Twist-Bend Nematic Phase from the Landau-de Gennes Perspective.

Generalized Landau-de Gennes theory is proposed that comprehensively explains currently available experimental data for the heliconical twist-bend nematic (NTB) phase observed in liquid crystalline systems of chemically achiral bent-core-like molecules. A bifurcation analysis gives insight into possible structures that the model can predict and guides in the numerical analysis of relative stability of the isotropic (I), uniaxial nematic (NU), and twist-bend nematic phases. An estimate of constitutive parameters of the model from temperature variation of the nematic order parameter and the Frank elastic constants in the nematic phase enables us to demonstrate quantitative agreement between the calculated and experimentally determined temperature dependence of the pitch and conical angle in NTB. Properties of order parameters also explain a puzzling lack of a half-pitch band in resonant soft X-ray scattering. Other key findings of the model are predictions of I-NTB and NU-NTB tricritical points and insight into biaxiality of NTB.

Introduction

Undoubtedly, the short-pitch heliconical structure formed by an ensemble of achiral bent-core-like mesogens and commonly referred to as the nematic twist–bend is one of the most astonishing liquid crystalline phases. It is the first example in nature of a structure where the mirror symmetry is spontaneously broken without any support from a long-range positional order. The structure itself is a part of an over 130 year-old tradition of liquid crystal science, demonstrating that even a minor change in the molecular chemistry can lead to a new type of liquid crystalline order, which differs in the degree of orientational and translational self-organization, ranging from molecular through nano- to macroscales.[1−3]

The most common of all known liquid crystalline phases is the uniaxial nematic phase (NU), where anisotropic molecules or molecular aggregates orient, on the average, parallel to each other. Their local, mean orientation at the point r̃ of coordinates (x̃,ỹ,z̃) is described by a single mesoscopic direction n̂(r̃) (|n̂(r̃)| = 1) known as the director. Because of the statistical head-to-tail inversion symmetry of the local molecular arrangement, the director states n̂(r̃) and −n̂(r̃) are equivalent. With an inversion symmetry and with a rotational symmetry of molecular orientational distribution about n̂(r̃), the existence of the director is a basic property that distinguishes the uniaxial nematics from an ordinary isotropic liquid. That is, the NU phase is a nonpolar 3D liquid with a long-range orientational order characterized by the point group symmetry.

One important consequence of n̂(r̃) being indistinguishable from −n̂(r̃) is that the primary order parameter of the uniaxial nematics is the second-rank (3 × 3) traceless and symmetric alignment tensor (the quadrupole moment of the local angular distribution function of the molecules’ long axes)having the components Q̃ U,αβ; is the scalar order parameter describing the degree of (local) molecular orientational ordering along n̂(r̃), and I denotes the identity matrix.

Beyond conventional uniaxial nematics, further nematic liquid phases, that (by definition) have only short-ranged positional ordering, were recognized. They involve symmetric biaxial nematics (NB) for nonchiral materials and cholesteric (N*) along with blue phases (BP) for chemically chiral mesogens, characterized locally by the point group symmetry. In order to account for their local orientational order, we need a full, symmetric, and traceless alignment tensor Q̃(r̃) with three different eigenvalues, as opposed to the uniaxial nematic 1, where only two eigenvalues of Q ≡ Q U are different.

This four-member nematic family is ubiquitous in nature and it has not been expanding for many years.[1] However, very recently, the situation has changed with the important discovery of two fundamentally new nematics: the twist–bend nematic phase (NTB)[4−7] and the nematic splay phase (NS);[8] it seems that these discoveries only mark the beginning of a new, fascinating research direction in soft matter science.[3,9−12]

Without any doubt, the discovered NTB phase is different than 3D liquids known to date because it exhibits a macroscopic chirality while formed from chemically achiral, bent-core-like molecules. A direct manifestation of chirality is an average orientational molecular order that forms a local helix with a pitch spanning from several to over a dozen of nanometers, in the absence of any long-range positional order of molecular centers of mass. NTB is stabilized as a result of (weakly) first-order phase transition from the uniaxial nematic phase or directly from the isotropic phase,[13,14] and therefore, (as already mentioned) its emergence is probably one of the most unusual manifestation of spontaneous mirror symmetry breaking (SMSB) in three-dimensional liquids.

At the theoretical level, the possibility of SMSB in bent-shaped mesogens has been suggested by Meyer already in 1973. He pointed out that bend deformations, which should be favored by bent-shaped molecules, might lead to flexopolarization-induced chiral structures.[15] About 30 years later, Dozov[16] considered the Oseen–Frank (OF) free energy of the director field n̂(r̃),[17,18] wherewhere K11, K22, and K33 are splay, twist, and bend elastic constants, respectively. He correlated the possibility of SMSB in nematics with the sign change of the bend elastic constant, K33. In this latter case, in order to guarantee the existence of a stable ground state, some higher order elastic terms had to be added to fOF. Limiting to defect-free structures, Dozov predicted competition between a twist–bend nematic phase, where the director simultaneously twists and bends in space by precessing on the side of a right circular cone, and a planar splay–bend phase with alternating domains of splay and bend, both shown in Figure . If we take into account the temperature dependence of the Frank elastic constants, then, the uniaxial nematic phase becomes unstable to the formation of modulated structures at K33 = 0, which is the critical point of the model. The behavior of the system depends on the relationship between the splay and twist elastic constants. As it turns out, the twist–bend ordering wins if K11 > 2K22, while the splay–bend phase is more stable if K11 < 2K22. Assuming that the wave vector k̃ of NTB stays parallel to the ẑ-axis of the laboratory reference frame (k̃ = k̃ẑ), the symmetry-dictated, gross features of the heliconical NTB structure are essentially accounted for by the uniform director modulationwhere n̂(0) = [sin(θ),0,cos(θ)] and is the homogeneous rotation about ẑ through the azimuthal angle ϕ(z̃) = ±k̃z̃ = ±2πz̃/p, where p is the pitch. The ± sign indicates that both left-handed and right-handed chiral domains should form with the same probability, which is the manifestation of SMSB in the bulk. Note that the molecules in NTB are inclined, on the average, from k̃ by the conical (tilt) angle θ−the angle between n̂ and the wave vector k̃ (Figure ). The symmetry of NTB also implies that the structure must be locally polar with the polarization vector, P̃, staying perpendicular both to the director and the wave vector

Figure 1

Schematic depiction of modulated nematic phases formed by achiral bent-shaped molecules. Pure bend distortion in 2D leads to the emergence of defects (red sphere). Their appearance can be circumvented by alternating the bend direction periodically or allowing nonzero twist by lifting the bend into the third dimension. These possibilities, respectively, give rise to the two alternative nematic ground states: splay–bend (NSB) and twist–bend (NTB). The twist–bend nematic has been first observed in the phase sequence of the liquid crystal dimer 1″,7″-bis(4-cyanobiphenyl-4′-yl)heptane (CB7CB), where two identical cyanobiphenyl mesogenic groups are linked by a heptane spacer (the CB7CB molecule can be viewed as having three parts: two identical rigid end groups connected by a flexible spacer). Schematic representation of molecular organization in the NTB with right and left handedness (ambidextrous chirality) has been depicted at the bottom of the image. The right/left circular cone of conical angle θ shows the tilt between the director n̂ and the helical symmetry axis, parallel to the wave vector k̃. The red arrow represents polarization P̃, where P̃∥n̂ × k̃. Note that NTB has a local symmetry with a two-fold symmetry axis around P̃.

