5.1 Heat flow algorithm

A temperature gradient can be maintained by keeping different regions, 1 and 2, of a system at different temperatures, see Figure 7. Mathematically, this can be brought about by adding thermostatting terms for each of the regions 1 and 2 to the ordinary Newtonian equations of motion [29],

$$m\ddot{\mathbf{r}}\_i = \mathbf{F}\_i - \hat{w}\_{1i}a\_1m\dot{\mathbf{r}}\_i - \hat{w}\_{2i}a\_2m\dot{\mathbf{r}}\_i - \zeta\_i \tag{14}$$

### Figure 7.

A temperature gradient is maintained by thermostatting one region (dark gray) of the system at a high temperature and another region (light gray) at a low temperature, whereby heat will flow from the high temperature region to the low temperature region. Reproduced from Ref. [15], with permission from the PCCP Owner Societies.

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

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 using the fact that the weighted kinetic energies are constant:

$$\frac{1}{2} \sum\_{i=1}^{N} \hat{w}\_{1i} m \dot{\mathbf{r}}\_i^2 = E\_{k1} \tag{15a}$$

and

5. Heat conduction

Figure 6.

Figure 7.

132

Owner Societies.

5.1 Heat flow algorithm

Non-Equilibrium Particle Dynamics

ordinary Newtonian equations of motion [29],

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

A temperature gradient can be maintained by keeping different regions, 1 and 2, of a system at different temperatures, see Figure 7. Mathematically, this can be brought about by adding thermostatting terms for each of the regions 1 and 2 to the

A temperature gradient is maintained by thermostatting one region (dark gray) of the system at a high temperature and another region (light gray) at a low temperature, whereby heat will flow from the high temperature region to the low temperature region. Reproduced from Ref. [15], with permission from the PCCP

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

mr€<sup>i</sup> ¼ F<sup>i</sup> � w^ <sup>1</sup><sup>i</sup>α1mr\_<sup>i</sup> � w^ <sup>2</sup><sup>i</sup>α2mr\_<sup>i</sup> � ζ, (14)

$$\frac{1}{2} \sum\_{i=1}^{N} \hat{w}\_{2i} m \dot{\mathbf{r}}\_i^2 = E\_{k2},\tag{15b}$$

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

### 5.2 Heat flow in nematic liquid crystals

The heat flow in an axially symmetric system such as a nematic liquid crystal or a cholesteric liquid crystal is given by

$$
\langle \mathbf{J}\_Q \rangle = - \left[ \lambda\_{\parallel} \text{ } \mathbf{m} \mathbf{n} + \lambda\_{\perp \perp} (\mathbf{1} - \mathbf{m} \mathbf{n}) \right] \cdot \frac{\nabla T}{T}, \tag{16}
$$

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

$$\begin{split} \dot{\boldsymbol{w}}\_{irr} &= -\left< \mathbf{J}\_{Q} \right> \cdot \frac{\nabla T}{T} = \frac{\mathbf{1}}{T^{2}} \left[ \boldsymbol{\lambda}\_{\perp \perp} \nabla T \cdot \nabla T + \left( \boldsymbol{\lambda}\_{\parallel \parallel} \parallel - \boldsymbol{\lambda}\_{\perp \perp} \right) (\mathbf{n} \cdot \nabla T)^{2} \right] \\ &= \left| \frac{\boldsymbol{d}\_{r} T}{T} \right|^{2} \left[ \boldsymbol{\lambda}\_{\perp \perp} + \left( \boldsymbol{\lambda}\_{\parallel \parallel} - \boldsymbol{\lambda}\_{\perp \perp} \right) \cos^{2} \theta \right], \end{split} \tag{17}$$

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

### Figure 8.

A schematic view of a nematic liquid crystal subject to a temperature gradient is shown. The temperature gradient T points in the z-direction, and the director n lies in the zx-plane forming an angle θ to the z-axis. Then a torque Γ arises in the direction of the y-axis. Reproduced from Ref. [15], with permission from the PCCP Owner Societies.

