Mobility expression correctiveness

III. Mobility Expression Correct to Order (Henry s Formula). 28... 

III. MOBILITY EXPRESSION CORRECT TO ORDER f (HENRY S FORMULA)... 

Henry s equation (2.6) assumes that is low, in which case the double layer remains spherically symmetrical during electrophoresis. For high zeta potentials, the double layer is no longer spherically symmetrical. This effect is called the relaxation elfect. Henry s equation (2.6) does not take into account the relaxation effect, and thus this equation is correct to the first order of Ohshima et al. [19] derived an accurate analytic mobility expression correct to order 1/ka in a symmetrical electrolyte of valence z and bulk concentration (number density) n with the relative error less than 1% for 10 < Ka < 00, which is... 

An approximate analytic mobility equation apphcable for arbitrary values of was daived by Levich [28], and Ohshima et al. [30] derived a more accurate mobility expression correct to order 1 /ka for the case of symmetrical electrolytes of valence z. The leading term of their expression is given by [30]... 

Boundary effects on the electrophoretic migration of a particle with ion cloud of arbitrary thickness were also investigated by Zydney [46] for the case of a spherical particle of radius a in a concentric spherical cavity of radius d. Based on Henry s [19] method, a semi-analytic solution has been developed for the particle mobility, which is valid for all double layer thicknesses and all particle/pore sizes. Two integrals in the mobility expression must be evaluated numerically to obtain the particle velocity except for the case of infinite Ka. The first-order correction to the electrophoretic mobility is 0(A3) for thin double layer, whereas it becomes 0(A) for thick double layer. Here the parameter A is the ratio of the particle-to-cavity radii. The boundary effect becomes more significant because the fluid velocity decays as r l when the double layer spans the entire cavity. The stronger A dependence of the first order correction for thick double layer than that obtained by Ennis and Andersion [45] results from the fact that the double layers overlap in... 

For a cylindrical particle oriented at an arbitrary angle between its axis and the applied electric field, its electrophoretic mobility averaged over a random distribution of orientation is given by pav = /i///3 + 2fjLj3 [9]. The above expressions for the electrophoretic mobility are correct to the first order of ( so that these equations are applicable only when C is low. The readers should be referred to Ref. [4-8, 10-13, 22, 27, 29] for the case of particles with arbitrary zeta potential, in which case the relaxation effects (i.e., the effects of the deformation of the electrical double layer around particles) become appreciable. 

Equation (2.16) is applicable for all for 10 < ka < oo. To obtain an approximate mobility expression applicable for ka < 10, it is convenient to express the mobility in powers of and make corrections to higher powers of in Henry s mobility equation (2.6), which is correct to the first power of Ohshima [25] derived a mobility formula for a spherical particle of radius a in a symmetrical electrolyte solution of valence z and bulk (number) concentration n under an applied electric field. The drag coefficient of cations, A+, and that of anions, A, may be different. The result is... 

Concentrations of moderator at or above that which causes the surface of a stationary phase to be completely covered can only govern the interactions that take place in the mobile phase. It follows that retention can be modified by using different mixtures of solvents as the mobile phase, or in GC by using mixed stationary phases. The theory behind solute retention by mixed stationary phases was first examined by Purnell and, at the time, his discoveries were met with considerable criticism and disbelief. Purnell et al. [5], Laub and Purnell [6] and Laub [7], examined the effect of mixed phases on solute retention and concluded that, for a wide range of binary mixtures, the corrected retention volume of a solute was linearly related to the volume fraction of either one of the two phases. This was quite an unexpected relationship, as at that time it was tentatively (although not rationally) assumed that the retention volume would be some form of the exponent of the stationary phase composition. It was also found that certain mixtures did not obey this rule and these will be discussed later. In terms of an expression for solute retention, the results of Purnell and his co-workers can be given as follows,... 

The mole fraction of the monomer units that are cross-linked in the polymer is X,., and nt is Ihe number-average number of atoms in the polymer backbone between cross-links. The temperature should be expressed in absolute degrees in this equation. The constant K is predicted to be between 1.0 and 1.2 it is a function of the ratio of segmental mobilities of cross-linked to uncross-linked polymer units and the relative cohesive energy densities of cross-linked and uncross-linked polymer (88). The theoretical equation is probably fairly good, but accurate tests of it are difficult because of the uncertainty in making the correction for the copolymer effect and because of errors in determining nf. 

In the traditional interpretation of the Fangevin equation for a constrained system, the overall drift velocity is insensitive to the presence or absence of hard components of the random forces, since these components are instantaneously canceled in the underlying ODF by constraint forces. This insensitivity to the presence of hard forces is obtained, however, only if both the projected divergence of the mobility and the force bias are retained in the expression for the drift velocity. The drift velocity for a kinetic interpretation of a constrained Langevin equation does not contain a force bias, and does depend on statistical properties of the hard random force components. Both Fixman and Hinch nominally considered the traditional interpretation of the Langevin equation for the Cartesian bead coordinates as a limit of an ordinary differential equation. Both authors, however, neglected the possible existence of a bias in the Cartesian random forces. As a result, both obtained a drift velocity that (after correcting the error in Fixman s expression for the pseudoforce) is actually the appropriate expression for a kinetic interpretation. 

It can be clearly seen from equation (9) that the expression for the retention volume of a solute, although generally correct, is grossly over simplified if accurate measurements of retention volumes are required Some of the stationary phase may not be chromatographically available and not all the pore contents have the same composition as the mobile phase and, therefore, being static, can act as a second stationary phase. This situation is akin to the original reverse phase system of Martin and Synge where a dispersive solvent was absorbed Into the pores of support to provide a liquid/liquid system. As a consequence a more accurate form of the retention equation would be,... 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

Every equation must be dimensionally correct. The dimensions on the right-hand side must be the same as those on the left-hand side. Suppose we know that the conductivity, a, of a material may be expressed as a = nq/i, where n is the number of carriers per volume, q is the charge per carrier (coulombs/carrier), and //. is the mobility, but we do not know the units of mobility. If conductivity, a, is expressed in (ohm m)-1, the units of mobility must be [drift velocity, (m/s), divided by the voltage gradient, (V/m). 

We can express the experimentally measured (corrected) mobility in relation to the mobility of the free species A as a function of the varying concentration of the ligand, B, and obtain... 

The transposition of the equivalent height of the mobile packed bed (H J) to a normal working unit is carried out through a correction function, which is applied to a bed height corresponding to a MWPB operated with air and water. If we consider that the major contributions to the correction function expression are given... 

Here and t] are, respectively, the relative permittivity and the viscosity of the electrolyte solution. This formula, however, is the correct limiting mobility equation for very large particles and is valid irrespective of the shape of the particle provided that the dimension of the particle is much larger than the Debye length 1/k (where k is the Debye-Htickel parameter, defined by Eq. (1.8)) and thus the particle surface can be considered to be locally planar. For a sphere with radius a, this condition is expressed by Ka l. In the opposite limiting case of very small spheres (Ka 3> 1), the mobility-zeta potential relationship is given by Hiickel s equation [2],... 

The net retention volume V is equal to the retention volume minus the column dead space, V — where both V and are corrected for the pressure-gradient correction factor (see below). If the noncolumn contributions to the dead space are insignificant relative to the total dead space, and if the sample size is small enough so that the velocity of the mobile phase within the band is not increased by the solute in the mobile phase, then the net retention volume is equal to kV, and the partition ratio is (V — F )/F . The relative retention for two solutes is the ratio of their net retention volumes (V2 — or Fj/Fj this is usually expressed by a... 

---