Christian Blaźquez

Christine Dalmazzone

Eliane Emond

Sophie Schneider

Crude Oil Foams: Testing and Ranking of Antifoams with the Depressurization Test

The addition of chemicals is the most widely applied solution to prevent the formation of foam or to destroy it immediately after its generation as a result of its simplicity and efficiency. Among the different chemicals that can be used as antifoams or defoamers, polydimethylsiloxane (PDMS) oils are the most common, followed by fluorosilicone oils for the most severe cases. Nonetheless, there is no clear management on the selection of these additives; therefore, it is still based on a trialand-error basis. For this reason, we have studied the properties and effectiveness of different chemical additives by defining two parameters based on the logistic model developed for the study of the defoaming kinetics of crude oil foams formed by depressurization: the effect on the foamability or antifoamability effect (AE) and the effect on foam stability or destabilization effect (DE). Finally, we have tried to go further in the understanding of the mechanisms involved in foam breaking, looking for similitudes on the defoaming behavior in the different crude oils tested.

INTRODUCTION

Dealing with unwanted foams during crude oil exploitation is a major issue. Big efforts have been carried out to prevent or destroy foams formed in the gas-liquid separator because the presence of foam reduces the production capacity. Furthermore, the presence of foam may damage the separation unit and the equipment downstream, leading to a shutting down of the refining process, causing a big economic loss. [START_REF] Callaghan | Crudeoil foaming problems at the Sullom Voe terminal. JPT[END_REF][START_REF] Chin | Chemical Defoamer Reduction with New Internals in the Mars TLP Separators[END_REF][START_REF] Shaban | A study of foaming and carry-over problems in oil and gas separators[END_REF] Several techniques have been developed to prevent foam formation or to improve its breaking up, such as spraying water onto the surface, mechanical breaking, increasing of the operational temperature, removal of the foamy agents, or using specially designed units. [START_REF] Chin | Chemical Defoamer Reduction with New Internals in the Mars TLP Separators[END_REF][START_REF] Barigou | Foam rupture by mechanical and vibrational methods[END_REF] However, the most common method for foam control is still the addition of chemicals because of its ease of use as well as its effectiveness. [START_REF] Bergeron | Antimousses et agents deḿoussantsMećanismes d'action[END_REF][START_REF] Bergeron | Polydimethylsiloxane (PDMS)-based antifoams[END_REF][START_REF] Colbert | Foam and Emulsion Control Agents and Processes: Recent Developments[END_REF][START_REF] Karakashev | Foams and antifoams[END_REF][START_REF] Rezende | Evaluation of the efficiency of polyether-based antifoams for crude oil[END_REF][START_REF] Stevenson | Foam Engineering: Fundamentals and Applications[END_REF][START_REF] Wylde | Successful field application of novel, non-silicone antifoam chemistries for high foaming heavy oil storage tanks in Northern Alberta[END_REF] In this case, special chemical agents are added to prevent foam formation (antifoamers) or to destroy the foam already created (defoamers).

Industrial experience from the oil industry has shown that, among the different molecules that can be used as antifoaming or defoaming agents, silicone oils are the most effective molecules. Actually, polydimethylsiloxane (PDMS) oils are the most common antifoam agents used in the oil industry, followed by fluorosilicone oils for the most severe cases. [START_REF] Colbert | Foam and Emulsion Control Agents and Processes: Recent Developments[END_REF][START_REF] Prud'homme | Foams: Theory, Measurements and Applications[END_REF][START_REF]Foams: Fundamentals and Applications in the Petroleum Industry[END_REF][START_REF] Blaźquez | Non-Aqueous and Crude Oil Foams[END_REF] Actually, some recent works are focused on the evaluation of these chemicals as antifoams in crude oil samples. [START_REF] Fraga | Method to evaluate foaming in petroleum[END_REF][START_REF] Fraga | Evaluation of the Efficiency of Silicone Polyether Additives as Antifoams in Crude Oil[END_REF][START_REF] Fraga | E Development and Evaluation of Oil in Water Nanoemulsions Based on Polyether Silicone as Demulsifier and Antifoam Agents for Petroleum[END_REF] Nevertheless, there are other kinds of molecules that can be used as antifoams, principally phosphate esters, metallic soaps of fatty acids, sulfonate compounds, amides, polyglycols, glycol ethers, and alcohols. These molecules usually exhibit less antifoaming effect than silicone oils, but they biodegrade faster than PDMS. In addition, their impact on the refining crude oil catalytic processes is negligible, contrary to silicone and silica, which are usually catalytic poisons. For instance, several nonsilicone agents have been tested in a heavy crude oil from north Alberta (diluted with 10% heptane), showing that a sulfosuccinate of sodium and a dispersion of long-chain fatty alcohols in water have a significant antifoaming effect. [START_REF] Wylde | Successful field application of novel, non-silicone antifoam chemistries for high foaming heavy oil storage tanks in Northern Alberta[END_REF] Despite the efforts carried out to understand the nonaqueous foam breaking mechanism, the selection of industrial antifoaming agents is still based on a trial-and-error basis, with no clear management. There are other considerations to take into account apart from the chemical structure of the surfactant: the solubility of the antifoaming agent, the presence of solids or gels on the gas-liquid interface, the oil and gas composition and properties, the operating conditions (thermal stability of the agent), or the impact that the agent may have in the post-treatment of crude oil.

It is noteworthy that most of the literature on defoaming and antifoaming chemicals is focused on the breaking of aqueous foams. Therefore, the theoretical developments concerning the mechanisms of defoaming were mainly performed on these systems. [START_REF] Garrett | Defoaming: Theory and Industrial Applications[END_REF] Two different mechanisms have been proposed for aqueous foam breaking by non-soluble oils: fluid entrainment or pinchoff. [START_REF] Bergeron | Antimousses et agents deḿoussantsMećanismes d'action[END_REF][START_REF] Bergeron | Polydimethylsiloxane (PDMS)-based antifoams[END_REF] In the first mechanism, an antifoam droplet spreads onto the gas-liquid surface, entraining the subjacent fluid with it and, by this way, creating a local film thinning and rupture. On the other hand, in the second mechanism, an antifoam droplet bridges the interface, creating a capillary instability, which destroys the liquid film. In both cases, the first condition needed is the entering of the antifoam droplet into the film lamellae. The definitions of the entering coefficient (E a/i ), the spreading coefficient (S a/i ), and the bridging coefficient (B a/i ) have been made based on thermodynamic criteria for capillary

where σ is the surface tension and the subscripts w, g, and a refer to water, gas, and antifoam, respectively. On the other hand, Denkov differentiates between two types of antifoams: "fast" and "slow". 19 The main property that differentiates between both types is the barrier to drop entry. This entry barrier has been defined as a function of the critical capillary pressure or the disjoining pressure, [START_REF] Bergeron | Polydimethylsiloxane (PDMS)-based antifoams[END_REF][19][START_REF] Hadjiiski | Effect of Oily Additives on Foamability and Foam Stability. 2, Entry Barriers[END_REF][START_REF] Arnaudov | Effect of Oily Additives on Foamability and Foam Stability. 1, Role of Interfacial Properties[END_REF] where the threshold value of the entry barrier is somewhere between 15 and 20 Pa, which separates the fast antifoam region from the slow antifoam region. These mechanisms have been developed for three immiscible phases, i.e., water, gas, and an oily phase containing the antifoam. However, having positive values of these coefficients does not guarantee foam destruction, but they provide a basis for using these types of antifoams. With regard to crude oil foams, other considerations must be taken into account. First of all, the antifoam is usually injected, dissolved, or dispersed in an organic phase instead of an aqueous phase; therefore, it is hard to predict the aggregation state of the antifoams and, thus, the validity of these mechanisms. Furthermore, the solubility of the antifoaming agent, the presence of solids or gels on the gas/liquid interface, or the effect of the additive on downstream processes are also important to the selection of the additive. However, some authors have hypothesized that antifoams should work by spreading onto the gas-liquid interface. [START_REF] Wylde | Successful field application of novel, non-silicone antifoam chemistries for high foaming heavy oil storage tanks in Northern Alberta[END_REF] As highlighted previously, the selection of additives and the prediction of foam behavior after chemical addition are quite complex and no mechanism has yet been clearly established for its understanding. The scope of this work was to develop a methodology for the study and ranking of chemical additives. This methodology is mainly based on the depressurization foaming test that has been presented and thoroughly described in a previous paper. [START_REF] Blaźquez | Crude oil foams. Part 1A novel methodology for studying non-aqueous foams formed by depressurization[END_REF] Furthermore, we propose a new way to study and compare the effect of the antifoaming chemical additives in terms of impact on both foamability and foam stability.

MATERIALS AND METHODS

Fluid Selection. Four crude oils with different American Petroleum Institute (API) gravities have been chosen to perform antifoaming tests. Their properties are summarized in Table 1. In all cases, crude oils were dehydrated (<0.5% H 2 O). A rheological characterization was performed using a rheometer with imposed stress (Haake RS150). In this work, we have used coaxial cylinders (statortype Z40 smooth with an inner diameter of 43.4 mm and rotor-type Z40 smooth with an inner diameter of 40 mm diameter) with shear rates between 0 and 1000 s -1 at different temperatures (from 20 to 40 °C). The density was measured with a densimeter Anton Paar DMA 4500M between 20 and 60 °C. The checking of the water content was made using a Metrohm 787 Titrino, which measures the amount of water by the Karl Fischer method. The surface tension was obtained using a drop tensiometer Kruss DSA 25 at a laboratory temperature (20 ± 2 °C).

Several silicone oils have been used as antifoam agents: four different PDMS silicones with different kinematic viscosities, two modified PDMS silicones, three hydroxyl-terminated PDMS, and two fluorosilicones. The antifoam solutions were created by diluting the selected silicone in an adequate solvent at 30% in volume. The properties of the different silicones and the solvent used are gathered in Table 2.

Crude Oil Foam Tests. The depressurization foaming test that we have developed in our laboratory has been thoroughly described in a recent paper. [START_REF] Blaźquez | Crude oil foams. Part 1A novel methodology for studying non-aqueous foams formed by depressurization[END_REF] This test is similar to the test developed by Fraga et al., [START_REF] Fraga | Method to evaluate foaming in petroleum[END_REF][START_REF] Fraga | Evaluation of the Efficiency of Silicone Polyether Additives as Antifoams in Crude Oil[END_REF][START_REF] Fraga | E Development and Evaluation of Oil in Water Nanoemulsions Based on Polyether Silicone as Demulsifier and Antifoam Agents for Petroleum[END_REF] but with some modifications in the design of the test, as in the data treatment and analysis. The principle is shown in Figure 1. The system consists of a high-pressure-high-temperature (HP-HT) aging cell made of stainless steel. After saturation of the crude oil with gas at a fixed pressure under agitation, the cell is depressurized by the instant opening of the ball valve. The mixture of oil and gas flows through a needle valve and a coil tube and fills a graduated glass tube. When the foam volume reaches the desired level (50 mL in our tests), the ball valve is immediately closed and the evolutions of both the air/foam and foam/liquid interfaces are recorded with a camera. The presence of the needle valve is the major difference between this test setup and the setup by Fraga et al. The purpose of this valve is to create a sufficient pressure drop, allowing for the increase of the test repeatability because the system is less impacted by any uncertainty caused by friction differences during the closing of the ball valve. The registered data (total volume V t and liquid volume V l ) are then processed by means of an image analysis software. In the case of antifoaming tests, the chemical additives to be tested are first diluted in a suitable solvent before being added in the crude oil by means of a precision syringe.

