Jean Colombani
email: jean.colombani@lpmcn.univ-lyon1.fr

Jacques Bert

Holographic interferometry study of the dissolution and diffusion of gypsum in water

We have performed holographic interferometry measurements of the dissolution of the (010) plane of a cleaved gypsum single crystal in pure water. These experiments have provided the value of the dissolution rate constant k of gypsum in water and the value of the interdiffusion coefficient D of its aqueous species in water. D is 1.0 × 10 -9 m 2 s -1 , a value close to the theoretical value generally used in dissolution studies. k is 4 × 10 -5 mol m -2 s -1 . It directly characterizes the microscopic transfer rate at the solid-liquid interface, and is not an averaged value deduced from quantities measured far from the surface as in macroscopic dissolution experiments. It is found to be two times lower than the value obtained from macroscopic experiments.

INTRODUCTION

Pressure Solution Creep (PSC) constitutes a major plastic strain process of immersed solids under low stress. A crystal wetted by a solvent (generally water) and saturated by its components is at chemical equilibrium. But if that crystal is then submitted to an external stress, it experiences dissolution. This originates in the chemical potential change of the solid induced by the stress. The dissolved species then diffuse away from the high stress region and precipitate in a stress-free zone. This mechanism is known to play a chief role in the upper crust and contributes, for instance, to upper crust deformation and to the diagenesis of sedimentary rocks.

The study of PSC in gypsum is of double interest. First it controls the upper crust strength in numerous geological situations because of its high ductility [START_REF] Demeer | Uniaxial compaction creep of wet gypsum aggregates[END_REF]. Additionally, PSC in gypsum also likely plays a role in the high ductility and low strength of plaster, which is essentially made of gypsum needles, in the presence of moisture [START_REF] Chappuis | A model for a better understanding of the cohesion of hardened hydraulic materials[END_REF].

To understand the mechanisms involved in PSC in gypsum, precise values of the involved quantitative parameters are needed. Among them, the diffusion coefficient and the dissolution rate constant are primordial. Furthermore, these values are also of interest in other fields. Wide areas of gypsum karsts exist world-wide The instability of these karsts and potential collapse undermines the importance of understanding karst evolution and subsequently, the importance of knowing the precise values involved in the water-gypsum interaction [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF]. The presence of the dissolved species of gypsum in water influences the dissolution of minerals containing pollutants (especially in uranium mines). Hence the study of such contamination also necessitates values regarding gypsum dissolution [START_REF] Kuechler | Investigation of gypsum dissolution under saturated and unsaturated water conditions[END_REF]. The existence of a large quantity of dissolved mineral may also alter the quality of the drinking water [START_REF] Raines | Mixed transport / reaction control of gypsum dissolution kinetics in aqueous solutions and initiation of gypsum karst[END_REF]. Lastly, knowledge of gypsum dissolution/precipitation behaviour is required in the oil and gas industry where gypsum is a common annoying "scale" mineral [START_REF] Raju | The thermodynamics of "scale" mineral solubilities. 3. Calcium sulfate in aqueous NaCl[END_REF].

Surprisingly, no diffusion coefficient measurements of gypsum in water are available in the literature [START_REF] Lobo | Physical sciences data 41, Handbook of electrolyte solutions[END_REF][START_REF] Zaytsev | Properties of aqueous solutions of electrolytes[END_REF]. Recent studies of gypsum dissolution either estimate the interdiffusion coefficient of the aqueous components [START_REF] Raines | Mixed transport / reaction control of gypsum dissolution kinetics in aqueous solutions and initiation of gypsum karst[END_REF] or make use of a numerical value computed in 1971 [START_REF] Barton | Dissolution rates of polycristalline samples of gypsum and orthorhombic forms of calcium sulphate by a rotating disc method[END_REF]) with a Nernst-Hartley equation from the tracer diffusivity of the ions [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF]. Classical dissolution measurements typically use rotating-disk setups, batchexperiments or column-experiments with powder or crystal samples. In these apparatuses, the concentration during dissolution is monitored by techniques such as titrimetry, conductimetry, atomic absorption spectroscopy, mass spectrometry or colorimetry. During these experiments, water is usually flowing and the overall measured concentration stems from a combination of diffusion, forced convection and dissolution. If empirical equations always enable one to describe the measurements, the link with pure phenomena is not straightforward and needs accurate knowledge of the surface morphology and of the transport properties of the system to be reliable (Jeschke and Dreybrodt, 2002a). To overcome these experimental limitations, we have carried out local, instead of global, experiments of gypsum dissolution in water with an alternative technique, holographic interferometry. This method allows simultaneous determination of the dissolution rate constant of gypsum in water at the dissolving interface and of the diffusion coefficient of its components in bulk water. The fundamental difference between our measurements and standard ones lies in the fact that we look directly at the surface behavior instead of deducing this behavior from quantities measured far from the surface.

EXPERIMENTS

Historically, real-time holographic interferometry has been recognized as a valuable tool in solution chemistry [START_REF] Knox | Holographic interferometry in electrochemical studies[END_REF]. One of its advantages is that defects along the optical path (except in the working cell) compensate between the two light expositions of the hologram (see below), thus requiring less demanding setups than classical interferometry devices [START_REF] Colombani | Holographic interferometry for the study of liquids[END_REF]. Furthermore the observation of the two-dimensional concentration field in the liquid enables the identification of any non-diffusive fluxes (natural convection, gravitational instability, . . . ), thus providing a guarantee of the reliability of the measurements [START_REF] Colombani | Hydrodynamic instabilities and Soret effect in an aqueous electrolyte[END_REF].

Our setup is designed as follows (see Fig. 1). The beam of a Diode Pumped Solid State laser (λ = 532 nm wavelength) is divided into a reference beam and an object beam by a 50/50 beam splitter. The two beams cross a half-wave plate and a Glan-Taylor prism (vertically polarizing), which act as a polarizer/analyzer system controlling the intensity of the beam. The two beams are both expanded and spatially filtered by a microscope objective/pinhole/convergent lens set. A parallel plate installed on a rotating mount is inserted between the pinhole and the convergent lens of the reference beam. This device allows the removal of eventual parasitic fringes on the initial interferogram. The two beams interfere on the holographic plate, their polarization vectors being parallel and colinear to the plate. The entire setup is located on a vibration-damping structure.

Air motion in the laboratory causes a random change with time of the path length along the beams, creating parasitic vibrations of the fringes. To reduce this source of uncertainty, the entire optical table is covered with a closed shelter and the real-time optical data acquisition is performed in a contiguous room instead of in the laboratory.

The primary interest of holography lies in the fact that the interference pattern of the object and the reference beams contains the amplitude (as in classical photography) and the phase of the object beam. In this way, the amplitude and phase of the object at time t 0 are recorded on the hologram. Therefore, when both the object and the hologram (containing the memory of the object at time t 0 ) are enlightened at time t by the laser, the phase difference ∆φ of the object between times t 0 and t are visualized through N = ∆φ/(2π) interference fringes (interferograms). The interferograms are visualized and recorded with a Charge-Coupled Device camera, a monitor and image acquisition software.

This device, thus, gives access to any evolution of the phase ∆φ = 2πe∆n/λ (e path length in the optical cell) and consequently to any evolution of the refractive index ∆n of the liquid (all other optical components being kept unchanged). Therefore, the change of solute molality ∆m = ∆n/(∂n/∂m) = N λ/(e(∂n/∂m)) in the solution can be registered (∂n/∂m is the derivative of the index of refraction n of the solution with the solute molality m).

The crystal/solution system is located in an optical cell inside a copper structure with circulating thermostated water. The temperature of the water bath is regulated by a PID regulator, which probes the temperature with a platinum thermistor inserted into the copper structure. Indeed a temperature variation may be the source of a solubility, diffusivity and dissolution rate constant change. Therefore, constant temperature must be guaranteed (±0.01 K in our experiments).

An experiment proceeds as follows: The optical cell is filled with ultrapure water. The liquid is left in the copper structure for a few hours in order to obtain a homogeneous temperature in the system. Subsequently, a reference hologram is taken. Then a cleaved gypsum single crystal (CaSO 4 ,2H 2 O) from the Mazan mine (Vaucluse, France) of rough dimensions 5×5×1 mm 3 is introduced into the cell, this time being considered as the time origin of the reaction. Interferograms are then recorded periodically (see Fig. 2). The white homogeneous zone above the fringes in the first interferograms is considered to be pure water (the inhomogeneous grey zone is a parasitic fringe). Indeed, no change of color is observable in this part of the cell compared to the reference hologram, so the phase has remained unchanged and we may infer that no discernible gypsum has migrated yet to this zone. Proceeding downwards, an initial black fringe is encountered, revealing a change of π in the phase of the transmitted beam, compared to the flat-colored zone just above. Hence, we can conclude that the solute molality has increased at this point until it has reached a value for which the resulting change of phase is π. This corresponds to a molality change ∆m = λ/(2e(∂n/∂m)) (cf. above). The liquid just above the fringe has been identified as pure water therefore the molality all along this fringe is merely m = ∆m. The adjacent white fringe, immediately below, corresponds to a π change of the phase compared to the black fringe, and a 2π evolution compared to the flat-colored zone. The molality at this level is then m = 2∆m. The adjacent black fringe, just below, corresponds to a π change of the phase compared to the white fringe, and a 3π evolution compared to the flat-colored zone. The molality at this level is then m = 3∆m. The molality in the whole cell is then gradually reconstructed (see Fig. 3).

The limiting factor of the spatial resolution of the interferogram is the number of pixels in the camera sensor. The larger the pixel number, the smaller the area depicted by one pixel. The size of the chip of our camera is 752×582 pixels and our resolution 40 µm. This value is small enough to provide us with a correct discretization of the molality in the cell.

