Olivier Herbinet
email: olivier.herbinet@ensic.inpl-nancy.fr

Pierre-Alexandre Glaude

Valérie Warth

Frédérique Battin-Leclerc

Experimental and modeling study of the thermal decomposition of methyl decanoate

Keywords: Methyl decanoate, Biodiesel, Jet-stirred reactor, Thermal decomposition, Pyrolysis, Kinetic modeling

The experimental study of the thermal decomposition of methyl decanoate was performed in a jet-stirred reactor at temperatures ranging from 773 to 1123 K, at residence times between 1 and 4 s, at a pressure of 800 Torr (106.6 kPa) and at high dilution in helium (fuel inlet mole fraction of 0.0218). Species leaving the reactor were analyzed by gas chromatography. Main reaction products were hydrogen, carbon oxides, small hydrocarbons from C 1 to C 3 , large 1-olefins from 1-butene to 1-nonene, and unsaturated esters with one double bond at the end of the alkyl chain from methyl-2-propenoate to methyl-8-nonenoate. At the highest temperatures, the formation of polyunsaturated species was observed: 1,3-butadiene, 1,3-cyclopentadiene, benzene, toluene, indene, and naphthalene. These results were compared with previous ones about the pyrolysis of n-dodecane, an n-alkane of similar size. The reactivity of both molecules was found to be very close. The alkane produces more olefins while the ester yields unsaturated oxygenated compounds.

A detailed kinetic model for the thermal decomposition of methyl decanoate has been generated using the version of software EXGAS which was updated to take into account the specific chemistry involved in the oxidation of methyl esters. This model contains 324 species and 3231 reactions. It provided a very good prediction of the experimental data obtained in jet-stirred reactor. The formation of the major products was analyzed. The kinetic analysis showed that the retro-ene reactions of intermediate unsaturated methyl esters are of importance in low reactivity systems.

Introduction

Fuel thermal decomposition reactions are of importance in kinetic studies as the interpretation of experimental phenomena is much simpler than in the case of oxidation reactions for the analysis of the primary reaction pathways of a reactant. Furthermore, the sensitive reactions are not quite the same due to the differences in experimental conditions (e.g., no oxygen molecule, higher temperatures in pyrolysis studies). These conditions are of importance in some practical cases such as diesel engines. The first reactions of diesel fuels in the combustion chamber are close to a pyrolysis leading to a high amount of unsaturated products and soot that are later oxidized in the flame front. Thus thermal decomposition and oxidation studies are complementary and both are needed to perform the validation of detailed kinetic models [START_REF] Musculus | SAE Technical Paper Series[END_REF].

Amongst new transportation fuels, the interest in biodiesel is increasing because of their use along with conventional diesel fuels in internal combustion engines [START_REF] Bozbas | [END_REF], [3], [4], [5] and [6]. Biodiesel fuels are composed of methyl esters deriving from the trans-esterification of vegetable oils and animal fats. These esters are made of an alkyl chain attached to an ester group, the main differences being the number of carbon atoms and the number of double bonds in the alkyl chain. As an example, palm, rapeseed and soybean biodiesels are mainly composed of C 17 . Methyl palmitate and methyl stearate have no Cdouble bond; length as m-dashC double bond whereas methyl oleate, methyl linoleate and methyl linolenate have one, two and three Cdouble bond; length as m-dashC double bonds, respectively [7]. The oxidation of large methyl esters has been the subject of numerous experimental and modeling studies during the last years [7], [8], [START_REF] Vaughn | Ignition Delay of Bio-Ester Fuel Droplets[END_REF], [10], [11], [12], [13] and [14], but the thermal decomposition of this type of molecules was very little investigated in the past and there is very little data that can be used for the validation of detailed kinetic models.

Hurd and Blunck [15] performed a study of the pyrolysis of seven esters (ethyl, phenyl, i-propyl, i-butyl and t-butyl acetates and methyl and ethyl phenyl-acetates) in sealed tubes (temperatures between 633 and 923 K). The nature of the reaction products (which were analyzed by titration) depends on the initial reactant. In the case of ethyl acetate, reaction products were hydrogen, carbon oxides, methane, ethylene, acetic acid, formaldehyde, acetaldehyde and acetic anhydride. The main result of this study is that esters mainly decompose into organic acid and olefin if there is an H-atom in the β position in the alkyl portion of the ester. Arnold et al. [16] studied the pyrolysis of cis and trans-2-methylcyclohexyl acetates in sealed tubes at 768 K. They also observed the formation of olefins: 1-methyl and 3-methyl-cyclohexene. These authors [15] and [16] proposed different possible mechanisms accounting for the formation of organic acids and olefins, one of them being a concerted elimination through a six membered ring transition state. Blades [17] studied the thermal decomposition of ethyl formate, isopropyl formate, ethyl acetate, and isopropyl acetate in a continuous flow reactor using toluene as carrier gas. Toluene was used as radical scavenger in order to favor molecular reactions. This study confirmed that the mechanism responsible for the formation of organic acids and olefins is a purely intramolecular decomposition reaction and kinetic parameters were derived from experimental results.

More recently, Archambault and Billaud [18] studied the pyrolysis of rapeseed oil methyl esters (enriched in methyl oletate -82.1 wt.%) in a tubular flow reactor over the range of temperatures 773-973 K and residence times 0.6-1.2 s. Species at the outlet of the reactor were analyzed by gas chromatography. The conversion of fuel was measured as a function of the temperature. Reaction products were hydrogen, methane, carbon oxides, linear alkanes and olefins form C 2 to C 18 , aromatics (benzene and toluene), C 3 to C 16 unsaturated methyl esters, saturated methyl esters (methyl acetate and methyl propanoate), coke and some other minor products (diolefins, polyunsaturated esters). Unfortunately, only the evolution of the selectivity of groups of species was given in this work. In the same area, Seames et al. [19] studied the pyrolysis of canola and soybean methyl esters in a batch reactor at temperatures around 713 K. They observed the formation of the same classes of products as Archambault and Billaud [18] but individual amounts of each reaction products were not given.

Farooq et al. [20] performed an experimental and computational study of the thermal decomposition of three small methyl esters (methyl acetate, methyl propionate and methyl butanoate) in a shock tube over the range of temperatures 1260-1653 K, pressures of 1.4-1.7 atm and reactant concentrations of 2-3% (dilution in argon). Measurements of CO2 time-histories were carried out. They showed that the formation of CO 2 was very little dependent on the size of the alkyl chain and that the three detailed kinetic models from the literature [21], [22] and [23] used for the computation tended to underestimate CO 2 yields. The largest underestimation was obtained with the model from Metcalfe et al. [22], the smallest one with the model of Gaïl et al. [23], with the prediction by the model of Fisher et al. [21] lying in between. An improved model, based on the work of Fisher et al. [21] and the reaction rate calculations from Huynh and Violi [24] provided the best prediction except at the highest temperatures. This paper presents new detailed experimental results of the thermal decomposition of methyl decanoate (Fig. 1) in a jet-stirred reactor. Methyl decanoate (C 11 H 22 O 2 ) was chosen because it has been proposed as a possible surrogate for large species found in biodiesel [START_REF] Vaughn | Ignition Delay of Bio-Ester Fuel Droplets[END_REF] and because a study of the oxidation of this compound was already performed in a jet-stirred reactor [12]. These new results were compared with previous ones obtained for n-dodecane [25], an n-alkane of similar size, in order to highlight the similarities and the differences between the two types of molecules. A detailed kinetic model for the thermal decomposition of methyl decanoate was generated using the software EXGAS [26], [27] and [28] which was recently used to simulate the oxidation of saturated methyl esters [12]. Computed and experimental data were then compared showing a good agreement.

Experimental method

The experimental study of the thermal decomposition of methyl decanoate was performed in a jet-stirred reactor operated at constant temperature and pressure. The experimental apparatus has already been described in previous papers [8] and [25]. This type of reactor was already used for numerous pyrolysis and oxidation gas phase studies [7], [8], [START_REF] Vaughn | Ignition Delay of Bio-Ester Fuel Droplets[END_REF] and [25]. Its main advantage is that it can be easily modeled as a 0 dimensional ideal perfectly stirred reactor as the temperature and the composition in the reactor are homogenous. Figure 2

Gas mass flow controller

Liquid mass flow controller

Gaseous product sampling line

Inert gas supply

Liquid nitrogen bath

Vent

Ice bath

On-line GC analysis

The jet-stirred reactor is made of fused silica to minimize catalytic reactions at the wall. It consists of an annular preheater (for a progressive heating of the reacting mixture up to the reaction temperature in a very short time -heating rates of 106 K s -1 can be reached according to Houzelot and Villermaux [29] and [30]; this is very important to avoid strong temperature gradients in the reactor) and of a sphere in which the reaction takes place (with an injection cross located at its center). The reactor (volume of 90 cm3) and the injection cross were designed in order to obtain four turbulent jets providing a good mixing of the gas phase (thus concentrations at the outlet and inside the reactor are the same) [31]. The heating of the preheater and of the reactor is achieved by the mean of Thermocoax resistances rolled around the different parts and controlled by Eurotherm 3216 controllers and type K thermocouples (temperature precision better than 1%). The heating of the preheater is divided in two zones, the first one being heated at a temperature 150 K lower than the reaction temperature and the second one being heated at the same temperature as the reactor. The reaction temperature was measured by another type K thermocouple located in the intra annular part of the preheater with its extremity at the level of the injection cross in the center of the reactor.

