P Oswald

Role of an oscillatory electric field on the Lehmann rotation of cholesteric droplets

Keywords: . 61.30.Jf Defects in liquid crystals -61.72.Lk Linear defects: dislocations, disclinations -81.40.Lm Deformation, plasticity, and creep

This paper deals with the Lehmann rotation of banded cholesteric droplets subjected to a temperature gradient when they coexist with their own isotropic liquid. I show that their angular rotation velocity increases -in absolute value-when they are subjected to an additional AC electric field in the conducting regime. This velocity increase is correlated with a prolate distortion of the droplets and the probable presence of electrohydrodynamical toroidal circulation flows inside and outside the droplets. I propose that the coupling between these flows and the director field is responsible for the increase of the angular velocity of the texture. The origin of these flows is discussed qualitatively in the framework of the leaky dielectric model by taking into account the generation of charges both in the bulk via a Carr-Helfrich mechanism (Tarasov, Krekhow and Kramer model) and at the surface of the droplet (Taylor-Melcher model).

Introduction

A cholesteric phase is a chiral nematic phase spontaneously twisted in one space direction [START_REF] De | Gennes The physics of liquid crystals[END_REF][START_REF] Oswald | Pieranski Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]. The Lehmann effect is the continuous rotation of cholesteric droplets subjected a e-mail: patrick.oswald@ens-lyon.fr to a temperature gradient. Discovered in 1900 by Otto Lehmann [START_REF] Lehmann | [END_REF], this effect was re-observed recently both when the droplets coexist with their own isotropic liquid [4][5][6] and when they are dispersed in a different liquid that partly dissolves the liquid crystal (LC) [7]. The first explanation of the Lehmann effect was given by Leslie in 1968 in terms of thermomechanical coupling [8]. Unfortunately, this explanation fails to explain the more basic experimental observations both in the coexistence region [9,10] and in the emulsions (for a review see Ref. [11]).

Other models were recently proposed such as the meltinggrowth model to explain the rotation in the coexistence region [12] and a pure hydrodynamical model to explain the rotation in the emulsions [13]. Despite these efforts, the Lehmann effect is not yet completely understood [11] and further experimental investigations are still desirable.

For this reason, I studied the role of an electric field on the Lehmann rotation when the latter is applied in addition of the temperature gradient. My motivation came from a recent observation by Auernhammer et al. about the deformation of nematic droplets in an AC electric field [14]. These authors showed that 'strong' flows can develop within the isotropic liquid around the droplets, leading to their deformation. These flows resemble very much the Marangoni flows observed around the droplets in the Lehmann experiment with emulsified cholesterics [11,15]. Because these flows are responsible for the droplet rotation in that case, one can expect that a similar effect exists in the coexistence region when an AC electric field is applied to the droplets. The goal of this paper is to check this prediction. I mention that, surprisingly and to the best of my knowledge, no systematic experiments were performed so far with an AC electric field. On the other hand, several experiments were conducted with a DC field, first in In-dia [16,17] and more recently in Russia [18,19]. The result of all these experiments was that a DC electric field can make the cholesteric droplets rotate provided it is large enough. The rotation under DC field was initially explained in terms of electromechanical coupling [16,17] by analogy with the Leslie thermomechanical coupling [START_REF] De | Gennes The physics of liquid crystals[END_REF] but it was shown later that this coupling is much too small to explain the observations [20]. Another explanation was proposed recently by Tarasov et al. [21][START_REF] Tarasov | Rotation of cholesteric drops under dc electric field[END_REF][START_REF] Tarasov | Stuctural Transitions and Dynamics of Liquid Crystals under Flows and Electric Fields[END_REF][START_REF] Skaldin | [END_REF] in the frame-

Experimental details

The experimental setup used to impose the temperature gradient is described in Ref. [4]. In brief, the sample is sandwiched between two transparent ovens regulated in temperature within 0.01 • C. A Leica Laborlux 12 Pol polarizing microscope equipped with a Hamamatsu C4742-95 digital camera is used to visualize the droplets. The temperature gradient G across the LC layer can be calculated from the temperature difference ∆T between the top and bottom ovens by using the formula G(K/m) ≈ 1750 ∆T [4].

The samples are prepared between two transparent ITO (Indium Tin Oxide) electrodes covered with a ∼ 100 nm-thick polymercaptan layer. This layer imposes a sliding planar anchoring to the cholesteric phase [28][29][30] which almost completely dewets in the coexistence region in the presence of the isotropic liquid. This dewetting favors the formation of droplets that have the shape of slightly truncated spheres as revealed by confocal microscopy [6]. Two Nickel wires are used as a spacer to fix the sample thickness which is precisely measured with a USB4000 Ocean Optics spectrometer. The LC used is the 8CB (4-Octyl-4-biphenylcarbonitrile from Frinton Laboratories, USA) doped with 1.5wt% R811 (R-(+)-4-(2-methyl butyl)phenyl-4-hexyloxybenzoate from Merck, Germany).

The cholesteric pitch at the solidus temperature is measured by using the Cano-wedge technique [START_REF] De | Gennes The physics of liquid crystals[END_REF][START_REF] Oswald | Pieranski Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]: P ≈ 6 µm (right-handed cholesteric phase). The ionic impurity TBAB (tetrabutylammonium bromide from Sigma-Aldrich) is used to dope the LC and increase its conductivity and its charge relaxation frequency. A LCR meter HP 4284A is used to measure the electric impedance of the samples from which the dielectric constants and the charge relaxation frequencies are deduced (Appendix A). Measurements are performed with 5 different samples. Only sample 4 is doped with 0.11wt% TBAB. For each sample the thickness and the charge relaxation frequencies in the isotropic (f I r ) and cholesteric (f ⊥ r , f r ) phases are given in Table 1. All the samples are sealed with NOA 61 UV glue. In spite of this precaution, the samples degrade in contact with the glue and the polymercaptan, their freezing range slowly increasing in time, mainly during the first two weeks. On the other hand, their charge relaxation frequencies do not change much (by less than a factor of 2), even after three months, meaning that the polymercaptan layer and the UV glue do not introduce large amounts of ionic impurities.

Results

In this paper, I only analyze the rotation of the banded droplets which are by far the most numerous. In these droplets the helical axis is perpendicular to the bands and to the temperature gradient (for a complete description of their texture, see [31]). These droplets form spontaneously in the region of coexistence with the isotropic liquid, when the temperature is close to 38 -39 • C. Under the action of the temperature gradient, the droplet (more rigorously, the droplet texture) rotates at constant angular velocity ω, clockwise (ω < 0) when the temperature gradient points upwards (∆T > 0) and anticlockwise (ω > 0) when the temperature gradient is reversed (∆T < 0), which is expected for a right-handed cholesteric LC (P > 0). Experimentally, ω depends on many parameters: the temperature gradient G (or the difference temperature ∆T ), but also the droplet radius R and the sample thickness d The three white dashed circumferences have a radius of 9.9 µm and the black dashed circumference has a radius of 10.7 µm.

These photos show that the droplet radius increases when a low-frequency field is applied and does not change, within the experimental accuracy, when a high-frequency field is applied.

Note in addition that the contrast and the periodicity of the bands change little when the field is applied, which indicates that the texture remains globally unchanged. The fact that the droplet recovers its initial radius when the field is switched off also indicates that the volume of the droplet does not change when the field is applied. Photos taken with sample 3.

(for a review, see [11]). In the present experiments, an AC

electric field E = √ 2V
d cos(2πf t) where V is the applied voltage given in Vrms and f the frequency, is also applied parallel to the temperature gradient. In this case, I show that the droplet rotation also depends on the electric field under certain conditions. I note now that in all of my ex-periments, the applied voltage is smaller than the critical voltage necessary to unwind the helix (around 6.5 Vrms in a 20 µm-thick sample and twice more in a 40 µm-thick sample) [START_REF] De | Gennes The physics of liquid crystals[END_REF][START_REF] Oswald | Pieranski Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]. This is important to not change too much the internal texture of the droplets and the band periodicity inside. In the following, I analyze separately the effects of the different parameters, by starting with the electric field which is the focal point of my study.

Role of the electric field

The main observation of this paper is that the droplets flatten and rotate faster when they are subjected to a low-frequency AC electric field. By contrast, they remain unchanged and rotate at the same velocity as with no field when a high-frequency field is applied. In the following, this velocity is denoted by ω. The change of shape is evidenced in Fig. 1

D s = (R 1 -R 2 )/(R 1 + R 2 ) (with R 1 and R 2
being the principal radii of the droplet in the vertical and horizontal directions) as a function of frequency f under constant voltage (4 Vrms) for droplets of radius 10 µm in sample 3. The deformation is calculated from formula

D s = 1 -(R/R ∞ ) 3 1 + (R/R ∞ ) 3 (1)
where R ≡ R 2 is the apparent radius of the droplet measured on the photos and R ∞ its radius measured at high frequency. This formula is obtained by noting that 1) the droplet volume V does not change when the field is applied and 2) the droplet remains quasi-spherical [6] with the same radius as with no field at high frequency [31] (see Fig. 1). As a consequence 1) is deduced. To confirm that the cutoff frequency is well given by the charge relaxation frequency, I perform similar mea- the conducting regime. I find that these two quantities increase (in absolute value) quadratically with the applied field. This is shown in Fig. 4 for droplets of radius R = 10 µm subjected to a temperature gradient ∆T = 2 • C.

