3.2.3.1. Usual viscous terms

With the Leslie notations, the viscous force and the symmetric part of the viscous stress tensor reads:

[3.20]

In this expression, the coefficients γ₁, α₁, α₄ (which appears as an “ordinary” viscosity) and represent the direct effects represented by the blue left and right arrows in Figure 3.1, while γ₂ describes the crossed effect represented by the green arrows in this figure.

This gives the following well-known expression for the complete viscous stress tensor:

[3.21]

where From these relations, we calculate γ₁ = α₃ − α₂ and γ₂ = α₂ + α₃ with α₂ + α₃ = α₆ − α₅, an equality known as the Parodi relation (1970). This relation shows that the six viscosity coefficients α_i, initially introduced by Leslie, are not independent because of the reciprocal Onsager relation that applies to the crossed effect “γ₂”, represented by the green arrows in Figure 3.1. We underline that all the viscosities (including γ₂) enter by construction into the expression of the complete viscous stress tensor.

3.2.3.2. Leslie TM and TH terms

These terms are only present in cholesterics in which no mirror symmetry exists because of the chirality of the phase. According to Leslie, these terms read (Leslie 1968):

[3.22]

Here, ν is the TM Leslie coefficient⁵ and μ is the TH Leslie coefficient. From the previous equations, the Leslie contribution to the total non-equilibrium stress tensor can be calculated:

[3.23]

with μ₁ = (μ + ν)/2 and μ₂ = (μ − ν)/2. As for the viscosities, we note that the two coefficients μ and ν enter into the expression of the total stress tensor, while only ν enters into the expression of the thermomechanical force. Another important point is to note that μ and ν (and thus μ₁ and μ₂) are pseudoscalars that must vanish in a nematic phase for symmetry reasons.

3.2.3.3. Akopyan and Zel’dovich TM and TH terms

These terms appear in both cholesteric and nematic LCs when the director field is distorted. These terms linear in are much more complicated. By using the Akopyan and Zel’dovich notations, they read:

[3.24]

where . In these equations, the coefficients are the thermomechanical Akopyan and Zel’dovich coefficients and the coefficients are the thermohydrodynamical coefficients.

Actually, the four TM coefficients can be associated with the four fundamental deformations of the director field. This is not so readily apparent when one looks at the previous equation. For this reason, we proposed in Oswald et al. (2017) to rewrite this force in the equivalent form:

[3.25]

where we defined the splay, twist, bend and Gauss contributions to the thermomechanical effects as:

[3.26]

[3.27]

[3.28]

[3.29]

The correspondence between the new TM coefficients and the ξ₁₋₄ of Akopyan and Zel’dovich is as follows:

[3.30]

A similar procedure can also be applied to the eight TH coefficients ξ₅₋₁₂ and leads to the equivalent form for σ^(s,AZ):

The correspondence between the new TH coefficients ζ₁₋₈ and the ξ₅₋₁₂ of Akopyan and Zel’dovich is as follows:

From the previous equations, the Akopyan and Zel’dovich contribution to the total non-equilibrium stress tensor and force can be obtained. Its general expression is very complex and will not be given here. A simplified expression can be found by assuming that the new TM coefficients are all equal and by setting ζ₁₋₆ ≡ 2ζ − ζ′ + ζ″, ζ₇ ≡ ζ′ − 2(ζ + ζ″) and ζ₈ ≡ ζ′ for the new TH coefficients. With this choice, we find:

[3.31]

The definitions of the coefficients ζ, ζ′ and ζ″ may feel a bit convoluted for now, but we will see later that, for some specific geometries, the terms in ζ′ and ζ″ do not contribute to the Navier–Stokes equation, thus motivating the convention adopted here. Again, we see that both TM and TH coefficients ξ₁₋₁₂ enter into the expression of the total stress tensor, whereas only the TM coefficients ξ₁₋₄ enter into the expression of the TM force. Last but not least, it is important to note that the ξ_i are true scalars, contrary to the Leslie coefficients μ and ν, which are pseudoscalars.

3.2.3.4. Heat conductivity tensor

The tensor of heat conductivity is given by

[3.32]

where is the transverse Kronecker delta. In this expression, κ_⊥ and κ_‖ are the thermal conductivities perpendicular and parallel to the director. To these usual terms add the TM and TH terms that are extremely small in comparison with the latter of the order of 10⁻¹² − 10⁻¹¹ in relative value (Dequidt et al. 2016). For this reason, they can be neglected and we shall not give them here. Their complete expression is however given in Poy and Oswald (2018).

3.4.1.1. A bit of history

As we already mentioned, it has long been believed that the Leslie TM effect was entirely responsible for the Lehmann effect (Lehmann 1900; Dequidt 2008; Oswald and Dequidt 2008b; Yamamoto et al. 2015; Ito et al. 2016). But this is wrong, as demonstrated by several experiments (Oswald 2012b; Poy 2017; Oswald and Poy 2018; Oswald et al. 2019b). For a review, see Oswald et al. (2019a). However, the first irrefutable experimental evidence of the Leslie TM effect in a cholesteric LC was reported by Éber and Jánossy (1982). This experiment was performed at the compensation temperature of a cholesteric phase, which raised a strong controversy with theorists, the latter claiming that the Leslie TM effect should disappear at this temperature (Pleiner and Brand 1987, 1988), while Éber and Jánossy found a non-zero effect at this temperature (Éber and Jánossy 1984, 1988). To make a decision about this issue, Padmini and Madhusudana (1993) conducted a new experiment to measure the electric analog ν_E of the thermomechanical coefficient ν in a compensated cholesteric mixture⁶. In doing this, they found that ν_E vanishes and changes sign at the compensation temperature, as proposed by theorists. As the same behavior should hold for the thermomechanical coefficient, they logically concluded that the Éber and Jánossy result was due to an artifact of measurement, thus putting a temporary end to the debate.

Temporary, because we redid these two experiments and found the same results as Éber and Jánossy on one side (Dequidt and Oswald 2007b; Dequidt et al. 2008) and Padmini and Madhusudana on the other side (Dequidt and Oswald 2007a), raising again a contradiction. To solve it, we analyzed in more detail the results of the Padmini and Madhusudana experiment and showed that they could not be explained in terms of electromechanical coupling, but resulted from a flexoelectric effect (Dequidt and Oswald 2007a). As for the Éber and Jánossy results, we realized that they were indeed compatible with the Leslie theory after noticing that the symmetry group of the cholesteric phase at the compensation temperature is D_∞ (the phase remains chiral, not only at the microscopic scale, but also at the macroscopic scale), and not D_∞h as in an usual nematic phase, in spite of the fact that the director field is unwound (Dequidt et al. 2008).

