Asphaltene Aggregation: A Molecular Thermodynamic Approach

J. H. Pacheco-Sánchez, I. P. Zaragoza, and J. M. Martínez-Magadán ... Luis Alberto Alcazar-Vara , Jorge Alberto Garcia-Martinez , Eduardo Buenrostr...
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Langmuir 2002, 18, 1928-1937

Asphaltene Aggregation: A Molecular Thermodynamic Approach Estrella Rogel* Departamento de Produccio´ n, PDVSA-INTEVEP, Apartado 76343 Caracas-1070A, Venezuela Received June 21, 2001. In Final Form: November 16, 2001 The aggregation behavior of asphaltenes in apolar solvents is studied using a molecular thermodynamic approach. The theory is based on a molecular model for asphaltene aggregates that describes them as aromatic cores, composed of stacked aromatic sheets, surrounded by aliphatic chains. Using this simple molecular model, an analytical expression is developed for the free energy of aggregation that incorporates five contributions due to (1) transfer of the polyaromatic rings from the solvent into the aromatic core, (2) mixing of the aliphatic chains with the solvent, (3) deformation of the aliphatic chains, (4) steric repulsion among the aliphatic chains, and (5) aggregate core-solvent interactions. The proposed approach provides a qualitative description of the main experimental trends observed for asphaltene aggregation. Specifically, the experimentally observed variation of cmc values and aggregate size with (1) asphaltene molecular characteristics, (2) asphaltene concentration, (3) solvent composition, and (4) temperature has been qualitatively reproduced by the theory. In addition, the thermodynamic molecular model developed does not utilize any information derived from experiments on asphaltene solutions, and therefore, it is strictly predictive.

Introduction Asphaltenes are a well-known problem for the oil industry. Their tendency to form deposits on well sites, tubing, and piping, as well as during refining processes causes heavy losses to the oil industry every year.1 Asphaltenes are the heaviest fraction of crude oils. They are complex mixtures of molecules composed of polyaromatic condensed rings, aliphatic chains, and heteroatoms such as nitrogen, oxygen, sulfur, and various metals.2 Asphaltenes are defined by their insolubility in low-boiling alkanes such as n-pentane or n-heptane and their solubility in toluene or benzene.3 The tendency of asphaltenes to form deposits is closely related to their capacity to self-aggregate. In fact, according to numerous investigators, the first step in the formation of precipitated asphaltene particles is the self-aggregation of asphaltenes to form small aggregates or pseudomicelles.4-8 This phenomenon occurs in different organic solvents, and it has been considered analogous to the formation of reverse micelles of surfactants in nonpolar solvents.9 Even though values for critical micelle concentrations (cmc’s) have been reported in the literature for asphaltenes using different techniques,4,7,8,10,11 there are some doubts about the existence of a cmc similar to the * Present address: 15315 SW 78th Court, Miami, FL 331572363. E-mail: [email protected]. (1) Thawer, R.; Nicoli, D.; Dick, G. SPE Prod. Eng. 1990, Nov, 475. (2) Speight, J. G. In Asphaltenes and Asphalts 1. Developments in Petroleum Science; Yen, T. F., Chilingarian, G. V., Eds.; Elsevier: Amsterdam, 1994. (3) Speight, J. G. In Catalysis on the Energy Scene; Kaliaguine, S., Mahay, A., Eds.; Elsevier: Amsterdam, 1984. (4) Rogacheva, O. V.; Gimaev, R. N.; Gudaidullin, V. Z.; Danil’yan, T. D. Colloid J. USSR 1980, 42, 490. (5) Rao, B. M. L.; Serrano, J. E. Fuel Sci. Technol. Int. 1987, 4, 483. (6) Maruska, H. P.; Rao, B. M. L. Fuel Sci. Technol. Int. 1987, 5, 119. (7) Andersen, S. I.; Birdi, K. S. J. Colloid Interface Sci. 1991, 142, 497. (8) Andersen, S. I.; Speight, J. G. Fuel 1993, 72, 1343. (9) Thiyagarajan, P.; Hunt, J. E.; Winans, R. E.; Anderson, K. B.; Miller, J. T. Energy Fuels 1995, 9, 829. (10) Sheu, E. Y.; De Tar, M. M.; Storm, D. A.; DeCanio, S. Fuel 1992, 71, 299. (11) Rogel, E.; Leo´n, O.; Torres, G.; Espidel, J. Fuel 2000, 79, 1389.

one shown by surfactants in aqueous solutions.12,13 Nevertheless, experimental evidence confirms that the aggregation of asphaltenes in different solvents occurs even at low concentrations. Several thermodynamic models have been proposed to describe asphaltene precipitation in crude. Among them, only a few describe the structure of asphaltene micelles taking into account the colloidal nature of the crude oil.14-16 These models have improved the understanding of the mechanism of asphaltene aggregation, including the effect of resins and, rather recently, the effect of asphaltene inhibitors.17 Despite the significant progress that has resulted from these works, no theoretical approach is capable of describing asphaltene aggregation on the basis of the characteristics of the asphaltenes. To describe the differences in aggregation,11 and precipitation18-20 found for asphaltenes from different origins, the current models generally employ a process of fitting the experimental data.21 Thus, the vast majority of these models cannot provide a priori predictions of the aggregation behavior of asphaltenes starting from the molecular structure of the asphaltenes and the characteristics of the organic solvent. The main goal of the present paper is to develop a molecular thermodynamic treatment that allows for the a priori prediction of asphaltene aggregation in different solvents. To this end, a molecular aggregation model is (12) Yarranton, H. W.; Alboudwarej, H.; Jakher, R. Ind. Eng. Chem. Res. 2000, 39, 2916. (13) Andersen, S. I.; Del Rı´o, J. M.; Khvostitchenko, D.; Shakir, S.; Lira-Galeana, C. Langmuir 2001, 17, 307. (14) Leontaritis, K. J.; Mansoori, G. A. Presented at the SPE International Symposium on Oilfield Chemistry, San Antonio, TX, Feb 4-6, 1987; Paper SPE 16258. (15) Victorov, A. I.; Firoozabadi, A. AIChE J. 1996, 42, 1753. (16) Pan, H.; Firoozabadi, A. SPE Prod. Facil. 1998, May, 118. (17) Pan, H.; Firoozabadi, A. AIChE J. 2000, 46, 416. (18) Carbognani, L.; Orea, M.; Fonseca, M. Energy Fuels 1999, 13, 351. (19) Kaminski, T. J.; Fogler, H. S.; Wolf, N.; Wattana, P.; Mairal, A. Energy Fuels 2000, 14, 25. (20) Rogel, E.; Leo´n, O.; Espidel, J.; Gonza´lez, J. SPE Prod. Facil., in press. (21) , Andersen, S. I.; Speight, J. G. J. Pet. Sci. Eng. 1999, 22, 53.

