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Kisliuk adsorption
In other instances, molecular interactions between gas molecules previously adsorbed on a solid surface form significant interactions with gas molecules in the gaseous phase. Hence, adsorption of gas molecules to the surface is more likely to occur around gas molecules that are already present on the solid surface, rendering the Langmuir adsorption isotherm ineffective for the purposes of modelling. This effect was studied in a system where nitrogen was the adsorbate and tungsten was the adsorbent by Paul Kisliuk (b. 1922-d. 2008) in 1957. To compensate for the increased probability of adsorption occurring around molecules present on the substrate surface, Kisliuk developed the precursor state theory, whereby molecules would enter a precursor state at the interface between the solid adsorbent and adsorbate in the gaseous phase. From here, adsorbate molecules would either adsorb to the adsorbant or desorb into the gaseous phase. The probability of adsorption occurring from the precursor state is dependent on the adsorbate’s proximity to other adsorbate molecules that have already been adsorbed. If the adsorbate molecule in the precursor state is in close proximity to an adsorbate molecule which has already formed on the surface, it has a sticking probability reflected by the size of the SE constant and will either be adsorbed from the precursor state at a rate of kEC or will desorb into the gaseous phase at a rate of kES. If an adsorbate molecule enters the precursor state at a location that is remote from any other previously adsorbed adsorbate molecules, the sticking probability is reflected by the size of the SD constant.

These factors were included as part of a single constant termed a “sticking coefficient,” kE, described below:

kE =

As SD is dictated by factors that are taken into account by the Langmuir model, SD can be assumed to be the adsorption rate constant. However, the rate constant for the Kisliuk model (R’) is different to that of the Langmuir model, as R’ is used to represent the impact of diffusion on monolayer formation and is proportional to the square root of the system’s diffusion coefficient. The Kisliuk adsorption isotherm is written as follows, where  is fractional coverage of the adsorbent with adsorbate, and t is immersion time:

Solving for  yields:

Henderson-Kisliuk
This adsorption isotherm was developed for use with the new field of Self Assembling Monolayer (SAM) adsorption. SAM molecules adsorb to the surface of an adsorbent until the surface becomes saturated with the SAM molecules’ hydrocarbon chains lying flat against the adsorbate. This is termed “lying down” structure (1st structure). Further adsorption then occurs, causing the hydrocarbon chains to be displaced by thiol groups present on the newly adsorbed SAM molecules. When this adsorption step takes place, electrostatic forces between the newly adsorbed SAM molecules and the ones previously adsorbed, causes a new structure to form, where all of the SAM molecules are occupying a “standing up” orientation (2nd structure). As further adsorption takes place, the entire adsorbent becomes saturated with SAM in a standing up orientation, and no further adsorption takes place.

The SAM adsorbate is usually present in a liquid phase and the adsorbent is normally a solid. Hence, intermolecular interactions are significant and the Kisliuk adsorption isotherm applies. The sequential evolution of “lying down” and “standing up” mercaptopropionic acid (MPA) SAM structures on a gold adsorbent, from a liquid MPA-ethanol adsorbate phase, was studied by Andrew P. Henderson (b. 1982) et. al. in 2009. Henderson et al used electrochemical impedance spectroscopy to quantify adsorption and witnessed that the 1st structure had different impedance properties to the 2nd structure and that both structures evolved sequentially. This allowed four rules to be expressed:

1.	That the amount of adsorbate on the adsorbent surface was equal to the sum of the adsorbate occupying 1st structure and 2nd structure.

2.	The rate of 1st structure formation is dependent on the availability of potential adsorption sites and intermolecular interactions.

3.	The amount of 1st structure is depleted as 2nd structure is formed.

4.	The rate of second structure formation is dictated by the amount of adsorbate occupying 1st structure and intermolecular interactions at immersion time, t.

From these statements, Henderson et al used separate terms to describe rate of fractional adsorption for 1st structure [1(t)] and 2nd structure [2(t)] as a function of immersion time (t). Both of these terms were dictated by the Kisliuk adsorption isotherm, where constants with a subscript of 1 relate to 1st structure formation and a subscript of 2 relates to 2nd structure formation.

These terms were combined in the Henderson adsorption isotherm, which determines the total normalised impedance detection signal strength caused by the adsorbate monolayer (zt) as a function of 1(t), 2(t), 1 and 2. Values of  are weighting constants, which are normalized signal values that would result from an adsorbent covered entirely with either 1st structure or 2nd structure. This isotherm equation is shown below:

zt = 1(t).[1.(1-2(t)) + 2.2(t)]

Although the Henderson-Kisliuk adsorption isotherm was originally applied to SAM adsorption, Henderson et. al. hypothesised that this adsorption isotherm is potentially applicable to many other cases of adsorption and that 1(t) and 2(t) can be calculated using other adsorption isotherms, in place of the Kisliuk model (such as the Langmuir adsorption isotherm equation).