DATA ANALYSIS

The investigated chemical reaction can be written as :

CaSO 4 • 2H 2 O(s) ⇌ Ca 2+ (aq) + SO 2- 4 (aq) + 2H 2 O(l) (1)
It takes place at the lower end of our cell. Considering the symmetry of the experiment, it can be reduced to a one-dimensional diffusion problem along the vertical coordinate z. In the absence of other added salts, the two aqueous species are present at each z value with the same molality to preserve local electroneutrality. Therefore,

m Ca 2+ (z, t) = m SO 2- 4 (z, t), m Ca 2+ (z, t) and m SO 2- 4
(z, t) being the molalities of Ca 2+ (aq) and SO 2- 4 (aq), respectively, at position z and time t. Our investigation method is solely sensitive to the change of refractive index n, regardless of the species causing this change. In fact, both ions induce the modification of n, plus a possible contribution of ion pairs and impurities. Therefore, we experimentally access an effective molality of dissolved gypsum m(z, t)

≈ m Ca 2+ (z, t) = m SO 2- 4 (z, t).
To evaluate m along the vertical coordinate as a function of time, Fick's second law is used. if the diffusion coefficient is considered as constant in our molality range (see Section 4), this law reads:

∂m ∂t z = D ∂ 2 m ∂z 2 t (2)
with D as the interdiffusion coefficient of the solution, and must be solved in space and time with the following boundary conditions (all the symbols used are summed up in Table 1):

• Mass balance at the dissolving interface implies the amount of gypsum leaving the solid and entering the liquid to being equal:

F dissolution (0, t) = F diffusion (0, t) (3)
where F is the flow rate and the origin of coordinate is the geometric solid-liquid interface, considered as fixed (actually moving of less than one pixel in the course of a typical experiment).

The expression of the dissolution flow rate is:

F dissolution (0, t) = dξ dt (4)
with ξ the advancement variable (amount of transformed reagent). If we follow a transition states theory [START_REF] Shiraki | Kinetics of near-equilibrium calcite precipitation at 100 • C: An evaluation of elementary reaction-based and affinity-based rate laws[END_REF], the rate of advancement dξ/dt takes the form:

dξ dt = ks r a sol 1 -exp(- N A RT ) ( 5
)
where k is the dissolution rate constant of gypsum in water, s r is the total dissolving gypsum-water interface, a sol is the activity of the reagent (conventionally taken as one for a solid), A is the chemical affinity of the reaction, R is the gas constant, T is the absolute temperature and N is a constant. The chemical affinity of dissolution, in other words the change of Gibbs free energy when gypsum changes its thermodynamic state from solid to dissolved, is A = -RT ln Ω. In this expression, Ω stands for the supersaturation:

Ω = a Ca 2+ a S0 2- 4 a 2 H2O K sp . ( 6
)
a X is the activity of the subscripted aqueous species and K sp is the solubility product of the reaction of Eq. 1 † . The solubility product can be written as:

K sp = a sat Ca 2+ a sat S0 2- 4 (a sat H2O ) 2
, where a sat X represents the activity of the subscripted species at chemical equilibrium. As the solubility of gypsum is small, the activity of the solvent a H 2 O is always considered to be equal to one. If we introduce the mean ionic activity coefficient γ ± , geometric mean of the two single ion activity coefficients, the supersaturation can be rewritten:

Ω = γ 2 ± m Ca 2+ m S0 2- 4 (γ sat ± ) 2 m sat Ca 2+ m sat S0 2- 4 . (7)
m sat X is the molality of the aqueous component X at chemical equilibrium, and γ sat ± is the mean ionic activity coefficient at chemical equilibrium. Here we make the reasonable assumption that at all the ionic molalities achieved in our experiments, the ionic strength remains small enough so that the mean ionic activity coefficient does not depart from its value in pure water. This term can thus be suppressed from the expression of Eq. 7.

In the absence of experimental or theoretical determination of the value of the constant N in Eq. 5, we have used the value successfully used for fitting calcite dissolution measurements [START_REF] Rickard | Mixed kinetic control of calcite dissolution rates[END_REF][START_REF] Shiraki | Kinetics of near-equilibrium calcite precipitation at 100 • C: An evaluation of elementary reaction-based and affinity-based rate laws[END_REF]: N = 1/2. This choice results in the dissolution flow rate at the solid-liquid interface:

F dissolution (0, t) = ks r   1 - m Ca 2+ (0, t)m SO 2- 4 (0, t) m sat Ca 2+ m sat SO 2- 4   . ( 8
)
We will identify the geometric mean of the ionic molalities with our effective molality m. The geometric mean of the ionic solubilities is well known and will be written m sat . Finally, the dissolution flow rate at the surface reduces to:

F dissolution (0, t) = ks r 1 - m(0, t) m sat . ( 9
)
The diffusion flow rate is linked to the diffusion flux at the dissolving interface J diffusion (0, t) merely through:

J diffusion (0, t) = 1 s F diffusion (0, t) ( 10
)
where s is the cell section perpendicular to the mass transport. This flux derives readily from Fick's first law:

J diffusion (0, t) = -Dρ ∂m(0, t) ∂z t ( 11
)
with ρ as the density of the solution. The introduction of Eq. 9, 10 and 11 into Eq. 3 induces for the boundary condition at the lower end of the cell:

∂m(0, t) ∂z t = - kβ Dρ 1 - m(0, t) m sat (12)
where we have introduced β = s r /s.

• At the upper end of the cell (water-air meniscus), the boundary condition is merely written J diffusion (L, t) = 0, L being the height of water in the cell.

To get a tractable analytical expression of m(z, t), we make a further assumption concerning the geometry of our experiment. Our cell is high enough and the investigated time short enough to consider that gypsum diffuses in a semi-infinite medium. This hypothesis has been experimentally verified, paying attention to the fact that the highest molality fringe has not reached the top of the cell at the end of the measurement. The solution of Equation 2 is in this case [START_REF] Crank | The mathematics of diffusion[END_REF]:

m(z, t) = m sat erfc z 2 √ Dt -exp kβz Dρm sat + ( kβ Dρm sat ) 2 Dt × erfc z 2 √ Dt + kβ Dρm sat √ Dt . ( 13
)
erfc is the complementary error function. The values of m sat , D and k are obtained through a fit of our experimental m(z, t) curves with the above expression. The shift to the right of the molality curves (Fig. 3) can be viewed as a signature of diffusion, and the shift of the molality curves upward can be viewed as a signature of dissolution.

RESULTS

∂n/∂m is a value of primary importance, but unfortunately no experimental determination is available. Therefore, we have carried out refractive index measurements with an Abbe refractometer in pure water and water where gypsum was dissolved until saturation at ambient temperature (m sat = 15 mmol kg -1 corresponding to 2 g l -1 [START_REF] Raju | The thermodynamics of "scale" mineral solubilities. 3. Calcium sulfate in aqueous NaCl[END_REF]). These experiments were not far from the resolution limit of the apparatus. A (0.019 ± 0.007) kg mol -1 value was found, which was used for the refractive index into molality conversion.

To guarantee the validity of the data analysis , Equation 2 requires D to remain constant with molality. Unfortunately no experimental determination of D exists, and furthermore of the dependance of D on molality. To obtain an initial, crude idea of the variation ∆D of the diffusion coefficient of gypsum in water in the molality range ∆m between pure water and saturated solution (∆m = m sat = 15 mmol kg -1 ), we have collected the ∂D/∂m values of other calcium and sulfate salts in water [START_REF] Lobo | Physical sciences data 41, Handbook of electrolyte solutions[END_REF][START_REF] Zaytsev | Properties of aqueous solutions of electrolytes[END_REF] and computed their ∆D for the same molality range. The worst case is ZnSO 4 at low molality with ∆D/D ∼ 20%. Evidently, no strict conclusion can be inferred for gypsum from that value, but a relatively weak dispersion of D in our experimental cell can be expected.

Precipitation and dissolution of minerals are known to be strongly influenced by the presence of impurities. To evaluate the purity of our single crystals, we carried out the Electron Probe Microanalysis of a sample of our material. No elements, other than the atoms constituting gypsum, have been detected with a concentration greater than 0.1%. The known impurities found in the gypsum of the Mazan mines, possibly present with a lower concentration, are dolomite rocks, quartz, anhydrite and celestite.

For the computation of β = s r /s used in Equation 13, the cross-diffusional section area s is taken as the horizontal section of our optical cell (s = 0.95 cm 2 ) and s r is taken as the dissolving surface area of the single crystal (s r ≈ 0.3 cm 2 ).

We have measured the pH of our solution at the end of the experiment and observe a pH of 5.7, which roughly corresponds to the pH of the CO 2 /HCO - 3 system stemming from the dissolution of the atmospheric CO 2 in water.

Seven measurements, all at 20.00 • C, have been performed to check the reproducibility. The fit of the experimental points by the theoretical law of Eq. 13 brings a solubility m sat = 15.6 mmol kg -1 , which is very close to the expected value of 15 mmol kg -1 [START_REF] Raju | The thermodynamics of "scale" mineral solubilities. 3. Calcium sulfate in aqueous NaCl[END_REF]. This agreement validates our analysis procedure and particularly the chosen ∂n/∂m value. We find a diffusion coefficient D = (1.0 ± 0.1) × 10 -9 m 2 s -1 , a value very close to the 0.9 × 10 -9 m 2 s -1 value derived from Nernst-Hartley equation, generally used in dissolution studies [START_REF] Barton | Dissolution rates of polycristalline samples of gypsum and orthorhombic forms of calcium sulphate by a rotating disc method[END_REF]. We find a dissolution rate constant k = (4 ± 1) × 10 -5 mol m -2 s -1 . The statistical uncertainty is the standard error, computed from the seven measurements.

