Soft condensed matter: the partition function for cluster formation

Question

I am stuck on understanding the form of the partition function presented in my lectures, for self assembly of clusters from monomeric molecules. If there are $N_T$ molecules that can form clusters of a variable size $\alpha$ molecules each (where $\alpha$ can rage from 1 to $N_T$. An $\alpha$ of 1 corresponds to a monomeric species and $\alpha$ of $N_T$ means all molecules are forming a single cluster.

Let $N_\alpha$ be the number of molecules that are encoorporated in clusters of size $\alpha$, and $n_\alpha = N_\alpha / \alpha$ is te number of clusters of this size. Clearly $\Sigma_{\alpha=1}^{N_T} N_\alpha = N_T$.

Apparetly in the low cluster concentration limit, and where the agregates do not interact, the partition function is

$Z = \Pi _{\alpha=1}^{N_T}\frac{1}{n_\alpha !}(\frac{Vz_\alpha}{\lambda_\alpha ^3})^{n_\alpha}$

were $\lambda_\alpha ^3$ is the volume of an alpha cluster, and $z_\alpha$ is the number of internal degrees of freedom in an alpha cluster.

I am used to seeing partition functions as a SUM over the microstates. This becomes a product if you have N independent ensembles (say, trying to find e number of lattice sites with a defect, if te occurance of a defect at one lattice site does not affect the presence of a defect on another lattice site). But in that case, the total partition function is just a single power of a single lattice site partition function. Here we take a product over varying n_alpha.

Clearly, we have made an assumption of relatively sparse cluster aggregates so that excluded volme is not accounted for. It seems to me like we have taken 'the formation of clusters of a specific size alpha' as independent "systems". In that case, I coulde see

$\frac{1}{n_\alpha !}(\frac{Vz_\alpha}{\lambda_\alpha ^3})^{n_\alpha}$ as being the partition function for this sstem, EXCEPT THAT IT DOES NOT ACCOUNT FOR THE PERMUTATIONS OF THE MONOMERIC SOLUTES, though it does accunt for the permutations of identical agregate clusters (and permutations of the monomers within each cluster via the internal degrees of freedom z) but not i) exchange of the monomers not inside clusters or ii)exchanging the monomers in clusters with those in solution.

To me this is problematic because we do not know the effect of other cluster agregation on the remaining monomer number, if considering each 'system' in turn. Wile one ma argue that the number of monomers largely swamps the number of aggregates, we ave in fact also allowed for te possibility of a cluster of all of the available monomers so this assumption breaks down. I can only see this working by imposing a cut off cluster size that is much less than the numb of available mo;lecules.

Secondly, in any case, the expected remaining number of monomer molecules would vary for each of the cluster sizes, so I think would be a non-trivial contribution to the partition function above, and would affect how we treat the 'systems' on non-=equal footing.

I am also comparing this with the study of polymers, where the effect of solutes and the solute entropy in permuting these molecules on the 'lattice sites' was important in the behavior of polymers, because of the different size of polymer and solute molecules (Flory Higgins theory). Therefore, the entropy contribution to the number of each of these would be different. This is certainly the case here where we treat varying size aggregates.

If anyone could shed light on this or provide a link, it would be much appreciated.

Answer 1

That partition function comes from considering an ensemble of non-interacting cluster. Under this assumption, you can use formula for the partition function of an ideal gas.

The canonical partition function for an ideal gas of $N$ particles contained in a volume $V$ is

$$ Z_N= \frac {V^N} {N! \lambda^{3N}} = \frac{Z_1^N}{N!} \tag{1} $$

where $Z_1=V/\lambda^3$, with $\lambda=(2 \pi \beta \hbar /m)^{1/2}$, is the partition function of a single particle. Now imagine to have in the same volume $N_1$ particles of type $1$, $N_2$ particles of type $2$, ...$N_n$ particles of type $n$ where $\sum_{k=1}^n N_k =N$, and that all the particles are still all non-interacting.

Then you would get

$$ Z_N = \prod_{k=1}^N \frac{Z_{1,k}^{N_k}}{N_k!} \tag{2} $$

where $Z_{1,k}=V/\lambda_k^3$, with $\lambda=(2 \pi \beta \hbar /m_k)^{1/2}$, is the partition function of a single particle of type $k$.

Your formula for the partition function of an "ideal gas of cluster" is exactly $(2)$, but with the partition function of a cluster of $k$ particles instead of $Z_{1,k}$. Basically, you replace particles of type $k$ with clusters of $k$ particles ($N_k$ becomes the number of clusters of $k$ particles):

$$ Z_N = \prod_{k=1}^N \frac{Z_{k}^{N_k}}{N_k!} \tag{3} $$

with

$$ Z_k = \frac 1 {k! \lambda^{3k}} \int' dq^N \exp[-U(q^N)] \tag{4} $$

where the $'$ denotes the condition that the $k$ particles are part of a cluster.

As you can see, the permutations of particles are accounted for into the term $k!$ in $(4)$.

Suggested read: Walter H. Stockmayer, Theory of Molecular Size Distribution and Gel Formation in Branched‐Chain Polymers J. Chem. Phys. 11, 45 (1943)