# Diffusivity Maximum in a Reentrant Nematic Phase

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}, whose dominant contribution is coming from translational diffusion.

^{129}Xe NMR Bharatan and Bowers [21] studied LC mixtures and found Arrhenius behavior of the spin-lattice relaxation time in both nematic and RN phases. Also, the activation energies appear to be system-dependent. For a binary mixture, the activation energy in the RN is more than two times the value in the N phase, whereas no difference is detected in a ternary mixture [21].

_{⊥}[24]. A pronounced peak at r

_{⊥}≳ 1.0 develops when the system enters the RN phase but is absent otherwise. As P increases, this peak grows in magnitude and shifts closer to 1.0. At these intermolecular distances, the molecules are probing the beginning of the repulsive part of their interaction potential. Thus, considering a molecule and its first neighbors, the mutual interactions cancel out and translation motion can be enhanced. Molecules effectively “levitate” in the local, mean molecular potential. A very similar physical situation is encountered in the problem of diffusion in nanoporous materials, such as zeolites. Indeed, a diffusivity maximum is seen in such systems as the size of the diffusing particle reaches a value close to the width of the pore, so that a mutual cancellation of forces occurs [25–27]. This effect is called levitation [25–27].

## 2. Results

**n̂**. If T is further lowered, positional order breaks next. Layers form in the LC which is now in its smectic phase. Within any single layer molecules have no long-range positional order, which is typical of a liquid whereas in one direction the system exhibits long-range positional order. If the normal to the smectic layers coincides with the nematic director the LC is called smectic A.

**1**is the unit tensor. See Section 3 for the definitions of the symbols in this Section. Hence,

**Q**is a real, symmetric, and traceless second-rank tensor which can be diagonalized. Its largest eigenvalue λ

_{+}defines the Maier–Saupe nematic order parameter S = 〈λ

_{+}〉 [42–44] where the angular brackets represent a time average. The eigenvector associated with λ

_{+}corresponds to the nematic director

**n̂**. In a macroscopic, bulk isotropic phase S = 0 ideally because molecular orientations are randomly distributed while in an ideal nematic phase S = 1 because molecules are perfectly aligned with

**n̂**. In any finite size system S ≳ 0 for reasons explained elsewhere [45].

**n̂**. A natural definition of an order parameter for a smectic A phase is therefore the leading coefficient of the Fourier series expansion of the density [46]

_{S}(P) ≃ 5.5 for P = 13.0 (Figure 1d) and at T

_{S}(P) ≃ 6.5 for P = 15.0 (Figure 1e). Below we show that even though there is no intermittent smectic phase the dynamical behavior of the system changes strongly at T

_{S}(P) so that we can still call RN the phase at T < T

_{S}(P). In Figure 2 we show three representative snapshots of the confined LC system in the nematic, smectic A, and RN phase. From these snapshots it is clear that the RN phase has a much larger degree of orientational order than the nematic phase while lacking completely the positional order typical of smectics.

_{i}

^{||}≡

**û**

_{i}·

**r**

_{i}and the subscript t indicates an average over initial time origins, which is a consequence of the stationary character of temporal correlations in equilibrium systems [48]. Because we are considering phases for which there is already a preferential global orientation, this definition of MSD captures the motion along the nematic director.

^{1/2}. The MSD in the direction perpendicular to the molecular axis 〈Δr

_{⊥}

^{2}(τ)〉

_{t}is obtained by replacing r

_{i}

^{||}with

**r**

_{i}

^{⊥}≡

**r**

_{i}− (

**û**

_{i}·

**r**

_{i})

**û**

_{i}in Equation 3. In Figure 3b we show the root MSD for perpendicular versus parallel molecular displacements $\mathrm{\Delta}{\overline{r}}_{\perp ,\left|\right|}\equiv \sqrt{{\langle \mathrm{\Delta}{r}_{\perp ,\left|\right|}^{2}\hspace{0.17em}(\tau )\rangle}_{t}}$. It is apparent that both parallel and perpendicular MSD reach the diffusive regime, that is, when a molecule on average has moved many times its length in the parallel direction, it will have moved also a number of times its diameter in the perpendicular direction [24]. Hence, we do not observe any dynamical behavior consistent with the existence of a columnar phase.

