Nicolas Combe
email: combe@cemes.fr

Jean Roch Huntzinger
email: jean-roch.huntzinger@univ-montp2.fr

Joseph Morillo
email: morillo@cemes.fr

Surface Loving and Surface Avoiding modes

Keywords:

We theoretically study the propagation of sound waves in GaAs/AlAs superlattices focussing on periodic modes in the vicinity of the band gaps. Based on analytical and numerical calculations, we show that these modes are the product of a quickly oscillating function times a slowly varying envelope function. We carefully study the phase of the envelope function compared to the surface of a semi-infinite superlattice. Especially, the dephasing of the superlattice compared to its surface is a key parameter. We exhibit two kind of modes: Surface Avoiding and Surface Loving Modes whose envelope functions have their minima and respectively maxima in the vicinity of the surface. We finally consider the observability of such modes. While Surface avoiding modes have experimentally been observed (Phys. Rev. Lett. 97, 1224301 (2006)), we show that Surface Loving Modes are likely to be observable and we discuss the achievement of such experiments. The proposed approach could be easily transposed to other types of wave propagation in unidimensional semi-infinite periodic structures as photonic Bragg mirror.

I. INTRODUCTION

Propagation of waves in periodic structures have been studied for decades [START_REF] Brillouin | Wave Propagation in Periodic Structures[END_REF] . Indeed, the periodicity leads to band gaps where propagation is forbidden. This property is very general and could be observed in many different fields. Electrons in crystals experience the periodic atomic potential leading to electronic band gaps [START_REF] Ashcroft | Solid State Physics[END_REF] . Electromagnetic waves cannot propagate in a spatial periodic dielectric constant system if their frequencies fall in the band gap: Photonic crystals [START_REF] Yablonovitch | [END_REF]4 are based on this statement. Elastic waves in solids could also endure a periodic Young modulus and/or a periodic density. Studies on such systems are currently numerous [5][6][7] and lead to a new field of physics namely Phononics 8,9 .

In periodic structures, two kinds of vibrations can be pointed out: 1) extended eigenmodes that propagate through the system; their frequencies fall out of the gaps and they satisfy the Floquet-Bloch theorem; 2) localized modes that can be found around isolated defects including surfaces 10 (surface modes belong to this category). Due to their localized character, these modes are not so sensitive to the periodicity of the underlying Crystal and thus their frequency can take any value, in particular within the forbidden band gaps.

Extended acoustical vibrations in superlattices (SL) have been described quantitatively in the frame of the elastic model in the middle of the century 11 . The superlattice, with its long period compared to the underlying crystals periods, results in a much shorter reciprocal Brillouin zone and hence a multiple folding of the initial acoustical branches. Experimentally, observations of these folded acoustical vibrations have first been performed using Raman scattering experiments for which selection rules are now well understood 12,13 . Since the nineties, time-domain optical experiments have investigated folded acoustical vibrations [14][15][16][17] . They exhibit two types of modes: phonons with finite wave vectors (q = 0) and zone-center modes (q = 0) in the reduced Brillouin zone.

Recently, Trigo et al. 18 studied both theoretically and experimentally these zone-center modes in phononic semi-infinite GaAs/AlAs SL yielding two conclusions. First, modes near the Brillouin zone-center observed by pump-probe experiments have not rigorously a zero wave vector, but instead an almost zero wave vector q ≈ 0. Second, these modes present a slowly varying envelope wave function with an amplitude minimum in the vicinity of the surface: they claim these modes "have a tendency to avoid the boundaries, irrespective of the boundary conditions" and for this reason, refer to them as Surface Avoiding Modes (SAM). Note that these surface avoiding modes have also been encountered near photonic 19 and acoustic 20 band gap materials and their nature is very general. In this work, we study in detail acoustical modes in the vicinity of Brillouin zone center, focussing on their avoiding character near the GaAs/AlAs SL surface. Our work is based on analytical studies corroborated by numerical simulations. Analytical calculations are based on the analogy between the propagation of waves in a periodic structure and the spatial parametric oscillator. Since this analogy is not well-known, we devote Sect. II to prove the fruitfulness of such analogy to describe waves in periodic structures in the vicinity of gaps. Considering the propagation in an infinite SL of longitudinal modes in the vicinity of band gaps, we show that these modes are the product of a fast oscillating function and a slowly varying envelope function. We derive the dispersion diagram in the vicinity of band gaps using the parametric oscillator equations. Then considering the case of a semi-infinite SL in Sect. III, we give the analytical expression of longitudinal waves assuming free boundary conditions and considering the dephasing of the SL compared to its surface. We show the existence of Surface Avoiding and Surface Loving Modes depending if the envelope function is minimum or maximum near the SL surface. We then discuss in detail the avoiding or loving character of modes near the center of the Brillouin zone. In Sect. IV, we discuss on the electron-phonon coupling of these modes and show both Surface Avoiding and Loving Modes are likely to be observable using time-domain optical experiments or Raman and Brillouin scattering experiments. Finally, Sect. V will be devoted to a discussion on our results.

II. PARAMETRIC OSCILLATOR

A. Parametric oscillator analogy

We consider the propagation of a sound wave in an unidimensional infinite SL made of 2 solid materials: respectively named A and B. d A and d B are the width of the layers, and d = d A + d B is the period of the SL. If u is the atomic displacement, in each material sound waves are solutions of:

∆ u - 1 c 2 i ∂ 2 u ∂t 2 = 0 (1)
where c i is the sound speed in the materials (i = A, B).

To determinate sound waves in a SL from Eq. ( 1), one needs to apply at each interface, the continuities of the displacement field and of forces d F = σd S acting on each elementary interface d S, where σ is the stress tensor. In a SL, the displacements can be determined exactly by the transfer matrix method 21 . Though very powerful, and clearly grounded, this method suffers from its numerical nature. It is not easy to get analytical expressions for finite size superlattices.

In this work, we will use the analogy between the propagation of sound waves in periodic media and the parametric oscillator. As it will be shown, this analogy involves severals approximations we will justify. However, it has two benefits. First, it yields analytical expressions of the displacements in closed form, with relatively simple calculations. Second, and especially, it allows to point out and to understand the main physical phenomena near the Bragg reflexions in the SL. We stress that the present treatment is complementary to the transfer matrix method and does not pretend to challenge it for quantitative analysis.

Finally, one must note that the present method could be applied to the propagation of light in photonic crystals or electrons in crystals 22,23 since their wave propagation equations are similar.

From now on, we consider a harmonic, plane, longitudinal sound wave of angular frequency ω: u(z, t) = u(z)e -iωt e z along the z-axis, perpendicular to the SL layers. Continuity of forces acting on each interface imposes the continuity of C 11 (z) ∂u(z) dz where C 11 is the stiffness constant, for a longitudinal sound wave propagating along z. In the following, we will focus on GaAs/AlAs superlattices, because they are the most experimentally used superlattices. Stiffness constant contrast between GaAs and AlAs is very weak: ∆C11 C11 ≈ 0.014 so that the continuity of C 11 (z) ∂u(z) dz at each interface reduces to the continuity of ∂u(z) dz . In such a case, the determination of sound waves in the SL can be directly achieved by the resolution of Eq. (1) considering only the variation of the sound speed since u(z) becomes a C 2 function 29 . Consequently, since the sound speed in a GaAs/AlAs SL is a periodic function of z, we may expand 1 c 2 as a Fourier series 1

c 2 = a 0 + ∞ m=1 a m cos( 2πmz d ) + b m sin( 2πmz d )
, so that sound waves in GaAs/AlAs superlattices are solutions of:

d 2 u dz 2 +ω 2 a 0 1 + ∞ m=1 a m a 0 cos( 2πmz d ) + b m a 0 sin( 2πmz d ) u = 0 (2) We now introduce G = 2π
d the primitive vector of the reciprocal lattice, c m = am a0 and d m = bm a0 the reduced amplitudes of the mth harmonic, and k 0 (ω) = ω √ a 0 the zero order wave number i.e. the wave number a wave would have in the absence of any periodic modulation of the sound speed. k 0 (ω) = ω √ a 0 is the usual dispersion relation for acoustic phonons in an homogeneous medium with mean sound speed c 0 = 1 √ a0 . Eq (2) then reads like a parametric oscillator equation in spatial coordinates:

d 2 u dz 2 +k 2 0 (ω) 1 + ∞ m=1 c m cos(mGz) + d m sin(mGz) u = 0 (3)

B. Single Harmonic Approximation

Eq. ( 3) includes all harmonics of the inverse square sound velocity. In order to understand the effect induced by the different harmonics, let us first consider the effect of an isolated harmonic. Eq. (3), in this case, reduces to a Mathieu equation [START_REF] Blanch | Handbook of Mathematical Functions[END_REF] of the form:

d 2 u dz 2 + k 2 0 [1 + a cos(k e z)] u = 0 ( 4
)
k e is the wave number of the periodic excitation. General solutions of Eq. ( 4) are periodic except in some regions of the plane (k 0 /k e , a): they are then the product of an oscillating function with a linear combination of an increasing and a decreasing exponential. These non periodic solutions correspond to the resonances of the parametric oscillator or band gaps of the SL. calculated numerically using the resolvent of Eq. ( 4) and the Liouville and Floquet-Bloch theorems [START_REF] Blanch | Handbook of Mathematical Functions[END_REF] .

