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Chemical Leslie eﬀect in a chiral smectic C?ﬁlm: nonsingular target patterns

F´elix Bunel∗and Patrick Oswald†

Ens de Lyon, CNRS, Laboratoire de physique, F-69342 Lyon, France

(Dated: November 25, 2022)

We analyze experimentally and theoretically the winding and the unwinding of the ~c-director in

a chiral smectic C?ﬁlm crossed by an ethanol ﬂow. This leads to a target pattern under crossed

polarizers when the +1 defect imposed by the boundary conditions is pinned on the edge of the

ﬁlm. We show that the target is deformed at the center of the ﬁlm when it is subjected to a ﬂow of

ethanol because of the presence of two recirculation vortices of chemohydrodynamical origin. This

deformation and the two vortices disappear during the unwinding of the target when the ethanol

ﬂow is stopped. This unambiguously shows that the target deformation is due to the vortices and

not to the elastic anisotropy. These two points are conﬁrmed theoretically. An estimate of the two

so-called chemomechanical and chemohydrodynamical Leslie coeﬃcients is also derived from this

study.

PACS numbers: 61-30.Eb, 61-30.Dk

I. INTRODUCTION

The introduction of chiral molecules into a liquid crys-

talline phase can give it new properties since certain sym-

metries, including the mirror symmetries and the center

of inversion, are broken. This is the case in the cholesteric

phase which is a chiral nematic phase with a spontaneous

twist of its director ﬁeld, but also in the smectic C?phase

studied here, which is the chiral –and so twisted– version

of the smectic C phase. The ﬁrst experimental obser-

vation of this symmetry breaking dates back to the pio-

neering experiments of Lehmann in 1900, who observed

the rotation of the internal texture of cholesteric drops

when subjected to a temperature gradient [1]. Lehmann

also noticed that by changing the chiral impurity, it was

possible to change both the sign of the spontaneous twist

and the sense of rotation of the drops [2].

If Lehmann understood the role of chirality in the rota-

tion of the drops, he was however unable to explain this

eﬀect which bears his name today. It was only much

later, in 1968, that Leslie proposed the ﬁrst theoreti-

cal explanation for the Lehmann eﬀect. Based on the

symmetries of the cholesteric phase, Leslie demonstrated

the existence of a torque proportional to the local tem-

perature gradient acting on the molecules of the liquid

crystal (LC) [3]. According to Leslie, this torque makes

the molecules rotate, which causes the internal texture

of the cholesteric drops observed by Lehmann to also ro-

tate. This explanation was at the time – and until very

recently – unanimously accepted by the scientiﬁc com-

munity and became a veritable paradigm.

∗Present address: Centre d’´etudes du Ripault 37260 Monts; fe-

lix.bunel@CEA.com

†patrick.oswald@ens-lyon.fr

It was not less than 40 years after Leslie’s discovery

that new experiments on the Lehmann eﬀect were per-

formed, ﬁrst by one of us (PO) and Dequidt [4, 5], and

then a few years later by the groups of Tabe and Sano

[6, 7]. These authors reproduced Lehmann’s original ex-

periment, which they ﬁrst interpreted within the frame-

work of Leslie’s theory [4, 5]. After further quantitative

studies, it was nevertheless shown that the rotation veloc-

ity of the drops could not be explained by Leslie’s theory.

It was shown in particular that the velocities predicted

by this theory were much too small [8, 9], and sometimes

even of the wrong sign in certain cholesteric mixtures

[10]. But the ﬁnal blow to Leslie’s theory was given by

Ign´es-Mullol et al. [11] when they discovered that it was

also possible to induce rotation in drops of the nematic

–and so achiral– phase of a chromonic lyotropic liquid

crystal (provided that the director ﬁeld inside the drops

is twisted, as required by symmetries). We mention that

the same phenomenon was also observed recently in the

nematic phase of a thermotropic LC made of bent-shaped

molecules [12, 13].

If it is now proven that the Leslie torque cannot explain

quantitatively the rotation of the internal texture of the

Lehmann drops, it is nevertheless certain that this torque

exists. This was shown by one of us (PO) and Dequidt

by performing a sliding planar sample experiment [4], as

originally proposed by Leslie [3]. In this dynamic exper-

iment, the cholesteric was conﬁned between two plates

treated for planar sliding anchoring. The experiment

showed that in the presence of a thermal gradient per-

pendicular to the plates, the cholesteric helix was rotat-

ing at constant speed, as predicted by Leslie [3]. In this

experiment, the Leslie thermomechanical coeﬃcient was

directly deduced from the measurement of the rotation

velocity of the helix, knowing the rotational viscosity of

the LC. The Leslie torque was also measured indirectly in

a static experiment ﬁrst performed by Eber and Janossy

2

in 1982 with a compensated cholesteric phase [14–16] and

reproduced later by Dequidt and one of us (PO) [17, 18].

In this experiment, the director ﬁeld does not rotate con-

tinuously, but is simply distorted by the Leslie torque.

The problem here is that another torque, coming from

the variation with temperature of both the elastic con-

stants and the spontaneous twist of the cholesteric phase,

is added to the thermomechanical torque, which made in-

terpretations delicate and subject to controversy.

Let us conclude this revision on cholesterics by recall-

ing that Leslie’s theory also predicts the existence of a

thermohydrodynamic eﬀect. To this second eﬀect is asso-

ciated a stress proportional to the temperature gradient

that should cause ﬂows in the samples. To the best of

our knowledge, this eﬀect has not yet been observed. We

refer the reader to the review article [19] for the descrip-

tion of possible experiments that could be carried out to

observe this eﬀect in cholesterics.

We now recall, as noted by de Gennes, that similar

eﬀects could be observed in cholesterics when the ﬂow

of heat is replaced by a ﬂow of particles or an electric

current [20]. In this context, only one experiment was

attempted in 1993 by Padmini and Madhusudana [21] to

detect the Leslie electromechanical eﬀect. It was nev-

ertheless shown later by Dequidt and one of us (PO)

that the texture rotation observed by these authors un-

der electric ﬁeld was not due to a Leslie electromechanical

eﬀect, but to the more classical ﬂexoelectric eﬀect [22].

In practice, similar eﬀects should also exist in the smec-

tic C?phase since the director ﬁeld ~n inside is also twisted

as shown in Fig. 1(a). In this ﬁgure, ~n gives the average

orientation of the LC molecules, ~

kis the unit vector nor-

mal to the layers and ~c is the unit vector that gives the

orientation of the molecules in the plane of the layers.

With these deﬁnitions one has

~n = sin θt~c + cos θt~

k, (1)

where θtis the tilt angle of the molecules with respect to

the normal to the layers. Importantly, in a smectic C?,~n

and −~n are equivalent. As a result, all the equations must

be invariant under the transformation (~

k, ~c)→(−~

k, −~c).

To the best of our knowledge, no thermomechanical ef-

fects have been reported in the literature so far in this

phase. In contrast, there are several experimental evi-

dences for a chemomechanical eﬀect –and to some extent

for a chemohydrodynamical eﬀect– in this phase. These

eﬀects have been observed in two distinct systems: in

chiral Langmuir monolayers when the LC is deposited at

the surface of an isotropic liquid (water in general) and

in free-standing ﬁlms of the smectic C?phase. In the two

cases, the Leslie eﬀects are due to the presence of a ﬂow

of particules accros the ﬁlms and are thus of chemical

nature.

Historically, the chemical Leslie eﬀect was studied ex-

perimentally for the ﬁrst time by Tabe and Yokoyama in

2003 in Langmuir monolayers [23]. It should however be

emphasized that the ﬁrst observation of the rotation of

the molecules in such a monolayer is due to Adams et al.

k

n

φ

θt

c

x

y

z

(a)

(b)

FIG. 1. Structure of the Smectic C?phase (a) and deﬁnition

of the ~n and ~c-directors (b).

in 1993 [24]. However, these authors failed to identify

the cause of this rotation. In their experiments, Tabe

and Yokoyama deposited a chiral LC on the surface of a

water-glycerol bath. The molecules, by covering the sur-

face, formed a monolayer with long-range orientational

order as in a layer of a smectic C?phase. Observation

with a polarized Brewster angle microscope then revealed

the formation of concentric rings with the reﬂected in-

tensity oscillating in the middle of the pattern. These

phenomena are due to the rotation of the molecules in

large areas of the monolayer when it is subjected to the

evaporation ﬂow of the water contained in the sub-phase.

These target-like patterns were explained as due to the

Leslie chemomechanical torque, which is formally equiv-

alent to the thermomechanical torque. In addition, Tabe

and Yokoyama showed that the rotation velocity of the

molecules in the center of a target was proportional to

the water evaporation ﬂow rate and changed sign when

the water ﬂow direction was reversed.

The work of Tabe and Yokoyama prompted several re-

searchers to work on the chemical Leslie eﬀect. Starting

with Tabe’s group that carried out molecular dynam-

ics simulations emulating Langmuir monolayers with dif-

3

ferent chiral compounds [25, 26]. Tsori and de Gennes

also published an article on the motion of defects in the

target-like patterns [27] and Svenˇsek et al. performed

numerical simulations to show the inﬂuence of thermal

ﬂuctuations on the observed patterns [28]. In addition,

they predicted that spirals should form when the rota-

tion begins with a +1 defect in the center of the zone

where the molecules can freely rotate. This prediction

was later conﬁrmed experimentally by Gupta [29]. Nito

et al. also reproduced Tabe’s experiment with new chiral

compounds and found that the rotation was not system-

atic for all of them, contradicting Tabe’s initial statement

that any chiral liquid crystal [30, 31] should feature the

Leslie rotation. Finally, we reproduced ourselves these

experiments [32] and proposed a complete characteriza-

tion of the chemomechanical eﬀect in these monolayers

by using the Leslie theory.