Hence, in the nematic twist–bend phase, both n̂ and P̃ rotate along the helix direction k̃, giving rise to a phase with constant bend and twist deformation of no mass density modulation (Figure ).

In 2013, Shamid et al.[19] developed Landau theory for bend flexoelectricity and showed that the results of Dozov are in line with Meyer’s idea of flexopolarization-induced NTB. Their theory predicts a continuous N–NTB transition, where the effective bend elastic constant, renormalized by the flexopolarization coupling, changes sign for sufficiently large coupling. The corresponding structure develops a modulated polar order, averaging to zero globally as in eq . Dozov’s model is also supported by measurements of anomalously small bend elastic constant (compared to the splay and twist elastic constant) in the nematic phase of materials exhibiting NTB (see, e.g., measurements for the CB7CB dimer of Babakhanova et al.[20−22]).

The second most widely used continuum model to characterize orientational properties of nematics is the minimal coupling, SO(3)-symmetric Landau–de Gennes (LdeG) expansion in terms of the local alignment tensor. It allows us not only to account for a fine structure of inhomogeneous nematic phases but also shows important generalizations of the director’s description in dealing with orientational degrees of freedom (see, e.g., ref (23)). In a series of papers,[11,12,24] coauthored by one of us, we developed an extension of LdeG theory to include flexopolarization couplings. The extended theory predicted that the flexopolarization mechanism can make the NTB phase absolutely stable within the whole family of one-dimensional modulated structures.[11] A qualitatively correct account of experimental observations in NTB (see, e.g., ref (3)) was obtained, such as trends in temperature variation of the helical pitch and conical angle and behavior in the external electric field.[25] The theory also predicted weakly first-order phase transitions from the isotropic and nematic to nematic twist–bend phase, again in agreement with experiments.[14,26] Despite this qualitative success of the LdeG modelling, one important theoretical issue still left unsolved is associated with the elastic behavior of the uniaxial nematic phase for materials with stable NTB. A few existing measurements of all three elastic constants in the NU phase show that K11 ≳ 2K22 (K22 ≈ 3–4 pN), while K33 ≈ 0.4 pN near the transition into NTB.[20] That is, the splay elastic constant is about 20 times larger than the bend elastic constant. On the theoretical side, the LdeG expansion with only two distinct bulk elastic terms cannot explain this anomalously large disparity in the values of K11 and K33. Actually, it predicts that they both are equal in the OF limit,[27,28] where the alignment tensor is given by eq . Therefore, there are anomalously small bend and splay Frank elastic constants on approaching NTB in the LdeG model with flexopolarization.[12] Although this prediction might suggest dominance of the structures with splay–bend deformations over that of the twist–bend ones, the NTB phase, as already discussed before, can still be found to be more stable than any of one-dimensional periodic structures, including the nematic splay–bend phase.[11] Most probably, this is due to the remarkable (and unique) feature of NTB being uniform everywhere in space that makes the SO(3)-symmetric elastic free-energy density independent of space variables.[11]

Central to quantitative understanding of NTB and related phase transitions is the construction of generalized LdeG theory that releases the K11 = K33 restriction of the minimal coupling model and accounts for the experimental behavior of the Frank elastic constants in the vicinity of NU–NTB phase transition. We expect that such a theory will allow for a systematic study of mesoscopic mechanisms that can be responsible for chiral symmetry breaking in nematics. It will also give a new insight into conditions that can potentially lead experimentalists to the discovery of new nematic phases. Although the choice of strategy has already been worked out in the literature,[24,29,30] the main problem lies in a huge number of elastic invariants in the alignment tensor, contributing to the generalized elastic free-energy density of nematics. Here, we show how the problem can be solved in a systematic way if we start from a theory which holds without limitations for arbitrary one-dimensional periodic distortions of the alignment tensor that serve as ground states. These ground states form the most interesting class of structures for it obeys the recently discovered new nematic phases. An additional requirement for generalized LdeG theory is that its ground state in the absence of flexopolarization should be that corresponding to a constant tensor field Q̃. The theory so constructed will then be applied to characterize properties of NTB formed in the class of CB7CB-like dimers and its constitutive parameters will be estimated from experimental data known for the CB7CB dimers in the NU phase.

Theoretical Methods

Alignment Tensor Representation for Homogeneously Deformed Nematic Phases

In the NTB phase, the director n̂ and the polarization vector P̃ are given by eqs and 4, while the equivalent alignment tensor order parameter, Q̃ U,TB, is obtained by substituting 3 into 1. Although these models seem to account for gross features of the orientational order observed in NTB, they do not exhaust possible nematic structures that can fill space with twist, bend, and splay. A full spectrum of possibilities is obtained by studying an expansion of the biaxial alignment tensor Q̃ and the polarization field P̃ in spin tensor modes of L = 2 and L = 1, respectively, and in plane waves.[12] Within this huge family of states, an important class of nematic states is represented by uniformly deformed structures (UDSs) where the elastic, SO(3)-symmetric invariants contributing to the elastic free-energy density of nematics[24,30] are constant in space. For such structures, the same tensor and polarization landscape is seen everywhere in space. They are periodic in, at most, one spatial direction, say z̃, and uniformly fill space without defects. In analogy to the conditions 3 and 4 for n̂ and P̃, they are generated from the tensors Q̃(0) and P̃(0) for z̃ = 0 by the previously defined homogeneous rotation .[11,31] More specificallywhere ± labels left- (+) and right-handed (−) heliconical structures. Hence, the most general representations for UDSs that generalize eqs and 4 can be cast in the form (see Figure )[11,31]where c± = cos(±m + ϕ±) and s± = sin(±m + ϕ±) and nine real parameters x̃0, ±k̃, r̃± ≥ 0, p̃±1 ≥ 0, , ϕ±, and ϕ±p for each of the ± labels characterize the fine structure of the phases, especially its biaxiality. Indeed, an arbitrary symmetric and traceless tensor field Q̃ fulfills the inequalities (see discussion in ref (32))which are satisfied as equalities for locally prolate (w = 1) and oblate (w = −1) uniaxial phases. States of nonzero biaxiality are realized for −1 < w < 1, with maximal biaxiality corresponding to w = 0. In particular, the parameter w(Q̃(z̃)) for Q̃(z̃) given by eq reads