_{||}for the different isobars studied here. At P = 9.0 (Figure 4a) the diffusivity has a value of D

_{||}≈ 1.5 at high T in the nematic phase; then, it exhibits a discontinuous drop to very low values D

_{||}≈ 10

^{−3}upon entering the smectic phase. From the parallel plot in Figure 1a it is evident that this drastic drop occurs upon entering the smectic phase at T ≈ 4.5. It is clearly due to the hindrance to translational diffusion caused by the smectic layers. Figure 4b shows that the isobar at P = 10.0 has the same qualitative behavior as the isobar at P = 9.0, that is a value D

_{||}≈ 1.5 in the nematic phase followed by a drop at T = 5.0 (see Figure 1b) to very small values in the smectic phase. At P = 11.0 (Figure 4c) D

_{||}has a different T-dependence. In the nematic phase (large T) D

_{||}≃ 1.5 as at lower P; at T = 5.0 there is a drop in D

_{||}due to the formation of smectic layers. We note that D

_{||}≃ 0.5 indicating that the smectic phase at P = 11.0 has a higher mobility than the one at lower P. This observation is in agreement with the lower value of Λ at P = 11.0 (see Figure 1c) for the smectic phase with respect to its value at P = 9.0 or P = 10.0.

_{||}when the RN phase forms. We define this transition temperature T

_{D}(P). D

_{||}reaches a maximum value approximately equal to 6.5 which is considerably larger than the typical diffusivity in the nematic phase. As T is further loweredD

_{||}decreases monotonically. Below we analyze the T-dependence of D

_{||}in the RN phase. The dramatic increase in self-diffusion in the direction of

**n̂**in combination with nearly perfect nematic order prompted us to refer to liquid crystals in the RN phase as “supernematics” [23]. At P = 13.0 (Figure 4d) D

_{||}≈ 1.5 in the interval 6.0 ≤ T ≤ 7 which corresponds to a typical value in the nematic phase. At T = T

_{D}(P) ≃ 5.5 the diffusivity jumps to almost a value of 7.0, which is a typical value in the RN phase. Figure 4d is interesting because it shows a sharp and distinct jump in D

_{||}even though there is no intermittent smectic phase such that the system is always in a nematic state. If we compare with Figure 1d, we realize that S shows a small jump at T = T

_{S}≃ 5.5. The coincidence of T

_{S}(P) and T

_{D}(P) indicates that even though these two states are dynamically distinct, structural differences between nematic (at high T) and RN (at low T) are very subtle. Finally, Figure 4e indicates that at P = 15.0 D

_{||}has a qualitatively similar behavior compared with the case of P = 13.0. Also for P = 15.0, the small jump in S occurs at the T = T

_{S}(Figure 1e) which coincides with T = T

_{D}where D

_{||}has a large increase (Figure 4e).

_{||}at P = 11.0 and P = 15.0. At P = 11.0 and for T below the smectic A-RN transition the calculated D

_{||}appears to follow an Arrhenius dependence, that is

_{||}we obtain an activation energy E

_{A}≃ 8.64 at both P = 11.0 and P = 15.0. This value of E

_{A}corresponds to 29.2 kJ mol

^{−1}if we choose ε = 0.56127 × 10

^{−20}J [49] (see also Ref [50]). As discussed elsewhere [23] there are only a few experimental investigations of the dynamics of RN’s. For example, for the activation energy of the dielectric relaxation frequency Ratna et al. [51] found a value of 0.457 eV or higher, depending on the mixture; our value E

_{A}= 0.3 eV is slightly lower. In proton NMR experiments on a pure LC exhibiting a RN phase Miyajima et al. [17] measured an activation energy of 23 kJ mol

^{−1}which turns out to be a bit lower than our result. The fact that our calculation of E

_{A}falls in between these experimental measurements is gratifying given the simplicity of the geometrical shape and of the interaction between the mesogens.

## 3. Model

**û**

_{i}and

**û**

_{j}, respectively, and their distance

**r**

_{ij}≡

**r**

_{i}−

**r**

_{j}, that is

**r̂**

_{ij}≡

**r**

_{ij}/r

_{ij}, r

_{ij}≡ |

**r**

_{ij}|, and the function d

_{ij}

^{m}(

**r**

_{ij},

**û**

_{i},

**û**

_{j}) is the minimum distance between the central axes of two mesogens [57]. The orientation-dependent interaction strength in Equation 6 may be cast as

_{fs}= ε and ${\rho}_{\text{s}}=2\pi /\sqrt[3]{2}$ is the areal density of a single layer of atoms arranged according to the (100) plane of a face-centered cubic lattice, and d

_{ik}

^{m}is the minimum distance between molecule i and wall k. The diameter σ of these substrate atoms is taken to be the same as the diameter of a spherocylinder of the confined fluid phase. The function g(

**û**

_{i}) in Equation 9 is the “anchoring function”, which introduces a dependence of the fluid-substrate interaction on the molecular orientation relative to the wall. The functional form of Equation 9 allows to easily select a preferential anchoring, while maintaining computational simplicity (other choices are possible, of course; see, e.g., [38,58,59]). Specifically, we choose degenerate planar anchoring [60]

**ê**

_{x}and

**ê**

_{y}are the unit vectors of the x and y axis, respectively.

