Stable Assemblies of Topological Defects in Nematic Orientational Order

Abstract

We considered general mechanisms enabling the stabilization of localized assemblies of topological defects (TDs). There is growing evidence that physical fields represent fundamental natural entities, and therefore these features are of interest to all branches of physics. In general, cores of TDs are energetically costly, and consequently, assemblies of TDs are unfavorable. Owing to the richness of universalities in the physics of TDs, it is of interest to identify systems where they are easily experimentally accessible, enabling detailed and well-controlled analysis of their universal behavior, and cross-fertilizing knowledge in different areas of physics. In this respect, thermotropic nematic liquid crystals (NLCs) represent an ideal experiment testbed for such studies. In addition, TDs in NLCs could be exploited in several applications. We present examples that emphasize the importance of curvature imposed on the phase component of the relevant order parameter field. In NLCs, it is represented by the nematic tensor order parameter. Using a simple Landau-type approach, we show how the coupling between chirality and saddle splay elasticity, which can be expressed as a Gaussian curvature contribution, can stabilize Meron TDs. The latter have numerous analogs in other branches of physics. TDs in 2D curved manifolds reveal that the Gaussian curvature dominantly impacts the assembling and stabilization of TDs. Furthermore, a strong enough curvature that serves as an attractor for TDs is a respective field that could be imposed in a fast enough phase transition. Assemblies of created TDs created in such a disordered environment could be stabilized by appropriate impurities.

1. Introduction

Continuous symmetry breaking phase transitions are ubiquitous in nature and appear at all length scales, including particle physics, condensed matter systems, and cosmology.¹ To describe the essential properties of the resulting broken phase one should identify the broken symmetry and examine its elementary excitations and topological defects.² For this purpose, Landau-type approaches are particularly convenient, in which the broken phase configuration is presented by an order parameter field.³ The modeling is based on symmetry and topology, for which a microscopic system’s details are of secondary importance. This is fingerprinted in the emergent rich pallet of universal behaviors. Furthermore, it seems that physical fields might constitute fundamental natural entities.⁴ Interpreting natural phenomena from this perspective might unveil several unresolved fundamental problems in understanding nature. For example, hot topics of constant interest for decades include the intriguing behavior of neutrinos and the origin of dark matter and dark energy; physicists are still struggling to fully understand the proton.⁵ In these phenomena, localized field structures⁶ might play an important role. To study in detail universal behaviors, it is of interest to identify experimentally accessible systems where such features could be studied in detail. Gained knowledge might shed light on behavior of their analogs in related systems, which are otherwise hardly experimentally accessible.⁷

Landau-type modeling of symmetry-breaking phase transitions introduces order parameter fields describing order in the broken phase. In general, an order parameter field consists of two qualitatively different components:⁸amplitude fields and phase fields. Amplitude fields measure the strength of established order and are in bulk equilibrium characterized by a single value. In the absence of external ordering fields, they equal zero in the higher symmetry phase. Phase fields determine the symmetry breaking system’s choice among an infinitely degenerate set of competing states. This degeneracy enables long-range forces and topological defects (TDs). The latter corresponds to topologically protected localized order parameter configurations.⁹ TDs are classified using group theory approaches. For this purpose, order parameter manifold is introduced,⁸ consisting of all equilibrium order parameter phase configurations. For example, the existence of the topologically protected wall, line, or point TDs is reflected in nontrivial homotopic groups^7,10, , and , respectively. These defect configurations are commonly described by conserved topological charges which are topological invariants.

Liquid crystal (LC) phases represent an ideal laboratory system to study the general physics of order parameter excitations.^8,11 They consist of relatively weakly interacting anisotropic molecules. Weak mutual couplings allow fast noncollective fluctuations which efficiently average out microscopic details. The resulting mesoscopic orientational order is described by molecular fields exhibiting in general p-atic order^12,13 (e.g., cases p = 1 and p = 2 can be described by vector and line fields, respectively). The orientational order is in several LC phases accompanied also by translational order. Resulting configurations possess a unique combination of liquid character, orientational and/or translational order, softness (capability to respond strongly to even weak stimuli), optical anisotropy, and optical transparency. This extraordinary symbiosis of properties allows relative ease of experimental observations.^11,14 In particular, owing to optic anisotropy, order parameter excitations could be monitored using simple optic polarization microscopy.^10,15−20 Furthermore, a rich pallet of existing LC phases possesses analogs of almost all physical phenomena. For example, smectic A (SmA) configurations bear some resemblance to superconductors²¹ and Abrikosov lattices,²² coarsening dynamics of defect structures in a fast enough isotropic–nematic (I–N) quenches provides clues of the early universe evolution,^15,23 and studies of topological defects^{7,10,16,17,24} and knots^18,20 in LCs give insight into particle-like excitations in nature.

