Grand canonical partition function of 2d lattice

Question

Assume we put a gas, such as a simple atomic system (Argon), into contact with a two dimensional surface so that this gas acts as a reservoir of particles. Assume that the chemical potential of this reservoir is $\mu$. Furthermore, assume that on this surface there are $M$ lattice sites, arranged on a triangular lattice, which can adsorb molecules from the gas with a binding energy $\epsilon_B$.

Assume that the particles on the surface do not interact at all, i.e. multiple patricles can share a lattice site at no cost.

I would like to find the grand canonical partition function for the 2d surface.

I thought the following:

$$\Xi(\mu, V, T) = \sum^\infty_{N=0}\exp[-\mu \beta N]\binom{M}{N}\exp[-\beta \epsilon_B N]$$ However, I'm not entirely sure this is correct, since there shouldn't be a state in which $N=0$ (no particle is adsorbed), right? Shouldn't this be the grand canonical partition function if two particles CAN'T be on the same site, but all are adsorbed?

Any clarification is very welcome

Answer 1

The state in which $N=0$ is certainly a valid and necessary part of the partition function. While it would have a low Boltzmann weight compared to the occupied states, it contributes nonetheless.

Yes, as written, this sum would terminate once all sites are occupied. However, there isn't a need for a binomial coefficient at all. Since there are no interactions and particles can occupy the same site, the multiplicity of a given state is simply given by $M^N$. There's no difference energetically between one site and another, so each particle is independent of the others and has $M$ options for where to go. So, your partition function becomes

\begin{equation} \Xi(\mu,V,T) = \sum_{N=0}^{\infty} e^{-\beta \mu N} M^N e^{-\beta \epsilon_B N} = \sum_{N=0}^{\infty} (Me^{-\beta (\mu + \epsilon_B)})^N = \frac{1}{1-Me^{-\beta (\mu + \epsilon_B)}}, \end{equation}

provided that $Me^{-\beta (\mu + \epsilon_B)} < 1$. Since $\epsilon_B <0$, this means that $\mu$ needs to be positive and large enough to overcome the binding energy, or else the sum will diverge. The physical interpretation of this divergence is that without a large enough chemical potential that would push particles away and establish an equilibrium, the system will accumulate adsorbed particles without limit.