Prior to oil saturation, a desired quantity of antifoam solution is added to the crude oil and the mixture is then agitated (60 min) to homogenize it. All antifoam tests were carried out with 5 bar of saturation gas (CO 2 or CH 4 ) at a laboratory temperature (20 ± 2 °C). All tests were performed with 40 ppm of surfactant active matter, which is usually recommended by the providers to avoid an overdosing. In all cases, blank tests were performed to determine the effects of the solvent on the foamability and foam stability, showing that both cyclohexane and xylene have no measurable effect, while methyl isobutyl ketone has a slight effect on the foamability but no effect on the stability of the foam.

In the case of crude oils alone, the defoaming kinetic curves exhibit a sigmoidal shape that can be adequately fitted by the logistic model, as presented in a previous paper. [START_REF] Blaźquez | Crude oil foams. Part 1A novel methodology for studying non-aqueous foams formed by depressurization[END_REF] This model has also been used in the case of foaming tests with chemical additives. [START_REF] Dalmazzone | Formation and Breaking of Crude Oil Foams[END_REF] The evolution of the normalized volume of foam with time can be described by eq 4, with three adjustable parameters (A 2 , t 0 , and p). In that case, the foam volume is normalized with the maximum quantity of foam that the system can create at the test conditions (V f /V f 0 ) and varies from 100 to 0%. However, it is noteworthy that the volume of foam can also be normalized with respect to the total volume of liquid used in the experiment (V l inf , i.e., the volume of liquid recovered when all of the foam has broken), as shown by eq 5. Here, the normalized foam volume varies from A′ 1 (V f 0 /V l inf > 100%) to 0%. This parameter A′ 1 , which represents the ratio between the maximum foam volume that is possible to create in the test conditions and the total volume of liquid in the system (A′ 1 = V f 0 /V l inf ), can be considered as a foamability index. On the other hand, parameter A′ 2 results from the multiplication of A 2 by A′ 1 .

2 = - + + () V V A A 1 1 t t p f f 0 2 2 0 (4) = ′-′ + +′ () V V AA A 1 t t p f l inf 12 2 0 (5)
Antifoamability Effect and Destabilizing Effect Definitions. We have defined two criteria based on eqs 4 and 5 parameters to simplify the comparison between the different chemical additives in terms of effect on the foamability (i.e., the effect on the initial foam volume) and foam stability (i.e., the effect on the foam volume after a certain period of time).

The antifoamability effect (AE) can be defined as the capacity of a surfactant to prevent foam formation or to reduce the amount of foam created. In our case, we have related this property to the foamability index, defined as the maximum amount of foam that the system can create at the test conditions. Mathematically, it can be expressed as shown in eq 6.

This new index is expressed as a percentage and shows the reduction rate of the foam created compared to the oil system without additive. Therefore, antifoamers can be easily classified according to their effectiveness. If AE > 0, then the chemical shows a foaming effect. If AE = 0, then the chemical has no effect. If AE < 0, then the chemical prevents foam formation, with the effect being all the more important as the absolute value of AE is high. The destabilizing effect (DE) is defined as the effect of the chemical compound on the foam stability. It can be related to the end-life time of the foam but it is worth noting that the different additives may have ad i fferent impact on the different stages of the life of the foam. To take this into account, we have defined the effect on the foam stability as a function of two indexes, which compare both half-life (t 50% ) and end-life (t f ) times of foams with and without antifoam. It is noteworthy not to confuse this effect with a defoamer effect because a defoamer is added once the foam is already created, while in our tests, the additive is mixed with the crude oil prior to foam tests. These two indexes can be expressed as follows:

=

RESULTS AND DISCUSSION

Additive Efficiency in Crude Oils with CO 2 . All tests were carried out in the same conditions, i.e., with 5 bar of CO 2 at controlled room temperature (20 ± 2 °C) and with 40 ppm of active matter of silicone additive. Each test was performed at least 3 times, with a maximum deviation of 10% for the foam volume and 7% for total and free liquid volumes. In addition, for modeling, at least three valid tests have to be considered. However, as a result of oil sample availability, we were not able to test all silicone additives with the same oil. The different tests performed are described as follows: oil Z1, silicones S1, S2, S3, and S4 and fluorosilicones FS1 and FS2; oil M1, silicones S1, S2, S3, and S4 and fluorosilicone FS1; oil Z2, silicones SOH1, SOH2, and SOH3 and modified silicone S9; and oil M2, silicones SOH1, SOH2, and SOH3, modified silicones S5 and S9, and silicone S2.

Figures 2345show the best fitted curve given by eq 4 for each oil. The parameters of the model are displayed in Tables 3 and4. With regard to these figures, it is possible to observe that each additive has a different effect on the behavior of the foam. In all cases, it is possible to notice a destabilizing effect, which varies in function of the oil used. There are three principal behaviors: a complete destabilizing effect, where the foam breakdown is faster with the addition of the silicone, an initial destabilizing effect followed by a stabilization effect at the end of the lifetime of the foam (see Figures 2 and4), and the particular defoaming kinetic of silicone S9, which exhibits a completely different behavior from the other additives (see Figure 4 and5). This silicone tends to stabilize the foam at the first stages of the lifetime of the foam (even more than crude oil alone for the case of oil Z2; Figure 4), and then the destruction of the foams takes place in a nearly linear manner.

To compare the efficiency of each additive and the effect of the crude oil, we have estimated the values of the destabilizing effect as shown in eqs 7 and 8 (see Table 5). On the one hand, all additives have a value of DE I lower than 1, which shows that, in all cases, the half-life time of the foam is reduced. As said before, it is worth mentioning the case of silicone S9, for which the DE I values are 0.83 for both oils Z2 and M2. For these oils, this silicone shows less effect than the other silicones at the early stages of the lifetime of the foam. On the other hand, the effectiveness of the silicones varies a lot with regard to the destabilizing effect at the end-life time of the foam (DE II). Some of the additives have a small impact on the foam lifetime (silicone S4 in oil Z1), no effect (silicone S2 in oil M1 or S9 in oil M2), or the opposite effect (silicones SOH1 and SOH2 in oil M2 or FS1 in oil M1). In addition, it is possible to observe that the crude oil also has an impact on the additive efficiency. a √, the additive has an effect; ×, the additive has a very small or no effect. The best additive is in bold. For instance, silicone S2 is the most performing in terms of foam breaking in oils Z1 and M2 but not in oil M1 (no effect at the end of the lifetime of the foam). In the same way, fluorosilicone FS1 is quite performing in oil Z1 but increases the lifetime of oil M1 by 35%. This leads to the deduction that the crude oil properties and composition have a strong impact on the additive effectiveness.

With regard to the effect on foamability AE (eq 6 and Table 5), almost all additives exhibit a significant effect. Furthermore, if we compare the results oil by oil, we observe that the effectiveness is of the same order of magnitude for each additive, except S9 and FS2. Silicone S9 shows no effect on foamability and can be considered as a pure foam breaker, while fluorosilicone FS2 has a foaming effect despite the reverse effect of its solvent (AE MIBK = -9.5%).

When both effects (AE and DE) are taken into account, it is possible to select the most performing additive for each crude oil. For oils Z1 and M2, the most effective silicone is S2. For oil Z2, hydroxylated silicone SOH3 shows the strongest destabilizing effect. For oil M1, silicones S1 or S4 both have similar performances.

Effect of the Saturation Gas. The nature of the saturation gas may have an impact on the behavior of the foam. [START_REF] Blaźquez | Crude oil foams. Part 1A novel methodology for studying non-aqueous foams formed by depressurization[END_REF] For this reason, we have tested silicone S2 in crude oil M2 using methane as the saturation gas, because it is more representative of the associated gas encountered on the field than CO 2 . The effect of the nature of the gas on both defoaming curves with and without additive is displayed in Figure 6. With to the effect of the additive, it is possible to observe that silicone has a destabilizing effect with both gases. Moreover, the shape of both curves is similar. However, in both cases (with and without additive), foams with CH 4 are more stable than foams with CO 2 (see Figure 6 and Table 6). Also, it is worth mentioning that S2 decreases the foamability for both gases, with this effect being stronger for CO 2 than for CH 4 . Hence, these results confirm the importance of performing tests on additives using a representative field gas because it has a significant impact on the additive efficiency.

Classical Foam Breaking Mechanisms by Chemical Additives. As mentioned previously, two mechanisms have been proposed to explain foam breaking in aqueous foams by chemical additives: spreading and pinch-off. To check the relevance of these mechanisms in our experiments, we have calculated the entering, the spreading, and the bridging coefficients (eqs 1, 2, and 3, respectively) for crude oil M2 and a silicone additive of each family tested: a PDMS (S2), a modified PDMS (S5), a hydroxylated PDMS (SOH1), and a fluorosilicone (FS1). The surface and interfacial tensions were measured with a drop tensiometer (Kruss DSA 25). The values of each coefficient are displayed in Table 7.

With regard to these values, it is possible to point out that the entering, the spreading, and the bridging coefficients are very similar for silicones S2, SOH1, and FS1, while the values for silicone S5 are significantly lower. However, silicone S5 is the second most performing additive, even more than SOH1, despite its very low spreading coefficient. It is noteworthy that the lack of correlation between the values of E, S, and B coefficients and the actual antifoam activity has been reported for aqueous foams and explained with the role of the entry barrier. 19 Unfortunately, it is impossible to clarify here the mechanisms involved in the case of crude oil without using complementary direct methods, such as optical observations or model experiments with foam films and/or plateau borders, which were not performed in the framework of this study.

Defoaming Rates. As seen, several factors may affect the defoaming behavior of a crude oil, such as the composition and physicochemical properties of the oil, the gas forming the foam, but also the properties and nature of the antifoam additive. This makes the understanding of the mechanisms involved in defoaming by chemical additives extremely complex. For this reason, we decided to study the defoaming rates of the crude oils with and without additives.

The foam breaking rate or defoaming rate can be defined as the variation of the foam volume with time. Mathematically, it can be obtained by deriving the defoaming kinetic curve (Figure 7).