V = 4 3 πR 3 ∞ = 4 3 πR 1 R 2 , yielding R 1 /R = R 3 ∞ /R 3 from which Eq. (

Role of the droplet radius

In previous studies on the Lehmann effect, the rotation velocity of the banded droplets was found to decrease quadratically with their radius [11]. Does this dependence

Role of the temperature gradient

It is known that the velocity ω is proportional to the temperature gradient [11]. Is it the same for the velocity jump

∆ω? To answer this issue, I systematically measure ∆ω in samples 1-4 as a function of the temperature difference ∆T for droplets of radius R = 10 µm. The result is shown in Fig. 7. This graph shows that ∆ω is odd in ∆T , meaning that the rotation velocity reverses when the temperature gradient is reversed. On the other hand, the absolute value in the isotropic liquid around the droplets. The fact that all the droplets rotate in the same direction when a temperature gradient is applied now explains easily. Indeed, imposing a temperature gradient forces the droplets to nucleate on the cold plate. As a consequence, all the droplets are in contact with the bottom plate when ∆T > 0 and rotate clockwise, while they are all in contact with the top plate and rotate anti-clockwise when ∆T < 0.

I also find that the deformation D 0 s measured in the conducting regime is independent of the temperature gra- This shows that ∆ω increases quadratically with the electric field.

dient within the experimental errors. This is shown in Fig. 6 and in Fig. 10.

In conclusion, the velocity increase ∆ω and the deformation D 0 s are mainly due to the electric field, the temperature gradient only selecting the sense of rotation of the droplets by just imposing on which plate they are in contact.

Role of the sample thickness

It is already known that the velocity ω of banded droplets is independent of the sample thickness [32]. I observe the same in the present experiments. However, the velocity increase ∆ω should depend on the thickness if it is due to the flows observed by Auernhammer et al. [14]. Indeed, the thicker the sample, the stronger must be the flows, and the faster the droplets should rotate, if this interpretation is correct. To check this point, I prepare a sample twice thicker than the rest (sample 5 of thickness 40 µm, table 1) and I observe that ∆ω still changes quadratically with the applied voltage (Fig. 9). I then measure D 0 s as a function of for the increase of the rotation velocity of the droplets under the electric field in the conducting regime. The origin of these flows is discussed in the next section.

Discussion

The first remark concerns the symmetries. Indeed, it has been shown in Ref. [31] that the director field of a spherical banded droplet is invariant with respect to a π rotation about a horizontal axis parallel or perpendicular to the bands. For this reason, such a droplet cannot rotate in average under an AC electric field if it is at equal distance from the electrodes. The only possibility to observe a rotation is to break this C 2 symmetry. This is indeed the case experimentally because the droplets are always in contact and, for this reason, slightly truncated, with one of the two plates limiting the sample.

The next question concerns the origin of the rotation. For the pure Lehmann effect (velocity ω), I refer to Ref. [11,12] for a discussion on its origin. As for the velocity jump ∆ω observed under electric field in the conducting regime, the present experiments suggest that it is due to the EHD flows. This is quite plausible since the velocity field couples to the director field to make the texture rotate with the angular velocity:

ω = drop (γ 2 An + γ 1 [(v • ∇)n -Ω × n]) • δn δθ d 3 r drop γ 1 ( δn δθ ) 2 d 3 r . (2)
In this formula, δn δθ = ∂n ∂θ -e z × n, by denoting by e z the unit vector along the vertical z-axis and θ the polar angle in cylindrical coordinates (r, θ, z), Ω = 1 2 ∇×v is the local rotation rate, A is the symmetric strain rate tensor of component A ij = 1 2 (v i,j + v j,i ) and γ 1 ≡ α 3 -α 2 and γ 2 ≡ α 3 + α 2 are two viscosities of the cholesteric phase (with α 2 and α 3 two of the five Leslie viscosity coefficients [START_REF] De | Gennes The physics of liquid crystals[END_REF][START_REF] Oswald | Pieranski Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]).

This general formula is derivated in Ref. [13] and remains valid when an electric field is applied as can be easily checked by adding the electric energy to the elastic and surface anchoring energies in the derivation and by replacing the elastic molecular field h by h + h E , where

h E = ε 0 ( -ε ⊥ )(n × E)E
is the so-called electric contribution to the molecular field. Note that the flexoelectric terms disappear under AC electric field because they are linear in E, and so they vanish on average over time.

In practice, the flows can have two origins, depending of the type of charges considered. G Fig. 12. Sketch of the flow field around a droplet observed by Auernhammer et al. by using tracer colloids (from Ref. [14]).

In the model of Tarasov, Krakhov and Kramer [21,[START_REF] Tarasov | Rotation of cholesteric drops under dc electric field[END_REF] the negative and positive charges separate in the bulk of the cholesteric phase in the conducting regime because of the anisotropy of the electrical conductivity. The sample remains electrically neutral in average. This is the Carr-Helfrich effect, responsible for the convective instabilities in a nematic slab under electric field in the conducting regime [START_REF] De | Gennes The physics of liquid crystals[END_REF][START_REF] Oswald | Pieranski Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]. Under the action of the electric field, these charges experience a Coulomb force, which generates flows in the cholesteric phase. These flows are responsible for the drift of cholesteric fingers under AC electric field in the conducting regime [21,33] and could explain, at least in part, the velocity jump ∆ω in the present experiments.

This model is plausible as it predicts that v -and so ∆ω according to Eq. ( 2)-must change as [21,[START_REF] Tarasov | Rotation of cholesteric drops under dc electric field[END_REF] ∆ω

∝ E 2 1 + f 2 /f 2 r (3)
where f r is the typical charge relaxation frequency. Such a dependence on f is indeed observed experimentally (see Fig. 2(a)), as well as the quadratic dependence on the electric field (see Fig. 4(a) and Fig. 9).

On the other hand, this model ignores the deformation of the droplets. The latter is due to the presence of charges at their surface. Indeed, according to the Taylor-Melcher model, positive and negative charges also appear on the surface of the droplet, with a net charge remaining equal to 0 as in the bulk. The action of the field on these surface charges generate electric stresses across the cholestericisotropic interface, which not only leads to deformation of the droplets but also generates flows in and around the droplets, which should add to the flows created by the bulk charges.

The flow out of the droplets was observed experimentally by Auernhammer et al. [14] and is schematically sketched in Fig. 12 when the temperature gradient points upwards (droplet in contact with the bottom electrode).

In this case, the direction of flow is the same as in the cholesteric emulsion experiments reported in Ref. [13] when the temperature gradient points upwards, which was shown to produce a rotation in the same sense as in the usual Lehmann effect. In practice, the confinement of the droplets between the two electrodes must alter the distribution of the electric field and the associated flows. This is shown in recent numerical simulations of confined two-dimensional isotropic droplets [34]. This effect could explain why the deformation D 0 s and the velocity jump ∆ω depend on the thickness in my experiments.

D s = 9ε 0 ε I 16γ Φ s RE 2 (4)
valid for a non-confined droplet in a uniform electric field.

In this formula, ε 0 is the permittivity of free space, γ is the surface tension and Φ s is given by This good agreement is certainly a coincidence. However, there is no doubt that this model leads to the correct order of magnitude and is pertinent, as already underlined by Auernhammer et al. [14].

Φ s = 1 - T (11M + 14) + T 2 [15(M + 1) + S(19M + 16)] + 15T 2 (1 + M )(1 + 2S)f 2 /f I 2 r 5(1 + M )[(2T + 1) 2 + T 2 (S + 2) 2 f 2 /f I 2 r ] (5
This model also predicts that the maximum velocity at the surface of the droplet is frequency dependent, given by [27]

v M = 9ε 0 ε I E 2 R 20µ I (1 + M ) T (T S -1) (2T + 1) 2 + T 2 (S + 2) 2 f 2 /f I 2 r (6)
The curve of v M as a function of f is shown in Fig. 13(b).

This curve shows the same behavior in frequency as the experimental curve for ∆ω, which is expected since ∆ω is proportional to v M according to Eq. ( 2). The velocity is also of the order of a few µm/s, which is compatible with the observations of Auernhammer et al. [14]. As a consequence, the Taylor-Melcher model could also explain the rotation of the droplets under AC electric field in the conducting regime.

Another crucial point is that, in the two models, only the conductivity ratios enter into the calculations. If all the conductivities change in the same way with the concentration of ionic impurities, which is likely at small concentrations, this could also explain why the deformation D 0 s and the velocity jump ∆ω are experimentally independent of the concentration of ionic impurities (see Fig. 3).