Regions of non periodic solutions form in the plane (k 0 /k e , a) bands that, when a tends to 0, converge to points k0 ke = n 2 with n ∈ IN * . i.e. wave numbers k e = 2k 0 , k 0 , 2 3 k 0 , k0 2 , . . . etc. Moreover, for a ≪ 1, it can be shown [START_REF] Landau | Mechanics[END_REF] that the band width of the nth resonance is a decreasing function of n proportional to a n . As a result, the larger resonance is obtained for n = 1 i.e. for k e = 2k 0 , the most famous resonance of the parametric oscillator, corresponding to an excitation twice faster than the zero order wave number.

Hence, going back to Eq. (3), varying the value of k 0 , each harmonic m of the excitation at wave number mG, if treated independently, is expected to create band gaps around k0 mG = n 2 . Thus, band gaps occur for 2k0 G = nm = l ∈ IN * . Fig. 2 schematically presents the band gaps independently created by each of the first fifth harmonics. Excepted the first band gap (l = 1) due exclusively to the first harmonic (m = 1), all other band gaps result from at least two harmonics. However, as a first approximation, since the larger resonance due to the mth harmonic is for n = 1 i.e. around k0 mG = 1 2 , we consider that the lth band gap of Eq. ( 3) is essentially due to the first resonance (n = 1) of the lth harmonic (m = l). We call this approximation the Single Harmonic Approximation (SHA). This approximation is equivalent to treating the modulation of the sound speed as a first order perturbation: |c m | ≪ 1 and |d m | ≪ 1 ∀m ∈ IN * . Our main motivation for the SHA is the possibility to derive analytical calculations and the possibility to produce good analytical approximations of solutions of Eq. ( 3) around a given gap. Of course, the validity of SHA will be discussed below in the considered case. Note that values of k 0 for which there are no periodic solution, correspond to a range of frequencies ω which is the usual band gap of the SL. Indeed, k 0 has been defined by k 0 = ω √ a 0 in Sect. II A.

C. Infinite medium: band gap

In order to test the validity of SHA, we solve Eq. ( 3) near the resonance 30 at ω

BG = G √ a0 .
The superscript "(2)" in ω

(2)

BG refers to the second band gap. Using the SHA, this band gap is only produced by the harmonic at 2G in the Fourier series of 1 c 2 . We thus study a reduced equation taking into account only that harmonic:

d 2 u dz 2 + k 2 0 [1 + c 2 cos(2Gz) + d 2 sin(2Gz)] u = 0 (5)
Where we have simplified the notation of k 0 (ω) using k 0 . The SHA will be checked a posteriori. Switching to complex number representation and introducing

p 2 = c2-id2 2
, Eq. ( 5) reads:

d 2 u dz 2 + k 2 0 (1 + p 2 e i2Gz + p2 e -i2Gz )u = 0, (6)
where p2 denotes the complex conjugate of p 2 . Similarly to the resolution of the parametric oscillator equation by Landau and Lifchitz 25 , we change the coordinate system to A(z) and B(z) defined by:

u(z) = A(z)e iGz + B(z)e -iGz (7)
where A(z) and B(z) are slowly varying functions. Introducing Eq. ( 7) in Eq. ( 6) and neglecting fast oscillating terms (at wavevector 3 G) as well as the second derivatives of A(z) and B(z), the identification of coefficients of plane waves at G and -G reads:

dA dz dB dz = - k 2 0 -G 2 2iG - p2k 2 0 2iG p2k 2 0 2iG k 2 0 -G 2 2iG A B (8)
whose general solutions are:

A(z) = A + e iκz + A -e -iκz (9a)
B(z) = B + e iκz + B -e -iκz ( 9b
)
where κ is given by:

κ 2 = (k 2 0 -G 2 ) 2 4G 2 - |p 2 | 2 k 4 0 4G 2 = k 2 0 2G 2 δ 2 -|p 2 | 2 ( 10
)
and δ = k 2 0 -G 2 k 2 0 ( 11
)
δ characterizes the detuning between the excitation wave number G and the zero order wave number k 0 .

The system undergoes a bifurcation when κ 2 changes its sign as a function of ω:

• Either κ 2 < 0. Solutions of Eq. ( 9) are then hyperbolic: A(z) and B(z) are a linear combination of increasing and decreasing exponentials. And therefore, no propagation can occur in the SL. The condition κ 2 < 0 defines the band gap of the SL around ω

BG . • Or κ 2 > 0. Solutions of Eq. ( 9) are sinusoidal and the general solution u(z) is thus periodic. Note that in this case, the spectrum of u(z) will be composed of a mixing of G + κ and G -κ plane waves. u(z) is then the product of a quickly oscillating function at G, times a slowly varying functions at κ. Moreover, Eq. ( 10) establishes a relation between ω (through k 0 (ω)) and wave vectors G ± κ i.e. the dispersion relation of the SL around the considered gap.

Note that the band gap only depends on the squared magnitude of p 2 :

|p 2 | 2 = c 2 2 +d 2 2 4
i.e. on the amplitude of the second harmonics at 2G. Besides, we already pointed out the similarities of sound waves propagation in SL and electron waves propagating in crystals 22 . Usually, when studying the independent electrons in a weak crystalline potential, one applies the first order perturbation theory to determine the width and position of electronic band gap 2 : the expression of the band gap width is analogous to Eq. (9a) and could be derived using the parametric oscillator analogy and the SHA 23 .

D. Dispersion diagram: numerical study

In this part, we will numerically check our analytical results of Sect. II C and the SHA applying our predictions on a GaAs/AlAs SL with d A = d GaAs = 5.9 nm and d B = d AlAs = 2.35 nm. Such a SL has been used in an experimental setup by Trigo at al. 18 . For these materials, longitudinal sound speeds are 21 :

c A = c GaAs = 4726 ms -1 (12a) c B = c AlAs = 5630 ms -1 (12b)
These parameters will be used in all the numerical examples given in this article.

Fig. 3 compares the dispersion diagram obtained from Eq. ( 10) (SHA approx.) and the transfer matrix method 21 (exact solutions) for this SL. Fig. 3 is restricted to the two first band gaps respectively due to the harmonics at spatial frequencies G and 2G. We would like to emphasize that in Eq. ( 10), the first band gap is obtained using coefficient p 1 = c1-id1 2 (instead of p 2 ) and the second bang gap using only p 2 = c2-id2 2 whereas the transfer matrix method considers all harmonics. According to Fig. 3, our predictions of band gaps with the SHA are in good agreement with the exact calculation of the transfer matrix method. Moreover, dispersion diagrams around the gaps also agree very well. They slowly diverge when moving away from the gaps: the A and B functions are varying faster and thus their second derivatives are not negligible any more. In addition, the development in Eq. ( 7) neglecting harmonics of order higher than 2 and keeping only the main term around wave number G is no longer appropriate. Finally, the SHA, keeping only one harmonic in Eq. ( 3) naturally becomes inadequate when moving too far away from the gap. Fig. 3 validates our SHA for the first and second band gaps; and it is easy to convince oneself that it is also true for any of the SL band gaps in a first order approximation. Below, we will focus on periodic solutions around the second band gap: SL eigenmodes in the vicinity of this gap will thus be well described by solutions of Eq. ( 6).