The director orientation can be determined by simulating systems, where a temperature gradient and a heat flow are maintained by thermostatting different parts of the system at different temperatures by using the above simulation algorithm (14). Such simulations have shown that the director of nematic liquid crystals consisting of soft calamitic ellipsoids tends to align perpendicularly to the tempera-

The angular distribution, pð Þ θ , of the director of a nematic liquid crystal consisting of soft calamitic ellipsoids around the temperature gradient where the angle between the director and the temperature gradient is denoted

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

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

consisting of discotic ellipsoids tends to align parallel to the temperature gradient. Thus, the energy dissipation rate is minimal in both cases. Moreover, if the director is constrained to attain a fixed orientation between the perpendicular and parallel orientation relative to temperature gradient by applying the Lagrangian constraint algorithm (3), the torque exerted can be obtained. Then, it is found that this torque turns the director of a calamitic system toward the perpendicular orientation and the director of a discotic system toward the parallel orientation. The same orientation behavior of the directors of calamitic and discotic nematic liquid crystals relative to the temperature gradient was observed in an earlier work [21]. However, then the Evans heat flow algorithm [22] was applied where a fictitious external field under non-Newtonian equations of motion rather than a real temperature gradient drives the heat flow. Therefore, it was not possible to determine whether the orientation phenomena were a real effect or a consequence of the non-Newtonian

There are also some early experimental works on the orientation of the director of nematic liquid crystals relative to temperature gradients [9–14] that probably support the conclusions of these heat flow simulations. Unfortunately, it is very difficult to carry out these experiments because if the temperature gradient is too large, there will be convection in the system, and if the temperature gradient is too small, it will not be strong enough to overcome the elastic torques or the surface

Finally, one example where the director orientation relative to a temperature gradient definitely is the one that minimizes the irreversible energy dissipation rate is a cholesteric liquid crystal. In this system, the director rotates in space around the cholesteric axis forming a spiral structure. Then experimental studies, where a temperature gradient is applied, have shown that the cholesteric axis orients parallel to the temperature gradient, whereby the energy dissipation rate is minimized since the heat conductivity is greater in the direction perpendicular to the cholesteric axis than in the parallel direction. Moreover, the whole spiral structure starts rotating in time. This phenomenon is known as thermomechanical coupling [3, 4, 18–20, 30, 31].

torques. Therefore, these experiments are not fully conclusive.

ture gradient, see Figure 9, whereas the director of nematic liquid crystals

by θ. Reproduced from Ref. [15], with permission from the PCCP Owner Societies.

synthetic equations of motion.

135

Figure 9.

λk k>λ⊥⊥, as in a nematic liquid crystal consisting of calamitic molecules, the heat current density and thereby w\_ irr are maximal when the temperature gradient and the director are parallel and minimal when they are perpendicular to each other. Conversely, when λk k < λ⊥⊥, as in a nematic liquid crystal consisting of discotic molecules, the heat current density and the irreversible energy dissipation rate are maximal when the director is perpendicular to the temperature gradient and minimal when it is parallel to the temperature gradient.

The temperature gradient exerts a torque on the molecules around an axis perpendicular to itself and perpendicular to the director, see Figure 8. This torque must be zero in the parallel and perpendicular orientations due to the symmetry, but it is impossible to determine whether these orientations are stable or unstable. Unfortunately, there is no linear relation between the torque and the temperature gradient since they are pseudovectors and polar vectors, respectively, due to the axial symmetry of the system. However, a quantitative relation between them can be obtained by noting that a cross coupling between a pseudo vector and a symmetric second rank tensor is allowed. The latter quantity can be obtained by forming the dyadic product of the temperature gradient, giving the following relation [15],

$$\Gamma = \mu \mathbf{e} : \mathbf{m} \mathbf{n} \cdot \frac{\nabla T}{T} \frac{\nabla T}{T} = \mu \mathbf{n} \cdot \frac{\nabla T}{T} \mathbf{n} \times \frac{\nabla T}{T} = \mu \left| \frac{\partial\_x T}{T} \right|^2 \cos \theta \sin \theta \mathbf{e}\_\eta = \frac{1}{2} \mu \left| \frac{\partial\_x T}{T} \right|^2 \sin 2\theta \mathbf{e}\_\eta,\tag{18}$$

where Γ is the torque density, μ is a cross coupling coefficient, and ε is the Levi-Civita tensor. The third equality is obtained by assuming that the temperature gradient points in the z-direction, and the director lies in the zx-plane, see Figure 8, whereby θ becomes the angle between these two vectors. This relation fulfills the symmetry condition according to which the torque must be zero when the director is parallel or perpendicular to the temperature gradient. Moreover, the torque is proportional to the square of the temperature gradient for given angle θ. Note also that, this relation is analogous to the relation between the strain rate and the antisymmetric pressure in planar elongational flow (12).

