Harunori N Yoshikawa
email: harunori@unice.fr

Farzam Zoueshtiagh
email: farzam.zoueshtiagh@univ-lille1.fr

Hervé Caps
email: herve.caps@ulg.ac.be

Pascal Kurowski

Philippe Petitjeans

Bubble splitting in oscillatory flows on ground and in reduced gravity

Keywords: 47, 55, dd Bubble dynamics -47, 35, Pq Capillary waves -47, 20, Dr Surface-tension-driven instability

The stability of centimeter scale air bubbles is studied in quiescent suspending liquid under imposed oscillatory acceleration field. Experiments were performed in reduced and normal gravity environments. A strong acceleration resulted in an instability leading to the breakups of the bubbles in both gravity environments. The breakup onset was investigated and found to be characterized by a critical acceleration acr. The influence of the liquid viscosity and the gravitational environment was studied. Empirical correlations for the onset are presented and discussed with the intention to reveal splitting mechanism. The inertial mechanism often deemed to cause the breakup of drops subjected to a rapid gas stream is shown to give explanations consistent with the experiments. A breakup criterion for both gravitational environments is proposed through discussions from an energetic point of view.

Introduction

Flows including dispersed phase are ubiquitous in the nature and over the whole human activity. Blood circulation transporting cells [START_REF] Thiriet | Biology and Mechanics of Blood Flows: Part II: Mechanics and Medical Aspects[END_REF], atmosphere with falling raindrops [START_REF] Pruppacher | Microphysics of clouds and precipitation[END_REF], gas-liquid mixture in bubble column reactors and flows in rivers and sea with sand suspension [START_REF] Fredsøe | Mechanics of coastal sediment transport[END_REF] are only some examples of such flows. Behaviors of dispersed phase in different flows have been extensively studied as it is a key for understanding the global transfer of momentum, heat and mass. The size distribution of that phase which decides the dynamics and the associated surface area is fundamental for modeling the transfer mechanism in the flow. The distribution can vary through breakups and/or coalescences due to interactions with continuous phase. The stability of the dispersed phases is thus of great importance.

Fluid systems subjected to periodic excitations have been studied with interest in striking modification of their behavior by the amplitude and frequency adjustments [START_REF] Davis | [END_REF]. Faraday instability is one of the well-known subjects of these studies [5,6]. Such systems are also investigated with the intention of flow control, e.g., [7,8] and drop atomization [9,10].

Motion of bubbles in vertically oscillating liquid has been a subject of study for performance improvement of bubble column reactors in the process industries [11]. Bubble volume change caused by bubble oscillatory vertical displacement of typical frequency of the order of 100 Hz can yield a net vertical force component which modifies the bubble rise in the reactors [12]. The present work concerns bubbles in liquid vertically vibrated at frequency of the order of 10 Hz. Behavior of bubbles without volume change in the acceleration field induced by the vibration is studied with focus on vibration-induced bubble breakups. This breaking method can be applied to many industrial processes for controlling bubble sizes. In particular, applications are expected in factories in space, where bubble size is no longer limited to the capillary length scale because of the absence of gravity.

Bubbles subjected to an oscillatory acceleration field are set in translational motion in stagnant suspending liquid by the buoyancy due to the acceleration field. They can eventually break up into smaller bubbles when the field is strong enough. The stability of large bubbles in such a configuration was first experimentally studied by Zoueshtiagh et al. [13]. The oscillatory acceleration field was induced by oscillating a test cell vertically with a given amplitude A and frequency f . The authors reported on behavior of bubbles of volume V b ranging from 1.1 cm 3 to 8.1 cm 3 in various suspending liquids with different surface tensions and viscosities. It was found that the imposed acceler- ation could have a control over bubble size distribution through bubble breakups. For large A values, compared to the volume equivalent diameter D e = (6V b /π) 1/3 , the bubble size was shown to be limited by twice an effective capillary length, based on the imposed acceleration, l c = γ/ρAω 2 (γ: surface tension, ρ: density of the suspending liquid, ω: angular frequency).

The purpose of the present work is to better understand the physical mechanisms behind the breakup instability of large bubbles in such oscillatory flows. Experiments have been performed to complete the results presented in the preceding work [13,14]. Particularly, the influence of a solid wall have been considered, by experiments on ground in normal terrestrial gravity environment (later designated by G) where bubbles were always in the proximity of a container wall, as well as in a microgravity environment (designated by µG), where the bubbles were apart from any walls. A universal criterion for both gravity environments will be proposed.

Experimental set-ups and Control Parameters

Experiments were prepared with the intention to determine the regimes for which a centimetric bubble subjected to vertical vibrations becomes unstable and breaks up. For this purpose, a test cell, which consisted of a polycarbonate parallelepiped container of 6.0×8.0×6.0 cm 3 filled by a test liquid and an air bubble of a given volume, was set in vertical oscillation. The liquid was either deionized water or various silicone oils. The physical properties of those fluids are listed in Table 1. The cell was prepared by filling the container through a 0.2 cm diameter side hole. The container was first fully filled with liquid and then some of the liquid was removed by a graduated syringe to create a bubble of a controlled volume. The hole was then blocked with a screw having the same length as the wall thickness to seal the cell. The tested bubble volume V b ranged between 2 cm 3 and 7 cm 3 .

For exploring the excitation frequency-amplitude plane, two distinct set-ups have been used. Each of these was related to a particular range of frequency and amplitude. The first setup (ST1) produced large oscillation amplitudes (0.75 ≤ A ≤ 12 cm) but delivered low oscillation frequencies (0.1 Hz ≤ f ≤ 10 Hz), while the second setup (ST2) provided small amplitudes (A ≤ 1 cm) at much larger frequencies (5 Hz ≤ f ≤ 100 Hz).

The setup ST1 is shown in fig. 1. It consisted of a linear unit system connected to a SGMPH-04AA Fenwick servomotor. The system converted the rotating motion of the drive pin into a linear movement of the guide platforms 1 and 2. The test cell was attached to the lower sliding plate, i.e. platform 1, while on platform 2 a counterweight of the cell was mounted. The platforms were moved to the middle or apart simultaneously. This anti-phase movement was aimed at reducing vibration transmission to the frame structure of the setup. The motion of the drive was computer controlled. The accuracy of the amplitudes and frequencies were held to within ±0.1 cm and ±1/60 Hz, respectively. The drive motion, hence the movement of the unit, could provide sinusoidal motions at speeds up to 500 cm/s and accelerations up to ∼ 3500 cm/s 2 .

The setup ST2 consisted of a commercial electromagnetic shaker providing vertical vibrations to the test cell. It was driven by amplified electric signals of a function generator. An accelerometer attached to the cell measured the provided accelerations. The acceleration values between 800 and 10000 cm/s 2 .

Experiments in G were performed in the laboratory with either ST1 or ST2 . Bubbles of different volumes (2 ≤ V b ≤ 7 cm 3 ) in different test liquids were subjected to acceleration of widely ranged frequency (0.1 ≤ f ≤ 100 Hz). Experiments in µG were realized on aircraft in parabolic flights. At each parabola, microgravity state lasted 22 s and its quality was typically 10 -2 g (g = 981 cm/s 2 ). Bubbles of 3 cm 3 were accelerated in different test liquids by ST1.

Bubble behavior was observed from above with a digital high-speed video camera at typically 150 images per second with an exposure time of typically 1 ms. The optical axis of the camera was set parallel to the translation direction of the cell so that the bubble motion could be observed in a longitudinal view from the top. Side view of the cells was allowed by a 45 degree inclined mirror attached to the cell (see fig. 1). Thanks to the mirror, longitudinal and transversal views of the cell were simultaneously recorded on the same pictures.

Dimensionless numbers

Various time scales can be associated with bubble dynamics in present experiments. Besides the time scale of the oscillations t osci ∼ ω -1 , the viscous time t ν ∼ D 2 e /ν and the advection time t adv ∼ D e /Aω characterise the flow surrounding the bubble. Bubble's free oscillation is characterised by the capillary time t c ∼ ρD 3 e /γ 1/2 . These four time scales yield the following three dimensionless parameters: the dimensionless frequency Ω = t ν /t osci , the dimensionless amplitude K C = t adv /t osci that is identical to the Keulegan-Carpenter number and the Ohnesorge number Oh = t c /t ν . More explicitly, they are defined as:

Ω = ωD 2 e 2ν , K C = 2A D e and Oh = ρν 2 D e γ (1)
In the present experiments, the dimensionless amplitude K C was varied over a wide range from 0.018 to 14. The Ohnesorge number Oh was also varied widely but always had values smaller than unity, ranging from 8.8×10 -4 to 0.16. Regarding the frequency parameter Ω, its values were much larger than the unity in both G and µG when bubble breakups were observed (19 < Ω < 5×10 4 in G and 12 < Ω < 2600 in µG ). These ranges of Oh and Ω imply that viscous dissipation occurs slowly compared to the oscillation and capillary time scales. Furthermore,

Experimental Results

Bubble shape & motion

In micro gravity, µG

In µG, the cell's oscillatory translation, -A cos ωt, imposed an unsteady acceleration to the system. The resulting artificial buoyancy ρV b Aω 2 cos ωt set the bubble into motion. For small amplitude and frequency values, the bubble was animated by an up-and-down motion with the imposed oscillation frequency without breaking (see fig. 2 (a)). As A was increased, the amplitude of the bubble translational motion, A b , was also increased until a saturation caused by the cell's finite hight. During the translational motion, the bubble exhibited different shapes, from an ellipsoid to a spherical cap. For the tested experimental parameters, the observed bubble shapes are in agreement with dynamical regimes reported for buoyancy-driven ascending bubbles in a stagnant liquid [START_REF] Clift | Bubbles, Drops, and Particles[END_REF].

