Abstract
The purpose of the work presented in this chapter is to test a recently proven variational principle according to which the irreversible energy dissipation rate is minimal in the linear regime of a nonequilibrium steady state. This test is carried out by performing molecular dynamics simulations of liquid crystals subject to velocity gradients and temperature gradients. Since the energy dissipation rate varies with the orientation of the director of the liquid crystal relative to these gradients and is minimal at certain orientations, this is a stringent test of the variational principle. More particularly, a nematic liquid crystal model based on the Gay-Berne potential, which can be regarded as a Lennard-Jones fluid generalized to elliptical molecular cores, is studied under planar Couette flow, planar elongational flow, and under a temperature gradient. It is found that the director of a nematic liquid crystal consisting of rod-like molecules lies in the vorticity plane at an angle of about 20° to the stream lines in the planar Couette flow. In the elongational flow, it is parallel to the elongation direction, and it is perpendicular to the temperature gradient in a heat flow. These orientations are the ones where the irreversible energy dissipation rate is minimal, so that the variational principle is fulfilled in these three cases.
Keywords
- liquid crystals
- nonequilibrium molecular dynamics simulation
- shear flow
- elongational flow
- heat conduction
- alignment phenomena
- minimal energy dissipation rate
1. Introduction
For a system in thermodynamic equilibrium, there is a variational principle according to which the free energy is minimal, that is, the Helmholtz free energy when the volume, temperature, and number of particles are constant, Gibbs free energy when the pressure, temperature, and number of particles are constant, etc. On the other hand, for systems driven away from equilibrium by an external dissipative field such as a velocity gradient, temperature gradient, chemical potential gradient or an electrical potential gradient doing irreversible work that is converted to heat, there has not been any variational principle to date. However, a theorem originally proposed by Ilya Prigogine stating that a quantity, known as the irreversible energy dissipation rate,
This theorem is not only of basic scientific interest but also of technological and practical interest since shear fields, temperature gradients, concentration gradients, or chemical potential gradients and electrical potential gradients are common examples of external dissipative fields that are ubiquitous in industrial applications and in everyday life. For example, in a lubricated bearing, a planar Couette flow arises in the lubricant in the narrow space between two surfaces rotating at different angular velocities, and
One way of testing this principle is to perform molecular dynamics simulations of microscopic model systems, but then it is hard to find a suitable model system that is easy to analyze. However, liquid crystals are particularly interesting for this purpose because the transport properties and thereby
The simplest kind of liquid crystal is the nematic liquid crystal [3, 4]. It consists of rod-like or plate-like molecules oriented in a certain direction—the director—but there is no translational order, see Figure 1. A nematic liquid crystal cannot support shear stresses, so it is by definition a liquid, but it can support torques, which is the basis for various orientation phenomena relative to external fields. A special case of a nematic liquid crystal is the cholesteric liquid crystal, where the director rotates in space around an axis perpendicular to itself—the cholesteric axis or the optical axis. The spatial rotation period or the pitch is of the order of 1 μm or about 500 molecular diameters. A cholesteric liquid crystal is different from its mirror image, and it is formed by chiral molecules.
There is some theoretical and experimental evidence indicating that the director comes to rest in an orientation where the irreversible energy dissipation rate is minimal in accordance with the variational principle. More specifically, such orientation phenomena have been observed in simulations of shear flow or planar Couette flow [5, 6], in experimental measurements of the viscosity [7] in this flow geometry, and in simulations of planar elongational flow [8]. In the latter case, it is actually possible to prove that the energy dissipation rate must be either minimal or maximal in a steady state in the linear or Newtonian regime by using the linear phenomenological relations between the velocity gradient and the shear stress.
