Coarse-Grained Model and Boiling Point Prediction for Asphaltene

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Coarse-grained model and boiling point prediction for asphaltene model compounds via HMC-WL simulations. Caroline Desgranges, and Jerome Delhommelle Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b01862 • Publication Date (Web): 19 Sep 2017 Downloaded from http://pubs.acs.org on September 20, 2017

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Coarse-grained model and boiling point prediction for asphaltene model compounds via HMC-WL simulations. Caroline Desgranges and Jerome Delhommelle∗ Department of Chemistry, University of North Dakota, Grand Forks, ND 58202, USA (Phone: 701-777-2495) E-mail: [email protected]

∗ To

whom correspondence should be addressed

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Abstract The determination of vapor-liquid equilibrium data is a key parameter for the conditions of stability or precipitation of asphaltenes. In this work, using molecular simulation, we parametrize a coarse-grained force field for a series of alkyldipyrene compounds, that are often used as model compounds for asphaltenes, and predict their thermodynamic properties at coexistence. To achieve this, we combine a Wang-Landau approach with hybrid Monte Carlo simulations in the isothermal-isobaric ensemble to sample extensively the configurations of the system over a wide range of densities. This allows us to obtain the conditions of coexistence and to shed light on the impact on the alkyl chain on the thermodynamic properties. Most notably, we identify a quasi-linear relationship between the boiling temperature of these compounds and the length of the alkyl chain.

Introduction Asphaltenes have been the focus of intense research in recent years, as these compounds often tend to yield thick deposits that can cause oil production to stop. 1–3 Furthermore, the issues caused by asphaltenes are not restricted to oil fields but can affect systems downstream, e.g. in refineries. There are several tremendous challenges both in experiments and theoretical studies. Asphaltenes have long been defined in terms of their solubility behavior (soluble in toluene and insoluble in n-alkanes). However, it is only recently that a clearer picture of their properties has started to emerge through SARA analysis, 4 which defines asphaltenes as a solubility class, and for their molecular weight and size through mass spectrometry and molecular-diffusion measurements. 1 These studies have revealed several possible structures for these compounds, 5–7 highlighting the preponderance of grouped aromatic rings and alkane chains in asphaltenes. This has also provided a starting point for understanding how asphaltene molecules can aggregate, 8–11 and how the data obtained for the vapor-liquid equilibrium behavior can be used to better predict the conditions of precipitation. 1,12,13 The determination of such properties, including the boiling point of asphaltenes using simu2 ACS Paragon Plus Environment

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lation methods also requires the development of specific strategies and methods suited to these systems. First, unlike small molecules, asphaltenes involve a large number of atoms that make usual force fields that model each atom or each functional group with an interaction site often too computationally expensive to study e.g. the aggregation of several thousands of asphaltenes. Second, the accurate determination of thermodynamic properties often requires sampling of millions of configurations for these systems, which has prompted the development of coarse-grained models for asphaltenes. 14,15 In such models, large groups of atoms are often lumped together and modeled with a single interaction site, thereby greatly reducing the extent of the calculations. Several studies have focused on simulating the precipitation behavior of asphaltenes using coarse-grained models. Here, as the first goal of this work, we focus on parametrizing a coarse-grained (CG) model that allows for the prediction of the boiling point for a series of alkyldipyrene molecules, with an alkyl chain of varying length connecting the two pyrene groups. These compounds contain building blocks of asphaltenes (polycyclic aromatic hydrocarbons, alkyl substituent groups), and, as such, can provide a basis for understanding the impact of the molecular structure on the thermophysical properties. As a result, alkyldiprenes have been used in prior experimental work as model compounds for asphaltenes. 16–19 The CG model is parametrized by fitting the boiling points, obtained with a more detailed united-atom force field that has been thoroughly tested to polycyclic aromatic hydrocarbons (PAHs) and alkanes in previous work. 20–22 Using a CG model, instead of the more detailed united atoms model, necessarily results in a trade-off between computing time and transferability of the model, that requires additional testing before being reliably applied to conditions outside the range used during the parametrization process and for predicting other properties than the boiling point. The second goal of this work consists of determining the dependence of the boiling point and of the thermodynamic properties at this state point on the length of the alkyl chain of the model asphaltenes. We carry out this task by extending to the case of model asphaltenes a simulation method based on a Wang-Landau sampling 23–28 in the isothermal-isobaric (NPT) ensemble. This approach takes advantage of hybrid Monte-Carlo simulations and has the particular advantage of avoiding the insertion and deletion steps that are commonly used in methods such as