The feed of liquid methyl decanoate (provided by SAFC, purity: 99+) was controlled thanks a liquid mass flow controller provided by Bronkhorst. This device was associated to an evaporator (also provided by Bronkhorst) in which methyl decanoate was mixed with the carrier gas (helium IC, provided by Messer) and heated to 473 K. Helium flow rate was controlled by a Tylan gas mass flow controller. Helium was used to avoid the condensation of the carrier gas during the sampling at liquid nitrogen temperature at the outlet of the reactor. The pressure in the reactor was maintained constant at 800 Torr (106.6 kPa) using a valve located downstream of the reactor (a pressure of 800 Torr was required because of the pressure drops in the sampling valves of online gas chromatographs).

Reaction products and unconverted reactant exiting the reactor were analyzed by the mean of several gas chromatographs. The analysis was performed in two steps as light and heavy species could not be analyzed at the same time. Heavy species (including at least five heavy atoms such as carbon and oxygen ones) were condensed in a trap connected to the reactor outlet and maintained at liquid nitrogen temperature. After a known period of time (typically five minutes), the trap was disconnected. Once the trap warmed up to the ambient temperature, an internal standard (n-octane) and a solvent (acetone) were added in it. The content of the trap was then injected in an Agilent 6850 gas chromatograph fitted with a HP-1 capillary column and a flame ionization detector (FID). Light species (including less than six heavy atoms) were analyzed online using two gas chromatographs. Both were fitted with a sampling valve (loop volume of 250 μL) for the injection of the gaseous mixture from the reactor. The first gas chromatograph (Agilent 6850) was fitted with a Plot-Q capillary column (providing a good separation for light hydrocarbons from methane to 1-pentene and oxygenated compounds such as acetaldehyde) and a FID. The second gas chromatograph (Agilent 7890) was fitted with a Carbosphere packed column (providing a good separation for very light species such as methane, C 2 hydrocarbons, carbon oxides and hydrogen), a thermal conductivity detector (TCD) and a FID associated to a methanizer. The use of a methanizer fitted with the FID provides a better sensitivity for the quantification of oxygenated species such as carbon oxides which are usually quantified by TCD: carbon oxides were detected at concentrations less than 10 ppm in the present study. As hydrogen was the only product quantified using TCD (carbon oxides were quantified using FID), it was possible to use argon as a carrier and reference gas to enhance the sensitivity (the limit of detection of hydrogen was 90 ppm). For other hydrocarbons and oxygenated species (mainly unsaturated methyl esters) the limit of detection was between 1 and 10 ppm according to the size of the molecule.

Experimental results

The thermal decomposition of methyl decanoate was studied at a pressure of 106.6 kPa and an inlet fuel mole fraction of 0.0218. A first set of experiments was performed at a constant residence time of 1 s and at temperatures ranging from 773 to 1123 K. Two other sets of experiments were performed at 873 and 973 K by varying the residence time from 1 to 4 s.

Reaction products observed during this study can be classified in the following way:

• Small oxygenated compounds: carbon oxides, acetaldehyde.

• Hydrogen and small hydrocarbons: methane, acetylene, ethane, propyne, allene, propane, 1,3-butadiene and 1,3-cyclopentadiene. Figure 3 displays the evolution of the conversion of the fuel with the temperature at a constant residence time of 1 s (Fig. 3a) and with the residence time at 873 and 973 K (Fig. 3b). Figure 3a shows that the consumption of the fuel becomes significant above 873 K. [Fig. 4], [Fig. 5], [Fig. 6] and [Fig. 7] display the evolution of the mole fractions of reaction products with the temperature. An excel spreadsheet with mole fractions of the species detected in this study is available in Supplementary material. The formation of some reaction products (such as carbon oxides, olefins and unsaturated esters, with mole fractions between 1 and 10 ppm) is already observed at 773 K, meaning that the decomposition of the fuel is already occurring at the lowest temperature studied in this work. Mole fraction profiles of most reaction products (1-pentene and larger olefins, methyl-4-pentenoate and larger unsaturated esters) exhibit a maximum at about 950 K. Mole fraction profiles of propene, 1-butene, methyl-2-propenoate and methyl-3-butenoate also exhibit a maximum but it is located at higher temperature. These latter species are not only primary products but are partly formed during the decomposition of larger olefins and unsaturated esters. Profiles of small alkanes (ethane, propane) and of aromatic compounds such as benzene and toluene display also a maximum. Mole fractions of small unsaturated hydrocarbons (acetylene, allene, propyne) and larger aromatics (styrene, indene, naphthalene) are increasing in the whole temperature range of this study. Carbon oxides, which come from the decomposition of the ester group, have mole fractions increasing with the temperature. Mole fractions of hydrogen increase in a monotonous way with the temperature as this species comes from the dehydrogenation of hydrocarbons. The experimental results of the thermal decomposition of methyl decanoate (C 11 H 22 O 2 ) were compared with a set of already published data about the pyrolysis of n-dodecane (C 12 H 26 ) under similar conditions [25] in order to highlight the differences between large n-alkanes and methyl esters. The carbon content in both sets of experiments is almost the same as the inlet fuel mole fractions were 0.02 for n-dodecane and 0.0218 for methyl decanoate. It can be seen in Fig. 3 that n-dodecane and methyl decanoate have very close reactivities, n-dodecane being slightly more reactive. As far as reaction products are concerned, the formation of unsaturated methyl esters and oxygenated species was not observed in the n-dodecane study as this hydrocarbon does not contain oxygen atom. The comparison between both sets of experiments showed that the main differences are observed for 1-olefins: mole fractions of these species are lower in the case of methyl decanoate and the difference tends to disappear for smaller 1-olefins (there is almost no difference for ethylene and propene). This can be explained by the fact that ester-alkyl radicals can decompose either to 1-olefins or to unsaturated esters whereas alkyl radicals only decompose to 1-olefins. The difference tends to disappear for small species which are not only formed from the decomposition of alkyl and ester-alkyl radicals but also from the decomposition of larger 1-olefins and unsaturated esters. Note that mole fractions of 1-octene are similar with both fuels. This is due to the fact that the formation of 1-octene is favored in the case of methyl decanoate by the low energy of the C-C bond in the α position of the ester function. The comparison between the methyl ester and the alkane also shows that there is almost no difference for the smallest species (e.g., hydrogen, methane, ethane, acetylene, allene, propyne). This can be explained by the same reason as for the small olefins. For 1,3-cyclopentadiene and aromatic compounds, some differences are observed. With n-dodecane, mole fractions of 1,3-cyclopentadiene, benzene, indene and naphthalene are slightly larger whereas they are lower for toluene and styrene. There is no obvious reason that explains these slight differences.

Modeling

A model for the pyrolysis of methyl decanoate was generated using software EXGAS, which was initially developed for the oxidation of alkanes [26], [27] and [28] and which has been recently updated for the oxidation of saturated esters [12]. Some reactions, which were not included in the model generated using software EXGAS, have been added to account for the decomposition of unsaturated esters (this point is discussed later). This model contains 3231 reactions involving 324 species and is made of three parts:

• A comprehensive primary mechanism (1200 reactions) in which the only reacting molecule is the initial reactant (methyl decanoate) and the free radicals produced. The molecules formed in this part of the model are called primary products. Their fate is considered in the secondary mechanism. • A lumped secondary mechanism (805 reactions) which contains the reaction of the primary products. Lumping consists in considering that all the species having the same chemical formula and the same functional groups are gathered into one unique species (e.g., all C 7 H 12 O 2 unsaturated esters are represented by one species in the model, whatever the position of the C=C double bond). In order to keep the size of the mechanism compatible with simulations, the reactions of lumped species are not any more elementary steps but global reactions which represent some successive elementary reactions, with a rate parameter corresponding to that of the limiting step. Reactions are written in order to promote radicals which already exist in the primary mechanism. Lumping provides a significant decrease of the numbers of species and reactions. • A comprehensive C 0 -C 8 reaction base (1226 reactions) which contains all the reaction of small species having no more than two carbon atoms, unsaturated C 3 and C 4 unsaturated hydrocarbons and first aromatics (benzene, toluene and styrene) [32]. Two reactions of consumption of the radical •C(=O)-O-CH 3 , which is specific to esters, were added in the C 0 -C 2 reaction base. These reactions have already been described in [12]. The C 3 -C 4 reaction base [32] is an extension of the previous C 0 -C 2 reaction base [33]. It was built from a review of the literature. The reactions of the first aromatics come from the oxidation mechanism developed for benzene [34] and toluene [35]. This C 0 -C 8 reaction base was successfully tested against experimental data from the study of rich premixed laminar flames doped with 1,3-butadiene [32]. Some pericyclic molecular reactions, which were not originally included in the model, were added in the secondary mechanism to account for the consumption of unsaturated esters. These reactions are called retro-ene and consist in a decomposition into propene and smaller unsaturated esters through a concerted mechanism involving a six membered ring cyclic transition state. These reactions are well known in the case of olefins (Fig. 8). Thermal decomposition studies [25] and [36] showed that they play an important role in the consumption of these large olefins. Retro-ene reactions have been added in the case of unsaturated methyl ester (Fig. 8), with the same rate constant as that used for the alkenes, i.e. k = 8.0 × 10 12 exp(-56,500 cal mol -1 /RT) s-1 as proposed by Richard et al. [37]. The retro-ene reactions involve the transfer of an H-atom in the γ position of the C=C double bond and consequently cannot happen for small unsaturated esters (methyl-4-pentenoate, methyl-3-butenoate and methyl-2-propenoate).

