## Mechanism of Separation Mode

Classical chromatographic separation occurs because of the differences in partition of compounds between mobile and stationary phases. More generally, separation is the result of local differences in distribution of the sample compounds in the mobile phase. The partition coefficient K is related to the thermodynamic relationship: ln K = AG°/RT, which indicates the possible effects of three factors: enthalpy (AH) or entropy (AS) changes, and temperature T (Kelvin). Practically, separation is achieved under the effect of two forces (or fields) operating in one or two phases. One phase is necessary for transport and may have a physical or chemical role in the separation.

In HDC, separation results under the effect of one hydrodynamic field which is moving one mobile phase. The nature of the mobile phase is theoretically irrelevant, but differences in results obtained with different solvents have been observed. The separation is due to the existence of a flow-velocity profile in the channel, in which small particles tend to be closer to the external wall, where the flow is stagnant. If a packing is present, its only role is to decrease the capillary size. Voids between beads (diameter $p) in packed columns play the role of small channels of continuously variable diameter similar to a set of capillaries. Bird proposed the equation:

where dP and dL are increments of pressure and column length, R is the column radius, r is the particle radius and ¿u is the eluent viscosity. It is easy to see that u(0) is a maximum when u(R) is zero. The average fluid velocity, u, is:

which is half that of the maximum. A particle in the fluid is assumed to have the same velocity as the flow, in its gravity centre, and moving from the column to a distance R- r of the wall:

R~rr dr

Taking into account the definition of RF, we derive:

An additional term expresses the motion, so that the velocity profile is:

up -MSMR

rotational

where e is the ratio of interstitial volume to total column volume, i.e. about 0.35-0.40, so that R is around ®p/5-6. In consequence, the mechanism of HDC in packed or capillary columns may be described by the same parallel capillary model.

The velocity profile u(r) is parabolic, obeying Poiseuille's equation:

where y is a wall effect parameter, the value of which depends upon the radial position of the particle (from about 1 to 60). In capillary HDC RF must be independent of length L, but the coefficients of the equation are quite far from the theoretical values. This expresses the fact that RF is higher than expected and that particles move far from the wall.

For small rp/R (about 0.1), RF is a linear function of rp/R. In fact, the corresponding curve tends rapidly to a plateau value. This means that this equation is valid for one particle and is only the result of the hy-drodynamic effect. If allowance is made for nonzero stagnant volume, another term is required to complete the above equation. It includes the fraction K of stagnant volume available for polymer and VS and Vm, stagnant and mobile volumes, respectively:

The concentration profile of particles, Cp(r), resulting from Brownian diffusion and colloidal interactions, must also be taken into account:

where:

Table 4 Resolution of various fractionation methods

after a sufficient diffusion in the column. KB is the Boltzmann constant and $(r) an energy term depending on packing and particle interactions. $ is the resulting sum of repulsive double-layer and attractive van der Waals forces (Born repulsive forces are of negligible effect). A graphical representation of the overall profile of particle concentration is shown in Figure 2B. Van der Waals forces depend on the Hamaker constant and double-layer forces depend on latex surface potential and dielectric constant of particles and packing. The result is:

Table 4 Resolution of various fractionation methods

Parameter |
Capillary |
HDC |
DCP |
TFFF |
SFFF |
SFC |

HDC | ||||||

Rs |
0.15 |
0.15 |
1.5 |
0.7-2 |
2.2 |
1.2 |

Rt |
0.025 |
0.025 |
0.15 |
0.2-0.02 |
0.07 |
0.03 |

t(min) |
6 |
6 |
10 |
45-120 |
30 |
20-60 |

^ fam) |
> 1 |
< 1 |
< 60 |
<1 |
<2 |
0.05 |

Reprinted with permission from Revillon A (1994) Journal of Liquid Chromatography 17: 2991-3023. CHDC, Capillary HDC; DCP, centrifugation under disc; TFFF, Thermal FFF; SFFF, steric FFF; SFC, supercritical fluid chromatography.

The exponential term may be corrected by another one accounting for the particle migration under iner-tial hydrodynamic force and electrokinetic lift effects. Many experiments have been carried out to fit elution (volume and peak width) results and equations, by adjusting these values. Listing of the results is more relevant to colloid chemistry than HDC.

Another effect must be considered, described as early as 1836 when Poiseuille observed a corpuscle-free region near the wall in blood vessels. More precisely, Taylor observed an uneven distribution of erythrocytes in flowing blood: there was low concentration not only near to the wall but also near to the centre, provided the velocity was large enough. Further studies concluded that radial forces tend to carry a rigid sphere to an equilibrium position at approximatively 0.6R, depending on the velocity and on the ratio r/R. This was called the tubular pinch effect. Experiments were done with large spheres (0.16-0.85 mm radius in a tube of radius 5.6 mm, at a velocity from 5 to 90 cm s_1 and viscosity from 17 to 410 cP. Ploehn assumed that lateral migration of small particles is primarily due to diffusion, while large particles are focused by the inertial force at an equilibrium position, as observed in the tubular pinch experiment. This experiment may be responsible for separation in the capillary column (see above).

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