Mesoscopic Simulation of Aggregation of Asphaltene and Resin

Energy & Fuels 2006, 20, 327-338

327

Mesoscopic Simulation of Aggregation of Asphaltene and Resin Molecules in Crude Oils B. Aguilera-Mercado,† C. Herdes,† J. Murgich,‡ and E. A. Mu¨ller*,§ Department of Chemical Engineering, Imperial College London, SW7 2AZ, U.K., Centro de Quı´mica, Instituto Venezolano de InVestigaciones Cientı´ficas IVIC, Caracas 1020A, Venezuela, and Departamento de Termodina´ mica y Feno´ menos de Transferencia, UniVersidad Simo´ n Bolı´Var, Caracas 1080A, Venezuela ReceiVed August 23, 2005. ReVised Manuscript ReceiVed October 20, 2005

A molecular model for simulating the aggregation of asphaltenes and resins in crude oils on a mesoscale is proposed. The asphaltene molecules are treated as discotic seven-center Lennard-Jonesium molecules, the resins are modeled as single spheres, and the surrounding crude oil is modeled as a continuum, characterized by a screening factor, and defined using a combination of its Hamaker and dielectric constants. The parameters for the model are obtained by coarse-graining the potential energy surface obtained from model atomistic simulations of pairs of asphaltenes and resins. Canonical Monte Carlo simulations are performed with this model, and effects of temperature, asphaltene, and resin concentration are studied parametrically. The results agree with experimentally observed tendencies. The asphaltene is seen not to conform to a linear aggregation model, but exhibits a more complex multimodal aggregation pattern. The screening constant of the crude oil, which ultimately controls the aggregation, can itself be related to other measurable quantities such as the refractive index.

1. Introduction Obtaining a detailed description of a crude oil is both an experimental and theoretical challenge, mainly due to the amazing complexity of the mixture, both from the point of view of the number and size distribution of the components and from the point of the complex interactions involved. Most physically based approaches attempt to recognize the relevant dominating effects and focus on modeling a pseudo-fluid whose rough average properties are supposedly representative of the collective. Of particular difficulty in modeling are crude oils, which present, in some cases, a significant amount of resins and asphaltenes. Some of the long-standing controversies in the literature seem to stem from the persistence of some researchers to assign to these fractions a unique generalized structure. They oversee the fact that crude oil is a natural product where despite the lower molecular weight molecules appearing in discreet compositions in all crude oils, the heavier ones, due to the larger combination of possible molecular morphologies, do not necessarily follow through. It is clear from the most recent reviews1,2 on the subject that the asphaltene fractions of crude oils are far from being well-characterized. This lack of a precise characterization reflects a limited understanding on the actual physical phenomena leading to the aggregation and posterior precipitation of some of the asphaltenes. However, it is this open problem that is particularly important in crude oil production and transport, since the economic cost associated with either reservoir, production pipe, or pipeline blockage due to asphaltenes is significant.1 It is most likely that tailor-made approaches must be considered when tackling the problem of modeling these systems. * Corresponding author. Phone: +44 20 7594 1569. Fax: +44 20 7594 5604. E-mail: [email protected]. † Universidad Simo ´ n Bolı´var. ‡ IVIC. § Imperial College London.

As mentioned above, the difficulty in modeling crude oils stems from several linked aspects that are currently unresolved: (1) the lack of proper characterization of the asphaltene molecules, (2) the large size and energetic asymmetry and the multiplicity of the components involved, and (3) the uncertainty in the precipitation mechanism of asphaltenes. Even apparently simple properties such as the molecular weight of asphaltenes are still not clearly obtained from experiments. Individual asphaltene and resin molecules may form tight aggregates, which are sometimes confused as being a single molecule.1,2 The proper definition of an asphaltene (as the fraction precipitated by the addition of an excess of n-heptane, albeit soluble in toluene) is by itself somewhat misleading. A solubility class is a very improper definition in terms of chemical composition since molecules of quite different chemical compositions may be insoluble in exactly the same solvent. Then, this definition may encompass very different molecules with the consequential confusion when they are analyzed in relation with a specific chemical or physical property. Most equations of state and engineering-type models3,4 that are used for modeling chemical or gas industry mixtures are unsuitable for modeling asphaltenes unless a relatively important amount of ad hoc variables are fitted, converting them in essence into correlation tools, which, albeit useful, add little to our physical understanding of the phase behavior. More modern equations of state are promising in overcoming these difficulties (e.g., the SAFT approach).5-8 (1) Speight, J. G. The Chemistry and Technology of Petroleum, 3rd ed.; Marcel Dekker: New York, 1998. (2) Strausz O. P.; Lown, E. M. The Chemistry of Alberta Oil Sands, Bitumens and HeaVy Oils; Alberta Energy Research Institute: Calgary, 2003. (3) Firoozabadi, A. Thermodynamics of Hydrocarbon ReserVoirs; McGrawHill: New York, 1998. (4) Danesh, A. PVT and Phase BehaViour of Petroleum ReserVoir Fluids; Elsevier: Amsterdam, 1998.

