3.2 Shear flow of nematic liquid crystals

In a nematic liquid crystal undergoing shear flow, the alignment angle, θ, between the director and the streamlines is determined by a mechanical stability criterion, namely, that the antisymmetric pressure must be zero when no external torques act on the system, that is, that the torques exerted by the vorticity and the strain rate cancel out. This makes it possible to derive a relationship between the alignment angle and the viscosity coefficients in the Newtonian regime by using the linear relation between the pressure tensor and the strain rate, see Refs. [3, 4, 23] and Appendix 1,

$$
\langle p\_2^a \rangle = -\tilde{\gamma}\_1 \frac{\chi}{4} - \tilde{\gamma}\_2 \frac{\chi}{4} \cos 2\theta = 0,\tag{6}
$$

where the definitions of the viscosity coefficients, η, ~η1, ~η2, and ~η3, and the derivation are given in Appendix 1. If the values of the various viscosity coefficients are inserted, it is found that the functional dependence of w\_ irr on θ is similar to that given in Figure 3. This function (8) has been obtained by shear flow simulations applying the SLLOD equations of motion [22] to a nematic phase of calamitic soft ellipsoids, see Refs. [5, 6]. The energy dissipation rate is low close to the preferred alignment angle and high when the director is perpendicular to the streamlines and parallel to the velocity gradient. Then, if we study the distribution of the director, we find that it is centered close to the minimum of w\_ irr. This minimum has also been observed in simulations of shearing nematic phases of discotic soft ellipsoids [5] and when experimentally measured, viscosity coefficients are inserted in the Eqs. (7) and (8) and the resulting alignment angle is determined [7]. Thus the system seems to select the alignment angle that minimizes the irreversible energy dissipation rate in accordance with the variational principle. This also means that the energy dissipation rate (8) must be minimal at the preferred alignment angle, θ0, given by Eq. (7). Thus, the derivative of the function (8) with respect to θ must be zero for θ0, giving an additional relation between the viscosity coefficients and

Reproduced from Ref. [6] with the permission of AIP Publishing.

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

The irreversible energy dissipation rate, w\_ irr, Eq. (8), due to the strain rate of a nematic liquid crystal phase of calamitic soft ellipsoids as a function of the director alignment angle, θ, is obtained by using the Gaussian constraint algorithm (3) to fix the director at various angles relative to the streamlines. The preferred alignment angle attained when no constraints are applied is equal to about 20° which is close to the minimum of w\_ irr.

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

cos 2θ0� ¼ <sup>~</sup>η<sup>2</sup>

where θ<sup>0</sup> has been eliminated by using Eq. (7). The Eqs. (7) and (9) do not coincide but they must still give the same value of θ0. This provides an important cross check for the correctness of the simulation algorithms and experimental methods used to determine the viscosity coefficients and for the computer pro-

2~η<sup>3</sup>

2~η3~γ<sup>1</sup> þ ~η2~γ<sup>2</sup> ¼ 0, (10)

(9)

the alignment angle,

grams used to run the simulations.

or

129

Figure 3.

where ~γ<sup>1</sup> is the twist viscosity, ~γ<sup>2</sup> is the cross coupling coefficient between the antisymmetric pressure and the strain rate, and p<sup>a</sup> 2 is the antisymmetric pressure in the vorticity direction perpendicular to the streamlines and perpendicular to the velocity gradient. The angular brackets denote that the pressure tensor is the ensemble average of a phase function. Then, if pa 2 is equal to zero, we obtain

$$\cos 2\theta\_0 = -\tilde{\gamma}\_1 / \tilde{\gamma}\_2 \tag{7}$$

for the preferred alignment angle, <sup>θ</sup>0, provided that the ratio <sup>~</sup>γ1=~γ2<sup>j</sup> is less than one. Then the liquid crystal is said to be flow stable. This condition is fulfilled in many liquid crystals, and θ<sup>0</sup> is between 10 and 20° both in real systems and in simplified coarse grained model systems such as the soft ellipsoid fluid, see Refs. [5–7], Figure 1, and Appendix 2. Note, however, that for some systems, often near the nematic-smectic A phase transition, the ratio <sup>~</sup>γ1=~γ2<sup>j</sup> is greater than one. This means that there is no orientation angle where the antisymmetric pressure is zero, so that no steady state is attained. Then the liquid crystal is said to be flow unstable and the director will rotate forever [3, 4, 23, 24].

The connection with the variational principle can be made by using the fact that there is an algebraic expression for the irreversible energy dissipation rate, w\_ irr, of a flow stable nematic liquid crystal, given by the dyadic product of the symmetric traceless pressure, P, and the traceless strain rate, ∇u,

$$\dot{\boldsymbol{\nu}}\_{irr} = -\overline{\mathbf{P}} : \overline{\nabla \mathbf{u}} = \left( \eta + \frac{\tilde{\eta}\_1}{6} + \frac{\tilde{\eta}\_3}{2} \sin^2 2\theta + \frac{\tilde{\eta}\_2}{2} \cos 2\theta \right) \mathbf{y}^2,\tag{8}$$

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

Figure 3.

This expression is obtained by applying Gauss's principle of least constraint [22].

