0:00

[MUSIC]

Now, in the previous analysis we considered the non-interacting case.

That is, we neglected any interactions between absorbed molecules.

Now, this is an adequate description when the surface coverage is low.

However, when surface coverage becomes high, interaction between

absorbed molecules becomes important and this leads to correlated surface coverage.

Now, this is analogous to the spacial correlations that are found

in the Eisen model.

Let's adopt a simple model for the absorption energy on a lattice.

In this expression, ni stands for the site occupancy.

Now ni can take the value 0 or 1, 0 if it is unoccupied, 1 if it is occupied.

0:52

Now K in this expression describes the coupling constant

to describe the interactions between neighboring adsorbed molecules.

An F bind is the free energy associated with binding of the absorbed molecule.

Now in this system, we need to take into account

the changes in the particular number of absorbed molecules.

Now what do you think?

Which ensemble should we pick?

Now the most convenient ensemble to pick is the grand canonical ensemble.

The grind canonical quotation function can be written in a standard way.

Now rather than carrying out a more detailed analysis of this equation

we will take a simpler approach and cast this in the language of the Eisen model.

What we need to do to do this Well, we need a change of variables where we can

relate the occupancy number NI to the spin variable SI by a linear transformation.

Now the simple mapping, maps the occupancy variable to a spin value of one.

When occupied, and a spin value of minus one when not occupied.

Now, this variable transformation allows us to recast the problem exactly

in the Eisen model.

In this expression, Z stands for

the coordination number of the lattice given by the number of nearest neighbors.

Now we have to remember to not double count, so

we need to divide this number by a factor of two.

So, for a two dimensional square lattice, the Z value takes two.

Therefore, with this exact map mapping of the current problem to the Ising model.

We can now evaluate all of these things in an analogous way to what we found in

the Ising model.

Therefore, we can exactly map our current problem

onto the Ising model with a suitable set of parameters.

The following parameter mapping holds for

the absorption case to the standardizing model.

Owing to the exact mapping between our current and the icing model,

we can note a few identical physical phenomena that arises in both systems.

Adsorption with interactions, exhibit of phase transition.

3:06

Well, this phase transition occurs when the coupling constant is sufficiently

large and more importantly, the lattice is not one dimensional.

Now, adsorption exhibits hysteresis in the surface coverage

versus gas pressure associated with the phase transition.

Now, the surface coverage is not completely uncorrelated.

Rather, the adsorbed molecules exhibit correlated fluctuations

due to favorable interactions.

3:34

And finally, the critical point in the adsorption isotherm is associated with

a divergence of the correlation length of these spatial fluctuations.

Now, to summarize, in this video we learned a lot about surface adsorption.

We discuss two specific cases of surface absorption

without any surface interactions.

Now this is the case of localized absorption and mobile absorption.

We derived expressions for

the surface coverage in these cases under constant temperature.

And then we proceeded to describe a picture for

surface absorption that accounts for interacting aborb molecules.

We showed the equivalence of this problem to a problem model

using a suitable change of variables.

Now this system exhibits a phase transition under sufficiently large

coupling constant and when the lattice is not one dimensional.