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clude VLE data in its parametrization set. Here, since we focus on the prediction of boiling points of alkyl-dipyrenes, we reparametrize the CG model using as input parameters the boiling points for smaller molecules (pyrene and alkane chains), obtained either experimentally or from simulations using united atoms models. The CG model defined and used in this work can be quickly summarized through the scheme shown, in the case of BDP, in Fig. 2 and is detailed in the next paragraph. Specifically, we model each of the pyrene groups with 4 Lennard-Jones interaction sites that are held together by bonded interactions (stretching and bending interactions), with an improper dihedral angle potential ensuring that the pyrene group stays planar. The functional forms as well as the parameters for these bonded interactions are given in Table 2. Table 1: Molecular structure for the compounds studied in this work. Compound

Formula

Molecular weight

1, 1′ − (1, 4 − Butanediyl)dipyrene

C36 H26

458

1, 1′ − (1, 8 − Octanediyl)dipyrene

C40 H34

514

1, 1′ − (1, 12 − Dodecanediyl)dipyrene

C44 H42

570

1, 1′ − (1, 16 − Hexadecanediyl)dipyrene

C48 H50

626

C52 H58

682

1, 1′ − (1, 20 − Eicosanediyl)dipyrene

Molecular structure

To determine the parameters for the Lennard-Jones potential, we first carry out a series of HMC-WL simulations on pyrene with a united-atom potential that has been extensively tested on PAHs. This provides a set of simulation data (density of the coexisting phases, vapor pressure and enthalpy of vaporization) that serves as a reference data set to fit the parameters for the coarsegrained Lennard-Jones (CG LJ) sites. Since we are particularly interested in the determination of the boiling point, we emphasize data points in the vicinity of the boiling point for pyrene in the reference data set used to fit the coarse-grained model. The parameters for the CG LJ sites that match the pyrene reference data set best are provided in Table 3. Experiments indicate a temperature range of 653 − 693 K for the boiling point of pyrene. 33 Turning to the predictions of the united atoms model, for instance at T = 700 K, we find a vapor pressure of 0.958 bar, and 5 ACS Paragon Plus Environment

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densities at coexistence of 0.980 g/cm3 for the liquid and of 3.6 × 10−3 g/cm3 for the vapor. For

the CG model, we obtain a vapor pressure of 1.144 bar, and densities at coexistence of 0.977 g/cm3

for the liquid and of 6.9 × 10−3 g/cm3 for the vapor. This leads us to obtain a boiling point for the CG model for pyrene of about 690 ± 10 K, in reasonable agreement with the experimental

range 33 of 653 − 693 K, and slightly above the correlation data (666 K) of White. 34 We add that the exclusion diameter σ for the CG sites exceeds the stacking distance between polyaromatic

molecules, leading to an only qualitative account of aggregation in such systems. This limitation directly stems from the definition chosen for the coarse-grained model, which only includes 4 isotropic LJ sites. Then, to obtain a coarse-grained model for the alkyl chain, we adapt the approach followed for the MARTINI force field and lump the 4 methylene groups into a single CG LJ site. The parameters for this CG LJ site are then obtained by matching the phase equilibria data for n-butane and for longer n-alkanes, where n is a multiple of 4, resulting in the set of parameters given in Table 3. These CG parameters give a correct account of the increase in the boiling point with chain length, leading to simulated boiling points that within 30 K of the experimental data. 35 The CG LJ sites for the alkyl chain are then connected together through a series of stretching and bending interactions with the parameters also provided in Table 3. Before turning to the prediction of boiling points for the alkyl-dipyrene compounds studied here, we carry out an additional test on the ability of the model to predict the boiling point for butyl-pyrene, for which correlation data is available. 34 Using the CG model, we obtain a boiling point for butyl-pyrene of 770 ± 20 K, as

