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p<sup>33</sup>

and

� � <sup>¼</sup> <sup>2</sup> <sup>η</sup> <sup>þ</sup> <sup>~</sup>η<sup>1</sup>

� � <sup>¼</sup> <sup>2</sup><sup>η</sup> <sup>þ</sup> <sup>~</sup>η<sup>1</sup>

p31

2 pa 2 � � <sup>¼</sup> ^λ<sup>2</sup>

<sup>w</sup>\_ irr <sup>¼</sup> <sup>P</sup> : <sup>∇</sup><sup>u</sup> ¼ � <sup>1</sup>

<sup>¼</sup> <sup>η</sup> <sup>þ</sup> <sup>~</sup>η<sup>1</sup>

ated in a nonequilibrium ensemble at a finite shear rate.

B. Appendix 2: The Gay-Berne potential

commonly used Gay-Berne potential [16, 17, 21],

Uð Þ¼ r12; u^1; u^<sup>2</sup> 4εð Þ ^r12; u^1; u^<sup>2</sup>

<sup>1</sup> � <sup>χ</sup><sup>0</sup> 2

ð Þ u^<sup>1</sup> � u^<sup>2</sup> <sup>2</sup> h i�1=<sup>2</sup>

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

u^<sup>1</sup> � u^<sup>2</sup>

1 þ χ<sup>0</sup>

and range parameters are given by

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

and

140

energy dissipation rate (A.1), we obtain

Non-Equilibrium Particle Dynamics

for planar Couette flow and

<sup>3</sup> <sup>þ</sup> <sup>2</sup> ~η3 3

3 � �

If these expressions for the pressure tensor are inserted in the expression for

� �

~η1

for planar elongational flow. The subscript γ denotes that the average is evalu-

In order to evaluate the above expressions for the irreversible work in shear flow, elongational flow, and heat flow, we have simulated a coarse grained model system composed of molecules interacting via a purely repulsive version of the

where r<sup>12</sup> ¼ r<sup>2</sup> � r<sup>1</sup> is the distance vector from the center of mass of molecule 1 to the center of mass of molecule 2, ^r<sup>12</sup> is the unit vector in the direction of r12, r<sup>12</sup> is the length of the vector r12, and u^<sup>1</sup> and u^<sup>2</sup> are the unit vectors parallel to the axes of revolution of molecule 1 and molecule 2. The parameter σ<sup>0</sup> is the length of the axis perpendicular to the axis of revolution, that is, the minor axis of a calamitic ellipsoid of revolution and the major axis of a discotic ellipsoid of revolution. The strength

2

( ) " # <sup>2</sup> (A.8a)

<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>

u^<sup>1</sup> � u^<sup>2</sup>

1 � χ<sup>0</sup>

2

<sup>2</sup> ps <sup>33</sup> � ps 11 � � sin 2<sup>θ</sup> � <sup>p</sup>

<sup>6</sup> <sup>þ</sup> <sup>~</sup>η<sup>3</sup> 2 sin <sup>2</sup>

w\_ irr ¼ 4η þ 2

γ cos 2θ, (A.4c)

γ sin 2θ, (A.4d)

γ

γ<sup>2</sup> (A.6)

, (A.7)

(A.5)

� � ¼ �~γ2<sup>γ</sup> sin 2θ: (A.4e)

s <sup>31</sup> cos 2θ

γ2,

D E

cos 2θ

2θ

σ0 r<sup>12</sup> � σð Þþ ^r12; u^1; u^<sup>2</sup> σ<sup>0</sup> � �<sup>18</sup>

<sup>2</sup><sup>θ</sup> <sup>þ</sup> <sup>~</sup>η<sup>2</sup> 2

<sup>3</sup> <sup>þ</sup> <sup>2</sup>~η<sup>3</sup> sin <sup>2</sup>

� �

� �

Sten Sarman\*, Yonglei Wang and Aatto Laaksonen Department of Materials and Environmental Chemistry, Arrhenius Laboratory, Stockholm University, Stockholm, Sweden

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

© 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, provided the original work is properly cited.