3.4.1.2. Principle of the experiment and theoretical prediction

The original experiment by Éber and Jánossy was performed with a compensated cholesteric mixture. In such a mixture, there exists a temperature, called compensation temperature (T_c), at which the equilibrium twist vanishes and changes sign. Mixtures of usual nematics with the chiral dopant cholesteryl chloride (CC) often have this property. A typical example is shown in the phase diagram of Figure 3.4 where the nematic LC used is an eutectic mixture (EM) composed of 57 wt% of 8CB (4-n-octylcynaobiphenyl) and 43 wt% of 8OCB (4-n-octyloxycynaobiphenyl). Note that similar phase diagrams are observed with the mixtures 8CB + CC⁷ or 8OCB + CC, also used by Éber and Jánossy and ourselves. Experimentally, the compensated mixture is introduced between two parallel glass plates treated for strong homeotropic anchoring. This sample is then placed inside a temperature gradient parallel to the glass plates. In our own experiments, the directional growth apparatus described in Oswald et al. (1993) was used to impose the temperature gradient. In this geometry, the cholesteric helix unwinds when its pitch becomes larger than the sample thickness. As a result, a band of homeotropic “nematic phase”, centered on the compensation temperature, forms in the sample. This region is bordered by cholesteric fingers, which are very clear under the microscope as shown in Figure 3.5 (for a review about the helix unwinding in homeotropic samples, see Oswald et al. (2000) and Oswald and Pieranski (2005)).

In practice, the director field in the homeotropic “nematic” band is a little distorted because of the presence of the temperature gradient. Two effects add here. The first one is due to the temperature variations of the elastic constants K_i and the equilibrium twist q₀ of the cholesteric phase. This equilibrium effect is described by the contribution to the molecular field given in equation [3.14]. The second effect is due to the non-equilibrium Leslie, Akopyan and Zel’dovich thermomechanical forces given in equations [3.22] and [3.24], respectively. At equilibrium, the torque equation reads

[3.39]

where is the usual molecular field given by equation [3.13]. This equation is complicated, but can be simplified by noting that the director field is very little distorted. As a result, one can write that, to first order in distortion, by taking the x-axis along the temperature gradient and the z-axis perpendicular to the glass plates. In this limit, the previous equation becomes

[3.40]

which gives explicitly by noting that

[3.41]

with

In the previous equations, the second derivatives with respect to x can be neglected because the sample thickness d is always much smaller than the width of the nematic band (this can be checked a posteriori). Solving these equations by neglecting these derivatives gives (Dequidt 2008; Dequidt et al. 2008):

[3.42]

These formula generalize the solution given by Éber and Jánossy (1982) since they are still valid out of the compensation point T_c. In particular, they give back the spinodal limit for the nematic phase as n_x and n_y diverge when qd = π(K₃/K₂) or d/p = K₃/(2K₂) (Oswald et al. 2000; Oswald and Pieranski 2005).

The solution can be linearized in q₀ in the vicinity of the compensation temperature T_c (at which q₀ = 0), which gives:

[3.43]

This formula shows that measuring K₃ and n_y at T_c (where n_x = 0) gives the effective coefficient . From this measurement and that of the corrective term , the value at T_c of the thermomechanical Leslie coefficient ν can be obtained.

3.4.1.3. Experimental results

In practice, the distortion of the director field in the middle of the nematic band was obtained by measuring the phase shift Φ_H between the extraordinary and ordinary components of a thin laser beam (5 μm in diameter at the waist), crossing the sample at this place. A straightforward calculation gives

[3.44]

In this formula, n_o and n_e are the ordinary and extraordinary refraction indices, k = 2π/λ is the angular wavenumber of the laser, θ is the angle (very small experimentally, but impossible to exactly cancel) between the beam and the normal to the sample and φ is the azimuthal angle of the beam with respect to the temperature gradient. This equation shows that at normal incidence (θ = 0), Φ_H is proportional to G² and d⁵, a result already given by Éber and Jánossy (1982). However, an additional term linear in G appears when the laser beam is slightly misaligned, but the term in G² remains unchanged.

Experimentally, Φ_H was measured by using a rotating analyzer, a quarter-wave plate, a photodiode and a lock-in amplifier following the method of Lim and Ho (1978). A typical curve measured with the compensated mixture 8OCB + 50wt% CC is shown in Figure 3.6. This curve, obtained with a sample of thickness d = 100 μm, is well fitted by a parabola of type a_HG − b_HG², in agreement with equation [3.44]. Performing similar measurements with samples of different thicknesses showed that the fit parameter b_H was proportional to d⁵ in agreement with equation [3.44]. From this measurement, a value of was obtained. Finally, K₂ and were measured. From these data, it was found that in the mixture 8OCB + 50wt% CC, ν = (2.8 ± 0.6) × 10⁻⁷ kg K⁻¹ s⁻² at T_c. Note that in these experiments, great care was taken to determine the uncertainties by using the maximum-likelihood method each time a quantity was measured. This value of ν is close to the value obtained before by Éber and Jánossy in the mixture 8CB + 50 wt% CC. This confirmed that the Leslie thermomechanical coefficient does not vanish at the compensation temperature of a cholesteric phase.

3.4.3.1. TIC rotation velocity: a general formula

In practice, the LC sample of thickness d is sandwiched between two parallel glass plates of total thickness ⁸. The temperature gradient G is obtained by imposing a temperature difference ΔT between the two external top and bottom faces of the sample. As a result, the local gradient G in the LC is proportional to the imposed temperature gradient . We refer to Oswald and Dequidt (2008b) and Dequidt (2008) for a description of the setup used in experiments to impose the temperature gradient.

With the z-axis vertical and oriented upwards, the components of the director in a TIC are given by

[3.46]

where the zenithal and azimuthal angles α and ϕ only depend on z since the system is supposed to be translationally invariant in the horizontal plane (x, y).

In the following, we show that the director rotates at constant angular velocity ω when a temperature gradient is applied along the z-axis, provided that the anchoring is sliding on the two glass plates. This rotation is due to the non-equilibrium TM torque that must equilibrate with the elastic and viscous torques in the stationary regime according to the general torque equation [3.8].

Two methods can be used to calculate this velocity. The first, used by Leslie (1968) and ourselves in Dequidt (2008); Dequidt et al. (2008); Oswald and Dequidt (2008a) and Oswald (2012b) consists of directly solving the torque equation [3.8] subjected to the boundary condition 3.12. This is quite easy to do with Leslie geometry when α = π/2. In this case, the bulk torque equation reduces to and reads explicitly

[3.47]

where the three contributions – elastic, viscous and thermomechanical – are easily recognizable. If the anchoring is sliding planar on the plates, this second-order differential equation must be solved with the boundary conditions

[3.48]

at the bottom plate z = 0 and

[3.49]

at the top plate z = d, with γ_S being the rotational surface viscosity.

Following Leslie (1968), we can look for a solution of the type . After substitution in the previous equations, we obtain

[3.50]

where G is the local temperature gradient given by by denoting by κ_g the thermal conductivity of the glass⁹. Integrating over the sample thickness, the first equation in [3.50] is written by using the two other equations and by assuming that the helix is slightly distorted :

[3.51]

where is the effective Leslie coefficient, which is measured experimentally.

