Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics… DOI: http://dx.doi.org/10.5772/intechopen.80977

where the products involving the Levi-Civita tensor ε are defined in the following way: ε : A ¼ εαβγAγβ and A � ε � B ¼ AαβεβγδBδ. The quantities η, ~η1, and ~η<sup>3</sup> are shear viscosities, ~γ<sup>1</sup> is the twist viscosity, η<sup>V</sup> is the volume viscosity, ~η<sup>2</sup> is the cross coupling coefficient relating the difference between the rotation and the director angular velocity, and the symmetric traceless pressure. According to the Onsager reciprocity relations (ORR), this coefficient is equal to ~γ2=2, the cross coupling coefficient relating the traceless strain rate and the antisymmetric pressure. The trace of the strain rate and the symmetric traceless pressure are related by the cross coupling coefficient, ζ, which, according to the ORR, is equal to the cross coupling coefficient κ between the traceless strain rate and the difference between the trace of the pressure tensor and the equilibrium pressure.

Application of a planar Couette velocity gradient, <sup>∇</sup><sup>u</sup> <sup>¼</sup> <sup>γ</sup> <sup>e</sup>zex, where <sup>γ</sup> <sup>¼</sup> <sup>∂</sup>zux is the shear rate and fixation of the director in the zx-plane at an angle θ relative to the stream lines, see Figure 2, by application of an electric or a magnetic field gives the following relations between the pressure tensor components and the strain rate in a director based coordinate system ð Þ e1; e2; e<sup>3</sup> where the director points in the e3 direction:

$$
\langle \overline{p}\_{11} \rangle = \left( \eta + \frac{\tilde{\eta}\_3}{3} \right) \gamma \sin 2\theta,\tag{A.3a}
$$

$$
\langle \overline{p}\_{22} \rangle = \frac{1}{3} \left( \widetilde{\eta}\_1 + \widetilde{\eta}\_3 \right) \chi \sin 2\theta,\tag{A.3b}
$$

$$
\langle \overline{p}\_{33} \rangle = - \left( \eta + \frac{\tilde{\eta}\_1}{3} + 2 \frac{\tilde{\eta}\_3}{3} \right) \gamma \sin 2\theta,\tag{A.3c}
$$

$$
\langle \overline{p}\_{31} \rangle = \left( \eta + \frac{\tilde{\eta}\_1}{6} \right) \gamma \cos 2\theta + \tilde{\eta}\_2 \frac{\gamma}{2}, \tag{A.3d}
$$

and

Acknowledgements

Non-Equilibrium Particle Dynamics

and viscosity coefficients

<sup>σ</sup> ¼ � <sup>1</sup>

½ ∇u þ ð Þ ∇u <sup>T</sup> h i

P

and

138

We gratefully acknowledge financial support from the Knut and Alice Wallenberg Foundation (Project number KAW 2012.0078) and Vetenskapsrådet (Swedish Research Council) (Project number 2013-5171). The simulations were performed using resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC, HPC2N, and NSC.) We also acknowledge PRACE for awarding us

A. Appendix 1: Relation between the pressure tensor, velocity gradient,

The relation between the velocity gradient, ∇u, and the pressure tensor, P, is more complicated in an axially symmetric system such as nematic liquid crystal than in an isotropic fluid due to the lower symmetry. In order to derive the linear phenomenological relations between the velocity gradient and the pressure, it is appropriate to begin by identifying the thermodynamic forces and fluxes in the

� �

where T is the absolute temperature, and u is the streaming velocity. The various parts of the second rank tensor are denoted in the following manner: the symmetric traceless part is given by <sup>A</sup> <sup>¼</sup> <sup>½</sup> <sup>A</sup> <sup>þ</sup> <sup>A</sup><sup>T</sup> � � � ð Þ <sup>1</sup>=<sup>3</sup> Trð Þ <sup>A</sup> <sup>1</sup> and the pseudovector dual of the antisymmetric part is denoted by <sup>A</sup><sup>a</sup> ¼ �½ε:A ¼ �½εαβγAγβ, where <sup>ε</sup> is the Levi-Civita tensor. Three pairs of thermodynamic forces and fluxes can be identified by inspection of the irreversible entropy production, namely, the symmetric traceless pressure tensor and the traceless strain rate, P and ∇u, the antisymmetric pressure and the difference between the rotation and the director angular velocity, <sup>P</sup><sup>a</sup> and ½<sup>∇</sup> � <sup>u</sup> � <sup>Ω</sup>, and the difference between the trace of the pressure tensor and the equilibrium pressure of a quiescent liquid crystal, and the trace of the strain

