Confinement and Interaction Effects on the Diffusion of Passive Particles - Selected Topics of Computational and Experimental Fluid Mechanics - page 370

1 Introduction
Diffusion processes of molecules and particles are very common in nature. Breathing,
human metabolism, motion of viruses and bacteria, medical drug delivery, are only
some examples. The fundamental mechanism behind these processes was elucidated
for the first time in 1905 when a theoretical framework for difussion was proposed by
Einstein ( 1905 ). Since then, classical work concerning diffusion of non-interacting
particles includes the study of isotropic particles in the absence (Einstein 1905 ;
Chandrasekhar 1943 ; Batchelor 1977 ) and presence of external fields (Ferrari 1990 ;
Zagorodny and Holod 2000 ; Foister and Ven 1980 ; Jimenez-Aquino et al. 2008 ).
Additionally, anisotropic particles diffusing in the absence (Han et al. 2006 ; Hinch
and Leal 1972 ) and presence of external fields (Grima and Yaliraki 2007 )havealso
been considered. More recently, and based on the previous works, the impact of
thermal agitation on non-interacting active particles (driven by an assumed internal
mechanism) has received attention. For example, steadily-swimming self-propelled
bodies of simple shape, one sphere (Howse et al. 2007 ; Hagen et al. 2011 ; Sandoval
et al. 2014 ), multiple spheres (Lobaskin et al. 2008 ), or ellipsoids (Hagen et al. 2011 )
have been studied.
Another interesting aspect about diffusive processes is the effect of confinement
on particle displacement. In nature and in many technological applications, parti-
cles (ions, molecules, photons) generally move under the presence of boundaries,
like through ionic channels (Alberts et al. 2007 ), membranes (Hille 2001 ), artificial
nanopores (Siwy et al. 2005 ; Healy et al. 2007 ) porus media (Daniel and Astruc
2004 ) and carbon nanotubes (Berezhkovskii and Hummer 2002 ). As it can be seen,
confinement is mainly due to physical walls (although hydrodynamic confinement is
also possible (Alar Ainla and Jesorka 2012 )) hence the need of including wall effects
on particle diffusion. A theoretical framework that includes wall effects on particle
diffusion was achieved by Zwanzig ( 1992 ). Based on the idea that physical walls can
be seen as entropic potentials,and that effective diffusion coefficients depend on posi-
tion, he derived for the first time, a Smoluchowski equation for a confined particle.
By solving this equation, an effective analytical diffusion coefficient that includes
the influence of the walls can then be obtained. Recent work concerning confined,
non-interacting particles is given by Reguera and Rubi ( 2001 ), Reguera et al. ( 2006 ),
Kalinay and Percus ( 2005 , 2008 ). Confined active non-interacting particles have also
been computationally studied (Ghosh et al. 2013 ). A theoretical analysis of active
non-interacting confined particles has just recently been developed ( Sandoval and
Dagdug 2014 ).
The latter works do not consider particle-particle interactions, hence in this
research we study confined particles where interactions among elements of the
system (particle-particle and wall-particle) are allowed. These interactions in their
own provide to the system richer thermodynamic properties (like phase transitions).
Moreover, the effect of confinement on the physical properties of geometrically con-
strained fluids is not yet well understood. It has been shown that confinement deviates
a traditional bulk fluid phase transition (Evans 1990 ; Klein and Kumacheva 1998 ;