Overview of the mesoscopic theory of liquid crystals

What are liquid crystals and why are they interesting?

Before we concentrate on the thermodynamics of liquid crystals we shall give some brief introductory remarks. A liquid crystal consists of form-anisotropic particles which show long-range correlations with respect to the orientation of their molecular axes, but which are still able to flow and posses no or only weak long-range correlations with respect to their centers of mass. Both phenomena together lead to a fluid-like material possessing crystal-like properties. The first liquid crystal was reported in 1888 in a letter from Lehmann to Reinitzer, both of them becoming interested in these new materials. Why now should orientational order result in crystal-like properties like birefringence? The answer is connected with the molecular structure of the material. Let us look at a typical liquid crystals like MBBA – n-4'-methoxybenzylidene-n-butylanilin. This is maybe the most prominent liquid crystal (but not the first known), even if industrial use of liquid crystals is usually made of mixtures of many different substances. The space filling model is shown in Fig. 1.
Space filling model of the liquid crystal MBBA
Figure 1: Structural formula and space filling model of MBBA. Note the conjugated double bonds and the "smeared" system of pi-electrons along the long axis of the molecule.

First of all the molecule is rather stiff and possesses an axis ratio of 2.5 (accounting for the form-anisotropy). In addition the conjugated double bonds yield a large pi-electron system distributed along the long axis of the molecule. Thus polarizability differs considerably parallel to the long axes and perpendicular to it. Since the dielectric constant of the material is an ensemble average over all molecules in a volume element the dielectric constant depends clearly on the orientational distribution of the molecules and so does the refraction index. Hence the optical properties depend on direction and polarization of the incident ray, and the material shows birefringence. It is clear that such materials are not only of academic interest but provide lots of interesting applications (displays, optical switches, and much more). A typical phase sequence (this is rather simplified and not valid in general: some liquid crystal show a cholesteric phase or much more smectic phases) of a liquid crystal is shown in Fig. 2.

typical phase sequence of a liquid crystal
Figure 2: Schematical phase sequence of a liquid crystal. From left to right: smectic C phase (tilt angle between layer normal and mean orientation of the molecules), smectic A phase (layered structure, no tilt), nematic phase, isotropic phase

Above the clearing temperature Tc the liquid crystal becomes an isotropic liquid. These properties make liquid crystals an interesting object for the application of thermodynamical methods. However, the practical importance of these materials is due to applications which take advantage of the orientational order in the fluid: we shall come back to that later.

Configuration space and orientation distribution function

The additional degrees of freedom in a liquid crystal are the orientations of the particles. However, in most cases the distinction between head and tail of these constituents is artificial, and we end up with the unit sphere with antipodal points identified as parameter space. Thus we describe the liquid crystal with the help of the nematic space . On this space we introduce the orientational mass density and adopt the following convention: Since the orientation of a particle is described by a pair of unit vectors { n, - n} the orientational mass density is a symmetric (even) function in n. Then the mass of all particles confined in a spatial volume V having orientations in the solid angle is

Consequently we have

and the local orientational distribution can be specified by the orientational distribution function (ODF)

By its very definition f inherits some properties which we shall keep in mind: There is a remarkable similarity of our mesoscopic description of liquid crystals to the theory of mixtures. To emphasize this connection we compare (note that we introduce the particle number densities ):

Thus the orientational variable n plays the same role in the mesoscopic theory as the component index (up to the difference that is a discrete variable with a finite number of possible values) in the theory of chemical mixtures, and our aim is clearly to derive the "partial balance equations" for liquid crystals which we shall call orientational balance equations.

