4. Planar elongational flow

### 4.1 The SLLOD equations of motion for planar elongational flow

A planar irrotational elongational flow arises when an incompressible liquid expands in the x-direction and contracts in the z-direction, see Figure 4. Then, the velocity field and the strain rate are given by u ¼ γð Þ xe<sup>x</sup> � ze<sup>z</sup> and ∇u ¼ ∇u ¼ γð Þ exe<sup>x</sup> � eze<sup>z</sup> . Planar elongational flow can be studied by applying a special version of the SLLOD equations of motion. Then the problem is that, if the simulation cell is elongated in the x-direction and contracted in the zdirection, the simulation can only continue until the width in the z-direction is equal to twice the range of the interaction potential. However, if the angle between the elongation direction and the x-axis is set to an angle, φ, the periodic lattice of originally quadratic simulation cells is gradually deformed to a lattice of cells shaped like parallelograms. Then, it can be shown that for a special value of this angle, φ0, the lattice of parallelograms can be remapped onto the original quadratic lattice after a certain time period, so that the simulation becomes continuous, that is, the Kraynik-Reinelt boundary conditions, see Figure 5 and Refs. [8, 25–28] for details. Then, if the angle between the elongation direction and the x-axis is equal to φ0, the velocity gradient becomes ∇u ¼ γ e<sup>0</sup> xe0 <sup>x</sup> � e<sup>0</sup> ze0 z , where e<sup>0</sup> <sup>x</sup> ¼ e<sup>x</sup> cos φ<sup>0</sup> � e<sup>z</sup> sin φ<sup>0</sup> and e<sup>0</sup> <sup>z</sup> ¼ e<sup>x</sup> sin φ<sup>0</sup> þ e<sup>y</sup> cos φ<sup>0</sup> are the elongation and contraction directions. Inserting this gradient in the SLLOD equations of motion gives,

$$\dot{\mathbf{r}}\_i = \frac{\mathbf{p}\_i}{m} + \mathbf{r}\_i \cdot \nabla \mathbf{u} = \frac{\mathbf{p}\_i}{m} + \gamma \mathbf{r}\_i \cdot \left(\mathbf{e}'\_x \mathbf{e}'\_x - \mathbf{e}'\_x \mathbf{e}'\_x\right) \tag{11a}$$

4.2 Planar elongational flow of nematic liquid crystals

mechanically stable, and the 90° orientation is unstable.

w\_ irr ¼ �P : ∇u ¼ 4η þ 2

energy dissipation rate in the linear regime,

around the elongation direction is displayed.

131

pa 2 ¼ �~γ<sup>2</sup>

can continue. Reproduced from Ref. [8], with permission from the PCCP Owner Societies.

see Appendix 1,

Figure 5.

The director alignment angle is in the first place determined by the mechanical stability in the same way as in shear flow whereby the antisymmetric pressure must be zero. In the linear or Newtonian regime, the alignment angle can be found by using the following relation between the antisymmetric pressure and the strain rate,

The Kraynik-Reinelt boundary conditions. The original simulation cell is square 1. When the angle between the elongation direction and the horizontal direction is equal to φ0, square 1 is deformed to a parallelogram, which, after a given time interval, becomes the dashed parallelogram, partially covering the squares (1–6). Then the triangles a', b', and c' in the parallelogram are periodic copies of the triangles a, b, and c in square 1. If the primed triangles are moved to the corresponding unprimed triangles, a square is recovered and the simulation

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

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γ

where θ denotes the angle between the director and the elongation direction, and

~γ<sup>2</sup> is the cross coupling coefficient between the antisymmetric pressure and the strain rate. From this expression, it is apparent that the alignment angle must be either 0 or 90°, that is, where the torque exerted by the strain rate is equal to zero. For a flow stable calamitic nematic liquid crystal, the cross coupling coefficient ~γ<sup>2</sup> is

negative [6], so that the 0° orientation parallel to the elongation direction is

Just as in planar Couette flow or shear flow, the connection to the variational principle can be made by considering the algebraic expression for the irreversible