For the moment, it is difficult to say which mechanism is the most important. However, there is no doubt that the surface charges are important to explain the deformation of the droplets. The confinement effects are also important to explain the thickness dependence of both the deformation and the rotation velocity of the droplets.

In the future it would important to solve numerically this problem by taking into account these two mechanisms. This is a very difficult task because of the com- becomes sliding under UV [36]. It would also be interesting to make the dielectric anisotropy of the LC vanish, for instance by mixing MBBA with a small amount of 8CB (or 5CB [37]), to avoid texture changes under electric field. Finally, it would be interesting to visualize the flows by doping the LC with colloids, as in the experiments of Auernhammer et al. [14]. Such experiments are planed for the future. is then fitted with formulas temperature. In the latter case, measurements are performed at low voltage (0.5 Vrms) when the cholesteric phase is in planar orientation (director perpendicular to the field) and at large voltage (20 Vrms) when the cholesteric phase is completely unwound and in homeotropic orientation (director parallel to the field). From these measurements, I obtain the values of f r , ε and σ in the isotropic liquid (f I r , ε I and σ I ) and in the cholesteric phase under planar (f ⊥ r , ε ⊥ and σ ⊥ ) and homeotropic (f r , ε and σ ) orientations. For 8CB, I find that in all samples ε ⊥ ≈ 7.5, ε ≈ 13 and ε I ≈ 9.6. These values compare well with the ones given in the literature [38,39]. The values of f r for the different samples are given in Table 1. Contrary to the dielectric constants, they depend on the purity of the sample and its content in ionic impurities.

R = R IT O + R p 1 + 4π 2 R 2 p C 2 p f 2 + R D 1 + 4π 2 R 2 D C 2 D f 2 (A.2) X = - 2πR 2 p C p f 1 + 4π 2 R 2 p C 2 p f 2 - 2πR 2 D C p f 1 + 4π 2 R 2 D C 2 D f 2 (A.

  work of an electrohydrodynamical (EHD) model. In this model, charges are generated in the bulk of the droplets due to the local divergences of the flexoelectric polarization. Under a DC electric field, the Coulomb force acting on these charges generates flows inside the droplets, making their texture rotate. This model predicts a linear dependence between the rotation velocity and the field, which is observed experimentally, and leads to rotation velocities that are compatible with experiments. By contrast, this model based on flexoelectricity cannot lead to a continuous rotation of the droplets under AC field because of its linear dependence on the field. The plan of the paper is as follows. In Section II, I give the details about the experimental setup and the samples used in this study. The experimental results about the droplet rotation velocity are given in Section III. In this section, the role of frequency and amplitude of the electric field is analyzed as well as the role of the concentration of ionic impurities, droplet diameter, temperature gradient and sample thickness. These results are then discussed in Section IV in the framework of two models. The first one is due to Tarasov, Krehkov and Kramer and only considers the bulk charges generated by the Carr-Helfrich mechanism [1, 2] associated with the anisotropic conduction in the cholesteric phase. The second one is based on the Taylor-Melcher model [25-27] and only accounts for the surface charges. Conclusions are drawn at the end of this section and new experiments are proposed to test the relative importance of these two models.

Fig. 1 .

 1 Fig. 1. Same banded droplet photographed in natural light when I successively impose V = 0 (a), V = 4 Vrms, f = 10 000 Hz (b) and V = 4 Vrms, f = 100 Hz (c) and V = 0 again (d). Note that in the meantime the droplet has rotated.

  showing the same droplet before and after the application of a low and high-frequency electric field. The change of velocity is shown in video S1. At the beginning of the movie, V = 0 and the droplet rotates slowly clockwise under the sole action of the temperature gradient (∆T = 2 • C). At time t = 300 s, a low-frequency field of amplitude 4 Vrms and frequency 100 Hz is applied. In doing so, the droplet flattens and rotates faster, always clockwise. Finally, the frequency is switched from 100 Hz to 10 kHz at time t = 600 s while maintaining an amplitude of 4 Vrms. Under this condition, the droplet recovers its initial radius and rotates clockwise as if no field is applied. To better quantify this phenomenon, I systematically measure the angular velocity and the droplet deformation

Fig. 2 (

 2 Fig. 2(a) shows that the angular velocity is constant at high and low frequency (with a tendency to decrease at very low frequency, in particular in the sample doped with DTAB 1 ) and is almost 9 times larger at low frequency than at high frequency. In this experiment, the cutoff frequency clearly coincides with the charge relaxation frequency, the values of which in the two phases are indicated by the vertical arrows in the figure. Fig. 2(b) shows that this velocity change is also clearly correlated with a deformation of the droplets into a prolate shape (D s < 0) in the lowfrequency regime.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Angular velocity ω (a) and droplet deformation Ds (b) as a function of frequency f . ∆T = 2 • C, V = 4 Vrms and R = 10 µm and d = 21 µm. The three arrows from left to right indicate, respectively, the values of f r , f ⊥ r and f I r -measured the same day for more confidence. Sample 3 was used for these measurements. The solid lines are fits with a law of the type a + b/(1 + cf 2 ) where a, b and c are the fit parameters.

Fig. 4 .

 4 Fig. 4. Angular velocity ω (a) and droplet deformation D 0 s (b) of droplets of radius R = 10µm as a function of the applied voltage V in the conducting regime at f = 100 Hz. A small temperature gradient is applied (∆T = 2 • C). Sample 3 was used for these measurements. The solid lines are the best fits with a law of the type a + bV 2 , where a and b are the fit parameters.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Angular velocity as a function of the radius when V = 0 (ω) and V = 4 Vrms, f = 100 Hz (ω0). The solid lines are the best fits with a 1/R 2 law. Sample 1 with ∆T = 2 • C.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Velocity jump ∆ω as a function of the temperature difference ∆T measured in samples 1-4 of thickness 20 µm at V = 4 Vrms.

Fig. 9 .

 9 Fig. 9. Velocity ω measured with droplets of radius R = 10 µm in sample 5 of thickness ∼ 40 µm as a function of the applied voltage V at f = 100 Hz and ∆T = 5 • C. The best fit (solid line) with a power law of the type a+bV ν gives ν = 2.06±0.16.

Fig. 10 .

 10 Fig. 10. Deformation D 0 s as a function of the droplet radius R measured for three different temperature gradients at V = 8 Vrms and f = 100 Hz in sample 5 of thickness ∼ 40 µm. The solid red line is the best linear fit passing through the origin of the full data set for R < 18 µm. The vertical line indicates the value of the half sample thickness. The points on the right of this line correspond to confined droplets. The dashed blue line shows the same curve obtained with sample 3 of thickness ∼ 20 µm under the same electric field (V = 4 Vrms and f = 100 Hz) and similar temperature gradients.

RFig. 11 .

 11 Fig. 11. Absolute value of the velocity jump |∆ω| measured with droplets of radius R = 10 µm in sample 5 of thickness ∼ 40 µm as a function of the absolute value of the temperature difference |∆T | at V = 8 Vrms and f = 100 Hz. Within the experimental errors, |∆ω| is independent of |∆T |, of average value (solid red line) about twice larger than that measured in the 20 µm-thick samples (dashed blue line) at V = 4 Vrms and f = 100 Hz. Each error bar indicates the standard deviation over a dozen measurements.

Fig. 13 .

 13 Fig. 13. Deformation Ds (a) and maximum velocity vM (b) predicted by the Taylor-Melcher model by taking the values of the parameters corresponding to sample 3. In the graphs, the curves from top to bottom have been calculated by taking M = 0.5, 1, 2, 10. The three arrows from left to right indicate the values of f r , f ⊥ r and f I r , respectively.