III. SAM AND SLM

In the previous section, we have studied the general solution of Eq. ( 3) and derived the dispersion diagram of the SL. Since experiments always involve a finite SL, we now consider the effects of the presence of a free surface at z = 0 with definite boundary conditions on SL eigenmodes. FIG. 3: (color online) Dispersion diagram of a GaAs/AlAs SL calculated using the transfer matrix method (dashed black) and predictions of the gaps using the parametric oscillator analogy and the SHA: (red) first gap calculated from the first harmonic of the inverse squared sound speed and (blue) second gap calculated from the second harmonic. We report using green horizontal dotted lines the positions of the lth band gap ω

Wave vector (nm

(l) BG = lG 2 √ a 0 .
Parameters corresponding to the SL described in section II D have been used.

A. Semi-infinite medium: surface effects

More precisely, we intend to study periodic solutions of Eq. (3) in the vicinity of the band gap at ω

(2) BG using free boundary conditions i.e. u(0) = u 0 and du dz (0) = 0. In the following, we will use the abbreviation NBPM (Near Bang gap Propagative Mode) in reference to such modes. Figure 4 plots 1 c 2 as a function of z. The dephasing τ of the SL compared to z = 0 is defined on Fig. 4. As shown below, τ will be a key parameter in our study. Coefficients a 0 , c m and d m in Eq. ( 3) are calculated from the Fourier components of 1 c 2 :

a 0 = 1/c 2 0 = γ 1 c 2 A + (1 -γ) 1 c 2 B ( 13a
)
c m = 2 c 2 0 c 2 A - c 2 0 c 2 B πm cos(πm(2α + γ)) sin(πmγ) ( 13b
)
d m = 2 c 2 0 c 2 A - c 2 0 c 2 B πm sin(πm(2α + γ)) sin(πmγ) (13c)
or

p m = c m -id m 2 = c 2 0 c 2 A - c 2 0 c 2 B e -i2πmγ -1 -i2πm e -i2πmα (13d)
BG . Note that a gap might actually be opened through the second parametric resonance n = 2 produced by the first harmonic m = 1 (cf. Fig. 2).

For a given γ, the effect of α is merely to rotate the figure by -2πmα, as seen in Eq.(13d). Note that experimentally, most of the SL have α = 0 and that for α = 0, p 2 as a function of γ never cross the real axis except when p 2 = 0: we will illustrate below to what extend the case p 2 non null and real is relevant. Details of the resolution of Eq. ( 6) are given in Appendix A. Since we are focussing on periodic modes, i.e. outside the gap, κ 2 in Eq. ( 10) is positive. Let us introduce ψ the argument of p 2 :

p 2 = |p 2 | e iψ (14)
The solution u(z) satisfying the free boundary conditions u(0) = u 0 and du dz (0) = 0 reads

u(z) = 2A 0 [(1 -ζ) cos(Gz + ψ/2) cos(κz + φ) -(1 + ζ) sin(Gz + ψ/2) sin(κz + φ)] (15)
where ζ, φ and A 0 are defined by

ζ = δ |p 2 | - δ |p 2 | 2 -1 (16) tan φ = -tan(ψ/2) G(1 -ζ) + κ(1 + ζ) G(1 + ζ) + κ(1 -ζ) (17)
A 0 = u 0 2 (1 -ζ) cos ψ 2 cos φ -(1 + ζ) sin ψ 2 sin φ (18)
u(z) is real and u(z, t) is a stationary wave, as expected since the surface is a perfect mirror for acoustic waves. Eq. ( 15) is linear combination of two terms each involving a quickly oscillating function at G and a slowly one at κ corresponding respectively to Bloch and envelope wave functions. The parameter ζ governs the relative contributions of each term and hence their dominant character which we will discuss in Sec. III B 2. The parameter φ is a dephasing of the envelope function with respect to the SL surface: it will be discussed in Sect. III B 3. A 0 governs the global amplitude of the displacement field. There are in general beatings in the solution u(z), as emphasized by the analytical expression for its root mean squared envelope u rms , derived in Appendix B:

u 2 rms (z) = 2A 2 0 1 + ζ 2 -2ζ cos 2(κz + φ) (19)
where the fast oscillating terms at G present in Eq.( 15) have been smoothed out.

B. Surface Loving and Avoiding Modes

In this section, we discuss the effect of the reduced dephasing α of the SL compared to the surface z = 0 on NBPM near the second gap around ω BG . We especially focus on the phase of the envelope wave function at wave vector κ. Examining the general case requires a numerical study. However, two limiting cases where p 2 is real, are remarkable.

1. Remarkable cases. Pure SAM and pure SLM • p 2 is real and p 2 > 0 (α = α 2 or α 2 +1/2 on Fig. 5) .

Therefore from ( 14), ψ = 0 and from (17), φ = 0[π].

-Just above the gap, we have δ ≈ |p 2 | (cf Eqs. ( 10) and ( 11)), implying ζ ≈ 1(Eq. ( 16)). Thus from Eq. ( 15), the main contribution to u(z) is proportional to sin(Gz) sin(κz): the envelope of the vibration mode is null at the surface. Following the denomination of Merlin [START_REF] Merlin | [END_REF] , we call such a mode a Surface Avoiding Mode (SAM): these SL eigenmodes shy away from the boundaries.

-Just below the gap, δ ≈ -|p 2 | and thus ζ ≈ -1. The main contribution to u(z) is cos(Gz) cos(κz): The envelope amplitude is maximal at the surface. In view of that, we refer to such SL eigenmode as Surface Loving Mode (SLM).

• p 2 is real and p 2 < 0 (α = α 1 or α 1 +1/2 on Fig. 5), ψ = π and thus φ = π/2[π].

-Just above the gap, δ ≈ |p 2 |. The main contribution to u(z) is then sin(Gz + π/2) sin(κz ± π/2) = ± cos(Gz) cos(κz). Therefore, the SL eigenmode corresponds to a Surface Loving Mode.

just below the gap δ ≈ -|p 2 |. The main contribution to u(z) is cos(Gz + π/2) cos(κz ± π/2) = ± sin(Gz) sin(κz): Therefore, the SL eigenmode corresponds to a Surface Avoiding Mode.

Note that we have chosen to base the latter discussion on Eq. ( 15), but Eq. ( 19) could also be used and leads to the same conclusions.

If the distinction between SAM and SLM is obvious in the cases where p 2 is real, in the general case we need a more precise definition: we will speak about SAM or SLM depending if the envelope amplitude at the surface (z = 0) is smaller or higher than its value a quarter period later at z = λκ 4 = π 2κ where λ κ is the wavelength associated with the envelope wave function. With this definition, according to Eq. ( 19), knowing ζ and φ is enough to determine the SAM or SLM character.

Beatings contrast

Let us focus on ζ which, as previously quoted, governs the relative amplitude of the two contribution in Eq. ( 15) and hence drive the beatings contrast, which is defined in Eq.(B4) and plotted in Fig. 6 as a function of the frequency ν = ω 2π . From Fig. 6, the contrast reaches its maximum value of 1 only at the precise band gap edges. In such a case, the envelope function exactly vanishes at its minimum: one could then speak about perfect SAM or SLM. Moving away from the gap, the contrast tends to 0 giving the same weight to both terms in Eq. ( 15). These beatings are also present in an infinite SL, just like in a temporal parametric oscillator. Away from the resonance condition, the excitation goes successively in-phase and out-of-phase with the movement of the system, leading to an increase or a decrease in the amplitude of the oscillations.

SAM or SLM?

To get a strong SAM or SLM, a good beating contrast is needed. Then if the envelope has a minimum or a maximum at the surface, it is a SAM or a SLM, respectively. Indeed, from Eq.( 19) the phase of the envelope relative to the SL surface depends essentially on φ and on the sign of ζ. The latter is seen in Fig. 6 to be negative below the gap and positive above. In this section we focus on φ, the dephasing of the envelope function.

For a given type of SL (materials and thicknesses), φ depends essentially on the frequency ν = ω 2π and on the reduced dephasing α of the SL. As in the whole article, we use here the numerical parameters of the SL described in Sect. II D whose dispersion diagram is plotted in Fig. 3.

Figure 7a) presents in a 3D plot the variations of φ as a function of α and of the frequency ν, calculated using Eq. ( 17) around the second band gap. Figure 7b) reports the value of φ as a function of α for two frequencies above and below the gap: 0.5807 THz and 0.6182 THz. These two frequencies reported in Fig. 6, correspond to a 0.5 beating contrast C (Eq. (B4)), so that at 0.5807 THz In the gap, φ is not defined. b) φ as a function of α at 0.5807 THz (black) and 0.6182 THz (red). Dark curves separate SAM and SLM regions. These curves have been calculated looking for couples (α, ν) that equal the envelope wave function amplitude in z = 0 and z = π 2κ , and correspond to φ = ± π 4 (cf Eq. ( 19)). Parameters corresponding to the SL described in section II D have been used.