It must be emphasized that in the Langmuir monolay-

ers, no ﬂow is observed. This is mainly due to the large

viscosity of the sub-phase with respect to the viscosity

of the monolayer itself. For this reason, this system is

not suitable for studying the chemohydrodynamical ef-

fect, i.e. the ﬂows induced by a ﬂux of particles crossing

the LC. The situation is diﬀerent in the free-standing

ﬁlms of smectic C?in which the LC can easily ﬂow. This

was demonstrated by Seki and Tabe in 2011 [33, 34].

By subjecting a ﬁlm to a ﬂow of vapor of alcohol, they

observed that the rotation of the molecules was accom-

panied by ﬂows. They also observed the advection of

tracers (small solid particles) deposited on the ﬁlm and

measured the force acting on them using optical tweez-

ers. This method allowed them to estimate the eﬃciency

of the chemohydrodynamical coupling. Unfortunately,

their results were not interpreted within the framework

of Leslie’s theory and the lack of explanations and exper-

imental details on their measurements did not allow such

an analyzis to be done a posteriori.

For this reason, we redid experiments on the free-

standing ﬁlms of smectic C?, while seeking to quantify

the experimental results as best as possible. We then

applied the Leslie theory to explain our results. In prac-

tice, we observed two types of patterns. The target-like

pattern already widely mentioned, and another one con-

sisting of spiraling targets partially wound with a +1

defect in their center. In this paper, we focus on the

nonsingular target-like pattern, and we refer the reader

to a forthcoming paper [35] for the description of the sin-

gular spiraling-target-like patterns with a +1 defect in

the center of the ﬁlm. We mention here that similar pat-

terns were already observed in non-chiral smectic C ﬁlms

when the phase winding and the ﬂows are produced by a

ﬁber that pierces the ﬁlm in its center [36]. To complete

our historical survey, we also underline the existence in

cholesteric LCs of a diﬀuso-mechanical coupling that was

interpreted by using the analogy with the Leslie thermo-

or chemo-mechanical coupling [37].

This paper is divided in ﬁve sections: A preliminary

section “Material and methods” (Sec. II), an experimen-

tal section (Sec. III), a theoretical section (Sec. IV),

a section in which we compare the theoretical and ex-

perimental results (Sec. V) and a conclusion (Sec. VI).

In Sec. II, we give essential details on the LC chosen

and the setup used to stretch, observe and subject the

ﬁlms to a controlled ﬂow of ethanol vapor. In Sec. III,

we present the main experimental results concerning the

formation of a target pattern when the ﬁlm is subjected

to a ﬂow of ethanol and its disappearance when the ﬂow

of ethanol is stopped. Particular attention will be paid to

the ﬂows which are present in the ﬁlm during the phase

winding, but also in the stationary regime when the tar-

get is formed. In Sec. IV we will recall the equations

which govern the dynamics of the ﬁlm and we will solve

them numerically (and analytically when possible) in the

framework of a simpliﬁed model for the viscosity tensor

allowing us to satisfactorily reproduce the experimental

results. We will see that, even if the agreement is not

perfect, the Leslie theory explains in a convincing way

all the experimental results. In Sec. V, a ﬁrst estimate

of the ratio between the chemomechanical and chemohy-

drodynamical coeﬃcients will be given. Finally, we will

summarize our work in the conclusion (Sec. VI) and will

present some ways to improve the agreement between

theory and experiment.

II. MATERIAL AND METHODS

A. Liquid crystal mixture

All of our measurements were performed with the com-

mercial mixture FELIX M4851-100 (Merck, Germany).

This LC has the advantage of being in the Smectic C?

phase over a wide range of temperature as shown by the

phase sequence:

Cr<-20◦C–Sm C?–67◦C–Sm A–71◦C–Chol–76◦C–Iso

This makes this mixture very easy to use since it is in

the Smectic C?phase at room temperature. Its main

characteristics are given in Ref. [38] . At room tem-

perature (25◦C) its pitch is P≈5µm, its spontaneous

polarization is Ps≈20 nC/cm2and the tilt angle of the

molecules in the layers is θt≈28◦.

B. Experimental setup

All of our experiments were performed inside a custom-

made copper oven (see Fig. 2). Its temperature was ﬁxed

at 25◦C and regulated within ±0.1◦C thanks to a RKC

CB100 temperature controller (TC Direct, France). The

room temperature was kept at a slightly lower temper-

ature, around 21◦C. The ﬁlm was observed under a re-

ﬂection microscope (Laborlux 12Pol, Leica) through a

glass slide which ensured the thermal insulation of the

ﬁlm. This window was slightly tilted to prevent the light

4

FIG. 2. Oven used in our experiments.

FIG. 3. Scheme of the device used to produce the air-ethanol

mixture.

reﬂected on its surface from entering the microscope ob-

jective. A second box with a pinhole ﬁxed on its top

was placed inside the oven. The pinhole was purchased

from Edmunds Optics (France). Most of our experiments

were performed with a pinhole with a diameter of 0.6 mm

made in a stainless steel sheet 200 µm thick. As the ﬁlm

attaches to the frame via a meniscus, the result was a

ﬁlm of constant thickness with a slightly smaller diame-

ter, close to 0.5 mm.

To observe the chemical Leslie eﬀects, we chose to sub-

ject the ﬁlm to a ﬂow of ethanol. With this alcohol,

these eﬀects are very strong, which made them easy to

observe. Moreover, ethanol does not seem to degrade the

LC, even over long period of time, which was a valuable

asset. Note that we also observed Leslie eﬀects with wa-

ter, oxygen and nitrogen, but the intensity of the eﬀects

observed was much weaker, because these molecules are

much less miscible than ethanol in the LC. In contrast,

a Leslie eﬀect stronger than with ethanol was observed

with acetone, very soluble in the LC, but we did not study

it systematically because the ﬁlms broke very often. In

practice, the compositions of the atmospheres inside and

outside the box were controlled independently via two

gas circuits (Fig. 3). More precisely, an air-ethanol mix-

ture was injected inside the box (under the ﬁlm) and

dry air was injected outside (above the ﬁlm). The air-

ethanol mixture was prepared by bubbling dry air (here,

synthetic air supplied by a pressurized bottle) in a sealed

compartment ﬁlled with ethanol. The latter was placed

in a Julabo thermostated circulating bath to maintain

it at a constant temperature of 18◦C. In this way, we

obtained at the outlet air saturated with ethanol vapor

at its saturation vapor pressure Psat =5.16 kPa at 18◦C.

Note that using a temperature lower than the room tem-

perature (here always close to 21◦C) allowed us to avoid

the condensation of ethanol in the rest of the circuit.

The ethanol-saturated air was then mixed with dry air

in a mixing compartment. Two Bronkhorst EL Select

F-201CV mass ﬂow controllers driven by a LabView pro-

gram made it possible to control the respective ﬂow rates

of dry air and air saturated with ethanol at the inlet of

the mixer and therefore to precisely set the composition

of the mixture. Another Bronkhorst controller made it

possible to ﬁx the ﬂow of dry air injected above the ﬁlm.

With this device, it was therefore possible to control both

the ﬂow rates above and below the ﬁlm and the percent-

age of ethanol vapor in the air injected into the box under

the ﬁlm.

Finally, let us make two remarks on this setup.

First, its proper functioning was checked by replac-

ing ethanol with water and by measuring the humidity

at the outlet of the mixer using a precision hygrometer

Testoterm 6010. The expected values were in very good

agreement with the values given by the hygrometer.

Second and more importantly, we checked that our ex-

perimental results were independent of the ﬂow rates of

dry air and of ethanol-air mixture injected on either side

of the ﬁlm, at least between 10 and 100 mL/min. To

realize the importance of the ﬂow rate, note that the in-

ternal volume of the box under the ﬁlm is of the order

of 3 mL. With a ﬂow rate of 50 mL/min, its atmosphere

is completely renewed after approximately 4s. In prac-

tice, most of our experiments were performed using a ﬂow

rate of 20 mL/min. The box was also ﬁlled with cotton

in order to attenuate the air currents inside and therefore

limit the parasitic ﬂows in the ﬁlm as much as possible.

With this precaution, no signiﬁcant ﬂow was observed in

the ﬁlm when dry air was injected on both sides of the

ﬁlm, even at the highest ﬂow rates. These precautions

turned out to be fundamental and allow us to aﬃrm that

the ﬂows observed in the ﬁlms are only due to the Leslie

forces and/or to the backﬂow eﬀects and are not artifacts

due to air currents in the vicinity the ﬁlm.

Last but not least, we highlight the two reasons why

injecting dry air above the ﬁlm is important. First, be-

cause the ethanol vapors that ﬂow through the ﬁlm must

be ﬂushed to maintain the good concentration gradient

of ethanol across the ﬁlm. Second, it was necessary to

prevent the water contained in the air of the room from

creating an uncontrolled parasitic Leslie eﬀect by diﬀus-

ing through the ﬁlm.

In the following, we will use as a control parameter for

the chemical Leslie eﬀect the diﬀerence in ethanol vapor

pressure Pbetween the two sides of the ﬁlm. As P= 0

on top of the ﬁlm, the latter is expressed as

∆P= % × Psat ,(2)

where % is the percentage of ethanol vapor below the ﬁlm

and Psat the saturation vapor pressure of ethanol equal

to 5.16 kPa at 18◦C.

