Diffusion in the Stationary Phase

Theoretically, dispersion can take place by diffusion in the stationary phase but, as will be seen, in practice, is much less in magnitude than that in the mobile phase. The theoretical treatment is similar to that for dispersion in the mobile phase using equation (10). 

Marcel Dekker, Inc. 270 Madison Avenue, New York, New York 10016 

However, in this case, over the time period (tr), a fraction of time spent by the solute in the stationary phase and, thus, 

(Ho(s)), the contribution to the variance per unit length, will be 

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

---

In a packed column, however, the situation is quite different and more complicated. Only point contact is made between particles and, consequently, the film of stationary phase is largely discontinuous. It follows that, as solute transfer between particles can only take place at the points of contact, diffusion will be severely impeded. In practice the throttling effect of the limited contact area between particles renders the dispersion due to diffusion in the stationary phase insignificant. This is true even in packed LC columns where the solute diffusivity in both phases are of the same order of magnitude. The negligible effect of dispersion due to diffusion in the stationary phase is also supported by experimental evidence which will be included later in the chapter. 

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. 

In the above derivation of the mass balance equation, we assume that the column is radially homogeneous, the compressibility of the mobile phase is negligible, the axial dispersion coefficient is constant, and the temperature is unchanged. Furthermore, no diffusion in the stationary phase is assumed. 

In accordance with Habgood and Hanlan (13) the ratio of D-/K was conveniently replaced by e D.. For a non-adsorbing gas, Iike helium, the value of K is fowJeso that KC/(1-e) can be neglected (14). Further, accounting for the fact that molecular diffusivi-ties in the mobile phase are generally much larger than effective diffusivities in the stationary phase leads to the final form s... 

For those cases in which longitudinal diffusion in the stationary phase is also significant, a plate height term of the following form [1] should be added... 

FIGURE 24-2 Comparison of curves for plate height against velocity for the individual terms (upper) and for the overall value Oower) of Equation (24-14). Left, the values for a liquid mobile phase right, values for a gaseous mobile phase (liquid stationary phases in both cases). The numbered curves represent eddy diffusion (1), molecular diffusion in the mobile phase (2), and resistance to mass transfer in the stationary (4) and mobile (5) phases. The contribution of the term for molecular diffusion in the stationary phase is negligible at velocities near the optimum for both liquid and gas systems. 

For the case in which all mass transfer resistance is due to diffusion in the stationary phase and the stationary phase is uniformly distributed on the surface of a uniform spherical packing, the constant C is related to the solute diffusivity (22) by... 

The polymer is deposited as a uniform annular coating in a glass capillary column. A solute is injected into an inert carrier gas that flows through the column. The elution curve of the sample is then used with a model to determine the solute activity and diffusivity in the stationary phase. A detailed description of the equipment and the experimental procedure is given by Pawlisch (361. It is of value to present the model used to describe the process. The description provided by Pawlisch (361, given below, indicates how the model was developed. 

Depending on the column configurations (packed, capillary, etc.) several formulations of Eq. (30) have been suggested. In the present case column parameters must be designed so as to magnify the effects of slow diffusion in the stationary phase. This is quite easily achieved with polymer stationary phases since their diffusion coefficients are usually smaller by two orders of magnitude than those of low molecular weight liquids. It should also be noted that measurements must be performed under equilibrium conditions, i.e., at temperatures in excess of Tg + 50°. 

Changes in the concentration of the solute in the stationary phase occur due to transfer of the solute across the phase interface and longitudinal diffusion in the stationary phase. Thus, it may be written... 

Occjuionally the measured B term has been much larger than antidpated from an estimation of the diffusion codfident This can be explained by a contribution from the diffusion in the stationary phase, whkh cannot always be excluded. In this case, we need to take the time spent in the stationary plum into account as well, and a more complicated picture emerges than the one painted above. In general, the nature of the diffusion term has not been studied extensively in HPLC, since in most cases it has been ne gible in practical... 

D J is the solute Diffusivity in the mobile phase, (D ) is the solute Diffusivity in the stationary phase. 

---