MIXED KINETICS

There has been a strong debate to clearly determine the nature and the kinetics of the slowest step of the dissolution process, controlling the kinetics. If mass transport in the liquid (proportional to the diffusion coefficient D) is slow compared to the chemical reaction at the solid surface (proportional to the dissolution rate constant k), ions are rapidly unbound from the solid but slowly transported away in the solution, hence the dissolution global kinetics is controlled by molecular diffusion. Conversely, if mass transport proceeds quickly compared to the reaction rate, the kinetics is controlled by the slow ion detachment from the mineral. Both macroscopic studies -rotating-disk, column-experiments . . . [START_REF] Raines | Mixed transport / reaction control of gypsum dissolution kinetics in aqueous solutions and initiation of gypsum karst[END_REF][START_REF] Dreybrodt | Comments on: Mixed transport / reaction control of gypsum dissolution kinetics in aqueous solutions and initiation of gypsum karst by Michael A. Raines and Thomas A[END_REF][START_REF] Dewers | Reply to comment on: mixed transport / reaction control of gypsum dissolution kinetics[END_REF][START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF]-and microscopic studies -atomic force microscopy [START_REF] Bosbach | In situ investigation of growth and dissolution on the (010) surface of gypsum by Scanning Force Microscopy[END_REF][START_REF] Hall | Scanning Force Microscopy of gypsum dissolution and crystal growth[END_REF]-have tried to address the question.

One of the reasons of the diversity of the available results lies in the fact that the balance between transport and reaction strongly depends on the geometry of the system and on the thermodynamic conditions. [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF] conclude with mixed kinetics, where both effects are of comparable magnitude, with the linear kinetics of Eq. 9 (F dissolution ∼ m, except very close to the saturation) for the dissolution. This conclusion is drawn from a combination of batch dissolution and rotating-disk experiments. Therefore, the authors make use of the formalism of these kinds of methods to evaluate the two phenomena: the dissolution velocity is evaluated by k s , an empirical dissolution rate constant, and the diffusion velocity is evaluated by k t = Dc eq /ε, a transport constant (c eq stands for the molarity of aqueous calcium at saturation and ε for the thickness of the diffusion boundary layer surrounding the crystal). The authors find k t = 1.5 × 10 -3 mol m -2 s -1 from batch experiments and k s = 1.3 × 10 -3 mol m -2 s -1 from both experiments (for c/c eq < 0.94). These two values are so close that the mixed aspect of the dissolution kinetics is unambiguously assessed for this situation. Beyond the interest of this result, one can see that the selected evaluation quantities are strongly linked to the experimental methods.

Besides, [START_REF] Murphy | Surface reaction versus diffusion control of mineral dissolution and growth rates in geochemical processes[END_REF] have proposed the ratio of the maximum diffusive and reactive flow rates α = Dρm sat /(βkL) to be a dimensionless number used to discriminate between transport-controlled and reactioncontrolled dissolution. This number has a more universal impact, with no quantity intrinsic to the chosen experiments (like ε above). Pure transport control corresponds theoretically to α = 0, pure surface control to α = ∞ and mixed control to α = 1. These authors have numerically investigated realistic geological dissolution configurations of quartz and calcite in water in geometries comparable to ours (zero-flux boundary) and conclude that at values of α between 10 -2 and 10, the rate of evolution of the system is controlled by mixed surface reaction and diffusion kinetics.

Our experimental values lead, with L = 3 cm and ρ = 10 3 kg m -3 , to α = 0.05. This reveals mixed kinetics, according to the classification of Murphy et al. At ambient temperature and pressure, the mixed kinetics of dissolution of gypsum in water seems to be a robust feature, in our geometry (a reacting surface at one boundary and a zero flux at the other) as well as in the geometry of [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF] (a reacting surface at one boundary and a constant composition reservoir at the other).

COMPARISON BETWEEN LOCAL AND GLOBAL DISSOLU-TION MEASUREMENTS

The comparison of our dissolution rate constant k values with results of global measurements requires particular care. As has been often stated [START_REF] Rickard | Mixed kinetic control of calcite dissolution rates[END_REF][START_REF] Jeschke | Pitfalls in the determination of empirical dissolution rate equations of mineral from experimental data and a way out: an iterative procedure to find valid rate equations, applied to Ca-carbonates and -sulfates[END_REF], derivation of rate constants from these kinds of experiments is a difficult task and the procedure strongly depends on the device geometry and sample morphology.

To enter into the details of a global experiment, the overall flux R in a vessel of volume v is computed from the time evolution of the overall concentration c and is considered to be identically equal to the diffusion and dissolution fluxes: R = J dissolution = J diffusion = (v/s)(dc/dt). But as water is flowing in the vessel, this equality is only valid at the frontier between the diffusional boundary layer and the bulk liquid, in other words along the section called s above. Indeed this section corresponds to the geometrical locus where mass balance of Eq. 3 applies. This statement remains true exclusively for mixed kinetics. For pure reaction-controlled dissolution, mass balance applies at the solid-liquid interface, called s r above, and for pure transport-controlled dissolution, no information on dissolution coefficients can be obtained (Jeschke and Dreybrodt, 2002a).

Between the solid surface and the extremity of the boundary layer, there is no loss of solute, therefore mass balance imposes the flow rates at these two surfaces to equalize. At the solid-liquid interface (of area s r ), the flow rate F r can be deduced from an analog to Eq. 9: F r = s r k(1c/c eq ). At the top of the diffusional boundary layer (of area s), the flow rate can be written F d = sk s (1c/c eq ) where k s is the empirical dissolution rate constant introduced in Section 5 [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF]. Accordingly F r = F d implies k = sk s /s r and we are now able to compare our microscopic dissolution rate constant k with a rate constant k s deduced from a global dissolution measurement.

Recently, [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF] found a linear dissolution kinetics and k s = 1.3 × 10 -3 mol m -2 s -1 , as previously mentionned. The authors estimate the geometric specific surface area of their powders by optical microscopy to range between 60 and 73 cm 2 g -1 , depending on the gypsum type. We consider this surface to be similar to our diffusional section s. The BET-surface of these powders is 1100 cm 2 g -1 . We identify this surface with our dissolving interface s r ‡ . Therefore, one finds a s/s r ∼ 1/15 ratio, which gives a k = 9 × 10 -5 mol m -2 s -1 value.

Beside the uncertainty on the surface ratio, a factor of 2 or 3 between their global measurements (9 × 10 -5 mol m -2 s -1 ) and our local measurements (4 × 10 -5 mol m -2 s -1 ) seems to exist at first sight. But the large differences between the two methodologies could account for this discrepancy and we can consider these two values as being in fair agreement. One should mention that the k we have measured concerns exclusively the dissolution of the (010) plane, whereas the k s of global experiments is an average of the dissolution rate constants of all the reacting planes of the dissolving powder. Therefore, the two values are not strictly comparable.

As a summary, microscopic measurements make the data interpretation easier for three reasons. First, there is no uncertainty on the surface where mass balance must be applied. Second, there is no flow in the cell and therefore no hydrodynamical assumptions and computations are needed. And finally the study of a single crystallographic interface avoid to obtain a multifaceted average dissolution rate constant.

CONCLUSION

We have used a non-invasive interferometric method to access to the microscopic dissolution rate constant of a cleaved (010) surface of gypsum in water. This constant has been revealed to be two times lower than the same constant measured by macroscopic averaged methods, which can be considered to be in fair agreement. The interdiffusion coefficient of the aqueous species of gypsum in water has also been measured during these experiments and exhibits a value close to the theoretical value generally used for dissolution studies. Now, to deepen the comprehension of pressure solution creep, we plan to perform the same experiments with gypsum crystals under stress. The knowledge of the influence of an uniaxial pressure on the dissolution parameters may shed new light on the mechanisms of PSC. ‡ Obviously this is questionable because the adsorption sites in the BET method and the dissolving sites in the dissolution experiment may slightly differ. But this assumption should at least bring the correct order of magnitude.

  Figure 1:

† Our standard state is characterized by a unit activity for pure gypsum (solid reagent) and pure water (solvent) at any temperature and pressure. For the aqueous species of the solute, it is a unit activity in a 1 mol kg -1 solution representing an infinitely dilute solution, for any temperature and pressure.

Acknowledgements-We would like to thank Elisabeth Charlaix and Pierre Monchoux for fruitful discussions, Sylvain Meille (LCR Lafarge) for gypsum samples, Xavier Jaurand for the chemical analysis of gypsum, and Christophe Bineau, Alexandre Enderlin, Hervé Féret and Agnès Piednoir for experimental help. We are also very grateful to the anonymous referees and to the associate editor, Eric H. Oelkers, for their comments. Part of this study was funded by CNES (French spatial agency).

List of Tables

Evolution of the molality with vertical position during the experiment of Fig. 2,9,20,30,60,75,105,120,180,240 and 360 min. after the beginning of the experiment. Circles are for experimental points and lines for theoretical fit. The horizontal line is the solubility. . . . . . . . 14


Jean Colombani

Measurement of the pure dissolution rate constant of a mineral in water

We present here a methodology, using holographic interferometry, enabling to measure the pure surface reaction rate constant of the dissolution of a mineral in water, unambiguously free from the influence of mass transport. We use that technique to access to this value for gypsum and we demonstrate that it was never measured before but could be deduced a posteriori from the literature results if hydrodynamics is taken into account with accuracy. It is found to be much smaller than expected. This method enables to provide reliable rate constants for the test of dissolution models and the interpretation of in situ measurements, and gives clues to explain the inconsistency between dissolution rates of calcite and aragonite, for instance, in the literature.

INTRODUCTION

Among heterogeneous reactions, dissolution of minerals in water is encountered in a wide spectrum of fields, from geochemistry to materials science, soil science, environmental science or oceanography. It plays a leading role, for instance, in the weathering of rocks as much as in the durability of mineral materials, in soils amendment, in pollutant spreading, or in sediment-water interaction. In all these situations, quantitative kinetic models are required to describe the involved phenomena.

More recently, this phenomenon has been recognized as being a keypoint in the modelisation of the atmospheric P CO 2 variation. Indeed, among the mechanisms controlling the pH of the ocean, and influencing in turn the atmospheric P CO 2 , dissolution drives two major steps: the weathering of terrestrial carbonates and the deep-sea sediment-water interaction [START_REF] Anderson | Glacial-interglacial stability of ocean pH inferred from foraminifer dissolution rates[END_REF]. Therefore kinetic models of P CO 2 change require reliable in situ and laboratory dissolution rates [START_REF] Hales | Evidence in support of first-order dissolution kinetics of calcite in seawater[END_REF].