10.1021/ef050272t CCC: $33.50 © 2006 American Chemical Society Published on Web 12/09/2005

328 Energy & Fuels, Vol. 20, No. 1, 2006

The overall mechanism that governs the precipitation of asphaltenes is far from being understood, even though multiple (and generally incompatible) models have been suggested. There is some intriguing amount of work suggesting micellar models, in which the asphaltenes are considered inherently insoluble in crude oils, only to be stabilized by the presence of the appropriate surrounding resins (see, for example, ref 9). These models build on existing chemical engineering knowledge of micellar (oil/water/surfactant) systems. It is not at all clear if such an extension is physically sensible. In particular, the lack of experimental evidence of a critical micellar concentration hinders the acceptance of these models. Merino-Garcia and Andersen10 proved experimentally that the crude oils with asphaltenes do not behave as surfactant solutions because they do not form micelles but rather molecular aggregates of colloidal size. Colloidal models are nowadays generally accepted,11 although it has been strongly argued12 that the asphaltene behavior may be modeled from solution thermodynamics alone. This is an unsettled issue, which shall not be dealt with here. It suffices to say that this lack of fundamental understanding of the physical chemistry of the process and of the molecular details is the motivation for studies such as the one presented here. Molecular mechanics, understood as the modeling of one or more individual molecules detailed in an atomistic way,13 poses as a powerful theoretical alternative for the study of fluid systems such as those involving asphaltenes. A fully atomistic description of asphaltene molecules is both complex and computationally time-consuming. Despite this, studies have been performed14-21 on the equilibrium configuration and aggregation of atomistic detailed molecular models which represent individual asphaltene molecules with and without resins. Available computational power limits these studies to a few asphaltene molecules; hence, only information about particular molecular recognition aspects and formation mechanisms of aggregates can be obtained. Moreover, due to the heavy computational requirements involved, these calculations only cover very short actual times ( 0, eq 9 may be rewritten in a more convenient way

φi,m,j )

∑i ∑j 4i,j(Hi,m,j) em 2

[(

)

σi,j(Hi,m,jem)-1/6 ri,j

(

12

or in an equivalent way as

φi,m,j )

∑i ∑j

ri,j

12

-

ri,j

σi,m,j ri,j

)]

-1/6 6

[( ) ( ) ] σi,m,j

4i,m,j

3. Simulation Details

-

σi,j(Hi,m,jem)

(10)

6

(11)

where the effective size and energy parameters are now functions of the properties of the media. 2 i,m,j ) i,j(Hi,m,j)2em ) i,j f i,m,j

σi,m,j )

σi,j (Hi,m,jem)1/6

are observed for similar values of fi,m,j. However, if one is limited to a given family of species (as is the case in this work) in which there is no sharp variation of either the dielectric constant or the Hamaker constant, the system may be conveniently analyzed by use of the screening factor. For the parameter values used within this work, Hi,m,j is in the range between 0.1 and 0.4 while the values of em range from 1.5 to 2.0 for most hydrocarbons, with a notable exception for strongly polar substances.

(12) (13)

where fi,m,j ) Hi,m,jxem, called herein crude screening factor, can be used as a scaling factor for the potential energy function. For the cross parameters, the usual Lorentz-Berthelot combining rules are applied, that is, σi,j ) 0.5(σi,i + σj,j), i,j ) xi,ij,j. In practical applications with organic solvents, one would expect the Hi,m,j to be less than 1 and em larger than 1, leading to a decrease in the effective two body attractions (screening of the potential). The screening effect of em and Hi,m,j on the potential is seen in Figures 4 and 5. For comparison purposes, in both figures, the unscreened (em ) 1; Hi,m,j ) 1) LJ potential is plotted. The screening factor is also given in the caption for each curve. It is worthwhile noticing that although the screening factor is a unified measure of the deviation from the LJ potential, it cannot be taken as a unique characteristic parameter. Comparison of Figures 4 and 5 shows that different behaviors

The stable (nonprecipitating) Boscan heavy crude oils in Venezuela may contain more than 15% (w/w) of asphaltenes, while some unstable (precipitating) light Arabian oils have as low as 0.2% weight asphaltenes. Thus, as a base case, an asphaltene concentration of 2.23% (w/w) was used. The resin weight percent is, as a rule of thumb, of the same order as asphaltene weight percent, and a resin concentration of 2.8% (w/w) was used. The temperature for the base case was fixed at 25 °C. The volume of the simulation box is defined by specifying a crude oil density, fixed here to 0.83 g/mL. The molecular weights of the resins and asphaltenes were taken directly from the atomistic models, 198 and 794, respectively. We stress that these numbers are not representative of any particular system, but rather are commonly occurring values. The systems were studied by Monte Carlo simulations in the canonical ensemble, where the number of particles, the volume, and the temperature are fixed. Random translation and rotation of the molecules were performed and accepted or rejected according to well-established statistical mechanics recipes. The reader is referred to the standard references38,39 for the complete details of the method. A cubic simulation box of 19.23 nm per side with periodic boundary conditions was used. Cutoff radius was fixed at 9.6 nm. A randomly chosen molecule was randomly translated or rotated to produce a new configuration. Acceptance ratio was targeted at 50%. For the base case, 100 asphaltene molecules and 503 resin molecules were used. Minimums of 5 × 107 configurations were rejected until equilibration was attained. The energy profiles of the configurations were examined to confirm the approach to equilibrium. For the more fully aggregated systems, equilibration was much slower, and simulation runs were longer by a factor of 2 or more. Average (38) Allen, M. P.; Tildesley, D. J. The Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (39) Frenkel, D., Smit, B. Understanding Molecular Simulation, 2nd ed.; Academic Press: San Diego, CA, 2002.

