Violation of Ericksen inequalities in lyotropic chromonic liquid crystals

In a nematic liquid crystal, the molecules have orientational order along an axis, which is called the nematic director and is represented by the unit vector $\hat{\bm{n}}$. The molecules are equally likely to orient up or down along this axis, and hence $\hat{\bm{n}}$ and $-\hat{\bm{n}}$ represent the same physical state. The axis of orientational order normally depends on position, so that the director becomes a position-dependent field $\hat{\bm{n}}(\bm{r})$. One fundamental problem of liquid-crystal elasticity theory is to determine how the free energy depends on spatial gradients of the director field. In early studies of liquid crystals, this problem was addressed by Oseen Oseen1933 and Frank Frank1958 , and further by Nehring and Saupe Nehring1971 ; Nehring1972 . Their work led to the Oseen-Frank free energy density, which includes terms representing three deformation modes—splay, twist, and bend—as well as a fourth term called saddle-splay. Hence, it has four independent elastic coefficients.

In any elasticity theory, there must be certain inequalities on the elastic coefficients so that the free energy is thermodynamically stable; i.e. no deformation can make the free energy go to negative infinity. For example, in conventional elasticity for an isotropic solid, the bulk and shear moduli must both be positive. Soon after the development of the Oseen-Frank free energy density for liquid crystals, Ericksen Ericksen1966 derived a corresponding set of four inequalities on the liquid-crystal elastic coefficients. These four inequalities will be presented in detail below.

The purpose of this paper is to re-examine the Ericksen inequalities. There are both theoretical and experimental reasons for this re-examination.

The theoretical reason is that our group has recently developed a reformulation of liquid-crystal elasticity theory Selinger2018 , which is based on a mathematical decomposition of the director gradient tensor by Machon and Alexander Machon2016 . This reformulation expresses the free energy density in terms of four director deformation modes—splay, twist, bend, and a less-well-known fourth mode represented as the tensor $\bm{\Delta}$. It is mathematically equivalent to the Oseen-Frank free energy density, but it provides a conceptually simpler way to describe many elastic phenomena in liquid crystals. Using this reformulation, several groups have investigated geometric compatibility constraints on director deformations Virga2019 ; Sadoc2020 ; Pollard2021 ; DaSilva2021 . As we will show, this theoretical progress provides new insight into the Ericksen inequalities.

The experimental reason is that a violation of the Ericksen inequalities has actually been observed in certain liquid-crystal materials. These materials are lyotropic chromonic liquid crystals, such as Sunset Yellow (SSY) and disodium cromoglycate (DSCG). In these materials, the molecules self-organize into long stacks in aqueous solution, and the stacks form nematic orientational order. Several experiments have put these liquid crystals into cylindrical capillaries Nayani2015 ; Davidson2015 , rectangular capillaries Fu2017 , or cylindrical shells Javadi2018 , with degenerate planar anchoring on the surfaces. In these geometries, the director field spontaneously forms a twisted structure rather than a uniform, achiral state. The twisted structure has been modeled by the experimental groups using a twist elastic constant that is anomalously small, violating one of the Ericksen inequalities. Based on this observation, one must ask: Is the violation consistent with liquid-crystal elasticity theory? If so, are there any constraints on the elastic constants?

In Sec. 2 of this article, we present the four Ericksen inequalities, using both the standard form of the Oseen-Frank free energy and our recent reformulation. We point out that these inequalities are excessively strict, because of the geometric constraints on the director deformation modes. In the following sections, we investigate what happens if the Ericksen inequality on the twist elastic constant is violated. In Sec. 3, we consider a severe violation ($K_{22}<0$), and show that the free energy can become arbitrary negative. This violation is forbidden for reasons of thermodynamic stability (unless the free energy includes higher-order terms). In Sec. 4, we consider the intermediate regime ($0<K_{22}<K_{24}$), as in the experiments on lyotropic chromonic liquid crystals, and show that the free energy does not become arbitrary negative. Rather, the liquid crystal has a well-defined, non-uniform ground state, which depends on the system size. Finally, in Sec. 5, we discuss the implications of these results for studies of lyotropic chromonic liquid crystals.

Oseen-Frank theory shows all the ways that the free energy density can depend on spatial gradients of the director field, as allowed by symmetry, up to quadratic order in the gradients. The free energy density is conventionally written as

Here, $S=\bm{\nabla}\cdot\hat{\bm{n}}$ is the splay, $T=\hat{\bm{n}}\cdot(\bm{\nabla}\times\hat{\bm{n}})$ is the twist, and $\bm{B}=-(\hat{\bm{n}}\cdot\bm{\nabla})\hat{\bm{n}}=\hat{\bm{n}}\times(\bm{\nabla}\times\hat{\bm{n}})$ is the bend. These three deformation modes contribute to the free energy density with elastic constants $K_{11}$, $K_{22}$, and $K_{33}$, respectively. The fourth term is called the saddle-splay term. This expression for saddle-splay has been used, for example, by Burylov Burylov1997 to model transitions among different director configurations in a cylindrical capillary.

