by many particles are independent of each other [2]. Here let us estimate the error of measurement for a typical case where sample particles 10 nm in diameter are contaminated by a small amount of larger particles 100 nm in diameter. Assume that the particle volume fraction (

*ρV*) for the sample particles is 10

^{-3}(i.e., 0.1%) and that for the larger particles (

*ρ’V’*) is 10

^{-6}. If

*V*and

*V’*represent the volumes of the sample particle and contaminant respectively, then

*ρ’V’/ρV*=10

^{-3}and

*V’/V*=10

^{3}. In this condition a comparison of errors can be estimated. The signal

*S*(for the grating method) and the signal

_{G}*S*(for the scattering method) can be written using constants

_{S}*C*

_{1}and

*C*

_{2}as

## 6. Conclusion

## References and Links

1. | ISO Reference, |

2. | M. Kerker, |

3. | H. G. Barth and R. B. Flippen, “Particle size analysis,” Anal. Chem. |

4. | C. F. Bohren and D. R. Huffman, |

5. | ISO Reference, |

6. | B. J. Berne and R. Pecora, |

7. | C. S. Johnson Jr. and D. A. Gabriel, |

8. | H. A. Pohl, |

9. | J. Voldman, R. A. Braff, M. Toner, M. L. Gray, and M. A. Schmidt, “Holding forces single-particle dielectrophoretic traps,” Biophys. J. |

10. | M. Washizu, S. Suzuki, O. Kurosawa, T. Nishizaka, and T. Shinohara, “Molecular dielectrophoresis of biopolymers,” IEEE Trans. Ind. Appl. |

11. | H. Watarai, T. Sakamoto, and S. Tsukahara, “In situ measurement of dielectrophoretic mobility of single polystyrene microparticles,” Langmuir |

12. | M. Terajima, “Translational diffusion of intermediate species in solutions,” Res. Chem. Intermed. |

13. |
For example, see
E. Hecht, |

14. | D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation Spectroscopy: The method of cumulants,” J. Chem. Phys. |

15. | R. Finsy, N. De Jaeger, R. Sneyers, and E. Gelade, “Particle sizing by Photon Correlation Spectroscopy. Part III: Mono and bimodal distributions and data analysis,” Part. Part. Syst. Charact. |

16. | R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation Spectroscopy. Part IV: Resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. |

17. | E. R. Pike and B. McNally, “Theory and design of photon correlation and light-scattering experiments,” Appl. Opt. |

**OCIS Codes**

(290.1990) Scattering : Diffusion

(350.4990) Other areas of optics : Particles

**ToC Category:**

Scattering

**History**

Original Manuscript: January 27, 2006

Revised Manuscript: May 12, 2006

Manuscript Accepted: May 16, 2006

Published: June 12, 2006

**Virtual Issues**

Vol. 1, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Yukihisa Wada, Shinichiro Totoki, Masayuki Watanabe, Naoji Moriya, Yoshio Tsunazawa, and Haruo Shimaoka, "Nanoparticle size analysis with relaxation of induced grating by dielectrophoresis," Opt. Express **14**, 5755-5764 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-12-5755

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### References

- ISO Reference, Particle size analysis -Laser diffraction methods-Part 1, ISO 13320-1 (1999).
- M. Kerker, The scattering of light, (Academic, New York, 1969), pp. 31-39.
- H. G. Barth and R. B. Flippen, "Particle size analysis," Anal. Chem. 67, 257R-272R (1995). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).
- ISO Reference, Particle size analysis - Photon correlation spectroscopy, ISO 13321 (1996).
- B. J. Berne and R. Pecora, Dynamic Light Scattering With Application to Chemistry, Biology, and Physics (General Publishing Company, Toronto, 1976).
- C. S. Johnson, Jr. and D. A. Gabriel, Laser Light Scattering, (Dover, New York, 1994).
- H. A. Pohl, Dielectrophoresis, (Cambridge University Press, 1978).
- J. Voldman, R. A. Braff, M. Toner, M. L. Gray, and M. A. Schmidt, "Holding forces single-particle dielectrophoretic traps," Biophys. J. 80, 531-541 (2001). [CrossRef] [PubMed]
- M. Washizu, S. Suzuki, O. Kurosawa, T. Nishizaka, T. Shinohara, "Molecular dielectrophoresis of biopolymers," IEEE Trans. Ind. Appl. 30, 835-843 (1994). [CrossRef]
- H. Watarai, T. Sakamoto, and S. Tsukahara, "In situ measurement of dielectrophoretic mobility of single polystyrene microparticles," Langmuir 13, 2417-2420 (1997). [CrossRef]
- M. Terajima, "Translational diffusion of intermediate species in solutions," Res. Chem. Intermed. 23, 853-901 (1997). [CrossRef]
- For example, see E. Hecht, Optics fourth edition, (Addison Wesley, San Francisco 2002) pp 449-457.
- D. E. Koppel, "Analysis of macromolecular polydispersity in intensity correlation Spectroscopy: The method of cumulants," J. Chem. Phys. 57, 4814-4820 (1972). [CrossRef]
- R. Finsy, N. De Jaeger, R. Sneyers, and E. Gelade, "Particle sizing by Photon Correlation Spectroscopy. Part III: Mono and bimodal distributions and data analysis," Part. Part. Syst. Charact. 9, 125-137 (1992). [CrossRef]
- R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, "Particle sizing by photon correlation Spectroscopy. Part IV: Resolution of bimodals and comparison with other particle sizing methods," Part. Part. Syst. Charact. 10, 118-128 (1993). [CrossRef]
- E. R. Pike and B. McNally, "Theory and design of photon correlation and light-scattering experiments, "Appl. Opt. 36, 7531-7538 (1997). [CrossRef]

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