compared to the correlation data 34 of 726 K. The deviation from the correlation data for butylpyrene is of about 40 K. This is consistent with what we observed for pyrene, both for the united atoms and CG model, for which the simulations predicted a boiling point near the top of the experimental range, and 30 − 40 K above the correlation data.


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Table 2: Bonded Interaction parameters for the Coarse-Grained model. Vbond (ri j ) = 12 kstretch (ri j − rstretch )2 kstretch (K/Å2 ) CG Pyrene 296500 CG Alkyl 150000 1 Vbend (θ ) = 2 kbend (cosθ − cosθ0 )2 kbend (K/rad 2 ) CG Pyrene 62500 CG Alkyl 12500 Vimp (ψ ) = 21 kimp ψ 2 kimp (K/rad 2 ) CG Pyrene 650000

rstretch (Å) 2.43 3.15

θ0 (deg) 60 180

Table 3: Coarse-Grained LJ parameters (the Lorentz-Berthelot combining rules are used to obtain parameters for the interactions between unlike sites).

σ (Å) ε (K) CG Pyrene 5.05 190.0 CG Alkyl 4.97 324.6

Molecular weight 50.5 56.0

Figure 2: CG model for BDP


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Wang-Landau simulations framework The use of Wang-Landau simulations 23,24 has been shown to give very good results for the determination of thermodynamics data, e.g. the coexistence properties at the liquid-vapor equilibria, surface tension, critical properties, and boiling points for a wide range of systems. 25,28,36–38 This type of simulation method is part of the flat histogram techniques, 26,27,39 that sample evenly all the configurations of a system along one or more order parameters. It differs from conventional Monte-Carlo simulations in the sense that a uniform sampling is achieved even if a phase transition occurs during the simulation. This method relies on using a biasing function to perform this uniform sampling. In this work, we take advantage of the Wang-Landau method, in which an iterative scheme is implemented to determine this biasing function during the course of the simulation. Since we are interested in the coexistence properties at the vapor-liquid transition, the natural choice for the order parameter is therefore the density, and, in practice, the simulations sample all configurations for the system for different volumes V . The idea is thus to perform a random walk along V , by dividing the range of volumes of interest into small volume bins. Within a Monte-Carlo framework, the system jumps from a old volume bin (o) to a new volume bin (n) according to the following acceptance criteria

acc (o → n) = min

"

# N exp − U(Γn ) V n kB T Q(N,Vo ,T ) . 1, Q(N,V n ,T ) V N exp − U(Γo ) o

(1)

kB T

where Q(N,V, T is the biasing function (canonical partition function), and U is the potential energy, calculated according to the CG model. To improve further the sampling of high-density configurations in a particular volume bin, 28 we perform MC moves based on short molecular dynamics trajectories, that allows for a concerted move of all the molecules in the system, as well as the relaxation of the alkyl chain. The probability of accepting a MC move based on a MD trajectory from an old configuration (o) to a new


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configuration (n) inside a given volume bin is then given as

acc (o → n) = min

h

i

1, exp(− (E(n)−E(o)) ) kB T

(2)

in which E is the sum of the potential energy U and of the kinetic energy K. As can be seen from Eq. 1, the uniform sampling along V implies that the biasing function is the canonical partition function of the system. At the end of the simulation, we are then able to obtain an estimate of Q(N,V, T ) and Q(N, P, T ), which in turn, allow us to evaluate the volume distribution p(V ) as p(V ) = where Q(N, P, T ) =

R∞ 0

Q(N,V, T ) exp (−PV /kB T ) Q(N, P, T )

(3)

Q(N,V, T ) exp(−PV /kB T )dV is the isothermal-isobaric partition function.