If the sample is very thin, the material constants are almost constant inside the sample. In this limit, the previous equation becomes simply by setting :

[3.52]

where the values of the material constants are taken at the average temperature of the sample.

The same calculation could be made in the general case, when but it is much more complicated. For this reason, we prefer another method, proposed in Dequidt et al. (2016), which does not need to explicitly solve the torque equation to find ω.

The starting point of this calculation consists of noticing that the elastic energy of the cholesteric phase is constant during the rotation. Mathematically, this condition reads

[3.53]

Replacing and C · in this equation by their expressions given in equation [3.9] and [3.12], after noticing that in a TIC¹⁰ yields:

[3.54]

Because , we obtain from this equation by using equation [3.46] and the expressions [3.22] and [3.24] of the Leslie and Akopyan and Zel’dovich forces

[3.55]

where

[3.56]

In these formulas, the temperature gradient G depends on z in general. It is obtained by writing that the heat flux is the same in the sample and in the glass plates. Using equation [3.32], this gives:

[3.57]

Equations [3.55]–[3.57] generalize to any TIC the formula [3.51], obtained in the Leslie geometry. They show that, in the general case, the rotation velocity of the helix only depends on the three non-equilibrium TM coefficients ν, . A very important point is that the rotation disappears when these coefficients vanish. This shows that the static equilibrium TM contribution to the molecular field cannot be responsible for the rotation of the helix in a TIC and contributes just to its deformation during the rotation. This is a major advantage with respect to the Éber and Jánossy experiment in which both effects – equilibrium STM and non-equilibrium TM – played the same role and should be separated.

3.4.3.2. Experimental results with the compensated mixtures in the Leslie geometry

The goal of these experiments was to directly evidence the TM effect predicted by Leslie in 1968. To perform them, a specific surface treatment of the glass plates, allowing a sliding planar anchoring of the LC molecules, was developed in collaboration with Żywoćinski (Oswald et al. 2008). This surface treatment – which consists of the deposition by spin coating of a thin layer of a viscoelastic polymer¹¹ – was fully characterized. In particular, its surface viscosity was measured (Oswald 2012a; Oswald and Poy 2013) and the problems of memorization that occur when the director stops rotating, or rotates very slowly, were described in Oswald (2014a).

In sliding planar samples, a great number of defects can form since the anchoring direction is degenerated. In practice, most of the defects are disclination lines, even if ±1 defects are sometimes observed. These defects usually nucleate in large number when the sample is cooled down from the isotropic liquid. As the anchoring is sliding, most of the defects annihilate (Oswald et al. 2008) but some of them can remain in the sample after annealing because they are pinned on a dust particle or a surface defect. In the following, we begin with a description of what happens in the samples far from the defects, in regions where the in-plane distortions of the director field are negligible. We will then study the director rotation in the vicinity of disclination lines and in the presence of a surface defect. Finally, we will show that it is possible to stop the rotation of the director by applying an electric field and we will describe the final state of the sample once the director has stopped rotating.

3.4.3.2.1. Rotation in a homogeneous sample

The first experiment in the Leslie geometry was performed with a compensated mixture of 8OCB + 50 wt%CC (Dequidt et al. 2008). This choice was driven by the Éber and Jánossy experiment that showed the existence of a TM Leslie effect at the compensation temperature of this mixture. In order to measure the rotation velocity of the director, a 10-μm-thick sample was prepared. This sample was then placed in the setup, described in Oswald and Dequidt (2008b), to impose a temperature gradient and was observed in the polarizing microscope. By doing so, it was found that, at the compensation temperature, the transmitted intensity between crossed polarizers was oscillating in time in a sinusoidal manner (Figure 3.8), revealing that the director was rotating at constant velocity in the sample. For the first time, this showed the existence of the TM Leslie effect at the compensation temperature, in a direct way.

The experiment was then performed at other temperatures from both sides of the compensation temperature (Oswald and Dequidt 2008a). An important point was the sense of rotation of the helix. It was determined by looking at the direction in which the polarizers should be rotated to maintain a constant intensity (another method will be described later). The result of these measurements is shown in Figure 3.9. As we can see, the sense of rotation is the same on both sides of the compensation temperature, counterclockwise (ω > 0) when the temperature gradient is directed downwards (ΔT < 0) and clockwise when the temperature gradient is directed upwards (ΔT > 0). The experiment also showed that ω is proportional to ΔT in agreement with formula [3.51]. From these measurements, the coefficient was deduced after the ratio κ_g/κ_⊥ and the viscosity were measured. At T_c, it was found that κ_g/κ_⊥ ≈ 7, γ₁ ≈ 0.075 Pa s and γ_S ≈ 3.2 × 10⁻⁷ Pa s m, which led to = ν ≈ 1 × 10⁻⁷ kg K⁻¹ s⁻². This value is compatible with that previously found in the Éber and Jánossy experiment. The curve shown in Figure 3.9(a) also suggests that changes little with temperature. Indeed, the viscosity decreases when the temperature increases, which could explain, in large part, why the velocity increases when the temperature increases. This was checked in minute detail in the mixture EM + 50 wt% CC in which the two viscosities γ₁ and γ_S were measured as a function of temperature. In this mixture it was found that, within experimental errors, was approximately constant, of the order of 1×10⁻⁷ kg K⁻¹ s⁻² in a temperature interval of 10^◦C below the clearing temperature. This value is comparable with that found at the compensation temperature in the 8OCB + 50 wt% CC mixture (Oswald 2012b), studied previously. The independence of with the temperature also suggests that the macroscopic Akopyan and Zel’dovich contribution to is negligible in these mixtures in which q₀ changes a lot with temperature. This point will be confirmed later.

More recently, we made similar measurements in diluted cholesteric mixtures¹² (Oswald 2014b). In these mixtures, the pitch was chosen to be of the same order of magnitude as in the previously studied zcompensated mixtures at the clearing temperature. Although the pitch was similar (between 5 and 10 μm, in practice), we found that was much smaller (typically, 50 times smaller) in these mixtures than in the compensated mixtures. An example of systematic measurements with a diluted mixture of 7CB (4-n-heptylcyanopbiphenyl) and R811 (R-(+)-octan-2-yl 4-((4-(hexyloxy)benzoyl)oxy)benzoate) is shown in Figure 3.10. These data show that is proportional to the concentration C of chiral molecules at low concentration. This result was expected since must vanish when C → 0 and led us to define the Leslie thermomechanical power

[3.58]

by analogy with the helical twist power defined to be

[3.59]

Typical values of the HTP and the LTP are given in Table 3.1 for the mixtures 7CB + R811, 7CB + CC, MBBA + R811 and MBBA + CC, where MBBA is the N-(p-methoxy-benzylidene)-p-butylaniline, the first known LC with a nematic phase at room temperature. This table shows that the LTP crucially depends on the chiral molecule chosen, but also on the host nematic LC. For instance, the LTP of the R811 is always positive and three times larger in 7CB than in MBBA. In comparison, the LTP of the CC is smaller in absolute value, although still larger in 7CB than in MBBA. On the other hand, the LTP of the CC is positive in 7CB and negative in MBBA. These results show that there is no direct relationship between the HTP and the LTP, i.e. between the spontaneous twist q₀ and the effective Leslie thermomechanical coefficient This appears more clearly still if we calculate the ratio R=LTP/HTP= which is very different from one mixture to another (Table 3.1). This is not surprising as the HTP is a static property of the phase, whereas the LTP is dynamic in nature.

