Diffusion, longitudinal

The jamming effect, i.e., the slowing down of the longitudinal diffusion of a polymer chain by the head-on collision with other chains, can be treated by a model similar to that proposed by Cohen and Turnbull [112] for self diffusion of small molecules in a fluid. This model assumes that if at least one surrounding polymer chain exists within the critical hole ahead of a test chain, both collide, and this prevents the test chain from diffusing longitudinally. With this assumption, we express the longitudinal diffusion coefficient Dp of the test chain as... 

The plate theory assumes that an instantaneous equilibrium is set up for the solute between the stationary and mobile phases, and it does not consider the effects of diffusional effects on column performance. The rate theory avoids the assumption of an instantaneous equilibrium and addresses the diffusional factors that contribute to band broadening in the column, namely, eddy diffusion, longitudinal diffusion, and resistance to mass transfer in the stationary phase and the mobile phase. The experimental conditions required to obtain the most efficient system can be determined by constructing a van Deemter plot. 

The major source of band-broadening in CZE is longitudinal diffusion. Longitudinal diffusion refers to the axial diffusive spreading of the solute from the solute zone into the bulk solution as it travels down the capillary. The variance in peak width contributed by longitudinal diffusion is given by... 

Optimum flow rates are achieved at the lowest value for H. The terms A, B and C are constants relating to eddy diffusion, longitudinal diffusion and to mass transfer of the analyte. 

Skoog and West describe three causes of zone broadening eddy diffusion, longitudinal diffusion, and nonequilibrium mass transfer [1], The Van Deemter equation was developed to relate the flow rate and plate height ... 

Considering a chromatographic process controlled by a partition equilibrium and neglecting extracolumn effects (i.e., band broadening caused by factors outside the column, e.g., tubings, detector etc.), several factors can contribute to the overall solute band broadening eddy diffusion, longitudinal diffusion, and resistance to mass transfer in mobile and stationary phase. 

Schematics illustrating the contributions to band broadening due to (a) multiple paths, (b) longitudinal diffusion, and (c) mass transfer.

To determine how the height of a theoretical plate can be decreased, it is necessary to understand the experimental factors contributing to the broadening of a solute s chromatographic band. Several theoretical treatments of band broadening have been proposed. We will consider one approach in which the height of a theoretical plate is determined by four contributions multiple paths, longitudinal diffusion, mass transfer in the stationary phase, and mass transfer in the mobile phase. 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

Kovat s retention index (p. 575) liquid-solid adsorption chromatography (p. 590) longitudinal diffusion (p. 560) loop injector (p. 584) mass spectrum (p. 571) mass transfer (p. 561) micellar electrokinetic capillary chromatography (p. 606) micelle (p. 606) mobile phase (p. 546) normal-phase chromatography (p. 580) on-column injection (p. 568) open tubular column (p. 564) packed column (p. 564) peak capacity (p. 554)... 

Fig. 8. Combined flow reactor models (a) parallel flow reactors with longitudinal diffusion (diffusivities can differ), (b) internal recycle—cross-flow reactor (the recycle can be in either direction), comprising two countercurrent plug-flow reactors with intercormecting distributed flows, (c) plug-flow and weU-mixed reactors in series, and (d) 2ero-interniixing model, in which plug-flow reactors are parallel and a distribution of residence times dupHcates that...

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

Diffusion plays an important part in peak dispersion. It not only contributes to dispersion directly (i.e., longitudinal diffusion), but also plays a part in the dispersion that results from solute transfer between the two phases. Consider the situation depicted in Figure 4, where a sample of solute is introduced in plane (A), plane (A) having unit cros-sectional area. Solute will diffuse according to Fick s law in both directions ( x) and, at a point (x) from the sample point, according to Ficks law, the mass of solute transported across unit area in unit time (mx) will be given by... 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

Thus, treating the diffusion process in a similar way to that shown in Figure 4 the total variance due to longitudinal diffusion in a column of length (1) is given by equation (7), viz.,... 

Equation (11) accurately describes longitudinal diffusion in a capillary column where there is no impediment to the flow from particles of packing. In a packed column, however, the mobile phase swirls around the particles. This tends to increase the effective diffusivity of the solute. Van Deemter introduced a constant (y) to account... 

Then, the contribution to the total variance per unit length for the column from longitudinal diffusion in the stationary phase will be... 

In summary, equation (13) accurately describes longitudinal dispersion in the stationary phase of capillary columns, but it will only be significant compared with other dispersion mechanisms in LC capillary columns, should they ever become generally practical and available. Dispersion due to longitudinal diffusion in the stationary phase in packed columns is not significant due to the discontinuous nature of the stationary phase and, compared to other dispersion processes, can be ignored in practice. 

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