# The Lattice Model of Particle Transport through Nano-Channels

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## Date

2014-06

## Authors

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## Publisher

Addis Ababa University

## Abstract

When membranes channels are very narrow so that their average diameters
are comparable to the average size of the permeating particles, then two particles
can’t pass each other inside the channel. These molecular-sized channels
are usually referred to as nano-channels. The research work presented
in this thesis deals with the modeling of nano-channels, description of their
transport properties and the channel gating mechanisms. As particles permeate
through nano-channels, they interact with multiple weakly attractive and
repulsive sites on the wall of the nano-channel, where the attractive sites act as
potential wells and the repulsive sites act as potential barriers. In this thesis,
the potential energy landscape of the internal wall of the entire length of the
nano-channel is modeled as a spatial distribution of potential wells and potential
barriers. Moreover, we considered the potential wells as the lattice sites of
a one dimensional lattice with s lattice sites and particles permeating through
nano-channels spend more time in the vicinity of these potential well sites. We
assumed that between these potential wells, there are particle repelling sites or
potential barriers that must be hopped by the permeating particles. Thus, the
height of the potential barrier between the potential wells or lattice sites controls
the rate of particle transport through nano-channels. The phenomenon of
particle transport or diffusion through nano-channels is modeled as hopping
of particles between nearest neighbor potential wells across the intervening potential
barriers.
The occupation numbers that denote the lattice sites in the emitter and collector
baths are e and c and the channel sites are i, where i = 1, 2, ..., s with
s denoting the total number of lattice sites. The average value of these internal
and external occupation numbers lies in the range between zero and one. In
this work, we employ the discrete or lattice model of particle transport through
nano-channels, which employs theMaster equation formulation of the rate equations
that describe the time rate of change of the average values of the occupation
numbers associated with each lattice site. The effects of the potential
barriers between the potential wells are taken into account via the forward and
backward transition rates. These transition rates describe the rate of forward
and backward hopping of a particle sitting at a given potential well across the
intervening potential barriers.
In the first chapter,we have proposed a simple lattice model of nano-channels
and simple expressions for the total rate at which particle are injected into and
out of the nano-channel at the two end sites. In chapter two, starting from
the Master equation of the rate equations and employing the steady state assumption
and the mean filed approximation of the effective transition rates
for both the internal and external sites, we have derived a general expression
of the average particle flux through a nano-channel at steady state. The expression
of the average particle flux is a multi variable function. That is, J =
J(ne, nc, e, c,
e,
c, 1, ..., s,
1, ...,
s, k+
e , k−
c , k−
1 , ..., k−
s , k+
1 , ..., k+
s ), where
i = 1−
i. The nano-channel is characterized by the channel parameters 1, ..., s,
1, ...,
s,
k−
1 , ..., k−
s , k+
1 , ..., k+
s . The effects of the emitter bath is taken into account via
the bath parameters ne, e,
e, k+
e and that of the collector bath via the bath parameters
nc, c,
c, k−
c . In chapter three and four, we have presented a detailed
description of the properties of a nano-channel with one and two internal potential
wells, respectively.
Our results show that the average particle flux through a nano-channel as
a function of n, for the specified values of the other parameters, is a rapidly
increasing function of n for small and intermediate values. When n = 0,
the average flux has a minimum value and as n increases to large values, the
average particle flux through a nano-channel with both one and two internal
potential wells increases slowly and finally enters into the saturation region in
which it no longer increase as the concentration gradient increases. For most
cases, the range of the concentration gradient in which the average particle flux
becomes saturated lies between 1.0 M and 10 M for the specified values of the
other parameters. Moreover, the average particle flux approaches more rapidly
xxxv
to its saturation value when the external potential well Ve is occupied and the
external potential well Vc is empty. The average particle flux as a function of n,
through a nano-channel with a single internal potential well is greater than the
average particle flux through a nano-channel with two internal potential wells,
that is J1( n| e, c) > J2( n| e, c).
Moreover, the average particle flux through a nano-channel, with one and
two internal potential wells, as a function of the external occupation number
e, for the specified values of the other parameters, is a rapidly increasing function
of e for small values of e, usually e 0.2 and after that it becomes saturated.
When e = 0, the optimized particle flux assumes a minimum value and
the nano-channel is in the inactivated state. When e = 1 the nano-channel
becomes fully activated and as a result the average flux approaches a maximum
positive value. Thus, e acts as a gating variable that controls the direction and
magnitude of particle transport through a nano-channel. Moreover, the average
particle flux as a function of c, for the specified values of the other parameters,
approaches to a maximum positive value as c ! 0 and decreases as c increases
and finally as c ! 1, the flux decreases to a small negative value. When c = 0,
the average particle flux has a maximum positive value and the nano-channel
is in the fully activated state and when c = 1 the nano-channel becomes fully
inactivated and as a result the flux approaches a small negative value. Therefore,
c also acts as a gating variable that controls the magnitude and direction
of particle transport through a nano-channel.
In this thesis, we have also shown that the average values of the occupation
numbers of the internal potential wells as a function of n, for the specified
values of the other parameters, are rapidly increasing functions of n for small
and intermediate values. When n ! 0, the average occupation numbers have
small positive values and as n increases to large values, the average occupation
numbers of the internal potential wells of a nano-channel with both one
and two internal potential wells increase slowly and finally approach the saturation
region in which they no longer increase as the concentration gradient increases.
For most cases, the range of the concentration gradient in which these
average occupation numbers become saturated lies between 1.0 M and 10 M
for the specified values of the other parameters. Moreover, the average occupaxxxvi
tion numbers approach more rapidly to the saturation value when the external
potentialwells Ve and Vc are occupied. For a nano-channel with two internal potential
wells, the average occupation number as functions of n of the internal
potential well V1 is greater than the average occupation number of the internal
potential well V2, that is 12( n| e, c) > 22( n| e, c).
The average value of the occupation number of the internal potential well
V1 increases more rapidly as e increases and slowly as c increases. Similarly,
the average value of the occupation number of the internal potential well V2 increases
more rapidly as c increases and slowly as e increases. As the average
value of the occupation number of V1 increases, the average value of the occupation
number of V2 also increases, and vice versa. The average value of the
occupation number of V1 increases as the values of the normalized potential
barriers u−
1 , u+1 and u+2 increase and as u+e , u−
2 and u−c decrease. Similarly, the
average value of the occupation number of V2 increases as the values of the normalized
potential barriers u−
2 , u+2 and u−
1 increase and as u+e , u+1 and u−c decrease

## Description

## Keywords

Transport through Nano-Channels