Unimolecular initiations are usually negligible in the secondary mechanism and these reactions were not written except in the case of methyl-4-pentenoate. This species has a particular C-C bond in the β position of the ester group with a relatively low energy. The breaking of this bond, yielding allyl and •CH 2 COOCH 3 radicals, requires only 70.7 kcal mol -1 .

A model, with no specific molecular reactions for the decomposition of methyl-4-pentenoate, methyl-3-butenoate and methyl-2-propenoate, was tested against the set of experimental data obtained in a jet-stirred reactor ([Fig. 3], [Fig. 4], [Fig. 5], [Fig. 6] and [Fig. 7]). It can be seen that the model accounts for the consumption of the reactant and the mole fractions of most reaction products except methyl-4-pentenoate and methyl-3-butenoate which are over-predicted (of about a factor of 2), meaning that a pure radical mechanism is not sufficient to account for the consumption of these particular species.

Thus some quantum calculations were performed in order to find other possible consumption routes for methyl-3-butenoate and methyl-4-pentenoate. Molecular reactions have been investigated in the case of methyl-3-butenoate and methyl-4-pentenoate. The calculations have been performed using Gaussian 03 (Rev. C.02) [START_REF] Frisch | Gaussian03, Revision B05[END_REF] and the composite CBS-QB3 method [START_REF] Montgomery | [END_REF]. Frequency analysis has made it possible to point out one imaginary frequency for each transition state (TS) and IRC calculations confirmed that the TS connected the reactant to the products. Figure 9 presents the different reaction channels and the energy barriers found in the case of methyl-3-butenoate. The pericyclic reactions involving the lowest energy barrier are the isomerizations to methyl crotonate through a four-centered H-transfer (reactions (1) and ( 2)). Trans and cis methyl crotonate can be obtained, the latter involving a slightly higher activation energy. Another four-centered reaction consists in the transfer of a H-atom from the α position of the ester group to the O-atom, yielding an acetal (reaction (3)). The transfer of the H-atoms in α position of the ester group and of the C=C double bond is easy: the C-H bond is particularly weak because of the stabilization induced by these two groups. By analogy with the reaction of the carboxylic acids, reactions of transfer or elimination of the methoxy group (reactions ( 4) and ( 6)) have been investigated. The energy barriers found, respectively 77.4 and 84.0 kcal mol -1 , prevent these reactions to play any role. The transfer of the methyl group producing CO 2 and 1-butene (reaction ( 5)) is very exothermic but involves also a high activation energy, showing that a direct production of CO2 through a molecular reaction should be negligible in the reaction of methyl esters. The rate constants of the three reactions with the lowest energy barriers were determined. High-pressure limit unimolecular rate constants were calculated using classical TS theory and the CHEMRATE software [START_REF] Mokrushin | Chemrate v.1.5.2[END_REF]. For reactions involving a H-transfer, a transmission coefficient has been calculated in order to take into account the tunneling effect using Eckart potential [START_REF] Johnston | [END_REF]. The kinetic parameters were obtained from a fitting between 700 and 1300 K, with a modified Arrhenius form. The rate constants are, with activation energy in cal mol (2)

C 5 H 8 O 2  CH 2 CHCHC(OH)(OCH 3 ) k = 1.23 × 10 -3 T 4.44 exp(-57,490 / RT) s -1 (3)
The product of reaction (3) decomposes easily since the O-CH 3 bond dissociation energy is only 42.8 kcal mol -1 . A global decomposition into CO 2 , methyl and allyl radicals was considered.

In the case of methyl-4-pentenoate, the four-centered H-transfer are more difficult since the C-H bonds on the alkyl chain are stronger that in methyl-3-butenoate. The only reaction with a low energy barrier is a six-centered isomerization presented in Fig. 10. The rate constant has been computed using transition state theory with calculations at the CBS-QB3 level of theory: k = 1.26 × 10 7 T 1.23 exp(-55,800/RT) s -1 . The reaction product is very unstable and decomposes easily into allyl radical and •OCH 2 COCH 3 radical, which is a mesomeric form of •CH 2 COOCH 3 , because of the low C-O bond dissociation energy (34.4 kcal mol -1 ).

Despite this effort, the inclusion of these new reactions has very little effect on the mole fractions of methyl-4-pentenoate and methyl-3-butenoate, which are still over-predicted. A-factors must be increased by more than a factor of 100 to observe a significant effect, meaning that there are likely other pathways for the consumption of these unsaturated esters. Note that the formation of methyl crotonate was not observed in the experiments, confirming the little importance of reaction ( 1) and ( 2) in the simulations due to their high activation energies.

Thermodynamic properties of molecules and radicals involved in the model were automatically calculated using THERGAS software [42], based on the group and bond additivity methods proposed by Benson [START_REF] Benson | Thermochemical Kinetics[END_REF]. These data are stored as 14 polynomial coefficients, according to the CHEMKIN II formalism [START_REF] Kee | Chemkin II: A FORTRAN Chemical Kinetics Package for the Analysis of a Gas-phase Chemical Kinetics[END_REF]. Bond dissociation energies of C-H, C-O, and -C bonds close to the ester function were updated from the values recently proposed by El-Nahas et al. [START_REF] El-Nahas | [END_REF]. These changes were described in detail in [12] and [46].

Discussion

The model presented in the previous section was used to compute the mole fractions of the reactant and of reaction products. The conversion of the reactant is fairly well reproduced (Fig. 3). The agreement is also satisfactory for most reaction products: hydrogen, methane, ethylene and larger olefins ([Fig. 4] and [Fig. 5]), and unsaturated esters in Fig. 6 (except methyl-3-butenoate and methyl-4-pentenoate, for which some decomposition pathways are likely missing as discussed in the model section). The general good agreement, which is observed for olefins and unsaturated esters, means that the rules used for the generation of the primary mechanism well account for the formation and consumption pathways of these species. The main discrepancies are observed for species with several unsaturations. Allene and propyne profiles are fairly well reproduced by the model (Fig. 4), whereas mole fractions of 1,3-butadiene, 1,3-cyclopentadiene, benzene and toluene are slightly under-estimated (Fig. 7). Mole fractions of styrene are slightly over-estimated at the higher temperatures. Styrene is the largest aromatic compound considered in the model and no consumption reaction has been included.

Consumption pathways of methyl decanoate

Figure 12 displays the main pathways of consumption of the reactant, methyl decanoate, at a temperature of 900 K and a residence time of 1 s. Under these conditions, the reactant is mainly consumed by H-abstractions by H-atoms, methyl and ethyl radicals. The main radical obtained through these H-abstractions is radical R28 in Fig. 12 with the radical center in position 2 in the alkyl chain (see Fig. 1 for the numbering of carbon atoms in the alkyl chain of methyl decanoate). The dissociation energy of this C-H bond is relatively low (94.3 kcal mol -1 whereas it is 98.8 kcal mol -1 for a secondary C-H bond [START_REF] El-Nahas | [END_REF]). Thus 43.7% of methyl decanoate is consumed to give radical R28. 12.6% of methyl decanoate is consumed to yield radical R27 (H-abstractions of an H-atom of the methyl group in the ester function) and 5.7% to radical R26 (H-abstractions of an H-atom of the other methyl group in the alkyl chain). The seven other possible radicals (from R29 to R35) are obtained through H-abstractions of secondary H-atoms. Each of them represents 5% of the consumption of the reactant.

Formation of carbon oxides and acetaldehyde

The use of a methanizer allowed the detection of the presence of small amounts (a few ppm) of carbon oxides at 773 K, the lowest temperature investigated in this study, for which a very low conversion of 0.07% was measured. The CO 2 to CO mole fraction ratio was calculated over the whole range of temperatures. It is around 0.7 at 773 and 793 K. It decreases in an abrupt manner from 0.7 to 0.3 between 793 and 833 K (the conversion increases from 0.17% to 0.81%) and remains around this value at higher temperatures. This means that the formation of carbon dioxide is enhanced at very low conversion and that a change in the reaction pathways is likely occurring in the temperature range 773-793 K. This change could be due to a homogeneous molecular reaction (with a low activation energy) forming directly CO 2 from the reactant or to a heterogeneous reaction at the wall of the reactor. Heterogeneous reactions are more probable than a molecular reaction at 773 K since the latter one should involve a very low activation energy. Brocard and Baronnet studied the pyrolysis of methyl tert-butyl ether in different closed vessels (packed and unpacked, coated and uncoated) [47]. They observed large changes in the reactivity of the system due to important wall effects. These reactions (molecular and heterogeneous) would be also important at higher temperature, but their effect would be masked by the chain radical mechanism occurring at higher temperatures. Note that the sequence of reactions proposed by Dagaut et al. [10] for the early production of carbon dioxide (unimolecular initiation by breaking of the CH 3 -O bond followed by the decomposition into CO 2 and an alkyl radical) is included in the model but does not play a significant role. This was expected as this bond has a relatively high dissociation energy (85.7 kcal mol -1 [START_REF] El-Nahas | [END_REF]).

The CO 2 to CO mole fraction ratio was also calculated from the data computed with the model. The computed values are similar in the range 800-900 K, but the computed profile is slightly different from the profile obtained from the experimental data: as expected, there is no abrupt decrease of the ratio at low conversion, since the model does not contain reaction forming CO 2 directly from the reactant.

