Page 150 - Book Hosokawa Nanoparticle Technology Handbook

FUNDAMENTALS CH. 3 CHARACTERISTICS AND BEHAVIOR OF NANOPARTICLES AND ITS DISPERSION SYSTEMS
References regarded as a continuum and the following convective
diffusion equation is solved:
[1] M. Fuji: Ceram. Soc. Jpn. Symp. Abst., 107 (2004).
[2] S. Tsukahara, T. Sakamoto and H. Watarai: Langmuir, C F
2
16 (8), 3866–3872 (2000). u C D C ext C (3.3.5)
t f
3.3 Brownian diffusion where C is the particle concentration, u the gas veloc-
ity, F ext the external force acting on particles.
Particles with a diameter smaller than 1 m exhibit Equation (3.3.3) cannot be used to predict the
irregular and random motion because their masses are microscopic structure of film formed by the particle
small enough to render fluctuation by the bombard- deposition. For obtaining the structure of particle-
ments of gas molecules. As a result of random motion, accumulated layer, discrete or stochastic model is
particles as a whole move toward to a low concentra- employed with the aid of computer simulation.
tion region from a high concentration region. This phe- Rosner et al. [1] introduced Diffusion Limited
nomenon, which is similar to gas molecules, is Aggregation model to simulate growth process of par-
referred to as Brownian diffusion of particles. The dif- ticle layer and studied the effects of Peclet number
fusion coefficient of particles both in liquid and air is and the mean free path of particles on the porosity,
given by the following Stokes–Einstein equation: thickness and surface roughness of the particle layer.
When an external force, F , acts on a particle, the
ext
equation of motion of a particle is given by the fol-
kT CkT
D c (3.3.1) lowing equation:
f 3 D p
dv fv () F ()
m F t t (3.3.6)
where f is the Stokes’ drag coefficient given by equa- dt D ext
tion (3.3.2), k the Boltzmann constant, T the tempera-
ture, the viscosity of fluid, and D the particle
p
diameter. where m is the mass of a particle, v the particle veloc-
ity, t the time, F is the fluctuating force acting on a
D
3 D particle by the bombardment of gas molecules.
f p (3.3.2) Equation (3.3.6) is integrated step by step with small
C
c time increments to obtain particle trajectories start-
ing at arbitrary positions. The direction of fluctuating
C is the Cunningham’s slip correction factor, which force acting on a particle at each time step is given by
c
is equal to unity for particles in a liquid and given by generating random numbers which follow Gaussian
equation (3.3.3) for particles in a gas. distribution with zero mean and the standard devia-
tion equal to 2D t for one-dimensional particle
⎛ ⎛ 11⎞ ⎞ diffusion.
.
C Kn 1 257 0 4exp ⎜ ⎝ Kn⎠ ⎠ (3.3.3) Equation (3.3.6) is based on the following two
⎟⎟
.
⎜
1
.
c
⎝
assumptions:
In the above equation, Kn is the Knudsen number 1. F is independent of particle velocity and F D
D
which is the ratio of mean free path of gas molecules averaged over many particles is equal to zero.
to particle radius, D /2. 2. Since F fluctuates in a very short time period,
p
D
the time scale of particle motion is considerably
2
Kn (3.3.4) large compared to the time scale of particle fluc-
D p tuation. Consequently, the particle velocity is
constant over a short time period of t and there
The mean free path of gas molecules is 0.065 m for is no correlation between F ’s at the start and
D
air at the normal temperature and pressure. As seen end of the time interval.
form equation (3.3.1), Brownian diffusion is more
significant for smaller particles and therefore the These assumptions do not seem to hold for practical
motion of nanoparitcles are governed by it. problems, but is verified through molecular kinetic
When a thin film is formed on a substrate by the theory [2]. These assumptions hold when t is
deposition of nanoparticles synthesized in gas phase, significantly large compared to the relaxation time
2
Brownian diffusion mainly determines the structure of particle, ( C D /18 ) and the particle’s
p
c p
and growth rate of the film. In order to predict the displacement during t is smaller than the mean free
deposition flux of particles, particle–gas system is path of particles.
126