The amplitude A b and phase θ of the bubble's relative motion to the cell have been measured by processing space-time diagrams representing translational motion of the bubble. Figure 3 shows a typical example of such diagrams. As shown by the smaller image in the figure, a narrow band of image passing through the bubble center is extracted along the direction of bubble's principle motion. These bands of image at different instants are arranged chronologically to construct a space-time diagram (the larger image in the figure). The oscillatory trajectory at the middle of the diagram represents the bubble motion, while the sinusoidal waves at upper and lower ends show cell's motion. It is seen that the bubble motion is sinusoidal and has well-defined amplitude and phase. Measured values of A b scaled by the amplitude of the cell's motion A and the phase θ are shown in fig. 4 (a) and (b), respectively. The ratio A b /A is seen to group around a single curve while plotted as a function of the dimensionless amplitude K C . For large K C values, it appears that the experimental bubble amplitude A b is independent of the frequency Ω. Regarding the phase shown in fig. 4 (b), the bubble motion always precedes the cell's motion. This arises from the liquid phase movement delayed by its inertia. Hence, the bubble motion is in advanced phase. The data also exhibit increasing tendencies with K C towards θ = π/2. Most of the experimentally determined phase values are beyond the maximum phase advance (∼ π/8) that a bubble can attain at small Reynolds numbers (cf. fig. 11.15 of Clift et al. [START_REF] Clift | Bubbles, Drops, and Particles[END_REF]). This observation suggests that bubble motions in present experiments are associated with large Reynolds numbers.

As seen in figure 2 (a), bubbles undergo large deformation during their oscillatory translation. This deformation will particularly act upon the bubble's associated drag coefficient and the resulting motion. Therefore, the amplitude A b /A and the phase θ should depend on the Ohnesorge number Oh to reflect the capillary effects. In order to investigate the influence of Oh, a least square fit of the power law Ω q Oh r K s C was applied to the experimental data on A b /A. It was found that q = -0.002, r = -0.150 and s = -0.517. The small value of q implies independency of the oscillation amplitude from Ω. The value of s nearly equal to -0.5 suggests that the bubble velocity

U b (= A b ω) is scaled by Aω 2 D e 1/2
, as spherical cap bubbles rising in stagnant liquid [START_REF] Davies | [END_REF]. Based on these observations, a power law A b /A ∼ Oh r ′ K -0.5 C is proposed and the value of r ′ is derived from a least square fit to the data. It is found that:

A b /A = 0.59 Oh -0.15 K -0.5 C (2)
U b = 0.416 Oh 0.15 Aω 2 D e (3)
This velocity is identical to the terminal velocity of a spherical cap bubble determined by Davies and Taylor [START_REF] Davies | [END_REF] but with a corrective factor proportional to Oh -0.15 .This correction represents the influence of bubble deformation during the oscillatory translational motion.

In normal gravity, G

In G environment, prior to the experiments the bubble lies below the upper wall due to buoyancy. It has a disklike shape of a thickness twice the capillary length ℓ c0 = (γ/ρg) 1/2 because of its large volume compared to the capillary length (D e > ℓ c0 ) [START_REF] De Gennes | Gouttes, bulles, perles et ondes[END_REF]. With small acceleration values of the cell, the bubble and the liquid are translated by the cell without any significant relative motion between them. A large acceleration leads to a relative motion between the two phases. The bubble is periodically squeezed on the top wall with the same frequency as the cell's motion without detaching from it (see fig. 2 (b)). Further increase of the acceleration can lead to partial or full detachment of the bubble from the wall and bubble splitting into smaller parts (for more detailed description cf. [13,[START_REF] Yoshikawa | [END_REF]).

In high frequency experiments carried out with set-up ST2, for excitation amplitude A smaller than twice ℓ c0 , two main differences have been observed in bubble dynamics compared to low frequency experiments (ST1). First, surface waves formed on the bubble lower surface above a certain acceleration value were standing waves and non axisymmetric, while in ST1 experiments they were propagative and axisymmetric (cf. [13]). Second, for large acceleration values, bubble breakups occurred by ejections of smaller bubbles from the mother one rather than the uniform splittings such as typically observed in the low frequency experiments. These phenomenological differences in low and high frequency experiments suggest that experiments carried out with an excitation amplitude A ℓ c0 fall in different dynamical regime. Standing wave formation in high frequency experiments suggests that Faraday instability probably plays a role in bubble breakups and ejections. The investigation of bubble breakup in such regime is out of scope of the present paper and will not be discussed further.

Bubble breakup thresholds

For large accelerations, bubbles are first pinched at one or more location before breaking into smaller parts. For breakups into more than two parts, splits can either occur simultaneously or successively until all bubbles reached sufficiently small sizes and the system reached a stable state. Breakups following bubble pinching usually happened within 0.1 s.

Breakup thresholds in micro gravity

In µG, the breakups usually occurred apart from the cell walls (see fig. 6). In longitudinal views, the bubble is clearly seen to first deform into a peanut shape before being pinched in the middle and break up. Due to the finite size of the cell, walls could sometimes influence the breakup mechanism. The initial deformation leading to the breaking could occasionally be initiated by an encounter event of the bubble with the wall. The critical frequency f cr of the breakup has been determined by varying the frequency f for given amplitude A values. This determination has been made with precisions of typically ±0.05 Hz. Results are presented in fig. 7 for µG experiments. A general tendency reveals that the acceleration a cr = A (2πf cr )

2 is approximately constant at the thresholds for a given suspending liquid. This finding is consistent with the preceding experiments [13] for ground experiments. However, the critical acceleration values found in µG are around four times smaller than those in G.

Figure 7 shows that a bubble in water (γ = 72 dyn/cm) breaks up at a critical acceleration a cr = 280 cm/s 2 in average, while it breaks at only a cr = 94 cm/s 2 in the silicone oil of 0.01 cm 2 /s (γ = 17.4 dyn/cm). This reveals the stabilizing effect of surface tension. Regarding the influence of the suspending liquid viscosity ν, results from different silicone oils with various viscosity values show a stabilizing effect. The critical acceleration a cr is, for example, 394 cm/s 2 in the oil of 1 cm 2 /s that is four times larger than in the oil of 0.01 cm 2 /s. The data for all the suspending liquids were used to find an empirical law of the breakup threshold in terms of the dimensionless parameters K C , Ω and Oh. As shown in fig. 8, it is found that all the data of the experimentally determined thresholds group around a single line extending over six orders of magnitude along the abscissa and three orders of magnitude along the ordinate. Data scatter and small deviations observed in the figure come from the difficulties encountered in evaluating the breakup threshold with precision in parabolic flights where µG phase could only last 22 second maximum. A least square fit to the data yields:

K C Oh 1.7 Ω 1.93 = 29.3 (4)
It implies that the critical acceleration a cr increases only slightly with the frequency (a cr ∝ Ω 0.07 ) and has a dependency on the liquid physical properties such that a cr ∝ ρ -0.85 ν 0.23 γ 0.85 .

Breakup thresholds in normal gravity

In G, the breakups are completed on the top wall. They begin with wave formation on the lower surface. The bubble forms a complete or incomplete torous before its pinchingoff leads to splitting. More details of the breakups in G are available in [13].

Critical frequency values in G experiments are shown in fig. 7 for bubbles of 3 cm 3 . Stronger oscillations were required to destabilize the bubble in G compared to µG experiments. The relation a cr = cst at the bubble breakup thresholds reported by [13] is confirmed in both set-ups except for the data with frequencies higher than 20 Hz (correspondingly, A 0.1 cm). The latter data will no longer considered in the present work, because these correspond to bubbles in a dynamical regime different from that in low frequency experiments, as described in sect. 4.1.2. The critical acceleration values are much larger than those in µG (e.g., in the oil of 0.01 cm 2 /s, a cr = 779 cm/s 2 in G while a cr = 94 cm/s 2 in µG) and these values are seen to increase with the viscosity and with the surface tension, as observed in the experiments in µG. A similar correlation to that for µG experimental data was derived for G data using dimensionless parameters K C , Ω and Oh. Figure 8 reveals that G data also group around a single line extending over several orders of magnitude along the abscissa and the ordinate. A least square fit to the data yields:

K C Oh 1.7 Ω 1.98 = 208 (5)
It appears, with quite surprise, that this power law is similar to that obtained for µG experiments (4) in spite of qualitatively different behavior of bubbles observed in G from in µG. As described above, wall influence is apparently important in bubble motion and breakup. The essential difference in the power laws of the breakup onset, however, concern only the constants appearing in equations ( 4) and ( 5), which is significantly larger in G due to larger a cr values for bubble breaking. This large difference will be explained energeticaly in the discussion of sect. 5.3.

Discussion on the breakup mechanism

In the literature, a large variety of criteria for the breakups of fluid particles (bubbles and drops) have been proposed under different flow configurations (see review papers, e.g., [19]). The criteria can be classed into three types [20]. (i) For particles in motion with small Reynolds numbers, viscous shear at the surface is responsable for the distortion and breakup of bubbles. Therefore, the criteria are expressed in terms of the capillary number, which reflects the relative importance of viscous to capillary forces. (ii) For particles in motion with large Reynolds numbers, inertia of the surrounding fluid flow is responsible for bubble deformation and breakups. The criteria write in terms of the Weber number, W e, that compares the flow inertia with the capillary pressure. (iii) For particles subjected to time periodic excitations of their natural frequency, resonance has been proposed as breakup mechanism. Besides

K C Oh 1.7 Ω K C Oh 1.7 Ω 1.93 = 29.3 K C Oh 1.7 Ω 1.98 = 208 K C Oh 1.7 Ω 2 = 34
-Normal gravitywater 0.01 cm 2 /s 0.1 cm 2 /s 0.2 cm 2 /s 0.5 cm 2 /s 1 cm 2 /s -Microgravitywater 0.01 cm 2 /s 0.1 cm 2 /s 0.2 cm 2 /s 0.5 cm 2 /s 1 cm 2 /s Fig. 8. Correlations of the breakup threshold in micro and normal gravity environments in terms of dimensionless numbers. Bubble volume is 3 cm 3 for all the data. these three classes, the Rayleigh-Taylor instability at the particle surface is often pointed out as destabilising mechanism for particles ascending or descending freely in stagnant suspending phase and for bubbles at the rebound stage of cavitation collapse [21,[START_REF] Brennen | Fundamentals of Multiphase Flow[END_REF]. For understanding the bubble breakups observed in present experiments, we examine the plausible mechanisms among them in this section. The Rayleigh-Taylor, resonance and inertial mechanisms are considered.

