The different flow velocities encountered in LC columns imply that stress is exerted on analyte molecules. To quantify the extent of stress, we can use a somewhat obscure dimensionless number from the vast toolbox of engineers. The Deborah number (\text{De}) can be expressed as
Equation 4.11: \text{De}=k_{\text{pac}} \cdot \frac{u_{\text{int}}}{d_{\text{p}}} \cdot \frac{6.12 \Psi \cdot \eta \cdot r^3_{\text{g}}}{R \cdot T}
Here, k_{\text{pac}} is a packing-structure factor with a typical value between 6 and 9, u_{\text{int}} is the interstitial linear velocity, d_{\text{p}} is the particle size, \Psi is the Flory-Fox parameter (2.5\cdot1023 mol-1 under typical SEC conditions), \eta the viscosity of the mobile phase, r_{\text{g}} the radius of gyration of the analyte molecules, R the gas constant, and T the absolute temperature. If we express u_{\text{int}} in m\cdots-1, d_{\text{p}} in m, \eta in Pa\cdots (equal to N\cdotm-2\cdots), r_{\text{g}} in m, R in J\cdotmol-1\cdotK-1 (equal to N\cdotm\cdotmol-1\cdotK-1) \text{De} is indeed dimensionless. \text{De} numbers below 0.1 are thought to represent negligible stress. When \text{De} > 0.5 deformation (e.g. elongation) of molecules is thought to occur, whereas severe deformation and breakage of bonds may occur when \text{De} > 2. In case of proteins, a high stress exerted on the molecules may result in changes to the secondary and tertiary structures
In terms of reduced velocity we may write
This shows that the \text{De} number (i.e. the stress on the molecules)
- Increases with increased velocity (an obvious conclusion)
- Increases strongly with a decrease in particle size
- Increases strongly with the size of the analyte molecule
Increasing the temperature may reduce the viscous stress slightly, because the absolute temperature varies across a narrow range in LC. If we consider the effect of the analyte molecular weight, we should realize that with increasing M_i the analyte diffusion coefficient decreases, but the solution viscosity increases. To a first approximation we may consider the product \eta \cdot D_{\text{m}} independent of molecular weight. If molecules in solution took the form of hard spheres r^3_{\text{g}} \propto M_i would be appropriate, but for the random coils encountered in practice r^3_{\text{g}} \propto M^2_i is a better approximation.
Figure 1. Calculated SEC Deborah numbers for polystyrene in THF. Numbers in the legend indicate the particle size of the packing material in micrometers. A reduced interstitial velocity of 1000 is assumed, which implies lower flow rates for larger molecules. The pink zones indicates the danger area, where significant stress is exerted on the analyte molecules.
Figure 1 helps estimate when the stress exerted on analyte molecules becomes a source of concern. The figure applies to high velocities (\nu_{\text{int}} = 100). It shows that serious deformation (the pink zone at the top of the figure) only occurs for very large molecules on columns packed with very small particles.
Incidentally, this is the domain of slalom chromatography, where molecular deformation underlies the separation mechanism.
SEE ALSO SECTION 4.2.2.4
Stress Exerted on Analyte Molecules
