^{1}and E. E. Tornau

^{1,a)}

### Abstract

The 4-state model of anthraquinone molecules ordering in a pin-wheel large-pore honeycomb phase on Cu(111) is proposed and solved by Monte Carlo simulation. The model is defined on a rescaled triangular lattice with the lattice constant a being equal to intermolecular distance in the honeycomb phase. The pin-wheel triangle formations are obtained taking into account the elongated shape of the molecules and anisotropic interactions for main two attractive short range (double and single dimeric) H-bond interactions. The long-range intermolecular interactions, corresponding to repulsive dipole-dipole forces, are assumed to be isotropic. Also, a very small (compared to short-range forces) isotropic attractive long-range interaction at the “characteristic” distance of a pore diameter is employed, and its effect carefully studied. This interaction is crucial for a formation of closed porous ordered systems, pin-wheel hexagons in particular. If each side of a pin-wheel hexagon is formed of n parallel molecules, the distance of this characteristic interaction is . The phase diagrams including different pin-wheel hexagon phases and a variety of other ordered structures are obtained. By changing the distance of characteristic interaction, different ordering routes into the experimental pin-wheel honeycomb phase are explored. The results obtained imply that classical explanation of the origin of the pin-wheel honeycomb phase in terms of some balance of attractive and repulsive forces cannot be totally discounted yet.

I. INTRODUCTION

II. MODEL

III. SIMULATION DYNAMICS

IV. RESULTS

A. Different pin-wheel hexagons

B. Phase diagrams with attraction at

C. Thermodynamics of the 333 phase

V. DISCUSSION

### Key Topics

- Hydrogen bonding
- 20.0
- Phase transitions
- 17.0
- Phase diagrams
- 14.0
- Chemical potential
- 9.0
- Monte Carlo methods
- 7.0

## Figures

Rescaling of AQ/Cu(111) lattice based on STM measurements ^{23} to a new lattice shown by dashed lines. Open circles – copper atoms, elongated molecules mark AQ.

Rescaling of AQ/Cu(111) lattice based on STM measurements ^{23} to a new lattice shown by dashed lines. Open circles – copper atoms, elongated molecules mark AQ.

(a) Schematic representation of a pin-wheel honeycomb structure 333 on a new lattice. Long arrows mark the characteristic interaction distance of the 333 phase. Small double arrows with numbers 1, 2, 3 denote the three bond vectors of our model; (b) Allowed and forbidden short-range interactions of our model; (c) Illustration of different short-range interaction energies of AQ molecule in a state s = a surrounded by two molecules in states a and b: −ε1 − ε2 (bond vector orientation α = 2, left); −ε2 + ∞ (α = 1, center), and −ε1 + ∞ (α = 3, right).

(a) Schematic representation of a pin-wheel honeycomb structure 333 on a new lattice. Long arrows mark the characteristic interaction distance of the 333 phase. Small double arrows with numbers 1, 2, 3 denote the three bond vectors of our model; (b) Allowed and forbidden short-range interactions of our model; (c) Illustration of different short-range interaction energies of AQ molecule in a state s = a surrounded by two molecules in states a and b: −ε1 − ε2 (bond vector orientation α = 2, left); −ε2 + ∞ (α = 1, center), and −ε1 + ∞ (α = 3, right).

Schematic representation of five porous pin-wheel phases: (a) 111, (b) 222, (c) 333, (d) 444, and (e) 555. Their concentrations are 3/4, 6/13, 9/28, 12/49, and 15/76, respectively.

Schematic representation of five porous pin-wheel phases: (a) 111, (b) 222, (c) 333, (d) 444, and (e) 555. Their concentrations are 3/4, 6/13, 9/28, 12/49, and 15/76, respectively.

Phase diagram in (A, μ) coordinates at ε d /ε1 = 0.047 and (a) k B T/ε1 = 0 and (b) 0.1. (c) Phase diagram in (A, ε d ) coordinates at μ/ε1 = 0.65 and k B T/ε1 = 0.1. The dots in (b) and (c) mark explored data points. The type of order in corresponding phases are shown above. The phases are distinguished by color and numbering (1–7).

Phase diagram in (A, μ) coordinates at ε d /ε1 = 0.047 and (a) k B T/ε1 = 0 and (b) 0.1. (c) Phase diagram in (A, ε d ) coordinates at μ/ε1 = 0.65 and k B T/ε1 = 0.1. The dots in (b) and (c) mark explored data points. The type of order in corresponding phases are shown above. The phases are distinguished by color and numbering (1–7).