Mechanism due to the Rayleigh-Taylor instability

In the present experiments, bubble breakups have occurred at a constant value of the acceleration field for a given suspending liquid. This might suggest that breakups are provoked by Rayleigh-Taylor instability as the latter originates in the acceleration, a, of the interface between two fluids of different densities. The breakup of fluid particles ascending or descending freely by the buoyancy in stagnant fluid has been studied by models based on this instability [21,[START_REF] Komabayashi | [END_REF]24]. According to theoretical study of Kitscha & Kocamustafaogullari [24], the breakup criterion for bubbles in such a configuration writes:

This model has been compared successfully with experiments when a was equal to the gravitational acceleration g. One can apply the model to the case of oscillatory acceleration field in the limit of small frequency; in other words, in the limit of large oscillation amplitude. In µG experiments, the interface acceleration a is given by subtracting the acceleration of the bubble motion in the cell's reference frame from the imposed acceleration field. This yields a = -Aω 2 cos ωt -A b ω 2 cos (ωt + θ). The experiments show that A b is small compared to A for large K C values (see fig. 4 (a)). The maximum value of a can then be estimated by Aω 2 . Introducing this estimate into the criterion (6) yields Aω 2 > 732.8γ/ρD 2 e for bubbles with small Ohnesorge numbers (i.e. in liquid of small viscosity). Although this result correlates a breakup at a constant acceleration, a comparison with experiments shows that a cr is largely overestimated. For instance, the model predicts for the silicone oil of 0.01 cm 2 /s a critical acceleration a cr = 4870 cm/s 2 which is, respectively, fifty and seven times larger than the corresponding a cr obtained in experiments in µG and G. This result suggests that the breakup regime is not governed by a Rayleigh-Taylor instability.

Mechanism due to resonance

Free oscillation of a bubble has natural frequencies f ℓ (ℓ = 2, 3, • • •) scaled by the inverse of the capillary time scale t -1 c = γ/ρD 3 e 1/2 . The explicit expression of f ℓ for a spherical bubble in inviscid liquid can be found in, e.g., [START_REF] Lamb | Hydrodynamics[END_REF]. Viscous correction to it was given by Prosperetti [START_REF] Prosperetti | [END_REF] for suspending liquids of any viscosity. For a bubble of volume 3 cm 3 in silicone oils of viscosity ranging 0.01 to 1 cm 2 /s, the fundamental mode (ℓ = 2) has natural frequencies ranging from 3.02 Hz to 1.56 Hz which are of the same order of magnitude as the imposed frequency in the present experiments. Although in fig. 8 no preferred frequency was observed which could suggest a resonance breakup mechanism, an interaction of the imposed frequencies with bubbles eigen modes was found in G experiments. Figure 9 plots the dimensionless critical acceleration a cr /G for silicone oil of viscosity 0.1 cm 2 /s as a function of the frequency scaled by the natural frequency f 2 calculated by the viscous theory [START_REF] Prosperetti | [END_REF]. In this figure, it is shown that a cr has a minimum when f /f 2 is equal to 1.5. This behavior of the critical acceleration suggests that the resonance can influence the principle breakup mechanism. This eventual influence of bubble's natural frequency could be explained by wall effects, since no similar finding has been made in µG. In G, bubbles are in contact with the upper container wall. Bubble deformations are required for any relative motion of the bubble to the liquid, while it is not the case in µG. In turn, perturbation flows destabilizing the bubble are affected by bubble deformations which become the most important with excitations at natural frequency.

0 0.5 1

Inertial mechanism

According to the results on the bubble velocity amplitude in µG (3), the bubble Reynolds number Re b is calculated as Re b = U b D e /ν = 0.59 Oh -0.15 √ K C Ω 2 . At the breakup onset, this gives typically large values as suggested earlier, except for the oil of 1 cm 2 /s. Using the correlation for the onset (4), one can find, e.g., Re b ∼ 2000 and 50 for the oil of 0.01 and 0.5 cm 2 /s, respectively. Hence, the inertial breakup would be one of the most credible mechanisms. Stability studies of drops subjected to air blasts and buoyancy-driven (ascending or descending) drops show the following criterion for inertial breakups in the limit of small viscosity [27,28]:

W e > 12 (7)
This criterion expressed in terms of the Weber number W e = ρU 2 b D e /γ means that a pressure perturbation, ρU 2 b , caused by flow inertia destabilizes drops against the stabilizing capillary pressure, γ/D e . This could be applied to the present bubbles, when the time scale of oscillation, t osci , remains large compared to the breakup time scale (∼ 0.1 s) and when the influence of cell's walls is absent (i.e. in µG). By considering the maximum velocity A b ω that a bubble can attain, the criterion (7) can be written for the present experiments as:

W e = ρ (ωA b ) 2 D e γ = (K C OhΩ A b /A) 2 > 12 (8)
where the third term just corresponds to Weber number written as a function of dimensionless quantities K C , Ω, Oh and A b /A. The latter amplitude ratio A b /A can be estimated from the measurements of bubble translation amplitude. Insertion of the derived empirical law (2), which takes into account bubble deformations, leads to the following breakup criterion:

K C Oh 1.7 Ω 2 > 34 (9)
Comparison of this criterion with the least square fit to the experimental data of eq. ( 4) shows very good agreement.

In fig. 8 the curve of eq. ( 9) is represented by a dashed line and is seen to fit very well the data obtained in µG.

A common breakup criterion in term of Weber number for both G and µG experiments

Although different empirical laws derived for experiments in G and µG were found to be similar (see eqs. ( 4) and ( 5)), they did not permit to define a universal breakup criterion for both environments. In this section, following the results from previous section, one such criterion will be proposed.

In G experiments, the bubble lower surface was observed to develop waves prior to its breakup, while its upper surface typically remained in contact with the top wall of the cell (cf. [13]). This suggests the destabilization of the lower surface is necessary for bubble breakups. Hence, flow inertia has to overcome not only the stabilizing effects due to surface tension but also those due to gravity. In other words, the inertial perturbation (ρU 2 b ) has to be strong enough to break up bubbles against the capillary pressure (γ/D e ) and the hydrostatic pressure due to the gravity (ρgD e ). Therefore, the criterion (7) should be modified accordingly. At the threshold, the relation ρU 2

b ∼ γ/D e + ρgD e is expected from an energetic point of view. The universal criterion in a dimensionless form will be written as:

W e > 12 + C Bo ( 10
)
where Bo = ρgD 2 e /γ is the Bond number and the constant 12 has been chosen to meet the criterion (7) in the limit of µG. In order to evaluate W e, one needs estimates of bubble velocity U b which are difficult to obtain in G as ground experiments are associated with the large bubble deformation near the top wall. Hence, we assume that U b = ωA b can be evaluated by the same law as that in µG (eq. ( 2)). The latter hypothesis is based on the observed similarity of the experimental breakup criteria (4) and (5) 1/2 to yield favourable agreements with the experiments. It is seen that the breakup criterion ( 10) is well satisfied by the experimental data of both environments. This suggests again that observed bubble breakups are caused by the inertial perturbation in suspending fluid. Note that, as the Bond number is relatively large in G experiments (44 ≤ Bo ≤ 240), the right-hand-side of equation ( 10) is roughly approximated by Bo itself. This implies that the breakup thresholds in G are characterized by the critical value of the Froude number F r = W e/Bo around the unity.

Conclusions

Stability of bubbles subjected to oscillatory acceleration field has been studied experimentally both in micro and normal gravity environments. In the attempt to reveal the physical mechanism behind the breakup, the breakup threshold was determined and discussed in detail with accompanied observations on the bubble motion under the excitation. Determined critical parameters were correlated by similar power laws K C Oh 1.7 Ω n = cst with the power n around 1.9 in both environments. Confrontations with different possible breakup mechanisms were performed. The possible mechanisms due to the Rayleigh-Taylor instability and the resonance with bubble eigen-modes were shown to be inconsistent with the experiments. The inertial breakup mechanism similar to that of drops in a uniform flow was found to give consistent explanation to experimentally determined breakup thresholds. From an energetic point of view, a universal criterion for experiments in G and µG was proposed in terms of the Weber number W e. Calculated W e values on the breakup thresholds with the help of the consideration on the bubble motion affected by fluid viscosity and capillarity showed the validity of the criterion in the experiments as seen in fig. 10. This suggests that inertial flow perturbation in suspending fluid is responsable for the observed breakups.

1 .

 1 Illustration of the first experimental set-up (ST1) for low frequency experiments used in micro and normal gravity environments.

Fig. 2 .

 2 Fig. 2. Side views of bubble motion under oscillatory acceleration field.

  the bubble Reynolds number Re b = U b D e /ν can be estimated by Re b ∼ K C Ω with the bubble velocity U b ∼ Aω. It is found that Re b is larger than 200 for the majority of experiments.

Fig. 3 .

 3 Fig. 3. Space-time diagram of bubble motion (the right larger image) in transversal view in µG. The suspending liquid is silicone oil of 0.5 cm 2 /s and the oscillation parameters are A = 1.5 cm and f = 1.6 Hz. The left smaller image shows how the diagram is constructed (see the text for details).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Amplitude and phase of bubble motion in µG.

Fig. 6 .

 6 Fig. 6. Bubble at the moment of breakup at the middle of the cell in µG (A = 4.2 cm and f = 1.3 Hz with silicone oil of 0.5 cm 2 /s). In each picture, the upper and lower half is in longitudinal and transversal view, respectively.

