Band broadening theory (Van Deemter equation)

It is well recognized now that column band broadening originates from three main sources:

  1. multiple path of an analyte through the column packing;
  2. molecular diffusion;
  3. effect of mass transfer between phases.

In 1956 J.J. Van Deemter introduced the equation which combined all three sources and represented them as the dependence of the theoretical plate height (HETP) on the mobile phase linear velocity. Originally, it was introduced for gas chromatography, but it happened that the same physical processes occurs in HPLC, and this equation is perfectly fit for liquid chromatography.

Multiple path

The velocity of mobile phase in the column may vary significantly across the column diameter, depending on the particle shape, porosity, and the whole bed structure. This is schematically shown below.

Variation of the zone flows

Band broadening is caused by differing flow velocities through the column, which may be written in form

          (8)

where Hp is the HETP arising from the variation in the zone flow velocity, dp is the particle diameter (average), and l is the constant which is almost close to 1.

This shows that Hp may be reduced (efficiency increased) by reducing the particle diameter (which will lead to the increasing of the column back pressure). Coefficient lambda depends on the particle size distribution. The narrower the distribution, the smaller l  (which actually lead to decreasing of the column backpessure also).

Analyte diffusion

It is well-known that molecules disperse or mix due to the diffusion. The longitudinal diffusion (along the column long axis) leads to the band broadening of the chromatographic zone. This process may be described by equation:

          (9)    
where Dm is the analyte diffusion coefficient in the mobile phase,
g  is the factor which is related to the diffusion restriction by column packing,
v  is the flow velocity.

It is obvious from the above equation that the higher the eluent velocity, the lower the diffusion effect on the band broadening. Molecular diffusion in the liquid phase is about five orders of magnitude lower than that in the gas phase, thus this effect is almost negligible at the standard HPLC flow rates.

Kinetics of the mass transfer

Mass transfer is the most questionable parameter. For the modern types of packing materials it may combine two effects: adsorption kinetics and mass transfer (mainly due to diffusion) inside the particles.

95% of all modern packing materials are the spherical, totally porous rigid particles with average diameter ~5 Ám and pore diameter ~100┼. Ratio of the particle to the pore diameter is 500/1. There is no pressure propelled flow inside the particle, and molecules can move there only by diffusion. It can be shown by analogy: if consider a tunnel in the mountain which has diameter of 2 meters and one kilometer length (same ratio 500/1), and if there is a storm outside with 200 km/h wind, there will be almost no wind in 50 meters from tunnel entrance.

Adsorption kinetics is almost negligible compare to the diffusion inside the particles, and band spreading of the peak may be written in form:

          (10)    
where: dp is the particle diameter,
Dm is the diffusion coefficient of the analyte in the mobile phase,
w   is the coefficient determined by the pore size distribution, shape,
      and also particle size distribution,
v   is the flow velocity.

The above equation describes the linear dependence of HETP on the flow rate. The slower the velocity, the more uniformly analyte molecules may penetrate inside the particle, and the less the effect of different penetration on the efficiency. On the other hand, at the faster flow rates the elution distance between molecule with different penetration depth will be high.

Total band broadening

Each term discussed above introduces its part in the total band broadening, therefore the sum of all of them will give the total column plate height.

	H = Hp + Hd + Hm             (11)    

or the expanded form will be

          (12)    

Experimentally measured dependencies of HETP on the eluent flow rate are shown below

HETP dependencies of (a) aniline, (b) benzene, (c) toluene on the eluent flow rate.

Different components have different dependencies of HETP on the flow rate on the same column. This shows that the components nature, types of the surface interactions and perhaps other parameters have an influence on the column efficiency related to the particular component.

Despite that, we also can notice, that the general dependence of the experimentally measured curves fit well to that described by the above equation. The theoretical graph below highlight the contribution of each terms of the main equation.

The most significant result is that we can find an optimum eluent flow rate where the column efficiency will be the best.