The next step is to determine whether it is the microscopic term of Leslie ν or the macroscopic term of Akopyan and Zel’dovich that is the more important in . The only way to answer this question is to measure the order of magnitude of the Akopyan and Zel’dovich TM coefficients . This problem will be discussed at the end of this section. But before this, we analyze the behavior of the defects in the Leslie geometry.

3.4.3.2.2. Rotation in the presence of disclination lines

In the previous section, we mentioned that disclination lines are often present in samples. In practice, they are almost impossible to completely eliminate. Hence the question as to whether their presence could disturb the velocity measurements.

To answer this question, we observed the rotation of the extinction branches of ±1/2 disclination lines that usually form in the samples and we simultaneously recorded the intensity transmitted between crossed polarizers at different places of the sample. In doing so, we observed that the director was rotating everywhere at the same velocity, as if the sample was homogeneous. An example of such measurements is shown in Figure 3.11. This photo was taken in a sample of the mixture EM+45 wt%CC at a temperature close to the compensation temperature under a temperature gradient ΔT = 40^◦C. In this case, the director was rotating clockwise so that the extinction branches of the -1/2 lines rotated clockwise while those of the +1/2 lines rotated counterclockwise¹³.

We now calculate the rotation velocity of the director in the presence of disclination lines. In this case, angle ϕ between the director and the x-axis not only depends on z, but also on x and y, which makes the problem very complicated. To simplify the calculations, we assume first that the material constants do not depend on z and that K = K₁ = K₂ = K₃ = −K₄ (isotropic elasticity). With these hypotheses, the elastic energy reads simply

[3.60]

and the bulk and surface torque equations [3.47]–[3.49] become

[3.61]

Integrating the first equation over the sample thickness yields an equation for the average angle by using the two boundary conditions:

[3.62]

where and . coordinates (r, θ), this equation reads

[3.63]

An obvious solution is . For integer or half-integer m, this solution has no singularity in the plane θ = 0 and represents a disclination of topological rank m.

This calculation shows that, in isotropic elasticity, the presence of a disclination line does not change the rotation velocity of the molecules, whatever the rank of the line.

This is indeed observed experimentally with the ±1/2 lines. However, the situation is different for ±1 lines, as shown in the sequence of photos in Figure 3.12(a). If the extinction branches of the -1 line still rotate regularly, the branches of the +1 line rotate in a very irregular way while distorting heavily close to the core of the line. This effect is also visible in Figures 3.12(b) and (c), where we can see that the intensity measured close to the +1 line varies in an irregular way, contrary to that measured close to the -1 line.

It turns out that this effect is due to the elastic anisotropy of the LC neglected so far in the calculations. This can be seen immediately by looking at how the energy of a disclination line of rank m varies during the rotation. To perform this calculation, we need the expression of the elastic energy per unit surface area of the sample. At the compensation temperature, the director field is very slightly twisted along z and we can neglect the twist energy with respect to the splay and bend energies of expressions

[3.64]

where ϕ is assumed not to depend on z. From these formulas, the energy E_m = of a disclination line of rank m, core radius r_c and external radius R can be calculated. By taking ϕ = mθ + ωt, we obtain:

[3.65]

This very simple calculation shows that for all disclination lines of rank m≠1 the energy does not change in time when the director rotates. By contrast, a +1 line oscillates between a radial configuration of energy πdK₁ ln(R/r_c) and a circular configuration of different energy πdK₃ ln(R/r_c) if K₁ K₃. This change of energy is responsible for the irregular rotation of the extinction branches of the +1 defect.

This result can be shown in a more compelling way by numerically solving the torque equation around a +1 line. By setting ϕ = θ + φ(r, t), the torque equation is written in the form

[3.66]

where is the elastic anisotropy factor. By choosing as the length scale and as the time scale ¹⁴, the previous equation is written in the dimensionless form

[3.67]

where r′ ≡ r/L and t′ ≡ t/τ. This equation in ϕ(r′,t′) was solved numerically with Mathematica by taking as boundary conditions ϕ(r′, 0) = 0 and ∂_r′ ϕ(r′,t′) = 0 at and r′ = 100 (these two boundary conditions reflect the fact that the director is free to rotate on the core of radius and far from the core at r′ = 100). The core radius can be estimated by assuming that the director escapes along the z-axis inside the core, in order to suppress the singularity. In doing so, the molecules are homeotropically anchored on the plates in the core, which costs anchoring energy. Minimizing the core energy with these assumptions yields where l_p = K₁/W_a is the extrapolation anchoring length of the sliding planar anchoring. Typical values for the compensated mixture used here are K₁ = 3 pN, K₃/K₁ = 1.7 (A = 0.41) and W_a = 3 × 10⁻⁵ J/m² (Oswald et al. 2008) and = 10⁻⁷ kg K⁻¹ s⁻², which gives, by taking G = 7 × 10⁴ K/m (ΔT = 40^◦C) and d = 7.5 μm, L ≈ 20 μm, τ 10 s and 0.02 (in dimensionless unit). From this numerical solution, the aspect between crossed polarizers of the line can be calculated at different times, as shown in Figure 3.13. This simulation reproduces the observations fairly well. It shows that the extinction branches are deformed and rotate jerkily inside a disk of typical radius 5L.

3.4.3.2.3. Rotation in the presence of an anchoring defect

In practice, it may happen that the polymercaptan layer dewets locally on the glass plates. This leads to a localized surface defect around which the extinction fringes of the director field form rings. One example is shown in Figure 3.14(a) where one can also see a -1/2 disclination line between crossed polarizers. This sequence of 10 photos, taken at a time interval of 10 s, shows that under the action of the temperature gradient, the rings continuously move inwards and collapse in the center of the defect, while the two extinction branches of the neighboring disclination line rotate at constant velocity.

In addition, a time recording of the local intensity measured in the vicinity of the defects shows that the director rotates at constant velocity, as if the sample was homogeneous. This observation indicates that the surface defect, just like the -1/2 disclination line, does not change the rotation velocity of the director.