A transport theorem and the orientational balance equations

The usual transport theorem of Reynolds can be extended to without any difficulties.
Proposition 1:Extended Reynolds transport theorem

Another proposition which is closely related tells us that there are no over-all fluxes on the unit sphere:
Proposition 2:

Now we can start considering a body in nematic space (The following ideas are strongly influenced by mixture theory and not "quite correct" in a sense that it is possible to use more strict methods to derive the same results (see PHYSICA_91). Since we do not want to complicate matters and the essential ideas become clear with the simple approach, too, we skip technical details.) and formulate global balances for particles in a given spatial region having orientations within a given solid angle. Since (by the very definition of bodies) the mass in such a region of the nematic space is conserved we obtain the equation

Using the extended Reynolds theorem we conclude that

holds true for all regions , and and are the material velocities in the nematic space. Hence we obtain by continuity the local form of the balance equation, the orientational balance of mass

Comparing this equation with the partial mass balance for a single component of a chemical mixture we see that the orientational flux plays the same role as the source term due to chemical reactions. This analogy becomes even more true, if we apply proposition 2 and remark that the "net orientational mass production" vanishes

Now we see a general principle: We can formulate the balance equations we are interested in on the nematic space in the same manner as we did for the global balances for certain components of a mixture. However, the "production terms" due to orientational changes are already accounted for by the extended Reynolds theorem and the concept of bodies in the nematic space, and we only have to concentrate on the supplies. Thus the general structure of orientational balance equations reads

with denoting the balanced quantity, and denoting the conductive (non-convective) fluxes of with respect to spatial and orientational changes, resp., and is the supply rate of . Thus we obtain the orientational balances in the form listed below (see also PHYSICA 91, MCLC 91, Liquid Crystals 91):
mass:
momentum:
angular momentum:
energy:
The new quantities are the pressure tensors and , the couple stress tensors and , the heat fluxes and – all of them appearing in "doublets" concerning spatial and orientational fluxes, resp., – and different supplies as the force density , the torque density and the radiation absorption .

Macroscopic balance equations

The balances in the previous section are formulated on the nematic space. For physical applications, however, we need the balances on . To obtain them we simply integrate over all possible orientations which results in new balance equations and new definitions of constitutive fields. For this purpose we remember proposition 2 which tells us that the terms with do not contribute to the macroscopic balances. The resulting equations are those of a micro-polar medium with the wanted fields , v and s obtained as averages of their mesoscopic counterparts

Then the macroscopic balance equations are:
mass
momentum:
angular momentum:
energy:
These equations contain new fields which are – of course – obtained from the mesoscopic fields. In detail we have for the force density and the pressure tensor

These definitions are supplemented by

and for the heat flux density and energy supply

We remark that the pressure tensor is in general not the average of the mesoscopic pressure tensor but contains additional kinetic contributions due to the peculiar velocities . These terms dominate in the kinetic theory of gases. Here, however, experiments and computer simulations show that the kinetic part of the pressure tensor in a liquid is often negligible in comparison to the potential part. Thus we shall assume in the following that they can be cancelled (compare especially the section on constitutive equations). Another remark is due about the spin s. Taking into account the balances of mass and momentum the redundant parts in the balance of angular momentum can be removed and we end up with the spin balance

The involved fields s, and m do not appear in simple continuum models without internal degrees of freedom. Thus the spin balance results in a proposition about the symmetry of the pressure tensor: , i.e. the pressure tensor is a symmetric tensor. I addition to the balance equations there is the second law in the form of the dissipation inequality and we can use the methods of Coleman and Noll or Liu to obtain restrictions for the constitutive equations. A typical state space for liquid crystals, however, must contain some quantities describing the orientational order in the fluid. Since we introduced orientational fields up to now on the nematic space only and the validity of the dissipation inequality is restricted to there is some discrepancy. Thus it is natural to look for macroscopic fields (i.e. fields on space-time) specifying the orientational distribution in a volume element of our material.

Order parameters: The alignment tensor family

The orientational distribution of molecules defines the orientational distribution function f. If we want to derive a set of macroscopic fields which shall represent f on , it is a good idea to remember the multi-pole expansion of electrostatics where the electrostatic potential can be described by the moments of (the same situation we find in the kinetic theory of gases). Thus we would like to have a suitable basis on the orientational part of the domain of f and use the "Fourier coefficients" with respect to that basis (which are now macroscopic fields!) to represent f on . This can be done using the following
Proposition 3:

For our purposes the distribution function f is even in n and its coefficients of odd order vanish. We call the tensor coefficients of f the alignment tensors of f. They are the wanted macroscopic order parameters which can be included in the state space. Nevertheless, there is an infinite number of alignment tensors and we have to choose how many we want to take into account. Fortunately, the situations in which we are forced to use more than the first two alignment tensors (i.e. the second and fourth order tensor) are very rare. We shall see also that the tensors themselves can have a very special form in many relevant cases which simplifies the description. Before we continue a remark on the "quality" of our new fields is in order: Some variables of a chosen state space might not be directly controllable (the so-called internal variables) but fulfill special evolution equations. This is true in particular for the alignment tensors. From the orientational mass balance and the definition of the orientation distribution function we obtain a relaxation equation for f (see PHYSICA 91) for f

Since the alignment tensors are just the multi-pole moments of f it becomes clear that there are also relaxation equations for them. Thus the alignment tensors cannot be directly controlled but have to be considered as internal variables. In most cases in physics of nematic liquid crystals the orientation distribution function f possesses a symmetry axis such that

holds true. The vector d is called the macroscopic director of the distribution; this case is usually referred to as uniaxial symmetry (which must not be confused with the uniaxial symmetry of single particles – here the symmetry is due to the distribution of the particles). In this case the alignment tensors take on a rather simple form

with
.
The most important case of the alignment tensor of second order is then written as

and the scalar order parameter S is referred to as Maier-Saupe order parameter. We close this paragraph with the relaxation equations for the alignment tensors:

The left hand side of this equation is Jaumann's time derivative of a symmetric, l-th order tensor. The right hand side contains terms with the peculiar velocities (translation and rotation). Thus we can write for short

with the alignment production density . It is obvious that

i.e. for a rigid body rotation of the ODF () and common movement of all particles () there is no alignment production. The essential problem connected with the alignment tensor balance is that the differential equation for the l-th alignment tensor cannot be solved separately from those of the other alignment tensors which yields a very complicated scheme of equations. For practical purposes one has to cut down the system and to choose special approximations.

Constitutive equations

We used two levels to deal with liquid crystals. First, the balances were formulated on the nematic space (mesoscopic level). Second, the dissipation inequality and the "measured" fields are fields on (macroscopic level). Hence we can formulate constitutive equations with mesoscopic or macroscopic state spaces. In some cases it is even advisable to use a mixed description and include mesoscopic as well as macroscopic fields in the state space. We shall give two examples to elucidate the method.

Macroscopic constitutive theory, exploitation of the dissipation inequality

We saw already that a liquid crystal theory should contain some macroscopic order parameter. In its most simple form this is the (macroscopic) director d (for situations with uniaxial symmetry and nearly perfect order of the molecules) or in more complicated and general cases some alignment tensors. The inclusion of d alone, however, is not enough since a liquid crystal can be distorted and the elastic energy due to that distortion can not be neglected (see Fig. 3).

Figure 3: Elastic distortions of a nematic liquid crystal: (a) splay (b) twist (c) bend

The elastic part of the free energy due to these distortions is expressed by d .
Thus the state space must contain gradients of the macroscopic order parameters in addition to the order parameters themselves. A typical state space reads as follows:
.
It is easy to imagine that an exploitation of the dissipation inequality for such a state space is a tedious enterprise. The length of the calculations is even for the simpler method of Coleman and Noll surprising (nothing to say about the Liu procedure) but gives some interesting insight in the constitutive theory of liquid crystals. We cannot perform the calculations here, but present only some results (see e.g. Int. J. Engng. Sci. 92, Liquid Crystals 93) which appear in the form "that something does not depend on something else":

Thus the free energy F would be independent on if the excess heat flux ( is the entropy flux density) vanishes – which clearly is impossible as we saw above. Hence liquid crystals are one example where heat flux density and entropy flux density have a more involved connection as it is presupposed in the thermodynamics irreversible processes (TIP). This is not surprising: entropy transfer in a liquid crystal can be observed without heat conduction since reorientation of the molecules clearly affects the entropy, but is not necessarily connected with a temperature gradient. The restrictions imposed by the second law, however, are rather weak and do not yield specific expressions for the constitutive quantities. We shall study an explicit example in the next paragraph.