3

, and it is always symmetric. In a uniaxially symmetric nematic

h i Trð Þ P � peq ¼ �ηV∇ � u � κ nn:∇u, (A.2c)

liquid crystal, the relations between the pressure and the velocity gradient can be deduced by symmetry arguments, and they can be expressed in a few different equivalent ways [23, 36]. It has been found that a notation due to Hess [36] is the most convenient one for deducing Green-Kubo relations and NEMD-algorithms:

� � ¼ �2η∇<sup>u</sup> � <sup>~</sup>η<sup>1</sup> nn � <sup>∇</sup><sup>u</sup> � <sup>2</sup>~η3nn nn:∇<sup>u</sup> <sup>þ</sup> <sup>2</sup>~η2nn � <sup>ε</sup> � <sup>ð</sup>½<sup>∇</sup> � <sup>u</sup> � <sup>Ω</sup>Þ � <sup>ζ</sup>nn<sup>∇</sup> � <sup>u</sup>,

<sup>2</sup> ð Þ� <sup>½</sup><sup>∇</sup> � <sup>u</sup> � <sup>Ω</sup> <sup>~</sup>γ<sup>2</sup>

Trð Þ� P peq � �

∇�u

<sup>2</sup> <sup>ε</sup> : nn � <sup>∇</sup><sup>u</sup> � �, (A.2b)

, (A.1)

(A.2a)

access to Hazelhen based in Germany at Rechenzentrum Stuttgart.

expression for the irreversible entropy production [3, 4, 23, 36]:

<sup>T</sup> <sup>P</sup> : <sup>∇</sup><sup>u</sup> <sup>þ</sup> <sup>2</sup>P<sup>a</sup>�ð½<sup>∇</sup> � <sup>u</sup> � <sup>Ω</sup>Þ þ <sup>1</sup>

rate, 1ð Þ =3 Trð Þ� P peq and ∇�u. Note that the strain rate is defined as

<sup>P</sup><sup>a</sup> h i ¼ �~γ<sup>1</sup>

1 3

$$
\langle \mathbf{2} \langle p\_2^a \rangle = \langle \hat{\lambda}\_2 \rangle = -\tilde{\gamma}\_1 \frac{\chi}{2} - \tilde{\gamma}\_2 \frac{\chi}{2} \cos 2\theta,\tag{A.3e}
$$

where ^λ<sup>2</sup> is the external torque density acting on the system. From these equations, it is apparent that the various elements of the pressure tensor are linear functions of sin 2θ and cos 2θ, so the various viscosity coefficients can be evaluated by fixing the director at a few different angles relative to the stream lines and calculating the averages of the pressure tensor elements.

In a planar elongational flow [8, 26–28], where the elongation direction is parallel to the x-axis, the contraction direction is parallel to the negative z-axis, and the velocity field is equal to u ¼ γð Þ xe<sup>x</sup> � ze<sup>z</sup> , so that the velocity gradient becomes ∇u ¼ γð Þ exe<sup>x</sup> � eze<sup>z</sup> . Then the linear relations between the velocity gradient and the pressure become the following in a director-based coordinate system ð Þ e1; e2; e<sup>3</sup> where the director points in the e1-direction, and θ is the angle between the director and the elongation direction or x-axis, e<sup>2</sup> ¼ e<sup>y</sup> and e<sup>3</sup> ¼ e<sup>1</sup> � e2,

$$
\langle \overline{p}\_{11} \rangle = -2 \left( \eta + \frac{\tilde{\eta}\_3}{3} \right) \chi \cos 2\theta,\tag{A.4a}
$$

$$
\langle \overline{p}\_{22} \rangle = -\frac{2}{3} (\tilde{\eta}\_1 + \tilde{\eta}\_3) \chi \cos 2\theta,\tag{A.4b}
$$

$$
\langle \overline{p}\_{33} \rangle = 2 \left( \eta + \frac{\tilde{\eta}\_1}{3} + 2 \frac{\tilde{\eta}\_3}{3} \right) \chi \cos 2\theta,\tag{A.4c}
$$

$$
\langle \overline{p}\_{31} \rangle = \left( 2\eta + \frac{\tilde{\eta}\_1}{3} \right) \gamma \sin 2\theta,\tag{A.4d}
$$

<sup>σ</sup>ð Þ¼ ^r12; <sup>u</sup>^1; <sup>u</sup>^<sup>2</sup> <sup>σ</sup><sup>0</sup> <sup>1</sup> � <sup>χ</sup>

DOI: http://dx.doi.org/10.5772/intechopen.80977

<sup>κ</sup>01=<sup>2</sup> � <sup>1</sup>

ellipsoids.

relevant.