10.1021/la0109415 CCC: $22.00 © 2002 American Chemical Society Published on Web 02/08/2002

Asphaltene Aggregation

developed to describe the aggregation behavior of different molecules that can be considered as representative of the asphaltene fraction. The theoretical framework is based on the well-known aggregation model developed almost 40 years ago: the asphaltene aggregates are composed of an aromatic core surrounded by aliphatic chains.22 In this accepted view of asphaltene aggregates, the attraction of molecules is driven by the π-π interactions between the aromatic plates of the asphaltene molecules.23,24 The aggregation of the molecules is limited by the steric repulsion of the aliphatic chains surrounding the aromatic core of the asphaltene aggregates.25 These two factors are included in the molecular thermodynamic approach, which also incorporates the effects of solvent properties and asphaltene characteristics on aggregation and the interfacial effects due to the asphaltene aggregate-solvent interface. The proposed molecular model can qualitatively predict the critical micellar concentration, the aggregate size distribution, and the apparent molecular weight of asphaltenes as a function of the (1) asphaltene concentration, (2) temperature, and (3) solvent composition. Micellization Model General Aspects. Surfactant self-assembly in aqueous solution has been well-described using molecular thermodynamic approaches.26,27 Although the behaviors of aqueous and nonaqueous systems are very different, they can be described using a common thermodynamic approach.28 In fact, using the theories previously developed for aqueous solutions, the solution behavior of surfactants in nonaqueous media28 and, particularly, in ethylene glycol29 has been successfully explored. In the present paper, a similar approach is used to study the aggregation behavior of asphaltenes in nonpolar solvents using theories developed earlier for surfactants in aqueous solutions. On the basis of X-ray diffraction studies,22,30-31 asphaltene aggregates are currently described as aromatic cores surrounded by aliphatic chains.32,33 According to this model, the aromatic core is composed of aromatic sheets stacked, one above another, at repeat distances of 3.53.7 Å. The attraction of the asphaltene molecules is attributed to the π-π interactions between the aromatic structures that compose asphaltenes.23,24 However, evidence has also been found of H-bond interactions between asphaltene molecules.34-39 On the other hand, the aliphatic (22) Yen, T.; Erdman, J.; Saraceno, S. Anal. Chem. 1962, 34, 694. (23) Wong, G. K.; Yen, T. F. J. Pet. Sci. Eng. 2000, 28, 55. (24) Syunyaev, R. Z.; Safieva, R. Z.; Safin, R. R. J. Pet. Sci. Eng. 2000, 26, 31. (25) Fenistein, D.; Barre´, L. Fuel 2001, 80, 283. (26) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710. (27) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (28) Ruckenstein, E.; Nagarajan, R. J. Phys. Chem. 1980, 84, 1349. (29) Nagarajan, R.; Wang, C. C. J. Colloid Interface Sci. 1996, 178, 471. (30) Yen, T. F.; Erdman, J. G.; Pollack, S. S. Anal. Chem. 1961, 33, 1587. (31) Sadeghi, M. A.; Chilingarian, G. V.; Yen, T. F. Energy Sources 1986, 8, 99. (32) Speight, J. G. The Chemistry and Technology of Petroleum; Marcel Dekker: New York, 1991. (33) Whitehead, E. V. In Asphaltenes and Asphalts 1. Developments in Petroleum Science; Yen, T. F., Chilingarian, G. V., Eds.; Elsevier: Amsterdam, 1994. (34) Moschopedis, S. E.; Speight, J. G. Fuel 1976, 55, 187. (35) Ignasiak, T.; Strausz, O. P.; Montgomery, D. S. Fuel 1977, 56, 359. (36) Taylor, S. R.; Li, N. C. Fuel 1978, 57, 117. (37) Speight, J. G.; Moschopedis, S. E. ACS Prepr. Div. Pet. Chem. 1981, 16, 907. (38) Monin, J. C.; Vignat, A. Rev. Inst. Fr. Pet. 1984, 39, 821. (39) Acevedo, S.; Leon, O.; Rivas, H.; Marquez, H.; Escobar, G.; Gutierrez, L. ACS Prepr. Div. Pet. Chem. 1987, 32, 426. 23.