Figure 2

Visualization of helicity modes introduced in eqs and 7: indices m = 0, m = ±1, and m = ±2 correspond to subscripts 0, ±1, and ±2 of { x̃0, ṽ0 }, {r̃±1, p̃±1}, and {r̃±2, p̃±2}, respectively. Change of m into −m corresponds to replacement of k̃ by −k̃. Bricks represent the tensor Q̃(r̃) where the eigenvectors of Q̃(r̃) are parallel to their arms, while the absolute values of eigenvalues are their lengths. Red arrows represent the polarization field P̃(r̃).

Note that in agreement with definition 5, the parameter w(Q̃(z̃)) is position-independent and can take an arbitrary value within the allowed [−1, 1] interval, eq . In contrast, for the uniaxial tensor Q̃ U,TB, corresponding to x̃0 = , r̃±1 = , and r̃±2 = the parameter w(Q̃ U,TB) = Sign(S) = ±1 (θ is the conical angle).

We should mention that the fields in eqs and 7 are insensitive to the choice of the origin of the laboratory reference frame, which allows us to eliminate one of the phases ϕ± (i = 1, 2, p), independently for each of the two states with “+” and “–” subscripts. The coefficients in eqs and 7 are chosen such that the norms squared of the order parameters are sums of squares of the coefficients: Tr(Q̃2) = x̃02 + r̃±12 + r̃±22 and Tr(P̃2) = p̃±12 + . Together Q̃ and P̃ characterize a family of all defect-free uniformly deformed (polar) helical/heliconical nematic phases. They are gathered in Table .

Table 1

Family of Uniformly Deformed Nematic Structures (UDSs) That Can Be Constructed out of the Fields Q̃ and P̃a

structure	nonzero amplitudes	abbreviation
Nonpolar Structures
(a) uniaxial nematic	x̃0	NU
(b) biaxial nematic	x̃0, r̃1, r̃2, k̃ → 0	NB
(c) (ambidextrous) cholesteric	x̃0, r̃2, k̃NC = 2k̃ ≠ 0	NC
Locally Polar Structures
(d) locally polar cholesteric	as in (c), p̃1	NCl
(e) nematic twist–bend	x̃0, r̃1, r̃2, p̃1, k̃ ≠ 0	NTB
Globally Polar Structures
(f) polar (a)–(e)	any of (a)–(e),	add subscript “p” to (a)–(e)

Limiting cases of the constant Q̃ and P̃ are also included.

Generalized LdeG Expansion for 1D Periodic Nematics

In this section, we introduce a generalized LdeG free-energy expansion in Q̃ and P̃, capable of quantitative description of the systems with stable one-dimensional periodic nematics. The most important members of this family are the nematic twist–bend phase[3] and recently discovered nematic splay phase.[8] Our main effort in this and next section will concentrate on the general characterization of LdeG expansion. An example of the UDS with its prominent representative−the NTB phase−will be studied in great detail. Parameters entering the LdeG expansion will be estimated from experimental data for the CB7CB compound in the uniaxial nematic phase. Then, the properties of the NTB phase resulting from the so constructed LdeG expansion will be calculated and compared with available experimental data.

We assume that the stabilization of NTB is due to entropic, excluded volume flexopolarization interactions,[33] induced by sterically polar molecular bent cores. The direct interactions between electrostatic dipoles will be disregarded[33] and the long-range polar order will be attributed to the molecular shape polarity. With Q̃ and P̃, the general LdeG expansion reads[24,28]where r̃ is the position vector, is the system’s volume, and the free-energy densities and are constructed out of the fields X. They involve the bulk nematic part , the nematic elastic part , and the parts and responsible for the onset of chirality in the nematic phase. Although the general theory has plenty of constitutive lparameters, part of them, at least for CB7CB, can be estimated from existing experimental data for the NU and at the I–NU and NU–NTB phase transitions. One of the issues we would like to understand is whether the theory so constructed allows us to account for the quantitative properties of the nematic twist–bend phase, below NU–NTB phase transition.

Bulk Nematic Free Energy

According to phenomenological LdeG theory, the equilibrium bulk properties of nematics can be found from a nonequilibrium free energy, constructed as an SO(3)-symmetric expansion in powers of Q̃. There are only two types of independent SO(3) invariants that can be constructed out of Q̃, namely, Tr(Q̃2) and Tr(Q̃3). Hence, is a polynomial in Tr(Q̃2) and Tr(Q̃3) with the only restriction on the expansion being that it must be stable against an unlimited growth of Q̃. The experimental data for in the nematic phase of CB7CB fit well to a model where the expansion with respect to Q̃ is taken at least up to sixth-order terms. More specifically, in the absence of electric and magnetic fields, introducing = Tr(Q̃2) and = Tr(Q̃3), we take for the bulk free-energy density of the isotropic and the nematic phases

A full account of phases and critical and tricritical points that this theory predicts is found in ref (32).

The coefficients of the expansion generally depend on temperature and other thermodynamic variables, but in Landau theory, the explicit temperature dependence is retained only in the bulk part, quadratic in Q̃. In what follows, as a measure of temperature, we choose the relative temperature distance, Δt, from the nematic–isotropic phase transition, defined through the relationwhere a0Q > 0, T is the absolute temperature, TNI is the nematic–isotropic transition temperature, T* is the spinodal for a first-order phase transition from the isotropic phase to the uniaxial nematic phase, Δt = (T – TNI)/TNI ≤ 0, and ΔtNI = (TNI – T*)/TNI > 0 is the reduced temperature distance of nematic–isotropic transition temperature from T*. Additionally, b, c, d, e > 0, and f > 0 are the temperature-independent expansion coefficients. The last two conditions for e and f guarantee that is stable against an unlimited growth of Q̃.[32] The expansion, eq , generally accounts for the isotropic, uniaxial nematic, and biaxial nematic phases.[32,34]

We should mention that the fourth-order expansion, where c > 0 and d = e = f = 0, predicts that the NTB phase can be absolutely stable within the family of one-dimensional modulated structures,[11] but the theory does not give a quantitative agreement with the data for in the nematic phase of CB7CB unless an unphysically large value of ΔtNI is taken (Figure S1).