5

FIG. 4. Diagram of the optical device employed to visualize

the ﬁlm and its texture.

C. Film characterization

To visualize the ﬁlm and its texture, we used a po-

larized reﬂection microscope (see Fig. 4). The ﬁlm was

illuminated with a mercury vapor lamp equipped with a

green ﬁlter (546 nm) or a blue ﬁlter (436 nm). A black

absorbent pad, the surface of which is tilted with respect

to the horizontal plane, was placed under the ﬁlm in or-

der to eliminate all the reﬂections other than those com-

ing from the ﬁlm. The position of the ﬁlm in the oven

was adjusted with an XY stage, which made it easy to

select the study area. The ﬁlm could also be rotated at

will since the oven was placed on the microscope rotating

stage. The images were acquired with an sCMOS cam-

era (Zyla 4.2 MP, Andor) with a resolution of 0.51 µm

with a ×10 objective or of 0.25 µm with a ×20 objective.

To be sensitive to the orientation of the molecules, the

microscope was equipped with a polarizer and an ana-

lyzer. We measured the intensity reﬂected from the ﬁlm

as a function of the orientation of the ~c-director. For this

purpose, we used a sheet of mylar of thickness 200 µm

in which a hole of dimension 4×0.5 mm was drilled. On

the sheet, we glued two metal electrodes parallel to the

large side of the hole and separated by a distance of 8

mm. Once the ﬁlm was stretched and equilibrated, the

molecules were oriented by applying a DC electric ﬁeld

and the ﬁlm was then rotated using the microscope ro-

tating stage. As the smectic C?phase is ferroelectric

with its spontaneous polarization perpendicular to the ~c-

director [39, 40], the latter orients perpendicularly to the

electric ﬁeld. These measurements allowed us to calibrate

the reﬂected intensity I(φ) (Fig.5) making it possible the

0π/2π3π/22π

φ

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

Intensity

0π/2π3π/22π

φ

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

Intensity

(a)

(b)

FIG. 5. Reﬂected intensity as a function of the orientation of

the ~c-director when the polarizer and the analyzer are crossed

(a) and slightly uncrossed by 5◦(b).

determination of the orientation of the molecules on the

images.

In particular, we observed that the ~c-director aligns

parallel to the meniscus. This anchoring condition is in-

compatible with a director ﬁeld uniformly aligned and

shows us that the ﬁlm always contains a +1 defect or

several defects whose total topological rank is equal to

+1. In practice, the defects are often localized on the

edge of the meniscus as can be seen in Fig. 6(a) where a

single +1 defect is present or in Fig. 6(b) where two +1/2

defects diametrically opposed are visible.

More rarely, the +1 defect detaches from the edge of

the meniscus. In this case, the defect is placed in the

center of the ﬁlm when it is at rest, which minimizes the

elastic energy of the system [Fig. 6(c)]. In this paper, we

will only study the ﬁlm behavior when the defect is at-

tached to the meniscus and cannot move, which happens

when it is pinned on a dust particle.

The last important parameter is the ﬁlm thickness,

which can vary greatly depending on how the ﬁlm is

stretched. As the ﬁlm properties, and the Leslie eﬀect

in particular, are strongly dependent on it, this quan-

6

(a)

(b)

(c)

FIG. 6. The three conﬁgurations frequently encountered. (a)

With a +1 defect at the edge of the ﬁlm; (b) with two 1/2

defects at the edge of the ﬁlm; (c) with a +1 defect in the cen-

ter of the ﬁlm. Photos taken between polarizer and analyzer

slightly uncrossed.

tity must be measured accurately. The simplest method

is to measure the ﬁlm reﬂectivity under the microscope

without polarizers. The reﬂectivity of the ﬁlm reads:

R(λ) = Ir(λ)

Ii(λ)=sin2H

E+ sin2H(3)

where Iiis the incident intensity, Irthe reﬂected intensity

and:

H=2πnH

λand E=4n2

(n2−1)2(4)

with n(λ) the average refractive index of the LC, Hthe

0 5 10 15 20

Number of layers

0

10

20

30

40

50

Thickness (nm)

FIG. 7. Film thickness as a function of the number of layers.

ﬁlm thickness and λthe light wavelength in vacuum. To

measure the index dispersion, we measured with a motor-

ized Jobin Yvon monochromator the reﬂectivity curves

R(λ) of several thick ﬁlms and we ﬁtted them with Eq. (3)

by assuming that the optical index was given by a sim-

pliﬁed Cauchy law:

n(λ) = A+B

λ2(5)

From these ﬁts with three parameters (H, A, B), we ob-

tained

A= 1.503 ±0.005 and B= 6500 ±100 nm2(6)

We then measured the thickness of a large number of

ﬁlms. Assuming that each ﬁlm has a thickness multiple of

the layer thickness, we obtained the graph of Fig. 7, from

which we deduced the layer thickness at 25◦C: 2.60 nm.

These data being known, we then plotted in Fig. 8 the

ﬁlm reﬂectivity as a function of the ﬁlm thickness given

in number of layers at the two wavelengths λ= 436 nm

and λ= 546 nm. These two wavelengths correspond to

two intense lines of the mercury vapor lamp used in our

experiments to observe the ﬁlms. This graph shows that

measuring the reﬂectivity of a ﬁlm at these two wave-

lengths is now enough to measure its thickness without

ambiguity. This simpliﬁed procedure was used to mea-

sure the ﬁlm thickness in all of our experiments.

III. EXPERIMENTAL RESULTS

In this section, we separately describe the problem of

phase winding when a ﬂow of ethanol is established and

that of phase unwinding when this ﬂow is stopped.

7

0 10 20 30 40 50

Thickness (number of layers)

0.000

0.025

0.050

0.075

0.100

0.125

0.150

Reflectivity

λ= 436 nm

λ= 546 nm

FIG. 8. Film reﬂectivity as a function of its thickness given

in number of layers for two diﬀerent wavelengths.

A. Phase winding and target formation

In this section we describe the behavior of a ﬁlm of uni-

form thickness when it is subjected to a ﬂow of ethanol.

The control parameters are the ﬁlm thickness and the

percentage of ethanol vapor or, equivalently, the diﬀer-

ence in ethanol vapor pressure ∆Pacross the ﬁlm given

by Eq. (2). In all the experiments reported here, the de-

fect is trapped on the meniscus and immobile.

1. Transient regime

The ﬁrst thing to notice when a ﬁlm initially at rest

is subjected to a ﬂow of ethanol is a rotation of the

molecules. This is particularly visible in the center of

the ﬁlm between slightly uncrossed polarizers where the

reﬂected intensity begins to vary periodically over time.

This rotation is rapidly accompanied by the formation of

a target-shaped pattern as can be seen in the sequence

of snapshots shown in Fig. 9 and in the Supplementary

Movie SM1 [41].

The second observation is the presence of signiﬁcant

ﬂows in the ﬁlm, visible to naked eye, with velocities

that can be as large as 2 mm/s for a percentage of ethanol

vapor of 50% [42]. These ﬂows arise everywhere in the

ﬁlm, and even before the rotation of the molecules be-

gins. They are shown by the red arrows in Fig.9. As we

have already said above, but it is worth repeating here,

these ﬂows are not due to an artefact, i.e. to air ﬂows

on either side of the ﬁlm, but to the presence of a ﬂow of

ethanol to which is associated a chemohydrodynamical

Leslie eﬀect. This can be checked immediately since the

ﬂows and the rotation of the director stop almost instan-

taneously when the ﬂow of air with ethanol is replaced

by a ﬂow of dry air with the same rate of ﬂow. It must be

noted that these ﬂows strongly inﬂuence the rotation of

the director, in particular near the edge of the ﬁlm where

they can prevent the rotation of the director, at least at

the beginning of the winding process. This phenomenon

is clearly visible in Fig. 9(f) where we can see that the

bands do not reach the edge of the ﬁlm.

To characterize the transient regime, we measured the

winding angle of the ~c-director in the center of the ﬁlm

as a function of time for the ﬁlm of Fig. 9. The cor-

responding curve for a percentage of ethanol vapor of

50% is shown in Fig. 10. This curve shows that φin-

creases, meaning that the ~c-director rotates counterclock-

wise when the ﬂow of ethanol vapor is directed upwards.

We also see that the rotation velocity was maximum at

the beginning and remained constant as long as the num-

ber of revolutions of the director was small, typically less

than 5. This means that, in this regime, the elasticity

is negligible, and that the Leslie torque generated by the

ﬂow of ethanol equilibrates with the viscous torque. We

shall see in the theoretical section that the Leslie torque

is proportional to ∆Pwhile the viscous torque is propor-

tional to the rotation velocity of the director, provided

that the ﬂows can be neglected. As a consequence, the

initial rotation frequency fshould be proportional to ∆P

in ﬁrst approximation. To check to what extent this pre-

diction was veriﬁed, we measured fas a function of ∆P

in the same ﬁlm. The result is shown in Fig. 11 for the

ﬁlm of Fig. 9. We see that the curve presents an aﬃne

behavior, but does not pass through the origin, because

no winding was observed for a percentage of ethanol va-

por lower than 15%. For these low vapor rates, the ﬁlm

remained in the conﬁguration of Fig. 9(a). We did, how-

ever, observe ﬂows inside near the edge of the ﬁlm, which

we believe probably prevented the rotation of the director

in the current conﬁguration. Other conﬁgurations, with

a +1 defect in the center of the ﬁlm, in which ﬂows are

present while the director does not or very little winds,

will be presented in a forthcoming paper.

These preliminary observations show that the ﬂows

play an important role in the free-standing ﬁlms, at least

at the beginning of the winding, contrary to what was

observed in Langmuir monolayers [32].