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Figure 1. Schematic representation of (a) an asphaltene molecule and (b) an asphaltene aggregate.

chains induce steric repulsion between the aromatic sheets and prevent the flocculation of the aggregates. The balance between attractive and repulsive interactions has been suggested as the origin of the complex colloidal behavior exhibited by asphaltenes.25 Different techniques have been used to evaluate the colloidal state of asphaltene solutions and crude oils. The results obtained have shown that the colloidal behavior of asphaltenes can vary strongly depending on the solvent,40,41 the temperature,4,9,42 the origin of the asphaltene sample,43,44 and the asphaltene concentration.12,42,45 In particular, the size and shape of the aggregates under different conditions have been explored using small-angle X-ray and neutron scattering data (SAXS and SANS, respectively).9,41-47 In these experiments, the ratio of gyration of the asphaltene particles is around 40 Å, independent of the model used to fit the scattering data, although values as high as 252 Å have been reported.25 In contrast, the shape of the asphaltene aggregates is still an open question. Spheres,48 disks,49 and rodlike particles9 can fit the experimental scattering data. It seems that the shape of the asphaltene particles cannot be determined unambiguously using these techniques.41,50 The complex geometry of the aggregates and the high polydispersity of the aggregates and asphaltene molecules might be the main causes of the lower sensitivity of these techniques to the shape of the asphaltene aggregates.44 In the present work, to simplify the calculations, the asphaltene particles are considered as rodlike particles generated by the stacking of the asphaltene molecules. A scheme of an asphaltene aggregate can be seen in Figure 1. In the same figure, the geometric characteristics of a typical asphaltene molecule are also shown. As it was stated before, the asphaltene molecules will be considered as composed of two regions: aliphatic and aromatic. The aromatic region contains a polynuclear aromatic moiety like the ones shown in Figure 2. The length of the aliphatic chains varies from 6 to 12 carbon atoms. Thus, a molecule asphaltene NiCj is composed of a polyaromatic nucleus Ni, with i ranging from 1 to 7 (40) Sheu, E. Y.; Storm, D. A.; De Tar, M. M. J. Non-Cryst. Solids 1991, 131-133, 341-347. (41) Espinat, D.; Ravey, J. C. Presented at the SPE International Symposium on Oilfield Chemistry, New Orleans, LA, Mar 2-5, 1993; Paper SPE 25187. (42) Speight, J. G.; Wernick, D. L.; Gould, K. A.; Overfield, R. E.; Rao, B. M. L.; Savage, D. W. Rev. Inst. Fr. Pet. 1985, 40, 51. (43) Szewczyk, V.; Be´har, F.; Be´har, E.; Scarsella, M. Rev. Inst. Fr. Pet. 1996, 51, 575. (44) Barre´, L.; Espinat, D.; Rosenberg, E.; Scarsella, M. Rev. Inst. Fr. Pet. 1997, 52, 161. (45) Sirota, E. B. Pet. Sci. Technol. 1998, 16, 415. (46) Storm, D. A.; Sheu, E. Y.; De Tar, M. M.; Barresi, R. J. Energy Fuels 1994, 8, 567. (47) Savvidis, T. G.; Fenistein, D.; Barre´, L.; Behar, E. AIChE J. 2001, 47, 206. (48) Storm, D. A.; Sheu, E. Y.; De Tar, M. M. Fuel 1993, 72, 977. (49) Ravey, J. C.; Ducouret, G.; Espinat, D. Fuel 1988, 67, 1560. (50) Storm, D. A.; Sheu, E. Y. In Asphaltenes and Asphalts 1. Developments in Petroleum Science; Yen, T. F., Chilingarian, G. V., Eds.; Elsevier: Amsterdam, 1994.

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(3) deformation of the aliphatic chains, (4) steric interactions among aliphatic chains, and (5) creation of an aggregate core-solvent interface. (1) Transfer of the Aromatic Moeity from the Solvent to the Aromatic Cores. During aggregation, the aromatic regions of the asphaltene molecules are transferred from contact with the solvent to the aromatic core inside the asphaltene aggregate. This term can be calculated using data for the solubility of the polyaromatic ring in the solvent. However, experimental data for the solubility of aromatic compounds are very scarce and only available for a few aromatic compounds. Thus, the solubilities of the aromatic molecules in different solvents are estimated using the following equation, which has been successfully used to describe the solubility behavior of many solids, including aromatic compounds53,54 and also paraffins and asphaltenes55

Xa) exp[-(∆Hm/RT)(1 - T/Tm) - 1 + (Vl/Vs) - ln(Vl/Vs) - (Vl/RT)(δs- δl)2] (2)

Figure 2. Fused polyaromatic rings representative of the aromatic moeities of asphaltenes.

according to Figure 2, and various aliphatic chains Cj, where j represents the length of the aliphatic chains. In the model, the shape of the asphaltene molecules is modeled as a disk according to previous studies, which indicate that asphaltene molecules show a discotic shape with a thickness of approximately 3.7 Å.51,52 In all cases, the aromaticity (ratio of aromatic carbons to aliphatic carbons) of the asphaltene molecules is considered to be 0.60, which is quite common for asphaltenes extracted using n-heptane. Description of the Free Energy Aggregation Contributions. The free energy change associated with the asphaltene aggregation represents the free energy difference between an asphaltene molecule present in an aggregate of aggregation number N and shape Sh and one present in the singly dispersed state in the solvent26

∆G°mic(N,Sh) ) (µ°N - Nµ°1)/N

(1)

Here, µ°N represents the standard chemical potential of an asphaltene aggregate containing N asphaltene molecules, and µ°1 is the standard chemical potential of an asphaltene molecule free in the solvent. The main goal of any molecular aggregation model is to evaluate this free energy difference ∆G°mic (N,Sh). In the present work, it is supposed that the free energy difference associated with the asphaltene aggregation is composed of the following five contributions: (1) transfer of the aromatic moeity from the solvent to the aromatic cores, (2) mixing of the aliphatic chains with the solvent, (51) Herzog, P.; Thcoubar, D.; Espinat, D. Fuel 1988, 67, 745. (52) Acevedo, S.; Escobar, G.; Ranaudo, M. A.; Gutie´rrez, L. B. Fuel 1994, 73, 1807.