Author details

141

Sten Sarman\*, Yonglei Wang and Aatto Laaksonen

\*Address all correspondence to: sarman@ownit.nu

Stockholm University, Stockholm, Sweden

provided the original work is properly cited.

Department of Materials and Environmental Chemistry, Arrhenius Laboratory,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

� �<sup>=</sup> <sup>κ</sup>01=<sup>2</sup> <sup>þ</sup> <sup>1</sup>

2

ð Þ ^r<sup>12</sup> � u^<sup>1</sup> þ ^r<sup>12</sup> � u^<sup>2</sup>

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

1 þ χu^<sup>1</sup> � u^<sup>2</sup>

where the parameter <sup>χ</sup> is equal to <sup>κ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>=</sup> <sup>κ</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> and <sup>κ</sup> is the ratio of the

by side and end to end configurations of calamitic ellipsoids or the ratio of the edgeto-edge and face-to-face configurations of discotic ellipsoids, and ε<sup>0</sup> denotes the depth of the potential minimum in the cross configuration, where ^r12, u^1, and u^<sup>2</sup> are perpendicular to each other. The parameters κ and κ<sup>0</sup> have been given the values 3 and 5, respectively, for the calamitic ellipsoids and 1/3 and 1/5 for the discotic

The denominators in Eqs. (A.8a) and (A.8b) are never equal to zero because the absolute value of the scalar product u^<sup>1</sup> � u^<sup>2</sup> is less than or equal to one since u^<sup>1</sup> and u^<sup>2</sup> are unit vectors, and the absolute values of the parameters χ and χ<sup>0</sup> are less than one. The ordinary Lennard-Jones potential is recovered in the limit when κ and κ<sup>0</sup> go to one. Note that, the potential is purely repulsive, so there are no potential minima but the value of κ<sup>0</sup> optimized for the attractive Gay-Berne potential has been retained. The transport properties of this system of purely repulsive soft ellipsoids are similar to those of a system where the molecules interact according to the conventional Gay-Berne potential with attraction as well, so the results are still

axis of revolution and the axis perpendicular to this axis, χ<sup>0</sup> is equal to

2

� � and <sup>κ</sup><sup>0</sup> is the ratio of the potential energy minima of the side

( ) " # �1=<sup>2</sup>

<sup>þ</sup> ð Þ ^r<sup>12</sup> � <sup>u</sup>^<sup>1</sup> � ^r<sup>12</sup> � <sup>u</sup>^<sup>2</sup>

1 � χu^<sup>1</sup> � u^<sup>2</sup>

2

,

(A.8b)

and

$$
\mathcal{Z}\langle p\_2^a \rangle = \langle \hat{\lambda}\_2 \rangle = -\tilde{\gamma}\_2 \mathcal{Y} \sin \mathfrak{2}\theta. \tag{A.4e}
$$

If these expressions for the pressure tensor are inserted in the expression for energy dissipation rate (A.1), we obtain

$$\begin{split} \dot{w}\_{irr} &= \overline{\mathbf{P}} : \overline{\mathbf{Vu}} = -\left\langle \frac{1}{2} \left( \overline{p}\_{33}^{\circ} - \overline{p}\_{11}^{\circ} \right) \sin 2\theta - \overline{p}\_{31}^{\circ} \cos 2\theta \right\rangle\_{\mathbb{T}} \\ &= \left( \eta + \frac{\tilde{\eta}\_{1}}{6} + \frac{\tilde{\eta}\_{3}}{2} \sin^{2} 2\theta + \frac{\tilde{\eta}\_{2}}{2} \cos 2\theta \right) \eta^{2}, \end{split} \tag{A.5}$$

for planar Couette flow and

$$
\dot{w}\_{irr} = \left(4\eta + 2\frac{\ddot{\eta}\_1}{3} + 2\ddot{\eta}\_3 \sin^2 2\theta\right) \gamma^2 \tag{A.6}
$$

for planar elongational flow. The subscript γ denotes that the average is evaluated in a nonequilibrium ensemble at a finite shear rate.