  It is also important to note that the Taylor-Melcher model predicts the good order of magnitude for both D s and the velocity, of the order of a few µm/s according to Auernhammer et al.[14]. To check this point, I plot the curve of the deformation D s as a function of frequency f in Fig.13(a) for a droplet of radius R = 10 µm under an electric field E = V /d = 2 10 5 V/m (corresponding to V = 4 Vrms when d = 20 µm), by using the formula[27]

  ) where S ≡ ε chol /ε I , T ≡ σ I /σ chol , M ≡ µ chol /µ I with µ the viscosity, and f I r = σ I /2πε 0 ε I . To plot this curve I assume that ε chol ≈ (ε + ε ⊥ )/2, σ chol ≈ (σ + σ ⊥ )/2 and I use the values of the parameters measured in sample 3, viz. ε = 13, ε ⊥ = 7.5, ε I = 9.6 and σ = 6.37 10 -7 S/m, σ ⊥ = 4.8 10 -7 S/m and σ I = 6.14 10 -7 S/m. The surface tension is given by Faetti and Palleschi in 8CB : γ = 0.95 10 -5 N/m [35]. It turns out that the curve calculated with Eqs. (4) and (5) depends little on the value of the viscosity ratio M and is very close to the experimental curve of Fig. 2(b) in spite of the fact that these formulas do not take into account the confinement of the droplet.

  plexity of the equations. A way to simplify the problem would be to calculate separately the velocity fields due to the Carr-Helfrich and Taylor-Melcher mechanisms. The contribution of each of them to the rotation velocity could then be calculated from the general equation (2) and compared. In these calculations, an additional simplification would be to consider that the director field is the same as at equilibrium, without flow.It would also be interesting to perform the experiments with other liquid crystals. Indeed, the Taylor-Melcher model predicts that depending on the values of the ratios of the dielectric constants and conductivities, the electric field can deform a droplet into a prolate or oblate ellipsoidal shape. With 8CB, an oblate shape is observed and the flows make the texture rotate in the same direction as in the Lehmann effect. As a consequence the two effects add.By contrast, a prolate shape should produce an opposite effect since the velocity changes sign. In that case, the electric field should make the texture rotate in the opposite direction of the Lehmann effect. It turns out that I observe such a phenomenon with a sample of MBBA (pmethoxybenzylidene-p'-butylaniline) doped with 1.5 wt% R811. In this mixture, the droplets deform very little but slow down and start to rotate in the opposite direction when an increasing electric field is applied. Unfortunately, the experiments with MBBA are complicated because the texture of the droplets change under electric field, the helix tending to align along the field because of the negative dielectric anisotropy of the LC. For this reason, it is difficult to determine whether this effect is due to a texture change or to a reversal of the flows. In addition, the polymercaptan dissolves in MBBA, so that the transition temperature of the samples constantly decreases. For this reason, the droplets melt during the measurements, which become very imprecise. A solution to solve this issue could be to change the surface treatment, for instance by using the azobenzene polymer poly(ethylene oxide-b-11-[4-(4-butylphenylazo)phenoxy]-undecyl methacrylate which

Fig. A. 1 .

 1 Fig. A.1. Electric representation of the sample. The LC layer is modeled by a resistance Rp in parallel with a capacitance Cp. The resistance RD and the capacitance CD are introduced to simulate the effect of the Debye layers that form at low frequency close to the electrodes [40]. The two ITO layers are replaced by the resistance RIT O .

3 )Fig. A. 2 .

 32 Fig. A.2. (a) Complex impedance of sample 3 measured in the planar orientation just below the solidus temperature and their fits with Eqs. (A.2) and (A.3) using a global fit procedure. From the fit parameters RIT O = 100 Ω, Rp = 147 kΩ, Cp = 843 pF, CD = 726 nF, RD = 4.4 kΩ, I calculate f ⊥ r = 1280 Hz and ε ⊥ = 7.6, knowing that C0 = 111 pF; (b) Complex impedance of sample 4 (doped with TBAB) measured in the isotropic liquid just above the transition and their fits with Eqs. (A.2) and (A.3) using a global fit procedure. From the fit parameters RIT O = 80 Ω, Rp = 1.96 kΩ, Cp = 1074 pF, CD = 102 nF, RD = 0.5 kΩ, I calculate f I r = 75600 Hz and εI = 9.6, knowing that C0 = 112 pF.

Table 1 .

 1 Thickness of the 5 samples used and their charge relaxation frequencies measured one day after filling with the LC in the cholesteric and isotropic phases.

	Sample n • d(µm) f r (Hz) f ⊥ r (Hz) f I r (Hz)
	1	21	1700	1300	2150
	2	21.5	1150	881	1650
	3	21.6	1280	1000	1590
	4	21.8	66700	49400	75600
	5	39.8	1330	1000	1647

This is certainly due to the formation of Debye layers near the electrodes that screen the field, as evidenced by the dielectric measurements shown in the Appendix.

Acknowledgements P. O. warmly thanks G. Poy, A. Dequidt and J. Ignés-Mullol for their critical reading of the manuscript.

A Measurement of the charge relaxation frequency

In the leaky dielectric model, the liquid crystal is treated as a dielectric medium with an Ohmic resistance. Let ε be the dielectric constant and σ the conductivity of the LC. To obtain C p and R p , the complex impedance R + jX of the sample is measured as a function of frequency f and

The charge relaxation frequency


The impact of international proceedings for bypassing the State: the example of Madagascar he transitional crisis realism invented for Madagascar created an international precedent in humanitarian diplomacy, especially for Africa. As a result of the political and diplomatic crisis which began in 2009, international financial sanctions were imposed on the country and, from 2009 to 2014, the only international funding authorised was humanitarian aid1 . However, this aid was substantial enough to affect Madagascar's money supply and so international stakeholders at multiple levels employed unprecedented proceedings to ensure it bypassed the State. The systems put in place were maintained after the end of the diplomatic crisis and even after the presidential elections in 2014. The comparative political analysis 2 made here seeks to answer the following crucial question: what "governance situation" have these bypass systems created 3 and what is its impact on international financial relations and crisis recovery prospects?

The substitution of governance Madagascar's transitional government was not recognised by the international community. Consequently, international partners pursued strategies for bypassing the State that were clearly intended to limit its decision-making and budgetary capacity. In so doing, international governance engaged in a process of delegitimising the nation State's humanitarian function and de-institutionalising its humanitarian coordination, normally carried out by the Bureau National de Gestion des Risques et Catastrophes [National Office of Disaster Risk Management (BNGRC)]. Thus, from a "financial transition" 4 point of view:

"2009 [was] a turning point in terms of the loss of capacity of State institutions through the suspension or cancellation of development assistance. There were no longer any high-level political interactions for managing cooperation in Madagascar, only direct technical exchanges for implementation…" 5 .

Moreover, the budgetisation of authorised humanitarian funding transformed 30 % of what had become prohibited Official Development Assistance into humanitarian aid. 6 And lastly, regional funding was only authorised if it was reclassified as bilateral humanitarian assistance. As a result, humanitarian aid was allocated directly to non-State and/or community actors which then channelled it to the beneficiaries. In other words, the allocation of international humanitarian resources circumvented the State. Indeed, we could say that the diplomatic measure of imposing financial sanctions resulted in the substitution of State governance by international humanitarian governance.

Stakeholder proceedings and popular perception

It all started in 2009 when the United Nations created for Madagascar the "Special situation of development" (SSD). Two types of systems were put in place. One was institutional: "Direct Implementation" by the United Nations Development Programme (UNDP), which maintained the usual amount of aid but paid it directly to the communities. After the presidential elections in 2014, "Direct Implementation" was replaced by "National Implementation" (NIM), which was overseen country-wide by the UNDP. The other system resulted in a de facto shift in the role of the World Food Programme (WFP), which departed from its development agency role and moved into a "humanitarian aid agency mode". Mandated by the United Nations to undertake emergency action, it was solicited by public and private donors, such as the African Development Bank, for example, to act as a humanitarian aid and coordination agency, the role that the BNGRC is normally supposed to assure.

Public and private networks of humanitarian actors organised their own procedures for bypassing the State, putting in place public/private sector-based governance in the form of "clusters" 7 : the ensuing interaction between public and private sector-based aid actions led to the emergence of a non-State coordination structure.

As for international humanitarian networks, their own standard humanitarian governance proceedings circumvented the State-based coordination of the host country, Madagascar. The consequences of this are paradoxical. While international humanitarian networks bypass the Malagasy State, their member countries worked closely with the diplomatic governance of their own States. Stéphane Gompertz explains this paradox by the fact that "NGOs willing to work hand in hand with their respective embassies makes things easier8 ". Bernard Kouchner, the emblematic "French Doctor"9 who was France's Foreign Affairs Minister from 2007 to 2010, contributed to "this governance situation" which was the focus of negotiations of the ad hoc international contact group established by donors to Madagascar. On the other hand this sharpens the stakes in geopolitical rivalry and diplomatic competition. The United States then claimed to be Madagascar's principal donor and principal humanitarian partner -a place that France traditionally claimed for itself.

But what about the beneficiaries? Let us recall that the State bypass measures implemented by international stakeholders were maintained after the presidential elections of 2014 which put an institutional end to the crisis. In order to conduct a post-crisis analysis of the action of the BNGRC and its international networks, in 2016 we thus carried out an unprecedented perceived satisfaction survey of 650 households in three regions exposed to different types of risks and natural disasters 10 .

What comes out is a rather medium range level of satisfaction in the Analamanga region in the centre of the country and in the Atsinanana region to the East. In Androy, in the South, the region with the highest famine and insecurity levels and where the State's hold is weakest, the findings showed blatant dissatisfaction. Indeed, the action of the State and its international partners was seen to be irrelevant and ineffective. It was considered a failure, despite the South being home to the most vulnerable populations and receiving the most aid.