(resp. 0.6182 THz), the displacement u(z) (Eq. ( 15)) is dominated by the cos(Gz + ψ/2) cos(κz + φ) term (resp. sin(Gz + ψ/2) sin(κz + φ)).

From Fig. 7a) and b), φ is a periodic function of α with period [START_REF] Brillouin | Wave Propagation in Periodic Structures[END_REF] 2 at fixed frequency ν: indeed, we are looking at the second band gap due to the harmonic at wavevector 2G; by changing α from 0 to 1, the surface z = 0 sweeps two periods of that harmonic. For example, the two remarkable modes discussed above for which p 2 is real, could be obtained for two values of α: these values are obtained by cancelling the d 2 term in Eq. (13c) (n = 2). This has also been illustrated in Fig. 5. The case p 2 real and positive corresponds to α = α 2 = 3/4-γ/2 ≃ 0.39 or α = α 2 +1/2 = 5/4-γ/2 ≃ 0.89, whereas the case p 2 real and negative corresponds to

α = α 1 = 1/2 -γ/2 ≃ 0.14 or α = α 1 + 1/2 = 1 -γ/2 ≃ 0.64.
In Fig. 7a), in addition to the value of φ, we plot dark curves separating SAM and SLM regions. These curves have been calculated looking for couples (α, ν) that equal the envelope wave function amplitude in z = 0 and z = π 2κ , and correspond to φ = ± π 4 (cf Eq. ( 19)). Below the gap, Figure 7a) shows that regions of SAM dominate: especially, as the frequency closes with the gap, the SLM regions are reduced. This result is also coherent with Fig. 7b): for 0.5807 THz, the slope of φ as a function of α in the vicinity of α = 3/4 -γ/2 ≃ 0.39 is very steep, and would be steeper for a frequency closer to the gap. Above the gap, the same conclusions apply: SAM regions dominate the phase space.

SAM/SLM character

In order to summarize the results of the two previous sections, let us define a "SAM/SLM character" as:

χ = ln u rms (0)/u rms π 2κ (20)
which is positive for a SLM and negative for a SAM. The highest |χ|, the more pronounced is the SAM or SLM character of the mode. χ is plotted as a function of the frequency ν and the reduced dephasing α of the SL in Fig. 8. It is clear that the more pronounced SAM or SLM (highest |χ|) are localized close to the gap, and that SAM dominate the phase space.

Numerical Study

To check our analytical predictions based on the SHA, we numerically integrate the differential equation Eq. ( 3) using a 7th-order Runge-Kutta integrator around the 2nd band gap: we thus take into account all harmonics of the inverse square sound speed. We choose here to represent two different cases closed to the remarkable modes we point out above: p 2 real positive for α = α 2 = 3/4γ/2 ≈ 0.39 and p 2 real negative for α = α 1 = 1/2-γ/2 ≈ 0.14. In both cases, we compute solutions of Eq. (3) for 0.5807 THz and 0.6182 THz (0.5 beating contrast) with free boundary conditions at the surface, i.e. u(0) = 1 and du dz (0) = 0; these numerical results are plotted in Fig. 9. For p 2 real positive (α = α 2 ), we clearly observe a SAM above the gap and a SLM below, whereas for p 2 real negative (α = α 1 ), it is the contrary. With respect to the SAM or SLM character, the numerical results thus fully agree with our analytical analysis. Fig. 9 not only validates our calculations and approximations, it also confirms that the parametric resonance analogy is fruitful to get the main physics of wave propagation near band gaps in semi-infinite SL. However, please note that numerical values of the contrasts C and the envelope wavelengths measured in Fig. 9 differ from the ones predicted by Eq. (B4) and Eq. ( 10). We attribute these discrepancies to the SHA. Indeed, the SHA allows the prediction of the width of the 2nd band gap with an error of the order of |p 1 | 2 . Thus, the gap edges predicted by the parametric oscillator analogy slightly differ from the one calculated from the exact transfer matrix method (or by a direct integration of Eq. ( 3)). Due to the almost horizontal dispersion diagram curve near the gap edges, a small variation of frequency ω may induce a non negligible variation in κ. Consequently, while the parametric oscillator analogy gets the main physics and especially allows us to find out the parameters to see SLM, it remains based on the SHA and thus, some care should be taken in order to predict quantitative values, which will better be derived from an exact approach like the transfer matrix method.

IV. OBSERVATION OF SLM AND SAM

We now address the problem of the experimental observability of NBPM. Trigo et al. 18 claim they have shown the existence of SAM using pump-probe experiments coupled to a theoretical analysis. In the following, we would like to focus in detail on the optical activity of NBPM.

Colvard et al. 12 addressed the phonon-electron coupling in GaAs/AlAs SL. The reduced dephasing of phonons compared to layers of the SL is a key parameter for the strength of this coupling. They showed in agreement with experiments that, out of electronic resonance conditions, B 2 phonons, in phase with layers of the SL, do not create a strong electron-phonon coupling and are thus not observable. On the contrary, A 1 phonons, in quadrature phase with layers of the SL do create a strong electron-phonon coupling and are thus observable. These conclusions rely on the electron-phonon coupling mechanism: the deformation potential (in GaAs/AlAs SL), whose hamiltonian is proportional to the divergence of vibration modes.

In the case of NBPM around the band gap at ω

(2) BG , their wave vector G±κ is slightly different from the primitive vector G of the reciprocal lattice, thus their relative phase compared to layers of the SL slowly shifts with z as ±κz. Thus, such modes could be in phase or out of phase with the SL depending on the position z in the SL (as long as κ = 0). However, though the envelope amplitude is varying, we can assume that these modes will be observable if they have the A 1 symmetry (i.e. in quadrature phase with the layers of the SL) at their maximum envelope amplitude.

As demonstrated in Appendix C for the remarkable cases of Sect. III B 1, we find that, out of resonance conditions, observable modes are found below the gap independently on their SAM or SLM nature.

To check our latter analytical prediction, numerical solutions of Eq. ( 3) below (0.5807 THz) and above (0.6182 THz) the gap obtained in Sect. III B 5 are compared to the SL in regions of maximum envelope amplitude. Fig. 9 reports these results. In both cases p 2 real positive (α = 3/4 -γ/2) and p 2 real negative (α = 1/2-γ/2), phonons above the gap are in phase with the layers of the SL at their maximum amplitude envelope in agreement with our prediction: they are hardly observable. On the contrary, phonons below the gap are in quadrature with the layers of the SL and will thus induce a strong electron-phonon coupling in agreement with our predictions.

V. DISCUSSION

We have shown that the envelope amplitude of near Brillouin zone center acoustical phonon in the vicinity of the surface can be maximum (SLM) or minimum (SAM) depending on the reduced dephasing of the SL relative to the surface. For a given function 1 c 2 (z), Eq. ( 1) is a second order differential equation whose solutions are entirely determined by two boundary conditions we can reduce to the definition of (u(0), du dz (0)). Knowing one solution u 1 (z), u 2 (z) = u 1 (z + z ref ) is also a solution for the boundary conditions u 2 (0) = u 1 (z ref ) and We now address two discrepancies between our results and related works in the literature. NBPM "have a tendency to avoid the boundaries, irrespective of the boundary conditions" suggesting that, whatever the boundary conditions, only SAM exist (note that this work 18 also only considers SL with α = 0). If we have restricted our study to the case of free boundary conditions, of course, the study could easily be extended to the case of any boundary conditions. Actually, as suggested in the begining of the section, we found that fixing the reduced dephasing α = 0 and varying the boundary conditions produce the same qualitative conclusions as fixing the boundary conditions and varying the relative phase: both SAM and SLM exists and SAM regions dominate the phase space. We have chosen not to discuss in detail such effects of changing boundary conditions because the case of free boundary conditions is experimentally the most relevant. To illustrate the correctness of our analysis, and in disagreement with statements of Ref. 18, the reader can easily check the existence of a SLM by direct integration of Eq. (3) using the following parameters: u(0) = 1.5 and du dz (0) = 1 and ν = 0.5807 THz and the SL defined in Sect. II D with α = 0 (the same as Ref. 18).

Let's now consider experimental observations of SAM or SLM. Ref. 18 claims to observe SAM under the gap using a SL with α = 0 and free boundary conditions. This observation agrees with our analysis: from Fig. 8, NBPM under the gap for α = 0 are SAM and from Sect. IV they are likely to be observable.

