The Lattice Model of Particle Transport through Nano-Channels
| dc.contributor.advisor | Bekele, Mulugeta (PhD) | |
| dc.contributor.author | Elfagd, Yitagesu | |
| dc.date.accessioned | 2018-07-10T06:26:20Z | |
| dc.date.accessioned | 2023-11-09T11:25:15Z | |
| dc.date.available | 2018-07-10T06:26:20Z | |
| dc.date.available | 2023-11-09T11:25:15Z | |
| dc.date.issued | 2014-06 | |
| dc.description.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 | en_US |
| dc.identifier.uri | http://10.90.10.223:4000/handle/123456789/7470 | |
| dc.language.iso | en | en_US |
| dc.publisher | Addis Ababa University | en_US |
| dc.subject | Transport through Nano-Channels | en_US |
| dc.title | The Lattice Model of Particle Transport through Nano-Channels | en_US |
| dc.type | Thesis | en_US |