Fig. 10 .

 10 Fig. 10. Critical Weber number values in micro and normal gravity environments.

Table 1 .

 1 Physical properties of the considered liquids.

	Liquid	Density Viscosity Surface tension (g/cm 3 ) (cm 2 /s) (dyn/cm)
	Water	1	0.01	72
		0.816	0.01	17.4
		0.93	0.1	20.1
	Silicone oils	0.95	0.2	20.6
		0.959	0.5	20.7
		0.965	1	20.9

  Fig. 7. Critical frequency of bubble breakups in micro and normal gravity environments. Bubble volume is 3 cm 3 for all the data. Curves of constant acceleration are also shown for the values discussed in the text.

	100					-Normal gravity -
							water	0.2 cm 2 /s
							0.01 cm 2 /s	0.5 cm 2 /s
				7 7 9 c m /s 2			0.1 cm 2 /s	1 cm 2 /s
	f cr [Hz]	10	Different dynamical regime	2 8 0 c m /s 2 3 9 4 c m /s 2	Experiments with ST1
				Experiments with ST2
						9 4 c m /s 2
		1		-Microgravity -
				water	0.2 cm 2 /s
				0.01 cm 2 /s	0.5 cm 2 /s
				0.1 cm 2 /s	1 cm 2 /s
		0.01			0.1		1	10
					A [cm]
	10000
	1000
	100
		10
		1x10 -6	1x10 -5	1x10 -4	0.001	0.01	0.1	1

  Fig.9. Dimensionless critical acceleration for the breakups of a bubble of 3 cm 3 in silicone oil of viscosity 0.1 cm 2 /s. The bubbles of volumes V b = 2, 3, 4, 5 and 7 cm 3 have natural frequencies f2 of 3.66, 2.99, 2.60, 2.32 and 1.97 Hz, respectively.

		1.4
			Normal gravity, silicon oil 0.1 cm 2 /s
			2 cm 3	5 cm 3
			3 cm 3	7 cm 3
		1.2	4 cm 3
				f cr = f 2
	G	1
	cr /
	a
		0.8
		0.6
				1.5	2	2.5	3
				f cr /f 2

The authors gratefully thank D. Vallet for technical supports and µG experimentation. The financial support from CNES (Centre National d'Etudes Spatiales) is acknowledged with appreciation.


Eleonora Morganti
email: eleonora.morganti@unibo.it

Urban food planning and transport sustainability: A case study in Parma, Italy

Keywords: Urban food system, Urban sustainability, City logistics, Parma -Italy

Nowadays local authorities try to solve problems related to the environmental sustainability of the cities through various urban logistics measures, which directly and indirectly affect the urban food supply system. This paper presents an ongoing research on the urban food transport system and its impact on the city environmental sustainability. The research analyses the city logistics project for food transport implemented in Parma, Italy, in order to identify external costs indicators that help policymakers in evaluating the environmental sustainability of different logistics measures. According to this state of the research, the paper aims at suggesting a methodological framework to estimate the urban food demand, the urban food supply and the urban food transport. These sub-system provide the baseline scenario to evaluate the external costs reduction obtained by the Parma's project.

Introduction

At global level, the population is increasingly concentrating in the cities. In Europe, around 75% of the population lives in urban areas and it is foreseen to increase up to 80 % by 2020 (EEA, 2010). At the same time, the quality of life in the cities is declining and urban pollution keeps increasing in terms of carbon dioxide (CO 2 ) emissions, waste, noise, and lack of greenery [START_REF] Alexandre | Policies to improve the urban environment in Our Cities, Our Future: Price, C[END_REF]EEA, 2010). Many of European cities struggle to cope with social, economic and environmental problems resulting from pressures such as overcrowding or decline, social inequity, health problems related to pollution and traffic. Looking through the urban environmental sustainability assessment carried out by cities and local authorities, the major impacts are generated by food and transport1 , as confirmed by various "food miles" studies [START_REF] Pirog | Food Fuel, and Freeways: An Iowa Perspective on How Far Food Travels, Fuel Usage, and Greenhouse Gas Emissions[END_REF][START_REF] Pirog | Calculating food miles for a multiple ingredient food product[END_REF]Coley et al., 2007). Considering the whole food system, [START_REF] Tukker | Identifying Priorities for Environmental Product Policy[END_REF] identifies food as one of the top three contributors to the environmental impact in society, and food transportation as a major part of that impact. In terms of transport externalities, total road transport is responsible for 70% of CO 2 emissions, and more precisely urban traffic accounts for 40% of these emissions. The urban freight traffic performance (ton/km) weights for 30% of the total freight traffic and the equivalent-vehicles used for freight urban distribution accounts for 20% of the total urban road congestion [START_REF] Ecmt/Cemt | Freight transport and the city[END_REF]. Few analyses focused on determining and evaluating the urban food transportation impact (DEFRA, 2007;[START_REF] Tukker | Identifying Priorities for Environmental Product Policy[END_REF], however this issue is object of a growing attention, as crucial component in the urban food system and urban mobility. Indeed, despite increasing interest in urban sustainability and food carbon footprint, little is known about the multilevel interrelationships in urban food systems related to food transport, logistics, distribution, waste and access to water [START_REF] Sonnino | Feeding the City: Towards a New Research and Planning Agenda International Planning Studies[END_REF]. The methodological framework proposed in this paper applies to the "urban food metabolism" (Bohle, 1993), to describe the key elements and the functioning of the food supply system in urban environment, with special attention to the transport and logistics sub-systems. The proposed scenario outlines the complex interrelationships amongst the various actors, as city users and inhabitants, urban food suppliers, logistics providers and carriers, in order to evaluate the process of food distribution within the city and its contribution to the urban ecological sustainability. The urban food metabolism tool is finalized to assess the impact of emerging city logistics solutions for food on a basis a of environmental sustainability indicators, thus we analyze the case study of Parma, a relevant city logistics project in the North of Italy. The paper aims at providing tools for a critical discussion of the methodology adopted within the ongoing investigation. The expected results of the study are to provide an analysis detecting the main features of the urban food demand and supply with qualitative and quantitative description, and an evaluation of the urban food transport at the distribution stage, including corporate retail, independent channels, hotels, restaurants and cafés and institutional food services within the city limits. Furthermore it is foreseen to evaluate the environmental impacts generated by the "last food mile" in terms of CO 2 emissions, congestion, and social costs. The evaluation compares the scenario before and after the urban freight distribution optimization implemented by the Municipality and the Wholesale Produce Market.

Urban sustainability integrated approach

Urban environmental quality is the result of drivers in many areas at different scales and policy levels. Policy response on urban matters from all governmental levels -besides the local level also the regional, national and European level -is necessary. The complexity of urban systems (see Figure 1) requires integrated and balanced solutions rather than isolated measures.

"Improving the quality of the urban environment, making cities more attractive and healthier places to live, work and invest in, and reduce the adverse environmental impact of cities on the wider environment' is consequently the target of the Thematic Strategy on the Urban Environment" (EC, 2006).

Local authorities are looking for new policies in order to improve the environmental impact of urban food economy, especially in terms of energy use, transport and waste. The emerging need is to improve supply chain efficiency to reduce problems and costs caused by food distribution, which contributes to climate change. Nonetheless the urban delivery of food is only part of the whole urban food supply chain, therefore it should be considered from a broader perspective. In recent years more articulated strategies addressing the food supply sustainability have been identified by European, national, regional and local policymakers and other actors along the food supply chain. Many cities are implementing agro-food governance policies and started re-designing urban food, by provisioning through single initiatives or coordinated strategies. The contemporary urban food planning is more than just reduce food insecurity and implement "feed the city" policies: it deals with creating new economic opportunities for small farmers and retailers, fostering individual and communal health, enhancing environmental friendly programs at urban and peri-urban level [START_REF] Marsden | Between the Local and the Global: Confronting Complexity of the Agri-Food Sector Elsevier[END_REF]. These initiatives focus on an ecological approach and set objectives based on the triple-bottom-line: environmental, economic, and social sustainable development.

Concerning the food transport policies, Maunsell (2007) proposes to focus on the following areas of intervention:

 The potential impact of consistent implementation of existing best practice on food transport logistics efficiency;  Opportunities for reducing the external impacts of food transportation by improving local supply and distribution networks;  Opportunities for improving transport efficiency through step changes in transport methods or infrastructure; and  The potential impacts of changing regulations on logistics.

Related to the food transport and logistics issues, a growing concern for the environmental and social effects generated by the urban haulage of goods has led local and national authorities as well as companies and transport operators to promote actions for an improvement of the associate logistics. Studies, research activities and pilot projects have followed in order to try and minimize the negative impact of urban haulage of goods and its externalities on the standard of living [START_REF] Rios | La mobilità delle merci nell'area metropolitana milanese[END_REF]. The challenge is to provide efficient distribution of food to urban small retailers and consumers, reducing the associated environmental and social impacts (in terms of carbon emissions, congestion, mobility) generated by the urban food deliveries. Indeed, the small scale distribution of goods in urban environment, defined as "last mile" logistics, is one of the most problematic parts of the supply chain, due to the high atomization of recipients and to their increasing requirements [START_REF] Lenz | The Last Mile. Old problem, new options?[END_REF]. This is highly relevant in the case of inner city food transport, intended as last part of "food miles". Therefore the crucial issue, analyzed by this paper, focuses on the "last food mile"2 . On a literature review, only few European city logistics projects identified specific measures optimizing the perishable goods transport at urban level, i.e. London LowHub service, delivering fruit and vegetables from the New Covent Garden Market to city center businesses. Other examples on optimizing urban food logistics are represented by the model of "alternative food hub" [START_REF] Morley | Food hubs: The missing middle in sustainable local food networks?[END_REF]. In Italy, one of the most relevant case is presented by the city of Parma, where the urban goods haulage has been redesigned through renewed logistics services provided by the F&V wholesale market, Centro Agro Alimentare e Logistica (CAAL). The project has implemented an urban distribution center with delivery services of selected food products categories, using environmental friendly vehicles, and optimizing load factors and routes thanks to the ICT tools.