We now return to the curve of Fig.10. When the num-

ber of turns is larger than 5, the rotation velocity starts

to decrease. This is clearly due to an increase of the elas-

tic torque when the turns accumulate. In this regime,

the pattern looks more and more like a target between

slightly uncrossed polarizers. But contrary to what is ob-

served in Langmuir monolayers, the center of this target

is not circular, but oval as can be seen in the photos of

Fig. 12 taken during the transient regime. The question

that then arises is to ﬁnd the origin of this deformation.

Is it due to elastic anisotropy or to the presence of ﬂows?

To try to answer this question, we investigated whether

there were still signiﬁcant ﬂows near the center of the

target at this stage of the winding. For this purpose, we

followed the trajectory of a dust particle which had acci-

dentally fallen onto the ﬁlm. This trajectory is shown in

Fig. 12 and Supplemental Movie SM2 [43]. We see that

8

(a) (b) (c)

(d) (e) (f)

FIG. 9. Sequence of snapshots showing the winding of the ~c-director under a ﬂow of ethanol. At time t= 0 a 50% ethanol

ﬂow is imposed to the ﬁlm; (a) t= 9 s; (b) t= 12.1 s; (c) t= 13.2 s; (d) t= 14.1 s; (e) t= 14.8 s; (f ) t= 17.4 s. The red arrows

drawn on the snapshots (a) and (d) show the direction of the ﬂows. The blue arrows on the snapshot (b) show the sense of

rotation of the director. The ﬁlm is 10 layers thick.

0 50 100 150

t (s)

0

10

20

30

40

φ /2π

FIG. 10. Time course of the phase φin the center of the ﬁlm

for a percentage of ethanol vapor of 50%. The slope of the

dashed line gives the initial rotation frequency f. This curve

has been obtained from the images shown in Fig. 9.

the particle performs a back and forth movement in the

central area while rotating globally in the same direction

as the director in the center of the ﬁlm. More precisely,

during one turn, the particle goes back and forth three

0 1 2 3 4 5

Δ(kPa)

0. 0

0. 5

1. 0

1. 5

2. 0

2. 5

f (Hz)

0 10 20 30 40 50 60 70 80 90 100

%

FIG. 11. Rotation frequency as a function of the percentage

of ethanol vapor. These measurements have been performed

at the beginning of the winding with the ﬁlm shown in Fig. 9.

The solid line is the best linear ﬁt of the experimental points

ignoring the ﬁrst two. The error bars correspond to variations

observed over several measurements.

9

(a) (b) (c)

(d)

FIG. 12. Experimental evidence of a ﬂow in the center of the target during the winding of the ~c-director. (a-c) The circles

and the arrows indicate the position and the direction of the particle at diﬀerent times: (a) t= 0 s, (b) t= 1.8 s; (c) t= 3.5 s.

The ﬁlm is 10 layers thick and the percentage of ethanol vapor is 50%. (d) Trajectory followed by the particle during one turn

(about 20 s here).

times and practically returns to its initial position. Dur-

ing this sequence, the observed maximum velocity of the

particle is of the order of 0.02 mm/s and the director

takes about 20 s to rotate by 2π. This observation shows

that the ﬂows in the center of the pattern are complex

and not marginal at all. We now note that this trajec-

tory cannot be explained solely by the backﬂow produced

by the rotation of the director, because, in that case, we

should only observe circular ﬂows in the same direction as

the winding. This observation suggests that the chemo-

hydrodynamic Leslie force must be largely responsible for

these ﬂows and probably responsible for the deformation

of the target. This point will be conﬁrmed later.

2. Stationnary regime

At very long time (typically after a quarter of an hour),

a stationary regime is reached. In this limit, the target

remains unchanged under the microscope, meaning that

the director stops to rotate. A typical target at equilib-

rium is shown in Fig. 13 (a). We note that its center is still

deformed, which suggests that there are still ﬂows in this

steady state. In Fig.13 (b) we plotted the phase φmea-

sured along a radius as a function of r/R where Ris the

ﬁlm radius. This graph shows that the phase is well ﬁtted

with a parabolic law, as already observed in the Langmuir

monolayers [32]. Fig. 14 shows the equilibrium targets for

various values of the percentage of ethanol vapor. This

graph shows that the smaller the percentage, the less the

target is wound. On the other hand, the center of the

targets is always strongly deformed, which means that

the ﬂows play an important role whatever the percent-

age of ethanol vapor. Finally, we measured the number

of turns at equilibrium nmax =φmax/2π. Fig. 15 shows

that nmax is proportional to ∆Pand Fig. 16 shows that

it decreases when the ﬁlm thickness increases at ﬁxed ∆P

(corresponding here to a percentage of ethanol vapor of

10

(a)

(b)

r / R

φ / 2π

FIG. 13. Target pattern observed at equilibrium. The ﬁlm

is 10 layers thick. The percentage of ethanol vapor injected

under the ﬁlm is 50%. (b)Phase proﬁle measured along a

radius of this ﬁlm. The solid red line is the best parabolic ﬁt.

50%). Such a behavior is expected as the ﬂow of ethanol

must decrease when the ﬁlm thickness increases.

The dependence with the thickness is not trivial. In-

deed one might expect nmax to vary as 1/H since nmax

must be proportional to ∆P/H, by assuming that all

the material constants integrated over the ﬁlm thickness

are just proportional to the ﬁlm thickness. This depen-

dence is not observed experimentally as shown by the

1/H curve plotted in Fig. 16. This means that the mate-

rial constants integrated over the ﬁlm thickness, and in

particular the elastic constants, are probably not propor-

tional to the ﬁlm thickness. This interpretation is plausi-

ble because the ﬁlms studied are very thin, with less than

20 layers in general, meaning that the surface eﬀects are

certainly important as already noted by several authors

in other materials (for a review, see Ref. [39]). Another

FIG. 14. Images of the target pattern obtained with diﬀer-

ent percentages of ethanol vapor. From top to bottom, the

percentage is 10%, 20%, 50% and 100%.

complication could come from the formation of a thin

boundary layer of thickness δin the vicinity of the ﬁlm

in which the concentration of ethanol changes. In that

case, a simple calculation shows that nmax must vary as

1/(H+δ) by assuming that the material constants are

proportional to the ﬁlm thickness. The details of the

model are given in Ref. [44]. The best ﬁt of the experi-

mental data with this law leads to δ= 29 nm, but the ﬁt

is not yet perfect as can be seen in Fig. 16. In practice,

all these eﬀects must act at the same time, and it would

be necessary to measure all the material constants as a

function of the ﬁlm thickness to be able to disentangle

them. This is an enormous task that goes well beyond

the scope of this study. Note in passing that a best ﬁt

was obtained with a power law in 1/Hnwith n≈1/2 as

can be seen in Fig. 16, but we have no model to explain

this dependence. We ﬁnally add to conclude that the de-

fect sometimes detaches from the meniscus, in particular

at large percentage of ethanol vapor, and starts moving

along the edge of the ﬁlm. In that case, a new stationary

regime is reached when the defect and the director ro-

tate at the same angular velocity. An example is shown

in Supplemental Movie SM3 [45]. In this regime, the ﬁnal

number of turns is always less than that reached when

the defect is ﬁxed, but it can vary from one experiment

to another because it depends on the mobility of the de-

fect in the vicinity of the meniscus. For this reason we

did not study this regime in detail.

11

0 1 2 3 4 5

ΔP (kPa)

0

10

20

30

40

50

60

nmax

0 10 20 30 40 50 60 70 80 90 100

%

FIG. 15. Number of turns at equilibrium as a function of

the percentage of ethanol vapor for the 10-layers-thick ﬁlm of

Fig. 13. The error bars correspond to the variations observed

by making several measurements. The solid line corresponds

to the best linear ﬁt.

0 10 20 30 40

Thickness (number of layers)

0

10

20

30

40

50

max

n

1/H

1/(δ+H)

1/√ H

FIG. 16. Number of turns at equilibrium as a function of the

ﬁlm thickness for ﬁlms of 0.5 mm in diameter. The percentage

of ethanol vapor is 50%. The error bars have been calculated

from the variations observed with diﬀerent ﬁlms. The lines

correspond to the best ﬁts with the functions given on the

graph.

B. Phase unwinding

If the ﬂow of ethanol is stopped and replaced by a ﬂow

of dry air once the target is formed, the latter unwinds

to relax the elastic energy it has stored. In practice, it

is also possible to prepare a target by blowing air on the

ﬁlm with a small fan or by using a rotating electric ﬁeld.

After stopping the fan or the electric ﬁeld, the target

unwinds in the same way as when stopping the ﬂow of

ethanol. In all the cases, the only forces involved during

the unwinding are the elastic and viscous forces. We ob-

served that at the start of the unwinding, the relaxation

was not exponential and depended on the way the target

had been wound. For this reason, we did not study this

initial regime in detail. By contrast, we observed that

after a few turns, the unwinding process was always the

same, and no longer depended on the initial conditions.

More precisely we observed that the phase φmeasured

in the center of the ﬁlm relaxed exponentially over time

as can be seen in Fig. 17 and Supplementary Movie SM4

[46]. We also observed that in this regime, the center of

the target was much less deformed than during the wind-

ing process in the presence of the ethanol ﬂow. This can

be seen by comparing the targets in Fig. 14 and Fig. 17.

This observation is important because it seems to show

that it is indeed the ﬂows created by the Leslie force, and

not the elastic anisotropy, that are responsible for the

oval deformation of the target in the center of the ﬁlm.