Elastic Free Energy

A spatial deformation of the alignment tensor Q̃ in the nematic phase is measured by the elastic free-energy density of the Landau free energy expansion . For the description of elastic properties of nematic liquid crystals, usually is expanded into powers of Q̃ and its first derivatives ∂Q̃ ≡ ∂Q̃ /∂x̃ = Q̃ , where only quadratic terms in derivatives of the order parameter field are retained, in line with similar expansion for the director field, eq .

This standard, the so-called minimal-coupling LdeG expansion for , comprises only two bulk elastic terms: [L1(2)] = Q̃ αβ,γQ̃ αβ,γ and [L2(2)] = Q̃ αβ,βQ̃ αγ,γ. Although again the theory, eq , with containing only these two elastic terms accounts for absolutely stable NTB among one-dimensional modulated structures,[11] it is not sufficiently general to quantitatively reproduce, for example, elastic properties of bent-core systems in the parent nematic phase for it implies equality of splay and bend Frank elastic constants, which so far is not an experimentally supported scenario with stable NTB. Thus, we need to include higher order elastic terms in LdeG theory to account for the experimentally observed elastic behavior of bent-core mesogens. A general form of the LdeG elastic free-energy density has been studied by Longa et al. in a series of papers.[24,28,30] As it turns out, the most important are third-order invariants of the form Q̃∂Q̃∂Q̃, given explicitly in the Supporting Information, because they are the lowest order terms removing splay–bend degeneracy of second-order theory.[28] However, with quadratic and cubic terms alone, the elastic free energy f̃Qel is unbounded from below and hence cannot represent a correct theory of nematics. To assure that the nematic ground state is stable against an unlimited growth of Q̃ αβ and Q̃ αβ,γ, we need to add some fourth-order invariants.[28] In total, there are 22 deformation modes [L(] of Q̃ up to the order Q̃Q̃ ∂Q̃ ∂Q̃ (see the Supporting Information for details). The next step is to single out the relevant elastic terms [L(] that should enter the expansion . A considerable reduction in the number of independent terms is obtained if we limit ourselves to a class of ground states represented by one-dimensional periodic structures Q̃(z̃ + p̃) = Q̃(z̃).[11,35] Then, the only relevant linearly independent [L]-terms (nonvanishing for uniaxial Q̃) are

• ∂Q̃ ∂Q̃ terms: [L1(2)], and [L2(2)]

• Q̃ ∂Q̃ ∂Q̃ terms: [L2(3)], [L3(3)], and [L4(3)]

• Q̃Q̃ ∂Q̃ ∂Q̃ terms: [L2(4)], [L3(4)], [L5(4)], [L6(4)], [L7(4)], [L10(4)], and [L11(4)].

As mentioned before, the most important are third-order terms Q̃ ∂Q̃ ∂Q̃ linear in Q̃ and quadratic in ∂Q̃ because they remove splay–bend degeneracy.[28] Hence, in what follows, we will keep three third-order terms and add three stabilizing terms of the order Q̃Q̃ ∂Q̃ ∂Q̃. More specifically, for the elastic free energy , we take a sum of quadratic terms in deformations of the formwhere the coefficients, L(, denote temperature–independent elastic constants that couple with the invariant [L(] and λ2 = L2(3)/(2L14(4)), λ3 = L3(3)/(2L6(4)), and λ4 = L4(3)/(2L7(4)). Use of the [L14(4)] invariant, which is a linear combination of the invariants Q̃Q̃ ∂Q̃ ∂Q̃, allows us to write the stability criteria for in a simple form (see the Supporting Information). Indeed, the elastic free-energy density is a sum of positive definite terms if

The conditions 15 are sufficient ones for to be positive definite ( ≥ 0). For smooth tensor fields Q̃, the ground state of ( = 0) corresponds to a constant, position-independent Q̃, which represents an unperturbed isotropic, uniaxial, or biaxial nematic state. As we show later in the text, the elastic constants L( entering expansion can all be estimated from the data for Frank elastic constants in the uniaxial nematic phase. To conclude, the free energy 14 is a thermodynamically stable expansion of the LdeG free energy in the local alignment tensor, complete up to third order for deformations realized in one spatial direction and nonvanishing in the uniaxial limit for Q̃.

The elastic free energy 14 can still be written in a simpler form by further selecting terms that are relevant for the UDS. Indeed, substitution of eq into expansion induces extra relations between cubic and quartic elastic invariants, namely

Thus, in seeking for relative stability of the UDS, two elastic terms in 14 are still redundant. This redundancy becomes especially transparent in the parameterization where the elastic constants L2(3), L3(3), L6(4), and L14(4) are replaced by appropriate linear combinations of L7(3), L8(3), L15(4), and L16(4). They are given bywhere, in addition, the inequality L15(4) > |L16(4)| is required to fulfill stability conditions 15. Substitution of 17 into 14 now yieldswherewhere [L8(3)] and [L16(4)] terms vanish for the UDS, given by eq .

Coupling with Steric Polarization

According to the current understanding of the formation of the stable twist–bend nematic phase, its orientational order, being similar to that of smectic C*,[1] should be accompanied with a long-range polar order of molecular bent cores.[36−40] As already pointed out, the other direct molecular interactions, such as between electrostatic dipoles, are probably less relevant for the thermal stability of this phase. Up to the leading order in P̃, at least five extra terms must be included in and , eq . They read[11,12,28]Here, aP = a0P((T – TP)/TNI) = a0P(Δt + ΔtNI,P), where ΔtNI,P = (TNI – TP)/TNI > 0, A4 > 0, bP > 0, εP, and ΛQP are further temperature-independent constitutive parameters of the model. Again, limitations for A4 and bP stem from stability requirement of a ground state against unlimited fluctuations of P̃(r̃). The εP-term represents lowest order flexopolarization contribution, while the ΛQP-term is the direct coupling between the polarization field and the alignment tensor. The presently existing experimental data seem to be in line with this minimal coupling expansion for the (flexo)polarization part of the free energy.[11,12] A full structure of (flexo)polarization theory, along with some of its general consequences, can be found in ref (24).

