Page 157 - Book Hosokawa Nanoparticle Technology Handbook

3.5 INTERACTIONS BETWEEN PARTICLES FUNDAMENTALS
[15] S.P. Timoshenko, J.N. Goodier: Theory of Elasticity Thus, the force working on the particle of the charge
(3rd ed., chapter 12), McGraw-Hill, New York (1970). Q is given as,
[16] K.L. Johnson, K. Kendall and A.D. Roberts: Proc.
R. Soc. Lond., A, 324, 301–313 (1971). f Q E (3.5.20)
[17] Chemical Society of Japan (eds.): Chemical
Handbook (Kagaku-Binran Kisohen) (4th ed.), Even if a particle is neutral, a so-called gradient force
pp. II.26, Maruzen, Tokyo (1993). works on the particle when an applied electric field is
[18] R.G. Horn, J.N. Israelachvili and F. Pribac: J. Colloid. non-uniform on the size scale of the particle. This is a
force due to the non-uniform electric field working on
Interf Sci., 115, 480–492 (1987).
the polarization charge induced by the field. The
[19] B.V. Derjaguin, V.M. Muller and Yu.P. Toporov:
gradient force, f , working on a spherical particle is
J. Colloid. Interf. Sci., 53, 314–326 (1975). g
given by
[20] V.M. Muller, V.S. Yushchenko and B.V. Derjaguin:
J. Colloid. Interf. Sci., 77, 91–101 (1980).
2
[21] V.M. Muller, V.S. Yushchenko and B.V. Derjaguin: f 2 p 0 E , (3.5.21)
3
a
J. Colloid. Interf. Sci., 92, 92–101 (1983). g 0 p 2 0
[22] M. Fuji, K. Machida, T. Takei, T. Watanabe and
M. Chikazawa: Langmuir, 15, 4584–4589 (1999).
where a is the radius of the particle and the relative
[23] Y. Kousaka, Y. Endo and Y. Nishie: Kagaku Kogaku p
dielectric constant of the particle.
Ronbunshu, 18, 942–949 (1992). Note that it is rarely the case that an external electric
[24] A. Fukunishi, Y. Mori: J. Soc. Powder Technol., Jpn., field applied to a nanoscaled particle will be non-
41, 162–168 (2004). uniform on that size scale. However, it can also be
[25] Y.I. Rabinovich, J.J. Adler, M.S. Esayanur, A. Ata, noted that the local electric field becomes non-
R.K. Singh and B.M. Moudgil: Adv. Colloid. Interf. uniform when charged two particles of similar size
Sci., 96, 213–230 (2002). approach each other. In this case, the electric field
generated by each charge on the particles is indeed
[26] M. Fuji: J. Soc. Powder Technol., Jpn., 40, 355–363
non-uniform in the vicinity of the particle, and the
(2003).
non-uniform field generated by one particle results in
[27] Y. Endo, Y. Kousaka and Y. Nishie: Kagaku Kogaku
a gradient force on the other one. When the permittiv-
Ronbunshu, 18, 950–955 (1992).
ity of the particle is high, and if the distance between
two particles is less than the diameter of the particle,
this effect is not negligible compared to the Coulomb
3.5.1.2 Electrostatic interaction
interaction due to the true charge on the particle. The
(a) Coulomb’s law detailed calculation of this effect for two same-sized
The force f [N] working between two particles with spherical particles with opposite charges of the same
charges q [C] and Q [C] is given by Coulomb’s law as absolute value is available in reference [1].
For a spherical particle with uniform charge on its
surface, the electric field outside the particle is equiv-
1 qQ 1 qQ alent to that produced by a point charge located on the
f r r ˆ (3.5.18) center of the particle and with the same charge as the
4 r 3 4 r 2
total surface charge. This equivalence is only a first
order approximation if the charge is distributed
where [F/m] is the permittivity of the medium between unequally on the particle surface. The actual surface
the particles, and rˆ the unit vector co-directional with r. charge distribution on the surface of a nanoparticle is
In air (or any gas phase, in general), the permittivity not understood at this moment.
of free space, 8.854 10 12 F/m, can be used to
0
approximate the permittivity of the medium. (b) Amount of charge on a particle in the gas phase
The superposition principle holds for Coulomb’s A particle suspended in the gas phase gains charge
law. Therefore, when there are multiple point when an ion attaches to its surface. It is assumed that
charges q (i 1 N) around a particle with charge all the ions reaching the surface are captured by the
i
Q, and an external electric field E [V/m] due to particle. Impact charging (field charging) and diffu-
0
boundary conditions, the electric field generated at sion charging are two mechanisms to bring ions to the
the position of the particle, excluding the charge Q particle surface. The former is dominant for bigger
itself, is given as, particles than 1 m, and vice versa.
N
∑ 1 q i (i) Impact (field) charging
E 3 i r E 0 (3.5.19) Suppose a spherical particle with a diameter a and
0
i 1 4 r i with a charge Q is situated in a uniform electric
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