The reaction pathways from methyl decanoate to carbon monoxide and carbon dioxide were investigated from a rate of production analysis at a temperature of 900 K ([Fig. 13] and [Fig. 14]). Percentages given in [Fig. 13] and [Fig. 14] are related to the consumption of the reactant (as an example, 1.13% between R43 and R21 in Fig. 14 means that this reaction corresponds to 1.13% of the consumption of the initial reactant, methyl decanoate). As expected, CO and CO 2 come from the decomposition of the ester group of the molecule in chain propagation reactions. Carbon monoxide is mainly obtained from the decomposition of C 11 H 21 O 2 radicals with the radical center located in position 4 and 6. These radicals decompose by successive reactions of β-scission leading to a C 3 H 5 O 2 radical (R42 in Fig. 13). The reaction of β-scission leading to the formation of ketene (not shown in Fig. 13) is disfavored by a high activation energy and this radical mainly isomerizes by shifting an Hatom from the methyl group of the ester function to the methylene group. Then the new radical leads to CO through two successive reactions of β-scission. 13] and [Fig. 14]). This radical mainly isomerizes to R31 and R32 which then lead to CO and CO 2 as shown in [Fig. 13] and [Fig. 14].

Acetaldehyde is mainly formed from the addition of methyl radicals to formaldehyde, followed by a reaction of decomposition by β-scission. The main source of formaldehyde is the decomposition of the radical R61 in Fig. 13. The under-prediction of the mole fractions of acetaldehyde is likely due to missing pathway in the secondary mechanism (e.g., reactions of consumption of primary oxygenated products).

Role of retro-ene reactions

The kinetic analysis of the mechanism shows that the retro-ene reactions play a role in the consumption of 1-olefins and esters having one double bond at the extremity of the chain (Fig. 8). At 900 K, the retro-ene reaction is responsible for 60% of the consumption of 1-octene. This type of reaction also plays a role in the formation of propene at temperature higher than 830 K. Below 830 K, propene is mainly formed by the radical mechanism and retro-ene reactions play a minor role. At 1000 K, the retro-ene reactions are responsible for 53% of the formation of propene. At 1100 K, they only represent 27% of its formation.

Formation of aromatics

The kinetic analysis of the model shows that benzene is formed through two main pathways. The first one is the decomposition of 1,3-cyclohexadiene to benzene + H 2 (69% of the formation of benzene at 900 K) and the second one is the recombination of two propargyl radicals, C 3 H 3 (30% of the formation of benzene at 900 K). Propargyl radicals come from the H-atom abstractions on allene and propyne (C 3 H 4 ) and 1,3-cyclohexadiene comes from the reaction of C 2 H 3 radicals to 1,3-butadiene. Benzene is mainly consumed by H-atom abstractions by H-atoms, CH 3 and C 2 H 5 radicals to form phenyl radicals. Benzene also reacts through ipso-addition with C 2 H 3 radicals to form styrene + H. Phenyl radicals recombine with methyl radical to form toluene and add to ethylene to form C 8 H 9 radicals that easily decompose to styrene + H. Toluene is also formed from benzyl radicals (through H-atom abstractions) which comes from the reaction of cyclopentadienyl radicals (C 5 H 5 ) with ethylene.

Toluene is fairly well reproduced by the model, meaning that the concentrations of the phenyl and methyl radicals are likely well predicted. The correct prediction of the concentration of the methyl radical by the model is also confirmed by the good agreement which is observed for species such as methane, ethane and propane, which are produced by H-abstraction on the reactant, and combination of two methyl radicals and methyl and ethyl radicals, respectively.

Conclusion

In this study, new experimental results of the thermal decomposition of methyl decanoate were obtained in a jet-stirred reactor operated at constant temperature and pressure. Thirty-two reaction products were detected: they were mainly 1-olefins from ethylene to 1-nonene and methyl esters with one double bond at the end of the chain from methyl-2-propenoate to methyl-8-nonenoate.

The formation of carbon oxides and acetaldehyde was also observed because of the presence of oxygen atoms in the reactant. The evolution of the ratio of mole fractions of CO and CO 2 with the temperature showed that there is a change in the formation pathways of CO 2 occurring at low conversion.

These new results were compared with previous ones about the pyrolysis of n-dodecane, a linear alkane of similar size. This comparative study showed that both molecules have about the same reactivity, methyl decanoate being slightly more stable than n-dodecane. As far as products are concerned, the main difference is the formation of specific species due to the presence of the ester function in methyl decanoate: unsaturated methyl esters and oxygenated compounds. Mole fractions of olefins are generally lower in the case of methyl decanoate than in the case of n-dodecane due to the competition between the formation of olefins and unsaturated esters from ester-alkyl radicals from methyl decanoate.

A model for the thermal decomposition of methyl decanoate was generated using the software EXGAS. The model reproduces fairly well the conversion of the reactant and the mole fractions of the reaction products. The kinetic analysis showed that retro-ene reactions play an important role in the consumption of olefins and esters with one double bond at the extremity of the chain. The model does not account for the evolution of the CO/CO 2 ratio at low conversion and for the consumption of small unsaturated methyl esters. This is likely due to missing consumption pathways such as molecular reactions. Further investigations are needed to well understand the chemistry involved in the pyrolysis chemistry of methyl esters.

Fig. 1 .

 1 Fig. 1. Structure of methyl decanoate (C 11 H 22 O 2 ).

Fig. 3 .

 3 Fig. 3. Evolution of the conversion of the reactant with (a) the temperature and (b) the residence time (■ and up triangle, filled: methyl decanoate experiments; × and +: n-dodecane experiments [25]; line: methyl decanoate simulations). Inlet mole fractions of n-dodecane and methyl decanoate are 0.02 and 0.0218, respectively.

Fig. 4 .

 4 Fig. 4. Evolution of the mole fraction of the reaction products with the temperature at a residence time of 1 s (•: methyl decanoate experiments; +: n-dodecane experiments [25]; line: methyl decanoate simulations). Inlet mole fractions of n-dodecane and methyl decanoate are 0.02 and 0.0218, respectively. Part one.

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .

 567891011 Fig. 5. Evolution of the mole fraction of the reaction products with the temperature at a residence time of 1 s (•: methyl decanoate experiments; +: n-dodecane experiments [25]; line: methyl decanoate simulations). Inlet mole fractions of n-dodecane and methyl decanoate are 0.02 and 0.0218, respectively. Part two.

Fig. 12 .

 12 Fig. 12. Main pathways of consumption of methyl decanoate (τ = 1 s, T = 900 K).

  and C 19 methyl esters: methyl palmitate (C 17 H 34 O 2 ), methyl stearate (C 19 H 38 O 2 ), methyl oleate (C 19 H 36 O 2 ), methyl linoleate (C 19 H 34 O 2 ) and methyl linolenate (C 19 H 32 O 2

traces removal) Vacuum pump Liquid hydrocarbon supply Reactor pressure control valve Liquid hydrocarbon trap Jet Stirred Reactor Annular preheating zone Controlled Evaporator and Mixer Pressurized vessel with liquid methyl decanoate

  displays a scheme of the experimental apparatus used in this study. The main features of the experimental and analytical methods are described below. Fig.2. Scheme of the experimental apparatus used in this study.

	Nitrogen bubbling	(O 2
	Nitrogen

Table 1 .

 1 List of unsaturated esters observed in this study.

	Structure					Name	Global formula	Molar weight (g.mol -1 )
	O
	C H 2	O	CH 3			Methyl-2-propenoate	C 4 H 6 O 2	86
	O
	C H 2	O	CH 3			Methyl-3-butenoate	C 5 H 8 O 2	100
		O
	C H 2		O	CH 3		Methyl-4-pentenoate	C 6 H 10 O 2	114
		O
	C H 2		O	CH 3		Methyl-5-hexenoate	C 7 H 12 O 2	128
			O
	C H 2			O	CH 3	Methyl-6-heptenoate	C 8 H 14 O 2	142
			O
	C H 2			O	CH 3	Methyl-7-octenoate	C 9 H 16 O 2	156
				O
	C H 2				O	CH 3	Methyl-8-nonenoate	C 10 H 18 O 2	170

• Olefins from ethylene to 1-nonene. • Unsaturated esters with one double bond at the end of the alkyl chain from methyl-2-propenoate (C 4 H 6 O 2 ) to methyl-8-nonenoate (C 10 H 18 O 2 ). The structures of these species are given in Table 1. • Aromatic and poly-aromatic compounds: benzene, toluene, styrene, indene and naphthalene.

  The formation of CO 2 (Fig.14) mainly occurs through the decomposition by β-scission of C 11 H 21 O 2 radicals with the radical center located in position 3, 5 and 7. Note that the C 2 H 3 O 2 radical (R21 in Fig.14) which is involved in the successive β-scissions can decompose either into CO 2 or CO but that the decomposition into CO 2 is prevalent (the ratio of both rates is about 26 at 900 K). The C 11 H 21 O 2 radical with the radical center located in position 2 (the major one which is obtained by H-atom abstraction from the reactant) does not decompose by β-scission as other C 11 H 21 O 2 radicals R30, R31, R32 and R33 do (see [Fig.