We finally recall that the pressure at coexistence is determined when the two probabilities for the liquid and gas phases (or areas below the corresponding curves) are equal. From there, it simply follows that the densities at coexistence are calculated as

ρliq =

R Vb N N PcoexV dV 0 V V Q(N,V,T ) exp − kB T R Vb PcoexV 0

ρvap =

Q(N,V,T ) exp −

kB T

dV

(4)

R∞ N N PcoexV dV Vb V V Q(N,V,T ) exp − kB T R∞ PcoexV Vb Q(N,V,T ) exp

−

kB T

dV

Simulation details HMC-WL simulations 28,40 are performed on systems of 100 molecules for all compounds studied in this work. We choose to sample volumes corresponding to densities from 1 × 10−8 g/cm3 to

1.4 g/cm3 for BDP, ODP, DDP, HDP and EDP. We follow the same method as in prior work 20–22,41

and collect two histograms. The first histograms allows for the determination of the canonical partition function Q(N,V, T ). The second histogram, H(V ), monitors the number of visits for each of the 1000 volume bins sampled during the simulation. Since the Wang-Landau scheme is


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an iterative scheme, we need to introduce a convergence factor f which is used in the updating process for the different histograms. At the beginning of the simulation, we set ln f = 1. When a given volume interval or bin is visited, Q(N,V, T ) is multiplied by f and the histogram for the number of visits H(V ) is updated accordingly (incremented by one unit). Once all volume bins √ have been visited at least 500 times, f is reduced to f and H(V ) is reset to 0 for all volume bins. This procedure is repeated until ln f = 10−5 . For all systems, we implement the HMC-WL method within the Monte Carlo framework. The different types of MC moves are the following. The first type of MC move consists of short MD trajectories, integrated with a multiple time step integrator 22,28 over 10 time steps of 1.2 f s for the nonbonded interactions (a shorter time step of 0.012 f s is used for the bonded interactions). This type of move, which improves the sampling of the configurations of the system and favors chain relaxations, account for 25 % of the random moves. The other random moves were the random translation of a single molecule (37 %), the random rotation of a single molecule (37 %) and random volume changes of the simulation cell (1 %). Finally, for the calculation of the interaction energy, we use a spherical cutoff radius for the CG LJ interactions, set to 15.15 Å (i.e. 3 times the largest σ ), and use the conventional long-range corrections to the energy beyond this cutoff. 42

Results and discussions We start by discussing the results obtained for the vapor pressure of the compounds studied in this work. The HMC-WL simulations consist of sampling the whole volume range covering the densities for the coexisting liquid and vapor phases through volume changes of the system. We show in Fig. 3 snapshots of the coarse-grained model used for BDP obtained at different densities during the HMC-WL simulations. These indicate how the expansion of the system impacts its structure and thus its overall properties, leading to the determination of the partition functions and of the volume distribution given in Eq. 3. The resulting volume distribution for BDP at T = 800 K is shown in Fig. 4. This plot displays


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(a)

(b)

Figure 3: Snapshots taken during the HMC-WL simulations. (a) shows a typical liquid configuration of BDP (ρ = 1.37 g/cm3 ) and (b) a configuration for the vapor (ρ = 0.074 g/cm3 ). the probability for the system to be in a given volume interval and thus characterizes the likelihood of the system to exhibit a given density. More specifically, Fig. 4 shows two peaks which correspond to the two phases that exist at coexistence for this system. Through Eq. 3, we are able to determine accurately the pressure at coexistence by fine-tuning the value required to obtain equal probability for the two phases, or equivalently the same area under each of the two peaks. Repeating this process at different temperatures for BDP allows us to obtain the temperature dependence of the vapor pressure for this compound (we proceed similarly for each of the other compounds, ODP, DDP, HDP and EDP). We report in Fig. 5 the simulation results for the 5 compounds for vapor pressures around 1 bar. The vapor pressure is shown to exhibit the expected behavior as it increases with temperature. Furthermore the results indicate that for a given temperature, the vapor pressure is a monotonic function of the length of the alkyl chain (n) between the two pyrene groups, with a steady decrease of vapor pressure with n. This trend is consistent with what is observed e.g. for simpler systems like alkane chains. For instance at T = 463.15 K, the vapor pressure of n-hexadecane is only around 52 mmHg which is much lower than 892 mmHg measured for n-decane. 35 Next, we set out to determine the standard boiling point for the alkyl-dipyrene compounds. For this purpose, we fit the simulation results to the Antoine equation for each of the systems and plot the resulting fits in 11 ACS Paragon Plus Environment