From diplomacy to humanitarian geopolitics

Bypassing the State and imposing international financial sanctions put considerable pressure on Madagascar's transitional government to achieve an electoral outcome to the diplomatic crisis. Admittedly, it also provided a safety net for those populations most exposed to climate, security and political risks. But this approach, maintained throughout the international negotiations to produce a road map for exiting the crisis, continued to weigh heavily on Madagascar's money supply, while remaining a useful political tool for multi-level international actors to influence the governance of the State and the country's political life.

As a result, bypassing the State helped weaken it. Madagascar was categorised as a failed State by many international stakeholders who used this as a diplomatic and geopolitical argument. It also enabled national and international actors or groups of actors to transform humanitarian influence into cronyism. Lastly, this was used by certain groups to constitute the basis required to ensure their political future11 .

Many of these international actors, especially the humanitarian ones, were pleased with the reduction in State capacity, perceived as an opportunity for the population, NGOs and civil society organisations to take charge of their own destiny beyond the control of the State. But why did nobody ever ask the beneficiaries what their opinion was? In the findings of our satisfaction survey, and with no mention on our part of the word "sovereignty", this notion nonetheless occurred very frequently in the responses made by the households interviewed.

The households perceived what our workings brought to light: the new humanitarian funding situation and the status of this funding during the diplomatic crisis reversed the coordination hierarchy. The transitional government could bring no influence to bear on the quota plan (planning and basic stocks in the event of a disaster), so Assia Sidibe's adage became particularly pertinent: "When there's no plan, there's OCHA's plan [START_REF] Sidibe | head of[END_REF] ". According to OCHA's Rija Rakotoson, this doesn't relieve them of uncertainty: "We don't know whether our responses are right, but we try to make them fair [START_REF]Humanitarian Affairs Officer at OCHA[END_REF] ". So is ethics the only protection against the reversal of governance in the context of international financial sanctions and extreme dependence?

In the 1990s, there was much talk about the decline of the State and the new transnational perspectives in the world view and that of humanitarian action [START_REF] Badie | the example of Madagascar", Humanitarian Alternatives, n°5[END_REF] . And what is witnessed today in the case of Madagascar - and can also be seen in other contexts -, in crisis situations, populations still turn spontaneously to the State and local government bodies. This is where their expectations lie. Our survey revealed that the people of Madagascar deplore the weakening of the State's role in humanitarian action, both in terms of loss of proximity and, above all, national sovereignty. They perceive international humanitarian action as being only relatively effective, essentially palliative and also a source of vital dependence.

Heightened humanitarian geopolitical competition

The geopolitical consequences of the reversal in hierarchy between national and international coordination are an erosion of sovereignty and a weakening of State power, corollaries to the predominance of international actors in the humanitarian landscape and all along the allocation process of humanitarian resources. This is indeed a very strong political and financial power in terms of local and global legitimation and delegitimation.

Finally, this political and diplomatic crisis in Madagascar's recent history also revealed a heightening in geopolitical competition for funding between humanitarian actors, which we believe to be the exact opposite of what is required in essential "humanitarian transition", in the sense understood by Jean-François Mattei 15 .

We have seen international humanitarian funding transformed into "soft power" through topdown and bottom-up political and geopolitical competition. It has become this way a driver for the (de)mobilisation of institutional networks on the one hand and political capitalisation through humanitarian action on the other. In Madagascar's case, the regional context intensified this competition: the huge gas deposits discovered in 2008 in the Mozambique Channel 16 and confirmed in 2012, have since been exploited by Mozambique and Tanzania on the basis of their Exclusive Economic Zone (EEZ), while Madagascar is still struggling to recover from the crisis. The natural gas wells that Madagascar could pretend to in the same channel are located within the EEZ of the "Îles éparses" (Scattered Islands) disputed with France 17 .

However, as humanitarian actors have experienced locally and globally, even a weak, poor and dependant State can prevent them from taking action. If it refuses them access to the country, or to fields of action, aid workers are powerless. In other words, States will always have the capacity to determine the scale and effectiveness of international humanitarian aid. To avoid reaching such extremes, of which the populations are often the first to suffer, and to ensure the State plays its rightful humanitarian coordination role in partnership, it is necessary to practice the coconstruction of national and international public humanitarian policies. It is important, to this end, to establish procedures to exit from curtailment of the State so as not to be satisfied with a governance of crisis but to contribute to its resolution. This supposes a re-institutionalisation 18 geared to the fight against the violence of nature and society, to the governance of peace - a common and sovereign asset, and effectively sustainable development.

Christiane Rafidinarivo •

  Sciences Po Paris/Institut d'Études politiques de Madagascar/Université de La Réunion Madagascar was severely hit by a protracted political and diplomatic crisis triggered by the 2009 military coup. The transitional government, in power until 2014, was unable to prevent the State from being bypassed and its humanitarian sovereignty from being transferred to international actors. This has done nothing to improve the situation of a country which is among the world's Least Advanced, ranked the third most vulnerable to climate hazards, behind Bangladesh and India (and the first in Africa) and the second most food-insecure in 2014. Christiane Rafidinarivo explains the mechanisms put in place by these actors between 2009 and 2014, and which are still applied today. She provides a comparative analysis in order to draw political and diplomatic lessons for African countries…

4

  Christiane Rafidinarivo, « Transition politique, transition financière et transition humanitaire : le cas de Madagascar », speech at the Convergences World Forum, Paris, 2016. 5 Maherisoa Rakotonirainy, Vulnerability Assessment Officer for the World Food Programme (WFP), cited by Christiane Rafidinarivo (dir.), Hantsa Ralamboson and Marie-Donna Ranaivoarivelo, « Transition politique et transition humanitaire… », op. cit., p. 22. 6 World bank, « La Gestion des Risques et Catastrophes et l'adaptation au changement climatique », World Bank memorandum, Antananarivo, 2014. 7 "Sector-based groups" put in place in 2005 by OCHA, the UN's Office for the Coordination of Humanitarian Affairs, as part of the humanitarian reform programme. These clusters are groups of humanitarian organisations, both UN and non-UN, in each of the main sectors of humanitarian action, e.g. water, health and logistics. They are designated by the Inter-Agency Standing Committee (IASC). The Resident Coordinator and/or the Humanitarian Coordinator and the Country Team manages the humanitarian response through these clusters. See « OCHA d'une seule voix : l'approche Cluster », mai 2012, www.unocha.org/sites/unocha/files/dms/Documents/OCHA%20on%20Message_Cluster%20Approach_vFR.pdf [editor's note].

  Christiane Rafidinarivo (dir.), production Nianja Raonivelo and Jaotiana Rasolomamonjy, advised by Thierry Sabathier, Ingénieur divisionnaire des travaux géographiques et cartographiques de l'État ["State survey engineer", editor's note], scientific poster exhibition "Political transition and humanitarian transition 2009-2014", association océan Indien de Science politique, 2016.

  Translated from the French by Mandy DuretBiography • Christiane RafidinarivoA Doctor habilitated to conduct research in political science, Christiane Rafidinarivo is an invited researcher at Cevipof (CNRS UMR 7048), Sciences Po Paris. She is visiting professor for the Europe-Africa and Europe-America programmes of Sciences Po. She is president of the

15

Jean-François Mattei, L'humanitaire à l'épreuve de l'éthique, Les liens qui libèrent, 2014.

16

Arm of the Indian Ocean between Madagascar and the rest of Africa [editor's note]. 17 Christiane Rafidinarivo et Johary Ravaloson (dir.), « Regards croisés sur les Îles Éparses : ressources et territoires contestés », Revue Madagascar Conseil International, dossier spécial Îles Éparses, n° 72-73, 2016. 18 Charles Rambolarson, deputy executive secretary of the BNGRC, « L'État et le BNGRC, enjeux d'une institutionnalisation de la coordination », speech at the FCRF and IRD seminar, 2016.

th Issue -July 2017 p.[74][75][76][77][78][79][80][81][82][83][84][85]

Christiane Rafidinarivo (dir.), L'État dans les relations internationales. La communauté internationale face à la crise malgache actuelle, Report to the Institut des hautes études de Défense nationale, association régionale océan Indien, Saint-Denis,

2010, letter of congratulations from Richard Laborde, vice-admiral and IHEDN director. 2 Christiane Rafidinarivo (dir.), Hantsa Ralamboson and Marie-Donna Ranaivoarivelo, « Transition politique et transition humanitaire : analyse politique comparée de la transition financière », Fonds Croix-Rouge française, Les Papiers du Fonds, n° 10, February 2017, www.fondcrf.org/wp-content/uploads/2017/03/Transition-politique-et-transition-humanitaire_Analyse-comparee-de-latransition-financiere.pdf 3 Elinor Ostrom, Gouvernance des biens communs. Pour une nouvelle approche des ressources naturelles, Éditions de Boeck, Paris, 2010.T

Stéphane Gompertz, Head of the African desk at the French Ministry of Foreign Affairs from 2010 to 2012, Fonds Croix-Rouge française, Les états de la recherche, Paris, 15 mars 2017 (speech).

