Phase transition in two-dimensional dipolar fluids at low densities

Monte Carlo computer simulations of a quasi two-dimensional (2D) dipolar fluid at low and intermediate densities indicate that the structure of the fluid is well described by an ideal mixture of self-assembling clusters. A detailed analysis of the topology of the clusters, of their internal energy and of their size (or mass) distributions is used to obtain approximations to their partition functions. Within the scope of these approximations, the results of this work suggest that the 2D dipolar fluid undergoes a phase transition from a dilute phase characterized by a number of disconnected clusters to a condensed phase characterized by a network or spanning (macroscopic) cluster that includes most of the particles in the system.

Figure 1

Typical equilibrium configurations of the quasi 2D DHS as obtained from MC simulations with $N_p=5776$ at $m^{*}=2.75$ and three reduced densities.

Figure 2

Mean reduced density of chains [circles, ${\varphi }^{*}(N,2,0)$], rings [squares, ${\varphi }^{*}(N,0,0)$] and networks [triangles, ${\varphi }_n^{*}\left(N\right)$] of size $N$, as obtained from simulations with ${\rho }^{*}=0.1$. The lines are the theoretical predictions for the chains and rings. Inset: mean reduced density of networks at ${\rho }^{*}=0.05$ (inverted triangles), ${\rho }^{*}=0.0625$ (diamonds) and ${\rho }^{*}=0.1$ (triangles).

Figure 3

Mean fraction of ends ($⟨n_{1,n}⟩∕N$, open symbols) and junctions ($⟨n_3⟩∕N$, full symbols) in networks of size $N$, at ${\rho }^{*}=0.1$ (circles), 0.075 (squares) and 0.0625 (diamonds). Simulation results are for clusters with $N$ in the range 50–600. The statistical uncertainty of these results is of the order of the size of the symbols. The full line is the function $2∕N$, the fraction of ends in chains of size $N$.

Figure 4

General representation of the solutions of Eq. (5) for different values of $N_p$. Dotted lines: $\mu (N_p,\rho )$ for finite $N_p$. Full line: $\mu \left(\rho \right)\equiv {\lim }_{N_p\to \infty }\mu (n_p,\rho )$; (a) and (b) represent systems where ${\rho }_t\equiv {\sum }_{N=1}^{\infty }NF\left(N\right)$ [see Eq. (5)] is infinite and finite, respectively. In case (b), the discontinuity in the derivative of $\mu \left(\rho \right)$ at $\rho ={\rho }_t$ signals the existence of a phase transition; $\lambda$ (dashed line) is the free energy per particle of an infinite cluster.

Figure 5

Generalization of Fig. 4 for the free energy per particle of an infinite network that depends on density. Dotted lines: $\mu (N_p,\rho )$ for finite $N_p$ as given by Eq. (A1). Full line: $\mu \left(\rho \right)\equiv {\lim }_{N_p\to \infty }\mu (n_p,\rho )$, when ${\rho }_t\equiv {\sum }_{N=1}^{\infty }(G_{rc}+G_n)$ [see Eqs. (A2, A3)] is finite. Dashed line: free energy per particle of an infinite network, ${\lambda }_n\left(\rho \right)$. Dashed dotted line: free energy per particle of infinite chains and rings, ${\lambda }_0$.