In Sect. IV, we show that both SAM and SLM could in principle be observable. We would like to address here technical problems associated with the observation of SLM. Indeed, in Fig. 8, we show that SAM regions dominate the phase space. Thus, a high precision on the value of α is needed if one wants to observe SLM. This might prove challenging for SL growth. As an illustration, let's consider the precision on the thickness of layers produced by molecular beam epitaxy, which is about 2 monolayers (≈0.5 nm). In the case of the SL already studied (AlAs(2.35 nm)/GaAs(5.9 nm)) a precision of ±0.25 nm on the value of the dephasing τ corresponds to a precision of ±0.03 on the value of α. If α exactly matches 3/4 -γ/2 = 0.39, the NBPM is a SLM for any value of ν below the gap. However, if α lays in the range 0.39±0.03, The frequency ν has to be lower than 0.583 THz to get SLM. Thus, because of the lack of precision on the layers thicknesses, observations of SLM would require a protocol that selects NBPM with a low enough frequency (below the boundary SAM/SLM shown in Fig. 8): we think such a selection may be achieved by the presence of a substrate. Indeed, the length of the SL and the presence of a substrate is an important parameter that tunes the lifetime of observable phonons created in the SL by the excitation laser. A precise study of these effects will be detailed in a forthcoming publication.

VI. CONCLUSION

We have shown using a fruitful analogy with the parametric oscillator that NBPM could either avoid (SAM) or love the surface (SLM) depending on the relative phase of the SL compared to the sample surface. Moreover, we have shown that both modes should in principle be observable using judicious experimental parameters. Whereas SAM have already been observed, experimental evidences of SLM will be difficult in view of the technical challenge to achieve a convenient SL. Despite these difficulties, we think such experiments are possible choosing an appropriate SL length. The experimental distinction between SAM and SLM may be demonstrated using pump-probe experiments with different penetrating length lasers.

Finally, we would like to underline that our study of sound waves in SL could be generalised to any kind of waves. So that electronic 27,28 or electromagnetic 19 wave functions in electronic SL or Bragg mirrors also show a Surface Avoiding or Surface Loving character depending on the surface termination of the SL.

Fig. 1 FIG. 1 :

 11 FIG. 1: Phase diagram of the Mathieu equation. Parameters inducing non periodic solutions are in dashed regions ("parametric resonances" or "band gaps").

FIG. 2 :

 2 FIG. 2: Schematic illustration of band gaps produced by each harmonic.

( 2 )

 2 BG defined by k 0 (ω (2) BG ) = G i.e. the first resonance due to several harmonics (m = 1 and m = 2). The expression of ω

FIG. 4 : 1 FIG. 5 :

 415 FIG. 4: (color online) Schema of the SL compared to the surface z = 0

FIG. 6 :

 6 FIG. 6: Beating contrast C of the displacement u, defined in Eq.(B4) as a function of the frequency ν = ω 2π . C equals ζ above the gap and 1 -ζ below. Parameters corresponding to the SL described in section II D have been used. Dot circles mark a beating contrast of 0.5 for frequencies 0.5807 THz and 0.6182 THz.

FIG. 7 :

 7 FIG.7:(color online) a) Value of φ (color blue φ = -π/2, red φ = π/2) as a function of the frequency ν (THz) and the reduced dephasing α. In the gap, φ is not defined. b) φ as a function of α at 0.5807 THz (black) and 0.6182 THz (red). Dark curves separate SAM and SLM regions. These curves have been calculated looking for couples (α, ν) that equal the envelope wave function amplitude in z = 0 and z = π 2κ , and correspond to φ = ± π 4 (cf Eq. (19)). Parameters corresponding to the SL described in section II D have been used.

FIG. 8 :

 8 FIG.8:(color online) SAM/SLM character χ as defined in Eq.(20). A black curve delimits regions of positive and negative χ. Parameters corresponding to the SL described in section II D have been used.

FIG. 9 :

 9 FIG. 9: Numerical solutions of Eq. (3) for 0.5807 THz (below the gap) and 0.6182 THz (above the gap) with free boundary conditions u(0) = 1 and du dz (0) = 0. The reduced dephasing α of the SL compared to the surface z = 0 is respectively α = α2 = 3/4 -γ/2 ≃ 0.39 for figure a) and b) and α = α1 = 1/2 -γ/2 ≃ 0.14 for figure c) and d). For each graph, we plot both a zoom and overall picture of the mode. The overall picture exhibits the Loving or Avoiding character of the modes. The zoom allows the discussion on the phases of the mode compared to the SL when the amplitude of the mode is maximum. Dashed parts represent AlAs layers and blank ones GaAs layers.

  du2 dz (0) = du1 dz (z ref ), and for the function 1 c 2 (z + z ref ). Consequently, if SAM exists, a judicious choice of z ref allows to obtain SLM: one just needs to cut the SL at z ref chosen in the region where the envelope of the SAM is maximum. Assuming free boundary conditions, this choice imposes the reduced dephasing α of the SL, which precisely corresponds to our study. Reciprocally, choosing α = 0, the choice imposes the boundary conditions.

  The first one concerns conclusions of Ref.26: "In the physically important cases of a free and a clamped interface, we emphasize, surface repulsion is a property of all modes with wavevectors close to those of dispersion gaps", suggesting that only SAM exist. We have shown in this work, that in the case of free boundary conditions, both SAM and SLM exist. The discrepancy between our results and the cited work may be explained since calculations of Ref. 26 only consider a SL with a reduced dephasing α = 0 relative to the SL surface. In the precise case of free boundary conditions and α = 0, our results agree with conclusions of Ref.26. The second discrepancy concerns claims of Ref.18:

APPENDIX A: RESOLUTION OF EQ.(6) IN A SEMI-INFINITE SL

Finding the solutions (9a) and (9b) amounts to diagonalize the square matrix in Eq. (8). The eigenvalues are ±iκ, defined by (10). Since we are focusing to the outside of the gap, κ is real and we may choose it positive (Changing κ by -κ is equivalent to permute constant A(B) + and A(B) -in Eq. ( 9)):

from Eq. (11).

After diagonalization, the following relations are found between the constants in Eq. ( 9):

where ψ is the argument of p 2 , so that

Let us now impose a real amplitude u 0 at the surface, u(0) = u 0 . This boundary condition reads

We may try B -= A + , since their respective factors are complex conjugates, and u 0 is real. Please note that changing u 0 to be complex would merely amount to dephase the vibration in time. Then, introducing A 0 (real) and φ so that A + = A 0 e i( ψ 2 +φ) , the displacement reads

with

The boundary condition du dz (0) = 0 yields to the following expression which allows the determination of the integration constant φ:

Note that Eq. (A7) defines φ modulo π. However, u(z) is fully defined in Eq. ( 15) by defining the couple A 0 and φ.

We thus use the convention to choose A 0 > 0 in Eq. (A6): φ is now defined modulo 2π. Note that close to the gap, this procedure always yields a value of φ in ] -π 2 , π 2 ]. We found one solution (A5) of this second order differential equation. This is the only one satisfying the given free boundary conditions. Finally, Eq. (A5) can be written in the alternative form (15), more convenient for the interpretation.

APPENDIX B: ENVELOPE EQUATION

As seen in Fig. 9, the displacement consist in a fast oscillating term at wave vector G, modulated by a lower wave vector envelope. One way to define this envelope is to take the square of u in Eq.( 15)

and then to average on the characteristic time of the fast oscillating term. So, we define:

This procedure amounts to take a local mean, around every position z, on a 2π G range. In Eq. (B1), using κ ≪ G, the cos 2 (Gz + ψ 2 ) and sin 2 (Gz + ψ 2 ) yield 1 2 and the last term vanishes. Finally, we end up with a local "root mean square" positive amplitude u rms

which will also be called the envelope of the displacement in this paper. The contrast of the beatings in u may be defined as

which is equal to ζ above the gap and to To check if NBPM around band gap at ω

BG have the A 1 symmetry at their maximum envelope amplitude, we examine their relative phase compared to the layers of the SL. So that, whereas in Sect. III B we were interested in the phase φ of the envelope compared to the SL surface, we now get interested in the phase ψ 2 of the quickly oscillating function (at spatial frequency G) compared to the layers of the SL.

Eq. (15) gives the analytical expression of NBPM. The layers of the SL can be described using the inverse square sound speed as shown in Fig. 4. However, only the first harmonic (at G) of the SL is necessary to get a picture of the layers of the SL: the coefficient p 1 defined by Eq. (13d) determines the phase of layers of the SL.

