Boundary conditions, cylindrical pore

Simulations of water in synthetic and biological membranes are often performed by modeling the pore as an approximately cylindrical tube of infinite length (thus employing periodic boundary conditions in one direction only). Such a system contains one (curved) interface between the aqueous phase and the pore surface. If the entrance region of the channel is important, or if the pore is to be simulated in equilibrium with a bulk-like phase, a scheme like the one in Fig. 2 can be used. In such a system there are two planar interfaces (with a hole representing the channel entrance) in addition to the curved interface of interest. Periodic boundary conditions can be applied again in all three directions of space. 

FIG. 3 Setup of simulation cell of confined electrolyte with periodic boundary conditions, (a) Electrolyte bound by two infinitely long charged plates, representing a slit pore, (b) Electrolyte in a cylindrical nanopore. 

Strong preference to block copolymer. A cylindrical pore with a size of Rex x Lz was used in the simulation, where Rex is the exterior radius of pore and Lz = 50 is the length of pore. The periodic boundary condition was applied in the axial direction ab = 0.5/cgT is the interaction energy between A and B segments, while as = — bs =... 

Further, Imdakm and Matsuura [61] have developed a Monte Carlo simulation model to smdy vapor permeation through membrane pores in association with DCMD, where a three-dimensional network of interconnected cylindrical pores with a pore size distribution represents the porous membrane. The network has 12 nodes (sites) in every direction plus boundary condition sites (feed and permeate). The pore length / is assumed to be of constant length (1.0 p,m), however, it could have any value evaluated experimentally or theoretically [62]. 

The unit cell and coordinates are illustrated in Figure 10.14(b). As with the array of microdiscs model, the unit cell is cylindrically symmetrical about an axis that passes through the centre of the pore, perpendicular to the electrode surface. The problem may thus be reduced from a three-dimensional one to a two-dimensional one. As with the microdisc electrode, this is a two-dimensional cylindrical polar coordinate system, and Pick s second law in this space is given by Eq. (9.6). The simulation space for the unit cell with its attendant boundary conditions is shown in Figure 10.15. 

The generalization of the Helmholtz-Smoluchowski description of the EOF for cylindrical pores (Figure 8.15a) is carried out as follows. Let a and L be the radius and length of a uniform cylindrical pore with surface potential. Let the electric field be along the x-direction and r be the radial distance from the axis of the cylinder in the yz-plane orthogonal to the x-axis. As in the case for the planar surfaces, the EOF velocity is calculated from Equation 8.81 by implementing the appropriate boundary conditions. 

I have tried to force an organization on these examples as follows. In Section 6.4.1, I have discussed the simplest empirical methods of organizing experimental results. In Section 6.4.2, I have reviewed theories for solute diffusion in a solvent trapped within cylindrical pores in an impermeable solid. In this case, solute-solvent interactions still control diffusion and the solid only imposes boundary conditions. Cases where the interactions are between the diffusion solute and the pores boundaries are covered in Section 6.4.3. Finally, cases not of cylindrical pores but of other composite structures are described in Section 6.4.4. 

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