where ∆Hm is the enthalpy of fusion at the melting temperature Tm; Vl is the molar volume of the solute in liquid state; δl is the solubility parameter of the solute; and Vs and δs are the molar volume and solubility parameter, respectively, of the solvent. As in the case of solubility, these physical properties are not generally available for polyaromatics and must be estimated. In a homologous series of compounds, there is often a regularity of physical and chemical properties.56 However, this is not the behavior found for polyaromatics. In particular, the solubilities of polyaromatic compounds depend on three structural factors: molecular size, topology, and degree of planarity.57 This dependence on structural factors also makes it difficult to estimate the physical properties of these compounds, for instance, on the basis of group contributions. In this paper, the physical properties needed to calculate the solubility in eq 2 are estimated using the approach of molecular descriptors.58 These parameters, called WHIM (weighted holistic invariant molecular descriptors), capture the three-dimensional molecular information related to molecular size, shape, symmetry, and atomic distribution with respect to a reference system. The calculation of the molecular descriptors for each molecule is based on the program WHIM, which is available on the Web.59 The estimation of physical properties for aromatic molecules using WHIM begins with the building of the molecular structure using a molecular simulation program. Then, the optimized structure is transferred to WHIM, which uses it to calculate the molecular descriptors. In general, each molecule is characterized by 45 descriptors. Finally, the physical properties of the molecules are correlated with the minimum set of descriptors that can accurately describe each physical property. The physical properties determined using this methodology are the molar enthalpy (53) Barton, A. F. M. Handbook of Solubility Parameters and Other Cohesion Parameters; CRC Press: Boca Raton, FL, 1985. (54) Reid, R. C.; Prausnitz, J. M.; Sherwood, J. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill Book Company: New York, 1977. (55) Chung, T. H. Presented at the 67th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Washington, D.C., Oct 4-7, 1992; Paper SPE 24851. (56) Kemp, D. S.; Vellacio, F. Organic Chemistry; Worth Publishers: New York, 1980. (57) Zander, M. Fuel 1987, 66, 1459. (58) Todeschini, R.; Gramatica, P.; Provenzani, R.; Marengo, E. Chemom. Intell. Lab. Syst. 1995, 27, 221. (59) See www.disat.unimib.it.

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of fusion ∆Hm, the molecular volume v, the molar volume of the solid Vsol, the molar volume of the liquid Vl, the melting point Tm, the boiling point Tb, and the molar enthalpy of vaporization ∆Hv at temperature Tb. The last two variables are used to calculate the solubility parameter according to53

δl )

[(∆Hv298

0.5

- RT)/Vl]

(13)

The molar enthalpy of vaporization at 298 K ∆Hv298 was calculated from the enthalpy of vaporization ∆Hv at temperature Tb using the Watson correlation.54 A set of 31 aromatic molecules was used to establish the correlations between physical properties and molecular descriptors. The size of these molecules varies from one to seven aromatic rings. When the correlations found are reapplied to the original data set, the relative average error in the determination of the physical properties varies from 0.6% for molecular volume to almost 15% for melting point. Thus, one might anticipate that the error might be even greater when the correlations are applied to molecules not in the original data set such as those in the Figure 2. The free energy associated with the transfer of the aromatic moeties of the asphaltene molecules from the solvent into the aromatic core is given by

∆Gtrans/RT ) -ln Xa

(4)

For surfactants in aqueous solutions, a free energy contribution is typically included to take into account the deformation of the surfactant tails inside the hydrophobic core. This is so because one end of the surfactant tail in the aggregate is constrained to remain at the aggregatewater interface, while the entire tail has to assume a conformation consistent with the maintenance of a uniform density equal to that of liquid hydrocarbon in the aggregate core.27 In previous asphaltene aggregation models, a similar term was added for asphaltene molecules to take into account the transfer of the whole asphaltene molecule.15-17 In the present work, this term was neglected because only the aromatic moeties are transferred and the aromatic core behaves as a solid. Additionally, molecular simulations of asphaltene aggregates indicate that the stabilization energies obtained for the aggregates are almost completely due to the interaction energy between the molecules in the aggregates and are not caused by conformational changes of the molecules upon aggregation.60,61 (2) Mixing of the Aliphatic Chains with the Solvent. This contribution arises from the transfer of the isolated aliphatic chain to the aggregate shell. It is calculated using the mean-field theory of Flory62 according to an approach developed earlier for surfactants in aqueous solution.27 In this approach, the volume fraction occupied by the aliphatic chains in the shell is independent of the radial coordinate and is given by

φchain ) NchainNcVCH2/Vshell

(5)

where Nc is the number of carbon atoms in the aliphatic chain; Nchain is the number of aliphatic chains in the shell; VCH2 is the volume of a methylene group; and Vshell is the total volume of the shell region, which depends on the (60) Brandt, H. C. A.; Hendriks, E. M.; Michels, M. A. J.; Visser, F. J. Phys. Chem. 1995, 99, 10430. (61) Rogel, E. Energy Fuels 2000, 14, 566. (62) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, New York, 1962.

aggregate shape and on the thickness D of the region. For a cylinder, Vshell is

Vshell ) Vcore[(1 + D/Rc)2 - 1]

(6)

where Vcore is the aromatic core volume and Rc is the aromatic core radius. This radius is similar to the radius of the polyaromatic ring because highly symmetric pericondensed nuclei are considered. The mixing free energy is given by

∆Gmix/RT ) Ncφchain(0.5 - χchain-s)