								O O
									O O
			5%
	R 33	R 32	1.81%	5% O O	O O		5%	O	5% 5%	O
	R 45	O 1.12% O O O R 44 0.48% 1.08% R 31	O 1.42% O R 42 O	O 3.12%	R 30	R 29	O	O O
						R 43	O
										0.50%
								4.19% 1.13%
						O O	R 61 R 21
		CO + CH 3 O	0.06%		O O
								4.18% 1.56%
					O CO 2 + CH 3 + CH 2 O R 14
	Fig. 14. Route of formation of carbon dioxide (τ = 1s, T = 900 K). Percentages are relative to the initial
				reactant, methyl decanoate.
								4.18%
						CO + CH 3
	Fig. 13. Route of the formation of carbon monoxide (τ = 1 s, T = 900 K). Percentages are relative to
				the initial reactant, methyl decanoate.

While other alkenes profiles are very well simulated, mole fractions of 1-octene are slightly under-estimated by the model (Fig.5). 1-octene is mainly formed from the decomposition by β-scission of the C 11 H 21 O 2 radical (with the same skeleton as the reactant) with the radical center on the fourth carbon atom in the chain (Fig.11). The activation energy of this reaction is lower than the value used in the case of the breaking of a C-C bond due to the vicinity of the ester group and was calculated theoretically by[12] (Ea = 27,800 cal mol -1 instead of 28,700 cal mol -1 for other C-C bond breaking in alkyl ester radicals). According to the sensitivity analysis that was performed for 1-octene, the activation energy of the reaction of decomposition by β-scission to 1-octene should be lowered of only 0.5 kcal mol -1 so that the model reproduces well the mole fraction profile of 1-octene (dashed line in Fig.5). This value would be in the uncertainty range of the calculated rate constant and does not change the simulated amounts of other products.

Acknowledgment

This work was funded by the European Commission through the "Clean ICE" Advanced Research Grant of the European Research Council.


Keywords: Stylolites, Surface Roughening, Pressure Solution, Porosity, LIDAR 1

Stylolites are rough surfaces formed by localized dissolution, mostly in carbonates and sandstones. They often account for a large degree of dissolution, and their impact on porosity and permeability is well recognized. Understanding their formation mechanism can advance our ability to predict their occurrence and effect on flow, which has appreciable geological and economical implications. Still, many fundamental issues concerning their structure and evolution are still unresolved. This manuscript studies the roughening of long parallel stylolites, which are one of three types of stylolite populations identified by us in a separate paper. Here we report measurements of stylolite surface roughness at a scale larger than ever measured before (10 -2 -10 1 m). Measurements were performed using ground-based-LIDAR on 6 naturally-exposed surfaces of >km long stylolites in Northern Israel. The outcome of these measurements is a topography model of the surfaces, on which different techniques for calculating their roughness characteristics were used. Our results show that up to scales of ~10cm, the average deviation of the surfaces from a planar surface is related to the scale by a power-law with an exponent H. The surfaces are thus defined as self-affine only up to ~10cm with H~0.7.

Above this scale H decreases almost to zero. This observed upper-bound of self-affine roughness measured here for the first time has been predicted by theory [1, 2, 2bis]. Our measurements support these theoretical models and together with them present a scenario in which stylolites evolve from preferential dissolution along an existing surface that was initially smooth and progressively roughened with time. Such a mechanism of stylolites growth is different from previously suggested mechanisms for other classes of stylolite which might propagate sideways from an initial defect. Based on the theoretical roughening model that we adopted, the upper limit to fractality for this class of stylolites may be used as a measure of the amount of dissolution on stylolites. Indeed, the amount of dissolution of the stylolites in our field site which we calculated from the upper limit to fractality is comparable to our estimates of dissolution from two additional independent techniques.

Introduction

Stylolites are rough dissolution surfaces abundantly found in carbonates and sandstones [START_REF] Heald | Stylolite in sandstones[END_REF][START_REF] Park | Stylolites -Their Nature and Origin[END_REF][START_REF] Kaplan | Origin of stylolites[END_REF][START_REF] Stockdale | Stylolites: their nature and origin[END_REF][START_REF] Tada | Pressure Solution During Diagenesis[END_REF] . They are lined by insoluble material (mainly clay minerals, oxides, and organic matter) which accumulates as the more soluble rock constituents dissolved [START_REF] Kaplan | Origin of stylolites[END_REF][START_REF] Stockdale | Stylolites: their nature and origin[END_REF].

They are characterized by anomalous porosity and permeability with respect to the host rock, and thus play a crucial role in determining both the deformation and permeability of rocks. In spite of their importance and decades of research, their formation is still not well understood. Specifically, the propagation and development of stylolites, as well as the controls on their lateral extent and teeth-amplitude is of interest.

In [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF] we showed that sedimentary stylolites usually appear not as isolated features, but in three main types of "populations" or "stylolite networks": isolated, self-similar stylolites, which have clear edges; networks of long bedding-parallel stylolites with uniform dissolution along the seam; and networks of interconnected stylolites with or without fractures. The evolution of the various populations was successfully modeled in [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF] assuming that the initial distribution of heterogeneities (such as clay) in the rock determines which kind of network will form and the lateral extent of the stylolites in the network.

In many examples stylolite teeth amplitude displays multiscale roughness characteristics that can be evaluated as statistically self-affine [START_REF] Ebner | Stress sensitivity of stylolite morphology[END_REF][START_REF] Karcz | Stability of a sodium chloride indenter contact undergoing pressure solution[END_REF][START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF][START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF]. A self-affine 1-D profile remains unchanged under the scaling transformation x

x    , [START_REF] Mandelbrot | Fractals in Physics[END_REF].

For 2-D surfaces, this self-affinity property can be described for sets of 1-D parallel profiles extracted from the surface. The roughness characteristics of a surface, such as whether it is self-affine or not, or the value of its Hurst exponent H which characterizes its self-affinity, and whether this parameter H varies with scale and the scale ranges, can give physical indications on the formation mechanisms when confronted with theory.

This has been claimed with regard to stylolites by [2bis, [START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF][START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF][START_REF] Karcz | The fractal geometry of some stylolites from the Calcare Massiccio Formation, Italy[END_REF][START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF]. They found two self-affine regimes for scales smaller and larger than ~1.1mm and connected this to two competing mechanisms driving dissolution: surface energy for smaller scales and elastic strain energy for the larger scales. The largest surfaces analyzed by [2bis, [START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF][START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF][START_REF] Karcz | The fractal geometry of some stylolites from the Calcare Massiccio Formation, Italy[END_REF][START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF] were ~10cm long. Since stylolites are often larger than 10cm ( [START_REF] Safaricz | Pressure solution in chalk[END_REF][START_REF] Safaricz | Pressure solution in chalk[END_REF] reported 8.5-mlong stylolites in chalk and in [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF] we describe the >1km-long stylolites which are measured in this current paper), studying roughness of stylolites on larger scales could lead to further insight on their formation process.

z z H    
The present paper focuses on field measurements, characterization, and modeling of the formation and evolution of one of the three stylolite populations mentioned above: the long bedding-parallel stylolites. Roughness measurements were performed on long stylolite surfaces in the "Blanche cliff", Northern Israel. Our very unique long scale measurements (10 -2 -10 1 m) show that stylolite roughness is scale-dependent up to ~10 -1 m, where a clear cutoff to this dependence appears. To model and explain these unique observations of the statistical properties over many scales we use growth models of stylolites roughening from an initial flat surface [1, 2, 2bis, 10, 11, 14]. The similarity between these models and the field measurements supports our claim that long parallel stylolites form on existing, very long (practically infinite) bedding-parallel surfaces undergoing preferential dissolution (e.g., surfaces of higher clay content). This is in contrast to isolated, single stylolites which have been suggested to propagate sideways from an initial defect (e.g. [17] ).

We explain in this manuscript how these models can be used to calculate the amount of dissolution on stylolites. The amount of dissolution is calculated for stylolites in the Blanche site, and is compared to values obtained by two other methods. We discuss when roughness measurements can be used as a method for calculation of dissolution amount on stylolites.

2

Field observations: Large scale roughness measurements of stylolites Out of several field sites we identified in [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF] as belonging to the long-bedding parallel stylolite population class, we focus here on the "Blanche" cliff of the Ein El-Assad Formation (Albian) exposed in Northern Israel (Figure 1), due to its unique exposure length (>1km), and its well developed stylolites (teeth amplitude on the scale of cm). It is a roughly 50m thick shallow inner-platform biomicritic limestone formation, with very low porosity, that dips gently to the west [START_REF] Sneh | Geological Map of Israel. Sheet 2-II Metulla[END_REF]. The bedding-parallel stylolites can be traced through the entire outcrop (>1km), and most likely continue beyond the exposure. Such a tracing length for stylolites is remarkable, and is observed for the first time. It has implications for understanding scales of permeability units and their structural continuity. Several large oil-reservoirs of the same geological age also contain stylolites which are claimed to highly affect porosity [START_REF] Oswald | Controls on Porosity Evolution n Thamama Group Carbonate Reservoirs in Abu Dhabi[END_REF]. Detailed field-work conducted in this site [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF] showed that the stylolites are similarly developed along their whole length and keep their spacing. The stylolites were suggested to form on pre-existing layers, possibly with higher clay content compared to the surrounding rock.

