IGC Isotherms (IGC-FC) and AEDF

A full analysis of IGC desorption curves and isotherms can be performed via the method of Balard¹.

This simplified version captures the essence of the issue, but is deliberately vague about units. The main point is to slide the E2 slider to show the differences between Lo, Med and Hi absorption energies and to create complex Adsorption Energy Distributions Functions (AEDF) by changing E1:H1, E2:H2 and E3:H3 to create a 3-peak approximation to a complex surface.

The steps

The three curves are based on the fundamental curve generated from standard isotherm theory showing the distribution function, χ based on E which is based on the central value ε_c and the actual site energy, ε: `E=(ε_c-ε)/(RT)`:

`χ.RT = e^E/(1+e^E)^2`

The AEDF is just the normalized sum of the three peaks generated from three ε_c values and their relative heights. Note that the fundamental theory shows that these peaks have a fixed width, independent of ε_c. The normalization is for a constant total peak area, so as you move from one peak to two or three, the maximum of any peak is reduced.

The isotherm is then calculated via integration for each value of φ, which is P/P0 (P0 is the saturated vapour pressure) of the height, χ, of the AEDF function and an energy value, `E=e^[(ε-E_L)/(RT)]` which is corrected for the so-called Lateral energy, E_L input as E_lat. The BET assumption that it is at least a factor of 10 smaller than the absorption energy so it is automatically limited to 1/10 of whichever is the largest peak in the AEDF. The calculation is for N, the number of molecules adsorbed at each P/P0.

`N = Σ{(χEφ)/[(1-φ)(1-φ+Eφ)]}`

As a bonus, the Henry portion of the BET curve is plotted based on the linearized first few points of the isotherm.

For IGC isotherms the P/P0 plot typically has a maximum around 0.1 to 0.3 so start with the default P_max of 0.2.

The BET graph is simply these data re-plotted as `P/(P0)` versus `φ/[N(1-φ)]`

Finally, the desorption curve is plotted on the basis that the retention time, t is simply the derivative of the isotherm. This takes some getting used to!

What is striking about this reverse process (which is far easier than the real process of going from desorption to AEDF!) is that all the "interesting" information is contained in the apparently boring long tail, while the "boring" information is contained in the apparently interesting portion of the desorption curve where the signal drops dramatically. By playing with the sliders, you will start to see what this means. In practice it means that to get good AEDF curves from desorption data you need a very good machine with a very stable baseline and low noise - plus the ability to inject the required saturating amounts in a highly-controlled and reproducible manner.

Heterogeneity

The assumption in standard IGC calculations about surface energy is that there is single absorption mode from which the surface energy is derived. But just a small percent of higher-energy sites is enough (as described below) to give a large increase in the measured surface energy. It is, therefore, important to distinguish between two surfaces that might give the same surface energy. One might have a simple surface giving the measured value. The other might be mostly a surface that would give a lower value, mixed with a small amount of surface with higher energies. A numerical way to distinguish such surfaces is via the Heterogeneity - the ratio of the area under sub-peaks to the total area. If you start with the default system and slowly increase the H3 value you see the higher energy peak appear and see the Heterogeneity increase from 0%.

More information than a BET measurement

A standard BET machine produces nice numbers, but these may have rather little relationship to the real surface. BET relies on a number of assumptions, such as a molecularly smooth surface with probe molecules able to easily interact laterally as well as building a smooth layer on top of the original monolayer. For most powders, these assumptions are invalid, so the BET value is the "apparent specific surface area".

The IGC-FC approach performs an analysis that contains the same basic assumptions wrapped up in the Langmuir isotherm approximation, so the calculated AEDF values will not, in general, be accurate. However, the general shape of the AEDF already contains a lot of information that a BET analysis cannot provide. If you then use probes other than alkanes, you can start to get other AEDF shapes which tell you about, for example, acid-base interactions. For those who care about the nature of the surface, rather than just wanting an apparent specific surface area, IGC-FC is a major step up from standard BET.

Experimental extra feature

As an additional feature, not part of the Balard approach, the app takes seriously the assumption that the elution time of a solute depends exponentially on the energy of the various sites according to:

`t=t_0Σn_ie^(E_i/(RT))`

From the calculated t via a pseudo Vg it is possible to calculate a pseudo ΔG_CH2 from which γ_d is derived. You will find that the exponential effect of some higher energy peaks (captured in the Heterogeneity value) gives a large increase in γ_d for a small increase in the number of high-energy sites.