Reduced Form of Generalized LdeG Expansion

For practical calculations, it is useful to reduce the number of model parameters by rewriting eq in terms of reduced (dimensionless) quantities. It reveals the redundancy of four parameters in the expressions , 14, 20, and 21 and allows to set them to one from the start.[12,24,28,29] This reduction is a direct consequence of the freedom to choose a scale for the free energy, F̃ = ΛF, for the fields Q̃ = ΛQ and P̃ = ΛP, for the position vector r̃ = Λr, where Λ are nonzero scaling parameters. Taking this freedom into account, we introduce the reduced quantities F (ftot), Q (equivalently S, x0, r1, and r2), P (equivalently p1 and v0), r, k, tQ, ρ2,2 – ρ4,16, tP, ad, eP, λ, cb, db, and eb with the help of the equations

The remaining quantities (S, x0, r1, and r2) and (p1 and v0) are connected with their tilted counterparts by the same relations as Q with Q̃ and P with P̃, respectively. In addition, the definitions 19 now become reduced to

Consequently, the generalized LdeG free-energy expansion in terms of reduced variables Q and P readswherewhere I2 = Tr(Q2) and I3 = Tr(Q3). In this parameterization, terms proportional to ρ3,8 and ρ4,16 vanish for the UDS.

The expansions –28 are our LdeG theory of modulated nematics. If we limit ourselves to a family of periodic structures with periodicity being developed in one spatial direction, the nonzero cubic and quartic couplings ρ3,4, ρ3,7, ρ4,7, and ρ4,15 should admit the UDS as global minimizers. The remaining two couplings ρ3,8 and ρ4,16 are solely responsible for one-dimensional, nonuniform periodic distortions, which makes the corresponding elastic terms vanish for the UDS. This means that depending on the choice of ρ3,8 and ρ4,16, we should be able to eliminate inhomogeneously deformed one-dimensional periodic structures from the ground states of LdeG, leaving only the UDS. In the remaining part of this paper, we are going to concentrate exclusively on this simpler case.

Bifurcation Scenarios for Uniformly Deformed Structures

Here, we limit ourselves to the UDS given in Table and determine bifurcation conditions for various symmetry breaking transitions. Clearly, the isotropic-uniaxial nematic bifurcation temperature is given by T* 12, which represents spinodal, while the I–NU phase transition takes place at TNI, slightly above T*. Likewise, TP entering aP, eq , is transition temperature for a hypothetical phase transition from the isotropic to ferroelectric phase (P ≠ 0), in the absence of the nematic order (Q = 0). Both T* and TP are examples of bifurcation temperatures from the less-ordered phase (isotropic) to the more ordered one (nematic, ferroelectric). There are further bifurcations possible for the UDSs. Below, we give bifurcation conditions for all possible phase transitions between UDSs given in Table . The procedure is found in ref (12) and we only briefly sketch it below. For given material parameters, the equilibrium amplitudes, y ∈ {r1, r2, p1, x0, v0 }, are found from the minimization of the free energy F, eq , calculated for the UDSs (explicit formula for F is listed in the Supporting Information). They are solutions of a system of polynomial equations ψ(tQ, tP, {yα}) ≡ ∂F/∂y = 0. In order to employ a bifurcation analysis to {ψα}, we expand y, tQ, and tP in an arbitrary parameter εwhere nonvanishing y defines the reference, higher symmetric phase. For example, if the reference state is the NU phase, only y4,0 ≡ x0,0 is nonzero in eq . By substituting eq into {ψα = 0} and letting equations of the same order in ε vanish, we find equations for y and t, (m = Q, P). The leading terms, proportional to ε0, are equations describing the high-symmetric reference state. Terms of the order ε1 give conditions for bifurcation to a low-symmetric phase. An equivalent approach would be to construct from F the effective Landau expansion δf(yp) in a primary order parameter yp of a low-symmetric phase by systematically eliminating the remaining parameters {y}. A detailed procedure is given in ref (29). For example, in the case of the NU–NTB phase transition, we could take for the primary order parameter either y1 ≡ r1 or y3 ≡ p1 with the final formulas being insensitive to the choice. Before we start, it proves convenient to introduce the auxiliary variables κ1 and κ2and relative phases χ1 and χ2which simplify the free energy and consequently also bifurcation formulas.

Bifurcation Conditions for I–NTB Phase Transition

Bifurcation conditions for a phase transition from the isotropic phase to the nematic twist–bend phase can be written down as an equation connecting tP and tQ

If we permit the k-vector and χ1 to vary, the bifurcation temperature tP,I–TB from the isotropic to nematic twist–bend phases will be the maximal tP, fulfilling the condition 32. Solving eq for tP and maximizing with respect to k and χ give the bifurcation values for k, sin(χ1), and tP. They readwhere tNI > 0 is the isotropic-uniaxial nematic transition temperature. Using formalism of ref (29), one can also show that the a-term, eq , is actually the leading coefficient in the Landau expansion of the free energy of the NTB phase about the reference I phase in the primary order parameter p1

Generally, the nonzero value kI–TB of the k-vector at the bifurcation point (eP2 > 8tQ) indicates that the I–NTB phase transition is, at least weakly, first order. It can be classified as an example of weak crystallization introduced by Kats et al.,[41] with fluctuations that should be observed near the k = kI–TB sphere. Interestingly, for k = kI–TB = 0, a direct inspection of higher order terms of the expansion shows that can change sign for sufficiently large λ. That is, for c > 0, the I–NTB transition can be first order (a = 0, b < 0), second order (a = 0, b > 0), or tricritical (a = 0, b = 0).

Bifurcation Conditions for NU–NTB Phase Transtition

Bifurcation conditions from NU to NTB expressed in terms of an equation connecting tP and x0 read:where κ3 = ρ3,4 – 4ρ3,7 and κ4 = 5ρ4,7 + 8ρ4,15 and x0 is the nematic order parameter calculated from the minimization of fQb in the uniaxial nematic phase. For fixed tQ (x0), the bifurcation temperature corresponds to the maximal tP, fulfilling the condition of a = 0, eq , where the maximum is taken over k and χ1. It yieldsfor k = 0 and sin2(χ1) = 1. As previously for the I–NTB phase transition, eq , the a-term, eq , is the leading coefficient in Landau expansion of the free energy of the NTB phase about the reference NU phase in the primary order parameter p1:[29]. A direct inspection of this expansion shows that the NU–NTB transition can be first order (a = 0, b < 0, c > 0), second order (a = 0, b > 0, c > 0), or tricritical (a = 0, b = 0, c > 0). Our analysis in the next section shows that for the case of CB7CB, the predicted NU–NTB transition is weakly first order. The tricritical conditions for I–NTB and NU–NTB phase transitions will be studied in detail elsewhere.

Bifurcations to Globally Polar Phases

In a similar way, we can derive the bifurcation conditions for phase transitions from I, NU, and NTB to globally polar structures listed in Table . It readswhere x0 = p1 = 0 for I–NU,p bifurcation, p1 = 0 for NU–NU,p bifurcation, and both x0 and p1 are nonzero when bifurcation takes place from NTB to NTB,p. Now, the parameter a is the leading coefficient of Landau expansion Δfp = in v0−the primary order parameter for phase transitions to polar structures. Given the form of the expansion for fP, eq , the tricritical point can only be found for the NTB–NTB,p phase transition.