This paper studies stylolitic roughness as a recorder of stylolitic development. Thus roughness of exposed stylolite surfaces in the Blanche (such as in Figure 1b) was measured, using Ground-based LIDAR. We used the Leica HDS3000 Laser scanner which provides resolution of ~ 3 mm when a target is scanned from a distance of 20-30 m. A single scan can provide a cloud with millions of points that may be interpolated to generate a topographic map or hundreds of profiles (Figure 2). Thus, the technique allows a statistical approach when calculating roughness. The scan data are rotated so that the mean surface is parallel to the x-y plane. The cloud of points is then interpolated to an xy-z grid, in which the topography is the value of the z-coordinate. Four surfaces were scanned in-situ and two scans were performed on "rockfalls" -m-size blocks that detached from the cliff. Surfaces with minimum erosion were picked for scanning. The measured surfaces range in size from 2.1*0.2 m 2 to 9.3*3.0 m 2 (see Table 1). The obtained measurements of stylolite surface roughness range in scale between 10 -2 -10 1 m, larger than ever measured before (so far the highest resolution measurements reported in the literature were done up to maximum scales around 10 -1 m, as e.g. in [START_REF] Ebner | Stress sensitivity of stylolite morphology[END_REF]). The scanned surfaces have cm-scale teeth amplitude, are symmetric with respect to the surface average plane (in the z-direction) and demonstrate no anisotropy (similar roughness characteristics in the x and y directions), as was found by performing the analyses explained below on length-parallel and width-parallel profiles. Results are presented for length-parallel profiles (Figure 3 and Table 1) so that larger scales can be studied.

Characterization of roughness

Characterization of rough surfaces depends on the scale of measurement. One common method is to measure the surface width (in the z-direction) as a function of the window at which the measurement is made (l). Maximum width (w ∞ (l) according to [START_REF] Schmittbuhl | Scaling Invariance of Crack Surfaces[END_REF])

or average width w(l) are commonly used, with the latter calculated by the standard deviation (STD) of its height from the average surface height. STD is defined by the Root Mean Square method (after [START_REF] Barabasi | Fractal concepts in surface growth[END_REF]):

(1)

w(l)  1 l (h(i)  h) 2 i1 l 
The smallest possible value for l (the smallest window for measuring width) is the distance between two data points. In our case, it is the distance between two points on the interpolated x-y grid (either 2 or 3 mm, for each surface). The largest value for l is the length of the surface. When a surface is self-affine, w and l are related by a power law with an exponent H, termed the Hurst exponent [START_REF] Barabasi | Fractal concepts in surface growth[END_REF]. This exponent can be derived using other common methods for surface characterization, such as power spectral density (PSD) as performed on stylolites e.g. by Renard et al., [START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF] and on fault surfaces e.g. by Sagy and Brodsky [START_REF] Sagy | Geometric and rheological asperities in an exposed fault zone[END_REF]. Power spectral density measures the strength of the sinusoidal components of the topography over a range of wavelengths by performing a Fourier decomposition. We average the 1-D spectra of hundreds of profiles in the x and in the y direction for each surface in order to arrive at a spectral estimate with minimal variance. RMS, as evident in Table 1.

Existing models of stylolite roughening from an initially flat surface

Our roughness measurements can be compared to predictions offered by existing roughening models which assume an initially flat surface and predict how stylolite roughness will develop with time [START_REF] Ebner | Stress sensitivity of stylolite morphology[END_REF][START_REF] Koehn | Growth of stylolite teeth patterns depending on normal stress and finite compaction[END_REF][START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF][START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF][START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF]. Rolland et al.

[2bis] or Schmittbuhl et al. [START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF] offers an analytic formulation and solution to the temporal and spatial evolution of the surface height, h, with a governing equation:

(

2) ) , ( ) ' ( ) ( ) ( h x x x x h x d L a x h x t h x xx          
Three terms appear on the right side of equation ( 2) and these control the roughening processes (from left to right): a surface energy term, an elastic strain energy term, and a term representing the spatial solubility heterogeneity. Here, a and L represent respectively a characteristic grain scale and a scale derived from the far field stress components, the elastic moduli, and the surface tension. In this equation, x, t and h are reduced dimensionless quantities, with a unit length, and a unit time set by a/v, with v an average dissolution speed.

Numerical solutions to the evolution of stylolite roughness from an initial flat surface are presented by [START_REF] Ebner | Stress sensitivity of stylolite morphology[END_REF][START_REF] Koehn | Growth of stylolite teeth patterns depending on normal stress and finite compaction[END_REF][START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF][START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF]. They simulate two granular rock bodies pressed together with a predefined fluid-filled flat interface on which dissolution occurs. The solubility of the rock is not homogeneous and this "quenched noise" is introduced by assigning a lower dissolution rate constant to a certain fraction of the particles. This is seen in Figure 4 (from Ebner et al., [START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF]): darker, less soluble grains get pinned and stuck in the interface as dissolution evolves. The pressure solution process is discretized in steps of dissolution of entire particles, following a linear rate law (see equation [START_REF] Ebner | Stress sensitivity of stylolite morphology[END_REF] in [START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF]).

According to this rate law, the dissolution velocity depends on the local elastic and surface energies and on the local deviation of normal stress from its average value on the surface. Both the analytical and numerical models described above [START_REF] Ebner | Stress sensitivity of stylolite morphology[END_REF][START_REF] Koehn | Growth of stylolite teeth patterns depending on normal stress and finite compaction[END_REF][START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF][START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF][START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF] assume three controlling processes: The first is that the system has pinned noise, provided by the heterogeneous solubility of the rock, which is the main drive for roughening. The second is that the process has long-range interactions, here provided by elastic surface forces which are increased by the surface misalignment on the normal to the principal far field stress axis. The third process is diffusion and smoothing, here provided by surface energies, which cause smoothing of the interface. The models presented above quantify stylolite roughness evolution by evolving Fourier or wavelet transforms, or by the evolution of surface average width, w (equation 1) as a function of time, t, and of the sampling size, l.

The evolution of stylolite surface width with time and sampling size according to these models is illustrated schematically in Figure 5 (here the maximum width, w ∞ (t,l), is shown since average width is harder to show visually; both quantities evolve in a similar manner). At any given time there is a critical sampling size, above which w ∞ is constant.

For example, in Figure 5 this critical size at time t 3 is l 2 . This behavior of surface evolution belongs to a universality class of surface growth models of the Family Viscek type, i.e. where the surface dynamics shows a Family Vicsek scaling [START_REF] Family | Scaling of the active zone in the Eden process on percolation networks and the ballistic depositional model[END_REF], on which much has been written (e.g.: [START_REF] Barabasi | Fractal concepts in surface growth[END_REF]). Other phenomena such as flame fronts, viscous fingering, electrochemical crystals and the dynamics of bug colonies obey such scaling.

All these surfaces develop in a similar manner, thus the parameters used to describe stylolite roughening are conventional parameters.

Mathematically, all functions can be collapsed on a Family Vicsek scaling form as:

) , ( t l w (3)          l f l t l w H ) , ( ;
where is a universal scaling function, and

  u f
 , is a critical sampling size which is a simple function of t. The scaling function has the following asymptotic behaviors:

1 ) (   u For const u f
and:

, which reflects the fact that:

1 ) (    u For u u f H (4)    l For l t l w H ) , (
which means that for measurement samples smaller than a size of the correlation length χ, the width, w, is a power law of the sampling size, l, only, and is independent of time. The scaling exponent, H is the Hurst exponent. The width in these small windows will never grow anymore, even though the surface keeps on evolving as a whole, since these small wavelengths have saturated.

In contrast, for measurement windows larger than χ:

(5)                 l For t l l t l w Z H H H H ) , (
the width, w, is independent of the measurement scale, l. It is constant no matter how large the sampling window is, but instead it grows as a power law of time.

The conceptual evolution of w(t,l) for several times according to the models described above is plotted schematically on a log-log scale in Figure 6. For a given time, t i , w(l) increases as a power-law up to a value of l≈χ above which w(l) becomes constant. This critical sampling size, χ is termed the "correlation length", and can be viewed as a characteristic distance between two points on the surface that are influenced by one another. χ increases as a power-law with time following

(6) z t / 1 
where "z" is called the dynamic exponent (or growth exponent). The correlation length continues to grow according to (6) unless χ reaches the system size, at which time it stops growing. In the case of stylolite growth models, z can be evaluated from time-dependant models. In the model of [START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF] z≈1.2 and similar or slightly lower values are given by [START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF][START_REF] Schmittbuhl | Interfacial Crack Pinning: Effect of Nonlocal Interactions[END_REF]. Equations ( 4) and ( 5) are scaling equations. In order to relate w concretely to other length scales, we normalize these equations by a -a characteristic size of the system, which might be related to an average or a characteristic grain diameter scale, at which the out of plane and in plane amplitudes are of the same order, i.e.

. This scale characterizes the prefactor of scaling laws for self affine functions, and is also called the topothesy [START_REF] Schmittbuhl | Fracture morphology and viscous transport[END_REF][START_REF] Santucci | Fracture roughness scaling: A case study on planar cracks[END_REF][START_REF] Simonsen | Wave scattering from selfaffine surfaces[END_REF].

l l w  ) (

Amount of dissolution

We next wish to infer the unknown amount of dissolution, A, that occurred on the stylolite (in meters), from the width measured in models or in the field. This may be done if we assume the dissolution rate (v, in m/s) was constant, so that A=vt, where t represents the time during which dissolution has been active (between the start of the roughening up to when dissolution stopped). Therefore, with:

(6)           l For a l a
                 l For a A a vt a t l w Z H Z H ) , (
The relationship between  and A is then:

(8) Z a A a 1              
Thus in order to calculate how much dissolution A occurred on stylolites one basically needs to have roughness measurements on a scale larger than χ, so as to capture accurately χ and w(χ).