If the viscosity coefficient ~η<sup>3</sup> is positive, this expression is minimal when θ is equal to 45° but this orientation is excluded because of the mechanical stability (12). If ~η<sup>3</sup> on the other hand is negative, this expression attains the same minimal value when θ is equal to 0 or 90°, that is, the elongation or contraction direction. Simulations of a nematic phase of calamitic soft ellipsoids have shown that ~η<sup>3</sup> is less than zero [8], so that the energy dissipation rate is minimal in the stable orientation also in this case of planar elongational flow. This is in agreement with the variational principle [1]. See also Figure 6 where the angular distribution of the director

~η1

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

<sup>γ</sup><sup>2</sup>

2θ

: (13)

<sup>2</sup> sin 2θ, (12)

and

$$
\dot{\mathbf{p}}\_i = \mathbf{F}\_i - \mathbf{p}\_i \cdot \nabla \mathbf{u} - a\mathbf{p}\_i - \mathfrak{B} = \mathbf{F}\_i - \gamma \mathbf{p}\_i \cdot \left( \mathbf{e}'\_x \mathbf{e}'\_x - \mathbf{e}'\_x \mathbf{e}'\_x \right) - a\mathbf{p}\_i - \mathfrak{B},\tag{11b}
$$

where r<sup>i</sup> and p<sup>i</sup> are the position and peculiar momentum, that is, the momentum relative to the macroscopic streaming velocity, of molecule i, F<sup>i</sup> is the force exerted on molecule i by the other molecules, m is the molecular mass, u is the streaming velocity, γ is the strain rate, α is a thermostting multiplier and β is a constraint multiplier used to conserve the linear momentum.

#### Figure 4.

Schematic representation of a nematic phase of a soft ellipsoid fluid undergoing irrotational extensional flow. The system is elongated in the horizontal direction and contracted in the vertical direction while the volume is constant. The molecules tend to be aligned in the elongation direction.

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

Figure 5.

4. Planar elongational flow

Non-Equilibrium Particle Dynamics

∇u ¼ γ e<sup>0</sup>

and

Figure 4.

130

xe0 <sup>x</sup> � e<sup>0</sup> ze0 z , where e<sup>0</sup>

SLLOD equations of motion gives,

conserve the linear momentum.

<sup>r</sup>\_<sup>i</sup> <sup>¼</sup> <sup>p</sup><sup>i</sup> m

p\_ <sup>i</sup> ¼ F<sup>i</sup> � p<sup>i</sup> � ∇u � αp<sup>i</sup> � β ¼ F<sup>i</sup> � γp<sup>i</sup> � e<sup>0</sup>

constant. The molecules tend to be aligned in the elongation direction.

4.1 The SLLOD equations of motion for planar elongational flow

A planar irrotational elongational flow arises when an incompressible liquid

<sup>x</sup> ¼ e<sup>x</sup> cos φ<sup>0</sup> � e<sup>z</sup> sin φ<sup>0</sup> and e<sup>0</sup>

þ γr<sup>i</sup> � e<sup>0</sup>

xe0 <sup>x</sup> � e<sup>0</sup> ze0 z

xe0 <sup>x</sup> � e<sup>0</sup> ze0 z (11a)

� <sup>α</sup>p<sup>i</sup> � <sup>β</sup>, (11b)

are the elongation and contraction directions. Inserting this gradient in the

m

where r<sup>i</sup> and p<sup>i</sup> are the position and peculiar momentum, that is, the momentum relative to the macroscopic streaming velocity, of molecule i, F<sup>i</sup> is the force exerted on molecule i by the other molecules, m is the molecular mass, u is the streaming velocity, γ is the strain rate, α is a thermostting multiplier and β is a constraint multiplier used to

Schematic representation of a nematic phase of a soft ellipsoid fluid undergoing irrotational extensional flow. The system is elongated in the horizontal direction and contracted in the vertical direction while the volume is

<sup>þ</sup> <sup>r</sup><sup>i</sup> � <sup>∇</sup><sup>u</sup> <sup>¼</sup> <sup>p</sup><sup>i</sup>