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

Figure 9.

λk k>λ⊥⊥, as in a nematic liquid crystal consisting of calamitic molecules, the heat current density and thereby w\_ irr are maximal when the temperature gradient and the director are parallel and minimal when they are perpendicular to each other. Conversely, when λk k < λ⊥⊥, as in a nematic liquid crystal consisting of discotic molecules, the heat current density and the irreversible energy dissipation rate are maximal when the director is perpendicular to the temperature gradient and mini-

A schematic view of a nematic liquid crystal subject to a temperature gradient is shown. The temperature gradient T points in the z-direction, and the director n lies in the zx-plane forming an angle θ to the z-axis. Then a torque Γ arises in the direction of the y-axis. Reproduced from Ref. [15], with permission from the

The temperature gradient exerts a torque on the molecules around an axis perpendicular to itself and perpendicular to the director, see Figure 8. This torque must be zero in the parallel and perpendicular orientations due to the symmetry, but it is impossible to determine whether these orientations are stable or unstable. Unfortunately, there is no linear relation between the torque and the temperature gradient since they are pseudovectors and polar vectors, respectively, due to the axial symmetry of the system. However, a quantitative relation between them can be obtained by noting that a cross coupling between a pseudo vector and a symmetric second rank tensor is allowed. The latter quantity can be obtained by forming the dyadic product of the temperature gradient, giving the following

> ∂zT T

where Γ is the torque density, μ is a cross coupling coefficient, and ε is the Levi-

Civita tensor. The third equality is obtained by assuming that the temperature gradient points in the z-direction, and the director lies in the zx-plane, see Figure 8, whereby θ becomes the angle between these two vectors. This relation fulfills the symmetry condition according to which the torque must be zero when the director is parallel or perpendicular to the temperature gradient. Moreover, the torque is proportional to the square of the temperature gradient for given angle θ. Note also that, this relation is analogous to the relation between the strain rate and the

 

2

cos <sup>θ</sup> sin <sup>θ</sup> <sup>e</sup><sup>y</sup> <sup>¼</sup> <sup>1</sup>

2 μ ∂zT T 

 

2

sin 2θ ey,

(18)

mal when it is parallel to the temperature gradient.

relation [15],

Figure 8.

PCCP Owner Societies.

Non-Equilibrium Particle Dynamics

Γ ¼ με : nn �

134

∇T T

∇T <sup>T</sup> <sup>¼</sup> <sup>μ</sup><sup>n</sup> � ∇T T n � ∇T <sup>T</sup> <sup>¼</sup> <sup>μ</sup>

antisymmetric pressure in planar elongational flow (12).

The angular distribution, pð Þ θ , of the director of a nematic liquid crystal consisting of soft calamitic ellipsoids around the temperature gradient where the angle between the director and the temperature gradient is denoted by θ. Reproduced from Ref. [15], with permission from the PCCP Owner Societies.

The director orientation can be determined by simulating systems, where a temperature gradient and a heat flow are maintained by thermostatting different parts of the system at different temperatures by using the above simulation algorithm (14). Such simulations have shown that the director of nematic liquid crystals consisting of soft calamitic ellipsoids tends to align perpendicularly to the temperature gradient, see Figure 9, whereas the director of nematic liquid crystals consisting of discotic ellipsoids tends to align parallel to the temperature gradient. Thus, the energy dissipation rate is minimal in both cases. Moreover, if the director is constrained to attain a fixed orientation between the perpendicular and parallel orientation relative to temperature gradient by applying the Lagrangian constraint algorithm (3), the torque exerted can be obtained. Then, it is found that this torque turns the director of a calamitic system toward the perpendicular orientation and the director of a discotic system toward the parallel orientation. The same orientation behavior of the directors of calamitic and discotic nematic liquid crystals relative to the temperature gradient was observed in an earlier work [21]. However, then the Evans heat flow algorithm [22] was applied where a fictitious external field under non-Newtonian equations of motion rather than a real temperature gradient drives the heat flow. Therefore, it was not possible to determine whether the orientation phenomena were a real effect or a consequence of the non-Newtonian synthetic equations of motion.