Implementation of the roughening models to the Blanche stylolites

The measured roughness characteristics (section 2) are similar to those of the models (section 3) in the fact that both demonstrate self-affine behavior up to a certain length and a crossover transition towards a very low roughness exponent above that length. The roughness exponent in the theoretical model equals zero for l>>χ. Our measurements show a value for H which is very low and close to zero (0.10 ±0.06, Table 1), as expected for surfaces following a complete Family Vicsek scaling. We can thus consider the measured cutoff as the correlation length of the Blanche stylolites with cm 10   . Our system size L is >1km, which is the observable length of the stylolites, so L>>χ and we can use χ or w(χ) to calculate the average amount of dissolution that occurred on the Blanche stylolites By plugging into equation ( 6) the values of χ, w(χ) and H of the self-affine part of the plot for each surface we can calculate the characteristic size, a:

H H H w a a a w                   1 1 ) ( ) (    
We find that a on average is ~20μm (see Table 1). We can then use either equation [START_REF] Tada | Pressure Solution During Diagenesis[END_REF] or [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF] to calculate the amount of dissolution, A:

a a w A H Z        ) ( a a A Z        
With the range of literature values for the dynamic exponent z, we find: for z=1.2, A≈70cm (this is for the average value of z in Koehn et al., [START_REF] Koehn | Growth of stylolite teeth patterns depending on normal stress and finite compaction[END_REF]); for a slightly lower value of z=1.0, A=10cm. The detailed results of these calculations (Table 1) demonstrate that the variability of A from one surface to another is much smaller than the sensitivity of A to the selected value of z. E.g. for z=1.2, A varies between 50 and 100 cm, whereas A is 5-10 times higher for z=1.2 compared to z=1.0 for a given surface. With an average spacing, d , of ~30cm between stylolites in the Blanche cliff this translates to between 10cm/(10+30)cm=25% and 70cm/(70+30)cm=70% strain.

An alternative method to calculate A is by a simplified mass-balance. Here we assume that (1) stylolites are the only source for cement, and (2) we have a closed system, meaning that material dissolving on stylolites precipitates in the available porosity and that porosity at the end of the process is close to zero. Several thin-sections that we prepared and studied, taken from the stylolites and up to several cm away from them show no porosity at this scale and thus support our assumption of zero porosity around the stylolites. We estimate from these thin section that the initial porosity, φ, is in the range of 0.2-0.3. Thus, we basically assume that initial porosity was completely shut by cement originating from dissolution on the stylolites. Using d =30cm we can calculate A:

) 3 . 0 12 ( 5 . 7 ) 2 . 0 1 ( 30 * 2 . 0 1          for cm or cm cm d A
This translates to between 20% and 30% strain, and is comparable to our lower roughness-based estimates (with z=1.0).

A third independent method we used to evaluate the amount of dissolution is by maximum amplitude of stylolites (peak-to-peak) measured in 2D in the field, assuming it is a minimum estimate to the amount of dissolution. A max was also used by Schmittbuhl et al., [START_REF] Schmittbuhl | Scaling Invariance of Crack Surfaces[END_REF] who defined it as . Our field-measurements, which are described in detail by [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF], show A max ≈4cm for one stylolite and A max ≈10cm for another. These values are also comparable to the lower estimates of dissolution of the roughness-based method. All three independent methods give similar estimates within an order of magnitude, which we interpret as a good sign of their validity.

 w 5 Discussion

We have demonstrated in this paper how models of stylolite roughening from an initial flat surface can be used to explain measurements that were performed on long parallel stylolites in the Blanche cliff, Northern Israel. Our roughness analysis on surfaces of several different stylolites from this site showed that the stylolites are self-affine below ~10cm with H≈0.7. This upper cutoff to fractality was consistent among several surfaces from different stylolites and we correlate it with the model-predicted cutoff to fractality, also termed the "correlation length", χ. Above this length, measured roughness decreased significantly, as quantified by the decrease in H (to ~0.1 or even lower).

The life and death of the Blanche stylolites

The conformity between the roughening models and our measurements agrees with a picture of dissolution occurring preferentially along a surface. This surface was initially smooth and progressively roughened with time. Had stylolitization in the Blanche continued for a long enough time we would have expected to see a self-affine behavior on all scales. The Blanche stylolites are not self-affine in the large scale, which implies that the stylolitization process had stopped at some stage. The reason for termination of the stylolitization process could be related to porosity destruction resulting in no place available for precipitation, an option which is supported by the mass balance calculations for A and by the thin-section observations, as explained above. Already in 1975 Bathurst [START_REF] Bathurst | Carbonate sediments and their diagenesis[END_REF] suggested that the growth of a stylolite could stop when the permeability in the adjacent sediment has fallen so low that transport of ions away from the solution film is practically inhibited. The role of stylolites as a local source for cement is well accepted [START_REF] Tada | Pressure Solution During Diagenesis[END_REF][START_REF] Wong | Burial Cementation in the Devonian, Kaybob Reef Complex, Alberta, Canada[END_REF][START_REF] Walderhaug | The effect of stylolite spacing on quartz cementation in the Lower Jurassic Sto Formation, southern Barents Sea[END_REF][START_REF] Ehrenberg | Porosity destruction in carbonate platforms[END_REF][START_REF] Finkel | Stylolitization as Source of Cement in Mississippian Salem Limestone, West-Central Indiana[END_REF], but how much of the dissolved material precipitates locally and how much is transported out of the system depends on local conditions and their temporal evolution.

The frequency of open fractures and their connectivity as well as permeability of the bulk control how "open" or "closed" the system is. Thin-sections of the Blanche show that current porosity is close to zero both close to the stylolites and at mid-points between adjacent stylolites and that the initial porosity is filled by cement. Very few small fractures or veins were identified, which also supports the concept of a closed system in which pore fluid has no path out and precipitation occurs near-by. Thus we suggest that the life cycle of stylolites in our case-study was controlled by precipitation: once pore space was completely cemented -the process of stylolitization stopped.

The correlation length -measured for the first time

In the present work we measured for the first time the upper bound of self-affinity and the correlation length which were predicted by previous theoretical studies [START_REF] Ebner | Stress sensitivity of stylolite morphology[END_REF][START_REF] Koehn | Growth of stylolite teeth patterns depending on normal stress and finite compaction[END_REF][START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF][START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF][START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF]. Karcz and Scholz [START_REF] Karcz | The fractal geometry of some stylolites from the Calcare Massiccio Formation, Italy[END_REF] analyzed roughness of stylolites from several field sites on 2D traces and found them to be self-similar from scales of 10 microns to 2.5m. They found one regime with H=0.55. Other previous studies [START_REF] Renard | Threedimensional roughness of stylolites in limestones[END_REF][START_REF] Schmittbuhl | Roughness of stylolites: Implications of 3D high resolution topography measurements[END_REF][START_REF] Karcz | The fractal geometry of some stylolites from the Calcare Massiccio Formation, Italy[END_REF][START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF] showed fractal stylolite roughness over several orders of magnitude, up to ~10cm. These studies showed two self-affine regimes for scales smaller and larger than ~1.1mm, with roughness exponents of H S ~1.1 (for small scales) and H L ~0.5 (for large scales). The authors connected this to two competing mechanisms driving dissolution: surface energy and elastic strain energy. The larger scales they measure correlate to the smaller scales in our LIDAR measurements, and the part with H~1.1 is below our resolution. The two regimes discussed in these papers, are therefore not the two regimes we present, and the correlation length was not measured in these cases either.

These previous stylolite roughness measurements have not measured the upper bound to fractality before, either because the largest scale of measurement was smaller than the correlation length, or because the correlation length had reached the system size.

When stylolites are connected by a framework of fractures (such as in the case of Karcz and Scholz [START_REF] Karcz | The fractal geometry of some stylolites from the Calcare Massiccio Formation, Italy[END_REF]), it may indicate that the system was open, at least to some degree, thus allowing dissolution to continue and roughness to develop to very large correlation lengths. In such a case, it would require very large sampling size to capture χ, or χ may even have reached the system size.

An alternative explanation to why the upper cutoff to fractality has not been reported by others is that not all stylolites are formed from existing flat surfaces, as the Blanche stylolites. Other types of stylolite populations have been reported in the literature, as summarized in [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF]. For different types of stylolites roughness develops differently, especially for those which propagate laterally. Their roughness is expected to be more developed in the center and decrease towards the "younger" stylolite edges, as observed by Stockdale [START_REF] Stockdale | Stylolites: their nature and origin[END_REF].

5.3

Upper bound to fractality: is it a good measure for A?