According to the defoaming rate, we have found three main zones in the lifetime of the foam. These three zones are shown in Figure 7. Zone I: almost no change in the foam volume is observed in this zone. This is mainly due to the superposition of two effects of opposite nature: on the one hand, the creation and growing of foam by bubble nucleation and bubble ripening and, on the other hand, the processes of foam decline and destruction, such as film drainage, gas diffusion through the atmosphere, and coalescence. In this zone, the rate of foam formation is similar to that of foam destruction, resulting in a constant volume of foam. This zone is not necessarily observed in every foam system, but it is possible to find it in highly viscous oils (as M2 or M1), in high gas saturation oils, and in stabilized foams. Zone II: here, the destruction processes become dominant. Therefore, it is possible to observe an increase of the foam destruction rate until a maximum, which corresponds to the inflection point of the sigmoidal curve. Zone III: this is the last zone observed. Once the inflection point is reached, the destruction rate begins to decrease. This can be explained by the deceleration of the drainage rate as the foam becomes drier as well as the decreasing of the Ostwald ripening phenomenon because the bubble sizes are more homogeneous. Moreover, at this point, some film stabilization phenomena may take place (e.g., Gibbs-Marangoni effect, capillary suction, etc.). Each zone can be defined by a characteristic defoaming rate: the initial defoaming rate (v d,0 ) for the first zone, the maximal foam breaking rate for the middle zone (v d,max ), and the final defoaming rate for the third zone (v d,f ). We have represented in Figure 8 the defoaming rates of the oils without additive to analyze their different behaviors. Oils M1 and M2 show lower values of defoaming rates because they are more viscous than Z1. Furthermore, these oils have initial values of defoaming rates closer to 0, which corresponds to the presence of the first region described above (see Table 8 and Figures 3 and5). In addition, oil Z2 exhibits a quasilinear defoaming rate until the end of the lifetime of the foam. With regard to the impact of the chemical additives, the addition of a few parts per million (ppm) results in an increase of the foam breaking rate, as shown in the case of PDMS silicones with oils Z1 and M1 (Figures 9 and 10 and Figures S1 and S2 of the Supporting Information). It is noteworthy that the increase of the defoaming rate is higher in the case of the less viscous oil Z1.

Similar tendencies can be observed in both oils: the initial defoaming rate increases with the viscosity (hence, the size) of the silicone, while the trend reverses once the maximum rate has been reached (rates tend to decrease with increasing viscosity). Actually, an increase of the molecular size leads to the displacement of the maximal defoaming rate through the first stages of the lifetime of the foam, even reaching the initial defoaming rate (as in the case of silicone S4, where the initial defoaming rate corresponds to the maximal defoaming rate). This suggests that PDMS has a strong effect on the second region of the foam kinetics of destruction. The same tendencies have been observed for PDMS-OH (see Figures S3 andS4 of the Supporting Information); therefore, we can think that, in both cases, the additives have a significant effect on the second region of foam breakage kinetics. Finally, we want to point out the special behavior of silicone S9 (Figure S5 of the Supporting Information), which shows no effect on the stability or foamability of the initial foam but exhibits a strong destabilizing effect after a while, maintaining high defoaming rates for the whole life of the foam. This leads us to think that this additive acts as a pure destabilizer agent for foam.

CONCLUSION

The addition of chemical additives is the most widely used way to control foaminess in crude oil because of its simplicity and efficiency. Among the different chemicals that can be used, silicone-based antifoams are the most common (essentially PDMS and fluorosilicone). Nevertheless, there are no clear guidelines to select a specific chemical additive for a specific oil; therefore, the selection of the additive is still performed on a trial-and-error basis. For this reason, we have adapted the foaming laboratory test presented in a previous paper to test and rank the effect of different additives on crude oil foam formation by depressurization. To do this, a desired amount of additive is added to the oil prior to gas saturation. The experimental procedure as well as the mathematical models described in the previous part are still valid for this kind of test.

The defoaming kinetic curves were analyzed in terms of foamability reduction (AE) and foam stability reduction (DE) using two parameters based on the logistic model. The definition of these two parameters allows for the selection of the best performing additive for a given crude oil. It is first of all important to keep in mind that the effectiveness of an antifoam additive is not only a function of its chemical properties and nature but also the properties and composition of the crude oil and the nature of the saturation gas. As seen during this study, the best additive selected for one crude oil will not be necessarily adapted to treat another crude oil (as in the case of silicone S2 with oils Z1 and M1). Furthermore, the foaming gas also has a significant impact on the ability of the additive to prevent or destroy foam. For this reason, it is recommended to perform the foaminess tests using a representative field gas.

With regard to the defoaming curves, the behavior of silicones PDMS and PDMS-OH was similar. It was observed that the initial rate of breaking increases with the viscosity (hence, the size) of the silicone. Also, S9 silicone shows a very particular behavior, because it has no impact on the foamability or foam stability at the beginning but, after a while, destroys the foam at high rates (near its maximum rate) until almost the end of the breaking process.

To conclude, the foaming test by depressurization can be readily used to test and rank different chemical additives, leading to the optimal selection of the chemical in a meaningful and repeatable way. Future objectives will be to identify the relevant parameters that control the mechanism of action of chemicals and to develop consistent approaches to control foaming in complex crude oil systems and to optimize the design of chemicals.

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.energyfuels.6b02567.

Normalized defoaming rates of crude oil Z1 (black line) with 5 bar of CO 2 and 40 ppm of S1 (empty blue diamonds), S2 (empty green squares), S3 (empty orange triangles), and S4 (red crosses) (Figure S1), normalized defoaming rates of crude oil M1 (black line) with 5 bar of CO 2 and 40 ppm of S1 (empty blue diamonds), S2 (empty green squares), S3 (empty orange triangles), and S4 (red crosses) (Figure S2), normalized defoaming rates of crude oil Z2 (black line) with 5 bar of CO 2 and 40 ppm of SOH1 (dark green diamonds), SOH2 (dark red squares), and SOH3 (dark blue triangles) (Figure S3), normalized defoaming rates of crude oil M2 (black line) with 5 bar of CO 2 and 40 ppm of SOH1 (dark green diamonds), SOH2 (dark red squares), and SOH3 (dark blue triangles) (Figure S4), and normalized defoaming rates of silicone S9 in crude oil Z2 (empty purple circles) and M2 (purple circles) with 5 bar of CO 2 (Figure S5) (PDF)

Figure 1 .

 1 Figure 1. Principle of the foaming test by depressurization (left) and parameters recorded during the foaming test in the graduated test tube (right).

Figure 2 .

 2 Figure 2. Modeled defoaming kinetic of crude oil Z1 (black line) with respect to chemical additive defoaming kinetics with 5 bar of CO 2 : S1 (empty blue diamonds), S2 (empty green squares), S3 (empty orange triangles), S4 (red crosses), FS1 (brown dashes), and FS2 (greenish plus signs).

( 8 )

 8 Therefore, as in the case of AE, surfactants can be classified according to their destabilizing effect. If DE I < 1 and DE II < 1, then the chemical shows a significant destabilizing effect. If DE I < 1 and DE II > 1, then the chemical acts at early stages of the lifetime of the foam with a stabilizing effect at the end of its life. If DE I > 1 and DE II < 1, then the chemical stabilizes the foam at the early stages of the lifetime of the foam but has a destabilizing effect at the end. If DE I > 1 and DE II > 1, the chemical stabilizes the foam.

Figure 3 .

 3 Figure 3. Modeled defoaming kinetic of crude oil M1 (black line) with respect to chemical additive defoaming kinetics with 5 bar of CO 2 : S1 (empty blue diamonds), S2 (empty green squares), S3 (empty orange triangles), S4 (red crosses), and FS1 (brown dashes).

Figure 4 .

 4 Figure 4. Modeled defoaming kinetic of crude oil Z2 (black line) with respect to chemical additive defoaming kinetics with 5 bar of CO 2 : SOH1 (dark green diamonds), SOH2 (dark red squares), SOH3 (dark blue triangles), and S9 (empty purple circles).

Figure 5 .

 5 Figure 5. Modeled defoaming kinetic of crude oil M2 (black line) with respect to chemical additive defoaming kinetics with 5 bar of CO 2 : SOH1 (dark green diamonds), SOH2 (dark red squares), SOH3 (dark blue triangles) S2 (empty green squares), S5 (pink asterisks), and S9 (empty purple circles).

Figure 6 .

 6 Figure 6. Modeled defoaming kinetic of crude oil M2 without additive with CO 2 (dark blue line) and with CH 4 (pink line) and with 40 ppm of silicone S2 with CO 2 (empty dark blue squares) and with CH 4 (empty pink triangles).

Figure 7 .

 7 Figure 7. Schematic defoaming curve. The red point represents the inflection point of the curve.

Figure 8 .

 8 Figure 8. Defoaming rates of crude oils Z1 (red squares), Z2 (purple triangles), M1 (green circles), and M2 (blue diamonds) with 5 bar of CO 2 .

Figure 9 .

 9 Figure 9. Defoaming rates of crude oil Z1 (black line) with 5 bar of CO 2 and 40 ppm of S1 (empty blue diamonds), S2 (empty green squares), S3 (empty orange triangles), and S4 (red crosses).

Figure 10 .

 10 Figure 10. Defoaming rates of crude oil M1 (black line) with 5 bar of CO 2 and 40 ppm of S1 (empty blue diamonds), S2 (empty green squares), S3 (empty orange triangles), and S4 (red crosses).

1

  stability of the foam lamellae as a function of the different surface energies in the system

	E a/i	σσσ =+-w/g w/a a/g	(1)
	S a/i	σσσ =--w/g w/a a/g	(2)
	B a/i	σσσ =+-w/g w/a a/g 2 2	2

Table 1 .

 1 Crude Oil Properties: Density (ρ), API Gravity, Viscosity (μ), and Surface Tension (σ)

	crude oil	ρ at 20 (kg/m 3 ) °C	API gravity (deg)	μ at 20 (mPa s) °C	°C (mN/m) σ at 20
	Z1	890	27	37	25.4
	Z2	908	24	76	27.5
	M1	921	22	173	27.8
	M2	921	22	185	28.9

Table 2 .

 2 Silicone Properties (Provided by the Supplier) and Solvent Used for the Antifoaming Solution a

	silicone	type	ρ (kg/m 3 )	ν (cSt)	provider	solvent
	S1	PDMS	970	1000	Bluestar Silicones	cyclohexane
	S2	PDMS	970	12500	Bluestar Silicones	cyclohexane
	S3	PDMS	973	60000	Bluestar Silicones	cyclohexane
	S4	PDMS	973	500000	Bluestar Silicones	cyclohexane
	S5	modified PDMS			Bluestar Silicones	cyclohexane
	S9	modified PDMS	910	160	Momentive	xylene
	SOH1	hydroxylated PDMS	973	14000	Bluestar Silicones	xylene
	SOH2	hydroxylated PDMS	973	50000	Bluestar Silicones	xylene
	SOH3	hydroxylated PDMS	973	135000	Bluestar Silicones	xylene
	FS1	fluorosilicone	1120	20000	Momentive	cyclohexane
	FS2	fluorosilicone	1248	10000	Dow Corning	MIBK

a MIBK = methyl isobutyl ketone.

Table 3 .

