# Polar Vector Property of the Stationary State of Condensed Molecular Matter

^{*}

## Abstract

**:**

_{el}. The investigation has been extended to liquid droplets made of dipolar entities by molecular dynamics simulations. We demonstrate the development of an $\mathrm{m}\overline{\infty}$ quasi bi-polar state leading to a charged surface.

## 1. Introduction

_{el}has been analyzed [13–15]. As a matter of fact, we do not know much about the polar domain states of (non-ferroelectric) molecular crystals. By analyzing both the 180° orientational disorder of dipolar entities and the forms of twinning related to the spatial distribution of the polarization, we can, however, answer the aforementioned key question: for crystals expressing a polar morphology by their (hkl) faces, a mono-domain, i.e., a homogeneous polarization distribution, may be observed experimentally [5–12]. This is a kinetic effect due to the critical slowing down occurring in real growth that hinders the reversal transition into a bi-polar state, as predicted by statistical mechanics calculations [2,3] (Figure 1a). This very form of twinning process can, in fact, transform an initial mono-domain polar seed into a final object featuring macroscopic domains of opposite polarities. In this article, we focus on understanding the bi-polar state of condensed matter constituted by polar molecules by symmetry and statistical mechanics arguments. Because of the close analogy, the analysis will also include liquid droplets (Figure 1b) and crystals made thereof.

## 2. Symmetry and Conservation

_{mol}of a molecular system commutes with each of these operations. Similarly, this holds for molecular crystals made stable by inter-molecular potentials. In our context, parity P is of particular relevance: both the polar vector operators μ

_{el}(dipole moment of molecules) and P

_{el}(polarization of molecular crystals) change their direction (sign) under the action of P (so, C, not T) [17]. When attempting to calculate observable properties, such as μ

_{el}or P

_{el}, we encounter the issue of integrating a state equation over time and space, because for every experiment, a certain time-space domain is committed. In quantum mechanics, the dipole moment μ

_{el}is defined by $\int \mathrm{\Psi}*er\mathrm{\Psi}dV$ (e: charge of the electron; r: distance vector). Because of:

^{*}Ψ = |Ψ|

^{2}with parity even (+), and since the integral is calculated over the whole space, the expectation value is zero. This is a short hand proof of a statement we can find in textbooks on quantum mechanics [18]: “Quantum-mechanical systems in the stationary state do not have permanent electric dipole moments”. For most molecules, Born–Oppenheimer-based calculations provide a description that can attribute a rigid nuclear frame. This may be extended to a vibronic state; however, the amplitudes of nuclear motion will only cover a small part of the full space of coordinates. Consequently, such a time averaged single object of fixed nuclear conformation (transforming under a polar point group) shows a non-zero dipole moment μ

_{el}. Assuming quasi-rigid objects carrying a dipole moment, thermal energy, i.e., temperature, is the variable that will lead to averaging. Consequently, a vector property showing P = −1 for single objects will not be observed for a stationary state of an ensemble. However, droplets adopt a bi-polar type of state (Figure 1b), when projecting single dipole moments onto diameters for a summation to yield 〈P

_{el}〉

_{zone}. Preliminary molecular dynamics (MD) calculations for carbon monoxide in the 2D liquid state show that the oxygen atom of CO molecules is preferentially pointing out of the boundary, whereas towards the center, averaging of 〈P

_{el}〉 takes place.

_{el}〉 onto this unique axis. The axis itself represents ∞/m(C

_{∞h}) symmetry [20]. Because of the finite size of crystals, the mirror plane m gets localized in the middle of the object. This means that the crystal splits into two domains with m perpendicular to the unique axis. A mono-domain polar state is thus not allowed: a polar packing, i.e., the average polarization, is expected to develop a bi-polar state, described by ∞/mm(D

_{∞h}) symmetry [20] (Figure 1a). In each domain, a polar vector type property of ∞m symmetry is allowed. For a detailed discussion of the eigensymmetry and the composite symmetry of twinned crystals, see [21] and Figure 3.3.10.2. therein. Considering real crystal structures, e.g., nucleation processes leading to n/m (n = 2, 3, 4, 6) point groups and growth along the axis n, the stochastic process transforms pro-chiral faces into chiral sectors. Because of the mirror plane m passing through a seed, sectors involving the +n or the −n direction show an enantiomorphic relation (for a discussion of other point groups and low index faces, see [22]).