(7)

where χchain-s is the interaction parameter

χchain-s ) Vchain/RT(δs - δchain)2

(8)

Here, δchain and Vchain are the solubility parameter and molar volume of the aliphatic chain, respectively. (3) Deformation of the Aliphatic Chains. The deformation free energy of the aliphatic chains in the shell region of the aggregate is calculated using the analytical expressions given by Nagarajan and Ruckenstein.27 For cylinders, the expression is

∆Gdef/RT) 3/2(LRc/aφchain) ln(1 + D/Rc)

(9)

where a is the area of the aromatic core per aliphatic chain and L is defined as (VCH2)1/3. (4) Steric Interactions among Aliphatic Chains. This contribution is due to the steric repulsions among the aliphatic chains at the interface between the aromatic core and the shell. The simplest way to evaluate the steric repulsion is given by26,27

∆Gest/RT ) -ln(1 - NchainACH2/A)

(10)

where ACH2 is the effective cross-sectional area of the methylene group L2 and A is the total area of the aromatic core. (5) Creation of an Aggregate Core-Solvent Interface. According to the models previously developed for surfactants in aqueous solution, the free energy associated with the formation of this interface can be calculated via the macroscopic relation26,27

∆Gint/RT ) σagg(A - NchainAp)

(11)

where σagg is the macroscopic aggregate-solvent interfacial tension. The aggregate-solvent interfacial tension σagg is a function of the surface tensions of the aromatic moeity (σarom) and of the solvent (σs). It also depends on the concentration of the aliphatic chains (φchain) in the aggregate shell, as well as on the interfacial tension due to the aliphatic-aromatic interactions. To calculate σagg, the theory developed by Prigogine63 and adapted by Nagarajan and Ruckenstein27 is used. In the present work, the shell with thickness D is considered as a solution of aliphatic chains with a uniform concentration φchain. For each value (63) Siow, K. S.; Patterson, D. J. Phys. Chem. 1973, 77, 356.

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of φchain, there is a concentration of methylene units φs in the surface given by

ln[(φs/φchain)1/Nc/(1 - φs)(1 - φchain)] ) [(σarom-s σchain-s)Vs2/3]/RT + 3/4χchain-s[(1 - φchain) - φchain] 1

/2χchain-s[(1 - φs) - φs] (12)

Using φs, the interfacial tension σagg can be calculated from

[(σagg - σarom-s)Vs2/3]/RT ) ln[(1 - φs)/(1 - φchain)] + [(Nc - 1)/Nc](φs - φchain) + χchain-s [1/2(φs)2 3

/4(φchain)2] (13)

where σarom-s and σchain-s are the interfacial tensions due to aromatic core-solvent and aliphatic chain-solvent interactions, respectively, and are given by

Figure 3. Contributions to the free energy of aggregation ∆Gmic of N4C7 molecules in benzene at 25 °C as a function of the aggregation number N.

σarom-s ) σs + σarom - 2.0ψ(σsσarom)0.5

(14)

σchain-s ) σs + σchain - 2.0ψ(σsσchain)0.5

(15)

Note that eq 17 does not include contributions from interactions between species in solution. This implies that the model cannot be applied to concentrated solutions of asphaltenes where interaggregate interactions become important. From the size distribution, it is possible to obtain the average sizes of the aggregates using the relations

In these equations, σs, σchain, and σarom are the solvent, aliphatic chain, and polyaromatic ring surface tensions, respectively. The constant ψ depends on the nature of the interactions and, for compounds of similar polarity, is very close to unity. In the present case, this constant is considered to be equal to 1. The main problem in using eqs 13 and 14 is estimating a reasonable value for σarom. In the present work, σarom is calculated using the relationship53

δ2 ) RV-1/3σ

(16)

where R is a constant that varies only between certain limits according to the type of molecule. This relationship is reasonable as long as the area per molecule is proportional to V2/3.53 For the molecules shown in Figure 2, the calculated molar volumes V and the molecular areas are proportional. The linear relationship between the two values shows a correlation coefficient equal to 0.9860. A value of 58 for the constant R has been successfully used to describe the surface tension of a wide variety of organic compounds. Using this approach, the surface tensions obtained for the seven molecules shown in Figure 2 vary from 64 to 110 mJ/m2. As a comparison, experimental values for carbon (95.67 mJ/m2),64 graphite (100-120 mJ/ m2),65 and asphaltenes (41 mJ/m2)66 are given. Computational Approach. The aggregate size distribution is given by26,27

XN ) X1N exp(-N∆G°mic/kT)

(17)

where X1 and XN are the molar fractions of monomers and aggregates of size N, respectively, and the free energy of aggregation ∆G°mic is the sum of the various contributions discussed in the previous section, namely

∆G°mic ) ∆Gtrans + ∆Gmix + ∆Gdef + ∆Gest + ∆Gint (18) (64) Glass, A. S.; Wenger, E. K. Energy Fuels 1998, 12, 152. (65) Gonzalez-Martin, M. L.; Janczuk, B.; Labajos-Broncano, L.; Bruque, J. M. Langmuir 1987, 13, 5991. (66) Papirer, E.; Kuczinski, J.; Siffert, B. Fuel 1987, 66, 1691.