 3 Logistic Model Parameters (eqs 4 and 5) for Oil Samples Z1 and Z2 with 40 ppm of Silicone Additive and 5 bar of CO 2

	a
						eq 4				eq 5
	oil sample	silicone	gas	A 2	t 0	pR 2	χ 2	A′ 1	A′ 2	R 2	χ 2
	Z1		CO 2	-85 ± 12	142 ± 11	1.45 ± 0.15	0.967	31	179 ± 10	-152 ± 16	0.941	192
		S1	CO 2	-36 ± 75 5 ± 9	1.62 ± 0.21	0.965	31	150 ± 11	-53 ± 9	0.959	112
		S2	CO 2	-18 ± 63 6 ± 3	1.72 ± 0.14	0.981	18	139 ± 6	-25 ± 4	0.984	39
		S3	CO 2	-30 ± 44 6 ± 7	1.19 ± 0.12	0.964	25	145 ± 9	-44 ± 6	0.958	91
		S4	CO 2	-35 ± 45 7 ± 12	1.06 ± 0.11	0.961	24	153 ± 9	-54 ± 6	0.963	77
		FS1	CO 2	-64 ± 88 1 ± 7	1.17 ± 0.15	0.984	11	158 ± 11	-101 ± 12	0.975	61
		FS2	CO 2	-204 ± 127	186 ± 79	1.20 ± 0.15	0.982	16	192 ± 21	-388 ± 79	0.887	302
	Z2		CO 2	-432 ± 13	1055 ± 57	0.99 ± 0.07	0.992	7	221 ± 7	-950 ± 36	0.986	65
		SOH1	CO 2	-32 ± 10	90 ± 11	1.49 ± 0.13	0.966	29	173 ± 8	-55 ± 6	0.966	98
		SOH2	CO 2	-43 ± 8	104 ± 10	1.34 ± 0.07	0.989	9	175 ± 4	-75 ± 4	0.988	32
		SOH3	CO 2	-65 ± 6	107 ± 7	1.02 ± 0.10	0.972	19	180 ± 6	-116 ± 7	0.976	71
		S9	CO 2	-245 ± 48	259 ± 28	1.64 ± 0.17	0.964	35	222 ± 8	-534 ± 27	0.962	208

a R 2 and χ 2 are statistical parameters.

Table 4 .

 4 Logistic Model Parameters (eqs 4 and 5) for Oil Samples M1 and M2 with 40 ppm of Silicone Additive and 5 bar of Gas a

	eq 4	eq 5

a R 2 and χ 2 are statistical parameters.

Table 5 .

 5 Effects on Foamability (AE) and Foam Stability (DE) of Silicones on Different Crude Oils with 5 bar of CO 2 and 40 ppm of Active Matter a

	oil	additive	AE (%)	AE?	DE I	DE II	DE?
	Z1	S1	-16	√	0.55 ± 0.06	0.66 ± 0.10	√√
		S2	-22	√	0.42 ± 0.03	0.62 ± 0.09	√√
		S3	-19	√	0.44 ± 0.04	0.80 ± 0.16	√√
		S4	-14	√	0.49 ± 0.06	0.97 ± 0.27	√×
		FS1	-12	√	0.56 ± 0.06	0.75 ± 0.11	√√
		FS2	7	×	0.67 ± 0.28	0.65 ± 0.29	√√
	M1	S1	-39 ± 1	√	0.42 ± 0.02	0.65 ± 0.07	√√
		S2	-42 ± 1	√	0.55 ± 0.04	1.00 ± 0.18	√×
		S3	-41 ± 1	√	0.51 ± 0.08	1.05 ± 0.54	√×
		S4	-44 ± 1	√	0.36 ± 0.03	0.69 ± 0.14	√√
		FS1	-47 ± 1	√	0.44 ± 0.01	1.35 ± 0.45	√×
	Z2	SOH1	-22 ± 1	√	0.61 ± 0.09	0.81 ± 0.19	√√
		SOH2	-21 ± 1	√	0.62 ± 0.08	0.81 ± 0.14	√√
		SOH3	-18 ± 1	√	0.45 ± 0.04	0.69 ± 0.08	√√
		S9	0.7 ± 0.1	×	0.83 ± 0.18	0.63 ± 0.08	√√
	M2	SOH1	-40 ± 3	√	0.55 ± 0.08	1.18 ± 0.59	√×
		SOH2	-35 ± 2	√	0.59 ± 0.05	1.14 ± 0.24	√×
		SOH3	-39 ± 2	√	0.52 ± 0.04	1.04 ± 0.14	√×
		S2	-41 ± 3	√	0.40 ± 0.04	0.66 ± 0.17	√√
		S5	-33 ± 2	√	0.55 ± 0.07	0.80 ± 0.17	√√
		S9	-0.6 ± 0.1	×	0.83 ± 0.09	1.00 ± 0.11	√×

Table 6 .

 6 Effects on Foamability (AE) and Foam Stability (DE) of Silicone S2 on Crude Oil M2 with 5 bar of Gas and 40 ppm of Active Matter a

	gas	AE (%)	AE?	DE I	DE II	DE?
	CO 2	-41 ± 3	√	0.40 ± 0.04	0.66 ± 0.17	√√
	CH 4	-24 ± 1	√	0.55 ± 0.02	0.87 ± 0.09	√√

a √, the additive has an effect; ×, the additive has a very small or no effect.

Table 7

 7

	. Entering, Spreading, and Bridging Coefficients of
	S2, S5, SOH1, and FS1 in M2
	additive	E a/o (mN/m)	S a/o (mN/m)	B a/o (mN/m) 2
	S2	12.8	5.49	457
	SOH1	11.5	7.49	461
	FS1	11.2	8.08	465
	S5	7.9	0.06	228

Table 8 .

 8 Initial, Maximal, and Final Defoaming Rates of Oils Z1, Z2, M1, and M2 with 5 bar of CO 2

	crude oil	v d,0 (%/s)	v d,max (%/s)	v d,f (%/s)
	Z1	-0.290	-1.468	-0.414
	Z2	-0.544	-0.544	-0.217
	M1	-0.006	-0.374	-0.089
	M2	-0.004	-0.202	-0.081

ACKNOWLEDGMENTS

The authors thank the financial support of this project from Bluestar Silicones. The authors also acknowledge IFP Energies nouvelles and the staff for their experimental support and advice. The authors also thank Vance Bergeron for useful discussion.

AUTHOR INFORMATION

Corresponding Author *E-mail: christine.dalmazzone@ifpen.fr.

ORCID

Christine Dalmazzone: 0000-0002-2567-9385

Notes

The authors declare no competing financial interest.


Marius Feldmann

Juan A Fraire

Felix Walter

Scott C Burleigh

Ring Road Networks: Access for Anyone

Keywords: Ring Road Networks, Delay-tolerant Networking, Contact Graph Routing

come

I. INTRODUCTION

In its existing form, digital connectivity expansion is based on the goal of universal continuous connectivity. Indeed, underlying Internet technology looks more like a phone call than sending a letter: a client expects a server to reply immediately. We claim that the difficulty of profitably achieving this goal globally has been the core reason for having billions of unconnected or under-connected people across the globe. The implications of commitment to this model might not be evident unless we move beyond the frontier of terrestrial communication, into cis-lunar space and beyond. Due to the light propagation speed limit and planetary occlusion, continuous connectivity will become impractical or impossible in such conditions, even with the best possible technologies.

The vision argued in this paper is that by developing the delay-tolerant networking (DTN) technologies required to construct a Solar System Internet, we have acquired the means to reduce inequality right here on our planet by connecting remote places where the deployment of Internet infrastructure is economically (or technically) infeasible.

The core construct derived from this vision is framed in the "ring road network" (RRN) concept. In an RRN, overflying low-Earth orbit (LEO) satellites act as data mules to receive, carry, and deliver data from and to places that lack Internet connectivity. The main advantage of this concept is that it can be implemented with DTN protocols and inexpensive nano-satellites such as CubeSats. Because of its delay-tolerant nature, even a single nano-satellite can provide (extremely) high-latency connectivity services on a global scale. The addition of more satellites will enable the reduction of the endto-end delay and the increase of the overall system capacity. An operative RRN will thus allow currently disconnected users to run Wikipedia searches, receive remote medical assistance, participate in global market stores, among many other services we are used to in urban environments. The only difference is that replies and feedback might arrive minutes or hours after queries are issued.

In this paper, we present the fundamentals of RRNs and the supporting research. Next, we use the results of simulations to assess the expected performance of RRNs. Our opensource DTN protocol implementation, named 𝜇D3TN, is then introduced and evaluated in the first in-orbit experiment of the RRN concept, utilizing ESA's OPS-SAT. Afterwards, we briefly examine the categories of applications that are suitable for communicating via RRNs and those that are not. Finally, we discuss the cost advantages of the RRN architecture and the integration of RRN services with Internet services that will be provided by the large LEO satellite constellations currently under construction.

II. BACKGROUND

A. CubeSats in LEO

CubeSats are standardized nano-satellites initially developed for pedagogical purposes, designed in a context where historically each new space mission required a full re-engineering of space and ground systems. Due to the CubeSat standard's merits in providing a common nano-satellite deployment platform, it found wide adoption. At the time of writing, different form factors are commonly deployed, named after the number of units they are composed of, whereas a 1-unit (1U) CubeSat may have a maximum weight of 1.33 kilograms and dimensions of 10×10×11.35 centimeters. For example, 3U, 6U, and 12U CubeSat platforms are now popular in the industry. Additionally, commercial off-the-shelf components and launch opportunities for CubeSats are readily available. Typically deployed in LEO, CubeSats orbit below the altitude of 2000 km and exhibit trajectories that can be accurately predicted. With an orbital period of ≈90 minutes, a LEO satellite appears on one horizon and disappears over the other in typically less than 10 minutes. Depending on the latitude, any single LEO satellite will be usually visible from a given spot less than five times a day; obviously, increasing the number of satellites in the constellation increases the frequency of overflights at any single location.

B. Delay-tolerant Networks

Delay-tolerant networking (DTN) extends network functionality beyond the environment in which the Internet works well, to challenged environments in which, e.g., long signal propagation delays, arbitrary and intermittent interruptions or limited communication resources are present (see RFC 4838 for details of the architecture). DTNs are relevant in many application domains such as airborne, vehicular, and underwater networks. Maturing since the early 2000's, DTN provides the means for nodes to cope with link disruptions and delays. In order to deal with interrupted end-to-end paths, the core idea behind DTN is to transmit packets via a store-carryand-forward principle. Thus, intermediate nodes may store packets until an appropriate next-hop node is available. The topologies of DTNs cannot be represented as regular (static) graphs; instead, the topologies are described via time-variant graphs. The appearing and disappearing edges of these graphs can be described as so-called contacts. A contact is a period of time during which two nodes are able to transfer data unior bidirectionally.

The Bundle Protocol (version 6 defined in RFC5050 or version 7, which is currently in IETF's standardization process) harnesses the DTN principles by introducing a dedicated "bundle layer". This layer is operating on top of different underlying protocols, which provide their service to the bundle layer via a shim layer called convergence layer. The Bundle Protocol implements fundamental DTN features such as custody transfer (version 6), fragmentation, quality of service, and deadlines [START_REF] Fraire | Routing in the Space Internet: A contact graph routing tutorial[END_REF]. A DTN node does not assume continuous end-to-end connectivity; instead, it utilizes persistent storage to survive waiting periods of seconds, minutes, or days during which one or more links in the end-to-end path are inoperable. Several DTN protocols, including the Bundle Protocol, are currently discussed and specified in IETF (for Internet standardization) and CCSDS (for space communications standardization).

DTN protocols have been validated in space (UK-DMC, DINET, JAXA, ISS) [START_REF] Fraire | Routing in the Space Internet: A contact graph routing tutorial[END_REF], but never as yet in the context of RRNs.