In this paper, we consider excitations in relatively simple achiral and chiral nematic thermotropic LC phases. They can be reached on cooling from the isotropic, i.e., ordinary, liquid phase. For illustration ease, we consider rod-like LC molecules. Their local orientational order is at the mesoscopic level commonly described by the tensor nematic order parameter⁸Q. In bulk equilibrium it, can be expressed in terms of the scalar uniaxial order parameter S ∈ [−1/2,1] (amplitude field) and pseudovector n (phase field), where states ± n are physically equivalent. The unit vector n, also referred to as the nematic director field, points along the local uniaxial direction. In the achiral nematic phase, Q is spatially homogeneous. In the chiral LCs, various structures can appear, which include the helical cholesteric phase and different blue phases.⁸ The order parameter phase space⁸ equals the two-sphere with antipodal points identified (S₂/Z₂). Its homotopy reads and , while is trivial. Therefore, these systems can exhibit topologically protected line and point defects. On the other hand, the wall defect could be stabilized only due to energy reasons.

2. Mesoscopic Model

We focus on bulk thermotropic achiral and chiral uniaxial LCs. We also consider cases where LC is manipulated by an external electric field E. In the frame of the Landau–de Gennes mesoscopic model, one commonly expresses the free energy functional in terms of the traceless and symmetric tensor nematic order parameter⁸

1

Here, s_i stands for the amplitude fields (Q eigenvalues) and e_i are the phase fields (normalized eigenvectors of Q). In the case of uniaxial order, Q is simplified to

2

2.1. Free Energy

We express the free energy density f = f_c + f_e + f_f as a sum of condensation (f_c), elastic (f_e), and external ordering field (f_f) contributions:^8,25

3a

3b

3c

We included only the most essential symmetry allowed terms to describe phenomena of our interest. Numerical coefficients are introduced for later convenience. The condensation term determines the local amplitude S imposed by inherent LC properties. These are defined by positive phenomenological material parameters a₀, b, c, and T*. The term enforces first order phase transition at , where the equilibrium value of S is given by and S_eq (T > T_IN) = 0, where . LC elasticity is approximated by a single positive elastic constant L. q₀ represents the LC inherent chirality wave vector. ε₀ is the vacuum permittivity, and Δε measures the dielectric anisotropic response in an external electric field E. We limit to LCs exhibiting positive anisotropies, where n tends to be aligned along E. One commonly assumes that the material quantities described above are temperature independent.

The model is characterized by numerous material-dependent lengths. For later convenience, we introduce the uniaxial (ξ) and biaxial (ξ_b) coherence lengths.^8,11 In the nematic phase, they are expressed as

4

2.2. Nematic Director Field Distortions

Let us focus on the nematic director field excitations which are, in general, relatively easily (i.e., the corresponding energy costs are low) established. It has been recently shown that the tensor gradient ∇n can be decomposed into four fundamental modes,^26,27 referred to as the bend, double splay, double twist, and tetrahedral splay modes. Each of these modes could be separately excited.

The classical Oseen–Frank free energy^9,26 is expressed as a sum of symmetry-allowed terms:

5

weighted by temperature-dependent Frank splay (K₁₁), twist (K₂₂), bend (K₃₃), and saddle-splay (K₂₄) elastic constants. However, these terms are generally coupled. The saddle splay distortions in general includes also splay and twist distortion. In particular, this contribution plays a particularly important role concerning topology. Namely, the saddle splay term can be expressed⁸ as the Gaussian curvature K_g of a small surface patch whose surface normal points along n:

6

In pioneering studies, this term has been commonly discarded because it can be transformed into the surface enclosing the LC body. It has been expected that for strong enough surface anchoring the anchoring contribution is dominating at the surface, overshadowing the saddle-splay contribution. However, this term determines volume LC elastic properties. By neglecting it, we discard possibilities to stabilize topological defects,²⁸ as will be shown in the following. In particular, we will illustrate that the Gaussian curvature plays an essential role in the physics of TDs.