_{i}= ±1, i = 0,…, N, Equation (2) is the energy of that configuration; (ii) periodic boundary conditions on the transverse direction and a free boundary condition on the longitudinal one.

_{⊥}= E

_{par}− E

_{antip}, where E

_{par}(resp. E

_{antip}) is the energy of two next-neighbor spins parallel (resp. anti-parallel), due to the transverse interaction. The sums are taken over all lattice points, specifically, i and j run over the longitudinal and transverse direction, respectively; ∊ is a constant term. Whereas the bilinear terms (longitudinal and transverse interactions) can induce polar alignment of vectors, they do not provide a definite direction of polarity 〈P

_{el}〉. Absolute polarity (directionality) is introduced by the linear effective particle operators S

_{0}and S

_{N}in H. These terms are formally equivalent to the action of an electrical field E, representing and, thus, introducing ∞m [20,25], i.e., polar vector symmetry. The energy parameter ∆E

_{f}[2,3] associated with these two linear terms in H reflect the polar symmetry of interacting molecules: in a simple description of the interactions, ∆E

_{f}is equal to E

_{AA}− E

_{DD}, i.e., the difference in the energy for defect formation in a single chain: E

_{AA}: −A ⋯ A−; E

_{DD}: −D ⋯ D− (A: acceptor; D: donor). Therefore, at each border of the unique axis, an effective field (∞m) [25] is reducing the symmetry of the axis to yield a polar vector property (∞m; also chiral ∞) in each domain. It can be shown that the introduction of the ∆E

_{f}energy parameter is not restricted to nearest neighbor interactions, but can be extended to a lattice sum.

## 3. Summary and Conclusions

_{∞h}) [20] symmetry, i.e., a bi-polar state (Figure 1a), where domains related by a mirror plane m show a polar vector property (∞m, 〈P

_{el}〉) induced by the linear terms ∆E

_{f}S

_{i}(i = 0, N) in H (Equation (2)). This general behavior predicted by H was reproduced by different types of MC and MD simulations, assuming either a simple nearest neighbor interaction scheme [2,3] or long-range interactions of real molecules (Figure 2).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Bi-polar state ∞/mm showing domains (∞m). The polarization may be oriented “down”(as shown) or “up”; (

**b**) Quasi bi-polar state of a stationary spherical object $(\mathrm{m}\overline{\infty},{\mathrm{K}}_{\mathrm{h}})$.

**Figure 2.**Average polarization vs. lattice site for an MC 2D-simulation, showing the bi-polar state formation. The full line is the fit with an exponential function. The system is composed of 21 × 5 molecules (1-bromo-4′-cyano-tetrafluoro-benzene) arranged in five parallel chains.

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**MDPI and ACS Style**

Hulliger, J.; Cannavacciuolo, L.; Rech, M.
Polar Vector Property of the Stationary State of Condensed Molecular Matter. *Symmetry* **2014**, *6*, 844-850.
https://doi.org/10.3390/sym6040844

**AMA Style**

Hulliger J, Cannavacciuolo L, Rech M.
Polar Vector Property of the Stationary State of Condensed Molecular Matter. *Symmetry*. 2014; 6(4):844-850.
https://doi.org/10.3390/sym6040844

**Chicago/Turabian Style**

Hulliger, Jürg, Luigi Cannavacciuolo, and Mathias Rech.
2014. "Polar Vector Property of the Stationary State of Condensed Molecular Matter" *Symmetry* 6, no. 4: 844-850.
https://doi.org/10.3390/sym6040844