III. RING ROAD NETWORKS

Ring Road Networks are composed of low-cost, LEO satellites. Due to the limited deployment and operational costs, they are perfectly suitable for providing Internet access to disadvantaged populations while possibly also relaying data inexpensively for other classes of users. As data in an RRN is transported by the LEO satellites using DTN protocols in a store-carry-and-forward manner, high end-to-end communication delays are given (see [START_REF] Burleigh | Toward a communications satellite network for humanitarian relief[END_REF] for details). Building upon the initial idea of combining small satellites and DTN presented in 2008 [START_REF] Krupiarz | Using smallsats and DTN for communication in developing countries[END_REF], the technological and scientific landscape in this context has shown major advances that favor the deployment of RRNs before the end of this decade [START_REF] Burleigh | Toward a communications satellite network for humanitarian relief[END_REF], [START_REF] Fraire | Benefits and challenges of cross-linked ring road satellite networks: A case study[END_REF]- [START_REF] Walter | Prediction-enhanced Routing in Disruption-tolerant Satellite Networks[END_REF]. In contrast The bundle can then reach the server on the Internet (𝑆), which determines that the optimal return path is via hot spot 2 and cubesat 2.

to existing satellite missions which aim to support the conversational, synchronous exchange of data, RRNs aim to provide reliable epistolary data transfer. They enable asynchronous connectivity to otherwise isolated network nodes on Earth (called "cold spots" in RRN terminology).

As RRNs can be built incrementally starting with a single satellite (e.g., in contrast to mega-constellations such as Starlink based on thousands of ≈250 kg satellites) and as they do not require continuous connectivity among satellites, deployment costs are very low (see [START_REF] Burleigh | Toward a communications satellite network for humanitarian relief[END_REF] for details). By tolerating episodic contacts, RRNs may comprise solely nadirpointing CubeSats in LEO: each satellite retains in-transit data collected from cold spots in a queue in its own local storage medium, awaiting its "hot spot" overflights. In RRN terminology, hot spots are ground stations that have Internet access, by which data can be forwarded to and answered by an application server (see Fig. 1).

Responses are returned to cold spots, usually via other satellites, within minutes -or even hours.

Even an RRN comprising a single LEO satellite would enable users to run Wikipedia searches, receive remote medical assistance, and participate in global market stores, among many other services we are used to in urban environments. The only difference is that responses to ad-hoc queries might take minutes or hours to arrive; however, prior subscription to information sources that are known to be of future interest would enable queries to be satisfied by local interrogation of previously cached information and, thus, yield immediate results. Moreover, introducing additional satellites in the constellation will increase the frequency of all cold spot and hot spot overflights, hence, both reducing round-trip latency and increasing overall system capacity. We present a numerical assessment of the the expected performance for RRN satellite fleets of various configurations in Section IV.

A. Beneficiaries

One of the essential target beneficiaries of the RRN concept are unconnected populations, anywhere in the world, that are currently isolated because the deployment of synchronous connectivity infrastructure is not cost-justified. This can be either because the area is difficult to access (i.e., remote islands, mountain ranges, jungles) or because the area does not represent an attractive market (i.e., the world's poorest regions).

The nature of low-cost CubeSats and asynchronous DTN connectivity yields an infrastructure that is orders of magnitude cheaper than satellite Internet infrastructure, which maps to very low cost for the final user, sharply lowering the accessibility barrier. In particular, RRN programs are a perfect and realistic fit for governments' and humanitarian organizations' investments in providing free Internet connectivity to remote regions. Finally, the same infrastructure can be applied to benefit a plethora of business-oriented cases to reduce the cost of time-insensitive communications [START_REF] Burleigh | Toward a communications satellite network for humanitarian relief[END_REF], including secure and low-data-rate secondary and backup channels for sensitive data.

B. Supporting Research

This section summarizes the most relevant research in the context of RRNs.

a) Taxonomy and Tools: A comprehensive taxonomy of RRNs has been presented in [START_REF] Feldmann | Refining the ring road-delays and path lengths in a LEO satellite message-ferry network[END_REF]. In this research, an investigation of the delay and path length characteristics is discussed. The authors evaluate RRN fleets from 10 to 50 satellites. Inter-satellite linking (ISL) was evaluated for RRNs in [START_REF] Fraire | Benefits and challenges of cross-linked ring road satellite networks: A case study[END_REF]. Results show that the delivery ratio could be improved up to 10% in walker formations comprised of 12 satellites, even though congestion issues were identified. A vast number of different simulation tools have been used and developed to analyze algorithms in RRN, e.g., the ONE, ns-3 [START_REF] Cello | Hot spot selection in rural access nanosatellite networks[END_REF], [START_REF] Marchese | E-CGR: Energy-aware contact graph routing over nanosatellite networks[END_REF], DtnSim based on Omnet++ [START_REF] Fraire | Benefits and challenges of cross-linked ring road satellite networks: A case study[END_REF], aiodtnsim [START_REF] Walter | Prediction-enhanced Routing in Disruption-tolerant Satellite Networks[END_REF], and further custom toolchains (e.g., [START_REF] Feldmann | Tracking lunar ring road communication[END_REF]).

b) Routing: A routing trade-off regarding the amount of available topological knowledge coined spot of maximum knowledge has been evaluated in a simulated Lunar network comprising 6 satellites in Lunar orbit supported by NASA's Deep Space Network (DSN) ground stations [START_REF] Feldmann | Tracking lunar ring road communication[END_REF]. A Selection scheme for routing via specific hot spots was presented in [START_REF] Cello | Hot spot selection in rural access nanosatellite networks[END_REF] and further energy-aware routing schemes for RRN scenarios were explored in [START_REF] Marchese | E-CGR: Energy-aware contact graph routing over nanosatellite networks[END_REF].

c) Contact Prediction: The estimation of the characteristics of transmission opportunities in RRNs was discussed and analyzed in concert with an extended deterministic routing scheme based on several earlier publications in [START_REF] Walter | Prediction-enhanced Routing in Disruption-tolerant Satellite Networks[END_REF]. For the validation, multiple RRN scenarios with 6-9 satellites and 10-15 ground stations have been derived from data sets collected via a global ground station network.

IV. EVALUATION

This section presents a quantitative assessment of a representative RRN scenario we have performed leveraging the aiodtnsim simulator described in [START_REF] Walter | Prediction-enhanced Routing in Disruption-tolerant Satellite Networks[END_REF].

We consider 10 randomly distributed ground stations served by 1, 5, 10, 15, and 20 satellites randomly selected from a set of 20 CubeSats (Two-Line Elements files obtained from Celestrak). Contacts between ground stations and satellites are considered viable when the elevation is equal to or higher than 10 • . No ISLs are considered in this evaluation. A contact plan is computed for a scenario duration of 3 days (72 hours). Bundles of 2 MiB are generated for the initial 80% of the total scenario duration, from and to random ground stations. The bundle generation period is set to 1000 (seconds) divided by the number of satellites, with a standard deviation equal to half of the interval. Twenty runs are performed for each scenario with different transmission plans (i.e., varying sourcedestination pairs). Each node's buffer size is set to 2 GiB, and data rates are configured to 256 kbit/s (uplink) and 1 Mbit/s (downlink). A maximum of 5 concurrent links are allowed.

We evaluate the expected performance of RRN when bundles are routed using three categories of routing approaches. On the one hand, Epidemic routing forwards a copy of the bundle to each possible neighbor, up to a maximum of 6 hops. This is a trivial approach, with high delivery ratio and minimal computational cost, achieved at the expense of high resource consumption (memory and energy). Spray and Wait [10] is a more efficient replica-based scheme where a limited number of copies (6 in our case) are "spread" into the network, to then "wait" till one of these nodes meets the destination. In contrast, CGR exploits an adaptation of Dijkstra's algorithm to optimally deliver a single copy of the bundle via the fastest path [START_REF] Fraire | Routing in the Space Internet: A contact graph routing tutorial[END_REF]. To this end, a pre-computed contact plan is assumed to be available at every node in the RRN.

A. Results

Results are summarized in Fig. 2. In terms of delivery probability, results in Fig. 2-a) show that congestion due to multiple replicas in Epidemic prohibits adequate scalability. Indeed, although the maximum delivery ratio is obtained for 5 satellites, only 51.2% of the bundles are delivered for the 20 satellite case. We observe that Epidemic exhausts both satellite memory and contact capacity after 30, 40, and 50 hours for scenarios with 20, 15, and 10 satellites, respectively.

Spray And Wait improves on the delivery ratio of Epidemic due to the more efficient replication strategy, but CGR is notably the most efficient approach with a 100% delivery success for any fleet size. CGR also provides the optimal delivery delay (Fig. 2-b), as it leverages topological data from the contact plan. In particular, an average end-to-end delay of 7.7 hours (𝜎=4.5) for a single satellite is reduced to 1.3 hours (𝜎=0.9) for 20 satellites.

Epidemic and Spray And Wait solutions fail to consistently reduce delay for larger fleets. The energy efficiency (Fig. 2-c) is also optimal for CGR, as every transmission effort results in the successful delivery of the routed bundle as long as the contact plan data is accurate (i.e., no packets are dropped). Epidemic routing ranges from 11% to 4% (1 to 20 satellites), while Spray And Wait presents a consistent efficiency of 22% for any fleet size.

Finally, copy-based routing presents the highest memory demands (Fig. 2-d), which are required to store the required replicas ranging from 60 MiB to 1.97 GiB. CGR is the most efficient in terms of storage, occupying 6.2 to 11.4 MiB (𝜎=7.1 to 12.9) for the evaluated traffic shape. These evaluation results indicate that state-of-the-art DTN routing solutions exploiting the predictable nature of orbits could be deployed on cost-efficient nano-satellite platforms.

V. IMPLEMENTATION

To enable the flow of data in RRNs in the conditions identified by the simulations above, we developed Micro-DTN (𝜇D3TN). 𝜇D3TN is an open-source lightweight DTN protocol implementation (https://gitlab.com/d3tn/ud3tn) written in the C programming language.

The software has a modular design that offers several convergence layer adapters (CLA) through which µD3TN can transport data over many different underlying network protocols. Additionally, integration with a wide variety of DTN applications is made possible by µD3TN's application agent protocol (AAP). It provides a simple, language-independent way for interacting with the DTN node. Furthermore, µD3TN implements a routing component (which includes a deterministic next-hop determination approach), contact management, and buffer management, among other features.

𝜇D3TN was first validated in a challenged environment in the context of the COLDSUN project in cooperation with ESA in 2019, and was subsequently integrated into the OPS-SAT LEO satellite.

VI. IN-ORBIT VALIDATION

The RRN proof of concept using 𝜇D3TN was validated on ESA's OPS-SAT during December 2020 and May 2021. Our goal with the OPS-SAT experiment was three-fold: (i) to show that the RRN concept can be applied in concrete LEO use cases; (ii) to flight-test the new version 7 of the DTN bundle protocol; and (iii) to evaluate µD3TN features in the RRN context.

A. OPS-SAT Overview

The OPS-SAT satellite is a three-unit CubeSat operated by ESA. It was launched in December 2019 and is equipped with a MityARM 5CSX board with a dual-core 800 MHz ARM processor.On top of this hardware, a Linux-based software stack launches and controls the experiments. A TCP interface is offered on the spacecraft for experiments to send/receive data to/from the communications subsystem of the satellite. The satellite's communication subsystem enables wireless packets in the Space Packet Protocol (SPP) format to be transferred via the OPS-SAT radio interface to the ground segment. 𝜇D3TN was adapted to utilize this TCP/SPP interface as a CLA. On the ground segment, the ground station at ESA's SMILE lab provides a similar TCP socket interface enabling us to connect our simulated hot and cold spots using a reverse SSH tunnel. Two major experiments were performed in this context.

