Odd-even effect in the dominant order of self-assembly of cross junctions in space dimension $d \ge 3$

Abstract

We consider the self-assembly of cross junctions in a general space dimension ($d$) as an extension of the problem studied in a previous paper for $d = 3$. This problem is equivalent to constructing a $d$-dimensional hypercubic jungle gym, at all junctions of which $2d$ rods with different colours meet. The analysis reveals a unique feature of the $d = 3$ case: the forced presence of at least one perfectly-ordered (singly coloured) direction (axis), in contrast to the possible absence of such a direction in $d \ge 4$. However, we will show that the uniaxial order is overwhelming not only in $d=3$ but also for $4 \le d \le 7$ and odd $d \ge 9$ in a sufficiently large system. For even $d \ge 8$, isotropic states dominate, leading to the alternation of dominant states between the uniaxial and isotropic orders depending on the parity of $d \ge 7$.