This interpretation will be conﬁrmed later by our numer-

ical simulations. We also observed by following under

the microscope the movement of small smoke particles

deposited on the ﬁlm, that there were backﬂow eﬀects in-

side the ﬁlm. The orthoradial velocity proﬁle normalized

with the phase value measured in the center of the ﬁlm

is shown in Fig. 18(a). This graph shows that this ratio

is constant over time, which indicates that the velocity

decreases with the same exponential law as the phase as

a function of time. We also show in Fig.18(b) how the

maximum velocity measured in the ﬁlm decreases (in ab-

solute value) as a function of time. This value is reached

at a distance of the order of 0.6Rfrom the center of the

ﬁlm, by denoting by Rthe radius of the ﬁlm. As ex-

pected, this curve is well ﬁtted by an exponential law

with the same characteristic time as for the phase mea-

sured at the center of the ﬁlm. For the 20-layers ﬁlm

shown in Fig. 17 the half-unwinding time τ1/2is of the

order of 67 s and the maximum velocity measured when

the number of turns is equal to 10 is -0.06 mm/s.

IV. THEORETICAL SECTION

In this section, we will begin by recalling the fun-

damental equations that govern the dynamics of free-

standing ﬁlms in the presence of a chemical Leslie eﬀect.

Then we will see how to solve them, ﬁrst in the case of

the winding in the presence of a vapor ﬂow, and second

in the case of the unwinding when the ﬂow is removed. In

each case, we will treat the problem ﬁrst in a simpliﬁed

way by neglecting the elastic anisotropy and the ﬂows,

and then in a more complete way by taking into account

either the elastic anisotropy or the ﬂows.

12

0 50 100 150 200

0

2

4

6

8

10

t (s)

φ / 2π

FIG. 17. Angle φin the center of the ﬁlm as a function of

time measured during the phase unwinding when the ﬂow

of ethanol is cut oﬀ. The solid line is the best ﬁt with an

exponential law. The inset shows the target pattern observed

during the phase unwinding. The ﬁlm is 20 layers thick.

A. Governing equations

The hydrodynamic variables are the ~c-director deﬁned

by the angle φit makes with the x-axis, the velocity ~v

and the pressure Pin the ﬁlm. As the ﬁlms are always

very thin, we will assume that these quantities are ho-

mogeneous in the ﬁlm thickness and only depend on the

(x, y)-coordinates.

The two equations which govern the dynamics of a ﬁlm

are the torque equation and the Cauchy equation. The

ﬁrst one mainly governs the rotation of the director while

the second allows one to calculate the velocity. These two

equations are nevertheless coupled and strongly nonlin-

ear, which makes the problem diﬃcult to solve.

Formally, the torque equation reads

~

ΓE+~

ΓV+~

Γcm = 0,(7)

and the Cauchy equation reads

~

∇ · σV+~

∇ · σE+~

∇ · σch −~

∇P=~

0.(8)

In these equations, ~

ΓE,~

ΓVand ~

Γcm are the elastic, vis-

cous and chimiomechanical components of the torque

acting on the ~c-director and σV,σEand σch are the

elastic, viscous and chimiohydrodynamical components

of the stress tensor. As for P, it is the pressure in the

ﬁlm given by the incompressibility condition

~

∇ · ~v = 0.(9)

Note that we neglected the inertial term in the Cauchy

equation because the Reynolds number in our experi-

ments is always small with respect to 1, of the order of

10−3−10−2.

0. 0 0. 2 0. 4 0. 6 0. 8 1. 0

r/R

−0.006

−0.004

−0.002

0.000

vθ/nmax (mm/s/revolution)

0 50 100 150 200

t (s)

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0.00

vmax (mm/s)

(a)

(b)

FIG. 18. (a) Velocity proﬁle measured along a radius during

the phase unwinding. Each color corresponds to a diﬀerent

tracer. For each tracer, the diﬀerent points with the same

color correspond to diﬀerent times. The solid line is a ﬁt

with a law in r−r3. (b) Maximal velocity as a function of

time measured during the phase unwinding and its ﬁt with

an exponential law.

We now give the expression of these diﬀerent torques

and stresses as a function of the constants of the material.

The elastic torque and the elastic stress are equilibrium

quantities which derive directly from the expression of

the elastic free energy of deformation of the director ﬁeld

~c. For a ﬁlm whose the thickness is very small compared

to the helical pitch of the smectic C?, the elastic energy

is reduced to the following expression:

f[~c] = 1

2KS~

∇ · ~c2

+1

2KB~

∇ × ~c2

.(10)

In this expression, KSis the splay constant and KBis the

bend constant. Note here that KBis renormalized by the

presence of the polarization charges since the Smectic C?

phase is ferroelectric [47–49]. For this reason, one may

expect that KBis larger than KSin our ﬁlms since the

LC FELIX used in our experiments has a large sponta-

neous polarization (of the order of 20 nC/cm2according

13

to Ref. [50]). In practice, we can split the elastic energy

into two parts: an isotropic part that reads as a function

of φ:

fi[φ] = 1

2K~

∇φ2

,(11)

and an anisotropic part, of expression:

fa[φ] = 1

2K cos(2φ)φ2

,x −φ2

,y+ 2 sin(2φ)φ,x φ,y .

(12)

In these equations, K=KS+KB

2is the average elas-

tic constant and ε=KB−KS

KB+KSis a dimensionless coef-

ﬁcient that characterizes the elastic anisotropy. From

these equations the elastic torque and the elastic stress

can be calculated. For simplicity, we only give here their

expressions in the case of isotropic elasticity when ε= 0:

~

ΓE=K∇2φ~

k , (13)

where ~

kis the unit vector normal to the ﬁlm directed

upwards, and

σE=−Kφ2

,x φ,xφ,y

φ,xφ,y φ2

,y (14)

in cartesian coordinates (x, y).

In a smectic C?ﬁlm, the most general form of the

nonequilibrium viscous stress tensor reads [51]:

σV

ij =µ0Aij

+µ3(ckAklcl)cicj

+λ2(ciCj+cjCi) + λ5(ciCj−cjCi)

+µ4(cickAkj +cjckAki)

+λ2(cickAkj −cjckAki).

(15)

In this expression, ~

Cis the corotational time derivative

of the ~c-director with respect to time :

~

C=D~c

Dt−~

Ω×~c, (16)

where D/Dtdenotes the total derivative with respect to

time, ~

Ω = 1

2~

∇ × ~v and Aij is the strain rate tensor :

Aij =1

2∂vi

∂xj

+∂vj

∂xi.(17)

In practice, this tensor can be rewritten as in the nematic

phase in the form:

σV

i,j =α4Ai,j

+α1(ckAk,lcl)cicj

+α2cjCi+α3ciCj

+α5cjckAk,i +α6cickAk,j

(18)

with the correspondence :

α1=µ3

α2=λ2−λ5

α3=λ2+λ5

α4=µ0

α5=µ4−λ2

α6=µ4+λ2

.(19)

We will use this form in the following. Note that, as

in nematics, the viscosity coeﬃcients must satisfy the

Parodi relation [52]

α2+α3=α6−α5.(20)

From this expression, the nonequilibrium viscous torque

can be calculated by using the relation ΓV

i=−εijk σV

jk

where εijk is the Levi-Civita tensor. The calculation

gives as in nematics

~

ΓV=−γ1~c ×~

C−γ2~c ×A~c , (21)

where γ1=α3−α2and γ2=α3+α2=α6−α5are

the two rotational viscosities of the phase. In cartesian

coordinates, this torque reads ~

ΓV= ΓV~

kwith

ΓV=−γ1Dφ

Dt +1

2(u,y −v,x)(22)

−γ2

2[cos (2φ) (u,y +v,x) + sin (2φ) (v,y −u,x)]

by denoting by uand vthe two components vxand vyof

the velocity and their derivatives with respect to xor y

by a subscript xor yafter a comma. In the presence of

a gradient of chemical potential, which is caused in our

experiments by the ﬂow of ethanol, new terms appear

in the expressions of the nonequilibrium torque and the

stress tensor. These terms have been calculated for the

ﬁrst time by Leslie in cholesterics in the presence of a

thermal gradient [3], but they also exist in the smectic

C?phase because of its chirality and the absence of mir-

ror symmetries. A complete derivation of these terms in

the simpliﬁed case of a smectic C?ﬁlm, using a general-

ization of the Akopyan and Zel’dovich method [53, 54],

can be found in Refs. [44, 55]. When the ﬂow of particles

is normal to the ﬁlm, this calculation shows the existence

of a new nonequilibrium chemomechanical torque of ex-

pression

~

Γcm =ν∆P~

k , (23)

and of a new nonequilibrium chemohydrodynamical

stress of expression

σch =µ∆P

2([~c ×~

k]⊗~c +~c ⊗[~c ×~

k]) ,(24)

where the symbol ⊗denotes the dyadic product between

two vectors [(~a ⊗~

b)ij =aibj]. Here, ∆Pis the diﬀerence

14

in vapor pressure between the bottom and the top of the

ﬁlm. In cartesian coordinates (x, y ) this stress reads:

σch =µ∆P

2sin(2φ)−cos(2φ)

−cos(2φ)−sin(2φ).(25)

Note that a nonequilibrium chemomechanical stress of

components σcm

ij =−εijk Γcm

kis associated with the

chemomechanical torque (23). As this stress has zero

divergence, it does not enter into the Cauchy equation

and so it will not play any role in our study, knowing

that we will ﬁx angle (or phase) φat the edge of the ﬁlm

in all of our calculations.