There are also some early experimental works on the orientation of the director of nematic liquid crystals relative to temperature gradients [9–14] that probably support the conclusions of these heat flow simulations. Unfortunately, it is very difficult to carry out these experiments because if the temperature gradient is too large, there will be convection in the system, and if the temperature gradient is too small, it will not be strong enough to overcome the elastic torques or the surface torques. Therefore, these experiments are not fully conclusive.

Finally, one example where the director orientation relative to a temperature gradient definitely is the one that minimizes the irreversible energy dissipation rate is a cholesteric liquid crystal. In this system, the director rotates in space around the cholesteric axis forming a spiral structure. Then experimental studies, where a temperature gradient is applied, have shown that the cholesteric axis orients parallel to the temperature gradient, whereby the energy dissipation rate is minimized since the heat conductivity is greater in the direction perpendicular to the cholesteric axis than in the parallel direction. Moreover, the whole spiral structure starts rotating in time. This phenomenon is known as thermomechanical coupling [3, 4, 18–20, 30, 31]. There are quite a few experimental studies available on this phenomenon, where it has been found in a conclusive way that the cholesteric axis remains parallel to the temperature gradient, so this orientation seems to be stable, and thus the irreversible energy dissipation rate is minimal.

linear regime of a nonequilibrium steady state. Therefore, we have reviewed molecular dynamics simulations and experimental work on director orientation phenomena in nematic liquid crystals and in cholesteric liquid crystals under external dissipative fields such as velocity gradients and temperature gradients. A general observation that we have made is that in all the examples studied, the director of the liquid crystals seems to attain precisely that alignment angle relative to the external dissipative field that minimizes the irreversible energy

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

In a nematic liquid crystal, the director orientation is in the first place determined by a mechanical stability criterion, namely, that the external torques acting on the system must be zero at mechanical equilibrium. This makes it possible to derive an exact relation between the alignment angle relative to the streamlines and the viscosity coefficients in the linear or Newtonian regime of planar elongational flow and of planar Couette flow. Both simulations and experimental measurements imply that the irreversible energy dissipation rate is minimal at this mechanically

It can be shown that the elongation direction is the stable orientation of flow stable calamitic nematic liquid crystals undergoing elongational flow in the linear regime. It can also be shown that the value of the energy dissipation rate is the same in the contraction direction and in the elongation direction, and that this value is either the maximal or the minimal value by using the linear phenomenological relations between the strain rate and the pressure. Simulations of the calamitic soft ellipsoid fluid have shown that the irreversible energy dissipation rate is minimal in

In calamitic nematic liquid crystals, the heat conductivity is larger in the direction parallel to the director than in the perpendicular direction, and the reverse is true for discotic nematic liquid crystals. Thus, the irreversible energy dissipation rate due to the heat flow depends on the angle between the director and the temperature gradient. When a nematic liquid crystal is subjected to a temperature gradient, a torque is exerted on the molecules. Due to symmetry, this torque must be proportional to the square of the temperature gradient and it must be zero when

In simulations of nematic phases of soft ellipsoids under a temperature gradient, it turns out that the director of a calamitic nematic liquid crystal aligns perpendicularly to the temperature gradient, whereas the director of a discotic nematic liquid crystal attains the parallel orientation. In both cases, the irreversible energy dissipation rate is minimal. These simulation results are probably supported by some experimental measurements, but they are difficult to carry out in practice so they

Finally, one system where there is definitely a conclusive experimental evidence

Thus, the director orientation relative to a temperature gradient also follows the variational principle even though there is a quadratic coupling between the torque and the temperature gradient. However, the temperature gradients are rather low

for the fact that the director attains the orientation that minimizes the energy dissipation rate due to a temperature gradient is the cholesteric liquid crystal. The cholesteric axis of droplets of cholesteric liquid crystals orient parallel to a temperature gradient and the director rotates. This is a well-established phenomenon observed in studies of thermomechanical coupling, and since the heat conductivity is lower in the direction of the cholesteric axis than in the perpendicular direction,

the director is parallel or perpendicular to this gradient.

the energy dissipation rate is minimal in this case.

so we are still in the linear regime.

137

dissipation rate.

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

stable orientation.

the elongation direction.

are not fully conclusive.

We can consequently conclude that the orientation of the director relative to the temperature gradient is consistent with the variational principle [1] even though the coupling between the torque and the temperature gradient is quadratic rather than linear and the system is inhomogeneous. However, the temperature gradient is rather weak, so we still remain in the linear regime.