Philippe Ryfman, Une histoire de l'humanitaire, La Découverte, 2016.

Christiane Rafidinarivo (dir.), Hantsa Ralamboson and Marie-Donna Ranaivoarivelo, « Transition politique et transition humanitaire… », op. cit.

Maherisoa Rakotonirainy, in Christiane Rafidinarivo (dir.), Hantsa Ralamboson et Marie-Donna Ranaivoarivelo, « Transition politique et transition humanitaire… », op. cit., p. 22.


Maurice Margenstern

Pascal Mayer
email: pascal.mayer@neuf.fr

Sergey Verlan
email: verlan@u-pec.fr

String Displacement Systems and DNA Circuits Based on Isothermal Constrained Loop Extension DNA Amplification

In this paper, we first describe the isothermal constrained loop extension DNA amplification (ICLEDA), which is a variant of amplification combining the advantages of rolling circle amplification (RCA) and of strand displacement amplification (SDA). Then, we formalize this process in terms of the theory of formal languages, which yields two new operations of sticking and displacement on normal and circular strings. The obtained formalism is quite expressive and we show how to implement the computation of any boolean function on DNA strands using ICLEDA. We also discuss variants of the model that could eventually bring more computational power.

Introduction

The first attempt to use DNA for solving computational problems was done by Adleman [START_REF] Adleman | Molecular computation of solutions to combinatorial problems[END_REF]. Since that time several models of computation using DNA have been proposed, we refer to [START_REF] Amos | Theoretical and experimental DNA computation[END_REF] for an overview. The structure and the properties of DNA molecules allow their natural representation as strings and operations on strings. So, abstracting from the physical constraints, it is possible to perform a language-theoretical study of the obtained models. This study can reveal some interesting properties and limitations that could eventually be translated back to the original DNA molecules. The book [START_REF] Pȃun | DNA Computing: New Computing Paradigms[END_REF] contains several examples of such an approach.

In this article we follow the same line by formalizing the process of a special type of string amplification and displacement in terms of operations on strings. The most common in-vitro DNA strand replication method (also called "DNA amplification") is based on PCR. This method is based on a series of primer extension cycles with changing temperature conditions to allow for strand separation at the beginning of each cycle.

Rolling-circle amplification (RCA) is another method of strand replication based on circular DNA molecules [START_REF] Fire | Rolling replication of short DNA circles[END_REF] and is inspired from the natural replication mechanisms of several viruses [START_REF] Gilbert | DNA replication: the rolling circle model[END_REF]. The important observation is that this method does not require changing the conditions of the test tube for DNA amplification and produces long single stranded DNA molecules including multiple complementary copies of the circular template DNA fragment. This procedure was used in DNA computing as a basis for the simulation of the resolution refutation in [START_REF] Lee | RCA-Based Detection Methods for Resolution Refutation[END_REF].

The strand displacement amplification (SDA) is based on the ability of a restriction enzyme to nick a modified recognition site and the ability of a polymerase to initiate synthesis at the nick and displace a downstream DNA strand during replication [START_REF] Walker | Isothermal in vitro amplification of DNA by a restriction enzyme/DNA polymerase system[END_REF]. Both above methods allow us to produce DNA strands in isothermal conditions. There are methods using both RCA and SDA, for example ramification-extension method (RAM) [START_REF] Zhang | Amplification of target-specific, ligation-dependent circular probe[END_REF].

In this article we consider a new isothermal DNA replication method, called ICLEDA for "Isothermal Constrained Loop Extension DNA Amplification", described in [START_REF] Mayer | Méthode d'amplification d'acides nucléiques et ses applications[END_REF]. It makes it possible to produce short linear and single stranded DNA strands in isothermal conditions. Importantly in the perspective of a practical application, the amplification is also possible when the template molecules are immobilized on a support. We formalize the amplification process in terms of formal languages. Such a formal system is constructed from a number of circular strings, which we call amplification loop complexes (or simply loop complexes) that can be in two states: blocked or unblocked. A loop complex in unblocked state produces infinitely the corresponding DNA strand (signal). The transition from a blocked to an unblocked state is done by annealing and primer extension, formalized as a string displacement operation. The resulting model is quite powerful, for example it is possible to simulate a signaling cascade whose nodes correspond to AND and OR gates. More precisely, using an approach similar to [START_REF] Ogihara | DNA-based self-propagating algorithm for solving boundedfan-in Boolean circuits[END_REF][START_REF] Ahrabian | DNA algorithm for an unbounded fan-in Boolean circuit[END_REF], we consider asynchronous circuits with dual-rail encoding, i.e. a bit is encoded using 2 wires and determined by the transition happening on one of them, exclusively. The amplification to express the presence of signals which can further trigger the amplification of new signals following the circuit in an autonomous way.

A preliminary version of this article can be found in [START_REF] Margenstern | DNA Circuits Based on Isothermal Constrained Loop Extension DNA Amplification[END_REF].

The biological mechanism

In this section, we first describe the ICLEDA amplification process defined in [START_REF] Mayer | Méthode d'amplification d'acides nucléiques et ses applications[END_REF] on which the whole work is based. Then, we show how this mechanism allows us to devise a configuration, which we call the loop complex, which will later on allow us to implement logical gates in this context. Note that the word loop refers to the shape of the biophysical complex we consider rather than the computational device which is usually understood by this term, this is why the term complex is attached to loop in this denomination.

The amplification and the loop

The ICLEDA amplification method designed in patent4 [START_REF] Mayer | Méthode d'amplification d'acides nucléiques et ses applications[END_REF] is to some extend a combination of RCA and SDA amplification. We refer to this patent for more technical information.

The mechanism is represented on Fig. 1(a). The loop complex is a circular molecule composed of two parts: the amplifiable fragment (2) and the loop link [START_REF] Adleman | Molecular computation of solutions to combinatorial problems[END_REF]. The arrow represents the 3' end of the amplifiable fragment. We represent this molecule schematically as on Fig. 1(b). For the sake of commodity we split the amplifiable fragment in 3 parts (3,4,5 on the picture) corresponding to the 3' end, middle and 5' end of the amplifiable fragment. The amplification mix contains primers (101) which hybridize to the 3' part of the amplifiable fragment (3). They can be further extended by DNA polymerase (102) present in the mix, see Fig. 2(a). The loop link (1) length is small compared to the length of the DNA fragment. The DNA fragment is also short in regards to the stiffness of double stranded DNA. In conditions where the biochemical replication reactions can take place, double stranded DNA molecules shorter than 300 -500 nucleotides are too stiff for their extremities to come into close proximity. In other words, a circular DNA molecule shorter than 300 nucleotides cannot exist in full double stranded form, but is found as stretches of double stranded portions separated by single stranded portions. This is true also for the loop complexes used in ICLEDA. At some point the complex will be composed from a single stranded DNA having n nucleotides from the 5' end of the amplifiable fragment, a double-stranded DNA corresponding to the 3' part of the amplifiable fragment, the extended primer and the linking loop of special length.

Since the two extremities of the amplifiable fragment are linked to each other this gives a geometric constraint for the loop. In order to continue the reaction either the single strand part should be extended to the maximum or the double stranded part should open at the opposite extremity. At some level of tension the energetic preference will be to continue the extension of the primer by DNA polymerase, while the opposite end will detach by Brownian motion. So, at the same time the double stranded fragment will be opened at 5' part and one nucleotide will be added by DNA polymerase. However it should be noted that the number of nucleotides on the double stranded part remains unchanged, due to the geometric constraints of the loop. Since no more nucleotides are bound at the 3' end of the amplifiable fragment (3) at some point it becomes accessible for a hybridization with a new primer, see Fig. 2(b). The extension is blocked when it reaches the end on the amplifiable fragment (105) because of the presence of non-natural nucleotides in the link, see Fig. 2(c). Now we remark that if in the mix a fragment of a single stranded DNA that matches the 3' part of the amplifiable fragment is present, then it can stick to the amplifiable fragment as shown on Fig. 3(a). We call such a strand a trigger. When a trigger is attached to the loop complex, no amplification can be done. A trigger can be detached from the loop complex by an activator that matches by its 3' end a part of the trigger strand as shown on Fig. 3(b). Once bound to the trigger the activator can be extended by DNA polymerase and this will release the trigger, so the loop complex will be able to start the amplification process.