2.3. Topological Excitations

As the simplest illustration, we first consider an achiral two-dimensional (2D) nematic where LC behavior is dictated only by the condensation and elastic free energy penalties. In the (x,y) Cartesian plane, we use parametrization n = e_x cos θ + e_y sin θ and assume spatially homogeneous uniaxial order. The unit vectors (e_x,e_y) point along the Cartesian (x,y) coordinates. Solutions to the equilibrium Euler–Lagrange equation ∇²θ = 0 read:⁸

7

Here, , m stands for the winding number, and θ₀ is a constant. Structures with m = 0 describe equilibrium spatially homogeneous nematic configurations where the symmetry breaking direction is given by θ₀. Furthermore, cases m ≠ 0 correspond to topological defects centered at (x,y) = 0. Due to head-to-tail invariance of n, the winding number can exhibit half integers. In 2D, m plays the role of topological charge, which is a topological invariant. Conservation of m dictates merging, decaying, and transformations of TDs. TDs bearing m > 0 and m < 0 are commonly referred to as defects and antidefects, respectively. Figure 1a depicts an m = 1 defect. In general, TDs exhibiting a minimal value of m (i.e., |m| = 1/2) can be locally stable,^8,29,30 and defect pairs {−m,m} in the nematic phase tend to annihilate into a defectless state. In Figure 1b, we show a defect pair {1/2, −1/2}, which could be induced by a local thermal fluctuation, prior to annihilation. On increasing the temperature above T_IN, numerous pairs {1/2, −1/2} are formed, where their mutual interaction is overshadowed by thermal fluctuations. The resulting “gas” of TDs corresponds to the isotropic phase.³¹ A representative time snapshot, where several pairs are present, is shown in Figure 1d.

Figure 1.

Characteristic director field of nematic point defects in 2D. (a) m = 1 defect. (b) Pair of defects {1/2, −1/2}. (c) An assembly of three defects {−1/2, 1, −1/2}. (d) An example of numerous defects which are excited by strong enough thermal fluctuations. In cases b and c, the far nematic director field is essentially spatially homogeneous.

In 2D, only point defects are possible. In 3D, in addition, line defects (disclinations) are ubiquitous.^8,23,32 These can either form closed loops^9,33−37 or originate and terminate at an LC limiting substrate.^9,11,38,39 In 3D, one assigns to TDs in addition to winding number (which reveals the local structure of disclinations) also 3D integer topological charge q, which is an integer. It is defined^10,40,41 by an integral over a closed surface parametrized by variables u and v:

8

Note that the sign of q is not uniquely defined due to the head-to-tail invariance of n.

A closed loop of a wedge disclination, which is locally defined by winding number m = 1/2 or m = −1/2, bears topological charge |q| = 1. On the other hand, the twist disclination, in which the winding number switches the sign of m, holds q = 0 (i.e., it is chargeless). Therefore, the far director field of a closed wedge disclination resembles that of a point defect of charge |q| = 1. An example is schematically sketched in Figure 2a. On the other hand, the far field of an isolated closed twist disclination is essentially spatially homogeneous. Due to their chargeless character twist, disclination loops are not topologically stable.

Figure 2.

(a) The far field of a closed m = 1/2 disclination loop (red line) resembles the director field of the q = 1 hedgehog point defect. This nematic structure is commonly stabilized in LCs confined within a droplet, whose surface imposes a homeotropic anchoring condition (i.e., n is aligned along the local surface normal direction). (b) Sketch of the DT local structure close to the cylindrical symmetry axis of the excitation. (c) Schematic structure of a lattice of merons within the x/y plane. The dashed rectangle represents a chargeless region. It consists of an escaped m = 1 nonsingular structure and two singular m = −1/2 defects.

In nematics, wall defects could be stabilized only energetically (i.e., not topologically) by imposing contradicting orientational order on distances comparable to the biaxial coherence length.^42,43 Furthermore, to avoid singularity at the centers of defects their cores^35,44,45 locally enter either isotropic or biaxial states.

3. Results

In the following, we demonstrate key TD creation and stabilization mechanisms. Note that, in general, bulk nematic equilibrium tends to be defectless. Namely, the cores of TDs are, in general, energetically expensive, and furthermore q ≠ 0 fingerprints elastic distortions in n.

3.1. Chirality and Saddle-Splay Imposed Stabilization

In Figure 1c, we show a 2D assembly of three defects bearing topological charges {m = −1/2, m = 1, m = −1/2} so that the total charge of the system equals zero. Under what conditions could this or a similar assembly be stable? Note that this three-component “quasi-particle” roughly resembles a neutron (i.e., its electric charge is zero, and it consists of three charged quarks).

In the following, we illustrate that a similar assembly could be stabilized in chiral 3D LCs in merons,⁴⁶ members of the skyrmion^6,24,28 family. For this purpose, we consider the double twist^27,28 nematic director field excitations. Locally, the corresponding n is roughly approximated in the cylindrical coordinate system (ρ, φ, z) by⁴⁶

9

where the double-twist (DT) periodicity Q is a variational parameter. The nematic distortion of this elastic mode nearby the symmetry axis at ρ = 0 is depicted in Figure 2b. To estimate the equilibrium value of Q, we calculate the free energy penalty of the DT solution, eq 9, within a cylinder of length h and radius (i.e., n(ρ = R) = −e_φ). The corresponding free energy reads

10

The DT saddle-splay penalty is given by

11

Therefore, the saddle splay elasticity promotes DT excitation for K₂₄ > 0. Of interest to us is obtaining conditions that minimize the free energy penalty of DT solution per LC volume V = πR²h. In terms of dimensionless quantities , , , and , one calculates f_DT(μ), which is minimized for . It follows that

12

In Figure 3, we plot μ_min for different anisotropies of Frank elastic constants. One sees that for reasonable values of K₂₄ the DT periodicity monotonically increases upon increasing K₂₄.

