Infinitely Wide Channels

For an uncharged polymer in a good solution or a flexible polyelectrolyte chain in an aqueous solution with moderate amount of salt, the size exponent V is 3/5. For such cases, the radial extent of the confined chain in a channel geometry and the confinement free energy are given by 

We have mapped the translocation process to a nucleation and growth phenomenon, characterized by a free energy barrier. The principal source of this free energy barrier is conformational entropy of the polymer. Additional contributions to the overall free energy landscape arise from solvent conditions, polymer-pore interactions, geometry of the pore, and the electrochemical potential gradient driving the whole process. 

We have explored these factors analytically by considering a number of idealized situations such as a structureless hole, a narrow pore, and a wide channel. 

The initial step of placing one chain end at the pore entrance is the primary source of the entropic barrier for polymer translocation. The typical value of this entropic barrier is about IksT, which is within the range of energy associated with the hydrolysis of one ATP molecule. The free energy barrier for polymer entrance at the pore is further modulated by the specific interaction between the chain end and the pore. 

When the pore or channel is wide, the polymer undergoes non-single-file translocation. For such geometries, simple scaling formulas are derived for the confinement free energy and the spatial extent of the polymer, as functions of chain length, pore diameter or channel thickness, and the size exponent for the 

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For relatively wide channels with negligible electrical double-layer overlap (r/8 > 10), a nearly flat flow profile is expected. It has often been stated that when the channel size and the Debye length are of similar dimensions (r 8), complete electrical double-layer overlap occurs and the EOF is negligible. However, when r 8, a significant EOF can still be created the EOF velocity in the central part of the channel is approximately 20% of that in an infinitely wide channel. Only at conditions where r/8 1 is the EOF fully inhibited by double-layer overlap [25], It should be noted here that the approximations made by using the Rice and Whitehead theory at r/8 < 10 may lead to significant errors in the calculation of the velocity distribution and magnitude of the EOF [17] compared to more sophisticated models. 

Figure 5.1 Sketches of geometries for polymer translocation (a) a tiny hole in a thin planar membrane, (b) two spherical cavities connected by a bridge, (c) a cylindrical pore, and (d) an infinitely wide channel.

With the development of modern computation techniques, more and more numerical simulations occur in the literature to predict the velocity profiles, pressure distribution, and the temperature distribution inside the extruder. Rotem and Shinnar [31] obtained numerical solutions for one-dimensional isothermal power law fluid flows. Griffith [25], Zamodits and Pearson [32], and Fenner [26] derived numerical solutions for two-dimensional fully developed, nonisothermal, and non-Newtonian flow in an infinitely wide rectangular screw channel. Karwe and Jaluria [33] completed a numerical solution for non-Newtonian fluids in a curved channel. The characteristic curves of the screw and residence time distributions were obtained. 

It was noted that the velocity in a channel approaching a weir might be so badly distributed as to require a value of 1.3 to 2.2 for the kinetic energy correction factor. In unobstructed uniform channels, however, the velocity distribution not only is more uniform but is readily amenable to theoretical analysis. Vanonil has demonstrated that the Prandtl universal logarithmic velocity distribution law for pipes also applies to a two-dimensional open channel, i.e., one that is infinitely wide. This equation may be written... 

For the rectangular channels the diffusional deposition problem was solved only for an infinitely wide duct. Real channels have a finite ratio of the height to the width, bh/bw. If it is 1, the formula allows one to use the flow rate rather than the flow velocity. The general form of the solution for such channels with... 

The problem in a vertical plane is formulated for a wide channel, ideally for an infinitely wide one, of a constant depth assuming that the flow pattern is approximately the same in each vertical plane along the flow direction, as it has been depicted in Fig. 1.5. A usual vertical velocity distribution that follows formula (1.1) takes place only above the vegetation submerged in the water flow. This distribution becomes complex and uncommon within the vegetation layer where a significant motion of the water still takes place. A number of experimental data given in [69, 172, 347, 370, 617] confirms... 

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