But the establishment of reliable rate laws faces several difficulties. First, multiple lengthscales, and accordingly timescales, are involved during the dissolution process of a mineral, from the individual atomic detachment to the change of shape of the solid. Secondly, the number of parameters influencing the global dissolution kinetics is particularly large: surface topology, amount and nature of surface defects, adsorbed species, reactive surface area, pressure, temperature, composition of the solution, hydrodynamic behavior of the liquid, pH, . . . So a comprehensive dissolution model is still lacking and experimental results of dissolution rates show inconsistencies [START_REF] Morse | Calcium carbonate formation and dissolution[END_REF].

To obtain rate laws of the dissolution of minerals in water, two methodologies are employed. In bulk experiments, the concentration of chemical species in stirred water, where a sample dissolves, is measured and the dissolution rate deduced from the time evolution of this concentration. In local experiments by Atomic Force Microscopy (AFM) or Vertical Scanning Interferometry (VSI), the evolution of the surface topology at the molecular scale is followed during dissolution and the kinetics possibly deduced from atomic step velocity or surface-normal retreat measurement [START_REF] Vinson | Multiple length-scale kinetics: an integrated study of calcite dissolution rates and strontium inhibition[END_REF]. Agreement 1 Jean.Colombani@univ-lyon1.fr 1 between bulk and local techniques remains scarce (cf. [START_REF] Arvidson | Variation in calcite dissolution rates: A fundamental problem? Geochim[END_REF] for the case of calcite, [START_REF] Luttge | Interferometric study of the dolomite dissolution: A new conceptual model for mineral dissolution[END_REF] for dolomite , [START_REF] Luttge | Crystal dissolution kinetics and gibbs free energy[END_REF] for albite and [START_REF] Jordan | Organic and inorganic ligand effects on magnesite dissolution at 100 • c and ph=5 to 10[END_REF] for magnesite).

In both experiments the solvent is flowing past the mineral surface, in a laminar or turbulent regime. A mass transport boundary layer develops in the vicinity of the solid, in which the liquid velocity grows from zero at the interface (the no-slip condition), to its bulk value. In this context, dissolution proceeds through three steps (cf. Fig. 1). First the ions are unbound from the solid and solvated. Following a transition state theory the kinetics of the matter removal from the surface obeys a power law R diss = k s (1-c s /c sat ) n , where R diss is the dissolution rate, k s the surface reaction rate constant, c s the concentration of the dissolved species at the surface, c sat their solubility and n a constant [START_REF] Lasaga | Kinetic theory in the earth sciences[END_REF]. Then the ions migrate through the diffusional boundary layer (DBL). The concentration is generally considered as linear in this layer and Fick's law writes R diff = k t (c s -c), where R diff is the diffusion rate, k t = D/δ a transport rate constant, D the diffusion coefficient of the dissolved components, δ the DBL thickness and c the concentration in the bulk liquid [START_REF] Lasaga | Kinetic theory in the earth sciences[END_REF]. Finally the ions are advected by the flow toward the concentration measurement device. The kinetics of the phenomenon is driven by the slowest step, so minerals are classified according to their transport-controlled (e.g. rock salt), when R diff ≪ R diss , or reaction-controlled (e.g. quartz), when R diff ≫ R diss , character. Carbonate and sulfate minerals are considered to belong to a mixed kinetics class where both the transport and reaction flow rates are comparable [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF][START_REF] Rickard | Mixed kinetic control of calcite dissolution rates[END_REF].

When mass transport is identified as the limiting step, the pure dissolution coefficient k s is inaccessible and only k t = D/δ is obtained in bulk experiments. Though this transport coefficient has been measured in numerous dissolution studies and is sometimes relevant for the understanding of dissolution scenarii in the field or in industrial situations, it does not concern, from a conceptual point of view, the nature of the mineral and the physicochemistry of its surface: the boundary layer depth δ is a pure hydrodynamical quantity and the diffusion coefficient value of the common ions is always D ∼10 -9 m 2 s -1 . For mixed kinetics studies, also largely common, with the help of the above laws, an empirical surface rate constant k can be obtained "function of both surface reaction and mass transport" [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF], which is different from k s , whereas only this coefficient contains the chemistry of water-mineral interactions and varies by orders of magnitude from minerals to others.

In this paper, (i) we recall that the kinetics of matter removal during dissolution is characterized by a pure surface reaction rate constant, independent of the concentration field in the liquid and of the transport kinetics from the surface to the bulk liquid, and (ii) we claim that this pure dissolution rate constant can be measured unambiguously, even for transport-controlled and mixed kinetics, provided that the concentration field above the surface is known. To ascertain this claim, we have followed the subsequent procedure. First a working mineral and a relevant experimental technique have been selected (Section 2). Then reliable measurements of the pure surface reaction rate constant of this mineral have been performed in conditions where mass transport is proven to have no influence (Section 3). Subsequently most of the experimental dissolution rates available in the literature have been critically collected (Section 4). Finally the validity of these measurements and of our results is discussed (Section 5)

MATERIAL AND METHOD

To be able to measure the pure dissolution coefficient without ambiguity, i.e., without needing hydrodynamical assumptions, we need (i) to observe dissolution in absence of any convective flow and (ii) to access to the concentration field in the liquid soaking the solid. To achieve these two goals, we have carried out holographic interferometry measurements of the dissolution of a gypsum single crystal in water at rest.

Holography proceeds in two steps. First the hologram of an object is recorded by enlightening a photographic plate with both the beam coming from the object and a known reference beam. Whereas a classical photograph, where the plate is only enlightened by the object beam, exclusively contains infor-mations about the light amplitude, the interference pattern of the object and reference beams contains informations about both the amplitude and the phase. The phase itself is linked to the optical path, so to the third dimension and to the refractive index of the object. Secondly the hologram is enlightened only by the reference beam. This one is diffracted by the recorded interference pattern, which gives birth to a three-dimensional image of the object.

Here we use the potentiality of holography to record phase objects, i.e., transparent objects exhibiting only variation of their index of refraction (a transparent liquid, in our case). We record first the threedimensional state of the studied system in a hologram at time t 0 . Subsequently we enlighten both the hologram with the reference beam -which creates a 3D image of the object at t 0 -and the object. Thereby we have a superposition of the object at time t and of this object at time t 0 . If a variation has occurred between these two times, of concentration for instance, it is visualized through interference fringes. Holographic interferometry differentiates from classical interferometry in the fact that no external reference is needed, the object interfering virtually with a memory of itself.

In the course of an experiment, the hologram of a thermostated optical cell filled with pure water is registered. Then a thin mineral sample is introduced in the cell at a time considered as the time origin t 0 . Subsequently both the object and the hologram are enlightened and their interference pattern is recorded periodically as shown in Fig. 2. As the mineral dissolves the concentration of the dissolved species in water evolves, so does the refractive index, and the resultant optical path length difference between the object at time t and time t 0 (recorded in the hologram) induce interference fringes. From the topography of these fringes, the two-dimensional concentration field in the cell can be deduced. Parallely second Fick's law has been resolved with a first order chemical reaction at the solid-liquid interface in a semi-infinite one-dimensional approximation and brings the concentration evolution c(z, t) with vertical dimension z and time t. The best fit of the experimental curves deduced from holographic results with this analytical expression brings the pure surface reaction rate constant k s . Details on the experimental setup, procedure and data analysis have been given elsewhere [START_REF] Colombani | Holographic interferometry study of the dissolution and diffusion of gypsum in water[END_REF].

To validate our assertion on the hydrodynamical bias present in classical dissolution measurements, we need a mineral with the following requirements. First its dissolution rate must give a discernable amount of dissolved matter in laboratory times. Secondly, its kinetics must be either transport-controlled or mixed. Thirdly the chemical reaction must be as simple as possible, to focus on the hydrodynamical and topological aspects of dissolution. We have therefore discarded calcite, although most effort on mineral dissolution in water has concerned this mineral, because the various steps of the reaction make its kinetic law complex and strongly dependant on the pH and P CO2 values during the experiments. Gypsum (CaSO 4 , 2H 2 O) has been chosen because it fulfills these three criterions and because numerous literature results are available. Beside these methodological considerations, we must notice that gypsum dissolution liberates Ca 2+ , a cation liable to fix CO 2 , and is consequently of importance for the geological sequestration of this greenhouse gas.

RESULTS

We had measured the dissolution coefficient of the cleavage plane of gypsum from the Mazan quarry (France), in the frame of a study presenting the possibility of performing mineral dissolution measurement by holographic interferometry [START_REF] Colombani | Holographic interferometry study of the dissolution and diffusion of gypsum in water[END_REF]. This work was focussed on the experimental protocole and the cleaved Mazan gypsum sample was studied as an example. Our primary goal here is the demonstration of the experimental bias introduced by the liquid flow in the classical dissolution experiments (by solution chemistry or local probe). But one difficulty to confront the literature measurements and ours lies in the fact that the mineral geometrical and chemical properties differ from one setup to another. In global studies, samples are often powders, sometimes compacted, or polished crystals, whereas we use cleaved single crystals. So when only one identified interface dissolves in our case, dissolution in literature stems from multiple crystalline planes separated by steps and kinks. Furthermore the gypsum origin (so the impurities) differ between experiments. The only sample studied in our preceding work is therefore not sufficient to estimate the dispersion induced by this variation of gypsum samples. To capture the influence of these differences on k s we have compared some samples to our reference one, i.e., the (010) cleavage plane of gypsum from the Mazan quarry at 20.00 • C (k s = (4 ± 1)×10 -5 mol m -2 s -1 ) [START_REF] Colombani | Holographic interferometry study of the dissolution and diffusion of gypsum in water[END_REF]. We have carried out holographic interferometry experiments of the dissolution of gypsums of two other quarries: Tarascon-sur-Ariège in France (k s = 4×10 -5 mol m -2 s -1 ) and Mostaganem in Algeria (k s = 7×10 -5 mol m -2 s -1 ) and a synthetic commercial one from MaTeck GmbH (k s = 3×10 -5 mol m -2 s -1 ), to check the influence of the origin. Then we have carried out experiments with the (120) plane of Mazan gypsum polished with silicon carbide paper of grit size down to 15 µm (k s = 7×10 -5 mol m -2 s -1 ), to evaluate the behavior of non-cleaved planes.