In the case of a nematic liquid crystal subject to a temperature gradient, there are quite a few early experimental works [9, 10, 11, 12, 13, 14] that might imply that the director of a liquid crystal consisting of rod-like molecules orients perpendicularly to this gradient. This means that the heat flow is minimized, since the heat conductivity is minimal in this orientation. Unfortunately, the results of these works are not wholly conclusive because the underlying experiments are very hard to carry out. On the other hand, molecular dynamics simulation of nematic phases of calamitic and discotic soft ellipsoids [15, 16, 17] clearly show that the directors orient perpendicularly and parallel, respectively, to the temperature gradient, so that the heat flow and thereby
This chapter is organized in the following way: in Section 2, the basic theory is outlined, and in Sections 3, 4, and 5, molecular dynamics simulation results and experimental measurements on the director orientation and the irreversible energy dissipation rate are presented and discussed for shear flow or planar Couette flow, planar elongational flow and heat conduction, respectively. In Section 6, the effects of the thermostat are discussed, and finally in Section 7, there is a conclusion. Some background theory is given in the Appendices.
2. Basic theory
2.1 Order parameter, director, and director angular velocity
In order to describe transport properties of a liquid crystal, we must first define the order parameter, the director, and the director angular velocity. In an axially symmetric system such as a nematic or a smectic
where
The director angular velocity is given by
2.2 Director constraint algorithm
Since the molecules studied in the work presented in this chapter are modeled by the Gay-Berne potential, which can be regarded as a Lennard-Jones potential generalized to elliptical molecular cores, see Appendix 2, they are rigid bodies. Therefore, the Euler equations are applied in angular space. Moreover, since the purpose often is to find the stable orientations of the director relative to an external dissipative field, it is interesting to calculate the torque exerted on the liquid crystal, when the director attains various fixed angles relative to this field. This can be done by adding Gaussian constraints to the Euler equations [21],
where
3. Shear flow
3.1 The SLLOD equations of motion for shear flow
In order to study shear flow and to calculate the viscosity and director alignment angles relative to the streamlines, it is convenient to apply the SLLOD equations of motion [22]. The name SLLOD stems from the similarity to the Dolls equation of motion derived from the Dolls tensor Hamiltonian. They are synthetic equations of motion that can be used to calculate the viscosity in the linear regime. On the other hand, the idea behind the SLLOD equations of motion is very simple: The velocity of the molecules is divided into the streaming velocity and the thermal velocity. The thermal velocity is related to the temperature, and the streaming velocity is the macroscopic external velocity. The SLLOD equations of motion are an exact description of adiabatic planar Couette flow and a very good approximation of shear flow at constant temperature both in the linear and nonlinear regime. The SLLOD equations are expressed in the following way:
and
where
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 thermostat is applied.
3.2 Shear flow of nematic liquid crystals
In a nematic liquid crystal undergoing shear flow, the alignment angle,
where
for the preferred alignment angle,
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,
where the definitions of the viscosity coefficients,
or
where
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
and
where
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,
where
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,
If the viscosity coefficient
5. Heat conduction
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],
where
and
where
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
where
where the last equality has been obtained by assuming that the director lies in 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 relation [15],
where
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, 10, 11, 12, 13, 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, 19, 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.
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.
6. Effects of the thermostat
In the above simulations of shear flow and elongational flow, the velocity gradient does work on the system that is converted to heat, which must be removed in order to keep the temperature constant and to maintain a steady state. In a real macroscopic system, this takes place by heat conduction to the system boundaries and this could in principle be arranged in a microscopic simulation cell as well. Unfortunately, this is inconvenient because a temperature gradient of molecular dimensions would make the system inhomogeneous, and thus make it difficult to define the thermodynamic state. Therefore, the temperature is kept constant by forcing the kinetic energy to be a constant of motion by applying a Gaussian thermostat, see Eq. (5). This thermostat was originally devised independently by Hoover
The situation is different in the heat flow simulations because here we actually want a temperature gradient. This gradient is obtained by applying two bar thermostats at different temperatures acting over a limited range and separated by a distance that is long compared to this range, see Figure 7 and Eq. (14). Therefore, the movements of only a small fraction of the molecules are affected by the thermostats, whereas the movements of the majority of the molecules away from the bar thermostats are governed by the ordinary Newtonian equations of motion. Thus, it is reasonable to assume that the influence of the details of the thermostat on the ensemble averages of the phase functions is limited in this case too.
