Surfaces, charged cylindrical

FIG. 8 Salt exclusion as a function of surface charge in a cylindrical pore in equilibrium with a 0.1 molar electrolyte. The open circles are GCMC results for 1 1 RPM electrolyte in a pore ofR = 5d The circles with a centered cross are results for a 2 1 electrolyte in a pore of = 5d. The up-trian-gles are results for a 2 1 electrolyte in a pore ofR = lOd. The solid circles are results for a 1 1 SPM model with 0.3 solvent packing fraction in a pore of = 5d. The solid squares are the same results for a pore of R = Id. 

FIG. 10 Nomalized unbalanced surface charge in a cylindrical pore with R = 5din the presence of an external potential The results, from left to right, are for original surface charge densities of —0.001, —0.005, —0.01, —0.02, —0.04, —0.05, —0.07123 C/m respectively. The x-intercepts are values of the corresponding equilibrium Donnan potentials. 

Fig. 6.3. The calculated reduced surface potential = e(ri)/kT versus the logarithm of the amphiphile concentration C (M) with no salt added for a spherical, cylindrical and planar aggregate. The surface charge density has been chosen as fixed at a8 = 0.228 Cm- 2. The radii of the sphere and the cylinder are 1.8 nm...

The net charge density a on the chain cylindrical surface is the difference between the positive and negative surface charge densities... 

W.R. Bowen and A.O. Sharif, The hydrodynamic and electrostatic interactions on the approach and entry of a charged spherical particle to a charged cylindrical pore in a charged planar surface—with implications for membrane separation processes, Proc. R. Soc. Lond. A 452 (1996) 2121-2140. 

For the arbitrary potential case, we use the relationship between the surface charge density a and the surface potential i/ o derived in Chapter 1. In the following, we consider the cases of a plate-like particle, a spherical particle, and a cylindrical particle [3],... 

Consider a dilute suspension of parallel cyhndrical particles of radius a with a surface charge density a or total surface charge Q = 2naa per unit length in a salt-free medium containing only counterions. We assume that each cylinder is surrounded by a cylindrical free volume of radius R, within which counterions are distributed so that electroneutrality as a whole is satisfied. We define the particle volume fraction 6 as... 

Consider the electrostatic interaction between two parallel dissimilar cylindrical soft particles 1 and 2. We denote by di and d2 the thicknesses of the surface charge layers of cylinders 1 and 2, respectively. Let the radius of the core of soft cylinder 1 be fli and that for soft cylinder 2 be U2- We imagine that each surface layer is uniformly charged. Let Z and N, respectively, be the valence and the density of fixed-charge layer of cylinder 1, and Z2 and N2 for cylinder 2. 

The radial potential distribution inside the capillary, (r), is then obtained by solving the Poisson-Boltzmann equation for cylindrical symmetry (30). The resulting potential depends on a single adjustable constant which is fixed by the boundary condition on the potential which relates the p gential gradient at r=l/2Dp to the surface charge density, J c. Then we define... 

Chiral phospholipid molecules aggregate spontaneously to form tubes with diameters of 500 nm and lengths of 50-100 pm. Diacetylenic phosphatidylcholine structures were first coated in 1993 by Baral and Schoen [72] with silica nanoparticles. The tubule dispersion was mixed with Ludox (a silica sol with a particle diameter of 10-15 nm and negative surface charge at pH 8.2) and allowed to stand for up to 9 days, during which time a white precipitate formed. TEM analysis of the collected precipitate showed a film with a thickness of about 50 nm, composed of silica particles, on the hollow cylindrical templates. The adsorption of the nanoparticles to the headgroups of the phospholipid is believed... 

It is essential that the inphase component of the field at the low-frequency spectrum coincides with the inphase component of the field in a uniform medium with conductivity ff2- A similar result is obtained when the source is the vertical magnetic dipole. It means that surface charges, arising at interfaces between the formation and the surrounding medium at the low-frequency spectrum, influence the quadrature component of the magnetic field only. Respectively, at the late stage of the transient response in the same way as in a medium with cylindrical interfaces, the field hx does not depend on the orientation of the magnetic dipole. 

It is desired to estimate the electrophoretic velocity U of a long, nonconducting, charged cylindrical particle of length L and radius a and with a low surface potential I, as a result of the application of an electric field parallel to the symmetry axis. The Debye length is arbitrary but finite, and the flow is a low Reynolds number, inertia free one. 

Equation (9.1) shows that P is the dipole moment per unit volume, a vector quantity that has its direction parallel to Ei for an isotropie medium. Imagine a small cylindrical volume of length d/ parallel to the polarisation and of cross-sectional area dA. Let the apparent surface charges at the two ends of the cylinder be dq. It then follows that P = dq dl/dv, where do is the volume of the small cylinder. However, do = dl dA, so that P = dq/dA = a, where a is the apparent surface charge per unit area normal to the polarisation. (Note a stands for conductivity in section 9.3.)... 

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