B. Scenario 1: Web Services

The objective of the first experiment was to show that access to a web server from isolated cold spots is possible in an RRN scenario. For this experiment, we leverage the curl command-line tool to send an HTTP request to our µD3TN-based cold spot communication gateway (step 1 in Fig. 3). This instance then generates a bundle carrying the HTTP request (step 2) and sends it over the SPP CLA toward ESA's ground station (step 3). At the core of the setup, our dispatcher service provides a link that multiplexes to simulate different ground stations, even though we only connect to a single ground station from ESA. Based on a schedule, the packets are transmitted toward the ESA ground station (step 4), which then sends the bundle via its S-band radio uplink to the OPS-SAT (step 5). Onboard, we run another µD3TN instance, which processes and later forwards the received bundle via another downlink contact (step 6). Upon reception, the ground station then forwards the bundle again via the SSH tunnel to our dispatcher service (step 7), which sends it to the hot-spot instance, also running µD3TN (step 8). After unpacking the bundle's payload (step 9), the HTTP request is delivered to the webserver (step 10). The HTTP communication gateway then packs together the response in a bundle and returns it on a path equivalent to the one taken for the request.

We performed a simplified version of this test during our first test window in December 2020 and the full version of the test in May 2021. It was the first successful test of BPv7 in space, and it demonstrated that the 𝜇D3TN implementation can be used for the entire communication path, on the ground and on the satellite.

C. Scenario 2: Advanced Features

The goal of the second experiment, conducted in collaboration with the SPATIAM Corporation, was to demonstrate feature interoperability between different DTN protocol implementations within a scenario based on remote access to a Cloud service. In this context, 𝜇D3TN interacts with another DTN protocol implementation: Interplanetary Overlay Network (ION) [START_REF] Burleigh | Interplanetary overlay network: An implementation of the dtn bundle protocol[END_REF], developed by the Jet Propulsion Laboratory, NASA. Here, ION is deployed in the hot spot element (see Fig. 3). (ION is the most complete and featurerich DTN protocol stack. It interoperates with 𝜇D3TN, but its larger computational/memory footprint limits its applicability in processing-constrained CubeSats.) The user sends a picture to the Google Cloud Vision (GCV) API so that different features in the image may be identified, labeled, and reported back via the RRN. In the course of this experiment, we were able to test not only interoperability but also bundle fragmentation, as the large image file had to be split into multiple packets (bundles).

This test was performed successfully between the 21st to the 24th of May 2021. Due to COVID safety concerns, the experiment team could not sit together in a mission control room. The satellite connection was set up remotely to operate from a virtual machine via the SSH tunnel mechanism. The satellite's log files were evaluated in a video conference together with the ESA OPS-SAT team.

D. Outcomes

The most relevant outcome of the experiments is that the DTN protocols and our implementation of them in 𝜇D3TN worked without any serious issues. Additionally, we were able to show interoperability between ION and 𝜇D3TN, which worked well even when exercising advanced protocol features to support machine learning applications over RRNs.

There were also some challenges identified. Due to the large image files transmitted from the cold-spot, more than 200 packets per second were created, which provoked an overload on the CubeSat transmission system. To cope with this, ESA suggested introducing a rate-limit, which was finally set to 45 packets per second emitted by the 𝜇D3TN CLA at maximum to ensure a stable data flow. Also, additional packets destined for other subsystems of the satellite were received via the on-board socket connection, for which we had to implement a filtering mechanism so that not every packet would be interpreted as a DTN bundle.

Finally, the communication that we have performed was solely based on static time windows pre-configured on the satellite as well as on the ground segment. Once these time tags matched the local clocks, bundle transmissions were triggered. The first consequence of this is, of course, that we need very well-synchronized clocks. This is not an issue in OPS-SAT but might be for other spacecraft. Secondly, we had to make assumptions about whether a link is present, complicating debugging. Indeed, we cannot be entirely sure of an active link being available until having real feedback from the satellite. As a first workaround, we scheduled the contacts in such a way that they only occur when a satellite has a high elevation in relation to the ground station -thus, the probability is quite high that a connection will be present (for some contacts we only scheduled transmissions above 15 • ). Of course, this strategy reduces the amount of data that can be transferred. Additionally, we implemented a periodic ping-like trigger for our tests, which sends one bundle to the satellite, which is immediately returned and invokes the test run upon reception. This mechanism significantly improved the reliability of the experiment setup.

E. Next Steps

RRN challenges remain in evaluating the approach in larger systems, with larger hop-counts and with larger time spans. We plan to advance these aspects in further field tests planned for mid 2022. These will be carried on as part of the REDMARS project, funded by the federal Ministry of Education and Research of Germany. The goal is to transfer ideas from the recursive internetworking architecture (RINA) [START_REF] Day | Patterns in network architecture[END_REF] to the DTN domain. Here, the challenge is to move from the conceptual to the concrete level by implementing recursive and scalable routing via bundle-in-bundle encapsulation (BIBE) and dedicated extension blocks. These will be validated in orbit -in the OPS-SAT mission, if still available, or on other similar platforms.

VII. OUTLOOK A. Applications

The vision of RRN frames the Internet as a special case of network communications, a constrained environment in which delay is insignificant and prolonged disruptions are considered a topological anomaly; RRN addresses a more general network model in which long round-trip times slow down but do not impede end-to-end communications. The DTN network infrastructure is a necessary but not sufficient element of this model; applications themselves must be delay-tolerant.

Most types of communication fulfil this requirement. Though the massive use of synchronous real-time communication such as VoIP or Instant Messaging, oftentimes communication does not depend on ad-hoc availability of data at the moment of its creation. Accessing knowledge bases such as Wikipedia, watching movies streamed from a content distribution network (CDN), receiving weather forecast data, or transmitting backup information to a remote server -all these scenarios have in common that parts of the network between the initial data source and the destination are tolerant in regards to delays of even hours. In a large number of scenarios, delays are totally acceptable. More generally, emerging application technologies based on publish/subscribe operations, notably including information-centric networking [START_REF] Sarros | Intermittentlyconnected IoT devices: Experiments with an NDN-DTN architecture[END_REF], would be supported by RRN.

B. Satellite Broadband

It could be asserted that the global Internet access promised by upcoming constellations will render RRN unnecessary. However, because the DTN architecture on which RRN is based is designed to utilize underlying Internet services where they are available, expanded Internet service will extend the capabilities of an overlying RRN network; integration of the two technologies is not an obstacle.

More importantly, it remains possible that even these extremely large constellations will be unable to provide global network service at a price affordable for the world's leastadvantaged populations. The three major competitors in satellite Internet deployment at this time are the Kuiper, Starlink, and OneWeb projects, with 3236, 4408, and 648 satellites, respectively. OneWeb projects the total cost of its constellation to be between $5.5 billion and $7 billion. Optimistically guessing the correct figure to be $6 billion, we obtain a mean cost per satellite of about $9 million. This suggests that the total cost of deploying the Kuiper constellation would be about $30 billion, and the total cost of deploying the Starlink constellation would be about $40 billion; the total cost of the three constellations would then be approximately $76 billion.

According to ABI Research [START_REF]ABI Research: Broadband Services Will Propel Satellite Broadband Market Revenues[END_REF], the total market for satellite broadband currently comprises 3.6 million subscribers and is projected to increase at about 8% per year, reaching 7.56 million users by 2031. Assuming all three providers remain in operation at that time, the deployment cost per subscriber will have been just over $10,000. In order to amortize that cost, monthly service charges would exceed the resources of many potential users in economically-disadvantaged regions. For example, Starlink beta subscribers are currently charged $99/month.

A further consideration is the extension of terrestrial broadband. Outside of populated and relatively affluent areas, particularly cities, satellite broadband is the only available option; however, the proportion of the world's population that resides in cities is projected to increase from 55% to over 66% by 2050, see [START_REF]World Bank: Demographic Trends and Urbanization[END_REF]. The relative demand for satellite broadband may decrease over time.

In short, we suggest that RRN may emerge as a better cost-justified answer to the problem of providing network communication service to the economically or geographically disadvantaged.

ACKNOWLEDGMENTS

The results and materials presented in this paper emerged from work started during and pushed since a visit of Marius Feldmann at the Jet Propulsion Laboratory in 2013 hosted by Scott Burleigh. The results and materials were mainly brought to a consistent paper by Juan Fraire. Felix Walter contributed via the provided simulation and solid implementation support during the last years. The first OPS-SAT field-test has been initially presented during the IETF DTN working group meeting at the IETF 110 meeting; the second field-test during the 5th ESA CubeSat Industry Days 2021.

We would like to thank Maximilian Nitsch and Georg Murzik for their massive support before and during the ex-ecution of the OPS-SAT field-tests. Additionally, we would like to thank the Spatiam Corporation, Lara Suzuki, Vinton Cerf, and the ESA OPS-SAT team (specifically, David Evans, Tom Mladenov, and Vladimir Zelenevskiy) for the fruitful collaboration.

Fig. 1 .

 1 Fig.1. RRN diagram. Nodes connected to the isolated cold spot can relay bundles to cubesat 1, which carries the data until flying over hot spot 1. The bundle can then reach the server on the Internet (𝑆), which determines that the optimal return path is via hot spot 2 and cubesat 2.

Fig. 2 .

 2 Fig. 2. Evaluation results for scenarios with 10 ground stations (10g) served by 1 to 20 LEO satellites (1s to 20s)

Fig. 3 .

 3 Fig. 3. OPS-SAT experiment set-up.


Giulia Ferri

Severine Humbert

Jean-Marc Schweitzer

Mathieu Digne

Veronique Lefebvre

Maxime Moreaud

Mass fractal dimension from 2D-microscopy images via an aggregation model with variable compactness

Keywords: fractal dimension, morphological model, image analysis

Microscopy-image analysis provides precious information on size and structure of colloidal aggregates and agglomerates. The structure of colloids is often characterized using the mass fractal dimension d f , which is different from the two-dimensional fractal dimension d p that can be computed from microscopy-images. In this work we propose to use a recent morphological aggregation model to find a relationship between 2D image fractal dimension and 3D mass fractal dimension of aggregates and agglomerates. Our case study is represented by scanning transmission electron microscopy-images of boehmite colloidal suspensions. The behaviour of the computed d f at different acid and base concentration shows a fair agreement with the results of Small Angle X-Ray Scattering and with the literature, enabling to use the d f vs d p relationship to study the impact of the composition of the colloidal suspension on the density of colloidal aggregates and agglomerates.