7. Conclusion
The purpose of this work has been to test a recently proven variational principle according to which the irreversible energy dissipation rate is minimal in the 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 dissipation rate.
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 stable orientation.
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 the elongation direction.
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 the director is parallel or perpendicular to this gradient.
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 are not fully conclusive.
Finally, one system where there is definitely a conclusive experimental evidence 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 energy dissipation rate is minimal in this case.
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 so we are still in the linear regime.
Acknowledgments
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 access to Hazelhen based in Germany at Rechenzentrum Stuttgart.
The relation between the velocity gradient,
where
and
where the products involving the Levi-Civita tensor
Application of a planar Couette velocity gradient,
and
where
In a planar elongational flow [8, 26, 27, 28], where the elongation direction is parallel to the
and
If these expressions for the pressure tensor are inserted in the expression for energy dissipation rate (A.1), we obtain
for planar Couette flow and
for planar elongational flow. The subscript
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 commonly used Gay-Berne potential [16, 17, 21],
where
and
where the parameter
The denominators in Eqs. (A.8a) and (A.8b) are never equal to zero because the absolute value of the scalar product
References
- 1.
Evans DJ, Searles DJ, Williams SR. Fundamentals of Classical Statistical Thermodynamics: Dissipation, Relaxation and Fluctuation Theorems. Berlin: Wiley-VCH; 2016 - 2.
de Groot SR, Mazur P. Nonequilibrium Thermodynamics. New York: Dover; 1984 - 3.
Chandrasekhar S. Liquid Crystals. Cambridge: Cambridge University Press; 1992 - 4.
de Gennes PG, Prost J. The Physics of Liquid Crystals. Oxford: Clarendon Press; 1993 - 5.
Sarman S. Microscopic theory of liquid crystal rheology. The Journal of Chemical Physics. 1995; 103 :393 - 6.
Sarman S. Nonequilibrium molecular dynamics of liquid crystal shear flow. The Journal of Chemical Physics. 1995; 103 :10378 - 7.
Jadzyn J, Czechowski G. The shear viscosity minimum of freely flowing nematic liquid crystals. Journal of Physics: Condensed Matter. 2001; 13 :L261 - 8.
Sarman S, Laaksonen A. Molecular dynamics simulation of planar elongational flow in a nematic liquid crystal based on the Gay–Berne potential. Physical Chemistry Chemical Physics. 2015; 17 :3332 - 9.
Stewart GW. X-Ray diffraction intensity of the two liquid phases of para-azoxyanisol. The Journal of Chemical Physics. 1936; 4 :231 - 10.
Stewart GW, Holland DO, Reynolds LM. Orientation of liquid crystals by heat conduction. Physics Review. 1940; 58 :174 - 11.
Picot JJC, Fredrickson AG. Interfacial and electrical effects on thermal conductivity of nematic liquid crystals. Industrial and Engineering Chemistry Fundamentals. 1968; 7 :84 - 12.
Fisher J, Fredrickson AG. Transport processes in anisotropic fluids II. Coupling of momentum and energy transport in a nematic mesophase. Molecular Crystals and Liquid Crystals. 1969; 6 :255 - 13.
Patharkar MN, Rajan VSV, Picot JJC. Interfacial and temperature gradient effects on thermal conductivity of a liquid crystal. Molecular Crystals and Liquid Crystals. 1971; 15 :225 - 14.
Currie PK. The orientation of liquid crystals by temperature gradients. Rheologica Acta. 1973; 12 :165 - 15.