Nn ) ΣNXN/ΣXN

Nw ) ΣN2XN/ΣNXN

(19)

where Nn and Nw are the number-average aggregation number and the weight-average aggregation number, respectively, with N ranging from 2 to ∞. However, the experimental aggregation numbers obtained using many different techniques depend on the total number of species in solution. Therefore, these techniques determine the apparent aggregation numbers

Nn,app ) (X1 +ΣNXN)/(X1 + ΣXN) Nw,app ) (X1 + ΣN2XN)/(X1 + ΣNXN) (20) On the other hand, many theoretical, as well as experimental approaches are available for determining the critical micelle concentration (cmc) or the concentration at which aggregation begins.27 In the present work, the cmc is estimated as the value of X1 for which the concentration of the singly dispersed asphaltene molecules is equal to that of the asphaltene molecules present in the form of aggregates

cmc ) X1 ) ΣNXN

(21)

For any asphaltene molecule, the value of ∆G°mic depends on two independent variables, the aggregation number N and the thickness D of the shell region of the aggregate. The most favorable ∆G°mic is obtained as a function of N by minimizing with respect to D. From this result, the aggregate size distribution can be obtained as a function of X1 according to eq 17, and therefore, the average aggregation numbers and cmc values can be obtained according to eqs 19-21, respectively. Results and Discussion Free Energy Contributions. Free energy curves for aggregation of the molecule N4C7 in benzene at 25 °C are shown in Figure 3. These curves are similar for all of the molecules studied, derived from the aromatic cores shown

Asphaltene Aggregation

in Figure 2. The solvents employed were benzene, toluene, cyclohexane, and n-decylbenzene. As in the case of surfactant aggregation in aqueous and nonaqueous media,28 the transfer free energy contribution is responsible for the aggregation. In other words, the low solubility of the polyaromatic moeties of the asphaltene molecules is the main driving force for asphaltene aggregation. This contribution is negative and independent of the size of the aggregates. Its magnitude significantly influences the beginning of aggregation, and the lower the solubility of the polyaromatic moeity, the lower the cmc determined according to eq 21. The other four contributions are positive and dependent on size. Therefore, they influence the beginning of aggregation, as well as the aggregate size distribution. Depending on their variations with aggregate size, these positive contributions can influence the aggregation in different ways. The interfacial free energy contribution decreases as the size of the aggregate increases, which favors the growth of the aggregates through the screening effect of the aliphatic chains. The rest of the free energy contributions due to the deformation, mixing, and steric repulsion of the aliphatic chains increase as the size of the aggregate increases, an aspect that limits the growth of the aggregates. It is also interesting to observe that, in all of the cases studied in the present work, the total free energy of aggregation favors the formation of small aggregates. Even the formation of dimers is energetically favored according to the calculated curves. In comparison, the small aggregates of surfactants are unstable in aqueous solution but stable in nonaqueous media. The formation of small aggregates is characteristic of nonaqueous media.28,29 Critical Micelle Concentration and Aggregate Size Distribution. Although asphaltene aggregates cannot strictly be considered as micelles, the term critical micelle concentration (cmc) is used in the present work because of its widespread use in the asphaltene literature. Figure 4 shows a comparison of the aggregate size distribution as a function of the total concentration of asphaltene for the systems N1C7, N4C7, and N7C7 in benzene at 25 °C. The cmc values obtained using eq 21 are also shown in this figure. A comparison of the curves reveals different behaviors for the various molecules. There are significant differences in relation to the content of small aggregates. Figure 4 shows that, for N1C7, at concentrations significantly larger than the cmc, the number of small aggregates is large compared to the number of large aggregates. As a consequence of the significant presence of small aggregates, the properties of the solution that depend on the size of the aggregates change gradually as a function of the concentration. This means that the experimental identification of a cmc for this system would be ambiguous, if not impossible. Some size-dependent quantities for N1C7, N4C7, and N7C7 are plotted in Figure 5 as functions of the total asphaltene concentration. The experimental measurements used to determine the cmc depend on the size and number of particles in solution. Therefore, these size-dependent variables can be related to physical measurements. For instance, surface tension depends on the concentration of monomers as the total concentration changes. For N1C7, the size-dependent variables show a gradual increase as a function of concentration rather than a sharp transition at some concentration, which is the operational definition of a cmc. Thus,, the experimental identification of a cmc for this system is difficult. It has been determined using interfacial tension measurements that asphaltenes from Athabasca, Cold Lake, and Cerro

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Figure 4. Calculated aggregate size distributions of (a) N1C7, (b) N4C7, and (c) N7C7 in benzene at 25 °C as a function of different monomer concentrations.

Negro bitumens do not exhibit critical micelle concentrations, although aggregation apparently occurs.12 The size distributions of the aggregates formed from N4C7 and N7C7 indicate that, for these systems, the number of smaller aggregates is small compared to the number of larger aggregates (Figure 5). For N7C7, the larger aggregates are clearly favored, and the sizedependent curves for this molecule shown in Figure 5 exhibit sharper transitions. Experimentally, this means that a cmc can be identified because the solution properties display a sharp transition in behavior. Values for the critical micelle concentration (cmc) have been reported in

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Figure 6. Predicted cmc as a function of the solubility of the aromatic moieties for the series NiC7 in toluene at 25 °C.

Figure 7. Predicted apparent molecular weight as a function of the condensation index for the series NiC7 in toluene at 25 °C (10 g/L).