In the literature, there are some variations in the notation for the saddle-splay coefficient: it sometimes written as $K_{24}$, $\frac{1}{2}K_{24}$, $({K_{22}+K_{24}})$, or $\frac{1}{2}({K_{22}+K_{24}})$. We choose the notation of Eq. (1), because it seems to be the most common in recent articles, but our results can easily be translated into any of the other notations.

In Ref. Selinger2018 , we argue that the elastic free energy for nematic liquid crystals can more easily be understood in terms of four bulk elastic modes, rather than three. This argument is based on a mathematical construction of Machon and Alexander Machon2016 , who decompose the director gradient tensor into four different types of mathematical objects as

Here, the scalar $S$, pseudoscalar $T$, and vector $\bm{B}$ are the splay, twist, and bend modes defined above. The fourth mode $\bm{\Delta}$ is a symmetric, traceless tensor in the plane perpendicular to $\hat{\bm{n}}$. It indicates how the director splays outward in one direction and inward in the orthogonal direction in that plane. All four modes are visualized in Ref. Selinger2018 . In this theoretical approach, pure splay is double splay, and pure twist is double twist. Planar single splay is a combination of pure splay and $\bm{\Delta}$ mode, while cholesteric single twist is a combination of pure twist and $\bm{\Delta}$ mode.

Using the four bulk elastic modes, the free energy can be written as

We emphasize that Eq. (3) is mathematically equivalent to Eq. (1), so these expressions for the free energy can be used interchangeably. However, Eq. (3) has the special feature that it is a sum of squares of elastic modes. In the sum of squares, the elastic constant for splay is $(K_{11}-K_{24})$, the elastic constant for twist is $(K_{22}-K_{24})$, the elastic constant for bend is $K_{33}$, and the elastic constant for the $\bm{\Delta}$ mode is $2K_{24}$. In this perspective, those four elastic constants are the fundamental parameters of the theory, while $K_{11}$ and $K_{22}$ are less fundamental: $K_{11}=(K_{11}-K_{24})+K_{24}$ applies to a combination of splay and $\bm{\Delta}$, and $K_{22}=(K_{22}-K_{24})+K_{24}$ applies to a combination of twist and $\bm{\Delta}$.

Ericksen Ericksen1966 looked for conditions on the elastic constants such that the free energy would be positive definite. Translated into our current notation, his inequalities were

These inequalities can be rewritten as

Because Ericksen worked with the Oseen-Frank free energy in a form similar to Eq. (1), the inequalities were not obvious; they required a significant derivation. However, with the reformulated free energy of Eq. (3), the inequalities are almost trivial. They simply state that the free energy is a sum of squares with positive coefficients.

Clearly the Ericksen inequalities are sufficient for the free energy to be thermodynamically stable. If the inequalities are satisfied, then any director deformation has a positive free energy. The ground state is a uniform director field, with $F=0$. However, it is not clear whether the Ericksen inequalities are necessary for the free energy to be thermodynamically stable.

The main subtlety here is that the four director deformation modes $S$, $T$, $\bm{B}$, and $\bm{\Delta}$ are not independent of each other. Rather, they must all be derived from the same director field $\hat{\bm{n}}(\bm{r})$. Only certain combinations of $S$, $T$, $\bm{B}$, and $\bm{\Delta}$ can be derived from the same director field, and these combinations are called compatible, while other combinations are incompatible.

One way to see this issue is by counting degrees of freedom: $S$ has one degree of freedom (as a scalar), $T$ has one (as a pseudoscalar), $\bm{B}$ has two (as a vector perpendicular to $\hat{\bm{n}}$), and $\bm{\Delta}$ has two (as a traceless, symmetric tensor in the plane perpendicular to $\hat{\bm{n}}$), for a total of six degrees of freedom in the deformation modes. By comparison, the director field has only two degrees of freedom (as a unit vector). Hence, we would expect four constraints among the director deformation modes.

An analogous issue occurs in the elastic theory of solids. The strain tensor is derived from the displacement vector field, but the strain tensor has more degrees of freedom than the displacement vector field. Hence, there must be compatibility constraints on the strain tensor. Here, the director field of a liquid crystal is analogous to the displacement field of a solid, and the deformation modes $S$, $T$, $\bm{B}$, and $\bm{\Delta}$ are analogous to the strain tensor of a solid.