Results and Discussion

Estimates of Model Parameters from Experimental Data for CB7CB

Before we explore relative stability of the nematic phases given in Table , we estimate some of the material parameters entering the expansion from experimental data in the uniaxial nematic phase. This will allow us to study properties of NTB with only a few adjustable parameters. Indeed, nearly all of the parameters of the purely nematic parts the bulk and the elastic can be correlated with the existing data in the uniaxial nematic phase. The NU phase usually appears stable at higher temperatures and NU–NTB phase transition is observed as temperature is lowered.

The very first compound shown to exhibit the stable nematic and twist–bend nematic phase was the liquid crystal dimer 1″,7″-bis(4-cyanobiphenyl-4′-yl)heptane, abbreviated as CB7CB.[5−7] This compound is constituted of two 4-cyanobiphenyls (CB) linked by an alkylene spacer (C7H14). Currently, it is one of the best-studied examples with stable NTB. In particular, Babakhanova et al.[20] have carried out a series of experiments for this mesogen in the uniaxial nematic phase. They determined the temperature variation of the nematic order parameter (see eq ), the temperature variation of the Frank elastic constants K (i = 1, 2, 3), the nematic–isotropic transition temperature TNI, the nematic twist–bend–nematic transition temperature TN. An interesting fact is that the twist–bend nematic phase formed by CB7CB can be supercooled to about 304.15 K[5] and then, there is a glass transition at approximately 277.15 K.[42] We use the data presented in Table S1 to estimate some of the parameters of extended LdeG theory.

Bulk Part

Under the assumption that Q̃ is uniaxial and positionally independent (eq with n̂(r̃) = const.), the order parameter can be determined from the minimum of the free-energy density , eq , which becomes reduced to that of the uniaxial nematic phase

Now, from the necessary condition for minimum, ∂/∂ = 0, solved for T(), we determine the ratios b/a0 = b̃, c/a0 = , d/a0 = , and f/a0 = by fitting {,T()} to the experimental data of Babakhanova et al.[20] Independently, the scaling factor a0 can be estimated from the latent heat per mole ΔHNI = for the nematic–isotropic phase transition.[43] It readswhere NI is the nematic order parameter at the uniaxial nematic–isotropic phase transition. For overall consistency, a0 has been multiplied onward by ρC = ρ/Mw ≈ 2.22 × 103 mol/m3, which is the ratio of the mass density (ρ ≈ 103 kg/m3) to the relative molecular weight of CB7CB (Mw ≈ 0.45 kg/mol), yielding the value 6.88 × 104 J m–3 K–1. Thanks to this operation, it was possible to express all expansion coefficients in units J m–3. Figure depicts results of fitting to experimental data, whereas numerical values of the parameters are gathered in Table S2. Please observe that the expansion parameter e couples to a purely biaxial part and therefore, it cannot be estimated from the data in NU.

Figure 3

Experimental data from ref (20) representing the temperature dependence of in the uniaxial nematic phase of CB7CB. The green line illustrates the effect of fitting predictions of the theory 38 to the data. T* represents the maximal supercooling temperature of the isotropic phase. Our fit is carried out by taking as an ansatz the experimentally known value of TNI and T*. Then, NI for CB7CB is estimated from our fitted function. If we compromise the agreement of TNI and T* with experimental data, a better fit can be obtained for close to the transition temperature without affecting the one in the vicinity of NU–NTB.

Flexopolarization Renormalized Elasticity of Uniaxial Nematics

It is important to realize that although (flexo)polarization terms 20, 21 vanish in the uniaxial nematic phase, any local deformation of the alignment tensor induces deformation of P̃ because of the flexopolarization coupling εP ≠ 0. Such deformations effectively renormalize elastic constants L( in ordinary nematic phases. The effect cannot be neglected if we intend to estimate L( from experimental data. A mathematical procedure of taking into account such deformations in the NU phase is to minimize the free energy, eq , over Fourier modes of the polarization field for given fixed Fourier modes of the alignment tensor. Assuming that deformation Q̃(r̃) is small and slowly varying, we obtain with this procedure the Q̃-induced deformation of P̃(r̃) expressed in terms of Fourier modes, which when transformed back to the real space take the form of a series in Q αμ and Q αμ,μ and in higher order derivatives of Q αμ. The relevant terms are

Substituting eq back to and , we obtain effective elastic contributions expressed in terms of only Q̃ αβ and Q̃ γμ,μ. When added to , they give an effective elastic free energy of uniaxial and biaxial nematics with L( being replaced by L(, where relevant L(’s are

An important physical distinction between the bare constant L( and the renormalized constant L( is of the same sort as the one between renormalized and bare Frank elastic constants, as discussed by Jákli, Lavrentovich, and Selinger:[3]L( gives the energy cost of Q̃ αβ,γ deformations if we constrain P̃α = 0 during the deformation, while L( relaxes to its optimum value during the deformation. Assuming that major contribution to flexopolarization is of the entropic, excluded volume type, any realistic experiment to measure elastic constants should not put constraints on the polar field P̃ but rather allows it to relax. In this case, which is analysed here, L( is the relevant contribution to the elastic constants in eq .

Elastic Part

In the hydrodynamic limit where spatial dependence of is disregarded and Q̃ is given by 1, the elastic free energy turns into the OF free-energy density of the director field n(r̃), eq , with K11, K22, and K33 being polynomials in (28)

The coefficients, K(, are functions of L(,[28,30] fulfilling the splay–bend degeneracy relation in the second order: K11(2) = K33(2). For completeness, they are given in the Supporting Information. As it turns out, K( with n = 2, 3, 4 along with flexopolarization renormalization 41 is sufficient to obtain a good fit of eq to experimentally observed K for CB7CB.[20] Importantly, they also provide an estimate for the (flexo)polar couplings εP and ΛQP. In finding K(, we use the (T) fit obtained from the analysis of the bulk free energy, which is a prerequisite to have a consistent theory of the uniaxial nematic phase for this compound. Results of fitting are gathered in Table S2. Finally, as the number of relevant couplings Lα( (n = 3, 4), eq , equals that of K(, we can correlate Lα( with K( using the results of the Supporting Information and of ref (28). It yields

Results of fitting of Lα(, eq , obeying stability ansatz 15 to experimental data for CB7CB are given in Table S2. The quality of fit is displayed in Figure .

