ABSTRACT
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716
A. Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716
B. Diverse Spatial and Temporal Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
C. Discrete-Particles and Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
D. Cross-Scale Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
E. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
II. Discrete-Particles: Algorithms and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
A. Off-Grid and On-Grid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
B. Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
C. Dissipative Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730
D. Fluid Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
E. Multilevel Discrete-Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736
F. Smoothed Particle Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738
G. Thermodynamically Consistent DPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740
III. Computational Aspects of Discrete-Particle Simulations . . . . . . . . . . . . . . . . . . . . . 742
A. Calculation of Interparticle Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
B. Temporal Evolution of Fluid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746
1. Verlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746
2. Leap-Frog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746
3. DPD-VV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747
4. SC-TH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747
C. Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
D. Clustering Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
IV. Colloidal Dynamics Modeled by Discrete-Particles . . . . . . . . . . . . . . . . . . . . . . . . 752
A. Rayleigh-Taylor Instability in Atomistic and Mesoscopic Fluids . . . . . . . . . . 752
B. Thin Film Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
C. Phase Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
D. Colloidal Arrays, Aggregates, and Dispersion Processes . . . . . . . . . . . . . . . . . 759
E. Discrete-Particle Model of Blood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
V. A Concept of Problem-Solving Environment for Discrete-Particle Simulations . . . . 769
VI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
Mesoscopic features embedded within macroscopic phenomena in colloids and suspensions, when
coupled together with microstructural dynamics and boundary singularities, produce complex
multiresolutional patterns, which are difficult to capture with the continuum model using partial
differential equations, i.e., the Navier-Stokes equation and the Cahn-Hillard equation. The con-
tinuum model must be augmented with discretized microscopic models, such as molecular
dynamics (MD), in order to provide an effective solver across the diverse scales with different
physics. The high degree of spatial and temporal disparities of this approach makes it a computa-
tionally demanding task. In this survey we present the off-grid discrete-particles methods, which
can be applied in modeling cross-scale properties of complex fluids. We can view the cross-
scale endeavor characteristic of a multiresolutional homogeneous particle model, as a manifestation
of the interactions present in the discrete particle model, which allow them to produce the micro-
scopic and macroscopic modes in the mesoscopic scale. First, we describe a discrete-particle
models in which the following spatio-temporal scales are obtained by subsequent coarse-graining
of hierarchical systems consisting of atoms, molecules, fluid particles, and moving mesh nodes. We
then show some examples of 2D and 3D modeling of the Rayleigh-Taylor mixing, phase separ-
ation, colloidal arrays, colloidal dynamics in the mesoscale, and blood flow in microscopic
vessels. The modeled multiresolutional patterns look amazingly similar to those found in laboratory
experiments and can mimic a single micelle, colloidal crystals, large-scale colloidal aggregates up
to scales of hydrodynamic instabilities, and the macroscopic phenomenon involving the clustering
of red blood cells in capillaries. We can summarize the computationally homogeneous discrete par-
ticle model in the following hierarchical scheme: nonequilibrium molecular dynamics (NEMD),
dissipative particle dynamics (DPD), fluid particle model (FPM), smoothed particle hydrodynamics
(SPH), and thermodynamically consistent DPD. An idea of a powerful toolkit over the GRID can be
formed from these discrete particle schemes to model successfully multiple-scale phenomena such
as biological vascular and mesoscopic porous-media systems.