Figure 5. Size-dependent quantities as functions of the total concentration of (a) N1C7, (b) N4C7, and (c) N7C7 in benzene at 25 °C.

the literature for asphaltenes from different origins using different techniques.4,7,10,11 The main difference between the molecules N1C7, N4C7, and N7C7 is the solubility of the polyaromatic moeity. The results obtained indicate that, for asphaltene molecules, a cmc might or might not be observed experimentally depending on the nature of the aromatic sheets that form the asphaltene molecules. This finding agrees with the contradictory experimental evidence indicating that some asphaltenes exhibit cmc’s4,7,10,11 whereas others do not.12 Influence of Asphaltene Structure on the Aggregation Behavior. The cmc values predicted for the series of molecules NiC7 at 25 °C in toluene are shown in

Figure 6 as a function of the solubility of the aromatic moiety. There is a clear relationship between the two values. According to numerous authors, the characteristics of the aromatic zones of the asphaltenes play a main role in the flocculation behavior of asphaltenes. In particular, the aromatic condensation index has been correlated to the tendency of asphaltenes to flocculate. A high aromatic condensation index is characteristic of asphaltenes extracted from deposits or unstable crude oils (crude oils that show asphaltene precipitation problems during production) that usually have extended aromatic nuclei.18,20,67 The sizes of the aggregates formed by soluble and insoluble fractions of an asphaltene sample determined using X-ray scattering revealed that the aggregates that flocculate first are the largest in size.43 This study also indicated that the asphaltenes from the insoluble fractions exhibited high aromatic condensation. Figure 7 shows the apparent molecular weight obtained for the series NiC7 as a function of the aromatic condensation index. The solubility of the polyaromatic rings is also presented in Figure 7 as a function of the condensation index. In general, the apparent molecular weight increases and the solubility of the polyaromatic ring decreases as the condensation index increases. This behavior is primarily a consequence of the decrease in solubility of the pol(67) Carbognani, L.; Espidel, J.; Izquierdo, A. In Asphaltenes and Asphalts, 2. Developments in Petroleum Science; Yen, T. F., Chilingarian, G. V., Eds.; Elsevier: Amsterdam, 2000.

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Figure 9. Predicted cmc as a function of the solubility of the polyaromatic ring for the series NiC7 in different apolar solvents at 25 °C.

Figure 8. (a) Predicted apparent molecular weight as a function of the concentration and (b) predicted cmc values for the series N4Cj in benzene at 25 °C.

yaromatic rings as the condensation index increases. This predicted tendency is in agreement with the experimental evidence, which shows that asphaltenes with extended aromatic condensation, and therefore lower hydrogento-carbon ratios, form bigger aggregates at the same concentration43,68 and, as a consequence, flocculate first.20,67-69 For the other solvents studied, namely, benzene, decylbenzene, and cyclohexane, similar behaviors were predicted. The dependence of the apparent molecular weight on the length of the aliphatic chains of the series N4Cj in benzene at 25 °C as a function of the concentration is presented in Figure 8a. As expected, aggregation decreases as the length of the aliphatic chain increases. The predicted cmc values are also shown in Figure 8b. The calculated cmc’s increase as the length of the aliphatic chain increases. Although there is no experimental evidence of greater aggregation due to shorter aliphatic chains in the asphaltene molecules, some studies have reported a higher tendency toward flocculation for the asphaltenes with shorter aliphatic chains.70-74 In other words, the asphaltenes with the shorter aliphatic chains flocculate first. (68) Ali, M. F.; Saleem, M. Fuel Sci. Technol. Int. 1988, 6, 541. (69) Storm, D. A.; Decanio, S. J.; Edwards, J. C.; Sheu, E. Y. Pet. Sci. Technol. 1997, 15, 77. (70) Kappor, M. P.; Kothiyal, V.; Singh, I. D. Fuel Sci. Technol. Int. 1993, 11, 975. (71) Singh, I. D.; Kothiyal, V.; Kapoor, M. P.; Ramaswamy, V.; Aloopwan, M. K. S. Fuel 1993, 72, 751. (72) Kapoor, M. P.; Singh, I. D. Fuel Sci. Technol. Int. 1995, 13, 1. (73) Fainberg, V.; Podorozhansky, M.; Hetsroni, G.; Brauch, R.; Kalchouck, H. Fuel Sci. Technol. Int. 1996, 14, 839. (74) Lemke, H. K.; Stephenson, W. K. Pet. Sci. Technol. 1998, 16, 335.

Influence of the Solvent on the Aggregation Behavior. Four different solvents were selected to study the aggregation behavior: toluene, benzene, n-decylbenzene, and cyclohexane. Figure 9 presents a plot of the calculated cmc values for the series NiC7 as a function of the solubility of the polyaromatic ring in these solvents. The cmc decreases with decreasing solubility of the polyaromatic ring in the solvent. Additionally, the cmc decreases in the following order for all of the molecules studied: benzene > toluene > cyclohexane > n-decylbenzene. This tendency is in agreement with the experimental evidence.7,11 Two solvent mixtures were also studied: toluene/nheptane and toluene/n-hexadecane. For both mixtures, experimental data are available.7,41 For the mixtures studied, the solubility parameter, surface tension, and molar volume are calculated according to

δs ) φ1δ1 + φ2δ2

(22)

σs ) φ1σ1 + φ2σ2

(23)

Vs ) x1V1 + x2V2

(24)

respectively, where φ is the volume fraction and x is the molar fraction. The subscript indices 1 and 2 represent the two components of the mixture. Predicted cmc values for the N1C7 molecule in toluene/ n-heptane and toluene/n-hexadecane mixtures are shown in Figure 10. The cmc values decrease as the volume fraction of the n-alkane in the mixture increases. This tendency is in agreement with experimental cmc values for asphaltenes measured in the two mixtures using calorimetry.7 Figure 11 presents the predicted apparent molecular weight as a function of the volume fraction of n-heptane added to a toluene solution of N4C7 (2% w/w). The aggregation increases as the volume fraction of n-heptane increases. The effect of the addition of n-heptane to a 2% w/w solution of asphaltene in toluene measured using small-angle neutron scattering (SANS) also indicates an increase in the molecular weight.41 Influence of the Temperature on the Aggregation Behavior. An analysis of the contributions to the free energy of aggregation indicates that the solubility of the polyaromatic rings is the most important factor in determining the cmc values for molecules (Figures 6, 7,

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Figure 10. Predicted cmc values for N1C7 as a function of the volume fraction of toluene in the mixtures toluene/n-heptane and toluene/n-hexadecane.