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Figure 4: Determination of the pressure at coexistence for BDP and T = 800 K. The plot shows the two peaks, of equal area associated with the vapor phase (on the left) and the saturated liquid (on the right). Fig. 6, and the fitting parameters in Table 4. We then determine the boiling point by identifying the temperature at which the Antoine equation crosses the 1 bar line for each compound. Table 4: Coefficients for the fits to Antoine’s equation: log10 P = A − B/(C + T ), with P in Pa and T in degree Celsius. Compound BDP ODP DDP HDP EDP

A B C 7.50464 2951.9 153.95 9.84886 6773.7 319.843 10.4286 7982.36 319.677 10.4902 8273.12 313.932 10.8941 9849.23 390.607

We gather in Fig. 6(a) the boiling points obtained for all systems and plot them as a function of the alkyl chain length. We find the boiling temperature to increase steadily with n. More specifically, we observe that TB increases by roughly 60 K as the chain length gets longer by 4 methylene groups, prompting us to perform a linear fit to the boiling point data. This allows us to provide an estimate of the boiling temperature (in K) as a function of n through the following


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Figure 5: Vapor pressure for BDP, ODP, DDP, HDP and EDP. Simulation results are shown as filled circles (error bars are smaller than the symbol size), while the dashed lines are fits to the Antoine equation. correlation TB (n) = 1230(±12) + 15.7(±0.9) × n

(5)

The increase in TB with n is consistent with what is observed experimentally on other systems. For instance, Altgelt and Boduszynski 43 developed a correlation between the molecular weight and the boiling point in crude oil and found an incremental change in boiling point in good agreement with the slope obtained here for alkyl-dipyrenes. Furthermore, considering the case of n-alkanes, the experimental data 35 shows that TB increases by 91 K from C8 to C12 and 72 K from C12 to C16 . The smaller increase in TB with n observed for the dipyrene compounds when compared to n-alkanes can be accounted for by their much greater molecular weight since it is expected that adding 4 methylene groups to a dipyrene compound will have a much reduced impact than e.g. for n-octane. The very large absolute values for the boiling points, when compared to the correlation data obtained on crude oils, 43 emphasize the predominant role played by the polycyclic aromatic part of the molecules, through its two pyrene groups, since polycyclic aromatic hydrocarbons are expected to have a much larger boiling point than alkanes of identical molecular weight. We


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finally add that, while the difference between the boiling points of two successive alkyl-dipyrene decreases as we go from BDP to HDP, it picks up as the chain becomes larger (see between HDP and EDP). This likely results from the increased ability of the molecule to fold and have the pyrene groups undergo additional π − π stacking.

(a)

(b)

Figure 6: Variation of the standard boiling temperature (a) and of the enthalpy of vaporization (b) as a function of the length (n) of the alkyl chain between the two pyrene groups. To refine our analysis of the thermodynamic properties at the boiling point, we now turn to the results obtained for the enthalpy of vaporization ∆Hv . Fig. 6(b) shows the variation of ∆Hv as a function of n, the length of the alkyl chain. In line with the results obtained for the boiling temperature, we observe a steady increase in ∆Hv with the chain length. Carrying out a linear regression analysis of the simulation results leads us to the following law for ∆Hv (in kJ/mol) ∆Hv (n) = 91(±5) + 3.0(±0.4) × n

(6)