<sup>z</sup> ¼ e<sup>x</sup> sin φ<sup>0</sup> þ e<sup>y</sup> cos φ<sup>0</sup>

expands in the x-direction and contracts in the z-direction, see Figure 4. Then, the velocity field and the strain rate are given by u ¼ γð Þ xe<sup>x</sup> � ze<sup>z</sup> and ∇u ¼ ∇u ¼ γð Þ exe<sup>x</sup> � eze<sup>z</sup> . Planar elongational flow can be studied by applying a special version of the SLLOD equations of motion. Then the problem is that, if the simulation cell is elongated in the x-direction and contracted in the zdirection, the simulation can only continue until the width in the z-direction is equal to twice the range of the interaction potential. However, if the angle between the elongation direction and the x-axis is set to an angle, φ, the periodic lattice of originally quadratic simulation cells is gradually deformed to a lattice of cells shaped like parallelograms. Then, it can be shown that for a special value of this angle, φ0, the lattice of parallelograms can be remapped onto the original quadratic lattice after a certain time period, so that the simulation becomes continuous, that is, the Kraynik-Reinelt boundary conditions, see Figure 5 and Refs. [8, 25–28] for details. Then, if the angle between the elongation direction and the x-axis is equal to φ0, the velocity gradient becomes

The Kraynik-Reinelt boundary conditions. The original simulation cell is square 1. When the angle between the elongation direction and the horizontal direction is equal to φ0, square 1 is deformed to a parallelogram, which, after a given time interval, becomes the dashed parallelogram, partially covering the squares (1–6). Then the triangles a', b', and c' in the parallelogram are periodic copies of the triangles a, b, and c in square 1. If the primed triangles are moved to the corresponding unprimed triangles, a square is recovered and the simulation can continue. Reproduced from Ref. [8], with permission from the PCCP Owner Societies.

### 4.2 Planar elongational flow of nematic liquid crystals

The director alignment angle is in the first place determined by the mechanical stability in the same way as in shear flow whereby the antisymmetric pressure must be zero. In the linear or Newtonian regime, the alignment angle can be found by using the following relation between the antisymmetric pressure and the strain rate, see Appendix 1,

$$
\langle p\_2^a \rangle = -\tilde{\gamma}\_2 \frac{\chi}{2} \sin 2\theta,\tag{12}
$$

where θ denotes the angle between the director and the elongation direction, and ~γ<sup>2</sup> is the cross coupling coefficient between the antisymmetric pressure and the strain rate. From this expression, it is apparent that the alignment angle must be either 0 or 90°, that is, where the torque exerted by the strain rate is equal to zero. For a flow stable calamitic nematic liquid crystal, the cross coupling coefficient ~γ<sup>2</sup> is negative [6], so that the 0° orientation parallel to the elongation direction is mechanically stable, and the 90° orientation is unstable.

Just as in planar Couette flow or shear flow, the connection to the variational principle can be made by considering the algebraic expression for the irreversible energy dissipation rate in the linear regime,

$$
\dot{\boldsymbol{\omega}}\_{irr} = -\overline{\mathbf{P}} : \overline{\nabla \mathbf{u}} = \left( 4\eta + 2\frac{\tilde{\eta}\_1}{3} + 2\tilde{\eta}\_3 \cos^2 2\theta \right) \boldsymbol{\chi}^2. \tag{13}
$$

If the viscosity coefficient ~η<sup>3</sup> is positive, this expression is minimal when θ is equal to 45° but this orientation is excluded because of the mechanical stability (12). If ~η<sup>3</sup> on the other hand is negative, this expression attains the same minimal value when θ is equal to 0 or 90°, that is, the elongation or contraction direction. Simulations of a nematic phase of calamitic soft ellipsoids have shown that ~η<sup>3</sup> is less than zero [8], so that the energy dissipation rate is minimal in the stable orientation also in this case of planar elongational flow. This is in agreement with the variational principle [1]. See also Figure 6 where the angular distribution of the director around the elongation direction is displayed.

