Spin-1 aggregation model in one dimension

We studied a simple model of aggregation in one dimension that resembles the self-assembly of amphiphiles in an aqueous solution. We mapped the water and amphiphilic molecules by Ising spin variables for S=1. The zero component of spin represents the water molecules, while the remaining components (+/-1) account for the amphiphilic molecules. We defined an aggregate in one dimension by a set of spin components (+/-1) placed between two zero spin components. There is no difference between up and down components of the spins inside the aggregates. In this way what really matters is the square of the spin component. The grand-canonical partition function and the probability of formation of different aggregate sizes were calculated by the transfer matrix method. We have shown that for any value of the chemical potential and temperature, the system does not exhibit the typical aggregate size distribution which is observed in micellar solutions at low concentrations. The distribution curve for the aggregate size does not show the minimum and the maximum as a function of the concentration which is the signature of the appearance of micelles. We can say that this one-dimensional model does not present any phase transition nor a transition from the micellar to nonmicellar state.