This principle is essentially the same as the Lagrange's method for handling constraints. However, Gauss's principle is more general in that it in addition to constraints involving the molecular coordinates also allows handling of some constraints involving the molecular velocities. This is very useful because it makes it possible to keep the kinetic energy constant whereby the temperature also will be constant. It is possible to show that the ensemble averages of the phase functions and the time correlation functions are essentially canonical when a Gaussian ther-

In a nematic liquid crystal undergoing shear flow, the alignment angle, θ, between the director and the streamlines is determined by a mechanical stability criterion, namely, that the antisymmetric pressure must be zero when no external torques act on the system, that is, that the torques exerted by the vorticity and the strain rate cancel out. This makes it possible to derive a relationship between the alignment angle and the viscosity coefficients in the Newtonian regime by using the linear relation between the pressure tensor and the strain rate, see Refs. [3, 4, 23]

> γ <sup>4</sup> � <sup>~</sup>γ<sup>2</sup>

γ 4

where ~γ<sup>1</sup> is the twist viscosity, ~γ<sup>2</sup> is the cross coupling coefficient between

pressure in the vorticity direction perpendicular to the streamlines and perpendicular to the velocity gradient. The angular brackets denote that the pressure

for the preferred alignment angle, <sup>θ</sup>0, provided that the ratio <sup>~</sup>γ1=~γ2<sup>j</sup>

than one. Then the liquid crystal is said to be flow stable. This condition is fulfilled in many liquid crystals, and θ<sup>0</sup> is between 10 and 20° both in real systems and in simplified coarse grained model systems such as the soft ellipsoid fluid, see Refs. [5–7], Figure 1, and Appendix 2. Note, however, that for some systems, often near the nematic-smectic A phase transition, the ratio

 is greater than one. This means that there is no orientation angle where the antisymmetric pressure is zero, so that no steady state is attained. Then the liquid crystal is said to be flow unstable and the director will rotate forever

The connection with the variational principle can be made by using the fact that there is an algebraic expression for the irreversible energy dissipation rate, w\_ irr, of a flow stable nematic liquid crystal, given by the dyadic product of the

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

<sup>2</sup> sin <sup>2</sup>

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

<sup>2</sup> cos 2<sup>θ</sup>

γ2

, (8)

symmetric traceless pressure, P, and the traceless strain rate, ∇u,

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

cos 2θ ¼ 0, (6)

is the antisymmetric

is equal to

is less

2

2

cos 2θ<sup>0</sup> ¼ �~γ1=~γ2, (7)

mostat is applied.

Non-Equilibrium Particle Dynamics

and Appendix 1,

zero, we obtain

<sup>~</sup>γ1=~γ2<sup>j</sup>

128

[3, 4, 23, 24].

3.2 Shear flow of nematic liquid crystals

pa 2 ¼ �~γ<sup>1</sup>

the antisymmetric pressure and the strain rate, and p<sup>a</sup>

tensor is the ensemble average of a phase function. Then, if pa

The irreversible energy dissipation rate, w\_ irr, Eq. (8), due to the strain rate of a nematic liquid crystal phase of calamitic soft ellipsoids as a function of the director alignment angle, θ, is obtained by using the Gaussian constraint algorithm (3) to fix the director at various angles relative to the streamlines. The preferred alignment angle attained when no constraints are applied is equal to about 20° which is close to the minimum of w\_ irr. Reproduced from Ref. [6] with the permission of AIP Publishing.

where the definitions of the viscosity coefficients, η, ~η1, ~η2, and ~η3, and the derivation are given in Appendix 1. If the values of the various viscosity coefficients are inserted, it is found that the functional dependence of w\_ irr on θ is similar to that given in Figure 3. This function (8) has been obtained by shear flow simulations applying the SLLOD equations of motion [22] to a nematic phase of calamitic soft ellipsoids, see Refs. [5, 6]. The energy dissipation rate is low close to the preferred alignment angle and high when the director is perpendicular to the streamlines and parallel to the velocity gradient. Then, if we study the distribution of the director, we find that it is centered close to the minimum of w\_ irr. This minimum has also been observed in simulations of shearing nematic phases of discotic soft ellipsoids [5] and when experimentally measured, viscosity coefficients are inserted in the Eqs. (7) and (8) and the resulting alignment angle is determined [7]. Thus the system seems to select the alignment angle that minimizes the irreversible energy dissipation rate in accordance with the variational principle. This also means that the energy dissipation rate (8) must be minimal at the preferred alignment angle, θ0, given by Eq. (7). Thus, the derivative of the function (8) with respect to θ must be zero for θ0, giving an additional relation between the viscosity coefficients and the alignment angle,

$$\cos 2\theta\_0 - = \frac{\ddot{\eta}\_2}{2\ddot{\eta}\_3} \tag{9}$$

or

$$2\ddot{\eta}\_3\ddot{\gamma}\_1 + \ddot{\eta}\_2\ddot{\gamma}\_2 = 0,\tag{10}$$

where θ<sup>0</sup> has been eliminated by using Eq. (7). The Eqs. (7) and (9) do not coincide but they must still give the same value of θ0. This provides an important cross check for the correctness of the simulation algorithms and experimental methods used to determine the viscosity coefficients and for the computer programs used to run the simulations.