To demonstrate this result, we solved the torque equation [3.63] for the average angle with a surface defect placed at the origin. To model this defect, we assumed that the viscosity is a function of r of the type:

[3.68]

where r₀ represents the typical radius of the defect. Solving this problem with Mathematica yields

[3.69]

In this expression, C is the Euler constant, Γ the incomplete Euler function, and a, b and c are integration constants. In order that φ does not diverge at the center of the defect, we must take c = 0. Constant a corresponds to a phase shift. It may be removed by changing the time origin. In the following, we shall take a = 0. Finally, we must take b = γ₁ to cancel the torque ∂ϕ/∂r at infinity. With this choice of the constants, the solution reads

[3.70]

This formula shows:

1) that the director rotates everywhere with velocity whatever the “strength” of the defect, as if the sample was homogeneous;

2) that the director winds radially inwards around the defect, which explains the fringes observed between crossed polarizers around the defect, the number of which is proportional to .

To this solution, one can add the solution to the torque equation −θ_d/2 corresponding to the -1/2 disclination line visible in Figure 3.14(a), where θ_d is the polar angle measured from the core of this line. The texture between crossed polarizers calculated from the complete solution is shown in Figure 3.14(b). In this calculation, the “strength” of the defect was chosen in order that the number of rings was approximately the same as in the experiment.

3.4.3.2.4. Rotation in the presence of an electric field

In this section, we describe an alternative method to measure the effective Leslie coefficient with an electric field. This measurement was performed in the Leslie geometry with the compensated mixture EM + 45 wt% CC. This LC is of positive dielectric anisotropy. The AC electric field (f = 10 kHz) was imposed by applying a voltage difference ΔV between two ITO electrodes deposited on the bottom plate at 2 mm apart (Figure 3.15(a)). As a result, the electric field is parallel to the glass plates and perpendicular to the edges of the insulating band separating the two electrodes. A calculation shows that the field is constant to better than 5%, given by , in the central part of width ∼ W/4 of the insulating band. For this reason, all measurements were performed in this region of the sample. Because the LC used is of positive dielectric anisotropy, its director tends to spontaneously align along the electric field. As a result, the electric field can be used to stop the rotation of the helix when a temperature gradient G is applied. From the measurement of the minimal electric field E_stop, required to stop the rotation, the value of was deduced.

In practice, E_stop was measured by recording the intensity between crossed polarizers as a function of time. Without an electric field, the intensity varies in a sinusoidal manner as shown in Figure 3.15(b). When the field is applied, the rotation becomes irregular, slowing down when the director is parallel to the field and accelerating when the director is perpendicular to the field (Figures 3.15(c) and (d)). As a result, the period of rotation Θ increases to finally diverge at E_stop. In practice, E_stop was measured by extrapolating to zero the curve of average angular velocity . Knowing E_stop, the value of was then calculated by numerically

solving the torque equation in the static regime.

The procedure was as follows. In the static regime (when E > E_stop), the torque equation reads:

[3.71]

This equation is subjected to boundary conditions = q₀(d) at z = d. In this equation, q₀ and K₂ are functions of z because they depend on temperature. In the compensated mixture, K₂ is, within experimental errors, a linear function of T so that one can write After substitution in the previous equation and assuming , we obtain

[3.72]

The general solution of this equation has the form:

[3.73]

After substituting into equation [3.72], we obtain an equation for φ:

[3.74]

with = 0 at z = 0 and z = d. In this equation, K₂, ε_a and the integral of q₀ are known functions of z as their temperature variations are known. The value of is obtained by setting E = E_stop in this equation and by searching numerically for the largest value of for which this equation has a solution. This method confirmed that was constant between T_c and the melting temperature, of the order of 1.2 × 10⁻⁷ N/m/K, in good agreement with dynamical measurements.

We end this paragraph by noting that π-walls can form and propagate in the sample when the electric field is larger than E_stop. Each wall separates two zones in which the director is uniformly oriented and does not rotate. On the other hand, the wall propagation allows the director to rotate everywhere, even above E_stop. From this point of view, these walls (or solitons) behave like dislocations in plasticity. Such walls are shown in Figure 3.16 when the temperature is equal to the compensation temperature. In this particular case, the director makes a constant angle with the electric field in each domain and rotates by ±π across each wall. The experiment showed that these walls propagate with a constant velocity that decreases when the electric field increases.

To understand this behavior and predict the velocity of a soliton, we must solve the torque equation close to T_c for the average angle by taking into account the electric field. Because close to T_c, this equation reads by assuming isotropic elasticity:

[3.75]

When G = 0, this equation has a solution of type

[3.76]

where is the electric coherence length. This equation describes a stationary π-wall of width perpendicular to the x-axis. Note that here the x-axis can make an arbitrary angle with the electric field from which the angle is referred. For this static solution,

A similar solution still exists when but now the wall propagates (Dauxois and Peyrard 2006). In this case, the solution must be searched in the form where V is the wall velocity along the x-axis. In the frame of the wall (X = x − Vt), the torque equation rewrites in the from

[3.77]

Multiplying this equation by and integrating between −∞ and +∞ gives, by noting that and :

[3.78]

This gives the general formula

[3.79]

For a soliton-like solution such that , where is the static solution obtained by equilibrating the thermomechanical torque with the electric torque: , this formula gives:

[3.80]

At a small temperature gradient, the solution of equation [3.77] can be found in the form where is a small parameter (of the order of 0.2 in experiments) and is given in equation [3.76]. By using equations [3.80] and [3.76] and by limiting the calculation to first order in this gives:

[3.81]

Note that the sign of the velocity depends on the sense of rotation of the director across the wall. This formula predicts that the wall velocity decreases when the electric field increases, in agreement with observations. We can also compare the velocity of the wall measured experimentally, in the order of 0.8 μm/s when E = 3.210⁴ V/m and G = 70 000 K/m, with the value of 0.94 μm/s calculated from the previous formula by taking Pa s and ν = 1.2 × 10⁻⁷ N/m/K. These two values are close, which confirms the value of given above.

3.4.3.3. Experimental results with the diluted mixtures in the Leslie and mixed geometries

The experiments in the Leslie geometry just allow the measurement of the effective Leslie coefficient . In the compensated mixtures, we suspect that the AZ term coming from the macroscopic twist of the director field is negligible against the microscopic Leslie term ν coming from the chirality of the phase and the molecules. The situation is certainly different in diluted mixtures in which these two contributions could be of the same order of magnitude because of the small concentration of chiral molecules.

To check this point, we measured the rotation velocity of a TIC in the Leslie and mixed geometries. According to our previous calculations, the rotation velocities must be different in these two geometries at equal external temperature gradient Ĝ. This is for two reasons: the first being that the local temperature gradient is different in the two configurations because of the anisotropy of the thermal conductivity¹⁵; and the second being that ω depends on ν and in the Leslie geometry and on ν, and in the mixed geometry. Indeed, we recall that the rotation velocity in the Leslie geometry is given by

[3.82]

where and by

[3.83]

in the mixed geometry where we defined the new dimensionless integrals

[3.84]

where α and ϕ are defined in equation [3.46] and ϵ = 1 − κ_⊥/κ_‖ is the relative anisotropy of thermal conductivity. These expressions were obtained from equation [3.51] and equations [3.55]–[3.56] by assuming that the material constants do not depend on z and by neglecting the surface viscosity γ_S. By eliminating between equations [3.82] and [3.83], we obtain

[3.85]

This quantity gives the order of magnitude of the Akopyan and Zel’dovich coefficients ξ_i if we assume that they are all equal: ξ = ξ₂ = ξ₃ = ξ₄. In that case, which gives .