Figure 3.

μ_min = Q_min/q as a function of k₂₄ = K₂₄/K₂₂ for different values of k₃ = K₃₃/K₂₂. Full line, k₃ = 1; dashed line, k₃ = 2; dashed-dotted line, k₃ = 3.

Next, we illustrate how a simple Landau-type analysis reveals conditions promoting DT-like local excitations. We introduce scaled dimensionless periodicities

13

and integrate the free energy density given by eq 10 in the cylindrical volume of the height h and radius R and expand the resulting free energy expression ΔF_DT in powers of Q_DT. Therefore, we consider examples satisfying the condition QR < 1 (e.g., QR ∼ π/4 in the so-called half skyrmions realized in conventional blue phases⁴⁶). Landau expansion yields

14

where F₀ is independent of Q_DT. In achiral samples, it follows

15

Therefore, the double-twist excitation could be stabilized only for k₂₄ > 1, which is not realized in conventional LCs due to the Ericksen stability condition,⁴⁷ which limits the maximal value of K₂₄.

On the contrary, in chiral samples, the DT excitations are always (meta) stable. Minimization of ΔF_DT taking into account the expansion up to the quadratic term in Q_DT yields

16

and

17

Note that ΔF_DT < 0 for k₂₄ < 1.

We next discuss the stability of the Meron⁴⁶-type structure, which could be observed in thin films. Merons are determined by DT cylinders for which QR = π/2. They can form periodic structures which are enabled by a lattice of line disclinations m = −1/2 running along the cylinder axis as depicted in Figure 2c. Their presence is required topologically. Note that the central regions of DTs correspond to nonsingular 3D patterns, characterized by m = 1 winding number in the x/y plane. This structure avoids singularity via local escape of the nematic director field along the z axis. The total LC structure is topologically neutral (i.e., the total 3D charge and 2D charge in each plane equal to zero). Consequently, within the x/y plane, each DT’s imposed m = 1 “escaped” singularity must be compensated by two m = −1/2 disclinations.

Further pieces of evidence that suggest the stability of the separated local assembly of {−1/2, 1, −1/2} TDs results from the analysis of the saddle-splay elasticity within the Meron structure. The related free energy density contribution can be expressed as f₂₄ = −2K₂₄K_G (see eq 6). The Meron-type excitation exhibits regions displaying K_G > 0 (the central “escaped” volume) and K_G < 0 (the outer part). 2D studies suggest⁴⁸ that regions exhibiting K_G > 0 (K_G < 0) attract defects with m > 0 (m < 0). Therefore, if a Meron-like excitation is formed in the nematic fluid, one expects that regions exhibiting K_G > 0 (K_G < 0) attract m > 0 (m < 0) defects. In addition, the saddle-splay elasticity locally renormalizes temperature. To illustrate this, we focus on the quadratic condensation term and the saddle-splay elastic term (we label them as f₂) close to the I–N phase transition, where quadratic terms in S dominate. Taking into account⁸K₂₄ ∼ LS², it follows that f₂ = a₀(T – T*)S² – 2K₂₄K_G = a₀(T – T_eff^*)S², where

18

Note that in the absence of other elastic contributions, the resulting phase transition temperature reads . Therefore, LC regions exhibiting K_G > 0 (K_G < 0) exhibit effectively lower (higher) temperatures. Consequently, regions with K_G > 0 promote orientational order and consequently unsplit m = 1 local structure escaping⁴⁹ along the third dimension. On the other hand, in K_G < 0 regions, singular m = −1/2 structures are preferred.