Introduction

Image-analysis is frequently used to characterize colloidal particles of silica 1 , alumina [START_REF] Chiche | Accurate determination of oxide nanoparticle size and shape based on X-ray powder pattern simulation: Application to boehmite AlOOH[END_REF] , soot or carbon. These particles can exist as a powder or within liquid suspensions, and they find applications in different domains like electronics, pharmaceutic and catalysis [START_REF] Brune | Nanotechnology: Assessment and Perspectives[END_REF] . Such applications depend on particles size and structure that are the result of the particles aggregation or agglomeration mechanism 4;5;6 . For this reason the control of aggregation and agglomeration within colloids is of high importance. In literature, the terms aggregate and agglomerate are often used interchangeably. When it is necessary to distinguish between them, authors refer to the type of cohesive force that can be either strong or weak [START_REF] Nichols | A review of the terms agglomerate and aggregate with a recommendation for nomenclature used in powder and particle characterization[END_REF] . With regard to the colloidal boehmite suspensions characterised in this work, assemblies of maximum diameter lower than or higher than 80 nm are called aggregates or agglomerates, respectively.

The mass fractal dimension d f is often used to characterize the structure of aggregates and agglomerates. Assuming that an aggregate/agglomerate consists of N identical spheres of radius a, the fractal dimension relates N to the gyration radius R gyr 8;9;10 .

N = k f R gyr a d f (1)
where k f is the fractal pre-factor whose value varies typically between 1 and 1.2, depending on the fractal dimension itself [START_REF] Christopher | The prefactor of fractal aggregates[END_REF] . The mass fractal dimension is close to unity for linear clusters and increases up to a maximum value of 3 for clusters with several folded branches. d f can be measured by experimental scattering techniques such as Small Angle X-Ray [START_REF] Mircea | Small-angle scattering from fractals: Differentiating between various types of structures[END_REF] or Neutron [START_REF] Teixeira | Small-angle scattering by fractal systems[END_REF] Scattering. A change in d f of a colloidal aggregate/agglomerate modifies its density, involving different dispersibility and settling behaviour [START_REF] Ibaseta | Ultrafine aerosol emission from the free fall of tio 2 and sio[END_REF] . This changes the workability of a colloidal suspension, as well as the textural properties of the solid obtained after a precipitation or drying process. In the context of γ-alumina catalyst carriers synthesis, the porosity of the final solid depends on the size and shape of the aggregates (meso-porosity ≃ 10 nm) and agglomerates (macro-porosity ≃ 100 nm) resulting from colloidal inter-particle interactions during different stages of preparation. By varying the chemical parameters of the colloidal suspension, it is possible to change the aggregation and agglomeration processes [START_REF] Lawrence M Anovitz | Effects of ionic strength, salt, and pH on aggregation of boehmite nanocrystals: Tumbler small-angle neutron and x-ray scattering and imaging analysis[END_REF] , leading to different size distribution and fractal dimension. Our interest is to use microscopy-image analysis to study this behaviour, in order to determine the physical-chemical parameters that lead to specific structure change within boehmite suspensions.

In this work, a characteristic fractal dimension d p is computed from microscopy images of boehmite aggregates and agglomerates. d p is then related to the mass fractal dimension d f . A similar purpose has been addressed many times in literature. The fractal dimension computed on two-dimensional images depends on the calculation method used 16;17 . For this reason, in order to compare several images it is necessary to use the same computation algorithm. Other studies on the relationship between cluster mass fractal dimension and their projection report that for 2 < d f < 3 the difference d f -d p increases with d f [START_REF] Tence | On the measurement of the fractal dimension of aggregated particles by electron microscopy: experimental method, corrections and comparison with numerical models[END_REF] . In this work the projected fractal dimension d p is measured using the relationship between area and perimeter [START_REF] Benoit B Mandelbrot | Fractal character of fracture surfaces of metals[END_REF] . This method is extensively used 20;21 , enabling to obtain directly a two-dimensional fractal dimension from the slope of the graph log(P 2 ) vs log(A), where P and A are respectively the perimeter and the area of the projections. The slope tends to 1 for circular objects and to 2 for linear objects. Since the fractal dimension of the projection d p is different from the mass fractal dimension d f , a relationship between these two parameters 22;23;24;25;26 is needed.

The aim of our work is to numerically provide such relationship, using a recent morphological aggregation model [START_REF] Ferri | Aggregation morphological model with compactness: application to colloidal system[END_REF] . The model is used to build clusters with a statistical average fractal dimension, as well as their opaque projections.

Determination of 3D-fractal dimension from 2D-microscopy images

In this section, we consider Scanning transmission electron microscopy (STEM) images of colloids of a boehmite suspension (on the left in Fig. 1) and we explain the step-by-step strategy to extract d f from such images.

Image processing and determination of d p

To compute the perimeter and area on STEM images, several operations are applied. The image noise is reduced 28;29 by preserving the sharp edges of objects. Morphological black TopHat operator [START_REF] Jean | Introduction to morphological filters[END_REF] is applied consisting on difference between the image and a morphological opening γ of the image, and a threshold operator tr s such that

tr s (I(x)) = 1|I(x) > s 0|I(x) ≤ s .
Extracted solid suspension can be expressed for an image I as tr s (I -γ B (I)) where s is obtained by inter-class variance maximization on the histogram of I 31 , and B is the structuring element which in our case is a disc of radius 30 pixels. In order to reduce the remaining noise, the morphological operations of opening γ and closing φ are applied, defined for the image I and the structuring element C as γ C (I) = δ C (ε C (I)) and as φ C (I) = ε C (δ C (I)) where δ and ε are respectively morphological dilation and erosion [START_REF] Jean | Introduction to morphological filters[END_REF] , C is a disc of diameter 6 pixels. The last step of image cleaning consists in a hole filling and in the removal of the objects touching the edges of the image [START_REF] Jean | Introduction to morphological filters[END_REF] . The holes are not always related to the existence of a real hole but rather to variations in thickness. While objects touching the borders would provide a misleading perimeter and area. Fig. 1 reports an example of an experimental image and of the final binary image. On the binary images it is possible to calculate the perimeter P and area A of each object. The d p is estimated from the slope of the line interpolating the experimental points on the plot log(P 2 ) vs log(A) [START_REF] Benoit B Mandelbrot | Fractal character of fracture surfaces of metals[END_REF] . To estimate the d f corresponding to a given d p it is necessary to pass through a morphological model. It is worth considering that the structures observed via STEM images are affected by the sample preparation process (drying under IR lamp). Furthermore, the choice of structuring element B can have a significant impact on the calculation of d p . For this reason, in order to compare the results, it is important to prepare all samples following the same procedure and to perform the image analysis using the morphological operations in the same order and with the same structuring elements.

Morphological aggregation model

The morphological aggregation model 27 builds clusters consisting of N non-overlapping elementary objects arranged according to a given statistical average fractal dimension d f . Clusters are built by sequential addition, assigning a different sticking probability to the points on the cluster. In particular, "concave points", leading to compact clusters, are distinguished from "non-concave" points, leading to loose structures. In addition the concave points closest to the center of mass of the cluster are distinguished as well, in these points the maximum compactness is reached. The model enables to obtain clusters with a fractal dimension d f between 1.3 and 3. A fast scheme is available in case of spheres [START_REF] Ferri | Aggregation morphological model with compactness: application to colloidal system[END_REF] and is used in this present work. A new object P is stuck to a cluster C either in a point on the cluster dilation

X P = F (δ P (C)), (2)
either on a concave point of the dilation

X CV P = F (δ P (C)) ∩ F (φ r (δ P (C))/δ P (C)) (3)
or in the concave point closest to the center of mass of the cluster C

X M P = x ∈ X CV P |infd(x, M ). ( 4
)
The expression φ r (δ P (C))/δ P (C) means the difference between the two sets. F (X) is the set of points at the border of X, F (X) = {z : B z ∩ X ̸ = ∅}, with B unit ball, M is the center of mass of the cluster C and d(x, M ) is the euclidean distance between the position x of the concave point and M . Fig. 2 illustrates how the morphological operations enable to select the concave regions of a cluster's boundary. An implementation of this model is available on the plugim! platform [START_REF]plugim![END_REF] .

𝑃 𝑃 ∈ 𝑋 𝑃 𝑀 𝑃 ∈ 𝑋 𝑃 𝑃 ∈ 𝑋 𝑃 𝐶𝑉 Figure 2:
The dilated set X P (in gray), the set of concave regions X CV P (in blue) and X M P are illustrated for a cluster C during the simulation. P is the primary particle.

Construction of d f vs d p chart

In order to compute d p of a cluster generated by the model, it is necessary to generate the projections of numerous clusters for a given d f in the largest range for N (the number of elementary spheres). The mass fractal dimension d f of each cluster is determined with Eq.(1) using the plots reported in Appendix A. In the present study a fractal pre-factor k f of 1.2 is used, in order to establish reference values for d f of the clusters generated [START_REF] Ferri | Aggregation morphological model with compactness: application to colloidal system[END_REF] . This value has been chosen because it provides a mass fractal dimension less than or equal to 3 for simulated clusters consisting of spherical particles.

The model inlet parameters were chosen in order to obtain a target statistical average d f . Only the clusters with d f -d f ≤ 0.01 were used to compute the projection. For each cluster, one orthogonal projection is computed, since a large number of simulations is performed it is possible to assume that area and perimeter computed on the projection are anisotropic. Fig. 3 represents three-dimensional clusters with their projection for different number of elementary spheres N . A hole filling operation is performed to obtain projection comparable to experimental images [START_REF] Jean | Introduction to morphological filters[END_REF] . The computation of area and perimeter enables the construction of the log(P 2 ) vs log(A) plot, whose slope is d p [START_REF] Benoit B Mandelbrot | Fractal character of fracture surfaces of metals[END_REF] .

The coefficients of the polynomial minimize least-squares fit. Eq.( 5) is valid for the morphological aggregation model described in a previous work [START_REF] Ferri | Aggregation morphological model with compactness: application to colloidal system[END_REF] , using spherical primary particles. Since there is no one-to-one correspondence between the fractal dimension of a cluster d f and the dimension of its projection d p , it is not possible to conclude whether this relationship can also be valid for other random aggregation models.

Validity region

To compare experimental and model projections, the perimeter-area plots are represented as log(P 2 ) vs log(A/A 0 ), where A 0 is the projection of the elementary object forming the cluster.

Aggregation simulations were performed with a sphere of radius 10 pixels, corresponding to an opaque projection of area A 0 of 380 pixels. This is important so that the relationship shown in Fig. 4(a) does not depend on the size scale and unit of measurement. The Eq.( 5) is valid within limits, in terms of projected log(A/A 0 ), for a given d f . The lower and the upper limit (respectively l A,min and l A,max ) are reported as reachable limit ranges by the aggregation model. The upper limit is also due to the maximum simulation volume, which in our case is of 10 9 voxels for time computing constraints. The lower and the upper limits can be described as a function of d f fitting respectively the lowest and the highest log(A/A 0 ) of the opaque projections of the simulated clusters. 5) in (a) and its validity in terms of surface area A in (b).

l A,min = 0.67 • (d f ) 4 -5.12 • (d f ) 3 + 14.37 • (d f ) 2 -17.12 • d f + 7.49 l A,max = -7.17 • (d f ) 4 + 68.19 • (d f ) 3 -241.09 • (d f ) 2 + 377.70 • d f -220.91. (6)
The validity region and its limits are represented in Fig. 4(b). A 0 does not influence the results for d p , but only the domain of validity of the relationship in Eq.( 5) on the log(P 2 ) vs. log(A/A 0 ) plot.