Formalization

By a DNA-like alphabet V we mean an alphabet with 2n letters, n ≥ 1, of the form V = {a 1 , . . . , a n , ā1 , . . . , ān }. Letters a i and āi , 1 ≤ i ≤ n are said to be complementary letters. The terminology originates from the basic DNA alphabet {A, G, C, T }, where the letters A and G have their complements T and C. By h w we denote the letter-to-letter endomorphism of a DNA-like alphabet V mapping each letter to its complementary letter; h w is also called the Watson-Crick morphism. Let us denote by rev the function yielding the reverse of a word. Then the Watson-Crick complementarity function ρ :

V * → V * is defined as ρ(u) = rev(h w (u)).
We shall abstract DNA molecules as words over a DNA-like alphabet. By convention we will consider that they are read in 5 ′ to 3 ′ direction, i.e. the leftmost symbol of the word corresponds to the first nucleotide at the 5 ′ end of the molecule. For the purpose of this paper the considered words will be partitioned into two kind of parts: sensitive and neutral. Sensitive parts of different words can eventually interact with each other, while the neutral parts can never interact with other sensitive or neutral ones. So in some sense, the neutral parts are just a separator for sensitive parts. We shall mark this difference between sensitive and neutral parts of words in the notation: sensitive parts will be denoted by capital letters and neutral parts will be denoted by lower case ones. As mentioned before we are not interested by the concrete symbols in words, but rather by the interleaving of sensitive and neutral parts.

Let V be a DNA-like alphabet. We denote by Sensitive(V ) the letters that are used to label sensitive parts (by convention in latin uppercase) and by N eutral(V ) the letters used to label neutral parts (by convention in latin lowercase). We will assume that in any word two neutral letters (parts) cannot be adjacent to each other.

We shall formalize the partial annealing of DNA sequences. For this we consider that DNA molecules can bind together at some point if they share a complementary sequence of sensitive parts at that place. Now in order to be able to write the resulting object in a linear way, the binding points will be identified by a unique number and the corresponding sensitive parts of both molecules will be marked by this identifier. We shall use the notation [A] i to denote that A is marked by i. For example, x[A] 1 yAz, tBp[ Ā] 1 indicate that molecules xAyAz and tBp Ā are partially hybridized on the first sensitive part A of xAyAz. Since indices are unique for each binding point, each complex molecular structure corresponds to an equivalence class grouping several indices, hence we can identify it by using only one of its representatives. We denote by [V ] the infinite extension of the alphabet V containing normal and integer marked symbols. Now we can define the operation Glue as follows.

Glue(u, v) = { u ′ , v ′ | where u = α 1 A 1 . . . A k α 2 , v = β 1 Āk . . . Ā1 β 2 , u ′ = α 1 [A 1 ] i . . . [A k ] i α 2 , v ′ = β 1 [ Āk ] i . . . [ Ā1 ] i β 2 , i > 0, k ≥ 1, α 1 , α 2 , β 1 β 2 ∈ [V ] * , such that there is no α ′ 1 , α ′ 2 , β ′ 1 , β ′ 2 ∈ Sensitive(V ) * , α ′ 1 α ′ 2 β ′ 1 β ′ 2 ̸ = λ satisfying u = α ′′ 1 α ′ 1 A 1 . . . A k α ′ 2 α ′′ , v = β ′′ 1 β ′ 1 Āk . . . Ā1 β ′ 2 β ′′ 2 and having α ′ 1 A 1 . . . A k α ′ 2 = ρ(β ′ 1 Āk . . . Ā1 β ′ 2 ) }.
We remark that Glue(u, v) returns all possible molecules that are obtained by a partial annealing of u and v. The condition stated in the definition also requires the annealed part to be non-extensible, i.e. there should be no possibility to anneal more adjacent regions. We also remark that already marked symbols cannot be used for annealing. We will use the notation u ⊗ v to denote Glue(u, v) and we will also extend this notation to circular strings and languages in a traditional way. We observe that ⊗ operation is commutative and in some cases associative, e.g. when there is no competition for gluing sites (there is no intersection between sensitive parts of different words), so it can be naturally extended to any number of operands. This corresponds to the fact that ⊗ models a situation in which the components are independent and may freely combine or not and in all possible combinations. Now we shall define the operation that will simulate the strand displacement. We shall consider two cases: (1) when the displacing strand contains the full annealed sequence of the displaced one and (2) when the displacing strand starts with symbol(s) complementary to the free positions just before the attachment of the displaced strand. The latter case corresponds to a displacement by the sequence extended in the 5 ′ (left) direction.

First, we define the full displacement operation Displace f .

Displace f (u, v) = {u ′ , v ′ | where there exist i, j > 0 such that u = α 1 A 1 . . . A n [B 1 ] i . . . [B m ] i α 2 , β 1 [ Bm ] i . . . [ B1 ] i β 2 , v = γ 1 ρ(A 1 . . . A n B 1 . . . B m )γ 2 , u ′ = α 1 [A 1 ] j . . . [A n ] j [B 1 ] j . . . [B m ] j α 2 , γ 1 [ Bm ] j . . . [ B1 ] j [ Ān ] j . . . [ Ā1 ] j γ 2 , v ′ = β 1 B m . . . B 1 β 2 }.
Next, we define the extension displacement operation Displace e .

Displace e (u, v) = {u ′ , v ′ | where there exist i, j > 0 such that

u = α 1 A 1 . . . A n [B 1 ] i . . . [B m ] i α 2 , β 1 [ Bm ] i . . . [ B1 ] i β 2 , v = ρ(A 1 . . . A n )γ, u ′ = α 1 [A 1 ] j . . . [A n ] j [B 1 ] j . . . [B m ] j α 2 , [ Bm ] j . . . [ B1 ] j [ Ān ] j . . . [ Ā1 ] j γ , v ′ = β 1 B m . . . B 1 β 2 }.
Finally, we define the displacement operation Displace.

Displace(u, v) = Displace f (u, v) ∪ Displace e (u, v).
We shall use the notation u ⊕ v to denote Displace(u, v). We observe that ⊕ operation is commutative and in some cases associative (when there is no competition for the displacement positions), so it can be naturally extended to any number of operands.

The loop complex can be formalized as a circular word ∼ F u R where u is the neutral part and F with R are the sensitive ones, corresponding to the parts 4,3,5 on Fig. 1(b). Its functioning can be described by the following equation defining the unary operation P roduce:

P roduce( ∼ F u R) = RūF.
(

A trigger can be formalized as t ĀF w, where w and z are the neutral parts and Ā, F are the sensitive ones. We remark that glue and displacement operations can be applied for the case of circular strings exactly in the same manner as for the linear one. Hence, the attachment of the trigger to the loop complex can be described as follows:

∼ F u R ⊕ t ĀF w ⊢ ∼ F u R ⊗ t ĀF w.
(

) 2
where u, t and w are neutral. From subsection 2.1, the result of (2) blocks the application of (1). The loop can be unblocked by an activator of form xA F y or xA:

∼ F u R ⊗ t ĀF w ⊕ y F Ax ⊢ { ∼ F u R, t ĀF w ⊗ y F Ax}, ( 3
) ∼ F u R ⊗ t ĀF w ⊕ Ax ⊢ { ∼ F u R, t ĀF w ⊗ F Ax}. (4)
A string displacement system is the construct D = (V, A c , A l ), where V is a DNA-like alphabet, A l ⊆ (V ∪ {⊗}) * is the set of initial partially annealed linear strings and A c ⊆ ∼ V (V ∪ {⊗}) * is the set of loop complexes eventually partially annealed with linear strings. The configuration of the system is a pair C = (C c , C l ), where as for axioms C l is the set of current partially annealed linear strings and C c is the current set of circular strings eventually partially annealed with linear ones. We pass from configuration C to C ′ (denoted as

C ⊢ C ′ ) if C ′ c = Displace(u, v), u ∈ C c , v ∈ C l and C ′ l = Displace(u, v) ∪ P roduce(z), u ∈ C c ∪ C l , v ∈ C l , z ∈ C c .
The language generated by a displacement system D is the set of all "pure" strings (not being part of a complex) that can be obtained from the axioms:

L(D) = {w ∈ V * | (A c , A l ) ⊢ * (F c , F l ) and w ∈ F l }.
It is clear that for any string displacement system D we have that L(D) is finite.

Implementing boolean functions

In this section we show how it is possible to implement boolean functions using the operations introduced in the previous section.

We start by a remark that if a loop ∼ F u R is unblocked, then there will be an unbounded number of copies of RūF . This assumption results from the observation that once started, the amplification could produce a large enough number of resulting molecules, even if the loop is blocked again afterwards.

This implies that we can consider that initially all loops are blocked, otherwise we substitute them by a large number of DNA molecules corresponding to their result. So the computation in such a system consists in unblocking some loops in some order. This corresponds in a direct manner to asynchronous boolean circuits where the electrical impulses are propagated in the circuit. The signals we use are always identified by the part at the beginning of the molecule, i.e. a signal A will be given by the string Aw for some w ∈ V * .

It is known that every boolean function of n variables can be implemented by a boolean circuit using AND, OR and NOT gates. It is possible to eliminate the NOT gate by considering the dual-rail encoding, i.e. that only "true" signals can circulate in the circuit, corresponding to the transition over the wire that we call signal. This encoding also allows to consider asynchronous circuits, especially important for the DNA computing area. The positional information can be handled by distinguishing signals (wires) and considering that each gate waits for concrete input signals and produces a unique output signal when all input signals are present.