Figure 11. Effect of the addition of n-heptane on the predicted apparent molecular weight for N4C7 in a 2% w/w toluene solution at 25 °C.

and 9). Because the solubility of different polyaromatic compounds increases as the temperature increases,75-77 one might also expect that the cmc increases with the temperature. However, for a set of asphaltenes from different origins, it was found that the cmc in toluene and in aromatic fractions decrease as the temperature increases.4 This indicates that the other contributions can play an important role depending on the temperature. The free energy contributions due to the deformation, mixing, and steric repulsion of the aliphatic chains can be easily calculated as functions of the temperature using the thermodynamic data available for the aliphatic chains and solvents. The calculations indicate that the changes in these contributions with temperature do not substantially affect the cmc values. The interfacial free energy contribution depends on the surface tension of the aliphatic chains, solvent, and polyaromatic ring. The dependence of the surface tension on temperature for polyaromatic rings is not avalaible, and therefore, this contribution cannot be calculated with precision at the present time. However, it is possible to suppose a variation in the surface tension for polyaromatics similar to that shown by other smaller aromatic molecules. (75) Choi, P. B.; McLaughlin, E. AIChE J. 1983, 29, 150. (76) Coon, J. E.; Troth, M.; McLaughlin, E. J. Chem. Eng. Data 1987, 32, 233. (77) Domenska, U.; Groves, F. R.; Mclaughlin, E. J. Chem. Eng. Data 1993, 38, 88.

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Figure 12. Predicted cmc values for N1C7 and N4C7 in toluene as a function of temperature.

Judging from experimental data for smaller aromatic compounds,78 decreases of 11 and 18% in surface tension were considered for the polyaromatic rings at 50 and 70 °C, respectively. cmc values and aggregation number were estimated using this rough rule for the decrease of the surface tension of polyaromatics as the temperature increases. The temperature dependences of the cmc’s estimated for N1C7 and N4C7 molecules in toluene are plotted in Figure 12. The predicted cmc’s decrease as the temperature increases in agreement with the experimental behavior reported.4 It is important to point out that, if smaller changes in surface tension as a function of temperature are considered, the predicted cmc’s increase as the temperature increases. Thus, the model is quite sensitive to temperature changes. Unfortunately, very few experimental cmc data have been reported at different temperatures to allow for a more complete verification of the model. Several experimental techniques have provided information about the aggregation behavior of asphaltenes with temperature.9,12,42,44 These results indicate a decrease in aggregation as the temperature increases for a wide range of asphaltenes and solvents. The predicted apparent molecular weights for N1C7 and N4C7 in toluene at different temperatures as a function of concentration are shown in Figure 13a and b, respectively. These figures resemble the experimental molecular weight curves in toluene obtained by vapor pressure osmometry (VPO)12 and show a decrease of the aggregation as the temperature increases, in agreement with the experimental data. Conclusions The aggregation behavior of asphaltenes in apolar solvents was studied using a molecular thermodynamic approach. The theory developed is free of any information derived from experiments on asphaltene solutions and was used to make some qualitative predictions about the variation of the cmc value and aggregate size with (1) asphaltene molecular characteristics, (2) asphaltene concentration, (3) solvent composition, and (4) temperature. These qualitative predictions are consistent with experimentally observed trends. The theory also offers a plausible explanation for some contradictory information about asphaltene aggregation behavior previously re(78) Yaws, C. L. Thermodynamic and Physical Property Data: Comprehensive Thermodynamic and Physical Property Data; Gulf Publishing: Houston, TX, 1992.

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Figure 13. Predicted apparent molecular weights for (a) N1C7 and (b) N4C7 in toluene at different temperatures as a function of concentration.

ported such as the existence or absence of a cmc similar to the one exhibited by surfactant aqueous solutions. Judging from the model, the occurrence of such behavior (or lack thereof) for asphaltene solutions depends on the chemical and structural characteristics of each asphaltene sample.

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The theory is based on the well-known model for asphaltene aggregates that describes them as aromatic cores, composed of stacked aromatic sheets, surrounded by aliphatic chains. The attraction of asphaltene molecules is attributed to the attraction between the aromatic structures. In this model, the aliphatic chains induce steric repulsion between aromatic sheets and prevent the flocculation of the aggregates. Using this simple molecular model, an analytical expression for the free energy of aggregation was developed. The free energy of aggregation incorporates five contributions due to (1) transfer of the polyaromatic rings from the solvent into the aromatic core, (2) mixing of the aliphatic chains with the solvent, (3) deformation of the aliphatic chains, (4) steric repulsion among the aliphatic chains, and (5) aggregate core-solvent interactions. The results obtained show that the transfer free energy of the polyaromatic ring is mostly responsible for the aggregation and strongly influences the cmc. Thus,, the model predicts that the larger and more condensed the aromatic structures in the asphaltenes, the lower the cmc values and the higher the aggregation numbers. In fact, experimental reports indicate that the characteristics of the aromatic zones of asphaltenes play a main role in their flocculation behavior. The proposed molecular thermodynamics approach provides a qualitative description of the main experimental trends observed for asphaltene aggregation and can be expanded to consider other aggregate shapes, asphaltene molecular polydispersity, effects of resins, and also effects of asphaltenes in the solubilization of paraffins. Further work considering these aspects is now in progress. Acknowledgment. The author is grateful to Olga Leo´n, Lante Carbognani, and So´crates Acevedo for very useful discussions on asphaltene aggregation and molecular weight measurements. The author also thanks PDVSA-INTEVEP for financial support and for permission to publish this work LA0109415