Methodology

To understand the urban food distribution system, we base on the concept of "urban metabolism", hereafter UM, [START_REF] Wolman | The metabolism of cities[END_REF], which sets the analogy with the metabolic processes of organisms, and asserting that "Cities transform raw materials, fuel, and water into the built environment, human biomass and waste" [START_REF] Decker | Energy and material flow through the urban ecosystem[END_REF]. Bohle adapted the UM model to the urban food system and identified the "urban food metabolism" (1993), thus we apply the model to selected processes of urban food provisioning, transport and consumption. In this study, the examined sub-systems are the urban food demand, the urban food supply, the food transport and the environmental impacts of food transport. The crucial factors constituting the model are density, morphology, transportation technology [START_REF] Kennedy | The changing metabolism of cities[END_REF], city users flows, urban consumer habits, food suppliers managing mechanisms, food products flows. Figure 2 presents the urban food flows on the basis of the main functions representing the urban food system. Cities are social and economic centers of communities and generate a complex interrelation of functional, environmental and social aspects. Daily, people coming from the peri-urban and regional areas travel to the city for a variety of reasons, including work, study, shopping, business and socializing. These transitory flows are defined as city users [START_REF] Martinotti | Metropoli[END_REF] and, together with permanent and temporary inhabitants and tourists; they provide a complete picture of people creating the city life, by utilizing goods and services, and participating to the socio-economic city networks. The issues of feeding inhabitants and city users, while guaranteeing the food access to different groups of citizens, are highly depending on a mix of urban policies and local interactions, such as, commercial opportunities, regulatory conditions, and transport services and infrastructures. These elements contribute to shape the urban food system and the set the frame in which food supply chain actors operate. The urban food system is investigated from the demand/supply perspective, as showed in Figure 3.The demand of food considered in the study is the daily average amount of food consumed by the inhabitants and city users, and the amount of food waste generated by them3 . In the Italian urban areas, except for a very small amount of food grown with own urban vegetable gardens4 , most of the food is purchased, through a variety of commercial channels, depending on the at home (supermarkets and groceries) or out of home (Ho.Re.Ca. and institutional food services) consumption.

The full urban food supply includes the amount of products sold by the food suppliers, and the amount of unsold or wasted food [START_REF] Segrè | Il libro nero dello spreco in Italia: il cibo[END_REF]. The volume of food products managed by the urban food suppliers at the distribution stage provides the total amount of food which daily reaches and circulates in the city. The urban food demand is the basis to estimate the urban food supply (adjusted with unsold and wasted food). The urban food supply represents the total amount of food circulating in the urban area, and it is the basis to evaluate the food transport performance. These three subsystems depict the baseline scenario of the urban food transport system, then the environmental impact of urban food transport is evaluated through external costs indicators.

The urban food demand

In this and in the following paragraph, we estimate the total amount of food requested by and available to the population, drafting a sort of Annona for the city of Parma. In ancient Rome, the Annona was the plan of all means of subsistence, especially grain stored in the public granaries, for provisioning the city (Chisholm, 1911). The former scope of the Annona was to guarantee the supply and the distribution of corn requested by the urban population, as a planning and managing tool. In this analysis, we choose to draw a descriptive scenario, estimating demand and supply of food in the whole city.

The food demand is calculated on the per capita food by the total flows of people which daily live the city and use goods and services. The people flows are: permanent and temporary residents, city users (employees, business-persons, one day tourists etc.), nights sleeping tourists.

According to the Italian food purchase figures5 (ISMEA, 2007), the daily purchase of food6 is 1.137gr., including the eventual food waste generated at the consumption stage. The consumer patterns are different among the whole urban population, thus the main trends are investigated, distinguishing on population groups and consumption habits.

The purchase of food is finalized to at home and out of home consumption. In the first case, the consumer buys food in the chain retail, or in independent retail or in alternative channels, in the second case the suppliers are hotels, restaurants, and cafés, institutional food services and corporate cafeterias.

The urban food supply

To identify the existing urban food supply and describe the main market features, a general analysis of the urban commercial channels is drafted. Taking into account the highly complex and fragmented supply chain in the European urban environment, the resulting description provides a representation of the atomized and non homogeneous scenario.

The full food supply is provided by the distribution system, which can be grouped in three market models (see Table 3). The available data on the Italian food market share show the dominant position of the chain retailers at the distribution stage. Consumers prefer the mainstream channels, with high competition among hypermarkets, supermarkets and discounts, reaching the 70% of the market turnover. The independent retail system, as corner shops and grocery stores, has a secondary relevance, covering some niche (RER, 2008).

In the descriptive scenario, we choose to include self-production by home gardens and public gardens within the city limits. While detailed data are scarce on urban horticulture in developed and developing countries, FAO observes a growing quantity of fruit and vegetables production in the urban and peri-urban areas (2010). In our study, the urban food supply is described through the corporate retail, independent retail, Ho.Re.Ca., and Institutional food services. The amount of food managed by these suppliers is estimated considering the sold food and the unsold food (wasted and/or redistributed products) 7 . The first step of the study is to obtain an urban food supply estimation, that we assume is the amount of food transported by commercial vehicles (light and heavy vehicles) circulating in the city area. The next step is to identify the performance and the main features of the urban food transport, providing the necessary data to assess the environmental impact of urban food transport, on the basis of the identified indicators.

urban food transport system

To describe the urban food transport8 , we focus on selected components by characterizing city logistics and perishable goods logistics processes, on the basis of the technical guidelines issued the Regione Emilia Romagna, Mobility and Transport Department (Rosini et al. 2005).

The resulting logistic variables' framework provides the basis to describe the "last food mile".

According to the City Ports report ( 2005), the variables characterizing the performance of city logistics processes are:

• Logistics variables, that characterize the transport service related to the process, in particular:

-frequency: delivery frequency; -load unit: shape in which the goods are usually grouped and loaded on vehicles (pallet, roll, box, etc.); • Technological and organizational variables, that characterize the technical organization of the transport, particularly:

-typology of vehicles: dimensions and technical features (for example refrigeration) of the vehicles; -delivery period: period of the day in which the delivery of the goods is usually carried out; -other: other transport features, for example necessity to respect hygienic rules (HACCP), necessity of staff training, etc; • Level of logistics optimization, the optimization of the process is here understood as capacity utilization of the vehicle (in weight and/or volume); • Logistics management, which is the formal procedure of goods delivery, • Carriers typology, which are the type of carriers used for the transport (couriers, special transport, etc.); under this heading the possibility of work on own account is also indicated; • Nodes of the supply chain, are the departure points (producers, suppliers, warehouses of wholesalers, etc.) and arrival (retail shops, final consumers, etc.) of the goods.

Perishable goods logistics focuses on "fresh food products" (meat, dairy products and sausages, etc.) The supply chain can be grouped such as:  The fresh retail supply chain, intended for large food distribution areas (either hypermarkets or supermarkets) or to smaller areas associated to a large distributive chain; it is characterized by an upstream optimization of the deliveries and by greater constraints in terms of service (time schedule and number of deliveries) for the recipient of the goods;  The traditional fresh supply chain, intended for sale areas of food products generally of small dimensions (the so called neighborhood commerce), that is characterized by a greater incidence of deliveries from wholesalers or of self provision.  Dry retail, for foodstuffs at middle or long expiry dates, not bound to specific preservation conditions (paste, oil, canned food, etc.), and household products (detergents and soaps for the personal hygiene, for woven, for dishes, etc.);  Fresh retails, for fresh foodstuffs with close expiry date and bound by special preservation conditions;  Frozen retails, for frozen foodstuffs. The food delivery phenomenon in urban environment is described by a qualitative point of view and, where possible9 , quantitative, identifying more specific features of the goods movements. The actors interviewed are representative of haulage industry, business, local authorities and community/ environmental groups.

The environmental impact of urban food transport

Few countries have analytical tools and data for evaluating the effectiveness of their policy measures concerning urban goods transport, resulting in their measures as a cause of unexpected side effects (OECD, 2003). In the literature various typologies of costs are distinguished, but there are not unified definitions among the different authors. Following the solution proposed by "Amici della Terra" Italian national report (Friends of the Earth, 2005) and Regione Emilia Romagna, Mobility and Transport Department data (Rosini et al. 2005), the quantification of the total external costs of transport in urban environment will be calculated using the set of indicators shown in Table 2. Despite the influence of many variables, as provided in paragraph 3.3, on the total impact assessment, the available coefficient on the external costs impact of urban food transport is calculated on the basis of the impact of one main technological variable that is the typology of vehicles (light and heavy ones circulating in the urban area).

Data collection

The research seeks to gather data for each of sub-system and to identify critical gaps in data availability. Primary and secondary sources are used to collect data for this study: interviews with supply chain participants; news articles; websites; direct observation of urban distribution center, wholesale market, stores and business premises and the collection of secondary economic and demographic data. Relationships with food suppliers, wholesalers, carriers, managers, shop owners, public representatives, and other stakeholders are investigated.

Parma case study

The analysis focuses on an Italian case study, the Parma city logistics project, which developed an innovative urban food delivery system, Ecocity, since 2008. Parma is a city of 180.000 inhabitants, located in Emilia Romagna region, well-known for its traditional, high quality food products. The attention to the food supply system sustainability boosted policymakers to redesign the urban distribution of goods, with the support of CAAL, the wholesale produce market. CAAL is located in a strategic area and is experienced in perishable goods logistics. Ecocity includes a renewed logistics platform dedicated to food products implemented at CAAL, and a fleet of twelve natural gas powered vehicles. As well at institutional level, the local authority promoted a new act to regulate freight transport in the limited traffic area. Moreover, the freight mobility plan adopts an ICT platform for the optimization of routes, which provides dynamic routing and scheduling to reduce distances travelled. The figure 4 shows the functioning of the city logistic project.