All of these measurements deserve their own study and they were performed here only to estimate the variability of k s in various experimental conditions. But, as a first approach, we can notice that the dispersion among the k s of the (010) samples of various origins (k s = 3 to 7×10 -5 mol m -2 s -1 ) is greater than the standard error deduced from seven measurements on the (010) Mazan sample in our preceding work (∆k s = 1×10 -5 mol m -2 s -1 ). The choice of the quarry has therefore a non negligible influence on the dissolution rate constant. The ( 010) and ( 120) planes have different chemical and topological characters. The first one shows originally only water molecules and tends to be atomically flat [START_REF] Fan | Surface behavior of gypsum during dissolution[END_REF] whereas the second one is rough and exhibits water molecules, cations and anions. But despite these differences, there is less than a factor of two between their dissolution coefficients.

We are just searching for a k s range representative of the k s encountered in all the samples of the literature (cf. Section 5) so no average value was computed. Instead, if k max s and k min s are the maximum and minimum values, the nominal value of k s has been chosen as (k max s

+ k min s )/2 = 5 × 10 -5 mol m -2 s -1 with an uncertainty of (k max s -k min s )/2 = 2 × 10 -5 mol m -2 s -1 .

LITERATURE ANALYSIS

Dissolution rates of gypsum in water have been measured for various purposes (karst formation, soil amendment, marine biology, . . . ) and we have tried to gather and analyse these measurements to obtain a coherent view of the literature data. Fig. 3 collects the dissolution rates as a function of calcium concentration of most of the literature results. These include batch (dissolution in a reactor with stirrer), rotating disk, shaken tube, mixed reactor (rotating disk + flowing liquid) and flume experiments. Unfortunately flowing cell results could not be exploited because dissolution occuring all along the cell, a fixed concentration could not be ascribed to the value of the dissolution rate at the output of the cell [START_REF] Kemper | Dissolution rate of gypsum in flowing water[END_REF][START_REF] Keren | Gypsum dissolution and sodic soil reclamation as affected by water flow velocity[END_REF]. Gypsum dissolution has been also widely used in marine biology to estimate biological objects interaction with water. But as was extensively demonstrated by [START_REF] Porter | Gypsum dissolution is not a universal integrator of 'water motion[END_REF] hydrodynamic conditions are often badly or wrongly defined in these studies and dissolution rates cannot be ascribed to unamibiguous concentration and DBL thickness values.

In all these studies, the solvent is pure water and the working temperature is either 20 or 25 • C. The pH and ionic strength values are almost never given. But despite this lack, the gypsum solubility obtained in all these works is always c sat ≈ 15 mmol l -1 , identical to the reference value deduced from literature analysis by [START_REF] Christoffersen | The kinetic of dissolution of calcium sulfate dihydrate in water[END_REF] or [START_REF] Raju | The thermodynamics of "scale" mineral solubilities. 3. Calcium sulfate in aqueous NaCl[END_REF]. This proves the proximity between the chemical conditions of these experiments, and gives confidence in the possibility of comparing their dissolution rates.

At first glance, all these measurements do not provide a consistent description of the dissolution of gysum in water (cf. Fig. 3). Up to now, attention has been paid before all to the exponent of the power law of R(c) -considered as representative of the kinetics-, the prefactor being ascribed to the specificities of each experiment. But a correct consideration of the hydrodynamical condition of each device should bring a consistent and predictive knowledge of the dissolution process, whatever the experimental setup. So, first of all, we do not postulate that experiments are either transport-controlled of reaction-controlled and we make the assumption that all of them are influenced by both chemical reaction and diffusive transport in a variable proportion. The systems are always carefully maintained in a quasi-steady state, where the concentration is uniform in the whole bulk liquid and evolves slowly. So keeping in mind the dissolution steps shown in Fig. 1, we consider that matter is conserved during migration in the diffusional boundary layer. We apply mass conservation between the bottom and top of the DBL, equalizing the dissolution flow rate s r R diss and diffusion flow rate sR diff :

sD c S -c δ = s r k s (1 - c s c sat ) (1)
with s r the reacting surface area and s the outer surface area of the DBL (similar to the so-called geometric surface area of the solid, cf. Fig. 1). We have followed [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF] and considered that n=1 in R diss . From this equation, we compute the only unknown parameter, the surface concentration c s , and we can write the experimental dissolution rate R = R diff measured far from the solid after being advected:

R = 1 s srks + δ Dcsat - 1 scsat srks + δ D c (2)
The δ = 0 situation reflects an hypothetical absence of DBL where the kinetics is directly driven by the reaction, the transport time between the surface and the measurement device becoming negligible. We see that the pure surface reaction rate constant k s can be accessed via lim δ→0 1/(∂R/∂c) = sc sat /(s r k s ). For this purpose, we need an evaluation of c sat , δ and s/s r . The solubility is known for long: c sat = 15 mmol l -1 [START_REF] Raju | The thermodynamics of "scale" mineral solubilities. 3. Calcium sulfate in aqueous NaCl[END_REF]. The diffusional boundary layer thickness wrapping the dissolving solid for each device is more delicate to obtain This quantity is clearly identified in rotating disk experiments: δ = 1.61D 1/3 ν 1/6 ω -1/2 with ν the kinematic viscosity of the solution and ω the angular velocity of the disk [START_REF] Barton | Dissolution rates of polycristalline samples of gypsum and orthorhombic forms of calcium sulphate by a rotating disc method[END_REF]. For batch experiments, we have considered that at first order the liquid develops a layer as if flowing past a semi-infinite plate: δ = (2π/0.244) 1/3 ν 1/6 v -1/2 D 1/3 x 1/2 with v the unperturbed liquid velocity and x the distance from the edge of the solid [START_REF] Jousse | A method to dynamically estimate the diffusion boundary layer from local velocity conditions in laminar flows[END_REF]. We have identified x with the longest size of the dissolving crystals and we have taken v = αv stirrer where α, strongly dependant on the geometry, was chosen as 0.3. For shaken tubes experiments, the same expression was used with v = v tube . For flume experiments, this expression was also used with the downstream velocity of water for v.

At this stage we have plotted in Fig. 4 the inverse of the slope of the experimental R(c) curves in Fig. 3 for each experiment against the DBL thickness δ. Despite the uncertainty in the determination of δ for each experimental device, the resulting curve is linear, as predicted by Eq. 2. So each experiment, according to its geometry and flowing configuration, probes a particular value of the boundary layer thickness 2 . Thus the discrepancies between the dissolution rate constants in the literature stem mainly from the hydrodynamic differences between the experimental setups.

DISCUSSION

Now that a consistency has been recovered between the literature results, is it possible to get from these results a reliable pure surface raction rate constant, comparable to our holographic value? The extrapolation in Fig. 4 of a linear curve fitting all the literature data to δ = 0 gives sc sat /(s r k s ). To obtain k s from this value, the roughness factor ξ = s r /s is required, so we have to evaluate the mean reactive surface area s r of the crushed samples used in the literature experiments.

Two topological classes of objects are known to have a strong influence on s r : etch pits and steps/kinks. For the first ones, we benefit from a recent thorough investigation by Atomic Force Spectroscopy of the cleavage surface of gypsum during dissolution by [START_REF] Fan | Surface behavior of gypsum during dissolution[END_REF]. The main inference drawn from these experiments is the minor role of etch pits, remaining always shallow, in the process. The latter is governed by the fast dissolution of very unstable steps, whatever the solution undersaturation. This feature tends to enhance the similarity of gypsum with other minerals, where etch pits seem to contribute to dissolution not via the increase of reactive surface area but as source of monolayer steps inducing overall dissolution [START_REF] Luttge | Interferometric study of the dolomite dissolution: A new conceptual model for mineral dissolution[END_REF][START_REF] Gautier | Are quartz dissolution rates proportional to B.E.T. surface areas?[END_REF]. For the second ones, we have too few informations at the moment about the behaviour of stepped and kinked faces of gypsum during dissolution to estimate their influence. So taking into account the AFM study we consider that the whole surface -for example measured by a BET adsorption experiment-is reactive, i.e., s r = s BET . Only one paper gives both its BET and geometric surface area values [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF]. As the roughness factor ξ = s r /s should be quite similar for all crushed gypsum samples, we have used the value of ξ = 1100/60 deduced from this work for all literature experiments. We have thereby computed k s = (∂R/∂c) δ=0 c sat /ξ and found k s = 7 × 10 -5 mol m -2 s -1 , which is the pure dissolution rate constant. This value agrees fairly well with our k s = (5 ± 2) × 10 -5 mol m -2 s -1 holographic value. We can say that this coefficient had never been measured but can be deduced from the literature measurements a posteriori, and that it is much lower than all the rate constants proposed by the authors from their R(c) curves.

In a more graphical way, the resulting c sat /(ξk s ) of our holographic value has been added in Fig. 4 at δ = 0. Remembering the experimental variability of k s , the dispersion of the literature values and the imprecision of the roughness factor ξ, the agreement between the value of k s extrapolated from the literature results and the value deduced from the holographic interferometry measurements is striking and validates both our analysis of the solution chemistry experiments and the accuracy of k s .

CONCLUSION

We have shown that despite apparent discrepancies, the correct consideration of the hydrodynamic conditions of the global dissolution experiments in literature brings a consistent view of the dissolution kinetics of gypsum in water and a pure surface reaction rate constant can be computed from all these measurements, much lower from what was expected. We have parallely measured by holographic interferometry this pure surface reaction coefficient in water at rest for various gypsum origins and surface morphologies. These two values agree well, thus validating a method able to provide benchmark values of dissolution rate constants of minerals.