To perform this experiment, the LC CCN37 (4α,4’α-propylheptyl-1α, 1’α-bicyclo-hexyl-4β-carbonitrile) was chosen because it dissolves much less of the polymercaptan used for the sliding anchoring than MBBA, or the cyanobiphenyls used before. Because of this property, the anchoring continues to slide much longer (up to 3 days, instead of just a few hours), which greatly facilitates the velocity measurements. Two mixtures were used: the mixture CCN37 + 3 wt% CC (mixture 1) and the mixture CCN37 + 0.166 wt% R811 (mixture 2). These concentrations were chosen so that the equilibrium pitch is the same in the two mixtures at the melting temperature (P = 60 μm corresponding to q₀ = 10⁵ m⁻¹) and is larger than 4d, knowing that d ≈ 10 μm in these experiments. This last condition was required to avoid the fact that the TIC destabilizes into a banded texture in the mixed samples (Baudry et al. 1996), which would immediately stop the rotation. Figure 3.17 shows the evolution of the rotation velocities in mixture 1 as a function of temperature when = 10⁴ K/m). As expected, the two textures rotate at different velocities at all temperatures, with ≈ −0.012 rad/s and rad/s at the melting temperature. For comparison, we performed the same measurements with mixture 2 and found and at the melting temperature. To calculate and defined in equation [3.85], we then measured the three elastic constants K₁, K₂ and K₃, the conductivity ratios , and the viscosity γ₁ at the melting temperature. These values are reported in Table 3.2. With these values, we numerically solved the differential equations = 0 and = 0, giving α(z) and ϕ(z) in the mixed geometry and numerically calculated the integrals given in equation [3.84]: = 0.66 and = 0.5, = 0.19 and = −0.12¹⁶. From these measurements, by taking the uncertainties¹⁷ into account, for mixture 1 we found:

and for mixture 2:

These measurements show that the value of the measured Leslie coefficient (or of the ratio /q₀) depends on the chiral molecules, whereas the value of is the same for the two mixtures within the experimental errors. This is expected because is a nematic-like thermomechanical coefficient that should essentially depend on the LC used, here the CCN-37. It must be emphasized that is only 3.8 times smaller in the mixture with R811, than in the mixture with the CC, although the concentration of R811 is 18 times smaller than that of CC. This means that the Leslie effect is 18/3.8 = 4.7 times stronger with the R811 than with the CC at equal concentrations. In other words, the LRP of the R811 is 4.7 times larger than the LRP of the CC, although the HTP of the R811 is 18 times larger than that of the CC. This again shows that the HTP and the LRP do not have the same origin, with the former corresponding to an equilibrium property, while the second is dynamic in nature.

These results also show that in diluted mixtures ν and are of the same order of magnitude. This contrasts with the situation in compensated mixtures, in which the macroscopic term must be negligible with respect to ν, since it must be of the same order of magnitude as in diluted mixtures. This confirms our suspicion that ≈ ν in the compensated mixtures studied before.

Last but not least, we see that the experimental value of ξ, obtained from the previous value of by assuming that ξ = ξ₂ = ξ₃ = ξ₄, is of the order of -15 fN/K, which is about 10³ times smaller in absolute value than the value given by Akopyan and Zel’dovich in their theoretical paper (according to Akopyan and Zel’dovich (1984), ξ ∼ 10⁻¹¹ N/K).

3.4.4.1. Theoretical predictions

The director field inside of a CF1 is shown in Figure 3.18. It can be numerically calculated by the minimization of the elastic energy. On this topic, we emphasize the important role of the elastic anisotropy, demonstrated for the first time in Lequeux et al. (1989). Indeed, isolated CF1s only form if the transition nematic→fingers is of the first order, which requires the elastic anisotropy to be large enough, according to the diagram shown in Figure 3.19 (Ribière et al. 1991). This is the case in most LCs, including those used in this work, in which K₃ is larger than K₁ and K₂ (see Table 3.2, for instance).

Gil and Thiberge were the first (Gil and Thiberge 1997) to propose that a Leslie-like electromechanical effect could be responsible for the of the CF1s when they are subjected to a DC electric field (Gilli and Gil 1994). We know today that this drift is not due to this effect, but to a coupling between the director field and the flows associated with the motion of flexoelectric polarization charges (Tarasov et al. 2003). However, the Gil and Thiberge model applies the thermomechanical case by replacing the electric field by the temperature gradient. In this model, the authors solve the torque equation in a perturbative way, for small values of the field, by using a Melnikov-type analysis. It turns out that this method is complicated and not general as it neglects the temperature variations of the material constants. For this reason, we propose another method (Dequidt 2008; Dequidt et al. 2016), which is completely general and easier to understand.

The starting point of the calculation consists of noticing that the total energy of a finger (including the elastic energy and the anchoring energy on the glass plates ) does not change in time when the finger drifts perpendicularly to its long axis at constant velocity V. This condition reads

[3.86]

Replacing in this equation by their expressions given in equations [3.9] and [3.12] yields

[3.87]

If the finger propagates along the y-axis with velocity V, then and with

[3.88]

where V^(TM) is the drift velocity due to thermomechanical cross-coupling and V^(T) is the drift velocity due to the variation of the free energy with temperature:

[3.90]

[3.89]

[3.90]

with

[3.91]

Formula [3.90] shows that, in addition to the thermomechanical terms, the variation in temperature of both the elastic constants and the anchoring energy can lead to a drift¹⁸. An important point is that this drift is due to the transverse gradient The latter exists because of the anisotropy of the thermal conductivity of the LC.

We conclude this section by giving a simplified expression of V^(TM). It is obtained by neglecting the surface viscosity γ_S and the anisotropy of thermal conductivity, which is the same as assuming that the temperature gradient is little different from the average gradient . If G is not too large (linear regime), the spatial variations of the material constants can also be neglected, and the director can be replaced by the equilibrium solution (linear regime). Under these assumptions, V^(TM) can be calculated by using the expressions of and given in equations [3.22] and [3.24]:

[3.92]

where the dimensionless integrals J_i are defined by

[3.93]

This formula generalizes a formula already given in Gil and Thiberge (1997); Oswald and Dequidt (2008c), where the thermomechanical coefficients were neglected ¹⁹.

Several remarks are in order as follows. First, we note that q₀d in integrals J₁, J₂ and J₃ must be equal to q₀d_c to observe isolated fingers. As a result, these integrals only depend on the elastic anisotropy and are independent of the sample thickness.

Another important point is that the integrals J_ν,J_γ and J_i (1 = 1 − 3) do not change sign when q₀ changes sign. As a result, the AZ contribution to v^(TM) must change sign when q₀ changes sign, contrary to the Leslie contribution, the sign of which is given by the sign of ν (independent of q₀). Finally, we note that V^(T) also changes sign when q₀ changes sign, since changes sign when q₀ changes sign.