3.2. Curvature Driven Stabilization

The impact of curvature on TDs is well explored in 2D curved manifolds exhibiting in-plane ordering. Most studies^48,50−53 have been based on XY-type modeling. Note that there are several experimental systems where such predictions could be realized. Examples include biological membranes⁵⁴ and LC shells.^{51,52,55−57} Theoretical studies and simulations reveal that surface patches possessing K_G > 0 and K_G < 0 are an attractor for point TDs bearing m > 0 and m < 0, respectively. This could be intuitively guessed by considering the Gauss–Bonnet and Poincare–Hopf theorems.⁵⁸ According to them, the total winding number m_tot of TDs hosted by the ordering field on a closed surface equals

19

Here, dA stands for the infinitesimally small surface patch, χ = 2(1 – g) is the Euler characteristic of the closed surface, and the genus g counts the number of the holes that the surface exhibits. For example, for spherical topology, it holds g = 0 and m_tot = 2. The differential form of eq 19 suggests

20

Therefore, K_G > 0 (K_G < 0) locally enforces dm_tot > 0 (dm_tot > 0). According to the Effective Topological Charge Cancellation⁵⁹ (ETCC) mechanism, each surface patch, to which one assigns a characteristic spatially average value of K_G, tends to be topologically neutral. Therefore, the “smeared Gaussian curvature charge” (the right-hand of eq 20) is screened by winding number or “real” (the left-hand of eq 20). This local tendency can be realized by rearranging existing TDs or by creating additional pairs {defect, antidefect}. The latter process is possible if regions exhibiting strong enough local curvatures are present.

To illustrate this curvature driven assembling of TDs, we consider closed surfaces (shells) of revolution exhibiting in-plane nematic order. The corresponding 2D Landau–de Gennes–Helfrich model⁵⁴ is summarized in the Appendix. The orientational order is presented by in-plane nematic director field n and 2D order parameter field λ, which equals zero exactly at the center of a point defect. The relationship between 3D and 2D tensor nematic order parameters is described in ref (60). In modeling, we change the shape of shells and calculate the related nematic texture. The shape is determined by a relative volume of a shell. Here, V refers to the volume enclosed by an axially symmetric shape of total surface area A, and V₀ is the volume of the sphere having the same surface area.

In Figures 4–6, we plot a characteristic pattern of defects in different shell geometries which all exhibit spherical topology which imposes m_tot = 2. Consequently, in a large enough spherical shell (i.e., its radius is much larger than the nematic order parameter correlation length), such a structure would exhibit four m = 1/2 defects. These defects are mutually repelling and in a spherical shell occupy corners of a hypothetical tetrahedron^52,57 inscribed within the sphere. In prolate shells (Figures 4), which are stable for relatively large values of v, these TDs are shifted toward the poles of a shape. Namely, at its poles, the Gaussian curvature exhibits a maximal positive value, and consequently the poles represent attractors for TDs bearing m > 0. The position of defects reveals the compromise between the attraction of m = 1/2 defects to the poles and their mutual repulsion. In the regime of intermediate values⁵⁴ of the relative volume, disk-like shapes are established, see Figure 5. This geometry exhibits maximal Gaussian curvature at the equatorial line. Consequently, the four mutually repelling TDs are assembled along this line. Finally, for relatively low values of v, the so-called stomatocyte shapes are formed, see Figures.6. In these structures, neck-like regions exhibiting relatively large values of |K_g| exist, which enable local formation of an additional two {defect, antidefect}. The two m = −1/2 antidefects are assembled within the neck where K_g < 0. The remaining six mutually repelling m = 1/2 defects are assembled in regions where K_g > 0.

Figure 4.

Order parameter profiles of equilibrium prolate nematic shell shape. (a) The nematic director field is superimposed onto the order parameter profile λ/λ₀ in the φ/s plane (λ₀ describes the bulk equilibrium value of λ). (b) 3D shell shape. (c) Gaussian curvature (K_g), mean curvature (H_c), and deviatoric curvature (D_c) of the shape. v = 0.80, R/ξ = 7, k = κ. Positions of TDs are marked with capital letters. .

Figure 6.

Order parameter profiles of equilibrium stomatocyte nematic shell shape. (a) The nematic director field is superimposed onto the order parameter profile λ/λ₀ in the φ/s plane. (b) 3D shell shape. (c) Gaussian curvature (K_g), mean curvature (H_c), and deviatoric curvature (D_c) of the shape. v = 0.20, R/ξ = 7, k = κ. Positions of TDs are marked with capital letters.

Figure 5.

Order parameter profiles of equilibrium oblate nematic shell shape. (a) The nematic director field is superimposed onto the order parameter profile λ/λ₀ in the φ/s plane. (b) 3D shell shape. (c) Gaussian curvature (K_g), mean curvature (H_c), and deviatoric curvature (D_c) of the shape. v = 0.60, R/ξ = 7, k = κ. Positions of TDs are marked with capital letters.

3.3. Phase Transition and Disorder Driven Stabilization of TDs

The most ubiquitous source of TDs is fast enough symmetry breaking phase transitions, which is explained by the universal Kibble–Zurek mechanism.^1,61 Note that the mechanism was originally introduced in cosmology to model coarsening dynamics of topological defects in the Higgs field in the early universe. The only two conditions that need to be fulfilled for the KZ mechanism are continuous symmetry breaking and a finite velocity of information propagation. According to it, the following features are expected. If a phase change is realized on a short enough time scale with respect to the relevant amplitude order parameter relaxation time, then order in well-enough separated spatial regions is not correlated due to the finite speed of information propagation. Consequently, different symmetry breaking directions are in general selected. Domains are formed, within which the phase ordering field points along a similar direction. At domain interfaces, topological defects are created.