Experimental application to boehmite suspensions

In this section, we use the relationship between d f and d p to compute the mass fractal dimension from STEM images of different boehmite suspensions. Results are compared to Small Angle X-Ray Scattering and literature.

Boehmite suspensions preparation

Boehmite suspension are prepared with a highly dispersible powder from Sasol GmbH whose crystallites have the geometry of a rectangular beam with dimensions 5.2, 8.1 and 2.4 nm. These values are obtained from the X-Ray diffractogram using generalized Debye equation 2 . The colloidal suspensions are prepared with different concentration of 1M Fisher Scientific nitric acid (J/5550/PB15) and 1M Chem-Lab ammonia (CL05.0101.1000). The powder concentration is kept at 0.02 g/L in order to minimize overlapping in STEM images. The composition of the samples is shown in Tab.1. The table also reports the measured pH (Mettler-Toledo Seven2Go pH meter). A2 and B0 are the same suspension, which represents the initial conditions before ammonia addition (samples B1-3).

STEM Images Acquisition

A droplet of 10 µL of the suspension is deposited on copper grids with carbon membranes without holes and dried under IR lamp. After 2 min, the excess water is drawn off with a tissue paper. The samples are characterized with Scanning Transmission Electron Microscope (STEM), using a FEI Nova NanoSEM microscope operating at 15keV.

The different chemical parameters of the suspensions (acid and base concentration) of the samples causes the formation of structures ranging within a wide size range (10 nm -10 µm). For this reason, the magnification level for a given sample is chosen in order to visualize the largest objects in their entirety. In particular we use a magnification of X 50000 for the samples A0, A1, A2 and A3, and a magnification of X 1500 is used for B1, B2 and B3.

Small Angle X-Ray Scattering

The analyses were performed at the SWING beamline at synchrotron SOLEIL in Saint Aubin, France. The boehmite suspensions are placed in a stirred reactor, from which the sample is pumped continuously through a quartz capillary of 1.5 mm, crossed by the X-ray beam, whose incident energy is 10 keV. The experiments are realized on an initial boehmite suspension analogous to the A2-B0 at 1 g/L. The SAXS curves measured 30 min after the base injection are fit with Beaucage model 33 in order to compute the mass fractal dimension. The model is detailed in Appendix B.

Monitoring of boehmite dispersion upon acid addition

The aggregates (observed in the samples A0, A1, A2 and A3) have an average diameter D between 20 and 40 nm (Tab.1). This diameter is calculated by averaging the diameters of objects on binary microscopy images. The diameter of each object is calculated as the average of the minimum and maximum diameters. For these objects A 0 is the one of an elementary crystallite. The average surface of the opaque projection of the crystallite is 38.25 nm 2 , obtained by averaging on 1000 projections considering random 3D orientations 34 with a precision of 0.1 nm/voxels. Such estimation accounts for discretization issues, and it is close to the one computed using the Cauchy average projected area theorem, which is 37.02 nm 2 . We report the STEM images of the aggregates at different acid concentrations and their corresponding log(P 2 ) vs log(A)/ log(A 0 ) plots in Fig. 5.

The grey regions in these graphs refer to the validity limits of the d f -d p relationship (Fig. 4). The results of d p and d f (computed with Eq.( 5)) are reported in Tab.1. The dispersion in acid leads to an increase in d p , indicating the formation of loose structures. This is in agreement with d f , and quite visible from the STEM images. The objects present in A3 are more elongated than the ones in A0-1. We observe a change of around 0.3 units of d f . The difference between A3 and A2 is less noticeable, but the image processing enables to identify it. In conclusion, the acid content decreases fractal dimension of the boehmite suspension as it has been observed in other research works [START_REF] Speyer | Peptization of boehmites with different peptization index: An electron microscopy and synchrotron small-angle x-ray scattering study[END_REF] . 5), and measured with Small Angle X-Ray Scattering at different pH.

Monitoring of boehmite agglomeration upon alkaline addition

In a similar way, the relationship between d f and d p is used to characterize objects of a bigger scale, the agglomerates (in samples B1-3), obtained by ammonia addition to the suspension A2-B0.

The average diameter D is of several hundreds of nanometers (Tab.1). In this case, the elementary particle is the aggregate, and A 0 was taken as the average area of the dispersed sample A2-B0, which is 453 nm 2 . We now focus on the STEM images in Fig. 6. The effect of the base content seems to favour the formation of more compact structures with a decrease in d p . Between samples B1-2 and B3, we notice an evident change in the structure. Sample B3 presents highly compact objects Fig. 6(e), for which the mass fractal dimension is 2.9. For B3 the pH is close to the Point of Zero Charge (PZC) of the suspension [START_REF] Kosmulski | ph-dependent surface charging and points of zero charge. iv. update and new approach[END_REF] . In these conditions the agglomeration is typically favoured [START_REF] Johannes | Theory of the stability of lyophobic colloids: the interaction of sol particles having an electric double layer[END_REF] . We were able to observe an increase in the mass fractal dimension towards the PZC also with in-situ Small Angle X-Ray Scattering. Details on SAXS data are reported in Appendix B. The fractal dimension of SAXS agglomerates is compared to the one obtained from microscopy images in Fig. 7.

The behaviour observed in SAXS results suggests that the proximity to PZC, besides favouring faster kinetics, leads to more compact structures. Since the operating conditions required by each analysis technique are different, a comparison can only be made on the basis of a trend. Even if the concentration and the sampling is different, the STEM image processing also showed that a pH close to PZC lead to more compact structures.

For a further improvement of our approach, it would be interesting to investigate in the future the effect of drying (under IR lamp) on the fractal dimension of boehmite agglomerates. Or to compare the fractal size estimated from scattering data with that obtained from liquid phase microscopy techniques.

Conclusion

The presented work provides a strategy to compute the mass fractal dimension from two-dimensional microscopy images. A relationship between d p and d f is numerically found using a recent morphological aggregation model. The mass fractal dimensions we obtained are in fair agreement with a visual estimate, enabling to use STEM images to determine the impact of different operating conditions on the structure of aggregates and agglomerated formed within a colloidal suspension. This allows a better insight of the physical-chemical aspects that produce such structures In particular, we observed that an increase in acid concentration, causes the formation of more loose structures, with a decrease in d f . While the increase of base content leads to more compact structures, leading to higher d f . A complete validation of our approach is achieved by comparison of our results for boehmite suspension at different acid and base contents, with the literature and Small Angle X-Ray Scattering experiments.

Appendix A log(P) vs log(A) plots

This section shows the plots produced by relating the perimeter and area of the morphological model projections for

d f = 2, d f = 2.3, d f = 2.

B SAXS data B.1 Beaucage model

The general formula of the model for n structural levels is

I(q) = n i=1 G i • exp - q 2 R 2 gyr,i 3 + B i - q 2 R 2 gyr,i+1 3
•      erf q • k i • Rgyr,i √ 6 3 q      Pi (7)
• n equals to 1,2 or 3 for agglomerates, aggregates and crystallites respectively.

• G i is the Guinier pre-factor depending on the number concentration of the structural unit

N d • r 2 e • ∆ρ 2 • V i (N d
is the particles number density, r e is the electron radius, ∆ρ is the electron density difference between the sample and the environment, V is the average particle volume).

• R gyr,i is the gyration radius.

• B i is a constant pre-factor depending on the type of power-law scattering and on P i 38 .

B i = G i • P i R ( gyr, i) Pi • Γ P i 2 • 6 • P 2 i (2 + P i ) • (2 + P i ) P i 2 . (8)
• k i is an empirical constant that depends on P i 38 , its value is close to 1.

• P i is the slope of the curve in the structural level i, corresponding to the fractal dimension.

In our case, the SAXS curves present three scales. A nonlinear Least-Square Algorithm DN2FB 39 has been used for the curve fitting.

B.2 SAXS results

The scattering curve of a boehmite suspension at 1 g/L at pH close to the PZC (9.4) is reported in Fig. 9. The three contributions of the simulated scattered intentity by agglomerates, aggregates and crystallites are illustrated. The experimental data are fit within a range of 0.001255 Å-1 ¡ q ¡ 0.1998 Å-1 for the scattering vector. Corresponding to a maximum diameter ξ l between 3 and 500 nm. According to the model, the agglomerates scatter in a range of 0.0078 Å-1 ¡ q ¡ 0.1998 Å-1 , for a maximum diameter 80 nm ¡ ξ l ¡ 500 nm according to the relationship

ξ l = 2π q . ( 9
)
Due to the size limit of SAXS results, the comparison with microsopy images is made under the assumption that the fractal dimension is the same even for objects larger than 500 nm in diameter. This assumption can be confirmed in terms of d p by the microscopy images analysis. Fig. 10 shows a microscopy image of our boehmite suspension, its binary image and the relative plot log(P 2 ) vs log(A) in which the points relative to the objects with a maximum diameter higher and lower than 500 nm are highlighted in different colours. The points can be represented by the same line, and therefore by the same dimension d p .

  μm

Figure 1 :

 1 Figure 1: Experimental image (on the left) and its binary version (on the right).

Figure 3 :

 3 Figure 3: Illustration of three-dimensional clusters and their projections for α = β = 0.1 and N = 20, 40, 160. α and β are the compactness parameters of the morphological aggregation model, more details are in Ferri et al.27.

Figure 4 :

 4 Figure 4: d f vs d p relationship described in Eq.(5) in (a) and its validity in terms of surface area A in (b).

  STEM image of A0 (b) log(P) vs log(A) plot of A0 500 nm (c) STEM image of A1 (d) log(P) vs log(A) plot of A1 500 nm (e) STEM image of A2-B0 (f) log(P) vs log(A) plot of A2-B0 500 nm (g) STEM image of A3 (h) log(P) vs log(A) plot of A3

Figure 5 :

 5 Figure 5: STEM images and log(P 2 ) vs log(A) plots of boehmite suspensions for varying acid content.

  STEM image of B1 (b) log(P) vs log(A) plot of B1 30 µm (c) STEM image of B2 (d) log(P) vs log(A) plot of B2 30 µm (e) STEM image of B3 (f) log(P) vs log(A) plot of B3

Figure 6 :Figure 7 :

 67 Figure 6: STEM images and log(P 2 ) vs log(A) plots of boehmite suspensions for varying base content.

  5 and d f = 3 in Fig.8. The d p is obtained from the slope of these plots.

Figure 8 :

 8 Figure 8: Log(P ) vs log(A) plots obtained with the morphological model.

Figure 9 :

 9 Figure 9: Experimental and simulated scattered intensity of a boehmite suspension at 1 g/L and pH close to the PZC. The simulated curve is given by the sum of three terms corresponding to agglomerates, aggregates and crystallites.

Figure 10 :

 10 Figure 10: (a) STEM image of boehmite agglomerates obtained with a magnification of X 3000. Binary STEM image in (b). In (c) log(P ) vs log(A) plot where the blue and green dots refer to the objects whose maximum diameter is below and above 500 nm respectively.

Table 1 :

 1 Composition, fractal dimension and average diameter of the boehmite suspension samples Sample Name HNO 3 [mM] NH 3 [mM] pH d p

	d f	D [nm]

Acknowledgment

SAXS experiments were performed on the SWING beamline at SOLEIL Synchrotron, France. We are grateful to Thomas Bizien for assistance and to the SOLEIL staff for smoothly running the facility.