The main objective of the project is to rationalize and optimize the deliveries through an ecofriendly service. The specific objectives are:  Raise vehicles filling rate  Reduce environmental impact  Reduce the number of commercial vehicles in the city centre  Increase the effectiveness of the whole delivery of goods Starting from Ecocity project, we will provide a case study to assess an effective evaluation of according to the last food mile approach 10 . The available project data have been integrated with the last food mile approach in order to gather the missing data to describe each of sub-system (food demand, food supply, food transport system) and to provide an effective and reliable analysis of the external costs impact.

Materials

10 Previous feasibility and monitoring studies mainly focus on urban freight transport and logistics, without benchmarking the urban food system.

Dealers

Customers

Firms which deliver in on their own Others Urban Distribution Center

CAAL

Last Mile

City of PARMA Primary and secondary sources have been used to collect data for this study: interviews with supply chain participants; news articles; websites; direct observation of urban distribution center, wholesale market, stores and business premises and the collection of secondary economic and demographic data. Relationships with food suppliers, wholesalers, carriers, managers, shop owners, public representatives, and other stakeholders have been investigated.

Results

At this stage of the investigation, partial results are available up to the first stage of the investigation, that is the assessment of the freight transport weight according to the food demand, food supply estimation and the characteristics of the food transport system that have been outlined in the methodological framework.

Parma's food demand

To estimate the demand of food expressed by the city in a working day, the number of consumers is calculated through the official population figures issued by the Municipality register. Moreover the estimation of the number of city users (employees, business-persons, students, tourists, etc.) is obtained with the support of the Municipality officers11 . In the city of Parma, the total estimated number of inhabitants and users of the Parma's city area is about 240.000 people (see Table 1), eating in the city at least one meal. On the basis of the Municipality's available data, the Parma self production is highly limited and, thus negligible in this estimation. To represent the complex urban complex demand of food, we assume commuters purchase in the city only the half of the total amount of daily purchase of food (most likely they will consume lunch and, eventually, breakfast in the city, and dinner at home) 12 . Permanent and temporary residents and tourists purchase and consume the entire amount of food within the city.

Final remarks

The analysis on the urban food system is still incomplete; however some cues can be suggested as preliminary considerations:

-including in the urban food demand estimation the city users flow and the household waste phenomenon provides a more realistic picture of the daily amount of food requested by the cities; -identifying the urban food supply through the full representation of food commercial activities, as corporate retail, independent retail, Ho.Re.Ca. and institutional food services describes the existing high atomization of the distribution system in our cities; -the available data on the urban food transport in the last mile service depicts a scenario characterized by strong logistic inefficiencies and high potential of city logistics optimization.

Due to the ongoing elaboration of the data, at this stage of investigation paper could not facilitate a full comprehension of the expected benefits of the methodology adopted.

As well the risk related to lack and disaggregation of data needs to be carefully considered along the next research steps.

Urban areas now constitute the living environment of the vast majority of the population, and it is common interest raising the quality of life in these areas with integrated policies. Special attention is on food supply system and the food transport, that directly impact on the city sustainability. Although urban food freight system is very complex and heterogeneous, indeed the delivery and collection of goods in urban and metropolitan areas, especially in the core areas of cities with old and established centres, have a major impact on the local community as concerns its economic strength, quality of life, and the accessibility and attractiveness of the city. At present, balancing the need to access for goods and services with local environmental and social concerns is a crucial point. Rethinking urban mobility, as requested by the European Commission, involves managing the transport demand, optimizing the use of all the various modes of transport, providing innovative services and coordinated logistics actions, particularly needed for the perishable goods transport. Moreover the last food mile logistics can represent a starting point to redesign the urban mobility system and to renew the "urban culture" of people living and using the city [START_REF] Bologna | Documento del Gruppo "Città" CNEL sulla mobilità urbana delle merci per lo sviluppo di un trasporto multimodale sostenibile[END_REF]. Urban food planners and policy makers are looking for the most effective logistics strategies to booster the economic prosperity, quality of life and environmental protection, and for tools helping in evaluating these strategies in terms of environmental impacts. Our approach focuses on setting a tool that helps planners and policy-makers to understand the urban food system from a logistic point of view, and we expect to identify sustainability indicators for city logistics solutions related to the food supply.

Figure 1 .

 1 Figure 1. The urban environment in relation to areas and activities beyond cities and towns

Figure 2 .

 2 Figure 2. The urban food flow -Focus on distribution

Figure 3 .

 3 Figure 3. Urban food system

Figure 4 .

 4 Figure 4. Ecocity logistic project

Table 1 .

 1 Urban Food Supply

Residents and city users food sourcing

		Source: own elaboration on USDA, 2009
	Corporate retail Indipendent retail Alternative channels Self production	 Hypermakets  Supermarkets  Superettes  Discounts  Grocery stores  Health and Specialty stores  Corner shops  Food Co-ops  Flea market  On farm retail  farmers markets  PYO -Pick Your Own  CSA-Community Supported Agriculture  Home gardens  Public vegetable gardens

Table 2 .

 2 External costs assessment

	External costs indicators
	Greenhouse gas -CO2, CH4, N2O
	Atmospheric COVNM,SO2, NOX, PM10) pollution Noise	(CO,	Light vehicles	Heavy vehicles
	Accidents (Number of Accidents,
	killed and injured)
	Congestion (Total time lost)
	Source: Friends of the Hearth,2005

Table 3 .

 3 Parma's residents and city users

	Population group	People in a working day Percentage
	Residents	184.460	76,9
	Temporary residents	21.000	8,8
	Commuters*	33.000	13,8
	Turists	1.200	0.5
	Total	239.660	100
	* Net of commuters, hiving off residents commuting to the peri-urban area.
	Source: our elaboration on Municipality of Parma data, official statistics and secondary
	sources, 2009

Table 4 .

 4 Parma's daily food demand

	Estimated daily purchase	Aggregate daily purchase
	per capita (kg)		(kg)
	Residents and tourists	1,137	547.347
	Commuters	0,568	4.515
	Total		551.862
			Source: our estimation

A growing number of cities is drafting various types of environmental impact assessments, i.e. London (City Limits,

2002), Toronto(The Toronto Ecological Footprint, 2002).

Here we define the concept of "last food mile" and we use it in the study to facilitate the comprehension of our research.

In many countries studies have estimated the quantity of food wasted by households; it results be a consistent portion of the total purchased food, between 20% and 30%, that go to the disposal sites. The first report, The food we waste, has been drafted byWRAP in 2008.

In Italy, urban agriculture has recently gained new attention, but its role in the urban food provisioning is minimum, due to the limited green spaces available within the historical city centers(Galli et al., 2010).

Cfr. Institute of services for the agricultural and food market, ISMEA Domestic purchase: trends for food products 2007. The data is calculated on the basis of the Italian population (age > 3 years) in 2007.

This data is confirmed by the data resulting by the amount of consumed food(INRAN, 2008) and the amount of wasted food by the Italian households (ADOC, 2010).

The estimation bases on the Last Minute Market dataset, University of Bologna.

More in general, the OECD definition of urban goods transport (2003) is: "The delivery of consumer goods (not only by retail, but also by other sectors such as manufacturing) in city and suburban areas, including the reverse flow of used goods in terms of clean waste".

Feasibility studies and urban freight transport data are available for the city of Parma and Region Emilia Romagna. We select data describing the main relevant features in order to evaluate the urban food transport performance.

There are no available official data about city users in the most of Italian cities. Some research has been conducted for big cities as Milan and Turin. The last population census, dated 2001, does not provide relevant aggregated data on city users.

Due to limited available data about average meal weight, we assume at home and out of home meals have the same weight.

CAT Confesercenti, 2009, Emilia Romagna Food Consumer Pattern.

Based on the Confesercenti ER'study 13 , in Emilia Romagna region people (age > 3 years) eating one meal out of home are 33% , thus around 78.100 people daily consume food out of home. Therefore 8,3% (45.935 kg) of the total food demand is consumed out of home.

Parma's food supply

The food commercial services and shops in Parma in the city center and in the whole urban area are different for management structure, size, and food products selection. On a preliminary analysis of the District's Chamber of Commerce database, we observed the high share of registered food commercial activities on the city full number of shops and stores. Additional information are gathered by the RER studies on the food distribution system (Table 5). Although more detailed info are necessary to complete the study of the corporate and independent channels, and, particularly, of the Ho.Re.Ca. sector.

Parma's food transport

According to the Region Emilia-Romagna (RER) estimations, in urban areas 70% of trucks bring 30% of the volumes and it calculated a huge potential for consolidation near city centre. Parma Municipality and RER examined the urban traffic system in 2001,2004,2006 with various data collection methods. Specific analysis have been conducted in order to describe the freight transport volumes in two different zones: inner city and city limits areas. Available data deal with the following supply chains: fresh food, dry food, Ho.Re.Ca., and fruit and vegetables from wholesale market. The elaboration of the collected data is undergoing.

Parma's food transport impact

At present, only general data have been collected to evaluate the environmental impact of the food haulage. The available information estimates that 55% of air pollution is generated by urban freight transportation in the inner city (CAAL estimations, 2010). The evaluation of the food transport environmental impact at urban level results from the complete gathering of previous sections, our expectation is to evaluate the Ecocity environmental impact on the current freight transport scenario.