In the following, we solve these equations ﬁrst in the

winding case and then in the unwinding case. Each time,

we will proceed in a progressive way, by ﬁrst analyzing

the solution when the elastic anisotropy and the ﬂows

are neglected. We will then analyze separately the role

of elastic anisotropy and of ﬂows.

B. Phase winding

1. Analytical solution in isotropic elasticity and in the

absence of ﬂow

Phase dynamics is particularly simple to analyze when

the elastic anisotropy and the ﬂows are neglected (ε= 0

and ~v = 0). In that case the equations reduce to the

torque equation of simpliﬁed form:

γ1

∂φ

∂t =K∇2φ+ν∆P(26)

This equation was already used to study the phase wind-

ing in Langmuir monolayers [32]. It shows that, at the

beginning, when the ﬂow of ethanol is switched on, the

phase uniformly winds as

φ(~r, t) = 2πf t (27)

with an initial rotation frequency fgiven by:

f=ν∆P

2πγ1

(28)

This formula shows that fis proportional to ∆P. It is

important to note here that this formula remains valid

in the presence of a defect since Eq. (26) is linear. It

is indeed possible to add to solution (27) the stationary

solution to equation ∆φ= 0 corresponding to the de-

fect. In practice, this behavior is not strictly observed

experimentally (see Fig. 11) because of the complications

coming from the ﬂows and the elastic anisotropy. This

law must also rapidly fail because the director cannot ro-

tate on the edge of the ﬁlm. By assuming that φ= 0 at

the edge of the ﬁlm [56], this condition leads to an equi-

librium target pattern at time t→ ∞ when ∂φ/∂t = 0

given by

φ(~r, t → ∞)=2πn 1−r

R2(29)

0 5 10 15 20

ft

0

1

2

3

4

5

φ/2π

0.00.2 0. 4 0.6 0. 8 1. 0

r/R

0

1

2

3

4

5

φ/2π

−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

0

π

2π

(a)

(b)

(c)

FIG. 19. Solution of Eq. (26) for a ﬁnal winding of 5 turns.

(a) Winding angle in the center as a function of time scaled

using 1/f as the characteristic time. (b) Radial angle proﬁles

at diﬀerent times. Each colored curve corresponds to a point

of the same color in (a). The dashed curve corresponds to the

equilibrium curve obtained when t→ ∞. (c) Aspect of the

target between slightly uncrossed polarizers when n= 5. The

local intensity is calculated by taking I= sin2φ.

15

where we have set

n=ν∆PR2

8πK (30)

Here, ris the polar coordinate by taken the origin in the

center of the ﬁlm and Ris the ﬁlm radius and nrepresents

the number of turns at equilibrium. This quantity will be

chosen as the control parameter in the following, instead

of ∆P. This law shows that, at equilibrium, the phase

proﬁle along a radius is parabolic. This dependence is

again pretty well observed experimentally as shown in

Fig. 13.

In the intermediate regime, Eq. (26) must be solved

with all the terms. With the boundary conditions

φ(~r, 0) = 0 and φ(R, t) = 0, this equation has a solu-

tion that can be expressed using Bessel functions. We

show the results graphically in Fig. 19.

2. Numerical solution in anisotropic elasticity and in the

absence of ﬂow

The previous calculation leads to perfectly circular tar-

gets. However, experimentally, targets are deformed in

the center not only during the transient regime, but also

in the stationary regime, as can be seen in Figs. 12-14.

In this paragraph, we analyze which role plays the elas-

tic anisotropy in the deformation of the targets when the

ﬂows are negligible.

This problem is unfortunately much more complex

than in the isotropic case. To solve it, we could have

solved the torque equation in anisotropic elasticity, but

this is diﬃcult because φnow depends not only on rbut

also on the polar angle θ. For this reason, we adopted

another strategy consisting in minimizing an eﬀective en-

ergy.

To ﬁnd this energy, let us return to the torque equation

of general expression when ~v = 0:

γ1

∂φ

∂t = ΓE+ν∆P(31)

In this equation ΓE=δf /δφ is the elastic torque calcu-

lated by taking f[φ] = fi[φ]+ fa[φ] where fi[φ] and fi[φ]

are given by Eqs. (11) and (12).

By setting φ(t) = φtand φ(t+δt) = φ, and by approx-

imating Eq. (31) by an implicit Newton type scheme:

∂φ

∂t 'φ−φt

δt (32)

it is possible to create two new functionals corresponding

respectively to the temporal term:

t[φ, φt] = γ1

2δt (φ−φt)2(33)

and to the Leslie term:

l[φ] = −νGφ (34)

the derivative with respect to φof which gives back the

associated torques. The torque equation can then be

rewritten in the form:

δe

δφ = 0 (35)

where e[φ] is an eﬀective energy of the form :

e[φ, φt] = t[φ, φt] + l[φ] + fi[φ] + fa[φ] (36)

In the following, we will take Ras a unit length and τ=

R2/D as a unit of time, where D=K/γ1is the average

orientational diﬀusion coeﬃcient. All our calculations

will be made using dimensionless variables:

˜x=x

R,˜y=y

R

˜v=R

Dv

˜

P=R2

KP

(37)

and we will set

ai=αi

γ1

and X=µ

ν.(38)

With these variables, the eﬀective energy is written in

the dimensionless form as follows:

˜e[φ, φt] = 1

2˜

δt (φ−φt)2−8πnφ +1

2φ2

,˜x+φ2

,˜y

+1

2cos(2φ)φ2

,˜x−φ2

,˜y+ 2 sin(2φ)φ,˜xφ,˜y,

(39)

where nis deﬁned in Eq. (30) and corresponds to the

numbers of turns at equilibrium calculated by neglecting

the anisotropy and the ﬂows.

The next step to solve the torque equation for each

time step was to minimize this functional with respect

to φ=φ(t+δt) knowing φt=φ(t). In addition, it was

possible to directly calculate the ﬁnal equilibrium state

by just omitting the time term in this equation.

From a practical point of view, the energy functional

was developed on a Finite Element space using the C++

library deal.II [57]. This energy was then minimized

thanks to a conﬁdence region algorithm ensuring both a

global and rapid convergence towards the minimum en-

ergy. All the details on this numerical method are given

in Refs. [44, 58]. It should just be noted that the main

numerical limitations came from the execution time. In-

deed, it was necessary to have a suﬃciently ﬁne grid to

correctly describe the terms in cos(2φ) and sin(2φ) in the

anisotropic part of the energy. In particular, the use of

an adaptive grid could not help to solve this problem,

because the gradients of φare important everywhere in

the ﬁlm. To overcome this problem, we therefore limited

ourselves to low winding simulations (nranging from 2

to 20) and we worked with grids with the highest level

of reﬁnement possible. In practice, the equilibrium tar-

get calculations were made on a grid containing 5 million

16

0π2π

(a)

(b)

FIG. 20. Target pattern at equilibrium calculated for n= 5

and two diﬀerent values of the elastic anisotropy: (a) ε= 0.7;

(b) ε= 0.95.

0. 0 0. 2 0. 4 0. 6 0. 8 1. 0

ε

1.00

1.05

1.10

1.15

1.20

φmax /2

πn

FIG. 21. Number of turns at equilibrium divided by nas

a function of the elastic anisotropy when the ﬂows are ne-

glected. The curve has been plotted for n= 5 but it should

be emphasized that this curve is almost independent of n.

0. 0 0. 1 0. 2 0. 3 0. 4 0. 5

t

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

φ/φmax

ε= 0.0

ε= 0.2

ε= 0.4

ε= 0.6

ε= 0.8

FIG. 22. Angle φin the center of the ﬁlm scaled by φmax =

2πnmax as a function of the dimensionless time when the ﬂows

are nglected. Simulations were performed with n= 5 for dif-

ferent values of the elastic anisotropy ε. Angle φmax corre-

sponds to the ﬁnal value of φduring the winding. nmax is

given in Fig. 21.

0.0 0. 2 0. 4 0. 6 0. 8

ε

0.12

0.13

0.14

0.15

0.16

τ1/ 2

FIG. 23. Half-winding time as a function of the elastic

anisotropy when the ﬂows are neglected. These results have

been obtained by taking n= 5 but it should be emphasized

that they depend very little on this parameter.

cells on which a revolution of the director is described by

200 points (for n= 5). The computation time necessary

to solve the problem was typically 10 hours and increased

up to one day for the largest values of the anisotropy. As

for the dynamic calculations on the winding of a target,

there were performed using a coarser grid (50 points per

revolution for n= 5) so that each time step could be

calculated over a duration of the order of a minute.

Using this method, we calculated equilibrium targets

for diﬀerent values of the anisotropy. The result for

ε= 0.7 (KB/KS= 5.7) and ε= 0.95 (KB/KS= 39)

are shown in Fig. 20 when n= 5. These images show

17

that the targets are little deformed in the center, even

at very large anisotropy, and bear little resemblance to

the targets observed experimentally (see Figs. 13 and 14).

We also plotted in Fig. 21 the number of turns at equilib-

rium nmax divided by nas a function of the anisotropy.

This number increases, but not so much, since we ﬁnd

nmax ≈1.2nat ε= 0.95. We also studied the inﬂuence

of the anisotropy on the phase winding process. The

results are shown in Fig. 22 where we plotted the evolu-

tion of the angle measured in the ﬁlm center scaled to

its maximal value as a function of the dimensionless time

for diﬀerent values of the anisotropy. This graph shows

that the curves are not very diﬀerent. Additionally, we

plotted the dimensionless half-winding time as a function

of the anisotropy in Fig. 23. These two graphs show that

the winding process slows down when the anisotropy in-

creases, but this slowing down is not considerable, even

at the strongest anisotropies.