^{2}/ε)

^{1/2}, T in units of ε/k

_{B}where k

_{B}is the Boltzmann constant, P in units of ε/σ

^{3}, and diffusivity in units of (εσ

^{2}/m)

^{1/2}.

^{7}steps) are necessary to access the diffusive regime. However, microcanonical MD simulations are plagued by drifts in the total energy (which should strictly be conserved on account of the underlying physical principles) because of the accumulation of numerical errors. To limit this problem, we choose a rather small integration step Δt = 10

^{−4}. Further, to speed up our simulations we parallelize the computation of molecular forces in our algorithm with OpenMP directives.

_{z}= 19.

## 4. Conclusions

_{A}= 29.2 kJ mol

^{−1}for the diffusion process which falls within the range of the few values reported experimentally that are available for the dynamics of RN’s in pure compounds and binary mixtures. Furthermore, considering the absence of single-file diffusion in the dynamics of the system, we can exclude the possibility that the phase with enhanced diffusivity is a columnar phase rather than reentrant nematic.

- the local potential energy landscape must be rather flat with only shallow minima, thus disrupting the attraction that stabilizes smectic layers, and
- the mutual cancellation of forces between neighboring molecules leads to an effect analogous to levitation in porous media, which can explain the enhanced diffusivity characterizing RN’s. Hence, we conclude that repulsive interactions may explain reentrance in the different physical situations mentioned above.

## Acknowledgments

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**Figure 1.**Temperature-dependence of the nematic order parameter S (circles) and smectic order parameter Λ (squares) for different isobars. Lines are guides for the eye. All quantities are expressed in standard reduced units (see Section 3).

**Figure 2.**Snapshot of typical configurations of the confined LC system in different phases; grey areas represent the confining walls. From top to bottom we show the nematic (P = 11.0, T = 6.0), smectic A (P = 10.0, T = 3.2) and reentrant nematic phase (P = 13.0, T = 4.0). Graphics generated with the software package qmga [47]. Different colors indicate the degree of alignment to the director, that is blue indicates 0 °, red 90 °, and green intermediate. All quantities are expressed in standard reduced units (see Section 3).

**Figure 3.**(

**a**) Typical mean square displacement in the reentrant nematic phase. The curve shows the two characteristic regimes: ballistic regime at short time scales, where the curve has a slope of two, and the diffusive regime at long time scales, where the curve has a slope of one; (

**b**) Root mean square displacement in the direction perpendicular to the molecular long axis vs. the root mean square displacement in the direction parallel to the molecular axis in units of the aspect ratio κ. Data in both panels are from simulations at P = 13.0 and T = 4.0. All quantities are expressed in standard reduced units (see Section 3).

**Figure 4.**Temperature-dependence of the parallel diffusivity D

_{||}for different isobars. Lines are guides for the eye. All quantities are expressed in standard reduced units (see Section 3).

**Figure 5.**Arrhenius plot of the parallel diffusivity D

_{||}at P = 11.0 (top panel) and P = 15.0 (bottom panel). The dashed line is a fit of the low T values to an Arrhenius equations. Both fits yield an activation energy E

_{A}= 8.64 (corresponding to about 0.3 eV). Solid lines are guides for the eye. All quantities are expressed in standard reduced units (see Section 3).

**Figure 6.**Phase diagram of the GBK model of mesogens. State points belonging to different phases are indicated as follows: Smectic A with circles, nematic with squares and RN with triangles. Dashed lines are guides for the eye indicating the approximate position of phase boundaries. All quantities are expressed in standard reduced units (see Section 3).

© 2012 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Stieger, T.; Mazza, M.G.; Schoen, M. Diffusivity Maximum in a Reentrant Nematic Phase. *Int. J. Mol. Sci.* **2012**, *13*, 7854-7871.
https://doi.org/10.3390/ijms13067854

**AMA Style**

Stieger T, Mazza MG, Schoen M. Diffusivity Maximum in a Reentrant Nematic Phase. *International Journal of Molecular Sciences*. 2012; 13(6):7854-7871.
https://doi.org/10.3390/ijms13067854

**Chicago/Turabian Style**

Stieger, Tillmann, Marco G. Mazza, and Martin Schoen. 2012. "Diffusivity Maximum in a Reentrant Nematic Phase" *International Journal of Molecular Sciences* 13, no. 6: 7854-7871.
https://doi.org/10.3390/ijms13067854