Arjen T Dijksman

Benoit Dubertret
email: benoit.dubertret@espci.fr

Accurate energy-size dependence of excitonic transitions in semiconductor nanocrystals and nanoplatelets using a phase jump approach

Keywords: semiconductor nanocrystals, quantum dots, photoluminescence, electron model

Accurate energy-size dependence of excitonic transitions in semiconductor nanocrystals in the strong confinement regime using classical theoretical approaches such as effective mass approximation, tight binding, or empirical pseudo-potential is difficult. We propose a simple empirical expression with three fitting parameters that accurately relates the size dependence of most known excitonic transitions in CdSe and in InAs nanocrystals. We show that this empirical expression can be deduced from a phase jump approach if the charge carriers are consid-ered to travel on the atomic lattice of the nanocrystal and gain energy upon bouncing at the nanoparticle boundaries. This empirical expression is also tested on the atomically flat CdSe nanoplatelets without any adjustment of the parameters obtained with the CdSe spherical nanocrystals, and provides an estimation of the CdSe nanoplatelets thickness that matches exactly the experimental observations. These results suggest that a phase shift approach could be useful to describe the electronic transitions in semiconductor nanocrystals.

1 Introduction A growing appreciation of the exceptional physical properties of colloidal semiconductor nanocrystals has motivated extensive studies of these materials since their discovery in the early 1980s [START_REF] Ekimov | Quantum Size Effect in 3-Dimensional Microscopic Semiconductor Crystals[END_REF][START_REF] Efros | Interband absorption of light in a semiconductor sphere[END_REF][START_REF] Brus | A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites[END_REF]. Technological know-how, optimized colloidal syntheses, new experimental data and theoretical modeling steadily increased our understanding of the effects of the confinement of charge carriers in colloidal semiconductor nanocrystals, also called quantum dots (QDs) [START_REF] Alivisatos | Semiconductor clusters, nanocrystals, and quantum dots[END_REF]. They constitute without doubt a very promising research field for the design of innovative materials. Several theoretical approaches have been used to describe the physics of the electronic transitions in QDs. They include essentially effective mass (EM) approximation [START_REF] Efros | The Electronic Structure of Semiconductor Nanocrystals[END_REF], tight-binding (TB) [START_REF] Leung | Exciton fine structure in CdSe nanoclusters[END_REF][START_REF] Sapra | Evolution of the electronic structure with size in II-VI semiconductor nanocrystals[END_REF][START_REF] Allan | Confinement effects in PbSe quantum wells and nanocrystals[END_REF] and empirical pseudo-potential [START_REF] Wang | Pseudopotential calculations of nanoscale CdSe quantum dots[END_REF][START_REF] Franceschetti | Manybody pseudopotential theory of excitons in InP and CdSe quantum dots[END_REF][START_REF] Hu | Semiempirical pseudopotential calculation of electronic states of CdSe quantum rods[END_REF] approaches. These different approaches have led to a much better understanding of the confinement effect in the QDs, but in the case of the excitonic transition dependence upon the nanocrystal size, their predictions remain limited. For example, in the case of effective mass approximation, discrepancies between theory and experimental data are observed [START_REF] Norris | Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots[END_REF], due to our incomplete knowledge of such characteristics as finite barrier, inter band coupling, exchange interaction and size depend-ence of the dielectric constant. For practical applications the size of QDs is deduced from their absorption spectra using an empirical polynomial fit [START_REF] Yu | Experimental Determination of the Extinction Coefficient of CdTe, CdSe, and CdS Nanocrystals[END_REF][START_REF] Jasieniak | Reexamination of the Size-Dependent Absorption Properties of CdSe Quantum Dots[END_REF].

In this paper, we present an alternative way to investigate the absorption spectra of semiconductor nanocrystals in the strong confinement regime, which leads to empirical expressions for the energy-size dependence of excitonic transitions. It includes a phase jump term in the solutions of the Schrödinger equation. The model is probed against high quality experimental data of CdSe and InAs QDs of the literature and shows good agreement in the strong confinement range. It is further verified and specified on semiconductor nanoplatelets, which provide a good benchmark for the comparison with different approaches.

Theoretical basis

We propose the following sizedependent expression for the energy of the i th excitonic transition:

𝐸 𝑖 (𝑎) = 𝐸 0 𝑖 𝑎+𝛾 𝜙 𝑖 𝑎 +𝛾 𝑡 𝑖 (1)
where 𝐸 0 𝑖 is a constant reference energy gap for the i th excitonic transition and 𝛾 𝜙 𝑖 and 𝛾 𝑡 𝑖 are respectively metric phase and time delay parameters for the same transition. Their physical interpretation will be discussed later.

Figure 1 shows the high quality experimental data of the first excitonic transition of CdSe QDs published by Norris et al. [START_REF] Norris | Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots[END_REF] and their fit with Eq. ( 1). The fit is very good (coefficient of determination R=0.9997) and suggests that Eq. ( 1) could be used for size determination, equivalently to empirical polynomial fitting formulas of higher order [START_REF] Yu | Experimental Determination of the Extinction Coefficient of CdTe, CdSe, and CdS Nanocrystals[END_REF][START_REF] Jasieniak | Reexamination of the Size-Dependent Absorption Properties of CdSe Quantum Dots[END_REF]. Equation (1) can also be used to fit the higher energy transitions of CdSe (see Fig. 2). For each transition, the set of parameters has to be adjusted, but the fit with the experimental data is strikingly good. Equation (1) can also be used to fit the InAs excitonic transitions reported by Banin et al. [START_REF] Banin | Size-dependent electronic level structure of InAs nanocrystal quantum dots: Test of multiband effective mass theory[END_REF] with an accuracy equivalent to that of CdSe (see Fig. 3). This suggests that Eq. ( 1) can be used with success for various materials. In the following section we provide a framework that explains how we obtained Eq. (1). (2). The output values for parameters 𝐸 0 𝑖 (eV), 𝛾 𝜙 𝑖 (nm), 𝛾 𝑡 𝑖 (nm)

and the coefficient of determination R 2 are given in Table 1. Dotted lines denote weak transitions. (Experimental data, courtesy of D. J. Norris) [START_REF] Norris | Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots[END_REF].

Table 1 Parameters for the CdSe optical transitions of Fig. 2. 3 Phase jump approach We will now propose a semi-empirical deduction of Eq. ( 1).

Transition
We assume the electron to be a localized particle which is moving through the crystalline structure. As an approximation, we will consider the "classical" path where it travels in a right line from atom to atom. Let us represent the electron's phase by a rotating vector. As the electron's phase 𝜙(𝑡) is progressing, the direction of the vector is changing, equivalently to a quantum stopwatch [START_REF] Onorato | Low-dimensional nanostructures and a semiclassical approach for teaching Feynman's sumover-paths quantum theory[END_REF], see Fig. 4. Its angular velocity 𝜔(𝑡) = 𝑑𝜙/𝑑𝑡 determines its energy 𝐸(𝑡) = ℏ𝜔(𝑡). In a steady state, 𝜙(𝑡) = 𝜔 𝑎 𝑡, and the action 𝐴 advances linearly with time at a rate equal to the energy 𝐸 𝑎 of the electron:

𝐸 𝑎 = 𝐴 𝑡 2 -𝐴(𝑡 1 ) 𝑡 2 -𝑡 1 = ℏ∆𝜙 ∆𝑡 = ℏ𝜔 𝑎 (2)
Figure 4 Illustration of a rotating vector representing the phase progression of an electron travelling from left to right on its path from atom to atom. The phase progresses at π rad per atom-atom length.

Since the electron moves from atom to atom, Eq. ( 2) can also be written as 𝐸 𝑎 = ℏ𝜙 𝑎 /𝑡 𝑎 , where 𝜙 𝑎 is the phase shift and 𝑡 𝑎 is the delay needed when the electron travels over a distance 𝑥 𝑎 between two consecutive identical atoms.

We suppose that the particle's phase evolves in phase with the global standing wave solutions of the nanocrystal. One could speak of phase matching between a pilot wave and a guided quantum particle [START_REF] De Broglie | La Nouvelle Dynamique des Quanta[END_REF][START_REF] Bohm | A suggested interpretation of the quantum theory in terms of "hidden" variables. I[END_REF]. In order to stay in phase with its environment, the rotating vector of the guided electron must continuously adapt its angular velocity.

We now analyze the influence of an abrupt interface. When the electron reaches an interface, it may bounce or reflect on it, if not trapped. Upon this reflection, the electron's rotating vector must readapt its phase in order to match phases with the standing wave profile in the new direction. If this phase jump 𝜙 𝑠 is performed during a time interval 𝑡 𝑠 , the angular velocity of the rotating vector is temporarily changed. In a nanoparticle with small dimensions, the electron bounces with high frequencies on the nanoparticle surface, producing phase jumps at angular velocity 𝜔 𝑠 = 𝜙 𝑠 /𝑡 𝑠 .

In a steady state, the rotating vector's angular velocity oscillates between two values: the steady-state bulk value 𝜔 𝑎 = 𝜙 𝑎 /𝑡 𝑎 and the phase jump value 𝜔 𝑠 = 𝜙 𝑠 /𝑡 𝑠 at the surface. The action carried out by an electron that progressed over a distance 𝑛𝑥 𝑎 along nearest identical atoms and bounced on p interfaces is:

𝐴 = 𝑛𝜙 𝑎 + 𝑝𝜙 𝑠 ℏ (3)
and the energy associated to this path is:

𝐸 (𝑛, 𝑝) = 𝑛𝜙 𝑎 +𝑝𝜙 𝑠 𝑛𝑡 𝑎 + 𝑝𝑡 𝑠 ℏ. (4)
If this phase jump is periodical, such that in between two consecutive bounces on interfaces, the particle travels over a distance 𝑛𝑥 𝑎 , the average energy is:

𝐸(𝑛) = 𝐸 𝑎 𝑛+𝜌 𝜙 𝑛 +𝜌 𝑡 (5)
where 𝜌 𝜙 = 𝜙 𝑠 /𝜙 𝑎 and 𝜌 𝑡 = 𝑡 𝑠 /𝑡 𝑎 are the phase jump and time delay ratios at the surface with respect to the regular phase and time progression between two nearest identical atoms. Equation ( 5) and ( 1) are mathematically equivalent. In Eq. ( 5), the number n of atomic distances over which the electron progresses in between two reflections is proportional to the dimension of the nanoparticles.