All these results show that the elastic anisotropy is not

so crucial in our system. In particular, it cannot alone

explain the deformation of the targets in the center of the

ﬁlm during the winding. For this reason, and also to sim-

plify the calculations, we neglected the elastic anisotropy

in the study of ﬂows that we present now.

3. Numerical solution in isotropic elasticity and in the

presence of ﬂow

In the presence of ﬂows, the previous minimization

method can no longer be used and it is necessary to

directly solve the three fundamental equations of the

dynamics, namely the torque equation (7), the Cauchy

equation (8), and the incompressibility equation (9).

From a mathematical point of view, these equations

resemble Boussinesq’s equations which describe the ﬂow

of a ﬂuid whose density varies with temperature. In our

problem, the angle φplays the same role as the tempera-

ture in the Boussinesq problem since the torque equation

is a diﬀusion equation in which the Leslie forcing term is

equivalent to a heating term in the heat equation. Our

problem is nevertheless more complicated since it also in-

volves two coupling terms between the director and the

velocity proportional to γ1and γ2which are not present

in the usual heat equation. As for our Cauchy equation,

it is formally equivalent to the Stokes equation for the

velocity in the Boussinesq problem, the Leslie chemohy-

drodynamical force and the elastic force replacing the

buoyancy force in our case.

If this analogy seems a little artiﬁcial, it is neverthe-

less relevant from a numerical point of view. As for the

Boussinesq system, we are dealing with two coupled ﬁelds

φ(~r, t) and ~v(~r, t). The ﬁrst, scalar, is governed by a dif-

fusion equation. The second, vectorial, satisﬁes a Stokes

type equation. For this reason, the same general tech-

niques of resolution as in the Boussinesq problem can be

used here.

In particular, the fact of neglecting the inertial term

in the Cauchy equation makes it possible to treat it sep-

arately from the torque equation. In other words, the

absence of the inertial term in the Cauchy equation im-

plies that to a given orientation ﬁeld φ(~r, t) is associated

a single velocity ﬁeld ~v(~r, t). A method to solve a time

step is therefore to solve successively and independently

these two equations.

In practice, we will use the velocity ﬁeld of the previous

time step to solve the torque equation with φas the only

unknown. In this way, a new estimate of φis obtained,

which can be used in the Cauchy equation to calculate a

new, more precise estimate of ~v. Repeating this process

allows one to reﬁne the general solution (φ, ~v) until it

satisﬁes the two equations.

The analogy with the Boussinesq problem also allowed

us to simply answer complicated numerical questions. In

particular, it is well known that the choice of the ﬁnite el-

ement space is crucial to correctly solve the Stokes equa-

tion. Using a pair of unstable spaces for the velocities

and the pressure will result in a potentially false and

non-physical solution. In our case, and by analogy with

other similar numerical problems [59], we used the mixed

space Q2

2×Q1, that was proven to be stable for solving

the Stokes equation.

In practice, our equations were made dimensionless be-

fore to be solved numerically. With the variables given

in Eq. (37), the torque and Cauchy equations are written

in dimensionless form as

Dφ

D˜

t=˜

∆φ+ 8πn −1

2(˜u,˜y−˜v,˜x)

−γ

2[cos (2φ) (˜u,˜y+ ˜v,˜x) + sin (2φ) (˜v,˜y−˜u,˜x)] ,

(40)

and

˜

~

∇ · ˜σV+˜

~

∇ · ˜σE+˜

~

∇ · ˜σch −˜

~

∇˜

P=~

0.(41)

In these equations, we set γ=γ2/γ1and the ˜σare the

dimensionless stresses. The expressions of the compo-

nents of ˜σVas a function of φand ˜

~v = (˜u, ˜v) are given

in Appendix. In the rest of this Section and in the Ap-

pendix, and to simplify the notation, we will remove the

tilde symbol and assume that we are working with di-

mensionless variables.

In practice, our numerical code was implemented in

C++ using the deal.II ﬁnite element library. All the de-

tails about it are given in Ref. [44]. We just mention

here an important point used to accelerate its conver-

gence. Indeed we noticed that the elastic force could be

expressed in the following form:

~

FE=~

∇ · σE=−~

∇φ∆φ−~

∇fi(42)

where

fi=1

2(φ2

,x +φ2

,y) (43)

is the elastic energy in isotropic elasticity. The elastic

force can therefore be rewritten as the sum of the elas-

tic energy gradient and a term proportional to ∆φ. By

18

grouping the term in energy gradient with the pressure

term and by setting P=P+fi, and by replacing in the

Cauchy equation ∆φby its expression calculated from

the torque equation:

∆φ(u, v) = Dφ

Dt−8πn +1

2(u,y −v,x)

+γ

2[cos (2φ) (u,y +v,x) + sin (2φ) (v,y −u,x)]

(44)

it was possible to greatly improve the code convergence.

To perform the calculations, the values of the 9 con-

stants n, X, γ, and the six aimust be given, knowing

that only 6 of them are independent since a3−a2= 1

and γ=a2+a3=a6−a5. In practice, we did not

know the material constants, in particular the viscosities

which are diﬃcult to measure in smectic C?. For this

reason, we used a reduced set of viscosities allowing us

to qualitatively reproduce our experiments while making

the calculations as simple as possible. In the literature

two sets of viscosities have been proposed. The ﬁrst one

is due to Pieranski et al. [39, 60, 61] and consists of

taking a1=a5=a6=γ= 0, the “ordinary” viscos-

ity a4>0 and a2=−a3=−0.5. With this choice

the equations considerably simplify, but they did not al-

low us to reproduce the experiments, in particular the

complex trajectory of the particle during the transient

regime. For this reason, we will not use this model in

the following, referring the reader to Ref. [44] for more

information about it. The second set of viscosities was

proposed by Stannarius et al. [62] and consists of tak-

ing a1=a3=a6= 0, a2=−a5=−1, a4>0 and

γ= 1. This choice is more realistic in comparison to the

viscosities measured in usual nematics and we will use it

because it allowed us to reproduce qualitatively our ex-

periments. In practice, we varied a4between 0.5 and 4,

Xbetween -5 and +5 and nbetween 0 and 10.

We now describe our results. As before, we start by

studying the ﬁnal state of the system when the phase

winding has reached its equilibrium. Fig. 24 shows the

target patterns and the associated velocity ﬁelds calcu-

lated for n= 5 and a4= 1 by taking X= 2.5 and

X=−2.5. The ﬁrst obvious remark is that the num-

ber of turns and the shape of the target at equilibrium

strongly depends on the sign of X. This is the direct

consequence of the ﬂows induced by the Leslie chemo-

hydrodynamical stress in the ﬁlm. If X > 0, the ﬂows

are almost circular and do not deform signiﬁcantly the

target which remains circular in the center. As the ﬂows

are clockwise, they also favor the phase winding, lead-

ing to a number of turns larger than 5 in the stationary

regime. This is obviously not what we have observed

as the target remains almost circular in this calculation.

The situation is very diﬀerent when X < 0 and much

more interesting because the target is this time strongly

deformed in its center due to more complex ﬂows with

recirculation loops. In that case, the target deforms and

becomes oval near the center, in agreement with our ob-

servations. This is a clear evidence that the ﬂows induced

(a)

(c)

(b)

(d)

−0.4−0.2 0.0 0.2 0.4

−0.4

−0.2

0.0

0.2

0.4

2.0

4.0

6.0

8.0

10.0

12.0

−0.4−0.2 0.0 0.2 0.4

−0.4

−0.2

0.0

0.2

0.4

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

FIG. 24. Target patterns and corresponding velocity ﬁelds in

the stationary regime when n= 5, a4= 1, ε= 0, and X= 2.5

(a, b) and X=−2.5 (c, d). Only the central part of the ﬁlm,

where the ﬂows are the most important, is shown in (b) and

(d), with the texture in background.

19

−4−20 2 4

X

0

5

10

15

20

25

30

vmax

−4−20 2 4

X

0. 6

0. 8

1. 0

1. 2

1. 4

1. 6

1. 8

φmax /2πn

a4=0.5

a4=1.0

a4=2.0

a4=4.0

(a)

(b)

FIG. 25. Number of turns scaled to nin the center of the ﬁlm

(a) and maximal value of the norm of the velocity (b) as a

function of Xfor n= 5 and diﬀerent values of a4when the

stationary regime is reached. In these simulations, ε= 0.

by the Leslie chemomechanical stress are largely respon-

sible for the central deformation of the target and that

Xis negative in our system. For completeness, we plot-

ted in Fig. 25 the number of turns divided by nand the

magnitude of the maximal velocity in the ﬁlm in the sta-

tionary regime as a function of Xfor diﬀerent values of

the “ordinary” viscosity a4. As expected, the larger the

viscosity a4, the less important are the ﬂows and the less

the number of turns deviates from n.

In a second step, we studied the role of the ﬂows on

the transient regime. Fig. 26(a) shows the time evolution

of the phase measured in the ﬁlm center for X=±2.5

and X= 0 and by neglecting the ﬂows (~v = 0) when

n= 5. The ﬁrst remark is that the curve obtained by

taking X= 0 is not very diﬀerent from the curve calcu-

lated by neglecting the ﬂows (~v = 0). This shows that

the backﬂow eﬀects do not play a considerable role. By

contrast, the ﬂows induced by the Leslie chemohydrody-

0. 0 0. 1 0. 2 0. 3 0. 4 0. 5

t

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

1. 2

1. 4

φ /2πn

0. 0 0. 1 0. 2 0. 3 0. 4 0. 5

t

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

φ / φmax

X= −2.5

X=2.5

X

v

=0

=0

(a)

(b)

FIG. 26. Number of turns in the center of the ﬁlm scaled to

n(a) or to nmax (b) as a function of the dimensionless time.