This work has two methodological consequences. From a theoretical point of view, the holointerferometric tool may contribute to the evaluation of models. For instance, it permits to access directly to the prefactor K in the kinetic reaction law R = K exp [-∆G/(RT )] of the transition state theory or the dissolution plateau A of the more elaborated model of [START_REF] Lasaga | Variation in crystal dissolution rate based on a dissolution stepwave model[END_REF] of the dissolution of minerals, these both terms being merely equal to k s . From an experimental point of view, the underestimation of the transport contribution to the dissolution rate can help to explain the discrepancy between dissolution rates measured by local and global setups, for instance the very low value of the dissolution rate of calcite and dolomite in water measured by normal retreat of the surface via VSI, compared to values measured in bulk solution chemistry experiments [START_REF] Arvidson | Variation in calcite dissolution rates: A fundamental problem? Geochim[END_REF][START_REF] Luttge | Interferometric study of the dolomite dissolution: A new conceptual model for mineral dissolution[END_REF][START_REF] Luttge | Crystal dissolution kinetics and gibbs free energy[END_REF]. [START_REF] Christoffersen | The kinetic of dissolution of calcium sulfate dihydrate in water[END_REF], [START_REF] Liu | The kinetic of dissolution of calcium sulfate dihydrate[END_REF] and [START_REF] Lebedev | Dissolution kinetics of natural-gypsum in water at 5-25 •[END_REF], from rotating disk experiments by [START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF] and [START_REF] Barton | Dissolution rates of polycristalline samples of gypsum and orthorhombic forms of calcium sulphate by a rotating disc method[END_REF], from mixed flow/rotating disk reactor experiment by [START_REF] Raines | Mixed transport / reaction control of gypsum dissolution kinetics in aqueous solutions and initiation of gypsum karst[END_REF], from shaken tube experiments by [START_REF] Bolan | Dissolution of various sources of gypsum in aqueous solutions in soil[END_REF] (with powder or compressed powder pellets) and [START_REF] Gobran | Dissolution rate of gypsum in aqueous salt solutions[END_REF] and flume experiments by [START_REF] Opdyke | Mass transfer from smooth alabaster surfaces in turbulent flows[END_REF]. c sat is the solubility of the dissolved calcium ions in water.

Figure 1 :

 1 Figure 1: Scheme of the successive mechanisms occuring during the dissolution of a crystal grain in a bulk solution chemistry experiment.

Figure 2 :

 2 Figure 2: Interferograms of the dissolution of a Mazan gypsum single crystal 20, 65, 105, 200 and 377 min. after the start of the experiment. The crystal is a thin grey line at the bottom of the optical cell.

Figure 3 :

 3 Figure 3: Dissolution rates R of gypsum in water as a function of calcium concentration c for various setups. The R(c) curves are drawn from stirred batch experiments by Jeschke et al. (2001),[START_REF] Christoffersen | The kinetic of dissolution of calcium sulfate dihydrate in water[END_REF],[START_REF] Liu | The kinetic of dissolution of calcium sulfate dihydrate[END_REF] and[START_REF] Lebedev | Dissolution kinetics of natural-gypsum in water at 5-25 •[END_REF], from rotating disk experiments by[START_REF] Jeschke | Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics[END_REF] and[START_REF] Barton | Dissolution rates of polycristalline samples of gypsum and orthorhombic forms of calcium sulphate by a rotating disc method[END_REF], from mixed flow/rotating disk reactor experiment by[START_REF] Raines | Mixed transport / reaction control of gypsum dissolution kinetics in aqueous solutions and initiation of gypsum karst[END_REF], from shaken tube experiments by[START_REF] Bolan | Dissolution of various sources of gypsum in aqueous solutions in soil[END_REF] (with powder or compressed powder pellets) and[START_REF] Gobran | Dissolution rate of gypsum in aqueous salt solutions[END_REF] and flume experiments by[START_REF] Opdyke | Mass transfer from smooth alabaster surfaces in turbulent flows[END_REF]. c sat is the solubility of the dissolved calcium ions in water.

Figure 4 :

 4 Figure 4: Inverse of the R(c) slopes of Fig. 3 as a function of the diffusional boundary layer thickness δ for each setup.

We may just notice that the gap between[START_REF] Raines | Mixed transport / reaction control of gypsum dissolution kinetics in aqueous solutions and initiation of gypsum karst[END_REF] measurements and the fitted law seems to corroborate the doubt shed by[START_REF] Dreybrodt | Comments on: Mixed transport / reaction control of gypsum dissolution kinetics in aqueous solutions and initiation of gypsum karst by Michael A. Raines and Thomas A[END_REF] on this result in their Comment.

Acknowledgements-We thank Elisabeth Charlaix and Jacques Bert for fruitful discussions, Nicolas Sanchez for experimental help, and Lucienne Jardin, Sylvain Meille and Pierre Monchoux for gypsum samples. This work was supported by Lafarge Centre de Recherche, Région Rhône-Alpes and CNES (french spatial agency).


Ye-Qiong Song
email: song@loria.fr

Networked Control Systems: From Independent Designs of the Network QoS and the Control to the Co-design

Keywords: Networked control system, Network, Quality of Service, Co-design

Taking a network QoS designer's point of view, this paper firstly reviews some of the recent advances on the NCS (Networked Control Systems) design then analyzes its requirements in terms of network QoS guarantees and its capacity to tolerate network performance variation. Current deterministic QoS design approach (including traffic schedulability analysis) may lead to network resource overprovisioning problem since worst-case scenario is often dictated to network QoS designers by the control application. We show that integrated control and network QoS co-design consists in a better solution to this problem. Taking into account the current available results, we see that more research efforts remain to do on control and network QoS co-design and on the development of on-line adaptive QoS mechanisms.

INTRODUCTION

Process control is the most important class of real-time applications that industrial networks and especially fieldbuses support. Considering the distributed sensors, controllers and actuators interconnected by a network, and bearing in mind that deterministic or upper bounded end-to-end packet response time must be guaranteed, great efforts have been made since 1980 to develop various but IEC standardized solutions [START_REF] Thomesse | Feildbus technology in industrial automation[END_REF], [START_REF] Zurawski | The industrial communication technology handbook[END_REF]. They are mainly MAC protocols but also application layer standards allowing the application interoperability. Real-time communication is ensured essentially at MAC layer. The main paradigm used to conduct those developments is assuming that messages to be transmitted by the network have deadlines that a network must meet, otherwise the solution is considered as invalidate since the consequence of missing a deadline is considered catastrophic for the application. Although this scenario could be true for real-time critical alarm reporting for example, for the most of control applications, occasional violation of a message deadline constraint can often be tolerated thanks to the closed-loop control robustness.

So looking for absolutely guaranteeing the deadlines of all the control loop related data exchanges is neither necessary nor optimal from resource utilization point of view. New design approaches should be developed. The resulting system that one can imagine is an adaptive one. Not only the network should be able to dynamically allocate necessary resources to a control application whenever needed, but also the application should accept some network QoS degradation resulting in a degraded but still acceptable QoC (Quality of Control). For a control application, dynamic network resources allocation can be done using network QoS adaptation mechanisms (e.g. priority re-allocation) according to the observation of the application related parameters such as the sensor-to-actuator delay, process state output deviation or some control loop cost function [START_REF] Boughanmi | Online adaptation of the IEEE 802.15.4 parameters for wireless networked control systems[END_REF], [START_REF] Juanole | Networked control systems: definition and analysis of a hybrid priority scheme for the message scheduling[END_REF]. When the network QoS adaptation reaches its limit (especially during network overload periods), control applications must be adapted by adjusting for example the control loop sampling period [START_REF] Eker | A feedback scheduler for real time control tasks[END_REF], [START_REF] Cervin | Integrated control and real-time scheduling[END_REF], [START_REF] Marti | Managing quality of control in network based control systems by controller and message scheduling co-design[END_REF], [START_REF] Simon | Robust control/scheduling co-design : application to robot control[END_REF], [START_REF] Antunes | Dynamic rate and control adaptation in networked control systems[END_REF], or both the sampling period and control gain [START_REF] Jia | Graceful Degradation of the Quality of Control through Data Drop Policy[END_REF], [START_REF] Felicioni | Optimal on-line (m,k)-firm constraint assignment for real-time control tasks based on plant state information[END_REF]. This results in a more radical solution to network overload problem than the QoS adaptation since the application traffic is reduced by increasing the sampling period.

All those approaches are co-design ones as they consider at the same time not only the network QoS but also the control application quality (QoC). In fact, for most of control applications, relaxing hard real-time constraint is not only possible thanks to the closed-loop control robustness, but also desirable in order to optimize the network resources allocation when they are shared by several applications. The goal of this paper is to present some new trends toward the adaptive real-time NCS (networked control systems). In particular, integrated control and network QoS co-design approach is presented as an interesting direction.

The rest of the paper is organized as following. Section 2 gives a short review of the NCS and defines the co-design approach. Section 3 presents some control application-aware network QoS adaptation mechanisms. Section 4 gives some insights on the QoC adaptation for dealing with system overload (mainly on processor sharing but can be applied to network sharing). Finally Section 5 concludes the paper.

Networked control systems

Figure 1 shows a general NCS architecture in which the communication between sensors, actuators and controllers occurs though a multi-purpose network often shared by several control loops (Fig. 1)

ref N E T W O R K Controller Actuator
Sensor Process Other flows

Fig. 1. General architecture of NCS

It is not so different from the fieldbus systems we are used to deal with in the factory automation community. However the focus is quite different. In fact, the fieldbuses or more generally speaking the industrial networks aim at providing general purpose real-time communication without considering one specific control application. The NCS is a research topic initiated by the automatic control community.

There are two objectives: control over networks and control of networks. The former deals with control law design problems taking into account the network characteristics (e.g. delay and packet loss) while the latter focuses on the control of the network QoS by applying feedback control theory.

There is abundance of research works dealing with NCS.

Interested readers can refer to [START_REF] Antsaklis | Technology of networked control systems[END_REF] and [START_REF] Zampieri | Trends in networked control systems[END_REF] for further details.

In this section we only focus on the control over networks. Readers can refer to [START_REF] Antsaklis | Technology of networked control systems[END_REF] for rather complete details on NCS.

