Page 153 - Book Hosokawa Nanoparticle Technology Handbook

3.5 INTERACTIONS BETWEEN PARTICLES FUNDAMENTALS
cylindrical capillary as r, penetration rate can be spherical particles of diameter D and D p2 is repre-
p1
expressed by the next equation. sented by the following equation:

2
dx r
cos rg Ad
(3.4.1) F (3.5.1)
v
dt 4 x 8 12 z 2
where x is the height of liquid column, t the time,
where
the surface tension of the liquid, the viscosity of the
liquid, the density of the liquid, g the acceleration of d DD p2
p1
gravity and the contact angle. The first term of D D (3.5.2)
right-hand side of equation (3.4.1) is a contribution of p1 p2
capillary suction force, and the second term is a con-
tribution of gravity. When the gravity is negligible, The negative sign in equation (3.5.1) indicates that the
integration of equation (3.4.1) gives the following force is attractive. A is the Hamaker constant [3], z the
equation: separation distance between particles, and d the
reduced particle diameter defined by equation (3.5.2).
r
cos The Hamaker constant depends on the materials
2
x t (3.4.2) (see Table 3.5.1 [4, 5]). For a rough estimation, the
2 values of the constant are in the range of (4–10)
10 20 J for hydrocarbons, (10–15) 10 20 J for metal
The capillary radius r is determined in advance by the oxides and (15–50) 10 20 J for metals. For the inter-
2
slope of x –t plot of powder and well-spreading (con- action between two different materials 1 and 2 in
2
tact angle 0) liquid. Next, by plotting x vs. t for a gases is approximately given by:
particular liquid, contact angle of the powder and the
liquid is calculated. A A A (3.5.3)
Evaluation of wettability is also done by dispersion 12 11 22
test of powder into various liquids or mixed solvent.
Besides aforementioned methods, wettability is also If a medium 3 such as water exists between the sur-
evaluated by analysis of adsorption isotherm, heat of faces of material 1, the Hamaker constant is approxi-
adsorption, or heat of immersion [5]. These methods mated by:
are effective to evaluate wettability of nanoparticles
whose contact angle cannot be measured directly. 131 ( A ) 2
A A 33 (3.5.4)
11
References In the presence of medium 3 between materials 1 and
2, the Hamaker constant is given by:
[1] M. Takahashi, M. Oya, M. Fuji: Adv. Powder Tech.,
15(1), 97–107, (2004). 132 ( A )( A )
A A A (3.5.5)
[2] M. Takahashi, M. Oya and M. Fuji: J. Soc. Powder 11 33 22 33
Technol. Jpn., 40, 410–417 (2003).
[3] M. Fuji, H. Fujimori, T. Takei, T. Watanabe and M.
Chikazawa: J. Phys. Chem. B, 102, 10498–10504, (1998). Table 3.5.1
[4] M. Preiss, H.-J. Butt: J. Colloid Interf. Sci., 208, Hamaker constants [4, 5].
468–477, (1998).
[5] M. Fuji, T. Takei, T. Watanabe and M. Chikazawa: Material Hamaker constant, A (10 -20 J)
Colloid Surf. A, 154, 13–24, (1999). Methanol 3.1
Ethanol 3.9
Water 4.4
3.5 Interactions between particles Polystyrene 6.2–6.6
MgO 10.6
3.5.1 Interactions between particles in gases and Al O 3 15.5
2
Mica 9.5
control of adhesion
Diamond 28.4
3.5.1.1 Van der Waals force and liquid bridge force [1, 2] Graphite 47.0
Cu 28.4
Van der Waals force that is caused by the charge fluc- Ag 40.0
tuation of atoms or molecules acts between all sur- Au 45.5
faces in contact. The interaction force between two
129