Estimates of amount of dissolution on stylolites have been performed by various authors (e.g.: [START_REF] Heald | Stylolite in sandstones[END_REF][START_REF] Kaplan | Origin of stylolites[END_REF][START_REF] Stockdale | The stratigraphic significance of solution in rocks[END_REF]34]). The common methods used are summarized in table 1 of Tada and Seiver [START_REF] Tada | Pressure Solution During Diagenesis[END_REF]. These include maximum teeth amplitude, seam thickness, heavy mineral condensation, displacement of veins and reconstruction of fossils and oolites. We have presented here an additional method for estimating the amount of dissolution and found that its results are comparable (similar order of magnitude) to maximum teeth amplitude and to mass balance calculations. The common method used for evaluating amount of dissolution from cores is by measuring the maximum amplitude. Two issues should be discussed:

1. When is maximum amplitude, A max , a good measure of dissolution amount?

The answer to this question is that A max is a good measure of dissolution in case the sampling size is larger than the correlation length. If it is smaller it will probably give an underestimate. For example, if we measure A max for l=χ/2 (assuming A max (χ/2)/A max (χ) = w(χ/2)/w(χ), [START_REF] Schmittbuhl | Scaling Invariance of Crack Surfaces[END_REF]), the dissolution underestimation would be, from equation [START_REF] Tada | Pressure Solution During Diagenesis[END_REF]:

Z H Z H H Z H a a a a A A                    2 ) ( ) 2 / ( ) max( ) 2 / max(    
. For z=1 the value of measured A max will be 0.5 of the value measured when l=χ. In general, the underestimated value would be b -z for measurement size l=χ/b.

Another requirement for A max to be a good measure of the dissolution amount is that the measured stylolite evolved on a pre-existing surface. If it evolved from a seed and grew laterally, the position from where the sample is taken is expected to have major influence on roughness. It is unknown at this stage how such different stylolite groups can be differentiated when studied in a core, but this is one interesting question to investigate in the future.

Is dissolution amount based on roughness measurements a better estimate than other methods?

The answer to this question is currently no. The main reason is that the major parameter which is unknown and evaluated only from models is the dynamic exponent, z.

This parameter defines the time-dependence of the evolution of roughness. In our calculation we gave a lower and upper estimate of this parameter, which resulted in quite a large range for A (between 10 and 70cm). This parameter could be better constrained from field studies of stylolitization evolution with time, or from laboratory experiments but these are obviously not easy to conduct, as stylolites have never been produced in the lab (except on the sub-grain scale by [35]). is marked by grey rectangles and is between 7 and 12 cm. This value is the correlation length, χ, as explained in section 3. [START_REF] Koehn | Growth of stylolite teeth patterns depending on normal stress and finite compaction[END_REF] on which we added schematic illustration of how maximum amplitude, A max depends on time and on sampling size, l. For example: maximum amplitude of l 2 at t 2 is smaller than at t 3 . At a given time A max depends on l up to a critical size, after which it remains constant (e.g.:

Tables

for l> l 2 , A max =const). This dependence on l is a characteristic of self affine surfaces.

Figure 3a shows results

  Figure3ashows results of the average width of the six stylolite surfaces, w, as a function of measurement window, l, calculated following equation[START_REF] Ebner | Stress sensitivity of stylolite morphology[END_REF]. The six surfaces show very similar behavior with two scaling regimes: above and below l= 10 ±3 cm. Up to this value of l the surfaces are self-affine with Hurst exponents of 0.67 ±0.10 (the error is the standard deviation of H of the six measured surfaces). For larger values of l the Hurst exponent is not constant and decreases to almost zero (H=0.10 ±0.06 for scales above ~1m). Therefore we identify an upper cutoff to the self-affine behavior of these stylolite surfaces: they appear rough on small scales, but at large enough scales (i.e. viewed from far enough away) they are smooth. The PSD analyses presented in Figure3b, as well as an additional analysis using the wavelet technique (Determination of the Hurst exponent by use of wavelet transforms, Simonsen, I; Hansen, A; Nes, OM, PHYSICAL REVIEW E, 58, 3, 2779-2787, DOI: 10.1103/ PhysRevE.58.2779, 1998) both provided similar values of H as the ones calculated by

Figure 1 :

 1 Figure 1: Photos of the study area: (a) Blanche cliff; (b) an exposed stylolite surfacewhich was scanned using the LIDAR technique ("in-situ 1" in Table1, long axis of surface is ~2 m); (c) cross-section view of 2 parallel stylolites (see camera cover for scale).

Figure 2 :

 2 Figure2: Topography of one of the scanned surfaces ("rockfall 1", in Table1) after data interpolation to an xyz grid.

Figure 3 :

 3 Figure 3: (a) RMS and (b) PSD analysis of surface roughness of 6 stylolite surfaces in the Blanche cliff (see Table 1 for dimensions and detailed results for each surface). The x-axis of the PSD analysis is k [m -1 ], as costumed in the literature, and equals (2*l) -1 . Two scaling regimes are seen: (1) self-affine behavior with H~0.7 for scales up to l~10cm; and (2) decrease of slope (H decreases) at larger scales up to almost flattening (H~0.1). The range of values for which H starts decreasing, this is very soft definition ….

Figure 4 :

 4 Figure 4: Modified from Figure 2 of Ebner et al. [14]. Simplified sketch of the numerical model setup which simulates stylolite roughening from an initial flat interface (marked by a red line). Initial setup is seen in (a), while the configuration after a certain amount of compaction appears in (b). The darker spheres are "pinning particles" which have a lower solubility and drive roughening.

Figure 5 :

 5 Figure 5: Simulation results, from Figure 4 of Koehn et al. [2] on which we added

Figure 6 :

 6 Figure6: Schematic log-log plot of w(l) for several times according to roughening models of Ebner et al.[START_REF] Ebner | The influence of rock heterogeneity on the scaling properties of simulated and natural stylolites[END_REF] and Koehn et al.[START_REF] Koehn | Growth of stylolite teeth patterns depending on normal stress and finite compaction[END_REF]. For each time there is a power-law dependence of w on l with an exponent H (dotted black line) up to l=χ. χ increases with time until it reaches the system size, L (assuming the process has not stopped before). At χ=L (t=tc) roughness stops evolving though dissolution may still continue.

Figure 1

 1 Figure 1: a

Figure 4 :

 4 Figure 4:

Figure 5 :

 5 Figure 5:

Table 1 : Physical and statistical characteristics of six stylolite surfaces from the Blanche cliff, Northern Israel. Hurst exponents calculated using three independent methods are shaded in blue (RMS= root mean square; PSD=power spectral density). H is significantly different below and above χ, the correlation length. χ and w(χ) are yellow-shaded; a (characteristic size calculated by equation 6) and A (amount of dissolution) are green-shaded. z is the dynamic exponent, as explained in section 3. The last row shows average values of each column and its standard deviation.

 1

	# Sample	Dimensions		H, l< χ		H, l> χ	χ (m) w(χ)	a	A (m)
	Name		(m)							(m)	(μm)
		length width RMS PSD wavelet RMS PSD				z=1.2 z=1.0
	1 in situ 1	2.14	0.20	0.57	0.5	0.56	0.04 0.02 0.12 0.0050	74	0.53 0.12
	2 in situ 2	9.32	0.30	0.72	0.67	-	0.15	0.2 0.07 0.0040	5	0.47 0.07
	3 in situ 3	4.80	0.12	0.63	0.68	-	0.11 0.11 0.08 0.0047	1	0.84 0.08
	4 in situ 4	3.30	0.23	0.75	0.82	0.8	0.02 -0.24 0.11 0.0050	26	0.58 0.11
	5 rockfall 1	5.37	2.73	0.64	0.67	-	0.11 0.16 0.11 0.0050	2	1.00 0.11
	6 rockfall 2	2.50	2.14	0.7	0.79	0.62	0.15 0.18 0.11 0.0045	15	0.65 0.11
	Average			0.67	0.69	0.66	0.10	0.07	0.10	0.0047	20	0.68	0.10
	(std)			(0.07)	(0.11)	(0.12)	(0.06)	(0.16)	(0.02)	(0.0004)	(28)	(0.20)	(0.02)

Appendix Parameters used in this manuscript, their symbols and units.

	Symbol	Name	Units
	A	Dissolution amount	m
	L	Sampling size	m
	L	System size	m
	Χ	Correlation length	m
	t c	Critical time for which χ=L	s
	W	Average "width" of stylolite interface	m
	H	Hurst exponent
	Z	Dynamic exponent
	A	Characteristic size of the system	m
	V	Dissolution rate	m/s
	D	Spacing between stylolites	m
	 	Porosity

Summary and conclusions

In this paper we have presented 3D roughness measurements of stylolite surfaces at a scale larger than ever measured before. We have compared these measurements to predictions from a model of surface roughening from an initially flat interface. The six analyzed surfaces show self affine characteristics with H of 0.67 ±0.10 for scales of several mm and up to ~10cm. Above this scale, H decreases almost to zero, indicating that there is an upper cutoff to the self-affine behavior of these stylolite surfaces. The existence of this upper cutoff, termed the "correlation length" (χ), was interpreted as an indicator that the stylolitization process stopped before roughness has saturated. We proposed a scenario in which dissolution and hence roughening stopped when all available porosity around the stylolites was shut down by precipitation originating from the material dissolved on the stylolites. A method for estimating the amount of dissolution, on stylolites from their roughness characteristics in cases where χ is much smaller than the stylolites length was presented. Dissolution amount for the Blanche stylolites was calculated and compared to its estimates from two other methods. The agreement between the three independent methods indicates that the upper bound to fractality of stylolites' roughness can indeed indicate on the amount of dissolution. One very important implication of the agreement between the models and measurements is that it provides support for formation of stylolites, at least in this field site, from existing surfaces. These surfaces could be perhaps of higher clay content [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF]. This is in contrast to in-plane propagation of stylolites, as suggested for other cases (e.g. [START_REF] Ben-Itzhak | Stylolite Populations in Limestone: Field Observations and Formation Models[END_REF]36,37]).