In recent years, several groups have investigated how the compatibility constraints for the director deformation modes are related to the geometry of space. Virga Virga2019 found there are only two ways to fill three-dimensional (3D) Euclidean space with constant director deformations: twist and $\bm{\Delta}$, or bend, twist, and $\bm{\Delta}$. Sadoc et al. Sadoc2020 found that 3D non-Euclidean curved space can be filled with a single director deformation mode, with the right correspondence between the type of curvature and the type of deformation. Pollard and Alexander Pollard2021 and da Silva and Efrati DaSilva2021 developed general compatibility conditions for an arbitrary combination of deformation modes and an arbitrary Euclidean or non-Euclidean geometry.

The compatibility conditions derived in Refs. Pollard2021 ; DaSilva2021 are mathematically complex, and we will not attempt to use them directly. Instead, we will construct director fields in 3D Euclidean geometry, and determine what happens to these director fields if the Ericksen inequalities are violated.

For a first step, consider a director field with the cholesteric helical structure

As shown in Ref. Selinger2018 , this structure does not have pure twist; rather, it is a combination of the twist and $\bm{\Delta}$ deformation modes. It is one of the allowed constant combinations found by Virga Virga2019 . Inserting this director field into the free energy density of Eq. (1) or (3) gives $F=\frac{1}{2}K_{22}q^{2}$. It involves $K_{22}$, which is a combination of the elastic constant $(K_{22}-K_{24})$ for pure twist and the elastic constant $K_{24}$ for the $\bm{\Delta}$ mode. Hypothetically, suppose that $K_{22}<0$. In that case, minimizing the free energy would drive $q\to\pm\infty$, and hence $F\to-\infty$. That situation would be thermodynamically unstable. Hence, a necessary inequality is $K_{22}>0$. Note that this inequality is weaker than the Ericksen inequality on $K_{22}$.

Next, consider a director field with the twist-bend heliconical structure

This structure is a combination of the bend, twist, and $\bm{\Delta}$ deformation modes; it is the second allowed constant combination. Inserting this director field into Eq. (1) or (3) gives the free energy density

Hypothetically, suppose that $K_{33}<0$. In that case, we can choose a cone angle $\beta$ such that the coefficient of $q^{2}$ is negative. Minimizing the free energy then drives $q\to\pm\infty$, and hence $F\to-\infty$, which is thermodynamically unstable. Hence, another necessary inequality is $K_{33}>0$. That inequality is the same as one of the Ericksen inequalities.

When we construct director fields, we are not limited to constant combinations of deformation modes; we can also investigate non-uniform combinations. For example, consider the director wave

for small $\theta_{0}$. By putting that wave into the free energy density of Eq. (1) or (3) and integrating over $\bm{r}$, we obtain the average free energy density

If any of the three coefficients $K_{11}$, $K_{22}$, or $K_{33}$ were negative, then minimization of the free energy would drive the corresponding component of $\bm{q}$ to $\pm\infty$, and hence drive $F\to-\infty$, which would be thermodynamically unstable. Hence, all three of those coefficients must be positive, and the set of necessary inequalities becomes

At this point, some readers might object that there is actually an extensive literature on liquid crystals with $K_{33}<0$, beginning with the work of Dozov Dozov2001 . However, that situation is different because Dozov uses a free energy with higher-order terms, involving either second derivatives or higher powers of first derivatives of the director field. These higher-order terms prevent the thermodynamic instability, and give a twist-bend nematic phase with finite values of wavevector $q$ and cone angle $\theta$. Likewise, in recent work by our group Shamid2013 ; Rosseto2020 ; Selinger2022 ; Rosseto2022 , an effective renormalized elastic constant ($K_{33}^{R}$, $K_{11}^{R}$, or $K_{22}^{R}$) is driven negative by interactions with another order parameter, and the free energy would be unstable, but it is stabilized by some higher-order couplings.

In elasticity theory, it is always possible to add higher-order terms to stabilize a free energy. When we search for stability conditions, the issue is whether the free energy is stable without higher-order terms. For a free energy with the form of Eq. (1) or (3), without higher-order terms, the conditions of Eq. (11) are necessary.

Summarizing the results of this section, we have derived a set of necessary inequalities (11), which are weaker than the Ericksen inequalities (5). If the necessary inequalities are violated, then the Ericksen inequalities are severely violated, and the free energy may go to negative infinity. This severe violation is forbidden (unless there are higher-order terms to stabilize the free energy). However, there are intermediate regimes in which the Ericksen inequalities are violated but the necessary inequalities are satisfied. In the following section, we consider what happens in those intermediate regimes.