Figure 4

Temperature dependence of elastic constants acquired from ref (20). Continuous lines depict the adopted approach for elastic constants within the model. Note that the model cannot explain a slight increase in K33 in the vicinity of the NU–NTB phase transition.

Predictions for the Nematic Twist–Bend Phase of CB7CB-like Compounds

Within this section, we explore the relative stability of the UDS, listed in Table , for the model 24 with parameters (estimated in previous sections), which are gathered in Table S3. We limit ourselves to the temperature interval where the NTB phase appears stable in the experiment (Table S4).

The temperatures tP and tQ are connected with the absolute temperature T of the system studied (see eqs , 20, and 22). Because a0P > 0, a0Q > 0, T* > TP, and T > TP, any straight line in the {tQ, tP} plane with a positive slope and negative tQ-intercept represents a permissible physical system with no polar order for Q̃ = 0. Thus, we present the phase diagrams in the general {tQ, tP} plane for a broader view. In our case, the experimentally related line has the form:

Results of our in-depth analysis profoundly reduced the number of adjustable parameters for CB7CB-like compounds to solely four (λ, eP, ad, and eb). From considerations related to the elastic constants, Table S2, it turns out that ΛQP (ipso facto λ), responsible for globally polar structures, is negligible, that is, . Thus, we set λ = 0. Onward, we take eP = 7.1 and ad = 0.75 as the best values to reproduce the temperature dependance of k. For the bulk biaxial parameter eb, we take two values: eb = 0 and eb = 1/6. If we recall eq , there is a term eb(I23 – 6I32) + I32. If we set eb = 0, it reduces to I32; on the other hand, when we set eb = 1/6, we have only I23/6. In the following discussion, the first scenario (eb = 0) will be referred to as theory (I32) and second one as theory (I23). The (I32) theory will enhance phase biaxiality because of its tendency to lower the equilibrium value of the w parameter, eq , while the (I23) theory is promoting the w = ±1 states through cubic and fifth-order terms in 25.[32] In this latter case, the biaxiality of NTB can only be induced by the elastic terms.

Figure a–f depict phase diagrams combined with density maps of k, θ, w, r1, r2, and p1, which are outcomes of theory (I32). In the analyzed case, being consistent with the experiment, stable, apart from isotropic, is the uniaxial nematic phase and the twist–bend nematic phase. The dashed green curve denotes numerically determined phase transitions and the red continuous curve marks the results from the bifurcation analysis. Vertical, dashed white lines designate the temperature span of NU stability (experimental) mapped on tQ (see Table S4). The purple straight line described by eq represents the phase transition sequence I ↔ NU ↔ NTB based on the CB7CB data from ref (20). From points lying on this line, we have attained information about the behavior of the pitch (p), tilt angle (θ), and nematic order parameter () in NTB, alongside the insight into the NTB’s biaxiality parameter (w) and the remaining order parameters (r1, r2, and p1) (see Figure a–f). With regard to the w parameter, in the literature, there are no available results concerning the biaxiality of NTB; thus, it is hard to compare. Nevertheless, our model permits to estimate the span and magnitude of the effect on experimentally measurable parameters.

Figure 5

Phase diagram combined with density map of the wave vector k (a), tilt angle θ (b), biaxiality parameter w (c), and mode’s amplitudes: r1 (d), r2 (e) and p1 (f) within the theory (I32). The red continuous curve marks the bifurcation between N ↔ NTB and I ↔ NTB, whereas the dashed green curve outlines the numerical results. The magenta line (described by eq ) reflects the phase sequence associated directly with the experimental data for CB7CB. Vertical, dashed white lines designate the temperature span of NU stability (experimental) mapped on tQ (see Table S4).

Figure 6

Comparison between experimental data (hollow points) and theoretical predictions (continuous green and dashed red line) for CB7CB’s NTB. (a) Temperature dependence of the NTB’s pitch p, (b) tilt angle θ, and (c) order parameter . (d) Plot depicts the behavior of the biaxiality parameter w and the plot of (e) mode’s amplitudes r1, r2, and p1 as a function of temperature in the range of NTB stability. (f) Plot illustrates the temperature behavior of the relevant factor, parameterizing the relative magnitudes of intensities of two leading harmonics of the dispersion tensor that contribute to the resonant soft X-ray scattering (RSoXS).[52] All the data are drawn with respect to the multiplied by factor 100 reduced temperature Δt, whereas key temperatures, corresponding to given Δt, are designated above each plot in absolute temperature T.

We set together results of our model with available experimental data concerning the pitch p (Figure a[7,44]), tilt angle θ (Figure b[45−50]), and nematic order parameter (Figure c[20,49−51]). At the transition temperature from NU to NTB, the pitch length is ca. 54 nm and with decreasing temperature, it saturates at the level of ca. 8 nm (Figure a). As one can see, it goes fairly well with the experimental data. Within the literature, the methodology regarding the pitch measurements for CB7CB is consistent, that is, all indicate that the pitch value reaches plateau at ca. 8 nm,[6,7,44] in contradiction to measurements of θ and .

Such span of experimental data for θ, Figure b, originates from the adopted method of determination and sample treatment. In refs,[45,46,48] birefringence measurements were employed; however, the choice of the region in which they were taken varied across the aforementioned papers (see discussion in ref (46)). In refs (49) and (50), the data regarding the conical tilt angle were extracted from X-ray methods, wherein in ref (49), it was small/wide angle X-ray scattering (SAXS/WAXS) and in ref (50) X-ray diffraction (XRD). In turn, the conical tilt angle from ref (47) was determined i.a. from 2H nuclear magnetic resonance (NMR) quadrupolar splittings of CB7CB-d4. Similar to the tilt angle, discrepancies between the data related to the order parameter, Figure c, arise from the method of acquisition. In ref (20), it was extracted from diamagnetic anisotropy measurements, in ref (49), from SAXS, in ref (50), from XRD, and in refs (50) and (51), from polarized Raman spectroscopy (PRS). As one can see, data from ref (20) stand out from the rest of the data (Figure c), although it was the only source that provided simultaneously the data for the temperature dependence of the orientational order parameter and elastic constants.