The subsequent domain growth is enabled by the annihilation of defects and antidefects. This reduces the concentration of energetically expensive domain walls. In common cases and in the absence of “impurities,” the characteristic linear size of domains grows with time, exhibiting a universal scaling law^62,63 ξ_d ∝ t^γ, where γ is the scaling coefficient. Note that qualitatively different TDs appear for polar and axial (i.e., quadrupole) symmetry breaking fields. The former can exhibit topologically stable point defects. On the contrary, nematic-like order also allows line defects.

The presence of “impurities”, which often act effectively as random-field like disorder, can arrest the domain growth and consequently stabilize TDs. In particular, phases reached by continuous symmetry-breaking phase transition are particularly susceptible to disorder due to the existence of easily excitable Goldstone modes. According to the Imry–Ma theorem,⁶⁴ even an infinitesimally weak random-field type of disorder can destroy long-range order of a pure system. The resulting structure is expected to exhibit short-range order, characterized by the Imry–Ma domain size ξ_d^(IM) ∝ w^–2/(4–d). Here, w measures the disorder strength and d stands for the spatial dimensionality. However, several studies reveal⁶⁵⁻⁶⁸ that, for weak enough disorder, quasi-long-range order or even long-range order might be established.

In the following, we illustrate the derivation of the characteristic size ξ_d^(p) of protodomains that are nucleated via the universal Kibble–Zurek (KZ) mechanism. We show that created domains could significantly depress the resulting phase transition temperature, which has not yet been reported. As a demonstrating example, let us consider a fast enough isotropic–nematic phase transition. Let us describe the temperature variation by the dimensionless temperature r = (T – T*)/T*, where T* stands for the supercooling temperature. Namely, in a fast enough isotropic phase, supercooling is very likely. We characterize the fast quench by linear time dependence characterized by the quench rate τ_Q:

21

Here, τ_Q describes the time needed to increase the temperature from T = 0 to T*. Close to weakly first-order I–N phase transition, the characteristic amplitude order parameter relaxation length and relaxation time obey equations

22

τ₀ and ξ₀ determine characteristic responses deep in the symmetry broken phase, and in typical nematic LC it holds η ∼ 1 and v ∼ 1/2. These values of critical exponents are also predicted by the mean-field description.

To estimate the size of first formed domains, the so-called protodomains of size ξ_p, one originates from the disordered phase. The maximal size of fluctuations exhibiting locally (para) nematic order within the disordered isotropic “sea” is estimated by ξ (see eq 22). The temperature regime where |t| > τ is referred to as the “impulse” regime. In it, the dynamic of the system is fast enough to adapt to changes in temperature, and the system displays a roughly equilibrium-like order. The qualitative change in behavior is expected when the time to reach the phase transition becomes comparable to the relaxation time. This time is termed as the Zurek time and is defined by the condition |t_z| = τ. It follows

23

In the so-called adiabatic time regime −|t_z| < t < |t_z|, the order parameter dynamics are relatively slow. In approximate treatment, one assumes that the dynamics are frozen in and the system falls out of equilibrium. When the system exits the adiabatic regime at t = |t_z|, the dynamics unfreeze. The size of the largest fluctuation generated clusters in the isotropic phase, which “survive” the phase adiabatic time regime crossing, is estimated by

24

and ξ_p ∼ ξ^(max).

We suggest that the elastic free energy contribution should also be included in the analysis, which was neglected in the original estimates made by Zurek. Below, we present expected related consequences which could be experimentally tested.

We express the free energy density as f ∼ f_c + f_e, where (see eq 3)

25a

25b

We thus neglect biaxial states. We assume that the quench is realized fast enough so that domain structure in the director field is established after the Zurek time. With this in mind, we set |∇n|² ≈ 1/ξ_d², where ξ_d describes a typical domain size. As a reminder, the director relaxation is slower than that of S. It follows that

26

where . Note that in bulk equilibrium it holds that T_IN = T* + b²/(4a₀b). Therefore, the elastic distortions effectively decrease the local phase transition temperature:

27

For example, for the 5CB LCs request, ΔT_IN = |T_IN – T_IN^(eff)| ≈ 1K would suggest that , which is a sensible value.