Figure 1 shows schematic diagrams of possible regimes for the elastic constants. The first row is expressed in terms of the fundamental elastic constants ${(K_{11}-K_{24})}$, $(K_{22}-K_{24})$, and $K_{24}$. The second row provides the same information expressed in terms of the conventional elastic constants $K_{11}$, $K_{22}$, and $K_{24}$. In each case, we can see that there is a blue regime where the Ericksen and the necessary inequalities are both satisfied, and a red regime where the Ericksen and necessary inequalities are both violated. Between these regimes, there are intermediate regimes where the Ericksen inequalities are violated but the necessary inequalities are satisfied. In particular, there are three intermediate regimes, which we label as:

• •

  Twist intermediate regime: $0<K_{22}<K_{24}$

• •

  Splay intermediate regime: $0<K_{11}<K_{24}$

• •

  $\bm{\Delta}$ intermediate regime: $K_{24}<0$, $K_{11}>0$, $K_{22}>0$

In this section, we concentrate on the twist intermediate regime, because it occurs experimentally in lyotropic chromonic liquid crystals. At the end of the section, we briefly discuss the other two intermediate regimes, which have not yet been reported experimentally to our knowledge. (We do not consider regimes in which more than one of the Ericksen inequalities are violated simultaneously.)

In this article, we have identified three distinct regimes of elastic constants in nematic liquid crystals, which are indicated schematically in Fig. 1. In the regime where the Ericksen inequalities (5) are satisfied, all four director deformation modes cost some positive free energy. For that reason, the ground state of a bulk liquid crystal has a uniform director field. By contrast, in the forbidden regime where the necessary inequalities (11) are violated, some physically realizable director field has a negative free energy, and hence the free energy (1) or (3) is thermodynamically unstable. It can only be stabilized by extra terms with higher powers of the director gradients, or higher-order derivatives of the director field, or couplings with other order parameters. In this forbidden regime, the liquid crystal may form a modulated structure with no singularities in nematic order, such as a twist-bend nematic phase or a splay nematic phase, as studied in much recent research Dozov2001 ; Shamid2013 ; Rosseto2020 ; Selinger2022 ; Rosseto2022 .

Between the Ericksen regime and the forbidden regime, there is an intermediate regime where the Ericksen inequalities are violated but the necessary inequalities are satisfied. In this intermediate regime, one of the director deformation modes—twist, splay, or $\bm{\Delta}$—has a negative free energy. However, the total free energy is stabilized by geometric compatibility constraints, which require that any physically realizable director field must have a combination of the favored mode with other, unfavorable modes. The intermediate regime has surprising properties. In a finite cylindrical geometry, the liquid crystal minimizes its free energy by forming a nonuniform director field, as shown in Fig. 2, 5(a), or 5(b). As the cylinder radius increases, the total free energy increases super-extensively. The super-extensive growth can only be avoided if the system adds disclinations, as in Fig. 4, provided that the disclination energy is low enough.

Remarkably, the twist intermediate regime actually occurs in lyotropic chromonic liquid crystals, according to experimental measurements of the elastic constants Nayani2015 ; Davidson2015 ; Fu2017 ; Javadi2018 . Hence, the behavior discussed here is not just a theoretical speculation, but can be studied in the laboratory. In particular, we emphasize three consequences of this theory for experiments on lyotropic chromonic liquid crystals.

First, these materials should be sensitive to the geometry of their container. In this article, we have seen the effects of changing the radius $R_{\text{max}}$ of a cylindrical capillary. We would expect equally important effects of changing the shape of the container. The favorable twist deformation naturally fills up a region that is roughly circular in the plane perpendicular to the director. If the cross section of the cell is not circular, then the director field must adapt in a complex way, perhaps by forming circular domains of twist separated by untwisted regions. As an example, the experiments of Ref. Fu2017 put lyotropic chromonic liquid crystals into capillaries of rectangular cross section and find complex director configurations.

Second, these materials have an unusual relationship between the free energies of double twist, cholesteric single twist, and a uniform state. In typical chiral liquid crystals, double twist is preferred over single twist, and single twist is preferred over a uniform state. By contrast, in lyotropic chromonic liquid crystals, double twist is preferred over a uniform state, and a uniform state is preferred over single twist. Hence, the materials should particularly avoid cholesteric single twist. Indeed, the structures in Ref. Fu2017 include double-twist regions and monodomain uniform regions.

Third, we anticipate that these materials should be particularly compatible with impurities, such as dust or other colloidal particles. Any particles will break up the liquid crystals into smaller volumes, and will allow the director field to break into more regions of double twist. Effectively, the particles could play the same role as the disclinations in the structures of Fig. 4. Hence, the negative free energy of the twist domains could offset the positive free energy of contact between the particles and the liquid crystal. Distortions induced by such particles could make these materials difficult to align. A recent study suspends rod-like particles in a lyotropic chromonic liquid crystal, and finds an anomalous twisted alignment of the rods with respect to the director field Ettinger2022 . We speculate that the spontaneous twist discussed here may be involved in that experiment.

Acknowledgments

This work was supported by National Science Foundation Grant DMR-1409658.