Sarman S, Laaksonen A. Director alignment relative to the temperature gradient in nematic liquid crystals studied by molecular dynamics simulation. Physical Chemistry Chemical Physics. 2014; 16 :14741 - 16.
Gay JG, Berne BJ. Modification of the overlap potential to mimic a linear site?site potential. The Journal of Chemical Physics. 1981; 74 :3316 - 17.
Bates MA, Luckhurst GR. Computer simulation studies of anisotropic systems. XXVI. Monte Carlo investigations of a Gay?Berne discotic at constant pressure. The Journal of Chemical Physics. 1996; 104 :6696 - 18.
Éber N, Jánossy I. An experiment on the thermomechanical coupling in cholesterics. Molecular Crystals and Liquid Crystals Science and Technology. Section A. Molecular Crystals and Liquid Crystals. 1982; 72 :233 - 19.
Oswald P, Dequidt A. Measurement of the continuous Lehmann rotation of cholesteric droplets subjected to a temperature gradient. Physical Review Letters. 2008; 100 :217802 - 20.
Oswald P. Microscopic vs. macroscopic origin of the Lehmann effect in cholesteric liquid crystals. European Physical Journal E: Soft Matter and Biological Physics. 2012; 35 :10 - 21.
Sarman S. Molecular dynamics of heat flow in nematic liquid crystals. The Journal of Chemical Physics. 1994; 101 :480 - 22.
Evans DJ, Morriss GP. Statistical Mechanics of Nonequilibrium Liquids. London: Academic Press; 1990 - 23.
Leslie FM. Some constitutive equations for anisotropic fluids. The Quarterly Journal of Mechanics and Applied Mathematics. 1966; 19 :357 - 24.
Sarman S, Laaksonen A. Flow alignment phenomena in liquid crystals studied by molecular dynamics simulation. The Journal of Chemical Physics. 2009; 131 :144904 - 25.
Kraynik AM, Reinelt DA. Extensional motions of spatially periodic lattices. International Journal of Multiphase Flow. 1992; 18 :1045 - 26.
Baranyai A, Cummings PT. Nonequilibrium molecular dynamics study of shear and shear‐free flows in simple fluids. The Journal of Chemical Physics. 1995; 103 :10217 - 27.
Todd BD, Daivis PJ. Nonequilibrium molecular dynamics simulations of planar elongational flow with spatially and temporally periodic boundary conditions. Physical Review Letters. 1998; 81 :1118 - 28.
Todd BD, Daivis PJ. Homogeneous non-equilibrium molecular dynamics simulations of viscous flow: Techniques and applications. Molecular Simulation. 2007; 33 :189 - 29.
Ikeshoji T, Hafskjold B. Non-equilibrium molecular dynamics calculation of heat conduction in liquid and through liquid-gas interface. Molecular Physics. 1994; 81 :251 - 30.
Leslie FM. Some thermal effects in cholesteric liquid crystals. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 1968; 307 :359 - 31.
Leslie FM. Thermo-mechanical coupling in cholesteric liquid crystals. Symposium of the Faraday Society. 1971; 5 :33 - 32.
Hoover WG, Ladd AJC, Moran B. High-strain-rate plastic flow studied via nonequilibrium molecular dynamics. Physical Review Letters. 1982; 48 :1818 - 33.
Evans DJ, Hoover WG, Failor BH, Moran B, Ladd AJC. Nonequilibrium molecular dynamics via Gauss’s principle of least constraint. Physical Review A. 1983; 28 :1016 - 34.
Hoover WG. Computational Statistical Mechanics. Burlington, MA: Elsevier; 1991 - 35.
Evans DJ, Sarman S. Equivalence of thermostatted nonlinear responses. Physical Review E. 1993; 48 :65 - 36.
Hess S. Transport phenomena in anisotropie fluids and liquid crystals. Journal of Non-Equilibrium Thermodynamics. 1986; 11 :175