3.4.4.2. Experimental results

The drift of cholesteric fingers subjected to a temperature gradient has been observed in two very different systems: in a compensated mixture close to the compensation temperature and in a diluted cholesteric mixture close to a cholesteric-smectic A phase transition. In the first case, we will show that the drift is due to the Leslie thermomechanical coupling, while it is likely due to the divergence of the twist and bend constants in the second case.

3.4.4.2.1. Drift close to a compensation temperature

The compensated mixtures are particularly well suited to observe the drift of CF1s for three reasons. First, the pitch changes with temperature. As a result, it is easy to fulfill the condition d = d_c for a large range of thicknesses by just changing the sample temperature. Second, the Leslie coefficient is large, which is essential to observe a drift. Third, the experiment can be performed at two temperatures T₁ < T_c and T₂ > T_c, on either side of the compensation temperature, which allows the testing of the role of the sign of q₀ on the drift velocity.

In practice, short segments of CF1 form in homeotropic samples when d = d_c. These segments have two necessarily different ends because of the absence of mirror symmetry in a cholesteric. One of them is rounded, marked with the minus sign on the pictures, while the other is pointed, marked with the plus sign²⁰. Under the action of a temperature gradient, these segments drift perpendicularly to their axes, as shown in Figure 3.20. In this figure, the x-axis is oriented from the pointed tip to the rounded tip. With this choice, the experiments with the mixture 8OCB + 50 wt% CC show that V is positive (negative) when ΔT > 0 (ΔT < 0) at the two temperatures T₁ and T₂, independent of the sign of q₀. The velocity is also found proportional to the temperature gradient (linear regime) and proportional to d, as shown in Figure 3.21. These results – particularly the fact that V does not change sign at T_c – show that the drift is mainly due to the Leslie thermomechanical coupling. This interpretation was confirmed by numerically calculating the constant from which a value of ν was found by taking ν ≈ 1.3 × 10⁻⁷ kg K⁻¹ s⁻² at T_c. This value is very close to the values previously given, confirming that the drift is mainly due to the Leslie thermomechanical coupling.

[3.94]

It should also be noted that isolated CF1s can be stabilized by the application of an AC electric field, when the sample temperature is larger than T₂. In this case, the pitch becomes smaller than the thickness and the CF1s become unstable, growing from their ends while undulating until they invade the whole sample. The only way to stabilize the fingers again is to impose an electric field to favor the homeotropic nematic with respect to the fingers. The equilibrium is reached for a voltage denoted by V₂ in literature. Of course, the larger the temperature above T₂, the larger V₂ is and the thinner the CF1s are, whose width always remains close to 3.5P (Ribière and Oswald 1990). It is then possible to impose a temperature gradient to the fingers and to measure their drift velocity. Such measurements were performed in a 15-μm-thick sample of the mixture 8OCB + 50 wt% CC. The results, reported in Figure 3.22, show that the drift velocity strongly decreases when the temperature and the electric fields increase. This effect is expected since A must be roughly proportional to the width of the finger according to equation [3.94], a quantity close to 3.5P that strongly decreases when the temperature increases.

We conclude this section by noting that the isolated segments of CF1 always destabilize in time by forming spirals. This evolution is inevitable (as shown in Pirkl and Oswald (1996); Oswald et al. (2000)) and is akin to what is observed with wave fronts in weakly excitable two-dimensional media (Mikhailov et al. 1994). Figure 3.23 shows two single and two triple spirals observed above and below the compensation temperature. The reader will note that these two types of spirals do not rotate in the same sense when T > T_c and ΔT > 0. This is due to the fact that the rounded tips lie in the center of the single spirals and on the outer border of the triple spirals, showing that the fingers are oriented in opposite directions in these two types of spirals. These images also show that far from their center, the spirals tend to

Archimedian spirals of an equation in polar coordinates

[3.95]

where is the pitch of the spiral and ω_s is its angular rotation velocity related to the drift velocity by the relation

[3.96]

This equation is general and gives the product ω_s. On the other hand, ω_s (and, thus, ) depends on the anchoring conditions of the spiral at the center, which may change from one spiral to another.

3.4.4.2.2. Drift close to a cholesteric-smectic A phase transition

The same experiment was conducted in the diluted mixture 8OCB + 1.14 wt% R811 (Oswald 2008). In this mixture, the Leslie coefficient is much smaller than in the compensated mixture because of the small concentration of chiral molecules and the Akopyan and Zel’dovich terms entering into the expression [3.92] of the drift velocity, which are not larger than the Leslie term according to our previous estimate of the ¯_i. As a result, these fingers should not drift in a visible way. This is indeed what we have found experimentally, except very close to the smectic A phase transition where the CF1 spontaneously formed spirals, as shown in Figure 3.24. This result is very surprising as we know that the viscosity γ₁ diverges at the smectic A-to-nematic phase transition (de Gennes and Prost 1995; Oswald and Pieranski 2006). One possibility to explain this observation with the “classical” thermomechanical model would be that the Leslie coefficient diverges faster than γ₁. To test this idea, measurements in the Leslie geometry were performed with the mixture 8CB + 1 wt% R811. It was found that the rotation velocity continuously decreases when approaching the smectic A phase, which clearly indicates that does not diverge but remains approximately constant (Oswald 2016). These results show that this drift is not due to the classical thermomechanical coupling, but rather to the other mechanism described above, coming from the temperature variations of the free energy. The corresponding drift velocity V^(T) is given in equation [3.90]. This expression simplifies if the anchoring energy is very large and if the glass conductivity is much larger than that of the LC. In these limits, both tend to 0 on the glass plates, leading to

[3.97]

This formula shows that the drift velocity is proportional to the horizontal temperature gradient. By denoting the temperature at the bottom of the glass plate by T₀, it can be shown by using the heat equation, and by setting where g is the solution of

[3.98]

with g = 0 at z = 0 and z = d. This shows that the horizontal gradient is proportional to and is different from 0 as long as κ_a 0. Due to this temperature gradient, one side of the finger is colder than the other²¹. For this reason, the finger tends to drift toward its hotter side to minimize its elastic energy. To some extent, one can say that the finger surfs on a heat wave that it creates itself. From the previous expression, one can estimate the order of magnitude of the drift velocity by orienting the finger in the same way as in Figure 3.20. This gives, within a numerical factor impossible to estimate by hand (Dequidt et al. 2016):

[3.99]

where K is a combination of the elastic constants that diverges at the transition. This mechanism could explain the formation of spirals close to the transition. However, numerical simulations would be important to confirm this interpretation.

