Page 198 - Book Hosokawa Nanoparticle Technology Handbook

FUNDAMENTALS CH. 3 CHARACTERISTICS AND BEHAVIOR OF NANOPARTICLES AND ITS DISPERSION SYSTEMS
force with time correlation. This scheme is termed as dispersion system of fine particles with non-negligible
generalized Langevin dynamics, and applied for study- degree of Brownian motion. New development of
ing behavior of ions in aqueous solution [17]. For the methodology for the above difficulty would be desper-
purpose to determine the surface forces with complex ately desired, some of which will be briefly reviewed in
molecules, the above extent of preciseness seems not the next section.
necessary. On the other hand, this literature gives an
important and useful finding: By definition the PMF
must have concentration dependence, but use of that 3.8.3 Recent simulation methods including
from infinite dilution, which is usually the case, can hydrodynamic interaction
give fair degree of prediction even for concentrated
systems. Much difficulty exits, as explained above, in simulat-
ing concentrated dispersion of submicron particles.
(iii) Brownian dynamics (Overdumped Langevin) Further difficulty would add if the system is subjected
to a flow field. This kind of system must be, however,
With the surface force obtained from the LD, one
would be able to conduct the BD. The inertia term can one of the most important dispersion operations
be neglected for particles with diameters ranging applied for producing functional materials by nanopar-
from tens to several hundreds of nanometers, result- ticles. Thus a new approach beyond conventional ones
ing in the simple basic equation shown in Fig. 3.8.3, would be desired. A typical method of particulate
which is sometimes called as the overdumped dynamics that includes HI is firstly the Stokesian
Langevin equation. The random force and friction dynamics [1], as mentioned in the beginning of this
term are the same as those for the LD. In this case the chapter, which does not include fluid explicitly, based
velocity can be directly determined by the force bal- upon the idea of Ermak, but expresses its effect as a
ance, and the algorithm becomes quite simple. One resistance tensor including relative velocities of all the
should, however, note that time increment for integra- pair of particles. This method suffers from complexity
tion t must be set as an intermediate value that com- in coding, instability in computation and high compu-
promises both of the following two conditions: long tational cost, which would limits its applicability to the
enough to smear out momentary thermal motion of large-scale of concentrated dispersion. In the follow-
particles; short enough to allow only a small dis- ing some of recent approaches will be briefly
placement for particle movement so as to treat surface described.
force as a constant. (i) Dissipative particle dynamics
This method is applicable to relatively dilute sys-
tems such as the electrostatically stabilized colloidal One of the recent methods attracting attention would
dispersion. An example is seen in literature, which be the dissipative particle dynamics (DPD) proposed
studied adsorption of colloidal nanoparticles onto a by Hoogerbrugge and Koelman [19]. The details are
substrate [18]. given by literature [20, 21], but this approach does not
remove solvent contrary to the case with the BD and
LD. Fluid is treated as composed of many coarse-
(iv) Brownian dynamics with hydrodynamic interaction (BD
grained mesoscopic “particles” each of which means
with HI) a mass of fluid molecules. Both of the fluid and solid
In a concentrated dispersion, the motion of a particle “particles” are subjected to the random force and the
will affect those of other particles mediated by sol- friction attenuation. The existence of the fluid makes
vent, which is termed as the hydrodynamic interaction it possible to treat the HI honestly with its multi-body
(HI). Ermak et al. [2] quite beautifully established its nature, which is in principle impossible in the case of
basic equations, but the long-ranged and multi-body BD. Also, the computational load is linear with the N
nature of the HI, coupled with the existence of the number of particles
random force, makes its equations and the algorithm Since the principle comes from the fluctuation–
far more complicated than those for the BD and SD dissipation theorem, the basic equation for the DPD is
[2, 10]. The HI must be expressed as a matrix of 3N x the same as the LD. This point is of much interest,
3N for a three-dimensional system with N-particles, because much similarity in computational scheme
which needs large memory area, and whose computa- must exists between the two methods, which leads to
2
tional time must be proportional to N . Further the a possibility to establish a new systematized compu-
existence of the HI brings correlation between ran- tational platform that can handle broad range of scales
dom forces of particles, resulting in a huge computa- from nano to micron, which may be called as a “gen-
tional cost for generating special type of random eralized mesoscale simulation”. At present, however,
numbers, each of which relates to all other particles’ the DPD seeks general guiding principles for, e.g., the
random displacements. size of the coarse-grained fluid, time increment for
The difficulty, therefore, rises in the case where one integration, and boundary condition for surface of
has to conduct a large-scale simulation for concentrated colloidal particles. If these points are clarified, the

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