Temperature dependences of (a) energy, (b) specific heat, and (c) molecular concentration close to the phase transition into the 333 phase obtained at μ/ε1 = 0.32 and A/ε1 = 0.38 and four values of ε d /ε1. Only the curves obtained for temperature decreasing from the disordered phase are presented, except for the curves 2 (ε d /ε1 = 0.047) in (a) which are taken to illustrate the energy hysteresis at T c point. Inset in (a) shows energy histogram at T c typical for the first-order phase transition (ε d /ε1 = 0.061).

Temperature dependences of (a) energy, (b) specific heat, and (c) molecular concentration close to the phase transition into the 333 phase obtained at μ/ε1 = 0.32 and A/ε1 = 0.38 and four values of ε d /ε1. Only the curves obtained for temperature decreasing from the disordered phase are presented, except for the curves 2 (ε d /ε1 = 0.047) in (a) which are taken to illustrate the energy hysteresis at T c point. Inset in (a) shows energy histogram at T c typical for the first-order phase transition (ε d /ε1 = 0.061).

Phase diagram in (T, ε d ) coordinates for transitions into the 333 phase and neighboring phases. Other parameters are: μ/ε1 = 0.32 and A/ε1 = 0.38. The names of phases are the same as in Fig. 4 .

Phase diagram in (T, ε d ) coordinates for transitions into the 333 phase and neighboring phases. Other parameters are: μ/ε1 = 0.32 and A/ε1 = 0.38. The names of phases are the same as in Fig. 4 .

Percentage of AQ molecules forming pin-wheel triangles or being directly attached to the triangle as a function of normalized AQ molecular coverage. The experimental data ^{23} are shown by dashed curve (1). Other curves are obtained by MC simulation: (2) (approach to 333 from disordered phase); (3) (approach to 333 from wave phase); (4) ; (5) ; (6) assuming the 333 phase is obtained via all sequence of nnn phases (N pw /N = 100% for n = 3 and 4/n × 100% for n > 3). Other parameters are: for curves 2 and 3, they are the same as in Fig. 4 and for the curves 4 and 5, A/ε1 = 0.35 and ε d /ε1 = 0.05.

Percentage of AQ molecules forming pin-wheel triangles or being directly attached to the triangle as a function of normalized AQ molecular coverage. The experimental data ^{23} are shown by dashed curve (1). Other curves are obtained by MC simulation: (2) (approach to 333 from disordered phase); (3) (approach to 333 from wave phase); (4) ; (5) ; (6) assuming the 333 phase is obtained via all sequence of nnn phases (N pw /N = 100% for n = 3 and 4/n × 100% for n > 3). Other parameters are: for curves 2 and 3, they are the same as in Fig. 4 and for the curves 4 and 5, A/ε1 = 0.35 and ε d /ε1 = 0.05.

The snapshots of structures occurring in calculations with characteristic distance of attractive interaction at : (a) disordered-333 phase separation, c m = 0.151 and A/ε1 = 0.25; (b) waves-333 phase separation, c m = 0.281 and A/ε1 = 0.38. Other parameters: ε d /ε1 = 0.05 and k B T/ε1 = 0.09.

The snapshots of structures occurring in calculations with characteristic distance of attractive interaction at : (a) disordered-333 phase separation, c m = 0.151 and A/ε1 = 0.25; (b) waves-333 phase separation, c m = 0.281 and A/ε1 = 0.38. Other parameters: ε d /ε1 = 0.05 and k B T/ε1 = 0.09.

The snapshots of structures occurring in calculations with characteristic distance of attractive interaction at : (a) c m = 0.14; (b) 0.21, (c) 0.24, and (d) 0.285. Other parameters: A/ε1 = 0.35, ε d /ε1 = 0.05, and k B T/ε1 = 0.09.

The snapshots of structures occurring in calculations with characteristic distance of attractive interaction at : (a) c m = 0.14; (b) 0.21, (c) 0.24, and (d) 0.285. Other parameters: A/ε1 = 0.35, ε d /ε1 = 0.05, and k B T/ε1 = 0.09.

The structures obtained at A/ε1 = 0.35, ε d /ε1 = 0.05, and 0.09 ⩽ k B T/ε1 ⩽ 0.11 at different values of characteristic interaction length r d . The digits on the left denote the distance r d and those above the structure snapshot – the molecular concentration of this structure, c m .

The structures obtained at A/ε1 = 0.35, ε d /ε1 = 0.05, and 0.09 ⩽ k B T/ε1 ⩽ 0.11 at different values of characteristic interaction length r d . The digits on the left denote the distance r d and those above the structure snapshot – the molecular concentration of this structure, c m .

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