We end this section by mentioning the existence of another thermomechanical effect observed during the growth of cholesteric fingers. As we mentioned before, CF1s grow from their ends while undulating when the confinement ratio is too large, typically more than 1, and the applied voltage (if any) is less than V₂. We have observed that during the growth, the ends of the fingers move at constant speed in a straight line when no temperature gradient is applied. By contrast, they follow circular trajectories when a temperature gradient is applied, the radius of which decreases when the temperature gradient increases. An example is shown in Figure 3.25. This phenomenon is observed at all temperature, but has not yet been explained.

3.5.2.1. Principle of the measurement

The basic idea for measuring the order of magnitude of the TH coefficients ξ_i (i = 5 − 12) in a nematic phase was already given in the seminal paper by Akopyan and Zel’dovich (1984). Indeed, these authors note from the beginning of their paper that a flow should be induced by a temperature gradient in a mixed sample treated for planar unidirectional anchoring on one plate and homeotropic anchoring on the opposite plate (Figure 3.28(a)). The order of magnitude of the velocity v can be easily found by solving the momentum equation This equation considerably simplifies if one assumes that in the (x, y, z) reference frame, with ²²; (2) the temperature gradient is constant, equal to G ²³ and (3) the material constants do not change with temperature in the sample thickness. By using the simplified expression of σ^(AZ) given in equation [3.31], and by further assuming that the nematic phase behaves viscously as an isotropic liquid of viscosity η, we obtain:

[3.105]

The reader will note that the TM term in ξ and the TH terms in and do not enter into this equation when α varies linearly with z.

If the nematic phase is free to flow along the x-axis, then P = constant and the solution reads

[3.106]

As expected, the velocity is maximum in the middle of the sample and vanishes on the two glass plates. One will note that the velocity at z = d/2 is independent of the sample thickness because the TH and viscous stresses both vary as 1/d.

This calculation assumes that the LC can flow freely along the x-axis, which is difficult to realize in practice. Indeed, the flow disappears as long as the LC is in contact with an obstacle (for instance, one of the spacers used to control the sample thickness). In that case, the TH stress is equilibrated by a pressure gradient and no flow occurs.

To avoid this difficulty, Akopyan and coworkers proposed to perform the experiment in a sample treated for planar circular anchoring on one plate and homeotropic anchoring on the opposite plate (Akopyan et al. 1999, 2001, 2004). This geometry is represented in Figure 3.28(b).

3.5.2.2. Theoretical predictions in the mixed planar-circular/homeotropic geometry

Because of the axial symmetry, calculations can be done in cylindrical coordinates by taking To write the equations, we used Mathematica and assumed that (1) the director field is given by with in the referential frame (2) G is constant; (3) the material constants are independent of temperature and (4) the AZ stress tensor is given by the simplified expression [3.31]. However, we used the full expression of the viscous stress tensor given in equation [3.21], in which we made because this term is usually very small in nematics (de Gennes and Prost 1995; Oswald and Pieranski 2005). In spite of the approximations, the equations are very long and will not be given here. We will just focus on the equation in far from the core of the disclination (typically for ). In this region, v_r and v_z tend to 0, and so can be neglected as well as the terms in and which gives

[3.107]

The solution to this equation reads

[3.108]

As expected, it can be checked that in the limit η_b → η_c = η, this equation gives back the result of equation [3.106]. This equation shows that far from the core of the disclination (in practice at distance r > d), a circular flow develops with a velocity in the center of the sample at z = d/2, given by

[3.109]

The reader will note that, as under unidirectional planar anchoring, the constants ξ, ζ^’ and ζ” do not enter into the solution far from the core of the disclination. This is not the case close to the core where the two terms in ζ^’ and ζ” are important. On the other hand, the TM term in ξ vanishes everywhere if the angle between the director and the z-axis varies linearly with z, as assumed in this calculation. The numerical solution to the full equations is shown in Figure 3.29 and is compared to the approximate solution given in equation [3.109]. In this simulation, we chose the same values for the material constants as in Figure 3.27, and we took ζ = ζ^’ = ζ” = 10⁻¹² NK⁻¹.

As expected, these calculations confirm that a circular flow develops at distance r > d of the core of the disclination. An important point is that the velocity in the middle of the sample is proportional to G and ζ and is independent of r. The latter result contrasts with the numerical simulation of Akopyan et al. (2004), who found that the maximal velocity along z, was proportional to r. We think that this result is wrong as we found a similar result to ours by numerically solving their own equations with Mathematica.

We note that another variant of the experiment has been proposed in Akopyan et al. (2001). It consists of destabilizing, with an AC electric field, a sample treated for the circular configuration. If the applied voltage V is larger than the Fréedericksz critical voltage V_c, the director field distorts and reads in polar coordinates in the form with and provided that V < 1.3 V_c. The same calculations as before shows that a circular flow develops around the core of the disclination. At distance r > d, the only non-zero component of the velocity is v_θ. This component depends on all of the TH coefficients ζ, ζ^’ and ζ”, but also on the TM coefficient ξ, as shown in Figure 3.30. In this figure, we plotted v_θ(r, d/2) as a function of r and v_θ(r, z) calculated at r = 100 μm as a function of z for each contribution ξ, ζ, ζ^’ and ζ” and for the total contribution. In this calculation, we assumed that ξ = ζ = ζ^’ = ζ” = 10⁻¹² N/K²⁴ and we took θ_m = 0.8 rad and the same values as before for the other parameters. These calculations show that all of the terms contribute in a similar way and lead to typical velocities of the order of 0.15 − 0.3 μm/s.

3.5.2.3. Experimental results

The experiment was performed by Akopyan and coworkers in the two mixed geometries depicted in Figure 3.29. In the two cases, the authors observe a flow with a maximal velocity proportional to the temperature gradient and of the order of 1 μm/s for G ≈ 1.4 10⁵ K/m (Akopyan et al. 1997, 1999). This is in favor of a thermohydrodynamical effect with coefficients ζ, ζ’ or ζ” of the order of 10⁻¹² N/K, which is the value we have taken in our previous numerical simulations. However, the authors observe in the circular geometry of Figure 3.28 that the velocity increases linearly with the radius (see Figure 4 in Akopyan et al. (2001)), which is incompatible with our own numerical simulations. This is a real problem for which we have no solution to propose for the moment. In addition, these results suggest that the thermohydrodynamical coefficients ζ, ζ’ or ζ” are much larger than the thermomechanical coefficient ξ in the order of 10⁻¹⁵ N/K, according to our experiments. This is surprising and for this we do not have any explanation. Note that an even larger value was given by Lavrentovich and Nastishin (1987) to explain the flows observed in flattened droplets deposited on a bath of glycerol and submitted to a temperature gradient²⁵. The problem in this experiment is the presence of the Marangoni effects that can distort the measurements. The same problem occurs in the experiment by Choi and Takezoe (2016), showing the formation at the place where a laser beam is focused on a localized circular flow in a homeotropically oriented nematic layer with a free surface. In the paper by Choi and Takezoe, this effect is interpreted to be due to a Marangoni effect, but an alternative explanation was proposed by Zakharov and Maslennikov (2019) in terms of the thermohydrodynamical effect, provided the ζ-coefficients are large enough.