Mesoscopic constitutive theory, viscosity of a nematic liquid crystal

We want to describe the viscous behaviour of a nematic liquid crystal. For this purpose we choose a mesoscopic state space

with denoting the vorticity of the flow field. N is then the so-called co-rotational time derivative of the microscopic director n. This state space is sufficient to deal with viscous properties of an anisotropic fluid. If we restrict our attention to an ansatz for the pressure tensor which contains no quadratic or higher terms in the shear rate tensor , it is rather easy to write down a representation theorem for the pressure tensor. Before we do that we split the pressure tensor in its equilibrium part and the friction pressure tensor

where the dependence on is contained only in p. Now the ansatz for p is due to Leslie (who used it for the macroscopic director) and reads

The coefficients are usually referred to as Leslie coefficients. From our previous considerations we know that for negligible peculiar velocities the pressure tensor on is the average (integration with f over all orientations) of the mesoscopic pressure tensor. Thus we write

For uniaxial distribution functions the occurring averages can be explicitly calculated and we end up with an expression for the friction pressure tensor which involves the Leslie coefficients and the scalar order parameters and (see Phys. Rev. 95).

Figure 4: Geometry of a plane Couette flow (simple shear flow)

Considering a plane Couette flow (compare Fig. 4) we can distinguish 3 different cases: If we apply a magnetic field (which is capable of aligning the mean orientation, i.e. the macroscopic director, of the liquid crystal) the resistance of the fluid to the movement of the plates depends on the orientation of the director with respect to the flow. The highest viscosity is obtained for d parallel to the gradient direction (y-direction in Fig. 4), whereas alignment parallel to the flow direction (z-direction in Fig. 4) yields the lowest viscosity. These viscosities with fixed director aligned parallel to the three main axes of the flow are usually referred to as Miesowicz viscosities. Fig. 5 shows the theoretical prediction for a system of ellipsoidal particles. The theoretical curves are compared with measured data for some members of the family of cyanobiphenyls. The free parameter of the model is the axis ratio of the ellipsoidal particles.


Figure 5: Miesowicz viscosities of some cyanobiphenyls
Use the mouse (left button to change the apect ratio, right button - or left button together with meta key (ESCAPE on most systems) - to shift) to adjust the "ellipsoid" enclosing the space filling model and watch the theoretical curves fitting the measured data!

Displays: The TN-cell

A survey on anisotropic fluids is certainly incomplete, if the famous application of liquid crystals in display technology is not mentioned. Therefore we shall describe the twisted nematic (TN) cell which is the simplest type of display, cheap and widely used in many applications. The principle idea is simple: The mean orientation of a nematic liquid crystal is influenced by the boundary conditions and external (electric) fields. If we manage to combine these two orienting sources in such a way that we can switch the orientation by applying the electric field, we have changed the optical properties of the fluid. Use of polarization filters might produce a device with two states: dark and bright, according to the state of the electric field. The general physical setup is sketched in Fig. 6.

Figure 6: A liquid crystal is inserted between two crossed, orienting layers yielding a twist of the director field (b). The molecules reorient under the influence of an applied electric field (a).

Two plates are prepared in such a way (this can be done, e.g, by rubbing the surface of the plates in one direction) that the molecules align in a certain direction at the surface. Thus it is possible to prescribe a mean orientation in the cell by the arrangement of the two plates . If we ensure that the plates are crossed, the induced director field resembles a helical structure (see (b) in Fig. 6) which gives rise to the name twisted nematic cell. The plane of polarization of the incoming light is guided according to the local orientation of the liquid crystal ("wave guiding"), and performs therefore a rotation of 90° passing the cell. If we apply an electric field (see (a) in Fig. 6), the liquid crystal is forced to align along the field direction. In this configuration there is no (or hardly none) optical anisotropy for the passing light beam, and the polarization plane does not change. Thus we can switch between two states with different polarizations of the passing (or reflected) beam. We complete this setup by two polarizers and obtain a display which can switch from a dark to a bright state. This is the principle of most liquid crystal displays.

Figure 7: Structure of a TN (twisted nematic) cell

Some final remarks

The survey on liquid crystals was very brief. Nevertheless, we saw some essential principles at work: Thus the phenomenological concepts of thermodynamics can be used in the theory of liquid crystals even out of the situations they were originally designed for. Liquid crystals are only an example we used here to demonstrate (at least) some of the methods used in modern constitutive theory.
Harald Ehrentraut
Institut für Theoretische Physik
TU Berlin, March 6th 1997