The input of the circuit is not the true or false value for the same variable x, but rather a signal for x or for ¬x. The output is also modified: instead of a single output having one of the values true or false, there are two outputs (marked by true and false) and a presence of a signal in some of the outputs indicate that the output value of the circuit is true or false. If we consider that the two output nodes are combined into a fictive output node then such a circuit is a DAG with the root being the output node and the leaves being the input variables and their negations. For a boolean formula ϕ such a modified circuit can be constructed by a superposition of two circuits, one computing ϕ and the other one computing ¬ϕ.

We remark an important similarity between traditional electronic implementation of boolean circuits and our implementation: if a signal (represented by an electric charge in electronics and by DNA molecule in our case) appears at some moment during the computation, then it is sufficiently strong and does not disappear in the consequent steps. This allows us to make a direct analogy between two implementations and use similar construction techniques. This is different from other approaches of simulation of circuits by DNA computing, as we do not need additional amplification phase anymore.

More precisely, let f : B n → B, B = {0, 1} be a boolean function and let D = (V, E, F ) be the circuit implementing this function (where V = {1, . . . , n} is the set of vertices, E ⊆ V ×V is the set of edges and F : V → {x p , ¬x p , AN D m , OR k , N OT q , out}, 1 ≤ p, q, m, k ≤ n is the function that labels vertices of the circuit).

Then for any inner node y such that F (y) = OR m (resp. F (y) = AN D m ) we construct an OR loop (resp. AND loop) as discussed in 3.1 (resp. 3.2). The final gate will send the signal to the output node (the root of the circuit (F (x) = out)). A similar construction should be performed for ¬f . The final assembly is the union of these two circuits.

In order to compute the result in the initial configuration signals corresponding to X k (where X k is either x k or ¬x k ) should be introduced. We give below an example of such a construction for the following function:

f (x 1 , x 2 , x 3 ) = (¬x 1 ∧ x 2 ) ∨ (x 1 ∧ x 2 ∧ ¬x 3 ).
The asynchronous dual-rail circuit computing f is given below. We numbered the gates and labeled the edges going out from the same left node (corresponding to a concrete signal produced by the corresponding gate). Having in mind that the true value of some node is represented by the presence of the corresponding signal (that labels the edge), it becomes clear that this construction can be directly implemented using loop complexes and corresponding signals by 4 AND gates and 2 OR gates.

x 1 B B B B B B B B B B B B B B B B B B B B B S1 L L L L L L L L L L L L L L L L L ¬x 1 S2 AN D 1 S7 K K K K K K x 2 S3 i i i i i i i i i i i i i i i 7
The signals S 1 -S 6 correspond to the input values and the signals S 11 and S 12 to the output. So the computation starts by giving input signals (taking care of not having an input x and ¬x at the same time). Then the gates will act in cascade and one of two output signals (S 11 if f is true or S 12 if f is false) will be obtained.

The OR-gate

We base our construction of the fact that the outcome of rule ( 1) can be interpreted as the production of the signal R.

In this condition, if the trigger t ĀF w is initially attached to a loop complex ∼ F u R, forming ∼ F u R ⊗ t ĀF w, rule (3) tells us that introducing the molecule Ax we obtain ∼ F u R and t ĀF w ⊗ F Ax. As now rule (1) applies to ∼ F u R: accordingly, we get the signal R.

It is not difficult to obtain a similar sequence of deductions with ∼ Ḡv R and the trigger r BGs. Introducing the molecule By we shall also get R by applying the rules. The final construction for the gate is shown on Fig. 4. We remark that the construction above can be extended to an n-ary OR gate. This gives the possibility to simulate semi-unbounded fan-in circuits.

The AN D-gate

We can simulate an AND gate by considering two active regions on the loop, i.e. loops of form ∼ FA u FB v R, where FA and FB are active zones for triggers having A and B. Then the loop is blocked by two molecules as follows

∼ FA u FB v R ⊗ t ĀF A w ⊗ q BF B
s, see Fig. 5(a). Now if both activators Ax and By are present, then the loop complex can be unblocked:

∼ FA u FB v R ⊗ t ĀF A w ⊗ q BF B s ⊕ Ax ⊕ By ⊢ ⊢ { ∼ FA u FB v R ⊗ t ĀF A w ⊕ Ax, q BF B s ⊗ FB By} ⊢ ⊢ { ∼ FA u FB v R, t ĀF A w ⊗ FA Ax, q BF B s ⊗ FB By} (5)
In the above derivation the unblocking can start by the signal Ax, but following the commutativity of ⊕ it yields the same result. It is clear that if only one of the activators Ax or By is present, then the loop complex is only partially unblocked (it has either the form

∼ FA u FB v R ⊗ t ĀF A w or ∼ FA u FB v R ⊗ q BF B
s) and cannot produce the resulting signal.

The initial AN D-gate

Another variant of the AND gate is described in [START_REF] Mayer | Méthode d'amplification d'acides nucléiques et ses applications[END_REF]. Like in the previous case it is also a complex of 3 molecules, however the loop complex is bound in only one place. Its construction is done in two stages: the loop complex ∼ FA w R is blocked by the trigger F A Ā. After that the molecule BA is added into the solution and it will stick to the Ā site. Hence, the complex ∼ FA w R ⊗ ĀF A ⊗ BA will be formed, see Fig. 5(b). We remark that since this complex is formed during the preparation stage, we can insure that no molecules Ax, (x ̸ = B) or tF A Av are present in the solution. Now during the computation the loop complex can be unblocked as follows:

∼ FA w R ⊗ ĀF A ⊗ BA ⊕ By ⊕ Ax ⊢ ⊢ { ∼ FA w R ⊗ ĀF A , BA ⊗ ĀBy} ⊕ Ax ⊢ ⊢ { ∼ FA w R, BA ⊗ ĀBy, ĀF A ⊗ FA Ax}. (6)
We remark that unlike the previous case this construction is not symmetric, i.e. first the signal By is removing the molecule BA from the complex, freeing the site Ā, which can be bound after that by the signal Ax that finally unblocks the loop.

Conclusions

In this article we present a formalization of a new method of DNA string amplification. This process lead us to the introduction of two operations on strings that permit to express the partial annealing and the strand displacement using a linear notation. The obtained model is very promising and it presents many mathematical challenges, especially because the introduced operations are not unary. Numerous extensions are possible, we cite [START_REF] Pȃun | DNA Computing: New Computing Paradigms[END_REF] for a good overview of different possibilities. We would like to mention the possibility to use multisets or fuzzy sets instead of sets of strings. The motivation for such an extension is that the model allows us to produce a trigger and to stop the loop amplification. Another interesting idea is to consider molecules (strings) attached to a support and use a washing procedure that would eliminate strings that are not attached or that will move unattached strings to some other compartment/membrane, allowing to further apply techniques from [START_REF] Gh | Membrane Computing. An Introduction[END_REF].

Another interesting outcome of this article is a new method for the simulation of boolean circuits. The use of ICLEDA offers many advantages like a single volume and unchanged reaction conditions. This implies that the corresponding implementation will not need any additional intervention. Moreover, since the loop complexes can be easily attached to the support it is possible to reuse the circuit by washing the tube and by introducing trigger molecules to block the loops. Another advantage of the method is that the signal molecules (corresponding to the true value of some gate) are of a small length; moreover, by introducing compartments it is possible to share some of the signals.

Fig. 1 .

 1 Fig. 1. The loop complex (a): the amplifiable fragment (2) and the loop link (1). The schematic representation (b) highlighting the 3 components of the amplifiable fragment

Fig. 2 .

 2 Fig. 2. The amplification process: primer extension (a); maximum stretch of amplifiable fragment and the opening of the 5' end of the double strand (b); a second amplification started (c). Notation: link loop (1), amplifiable fragment (3,4,5), DNA polymerase (102), (extended) primer (101), single stranded fragment (104).

Fig. 3 .

 3 Fig.3. The loop complex blocked by a trigger (a) and the hybridization with further extension used to remove the trigger (b). Notation: link loop (1), amplifiable fragment[START_REF] Amos | Theoretical and experimental DNA computation[END_REF][START_REF] Fire | Rolling replication of short DNA circles[END_REF][START_REF] Gilbert | DNA replication: the rolling circle model[END_REF], trigger[START_REF] Pȃun | DNA Computing: New Computing Paradigms[END_REF], DNA strand used to release the trigger[START_REF] Walker | Isothermal in vitro amplification of DNA by a restriction enzyme/DNA polymerase system[END_REF]

Fig. 4 .

 4 Fig. 4. The simulation of the OR gate.

Fig. 5 .

 5 Fig. 5. The simulation of the AND gate with two active regions on the loop (a) and with one active region on the loop (b)

Presently, this patent is in the public domain.

OR 4 S11 out true ¬x 2 S4 X X X X X X X X X X X X X X X X X X X X X X X X AN D 3

Acknowledgements Sergey Verlan would like to acknowledge the support of the ANR project SYNBOTIC.