We thus, only have to compare the relative phase of the phonons (at their maximum envelope amplitude) to the one of the first harmonic of the inverse square sound speed.

We now examine the two remarkable cases mentioned in Sect III B 1.

Let us start with the cases p 2 real and positive obtained for α = 3/4 -γ/2 or α = 5/4 -γ/2: the SAM mode, above the gap, varies like sin(Gz) sin(κz), whereas the SLM, below the gap, varies like cos(Gz) cos(κz). From Eq. (13b) and Eq. (13c), we can see that c 1 = 0 and d 1 = 0 which implies that the first harmonic of the SL varies like sin(Gz). Thus, it turns out that the SAM mode is in phase with the SL at its maximum amplitude (kz = π/2[π])): the electron-phonon coupling is thus weak which makes it hardly observable. On the contrary the SLM mode is in quadrature with the SL at its maximum amplitude (kz = 0[π])): the electron-phonon coupling is high and so, this SLM mode is likely to be observable.

A similar analysis for the cases p 2 real and negative (α = 1/2 -γ/2 or α = 3/2 -γ/2) leads to the conclusion that in that case, the SLM mode, above the gap, will be hardly observable whereas the SAM mode just below the gap, is likely to be observable.

Hence, in both remarkable cases, observable modes are found below the gap.


A H Essadki
email: essadki@est-uh2c.ac.ma

I Nikov

H Delmas

Influence of Bubbles Characteristics on the Skin Friction and Velocity Gradient on Solid Sphere

Keywords: electrochemical probe, velocity gradient, frictional force, calibrated bubbles bubble column, statistical and spectral analysis

A detailed study of the effects of individual bubbles at high gas flow-rate has shown, that the dominant influence on skin friction over a solid sphere is the bubble volume in compared to bubble frequency. Nevertheless the bubble frequency is very important in case of low gas flow-rate. Referring to bubbles produced by a gas distributor, statistical and spectral analyses were performed to study the influence of bubbling on exposure time and magnitude of fluctuations. Referring to a calibrated bubble train, the existence of critical frequency is demonstrated. A bubble with larger volume and a mobile, oscillatory surface generates larger velocity gradient. In the case of gas distribution, histograms of the velocity gradient for a 2 mm glass sphere creating bubble coalescence reveal the maximum exceeds 48 000 s -1 in the front zone and 2000 s -1 in the rear zone ( = 180 • ). For 5 mm plastic spheres creating bubble break-up, the maximum of the velocity gradient is only 8100 s -1 for the front part of the sphere and 2000 s -1 in the rear zone.

INTRODUCTION

D

rag-velocity analysis on a particle in liquid fluidised bed system has shown that a major portion of the power draw of flow conforms to pressure drag [START_REF] Nikov | Mechanism of Liquid-Solid Mass Transfer and Shear Stress in Three-Phase Fluidized Bed[END_REF]. Only a small part of it, up to 20%, is due to skin friction in the external flow. However, in gas-liquid fluidised bed, the skin friction which increases rapidly with wall shear rate is related to bubble behaviour [START_REF] Essadki | Local Shear and Skin Friction Particles in Three-Phase Fluidized Beds[END_REF]. The shear rate on particles immersed in real media of food and bioproduct processing is an important component of it and should be considered. Thus, for good understanding of the role of bubbles, their properties as velocity, frequency and size have to be studied to correlate them to wall velocity gradients and the frictional force on a solid. [START_REF] Bakke | Biological nitrogen removal from wastewater; simulations and control strategies[END_REF] showed that operating with thin films (<50 m) may change the performance characteristics of the reactor. [START_REF] Stathopoulos | Shear Stress Effects on Human Embryonic Kidney Cells In Vitro[END_REF] showed that microorganisms attached to microcarriers are very sensitive to excessive agitation. This may lead to biomass destruction. [START_REF] Croughan | Hydrodynamic Effects on Animal Cells Grown in Microcarrier Cultures[END_REF] have shown that the velocity gradient magnitude adjacent to microcarriers can be determining factors for growth. [START_REF] Lertpocasombut | Carbon removal by thin biofilm in a three-phase fluidized bed[END_REF] found that the biofilm thickness decreased rapidly with increasing gas velocity. Knowledge of local pressure drop by friction to the solids of a fluidised bed in columns with two-or three-phase flows allows to determining the thickness of the fixed biomass layers. In most works reported in the literature, friction is deduced from global measurements such as total pressure gradient and void fraction which may not be representative for the local conditions.

Flow complexity in two-and three-phase beds do not lead to modelling and determination of friction. However, using electrochemical spherical probe, [START_REF] Essadki | Local Shear and Skin Friction Particles in Three-Phase Fluidized Beds[END_REF] present a simple local method to obtain the velocity gradients. This work is aimed at determining to friction force and shear rate on a particle in twoand three-phase fluidised beds. The electrochemical shear-rate measurement has been validated for single-phase flow. Correlations for frictional force in fluidised beds have been worked and recommended in cases of coalescence system and break-up systems. [START_REF] Vlaev | Shear and Skin Friction on Particles in Power-Law Fluids Agitated by Flat Blade and Fluid Foil Impellers[END_REF], using the same electrochemical method, have studied the effects of rheology and impeller design on the shear field. The skin-friction drag corresponding to the impellers has been determined and compared. [START_REF] Yuichi | Skin Friction Reduction by Large Air Bubbles in a Horizontal Channel Flow[END_REF] have studied the local skin friction profiles for individual bubbles. They concluded that the bubble size depend of the average skin friction in the intermediate bubble size condition. Groˇe and Schr öder (2008) have used the micropillar wall shear stress sensor to measure the dynamic wall-shear stress in turbulent pipe flow.

The present work studied the influence of individual bubbles on the frictional forces and the fluctuations of velocity gradient generated in bubble column (BC) and two-and three-phase fluidised beds.

MATERIALS AND METHODS

Experimental Apparatus

A plexiglass column of I.D. 9.4 cm and 157 cm high was employed. An electrochemical probe used as the working electrode, a nickel ring as an auxiliary electrode, and a reference electrode furnished with a Luggin's capillary were connected to potentiometer which was linked to a computer performing the acquisition of electrical current signals. Details of the apparatus have been described in a previous work [START_REF] Essadki | Local Shear and Skin Friction Particles in Three-Phase Fluidized Beds[END_REF].

The signal has been digitised via a 12-bit UEI-800 A/D card. A home-made FORTRAN program permitted the control of data acquisition duration, frequency, statistical and spectral analysis.

Part of the column was surrounded by a square duct in order to eliminate optical distortion for visualisation. The bubbles were generated by an orifice gas distributor equipped with 1, 2 and 4 mm tubes T 1 , T 2 and T 4 submerged in the liquid in the centre at the column bottom section. The tubes were placed 5 cm below the probe. The probe was placed 30 cm above the liquid distributor.

The distributor of gas was used for generating bubbles across the whole section area. It consisted of two-concentric circular tubes of 4 and 7 cm diameters with 90 regularly spaced holes (0.4 mm diameter).

Measurement of Velocity Gradient

Electrochemical technique has been adopted to measure the diffusion-limiting current, I l , during the reduction of electroactive species [START_REF] Essadki | Local Shear and Skin Friction Particles in Three-Phase Fluidized Beds[END_REF]. The reduction occurred on a small active circular surface flush with the surface of a polymer (plastic) sphere 5 mm in diameter. The sphere was equipped with an inside channel bent 90 • in which a gold wire of 1 mm diameter was introduced cut flush with the surface (Figure 1). A rigid tube served as support. The microelectrode should be positioned relative to the average direction of the main liquid flow by rotating the support. The graph 'electrical current intensity versus potential' exhibit plateau, the height of which is related to the intensity of the limiting current of the working electrode. The reduction of ferricyanide takes place at the working electrode, whereas, the oxidation of ferrocyanide takes place at the counter electrode. The physico-chemical properties of the solution are given in Table 1.