4. Discussion

In most familiar examples, TDs in nematic LCs are stabilized by appropriate boundary conditions at LC-confining body interfaces.^{10,14,30,34,38,39} In these cases, surface conditions impose topological constraints (embodied in nonzero topological charges) that stabilize various configurations of TDs. Within a region, the total topological charge q_tot (i.e., m_tot in 2D) is determined by topologically imposed constraints. Note that for a given value of q_tot there can be different configurations of defects, which however must obey the topological charge conservation law. Therefore, for instance, strong thermal fluctuations could trigger a transformation between two different competing structures hosting TDs; however, q_tot remains unchanged.

In the present paper, we consider first cases where the total topological charge of the system equals zero. In such a setting, TDs are not enforced topologically but can be stabilized on energy grounds. For illustration, we use a simple Landau-type analysis. It reveals conditions that set fertile ground to form TDs. We demonstrate that a combination of chirality and saddle-splay elasticity may promote strong enough double-twist fluctuations. The latter might stabilize localized phase field distortions which are in a relevant 2D cross-section characterized by winding numbers {−1/2, 1, −1/2}. Such an assembly is topologically neutral and could be embedded in a uniform director far-field. Furthermore, the resulting saddle-splay deformation changes local effective temperature. For K₂₄ > 0 (which is commonly realized⁸ in nematic LCs), the local director field exhibits a positive Gaussian curvature K_g profile which effectively decreases local temperature. Consequently, this region favors the escaped nonsingular m = 1 deformation, which does not require local LC melting. On the contrary, regions possessing K_g < 0 (and a concomitant local increase in T) are more favorable for m = −1/2 distortions whose cores require a local change of LC amplitude. Such assemblies are in LCs referred to as merons.⁴⁶ In confined geometries,^16,24 merons could be stable even as isolated entities, where their structure could be studied using relatively simple confocal polarizing microscopy.¹⁶ Furthermore, in such a setting, their structure could be enlarged, i.e., making it even better experimentally approachable, by approaching the critical point.⁶⁹ Namely, close to critical points, relevant order parameter amplitude correlation time and correlation length diverge.⁸ One could tune critical conditions⁷⁰⁻⁷³ simply by reducing the cell thickness, supposing that the surface-wetting interaction has a term that is linearly conjugated with the order parameter amplitude. Note that merons are members of the skyrmion family,⁴⁶ which has several analogs^6,74 in other branches of physics. Experiments in LCs evidence that strong enough chirality allows stabilization of quarter skyrmions, half skyrmions, or structures between the mentioned two configurations.⁷⁵ Indeed, skyrmions (the so-called full-skyrmions) were first introduced in particle physics to explain the structure of mesons and baryons.⁶ Therefore, LCs provide a particularly convenient testbed to study in detail the fundamental properties of such localized structures.

The importance of K_g in stabilizing TDs is evidently shown in 2D structure via Gauss–Bonnet and Hopf–Poincare theorems.⁵⁸ Note that these theorems are based on universal parallel transport.⁷⁶⁻⁷⁹ It was originally introduced to describe parallel lines in curved manifolds.^76,79,80 Based on that, general relativistic theory was developed in which the curvature of space–time emerges as a gravitational force. Parallel transport was also used in nematic LCs to determine the presence of defects^57,81,82 (i.e., regions exhibiting nonzero winding number) in a given surface patch hosting nematic order. Furthermore, using it, one can define intrinsic and extrinsic geometric potentials⁸² which determine regimes where defects are likely to appear without solving Euler–Lagrange equations. Extremes of the intrinsic geometric potential are in general attracting TDs. On the contrary, regions exhibiting a large absolute value of the extrinsic geometric potential repel TDs. To calculate the potentials, one essentially needs only information about the geometry⁵⁴ of the manifold which hosts nematic order. These and other studies reveal that nematic LCs are perfectly suited to study the impact of curvature on TDs. Furthermore, concepts developed in 2D, where phenomena could be relatively visualized, could be applied to higher manifolds exhibiting higher dimensionalities.⁵⁸

Efficient generators of TDs are also sudden changes in systems. Due to them, distant enough regions cannot synchronize their ordering. Resulting frustrations are often resolved by the creation of TDs. Consequently, in fast enough symmetry phase transition domains are formed. Note that the early I–N coarsening dynamic is dominated by chargeless (twist) disclination loops.³² Therefore, these TDs are most ubiquitous in the early regime. They weakly interact with the surrounding nematic structure, and they gradually vanish because they are not topologically protected.^83,84 Recent studies^39,85 suggest that these disclinations can act simultaneously like defects and antidefects. This remains somehow for Majorna particles⁸⁶ and neutrinos, the behaviors of which are still not fundamentally understood. Note that recent experimental^18,41,87 studies suggest that twist nematic loops could be stabilized by toroidal topology. In these studies, colloids of toroidal topology, where their surface enforces homeotropic anchoring (i.e., nematic director tends to be aligned along the local surface normal), are immersed in nematic LC. Note that in 3D the torus does not enforce the 3D topological charge.⁴¹ However, 2D analysis reveals that TDs must be present. Therefore, either pair or pairs {defect,antidefect} are formed, or chargeless loops. Namely, a torus possesses regions exhibiting K_g > 0 and K_g < 0, which individually attract different signs of winding number. Recent experiments⁸⁷ reveal that torus geometry efficiently stabilizes twist loops in chiral LCs. Hence, if neutrinos have roughly similar local structures, they could be stabilized by an appropriate topology of higher dimensional space (exhibiting alternating K_g > 0 and K_g < 0 regions) in the string theory description of nature.