In this model of strong confinement, we suppose that the energetical transitions of both electrons and holes can be described with Eq. ( 5), and that both charge carriers have the same time delay ratio 𝜌 𝑡 . It results that the electron and hole contributions can be summed up and Eq. ( 5) can be used for all excitonic transitions under strong confinement. This assumption holds if the masses of the charge carriers cancel out in the ratio 𝜌 𝑡 .

4 Discussion Equation ( 5) has three parameters that we now discuss.

E a in Eq. ( 5) corresponds to the energy of the transitions for 𝑛 → ∞, i.e. for the bulk material. For the first excitonic transition, this should correspond to the energy gap E g . However, the fitted reference energies 𝐸 0 𝑖 obtained by fits of Eq. ( 1) are considerably smaller than the bulk bandgap energy E g (except for transition (j)). This discrepancy means that the phase jump model can only be applied in the strong confinement regime.

The two other parameters in Eq. ( 5) are related to the phase jump both at the interfaces and between atoms, as well as the time necessary for these phase jumps. If this approach is valid, these parameters should be identical in identically structured materials whatever the shape of the nanocrystal, provided that the effect of surface chemistry on the phase jumps is identical. We tested this effect with CdSe nanoparticles of different shapes.

Fortunately, CdSe nanocrystals have been synthesized in different shapes including the spheres [START_REF] Murray | Synthesis and characterization of nearly monodisperse CdE (E = sulfur, selenium, tellurium) semiconductor nanocrystallites[END_REF][START_REF] Peng | Kinetics of II-VI and III-V Colloidal Semiconductor Nanocrystal Growth: "Focusing" of Size Distributions[END_REF] and more recently the nanoplatelets [START_REF] Ithurria | Quasi 2D colloidal CdSe platelets with thicknesses controlled at the atomic level[END_REF]. The advantage of the nanoplatelets is that they have been shown to be atomically flat with a thickness that is controlled with atomic preci-sion [START_REF] Ithurria | Quasi 2D colloidal CdSe platelets with thicknesses controlled at the atomic level[END_REF]. The crystal structure as well as the crystal orientation of the nanoplatelet facets have been documented, and they have been shown to have pure 1D confinement. They are thus a very interesting object to test this phase jump approach without different fitting parameters.

In a nanoplatelet, we hypothesize that the charge carrier bounces back and forth between the two extended (001) surfaces of the nanoplatelet, experiencing a phase jump at each reflection, in analogy to a photon in a Fabry-Perot resonator, as presented in Fig. 5. The path we propose for the first transition is the shortest path for a particle moving from Cd atom to Cd atom using the nearest atom at each jump. If we transpose this path to a spherical QD, with identical phase jump and time delay ratios, the charge carriers should circle inside the sphere, bouncing 4 times for a complete cycle, see Fig. 6. If such a path holds for spherical QDs, a nanoplatelet with a thickness of n monolayers should have the same transition as a spherical QD with a radius of n monolayers, provided that the two systems have the same nearest neighbour distance, the same composition (and thus the same 𝜙 𝑎 ) and similar surface chemistries (and thus the same 𝜙 𝑠 ).

In Fig. 7, we plot the experimental room temperature energy of the first excitonic CdSe transition for nanoplate-lets vs. their thickness in monolayers [START_REF] Ithurria | Colloidal nanoplatelets with twodimensional electronic structure[END_REF] and for QDs vs. their radius in shell monolayers [START_REF] Peng | Kinetics of II-VI and III-V Colloidal Semiconductor Nanocrystal Growth: "Focusing" of Size Distributions[END_REF]. The results coincide very well and can both be fit with the same parameters using Eq. ( 5). In addition, the NPL thicknesses we deduce from this model are exactly the ones measured on CdSe NPLs using HRTEM [START_REF] Mahler | Core/Shell colloidal semiconductor nanoplatelets[END_REF]. The experimental observation that a nanoplatelet with a thickness of n monolayers has its first absorption feature at the same wavelength as a spherical quantum dot with a radius corresponding to n CdSe monolayers is thus confirmed by this approach. Of course, this result only holds for the hypothesized path along the nearest cation-cation direction. Other transitions may result from other paths, phase jumps or time delays.

Figure 7

Comparison between first absorption peak of CdSe QDs (crosses) [START_REF] Peng | Kinetics of II-VI and III-V Colloidal Semiconductor Nanocrystal Growth: "Focusing" of Size Distributions[END_REF] respectively CdSe nanoplatelets (NPLs) (open figures) [START_REF] Ithurria | Colloidal nanoplatelets with twodimensional electronic structure[END_REF] vs. their radii respectively thicknesses expressed in terms of monolayers (the QD literature sizes in nm are divided by 0.304, which corresponds to one monolayer in the zinc-blende <100> direction). The solid line represents a close fit with E 0 = 1.52 eV, 𝜌 𝜙 = 3.2, 𝜌 𝑡 = 0.5.

Figure 8

Comparison between first absorption peak of CdTe QDs (crosses) [START_REF] Yu | Experimental Determination of the Extinction Coefficient of CdTe, CdSe, and CdS Nanocrystals[END_REF] respectively CdSe nanoplatelets (NPLs) (open figures) [START_REF] Ithurria | Colloidal nanoplatelets with twodimensional electronic structure[END_REF] vs. their radii respectively thicknesses expressed in terms of monolayers (the QD literature sizes in nm read from the graph of Ref. 13 are divided by 0.324, which corresponds to one monolayer in the zinc-blende <100> direction). The solid line represents a close fit with E 0 =1.39 eV, 𝜌 𝜙 = 3.2, 𝜌 𝑡 =0.5.

We have also compared the energy of the first transition of CdTe spherical QDs [START_REF] Yu | Experimental Determination of the Extinction Coefficient of CdTe, CdSe, and CdS Nanocrystals[END_REF] with CdTe nanoplatelets [START_REF] Ithurria | Colloidal nanoplatelets with twodimensional electronic structure[END_REF]. We first fit the CdTe QDs size dependence using Eq. ( 5), leaving out the 3 smallest sizes. We then use their parameters without any change to deduce the thicknesses of CdTe nanoplatelets. The results are reported in Fig. 8. While the exact thicknesses of CdTe nanoplatelets have not been obtained by HRTEM yet, we observe a difference of 3 monolayers in thickness estimation compared to EM approximation. Interestingly, the fitting of Eq. ( 5) for the first transition of CdSe and CdTe QDs shows that 𝜌 𝜙 and 𝜌 𝑡 are identical for the two materials. Only 𝐸 0 is different.

Conclusion

We propose a simple expression that relates the electronic transitions of semiconductor nanocrystals with their size. This expression has three parameters that are used to fit with a very good accuracy all the excitonic transitions of CdSe and InAs spherical nanocrystals as well as the excitonic transitions of atomically flat CdSe and CdTe nanoplatelets. We show that this simple expression can be deduced from a rotating vector representation of the charge carriers in the nanocrystal travelling from atoms to atoms and bouncing on the crystal facets. This simple phase jump model, although semi-empirical, provides a very precise relationship between the size of a nanocrystal and its electronic transitions. It provides a very simple description of all the electronic transitions of the nanocrystals we have investigated, that seems to be accurate at least in the strong confinement regime.

It can be used to predict successfully the transitions of some nanocrystals. For example, we show that the parameters used to fit the excitonic transitions of spherical CdSe nanocrystals can be used without adjustment to fit the excitonic transitions of CdSe nanoplatelets in the pure 1D confinement regime, as expected in this model. This approach considers semiclassical periodic paths in closed systems. It shows some similarities with other semiclassical treatments, like chaotic quantum systems considered with Gutzwiller or de Broglie-Bohm periodic trajectories [START_REF] Gutzwiller | Periodic Orbits and Classical Quantization Conditions[END_REF][START_REF] Stafford | Cohesion and Stability of Metal Nanowires : A Quantum Chaos Approach[END_REF][START_REF] Wisniacki | Dynamics of quantum trajectories in chaotic systems[END_REF]. It would therefore be interesting to study the application of these semiclassical methods to the QDs and NPLs we have investigated in this paper. We hope that the good agreement of this phase jump model with experimental observations will foster further investigation and identification with other approaches.

Figure 1

 1 Figure 1 Energy from the first excited state in spherical CdSe nanocrystals as a function of 1/a 2 . Equation (1) is fitted with a coefficient of determination 0.9997 for parameters E 0 =1.40 eV, 𝛾 𝜙 = 2.34 nm, 𝛾 𝑡 = 0.60 nm. (Experimental data and effective mass approximation from Ref. 12, courtesy of D. J. Norris).

Figure 2

 2 Figure 2 Energy from excitonic transitions (a) to (j) of CdSe (Ref. 12), versus 1/a 2 , with best fit functions determined from Eq.

Figure 5

 5 Figure 5 Schematic side view of a zinc blende CdSe 3 monolayer thick nanoplatelet. The excited electron starts along <101> direction. After a phase progression 3𝜙 𝑎 , it reflects at the (001) upper surface with phase jump 𝜙 𝑠 , continues along <1,0,-1> direction, reflects again, etc.

Figure 6

 6 Figure 6Schematic side view of a zinc blende CdSe 3 monolayer radius QD. Electron starts along <101> direction. After a phase progression 3𝜙 𝑎 , it reflects at (001) upper surface with phase jump 𝜙 𝑠 , continues along <1,0,-1> direction, reflects again, continues along <-1,0,-1> direction, reflects again, etc.

Table 2

 2 Parameters for the InAs optical transitions of Fig. 3.

	R²

Transition E 0 (eV) 𝛾 𝜙 (nm) 𝛾 𝑡 (nm)

Acknowledgements

The authors thank C. Ciuti, A. Bramati, M. Tessier and T. Pons for stimulating discussions. A.T.D. acknowledges funding by the Agence Nationale de la Recherche.