Control loop robustness and QoC

Control systems are often considered as typical examples of hard real-time systems where deadline violation is strictly forbidden. Ensuring deadline meet has been the main preoccupation of the real-time scheduling theory, where the deadline constraint is directly deduced from the sampling period. With NCS, ensuring strict deadline meet is much more difficult because of the network QoS fluctuation, especially in case that the network is shared by several different applications. Fortunately experiments show that this hard deadline assumption may be false for closed-loop control. In fact, any practical feedback control system is designed to obtain some stability margin and robustness with respect to the process parameters uncertainty. This also provides some robustness with respect to timing uncertainties. It means that closed-loop control systems are able to tolerate some sampling period deviations and occasional data loss without losing the stability. In general, a commonly shared idea is that lower is the control period, better is the control performance. In [START_REF] Cervin | Integrated control and real-time scheduling[END_REF], the control performance variation has been experimentally checked using an inverted pendulum with a LQ controller for different sampling period values. It has been shown that the control system remains stable although the performance degradation. In [START_REF] Moyne | The emergence of industrial control networks for manufacturing control, diagnostics, and safety data[END_REF], an illustrative chart (Fig. 2) has been given to show the importance of choosing a good sampling period which gives trade-off between the control performance and the related network load. Where x(t) represents the process (plant) state, y(t) the output signal, () ut its input (command signal); () vt is white Gaussian noise disturbing the process state, which assumed to be independent and with zero mean. A, B, C and G are constant matrix describing the process dynamic.

And consider a proportional controller with gain L:

( ) ( ) u t Lx t (2)
The stability is of primary important for a control loop. There exist several stability criteria [START_REF] Aström | Computer-controlled Systems: Theory and Design[END_REF].

However the main idea is that the system state variation should be kept within a certain limits.

Typical performance criteria for feedback control loops include overshot to a step reference, steady-state tracking error, phase margin or time-average tracking error. Variance of the state variables x(t) is often used as a cost function to which the variance of the control effort u(t) is also added as shown in equation 3.

0 1 ( ( ) ( ) ( ) ( )) H TT J x t Qx t u t Ru t dt H (3)
Where Q and R are quadratic weight matrix representing the importance of the elements in the vectors x(t) and u(t) respectively. H is the time horizon during which the cost function is calculated. A great value of J indicates thus either a great deviation of the process state to a desired state or a great control effort to bring the state to its reference value.

The system is instable when J tends to infinite.

In general, the QoC is defined as the control performance which can be measured using for example the cost function defined by equation 3.

NCS co-design

Most of the research works on NCS are from automatic control point of view and focus on the robust control loop design which takes into account the network induced delays and data loss. Network delays are assumed either constant (by input data buffering) or randomly distributed following a well-known probability law. Data losses are assumed following Bernoulli process. This paper does not have intention to review those results since very good reviews can be found in [START_REF] Antsaklis | Special issue on networked control systems[END_REF], [START_REF] Antsaklis | Technology of networked control systems[END_REF], [START_REF] Zampieri | Trends in networked control systems[END_REF].

From those works we can see that the assumptions on network delay or data loss pattern seldom consider the actual network characteristics and the possible QoS mechanisms which are specific from one type of network to another. Indeed, using a prioritized bus like CAN, a switched Ethernet or a wireless sensor network will result in fundamentally different QoS characteristics. This point is of primary importance especially when the network is shared by several control loops and other applications whose exact characteristics are often unknown at the control loop design step. In this case the traffic scheduling has a great impact on both delay variation and packet loss, which in turn impact on the quality of control (stability and performance) of the control loops.

One solution to this problem is tightly coupling the control with network during the design step which is called co-design approach [START_REF] Branicky | Networked control systems co-simulation for co-design[END_REF]. One important issue towards this objective is the co-design tool development capable of modelling and simulating both the control and network parts. Mathlab/Simulink based TrueTime [START_REF] Ohlin | TrueTime1.5-Reference Manual[END_REF]) is one of the useful tools for this purpose.

Control of networks

Control of networks has the same objective than the traditional network QoS design. However they are different in terms of the used approaches. Traditional network QoS design approach uses rather static resource allocation principle by scheduling the different resource demands of the different network data flows (or applications). The control of networks approach is based on the dynamic feedback control of the resource allocation to the network data flows for maintaining their desired QoS. This requires the monitoring and on-line modification of the network parameters, which are not easily implementable considering the current layered communication protocols. A typical example is the control of the network congestion by using a stochastic controller LQG [START_REF] Altman | Congestion control as a stochastic control problem with action delays[END_REF]. For a condensed review of the recent works, readers can refer to [START_REF] Zampieri | Trends in networked control systems[END_REF].

Using

Fig. 3. Control and network QoS co-design

In the next section, we choose to only review some simpler QoS adaptation mechanisms without necessarily using feedback control theory.

Application-aware dynamic QoS adaptation

Dynamic network QoS adaptation aims two objectives: adaptation to the network operating condition and to the change of the application requirements. QoS adaptation to the network operating condition is normally ensured by the existing QoS mechanisms and will not be reviewed in this paper.

NCS can use network QoS adaptation mechanisms (e.g. priority re-allocation) according to the observation of the application related parameters such as the sensor-to-actuator delay, process state output deviation or some control loop cost functions such as what is define by equation 3. There are several works on this direction which can give inspiration for further adaptive QoS mechanisms development.

Dynamic CAN message priority allocation according to the control application needs

In [START_REF] Juanole | Networked control systems: definition and analysis of a hybrid priority scheme for the message scheduling[END_REF], a hybrid CAN message priority allocation scheme is proposed. This scheme is inspired from the mixed traffic scheduling scheme defined by [START_REF] Zuberi | Scheduling messages on controller area network for real-time cim applications[END_REF]. The idea is to separate CAN identifier bits into two fields: one for dynamic priorities and one for static priorities. In the normal case, messages are assigned with static priorities. When there is a urgent transmission need, dynamic priority field can then be used giving thus higher priority to the urgent transmission. The main contribution of the work of [START_REF] Juanole | Networked control systems: definition and analysis of a hybrid priority scheme for the message scheduling[END_REF] resides in the exhibition of a link between the hybrid priority scheme and the control loop performance. The studied NCS is similar to that of Fig. 1. The process to control is a DCservo process and the controller is PD (Proportional Derivative). Following cost function is used which interests in the difference between the reference r(t) and the process state output signal y(t) :

2 0 1 ( ( ) ( )) H J t r t y t dt H (4)
The key issue is to define a relation to translate the control need in terms of the QoS (mainly message transmission delay) into a dynamic priority. In this work authors proposed to use the control signal u as the indicator to trigger dynamic priority allocation. The triggering is done by the controller node when it receives the CAN message that the sensor node sends at each sampling period. According to the sensor data, the controller calculates the new value of u, and decide to choose a new priority to send the command data if necessary.

So the important result in this work is that the priority to be used to send a command data through CAN network is a function of the control signal u so that the network QoS is dynamically adjusted to accommodate to the dynamic control application needs.

3.2 Dynamic priority in IEEE802.15.4/Zigbee according the NCS needs IEEE802.15.4/Zigbee based wireless sensor network technology is becoming attractive for industrial communication [START_REF] Willig | Recent and emerging topics in wireless industrial networks: A selection[END_REF], [START_REF] Mouney | On the implementation of one process control application type through a network considering three LANs: CAN, WiFi, ZigBee[END_REF].

Wireless NCS is a new and important topic that should be further investigated. In [START_REF] Boughanmi | Online adaptation of the IEEE 802.15.4 parameters for wireless networked control systems[END_REF] we proposed a simple dynamic priority scheme, which for the moment only provides two priorities. The key idea explored is that when the QoC of the controlled system is not sufficient, the quality of service of the network is dynamically adapted by giving higher priority (thus less delay) to the important control data transmission. This adaptation is realized by modifying the macMinBE parameter of the MAC protocol of the IEEE 802.15.4. For illustrating the approach, a cart position control system is considered (Fig. 4) and its control loop is given in Fig. 5. In this example, CSMA/CA MAC instead of GTS has been chosen. In order to make CSMA/CA support different priorities, in [START_REF] Boughanmi | Online adaptation of the IEEE 802.15.4 parameters for wireless networked control systems[END_REF]) particular protocol parameter configuration is suggested which consists in choosing different upper ranges of the backoff exponent for the control loop related nodes when the QoC begins to decrease. The network is shared by other applications generating an additional workload. In normal network operating condition (i.e. light workload), all the nodes have the same priority. By increasing the external workload, the traffic generated by the control loop is disturbed resulting more packet delay or loss, which in turn impact the QoC. In this case, it is preferable to increase the priority of the control loop related traffic.

For observing the QoC variation the control error e r y is used. The error should be bounded by a threshold to ensure the required QoC to the plant. This threshold depends on the controlled process and on the reference value if there is any.

If e > threshold, the system is considered to be in a critical situation and action has to be taken. The trouble is caused either by the controlled system itself or by the network (overloaded network). Action should be taken on the network for a certain period of time through offering more resources to the wireless NCS. If the situation is still not improved (there is a problem in the control loop), action should be taken on the control loop by changing the sampling period for example.

Fig. 6 and 7 show respectively the simulated system response with the same workload without and with the dynamic QoS adaptation. The effectiveness of the adaptation can be easily seen. It is worth noting that with IEEE802.15.4, it is indeed also possible to get real-time communication feature by using GTS (Guaranteed Time Slots) [START_REF] Koubâa | Improving the IEEE802.15.4 slotted CSMA/CA for time-critical events in wireless sensor networks[END_REF]) [START_REF] Koubâa | A time division beacon scheduling mechanism for IEEE 802.15.4/Zig-Bee clustertree wireless sensor networks[END_REF]), (Francome 2007). Further effort should be made to find a concrete way to dynamically allocate the GTS slots according to the QoC variation. In fact, the standard mechanism only allows the nodes to ask to the coordinator for obtaining the desired GTS slots by sending their demands via CSMA/CA. This may result in important demand latency.