The frictional drag force was obtained by integrating the velocity gradient at the sphere surface S [START_REF] Essadki | Local Shear and Skin Friction Particles in Three-Phase Fluidized Beds[END_REF]: Table 1. Physico-chemical properties of the solution used in the study Temperature ( • C) 25 l (×10 -3 kg m -3 ) 1.0204 l (×10 3 Pa s)

Ff = 2R 2 l 0 S() × sin 2 d (1)
1.17 D (×10 10 m 2 s -1 ) 5.17

The shear stress is defined by: = l × S, in which l is the dynamic viscosity (Newtonian liquid). The apparent mass transfer coefficient K a at any microelectrode position is given by:

K a = I l n × F × A e × C 0 ( 2
)
I l is the intensity of the limiting current of the working electrode and S is related to K a by the relationship:

S() = (1.16K a ) 3 × d e D 2
(3) D is the diffusion coefficient of the potassium ferrocyanide. D is measured by means of a turning disc using the relationship [START_REF] Levart | Analyse du Transport Transitoire sur un Disque Tournant en Régime Hydrodynamique Laminaire et Permanent[END_REF]:

J = D × C 0 Sc 1/3 ω 1/2 1 k k = 1.61173 + 0.480306Sc -1/3 + 0.23393Sc -2/3 + 0.113151Sc -1 (4)
Sc is the Schmidt number.

Signal Statistical and Spectral Analysis

A statistical analysis was applied to the temporal velocity gradient signals.

Discrete random variables can be plotted in a histogram which shows the frequency (on the ordinate) as a function of some measured parameter (on the abscissa for a given class width).

A probability density function (PDF) indicates the probability for a certain shear rate value to occur within a certain range conditions.

The relative standard deviation of the shear rate is defined as:

s = s 2 1/2 S ( 5
)
s 2 = 1 T N i=0 s 2 (t i ) i T S = S + s(t) s = 1 T N i=0 s(t i ) i T (6)
T is the time period acquisition time

t i = i * T
The signal treatment was developed by fast Fourier transforms. These later were used to determine the power spectrum of the fluctuations of the velocity gradient, s:

P s (f ) = 1 T |F s (f )| 2 = 1 T T 0 s(t)e -2ift dt 2 (7)
and F s is the Fourier transform of s(t).

The aim, for a given data set of P s computation, is intended to establish its frequency distribution and its relationship to the physical system involved.

The autocorrelation function gives evidence about the bubble passage process history and is obtained by an inverse Fourier transform, as suggested by Wiener-Kinchine theorem: The normalised autocorrelation function r(t) is obtained, as:

R(t) = ∞ 0 P s (f )e 2ift df ( 8
)
r(t) = R(t) s 2 (9)
To help evaluate the major frequencies in the spectrum, the average frequency of fluctuations of velocity gradient is defined, as follows:

f = P s (f )f df P s (f ) df (10)
The aforementioned integrals can be evaluated numerically, as:

f = Ps(f )f Ps(f ) (11)
All these tools allow correlation of the signal generated by the bubbles, as well as the stress distribution. The knowledge of different aspects of the signal, namely, histogram, PDF, autocorrelation, variance, etc. gives the information and uncovers the direct relationship between the signal and the prediction of shear rate or shear stress distribution.

Measurement of Volume, Velocity and Frequency of Bubbles

A high speed camera was used, for example 1000 images per second to visualise and analyse the images. It was observed that under a stationary flow-rate, the bubbles formed in line had the same shape and identical volume.

The bubble frequency was determined by counting the number of bubbles in a fixed location during of 10 s time interval, whereas, bubble rise velocity was deduced by observing the passage in locations x 1 and x 2 at time t 1 and t 2 .

The bubble volume was deduced from frequency and gas flowrate data.

Vb = Q g f b (12)
The electrochemical probe was used to measure the bubble frequency and bubble volume at low gas flow-rates [START_REF] Essadki | Electrochemical Probe for Bubble Size Prediction in a Bubble Column[END_REF]. Thus, the regular passage of bubbles close to the electrochemical probe at diffusionally limiting current or velocity

RESULTS AND DISCUSSION

Single Orifice: Bubble Train

The average velocity gradient is deduced from the apparent mass transfer coefficient as suggested by [START_REF] Reiss | Experimental Study of the Unsteady Nature of the Viscous Sublayer[END_REF] for single phase. [START_REF] Souhar | Wall friction in gas-liquid two phase flows in vertical ducts: bubble and slug regimes[END_REF] verified that the method is valid for two-phase flow conditions if the wall was totally wetted by liquid with a film thickness of at least 20 m. Previous experiments reported by [START_REF] Essadki | Local Shear and Skin Friction Particles in Three-Phase Fluidized Beds[END_REF] for BCs and fluidised beds showed that minimum film thickness is of about 30 m indicating that the condition was fulfilled. In Figure 3, the external wall velocity gradients are shown as functions of gas flow-rates corresponding to different angles between the electrode surface and the main fluid flow direction. The plots lines indicate that a clearly defined maximum obtained without gas at 45 • [START_REF] Essadki | Local Shear and Skin Friction Particles in Three-Phase Fluidized Beds[END_REF] was less accentuated with gas and the maximum was close to near 30 • . The presence of bubbles has an important influence concerning the flow behaviour around the solid sphere such as drastic reduction of the wake.

The goal of the section has been targeted to study the effect of bubble characteristics such as frequency, velocity and volume. Those three parameters are most important for controlling the frictional force. The results obtained are summarised in Table 2a-c. Bubble volume and frequency were selected as independent variables and to minimise the influence of liquid velocity, this later was fixed at low value (0.27 cm s -1 ). As shown in Figure 4 at low gas volumetric flow-rate (Q g < 3.3 cm 3 s -1 ), inducing a bubble frequency less than 17 Hz, the influence of bubble frequency on frictional force is better pronounced than the bubble volume. For example, the frictional force created by the train of bubbles 0.2 cm 3 at bubble frequency 5 Hz is about 8 × 10 -5 N; referring to bubble volume of 0.06 cm 3 and bubble frequency of 15 Hz, the frictional force induced is about 14 × 10 -5 N.

In contrary, bubble frequency can be neglected when the gas volumetric flow-rate increases (Q g > 4.9 cm 3 ).

In conclusion, a bubble with larger volume and a mobile, oscillatory surface generated high level of turbulence that induced high rate of liquid-solid mass transfer corresponding to high velocity gradient. The existence of a critical frequency has also been demonstrated.

Bubble Column and Fluidised Bed: Gas Distributor (Distribution)

Gas flow dynamics and solid content effect on the velocity gradient exposure

Figure 5 shows histograms of the velocity gradient in a BC at a 30 • angle position, representing the sphere front corresponding to various superficial gas velocities. The liquid velocity is fixed (U l = 0.27 cm s -1 ). The Figure 5 shows the gradual development in gradient patterns as the gas velocity increased, the ordinate value giving the time fraction exposed to stress.

The maximum value for the velocity gradient increased steadily with gas flow. This increase in maximum values could be attributed more to bubble-to-bubble interaction caused by the increased gas fraction and the collisions close to the probe, thus increasing local liquid shear. The maximum can reach 28 000 s -1 . In the rear zone the maximum value for U g = 5.3 cm s -1 is 2100 s -1 .

Another aspect is observed in these histograms corresponding to transition bubble regime at gas velocity between 2.5 and 5.2 cm s -1 . The maximum value in this case jumps from 7100 up to 28 600 s -1 .

In the case of fluidised bed, it is important to distinguish between a coalescence system and a break-up system. Biofilm reactors are frequently performed in fluidised or suspended bed. In this case, the decay or biomass growths are probably affected by the range of variation and the time of hydrodynamic stress exposure. In our previous work [START_REF] Essadki | Local Shear and Skin Friction Particles in Three-Phase Fluidized Beds[END_REF], the velocity gradient and so the frictional drag force increased much more with gas velocity in the coalescence regime than in the bubble break-up regime. For a glass sphere of 2 mm diameter, bubble coalescence occurs, whereas, for plastic spheres of 5 mm, a bubble break-up phenomenon predominates.

The same tendency is observed for glass beads creating bubble coalescence (Figure 6) in the BC. In addition, other values of velocity gradient appear and the maximum exceeds 48 000 s -1 in the front zone and 2000 s -1 in the rear zone ( = 180 • ). For plastic spheres creating bubble break-up, the maximum of velocity gradient is only 8100 s -1 (Figure 7) for the front part of the sphere and 2000 s -1 in the rear zone.

The results found for both BC and fluidised bed confirm the prevalence influence of bubbles size.