One can see that results of our model are generally in a very good agreement with experimental results, perhaps except an immediate vicinity of the NU–NTB phase transition where fluctuations, not included in the present analysis, may play a role. Predictions concerning the effect of intrinsic, molecular biaxiality on NTB seem interesting. Although the pitch, , and p1 are practically insensitive to w, the remaining observables are affected. In particular, for the tilt angle, the green continuous line associated with eb = 0 (I32 model) fits well in between the data from ref (45) (blue circles) and ref (48) (yellow squares), whereas the red dashed line, associated with the weakly biaxial case (I23 model), markedly departs from the abovementioned experimental data. Based on the results of Babakhankova et al.,[20] we can conclude that biaxiality of NTB, initially small at the NU–NTB phase transition, considerably increases on departing from the transition temperature (green line in Figure d). Figure e illustrates the behavior of the order parameters r1, r2, and p1 in the NTB phase of CB7CB, where the ratio σ = r1/r2 can be correlated with data acquirable from resonant soft X-ray scattering (RSoXS).[44,53,54] In order to make this correlation, we translated our formalism into the one presented in ref (52). Thanks to that, we could tie the results for σ with experimentally measurable scattering intensities through the following formulawhere Ξ = f1/f2 and θ is the conical tilt angle. The value of the parameter Ξ determines the intensity of the 2q0 peak (half-pitch band) with respect to the intensity of the q0 peak (full-pitch band) in the NTB phase, where q0 = 2π/p is the magnitude of the wave vector of the heliconical deformation with the pitch p. As it was stated in ref (52), if Ξ ≥ 1, then the intensity of the 2q0 peak is approximately 2 orders of magnitude lower than the intensity of the q0 peak and further strongly decreases with increasing Ξ. On the other hand, if Ξ < 1, then the intensity of the 2q0 peak escalates rapidly. As one can see in Figure f, which illustrates the temperature dependence of Ξ for both theories I32 and I23, all the data obey the relation Ξ ≥ 1, indicating a significant weakness of the 2q0 peak. Interestingly, although for I23 theory, the relative magnitude of the intensities should roughly differ by 2 orders of magnitude irrespective of the temperature, the I32 model predicts further strong reduction in the relative intensity with temperature. To the best of our knowledge, the 2q0 signal has not been detected so far in any of the examined NTB-forming compounds.[44,53−55]

Conclusions

The understanding of self-organization in the twist–bend nematic (NTB) phase is at the forefront of soft matter research worldwide. This new nematic phase develops structural chirality in the isotropic and uniaxial nematic phases, despite the fact that the molecules forming the structure are chemically achiral. Currently existing experimental data are in favor of the theory that the NU–NTB phase transition is driven by the flexopolarization mechanism. According to this theory, deformations of the director induce a local polar order which, in turn, renormalizes the bend elastic constant to a very small value relative to other elastic constants, eventually leading to the twist–bend instability. It is dictated by the term coupled with [L2(2)] (eq ), which changes sign from positive to negative at the bifurcation temperature between the nematic and twist–bend nematic.

However, a fundamental description of orientational properties of nematics based on minimal coupling LdeG theory of flexopolarization suggests that softening of K33 does not need to be a universal mechanism. Even when both K11 and K33 are simultaneously reduced because of splay–bend degeneracy (inherent to the minimal coupling LdeG expansion), the NTB phase can still become absolutely stable among all possible one-dimensional periodic structures. Because this case has not been observed experimentally to date, an important question that arises is whether the flexopolarization mechanism is indeed sufficient to explain the experimental observations at the level of “first principles” LdeG theory of the orientational order. To address this issue, we proposed generalization of mesoscopic LdeG theory of nematics, where higher order elastic terms of the alignment tensor are taken systematically into account, in addition to the lowest order flexopolarization coupling.

We demonstrated that the experimental observations involving the nematic twist–bend phase and the related uniaxial nematic phase can be explained if we generalize minimal coupling theory to the level where the properties of the high-temperature uniaxial nematic phase are properly accounted for. Especially, the constructed generalized free-energy density is in line with experimentally observed temperature variation of the orientational order parameter and the Frank elastic constants, except slight pretransitional increase in K22 and K33 on approaching the N–NTB phase transition. The origin of this pretransitional elastic response is still not fully understood. Babakhanova et al.[20] attributed increase in the elastic constants to the formation of clusters with periodic twist–bend modulation of the director, in analogy to similar pretransitional increase in K22 and K33 near the nematic-to-smectic A phase transition.[56−58] Shi et al.[59] in their Monte Carlo simulations of an augmented Lebwohl–Lasher lattice model predicted the temperature behavior of K33 to be in qualitative accordance with experimental results. Comparison of Figures 2f and 7 in ref (59) suggests that a probable cause of this pretransitional anomaly of elasticity is flexopolarization-induced short-range polar ordering in the uniaxial nematic phase. In the present model, none of the abovementioned scenarios are included. It would require a phenomenological theory with fluctuating Q̃ and P̃ fields, which is far beyond our ground-state analysis.

Our generalized theory of uniformly distorted nematics extends the elastic part of LdeG by additional two terms of third order. The added elastic terms are the only independent ones for the UDSs and various UDSs can become minimizers of the free energy, including the nematic twist–bend. This conclusion follows directly from the bifurcation analysis and the observation that the remaining four independent elastic terms of third order, not included in the theory, can always be written in such a way that they vanish for UDSs. It is worth noticing that only one more term is generally needed to extend the studies of the UDS class to all possible one-dimensional distortions of the alignment tensor. This sort of hierarchy between various elastic invariants indicates that the constructed theory can also serve as a starting point in seeking different mechanisms of softening of the nematic elastic constants. This can potentially lead to the discovery of new classes of modulated nematic structures.

The numerical analysis of the model quite well reproduces measured quantities for the NU and NTB phases of the CB7CB-like mesogens and gives numerical estimates for its constitutive parameters including otherwise difficult to access (flexo) polarization couplings. Overall, the NTB phase is predicted to be biaxial with theoretical support that major contribution to the phase biaxiality can originate from the bulk term(s) in the free energy. Although the phase transitions to NTB are weakly first order for CB7CB, the theory permits the transitions to NTB be second order with tricritical I–NTB and NU–NTB points.

Finally, very recent theories of Čopič and Mertelj[60] and Anzivion et al.,[61] also based on Q-tensor LdeG expansion, address the issue of relative stability of uniaxial, twist–bend, and splay–bend (NSB) nematic phases for thermotropic bent-core-like materials[60] and for lyotropic colloidal suspensions of bent rods.[61] Their theory uses minimal coupling expansion[12,24] extended to include one of the cubic invariants (eq S.2) that breaks splay–bend degeneracy. Because their thermodynamic potentials are unbounded from below for the general Q̃-tensor, they can only study uniaxial tensors, eq , of ≥ 0 to model twist–bend and splay–bend nematics, in contrast to our theory which is free of such limitations. They predict a sequence of phase transitions between N, NTB, and NSB with the modulated nematic order being observed upon increasing . Although these predictions are generally consistent with our bifurcation analysis of the N–NTB phase transition, eq , a more complex behavior can also be envisaged for nonzero λ, κ3, and κ4, if they all are allowed to vary independently.