Finally, if impurities are present, they could stabilize^88,89 TDs which are formed in a fast enough quench. The resulting phase might exhibit glass behavior,^89,90 which also represents an open problem in physics. Therefore, looking at the glass problem from the perspective of curvature-trapped TDs might yield some additional understanding in the field.

5. Conclusions

TDs in a physical field are inevitably formed in symmetry-breaking phase transitions. Such transitions allow the existence of competing configurations whose coexistence is enabled by TDs. Namely, their local structure permits realization of large changes in ordering on relatively short length scales, and for this reason they serve as mediators between “conflicting” regions. There are several pieces of evidence that curvature plays a crucial role in stabilizing TDs. An ideal testing bed to study curvature-driven stabilization of TDs is thermotropic nematic LCs. They exhibit temperature-driven isotropic–nematic phase transition in which continuous rotational symmetry is broken. Established orientational order is commonly described by the second rank traceless and symmetric tensor nematic order parameter, which is in bulk equilibrium determined by scalar order parameter (amplitude) field and nematic director (phase) field. The equilibrium degeneracy of the latter enables the existence of TDs. We have illustrated using a simple Landau-type analysis how the combining effect of chirality and saddle splay elasticity could stabilize TDs in a system whose total topological charge equals zero. In this case, TDs are stabilized due to inherent “defect-fertile” conditions within the system. The saddle-splay deformation is in turn related to the Gaussian curvature of hypothetical planes whose normal is pointing along local nematic director field n. In general, the impact of Gaussian curvature and curvature on TDs is well explored in 2D systems, and we have provided typical examples which manifest these effects. Furthermore, curvature in n patterns could be nucleated in a fast enough phase transion, and generated TDs could be stabilized by impurities present in the system. Similar phenomena are expected to appear in other systems reached via continuous symmetry-breaking phase transitions.

Appendix: 2D Landau–de Gennes–Helfrich Mesoscopic Approach

The geometry of the shape, whose local normal is given by v, is determined by the curvature tensor^90,91C, and the orientational ordering on the shape is described by the 2D nematic tensor order parameter⁸¹Q^(2D). These tensors are expressed in their eigenframes as

A1

A2

In eq A1, the unit vectors {e₁,e₂} determine a local principal curvature frame, where {C₁,C₂} are the corresponding principal curvatures. In eq A2, the nematic director fields n and n_⊥ correspond to the Q^(2D) eigenvectors, where v = n × n_⊥, and λ ∈ [0, 1/2] is the orientational order parameter. The free energy contributions f = f_H + f_c + f_e are expressed as follows:

A3

A4

A5

where f_H stands for the classical Helfrich contribution,⁹⁰ which resists surface bending deformations for a positive bending modulus κ. The nematic condensation contribution f_c is enforcing equilibrium nematic orientational order below a critical temperature T_c. The quantities α₀ and β are positive phenomenological constants. The elastic contribution f_e is weighted by the positive elastic constant k; ∇_s stands for the surface gradient operator.⁸¹

In simulations, we restrict to closed axisymmetric two-dimensional shells exhibiting spherical topology. The shell surface is constructed by the rotation of the profile curve about the z axis by an angle of φ = 2π in order to obtain a surface of revolution. A generic point on the surface is given by⁵⁹

A6

where ρ(s) and z(s) are the coordinates of the profile in the ρ/z plane and s represents the arc length of the profile curve. The total length of the arc of a profile is denoted by L_s. We define that the principal directions (e₁, e₂; see eq A1) point along meridians (φ = const) and parallels (s = const), since these are the lines of principal curvatures {C₁,C₂}. On any point on the surface, we can calculate the Gaussian curvature K_g = C₁C₂, the mean curvature , and the deviatoric curvature .

Author Contributions

S.K. initiated this study. S.K. and A.H. wrote the manuscript. L.M. and A.I. developed a Monte Carlo program for the calculation of nematic profiles, and L.M. performed the numerical simulations. A.H. and J.P. developed the Landau approach in studying merons and prepared the figures.

The authors declare no competing financial interest.