Gas flow dynamics and solid content effect on relative standard deviation and fluctuation frequency of the velocity gradient

The relative standard deviation of the shear rate s , shows the gas impact on the velocity gradient fluctuations which may be seen as a measure of the liquid phase turbulent fluctuations. In the BC, Figure 8 shows that the largest relative deviation was found at the lowest gas input. At low gas flow-rate, the large fluctuations were caused by single bubbles moving at a frequency low enough for the liquid to return to a semi-stable rest state between bubbles. At higher gas flow-rate the bubble-bubble interaction became stronger and the increase of the average velocity gradient was defined by the medium change and the increase of the maximum gradient values. It is known that, in case of a single-phase liquid, the velocity gradient fluctuations change with position on the sphere surface: the flow pattern on the front hemisphere showed a low fluctuation level, whereas, the rear zone showed higher relative fluctuation levels. With the introduction of gas, the tendency changed completely. In the fluidised bed case, Figure 9 shows the relative fluctuation. In contrast to the case of BC, the fluidised bed showed a continuous increase in fluctuation at gas flow variation. This increase was more important in the case of glass beads creating coalescence than the plastic creating break-up in which the increase of fluctuation was very slight. The existence of large bubbles in coalescent fluidised bed occurred when the superficial gas velocity exceeded than 0.6 cm s -1 . This is showed by the jump in the maximum values in the histograms, for example from 6100 s -1 at 0.6 cm s -1 to 16 100 s -1 at 1.3 cm s -1 in Figure 6.

In general, the fluctuation level was found to be important in a fluidised bed than the BC (Figure 10) the reason being the presence of solids. Indeed, the solids fraction in the bed was about 0.45 giving interstitial velocities about twice as high as those in the BC and this might be the main reason for the difference.

An example of a power spectrum is given in Figure 11 for = 60 • for the fluidised bed without gas. The same tendency is obtained with different superficial gas velocities and both for coalescence and break-up fluidised bed. The gas influence does not appear in the spectrum which present probably the solid collisions and mask the bubbles interactions.

The average frequency is different in the cases of break-up system and the coalescence system. As expected, Figure 12 shows that the average frequency is greater in the case of plastic spheres (break-up made) than in the case of glass spheres (coalescence made).

As expected by the study of the influence of individual bubbles, the influence of bubble volume was found to be more important than bubble frequency in fluidised bed.

CONCLUSION

The study of the instantaneous local velocity gradients at the surface of fluidised particles in gas-liquid-solid flow reveal that compared to the bubble frequency bubble volume has a predominant effect. The exposure time of shear rate or shear stress shows that in coalescence regime the three-phase fluidised bed generates high values of the velocity gradient at the surface particle. In contrast to the coalescence case, the three-phase fluidised bed operating in break-up regime generates relatively low values. In both systems, the front zone of the sphere is subject to larger deformation than the rear one.

The relative standard deviation of the velocity gradient at the wall in the case of fluidised bed promoting bubble coalescence shows that fluctuations increase with the superficial gas velocity and reaches 80%. In the case of fluidised bed promoting bubble break-up, the fluctuations remain almost constant and reach 60%. Expected, the spectral analysis confirmed that the average frequency in the 'break-up' fluidised case is greater than in the 'coalescent' fluidised case. bubble velocity (m s -1 ) U g superficial gas velocity (m s -1 ) V b bubble volume (m 3 ) U l superficial liquid velocity (m s -1 ) l dynamic viscosity of liquid (Pa s) cinematic viscosity of liquid (m 2 s -1 ) s relative standard deviation of shear rate polar angle with respect to front stagnation point ω angular velocity (s -1 )

Figure 1 .

 1 Figure 1. The electrochemical probe.

Figure 2 .

 2 Figure 2. a: History records of the velocity gradient at the wall: Effect of the regular passage of bubbles near the electrochemical probe on the wall velocity gradient, = 30 • . b: Power spectral density function: Effect of the regular passage of bubbles near the electrochemical probe on the wall velocity gradient, = 30 • . c: Autocorrelation function of the velocity gradient at the wall: Effect of the regular passage of bubbles near the electrochemical probe on the wall velocity gradient, = 30 • . [Color figure can be seen in the online version of this article, available at http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1939-019X]

Figure 3 .

 3 Figure 3. Velocity gradient at the wall versus for various gas flow-rate: Influence of calibrated bubbles generated by tube T 2 (d o = 2 mm), U l = 6.4 cm s -1 . [Color figure can be seen in the online version of this article, available at http://onlinelibrary.wiley.com/journal/10.1002/ (ISSN)1939-019X]

a

  T 1 (d o = 1 mm) and U l = 0.27 cm s -1 . b T 2 (d o = 2 mm) and U l = 0.27 cm s -1 . c T 4 (d o = 4 mm) and U l = 0.27 cm s -1 .

Figure 4 .

 4 Figure 4. Frictional force versus bubble frequency for various bubble velocities and volume bubble, U l = 0.27 cm s -1 . [Color figure can be seen in the online version of this article, available at http://onlinelibrary. wiley.com/journal/10.1002/(ISSN)1939-019X]

Figure 5 .

 5 Figure 5. Histograms of the velocity gradient in bubble column (BC) at the 30 • angle, U l = 0.27 cm s -1 . [Color figure can be seen in the online version of this article, available at http://onlinelibrary.wiley.com/journal/ 10.1002/(ISSN)1939-019X]

Figure 6 .

 6 Figure 6. Histograms of the velocity gradient in fluidised bed (glass spheres, 2 mm) at the 30 • angle, U l = 0.27 cm s -1 . [Color figure can be seen in the online version of this article, available at http:// onlinelibrary.wiley.com/journal/10.1002/(ISSN)1939-019X]

Figure 7 .

 7 Figure 7. Histograms of the velocity gradient in fluidised bed (plastic spheres, 5 mm) at the 30 • angle, U l = 0.27 cm s -1 . [Color figure can be seen in the online version of this article, available at http:// onlinelibrary.wiley.com/journal/10.1002/(ISSN)1939-019X]

Figure 8 .

 8 Figure 8. Relative standard deviation of the shear stress in the bubble column (BC). [Color figure can be seen in the online version of this article, available at http://onlinelibrary.wiley.com/journal/10.1002/ (ISSN)1939-019X]

Figure 9 .Figure 10 .

 910 Figure 9. Relative standard deviation of the shear stress in fluidised bed. fbv, glass beads (2 mm); fbp, plastic beads (5 mm). [Color figure can be seen in the online version of this article, available at http://onlinelibrary. wiley.com/journal/10.1002/(ISSN)1939-019X]

Figure 11 .Figure 12 .

 1112 Figure 11. Typical spectrum of the three-phase fluidised bed. [Color figure can be seen in the online version of this article, available at http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1939-019X]

  of s(t) (s -1 ) I l intensity of diffusional-limiting current (A) J molar flux species (mol m -2 s -1 ) K a apparent local mass transfer coefficient (m s -1 ) n number of electrons liberated during the course of the electrochemical reaction Q g volumetric gas flow-rate (m 3 s -1 ) P s the power spectrum of the fluctuation of the velocity gradient (s -1 ) R sphere radius (m) R(t) the autocorrelation function (m 2 s -2 ) r(t) the normalised autocorrelation function S velocity gradient at the wall (s -1 ) S average velocity gradient at the wall (s -1 ) s fluctuations of the velocity gradient at the wall (s -1 ) s 2 1/2 average fluctuations of the velocity gradient at the wall (s -1 ) Sc Schmidt number defined by /D t time (s) T duration of the experiment (s) shear stress (Pa) U b

Table 2 .

 2 Frictional forces on the electrochemical probe: influence of bubbles induced by different tubes diameters

	Q g (cm 3 s -1 ) a V b (cm 3 )	0.3 0.055	1 0.062	1.6 0.055	3.3 0.064	5 0.122	6.7 0.175
	U b (cm s -1 )	26.7	26.6	32.4	33.8	37.3	37.2
	f b (Hz) F f × 10 5 (N)	6 8.1	16 14.8	30 18.9	52 19.7	41 23.2	38 28.4
	Q g (cm 3 s -1 ) b V b (cm 3 )	0.3 0.11	1 0.198	1.6 0.155	3.3 0.208	5 0.199	6.7 0.202
	U b (cm s -1 )	25.3	26.1	29.1	32.7	35.7	38.5
	f b (Hz) F f × 10 5 (N) Q g (cm 3 s -1 ) c V b (cm 3 )	3 5.5 0.3 0.165	5 8.6 1 0.25	11 16.4 1.6 0.21	16 27.3 3.3 0.24	25 28.6 5 0.29	33 38.8 6.7 0.32
	U b (cm s -1 )	26.1	28.7	28.6	32	35.2	37
	f b (Hz) F f × 10 5 (N)	2 4.9	4 10.3	8 16.2	14 27.7	17 32.2	21 39.4