Discussions

A lot of other related works contribute to this topic they will not be reviewed here because of space limitation [START_REF] Walsh | Stability analysis of networked control systems[END_REF]. Considering the important position of Switched Ethernet in industrial communications, we just cite the work of [START_REF] Diouri | Accommodation of delays for NCS using classification of service[END_REF]) that dealt with the dynamic allocation of bandwidth share in Ethernet switches with WRR (Weighted Round-Robin) scheduler according to both observed delay and the QoC (difference between the reference and the process state). For this purpose, the bandwidth share (i.e., the weight assigned to each data flow or Ethernet switch port) is defined as a function of the sensor to actuator delay and the current QoC level.

Instead of looking for meeting deadlines by static network resource allocation, the works reviewed in this section present a new approach which consists in dynamically allocate network resource according to the control application needs. This leads to more effective resource utilization since network resource can be used by other applications during the period that a specific control application does not need. For embedded systems with resource constraint, this applicationaware QoS design approach allows also avoid resource overprovisioning problem as the resource is no longer dimensioned for the worse-case. In summary, dynamic QoS adaptation according to network operating condition and QoC requirements effectively improve the total QoC of the NCS. However for each type of network, one has to find not only the critical QoC condition for triggering the QoS adaptation, but also the corresponding mechanisms to act on the network parameters [START_REF] Mouney | On the implementation of one process control application type through a network considering three LANs: CAN, WiFi, ZigBee[END_REF]. This calls for more research effort.

QoC adaptation to network operating conditions

Network QoS adaptation can enhance the NCS performance until some limit. Beyond this limit, the control application should re-adjust its own parameters to continue to run even in a degraded mode. This can happen during network overload in NCS where a network is shared by several control loops. A common approach to deal with this problem is to dynamically change the sampling period. In fact the works reviewed in the previous section have in most of the case a control loop with variable sampling period because of the no constant network induced delay. The difference is that in previous section, we suffer from this sampling period variation and only act on the network QoS (by dynamic resource re-allocation) but not on the control loop itself. In this section we review some representative works that dynamically change the control loop parameters when, for example, the underlying network can no longer provide requested QoS. We choose to present two types of solutions: explicit sampling period adjustment and indirect sampling period adjustment which is based on selective packet drops according to (m,k)-firm model (Hamdaoui and Ramanathan 1995).

Explicit sampling period adjustment approach

The first work dealing with sampling period adjustment is reported in [START_REF] Seto | On task schedulability in real-time control system[END_REF]. The problem considered is the optimization of the QoC of a set of control loops sharing a same resource (processor). The QoC of each control loop is measured by the cost function (equation 5) which is similar to equation 3 but it is a function of the sampling period h.

( ) lim ( ( ) ( ) ( ) ( ))

H TT H J h

x t Qx t u t Ru t dt (5)

The problem to solve is then the calculation of the optimal sampling periods minimizing the weighted sum of the cost function of each control loop:

( ,..., ) 1 min ( )

n n h h i i i w J h (6)
under constraint of the schedulability:

1 / n ii i C h A (7)
Where w i is used for weighting the importance of a control loop with respect to the others. C i is the execution time of the control task i. 01 A is the processor utilization threshold guaranteeing the tasks schedulability and whose value depends on the used scheduling policy. This method has been extended in [START_REF] Eker | A feedback scheduler for real time control tasks[END_REF] and [START_REF] Cervin | Integrated control and real-time scheduling[END_REF]). On-line adaption has been proposed to accommodate to the application configuration change. This is achieved using a regulator monitors the current configuration of the application. As soon as a configuration change trend is detected by this regulator, the optimal periods for the new configuration are calculated according to an approximate method of the equations 5 to 7 (because of the high computing complexity of those equations). Furthermore, in [START_REF] Henriksson | Optimal on-line sampling period assignment for real-time control tasks based on plant state information[END_REF] the process state has been included in the optimal period calculation. This allows to favor control loops whose state variables is experiencing great changes comparing to those whose state variables are more stable.

Although current existing works only deal with control tasks sharing a cpu, similar principle should also be applicable to control loops sharing a network.

Sampling period adjustment with selective packet

dropping Adjusting sampling period could lead to implementation difficult for general purpose network since at network overload situation, the period change decision must still be transmitted to the sensor nodes via network, implying sometimes complicated mechanisms. In [START_REF] Jia | Graceful Degradation of the Quality of Control through Data Drop Policy[END_REF]) and [START_REF] Felicioni | Optimal on-line (m,k)-firm constraint assignment for real-time control tasks based on plant state information[END_REF] we proposed an indirect sampling period adjustment method which selectively drops some sampling packets in case of network (or processor) overload. The sampling period of a control loop is therefore increased although this alternative can only adjust the period by the multiple of the nominal one. The result is less accuracy than the direct sampling period adjustment method but it often implies less implementation complexity.

It is well known that data losses have a direct impact on the QoC. But most importantly is to avoid long term consecutive data losses [START_REF] Felicioni | Optimal control with packet drops in networked control systems[END_REF]. One way to avoid long term consecutive data losses is to use (m,k)-firm model introduced by (Hamdaoui and Ramanathan, 1995). This model can be used, for instance, to indicate that at least m out of any k consecutive sampling packets has to be sent to the controller within their deadline, where m and k are two positive integers with mk (the case where m=k is equivalent to the ideal case, which is noted by (k,k)-firm and corresponds to the hard real-time constraint). If a control system is designed to accept a control performance degradation until k-m deadlines misses (or equivalently packet losses) among k consecutive ones (this can be justified by the observation that most control systems can tolerate misses of the control law updates to a certain extent), the system can then be designed according to the (m,k)-firm model to offer the variable levels of QoC between (k,k)-firm (ideal case) and (m,k)-firm (worst case) with as many intermediate levels as the possible values there are between k and m. This results in a control system with graceful degradation of control performance [START_REF] Ramanathan | Overload management in Real-Time control applications using (m,k)-firm guarantee[END_REF].

In [START_REF] Felicioni | Optimal on-line (m,k)-firm constraint assignment for real-time control tasks based on plant state information[END_REF], we considered N physical plants with one dedicated controller implemented as a real-time task for controlling each plant. Each instance of a task is responsible for carrying out the control law computation and has a deadline by which it is expected to complete its computation. We consider a centralized implementation of all the controllers as shown in Fig. 8. Note that the same approach applies to the case where N control loops share a same network.

i i i am k a k m (8)
and as optional, otherwise.

The control tasks are scheduled using the fixed priority policy. The mandatory instances of all the tasks are assigned the rate-monotonic priorities. That is, the mandatory instances of i are assigned a higher priority than the mandatory instances of j if h i <h j . The optional instances are assigned the lowest priority. Following sufficient condition ensures the schedulability.

Given a task set ( 1, 2 … n ) such that h 1 <h 2 < ..<h n . Let:

j i ij j j m h n k h (9) If 1 1 i i ij j i j C
n C h for all 1 i n, then the (m i, k i )-firm constraint of each task i is satisfied.

In [START_REF] Jia | Graceful Degradation of the Quality of Control through Data Drop Policy[END_REF], the cost function defined by equation 5 has been adapted to include also the distribution of m i among k i instances and an optimization problem is formulated similar to that of [START_REF] Seto | On task schedulability in real-time control system[END_REF] but equation 9 as the schedulability constraint. Moreover a method is given to calculate the optimal control gain for each value of m i .

For giving a concrete insight, let's consider a system composed of four control loops for controlling four carts and sharing a same processor [START_REF] Jia | Graceful Degradation of the Quality of Control through Data Drop Policy[END_REF]). At a system configuration change, the task handler (Fig. 8) receives the information about the number of tasks sharing the processor and the actual execution time C i of each task, and deduce the new (m i ,k i )-firm constraint as well as the corresponding optimal control gain for each control task by resolving the optimization problem.

Fig. 9 shows the simulation trace of the system without instance dropping according to (m,k)-firm. At t = 2, 3 is turned on. Together with 1 and 2 , the pre-emption due to these tasks makes the execution of 4 impossible. As a result, the cart system Cart 4 becomes instable since all its instances miss their deadline. In this paper we presented the control and network QoS codesign approach. Two important points are presented. One is the control application-aware dynamic QoS adaptation. It allows providing the QoS to the applications only when needed, releasing thus network resource to other applications. Another is to make the control application adaptive to the network operating condition variation. It leads to more robust NCS. So the advantage of this integrated co-design approach is on the one hand the great improvement of the system robustness, and on the other hand the minimisation of the resources necessary for meeting the required QoC. This minimisation is of special interest for autonomous embedded systems.

Networked Control Systems form a growing field and call for the development of integrated approaches requiring multidisciplinary skills in control, real-time computing and communication protocols. More research efforts remain to be provided on control and network QoS co-design and on the development of on-line adaptive QoS mechanisms, especially for wireless NCS. Another interesting direction is applying feedback control theory to the network QoS control. Further research efforts are necessary for the mathematic modeling of pertinent network behaviors before applying to a control loop. Its interaction effect with the application control loop has also to be investigated.

Fig. 4 .

 4 Fig. 4. IEEE 802.15.4/Zigbee using CSMA/CA

Fig. 6 .

 6 Fig. 6. System response using CSMA/CA

Fig. 8 .

 8 Fig. 8. System architecture with N control loops sharing a same processor At any time, there are n activated plants (with n N), i.e. n tasks must be computed. This raises the problem of the schedulability of these n tasks. Each task is under (m i ,k i )-firm constraint. Assume that each task i has period h i (corresponding to the nominal sampling period of the related control loop) and the worst case execution time C i . The value of k i corresponds to the maximum number of consecutive non-execution instances that the ith control loop can accept before going to the instability. m i corresponds to the desired QoC. The instances of each task are partitioned into two sets: the mandatory instances and the optional instances. An instance of i , activated at time aT i , for a = 0,1,… is classified as mandatory if

  i

Fig. 9 .

 9 Fig. 9. Close-up of schedule at t=2 under traditional scheduling

Fig. 10 .

 10 Fig. 10. Close-up of schedule at t = 2 under scheduling approach with (m,k)-firm constraint regulation