In these simulations, n= 5, ε= 0 and a4= 1.

namic stress, in addition to changing the number of turns

at equilibrium also changes the shape of the curves, in

particular in the initial regime. Indeed, at X= 2.5, the

rotation velocity ﬁrst increases before decreasing when

the turns accumulate whereas, at X=−2.5, the ro-

tation velocity constantly decreases from the beginning.

Fig. 26(b) shows the same curves when the phase is scaled

with its maximal value reached in the stationary regime.

These curves show that the half-winding time is almost

the same for X= 0 and X= 2.5 and is shorter for

X=−2.5. In all the cases this time is shorter than when

the ﬂows are neglected, which shows again that they play

an important role in the winding process.

In a third step, we tried to reproduce the complex tra-

jectory of the particle observed during the transient pro-

cess. Experimentally, we have seen that the particle ro-

tates on average clockwise while going back and forth.

It is not diﬃcult to convince oneself that this back-and-

forth motion is impossible if the ﬂows are everywhere

circular as in Fig. 24(b) when Xis positive. Conversely,

20

(a) (b)

(c) (d) (e)

FIG. 27. Simulation of the complex trajectory of the particles. (a-d) The red dot shows the position of the particle at diﬀerent

times and the corresponding velocity ﬁeld. (e) Trajectory of the particle during a full rotation of the director in the center of

the ﬁlm.

the presence of recirculating vortices in the velocity ﬁeld

of Fig. 24(d) explains this motion when the particle is

trapped on one of these vortices. The average rotation of

the particle in this case results of a continuous rotation

of the velocity ﬁeld which accompanies the rotation of

the director during the transient state.

To reproduce our observation, it would be necessary to

calculate the velocity ﬁeld for a value of nmuch larger

than those we are able to calculate with our computer.

It was therefore not directly possible for us to reproduce

the experiment. On the other hand, we were able to

reproduce the observed trajectory in a qualitative way

starting from a numerically calculable less wound target.

By then rotating this target and the associated velocity

ﬁeld at a constant and well-chosen rate of rotation, it

became possible to numerically reconstruct a trajectory

that was very similar to the one observed experimentally.

The result is shown in Fig. 27 and in Supplemental Movie

SM5 [63] . This semi-quantitative calculation conﬁrmed

that it was well the recirculation vortices coming from the

conﬂict between the backﬂow eﬀects and the Leslie ﬂows

of chemohydrodynamic origin which were responsible for

the back-and-forth motion of the particle near the center

of the target. This again conﬁrmed that Xwas negative

in our product.

C. Phase unwinding

If the ﬂow of alcohol is stopped, the target unwinds

to decrease its elastic energy. The governing equations

for the unwinding process are the same as before with-

out the Leslie forcing terms since now ∆P= 0. As in

the previous section, we will proceed by successive ap-

proximations to analyze this problem and we will use the

dimensionless variables (without the tilde for simplicity).

1. Analytical solution in isotropic elasticity and in the

absence of ﬂow

In isotropic elasticity and if the backﬂow is neglected,

the torque equation reduces to

∂φ

∂t = ∆φ=∂2φ

∂2r+1

r

∂φ

∂r (45)

in cylindrical coordinates. By taking for initial condition

φ(r, t = 0) = φmax(1 −r2) and by imposing φ(1, t) = 0

on the edge of the ﬁlm, the solution is given by

φ(r, t) = φmaxJ0r

√τexp −t

τ,(46)

where J0is the Bessel function of order 0. This equation

shows that φdecreases exponentially over time with a

characteristic time τgiven by

J01

√τ= 0 .(47)

From this equation, the half-unwinding time was de-

duced: τ1/2= ln(2) τ= 0.12. In this calculation, the

target remains circular over time.

21

0. 0 0. 1 0. 2 0. 3 0. 4 0. 5

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

t

φ/φmax

ε= 0.0

ε= 0.2

ε= 0.4

ε= 0.6

ε= 0.8

FIG. 28. Angle φin the center of the ﬁlm scaled by φmax

as a function of the dimensionless time when the backﬂow

is neglected. Simulations were performed with n= 5 for

diﬀerent values of the elastic anisotropy ε. Angle φmax =

2πnmax corresponds to the initial value of φat the start of

the unwinding. nmax is given in Fig. 21.

0. 0 0. 2 0. 4 0. 6 0. 8

ε

0.12

0.13

0.14

τ1/ 2

FIG. 29. Half-unwinding time as a function of the elastic

anisotropy when the backﬂow is neglected. These results have

been obtained by taking n= 5 but it should be emphasized

that they depend very little on this value.

2. Numerical solution in anisotropic elasticity and in the

absence of ﬂow

Secondly, we investigated the role of the elastic

anisotropy during the unwinding, in the absence of back-

ﬂow. As the anisotropy makes the problem no longer

tractable analytically (angle φnow depends on rand the

polar angle θ), we solved the torque equation numerically.

The resolution was done by using the same method as for

the winding. We ﬁrst calculated the target at equilibrium

by taking n= 5 and we then calculated its time evolu-

tion when the ﬂux of ethanol was stopped. Calculations

0. 0 0. 1 0. 2 0. 3 0. 4 0. 5

t

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

φ/φmax

=0v

≠ 0v

−1.0 −0.5 0. 0 0. 5 1. 0

−1.0

−0.5

0.0

0.5

1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

(a)

(b)

FIG. 30. (a) Normalized phase as a function of the di-

mensionless time calculated when ~v = 0 (dashed line) and

~v 6= 0 (solid line) by taking a4= 1 and ε= 0. At time

t= 0, φ=φmax = 10π(5 turns); (b) Velocity ﬁeld when

φ= Φmax/2 = 5πwith the target pattern in background.

showed that the target remained almost circular during

the relaxation. The curves of the ratio φ/φmax measured

in the center of the target as a function of the dimension-

less time are shown in Fig. 28 for diﬀerent values of the

anisotropy ε. The corresponding half-unwinding times

τ1/2are reported as a function of εin Fig. 29. These

graphs show that the larger the anisotropy, the slower

is the relaxation. This eﬀect is nevertheless not so large

since τ1/2varies from 0.12 to 0.135 when εvaries from

0 (KB=KS) to 0.8 (KB= 9KS). Our conclusion is

that the anisotropy plays a negligible role during the un-

winding, as during the winding. For this reason, we will

neglect anisotropy in the following.

22

3. Numerical solution in isotropic elasticity and in the

presence of ﬂow

We have seen that ﬂows play an important role dur-

ing the winding of a target. In this case, however, the

ﬂow induced by the chemohydrodynamic force add to the

backﬂow induced by the rotation of the director. This can

lead, when they act in opposite directions, to recircula-

tion vortices responsible for the complex trajectory of the

particle observed in Fig. 12. The situation is simpler dur-

ing the unwinding because only the backﬂow is present.

This case was already analyzed in detail by others, in

particular by Stannarius et al. [62] and Pieranski et al.

[39, 60, 61]. In practice, we used the same code as before

and simulated the unwinding of a target by taking a4= 1

and starting from a target at equilibrium with n= 5. We

observed that during the unwinding, the initial oval de-

formation of the target in the center of the ﬁlm rapidly

disappeared as we can see in the simulation of Fig. 30(b)

showing the velocity ﬁeld and the target in background

when it is half unwound. This is well observed exper-

imentally (see the inset in Fig. 17) and due to the fact

that the ﬂows induced by the rotation of the director are

also circular. Because the ﬂuid and the director rotate in

the same sense (here clockwise), the unwinding process is

faster when the backﬂow eﬀects are present. This is clear

in Fig. 30(a) where the relaxation curves of the phase in

the center of the ﬁlm are plotted with and without back-

ﬂow. Note that in this example, τ1/2passes from 0.15 to

0.12 when the backﬂow is taken into account.

Rather than showing more numerical results, we show

in the next paragraph that it is possible to make a more

complete analytical calculation of this solution without

making any assumptions about the viscosities, on the

condition of assuming that the target and the stream-

lines are circular, as suggested by the numerics and the

experiments.

4. Approximate analytical solution

In this paragraph, we neglect the elastic anisotropy

(ε= 0) but we make no assumptions about the viscosi-

ties which we normalize with viscosity γ1as before. On

the other hand, we assume that the phase φand the ve-

locity ~v do not depend on the polar angle θand are only

functions of rand t. Under these conditions the general

equations simplify considerably. Again, we use dimen-

sionless variables in this paragraph.

The ﬁrst equation is the torque equation. In the gen-

eral case, it is written in the form

Dφ

Dt= ∆φ

+1

2vθ

r+∂vθ

∂r −γ

2cos [2(θ−φ)] ∂vθ

∂r −vθ

r(48)

If φand ~v are independent of θ, then Dφ

Dt=∂φ

∂t because

0. 0 0. 2 0. 4 0. 6 0. 8 1. 0

r

0

1

2

3

4

5

φ/2

π

β=−0.1

β=0

β= 0.2

β=∞

FIG. 31. Phase proﬁles analytically calculated at t= 0 when

φ= 10πat the center of the ﬁlm for diﬀerent values of ¯

β. The

elastic anisotropy is neglected (ε= 0). The proﬁles φ(r, t) at

time tare obtained by multiplying these proﬁles by e−t/τ .

the ﬂuid particles are advected in a zone where the direc-

tor keeps the same orientation. In addition, the term in

cos[2(θ