The behavior observed here is consistent with the experimental data on n-alkanes, which also exhibit a steady increase in ∆Hv at the boiling point as the chain length increases, 35 from ∆Hv = 34.6 kJ/mol for n-octane, to ∆Hv = 43.63 kJ/mol for n-dodecane and finally up to ∆Hv = 57.67 kJ/mol for n-eicosane. Furthermore, we find a slope for ∆Hv for the dipyrene series that is 14 ACS Paragon Plus Environment

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also consistent with that for the n-alkane (roughly 2 to 2.5 kJ/mol per methylene group added to the chain). Table 5: Thermodynamic properties at the boiling point (uncertainties are of the order of 10 K for TB , 5 kJ./mol for ∆Hv and 0.02 g/cm3 for ρl ). TB (K) BDP 1298 ODP 1362 DDP 1424 HDP 1466 EDP 1554

∆Hv ρl (kJ/mol) (g/cm3 ) 105 0.99 116 0.97 125 0.92 132 0.89 156 0.84

We gather the data for the thermodynamic properties at the boiling point in Table 5, together with the density of the liquid phase under those conditions. As a result of the presence of the pyrene groups, we find fairly large densities for the saturated liquid at the boiling point, when compared to n-alkanes. We also observe that the density of the liquid is found to decrease with the alkyl chain length, a feature that is consistent with the experimental data for n-alkanes that show a steady decrease in the density of the saturated liquid at the boiling point 35 (e.g. 0.61 g/cm3 for n-octane, as compared to 0.59 g/cm3 for n-dodecane and finally 0.57 g/cm3 for n-hexadecane). The large values obtained for the enthalpy of vaporization shown in Table 5 emphasize further the strong cohesion that exists in the condensed phases for asphaltene compounds, which is a key factor in their tendency to aggregate and precipitate.

Conclusions In this work, we focus on determining the thermodynamic properties of a series of alkyldipyrene compounds, that are often used as model compounds for asphaltenes. For this purpose, we develop a simulation framework, encompassing the parametrization of a coarse-grained model for these systems and with the extension of the Hybrid Monte Carlo-Wang-Landau method to these complex molecules, to shed light on the impact of the molecular structure on the thermodynamic properties at the vapor-liquid coexistence. In line with recent work in the field, 10 we build the 15 ACS Paragon Plus Environment

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coarse-grained model for alkyldipyrene compounds in the spirit of the MARTINI force field. 32 Since we focus here on predicting the conditions for phase coexistence, we include as input data, during the parameterization of the coarse-grained model, vapor-liquid equilibrium results, obtained with well-established united atoms models, for pyrene and n-alkanes. Then, using the coarsegrained model for alkyldipyrene compounds in the HMC-WL simulation approach, we determine the vapor-liquid equilibria for these compounds. This allows us to obtain a full picture for the dependence of the properties of the system at the boiling point, and to identify a quasi-linear relation of the boiling temperature on the length of the alkyl chain connecting to the 2 pyrene groups, in agreement with the incremental change in boiling point with the alkyl chain length given by the correlation of Altgelt and Boduszynski. 43 The framework developed in this work can be readily applied to other families of model compounds for asphaltenes, including systems with asymmetrical ring structures, archipelago (cross-linked) structures and island structures. 44 The HMC-WL method provides a way to predict phase-equilibria data for these systems, a key parameter in the determination of the asphaltene-precipitation envelope. The study of other closely related properties, including the solubility of these compounds, as well as their aggregation behavior will be the focus of future work.

Acknowledgement Acknowledgement is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research through grant 548002-ND10.

References (1) Akbarzadeh, K.; Hammami, A.; Kharrat, A.; Zhang, D.; Allenson, S.; Creek, J.; Kabir, S.; Jamaluddin, A.; Marshall, A. G.; Rodgers, R. P. et al. Asphaltenes - problematic but rich in potential. Oilfield Rev. 2007, 19, 22–43.


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Coarse-Grained Model and Boiling Point Prediction for Asphaltene

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