where m is the molecular mass, r\_<sup>i</sup> and r€<sup>i</sup> are the velocity and the acceleration of molecule i, and F<sup>i</sup> is the force exerted on molecule i by the other molecules. The thermostatting terms are w^ <sup>1</sup>iα1mr\_<sup>i</sup> and w^ <sup>2</sup>iα2mr\_<sup>i</sup> where w^ <sup>1</sup><sup>i</sup> and w^ <sup>2</sup><sup>i</sup> are two normalized weight functions. These terms are actually similar to the thermostatting term in Eq. (4b), but here region 1 and region 2 are thermostatted separately at different temperatures. This is achieved by letting the weight functions be Gaussian functions centered in region 1 and 2, respectively, and with decay lengths that are considerably shorter than the distance between these two regions. In this way, only the molecules in region 1 contribute to the temperature in region 1, and only the molecules in region 2 contribute to the temperature in region 2. The molecules far away from the centers of these regions move according to the ordinary Newtonian equations of motion. Note that, it is not necessary to use Gaussian weight functions; it is possible to use any function with a maximum and a rather short decay length. The parameters α<sup>1</sup> and α<sup>2</sup> are thermostatting multipliers in the same way as the multiplier α in Eqs. (4b) and (5), but here, they thermostat the regions 1 and 2 separately. They are determined by applying Gauss's principle of least constraints

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

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using the fact that the weighted kinetic energies are constant:

and

where J<sup>Q</sup>

133

w\_ irr ¼ � J<sup>Q</sup>

� � �

∇T <sup>T</sup> <sup>¼</sup> <sup>1</sup>

> <sup>¼</sup> <sup>∂</sup>zT T � � � �

1 2 ∑ N i¼1

1 2 ∑ N i¼1

whole system is constant. It goes to zero in the thermodynamic limit.

5.2 Heat flow in nematic liquid crystals

JQ

a cholesteric liquid crystal is given by

w^ <sup>1</sup>imr\_ 2

w^ <sup>2</sup>imr\_ 2

where Ek<sup>1</sup> and Ek<sup>2</sup> are the weighted kinetic energies for region 1 and 2, respectively. The algebraic expressions for the thermostatting multipliers are given in Ref. [15]. The parameter ζ is a multiplier determined in such a way that the linear momentum of the

The heat flow in an axially symmetric system such as a nematic liquid crystal or

� � is the heat current density, <sup>λ</sup>k k is the heat conductivity parallel to the

<sup>T</sup><sup>2</sup> <sup>λ</sup>⊥⊥∇<sup>T</sup> � <sup>∇</sup><sup>T</sup> <sup>þ</sup> <sup>λ</sup>k k � <sup>λ</sup>⊥⊥ � �ð Þ <sup>n</sup> � <sup>∇</sup><sup>T</sup> <sup>2</sup> h i

<sup>2</sup> <sup>λ</sup>⊥⊥ <sup>þ</sup> <sup>λ</sup>k k � <sup>λ</sup>⊥⊥ � � cos <sup>2</sup><sup>θ</sup> � �,

where the last equality has been obtained by assuming that the director lies in the zx-plane forming an angle θ with the temperature gradient, see Figure 8. When

� � ¼ � <sup>λ</sup>k knn <sup>þ</sup> <sup>λ</sup>⊥⊥ð Þ <sup>1</sup> � nn � � �

director of an ordinary achiral nematic liquid crystal or parallel to the cholesteric axis of a cholesteric liquid crystal, λ⊥⊥ is the heat conductivity perpendicular to the director of a nematic liquid crystal or perpendicular to the cholesteric axis of a cholesteric liquid crystal,T is the absolute temperature, and n is the director. Then, the irreversible energy dissipation rate of the system due to the heat flow becomes,

<sup>i</sup> ¼ Ek<sup>1</sup> (15a)

<sup>i</sup> ¼ Ek2, (15b)

∇T

<sup>T</sup> , (16)

(17)

Figure 6.

The angular distribution, pð Þ θ , of the director of a calamitic nematic liquid crystal consisting of soft ellipsoids around the elongation direction where the angle between the director and the elongation direction is denoted by θ. Reproduced from Ref. [8], with permission from the PCCP Owner Societies.