Asphaltene and Resin Molecules in Crude Oils

Energy & Fuels, Vol. 20, No. 1, 2006 333

Figure 5. Effect of the dielectric constant em on the screening of the reduced intermolecular potential φi,m,j/i,j of a Lennard-Jones site immersed in a continuum fluid. From bottom to top, the curves correspond to em ) 20, 10, 5, 1.5, and 1 (corresponding to values of fi,m,j ) 1.34, 0.95, 0.67, 0.36, and 0). The Hamaker constant, Hi,m,j, is kept fixed at Hi,m,j ) 0.3. Gray line is the in vacuo LJ potential.

statistics were gathered for 5 × 107 configurations after achieving equilibration. No cluster moves or similar techniques were used. The number of clusters and their conformations were closely monitored during the simulation runs. A method similar in spirit to that of Sevick et al.40 was employed.33 Clusters were identified by recognizing that two molecules form a cluster when any two parts of the molecules were closer than a given distance, namely 21/6 σi,m,j. Note that this distance is rather arbitrary, however, and the results did not seem to be sensitive to small variations of it. For an aggregation process such as that expected with asphaltenes and resins, a definition of clusters in a dynamical sense (i.e., taking into account the time that the molecules remain bonded) is probably unnecessary due to the expected stability of the clusters. In any case, the Monte Carlo method employed precludes any observation of dynamics. To obtain quantitative results in terms of a macroscopic property (in this case the Hamaker constant of the crude oil), one must assign values to the Hamaker constants of the asphaltenes and the resins in vacuo (cf. eqs 5-7). We have treated these values as adjustable parameters, calibrating them in such a way that for a pure asphaltene (without resins) we have an onset of aggregation upon placing the asphaltene molecules in a media corresponding to pure n-heptane (Am ) 4.1 × 10-20 J, e ) 1.92) but a dispersed phase in a media corresponding to benzene (Am ) 5.0 × 10-20 J, e ) 2.28). Taking the break point to be fi,m,j ≈ 0.3197, one obtains a value of Hi,m,j ) 0.2117, which when used in conjunction with eqs 6 and 7 gives as a result Aasphaltene ) 17.15 × 10-20 J. Similarly, the resin Hamaker constant was calibrated by imposing a system of pure resins to aggregate when placed in solvents lighter than propane (or acetone) but to disperse in n-pentane. This gave as a result Aresin ) 13.25 × 10-20 J. These should not be taken as absolute measurements of real asphaltene and resin Hamaker constants (cf. Table 3), but rather as parameters that allow the fine-tuning of the model to the expected situations; however, the values obtained are in the expected range for these systems. Buenrostro-Gonzalez et al. have presented a method6 for determining Hamaker constants from precipitation curves. Their results for the Hamaker constants of asphaltenes (Aasphaltene ) 15.56 × 10-20 J) compare favorably with the ones reported here. Obviously, if the experimental values were available, they could be used directly in our model. (40) Sevick, E. M.; Monson, P. A.; Ottino, J. M. J. Chem. Phys. 1988, 88, 1198.

Table 2. Dielectric Constants, e, the Main Electronic Absorption Frequency in the UV Region, We, Refractive Index of the Medium in the Visible Spectra, ni, and Hamaker Constants, A, for Some Solvents at 25 °Ca substance

e

ni

Ve (1015 s-1)

A (10-20 J)

n-propane (liquid) n-butane (liquid) n-pentane n-hexane n-heptane n-octane n-nonane n-dodecane n-tetradecane n-hexadecane benzene toluene carbon tetrachloride acetone pyridine

1.610 1.40 1.844 1.890 1.920 1.948 1.972 2.014 2.030 2.050 2.280 2.4 2.240 21.0 12.3

1.290 1.333 1.349 1.365 1.378 1.387 1.395 1.411 1.418 1.423 1.501 1.496 1.460 1.359 1.510

3.00b 3.00b 2.99 2.98 2.98 2.97 2.97 2.99 2.94 2.94 2.10 3.00b 2.70 2.90 3.00b

2.7 3.4 3.75 4.07 4.32 4.5 4.66 5.04 5.1 5.26 5.0 5.4 5.5 4.1 7.5

a Data from refs 34, 35 and 54 or calculated from eq 25. b Estimated value.

4. Results 4.1. Asphaltene Aggregation Limits. As the Hamaker constant of the solvent is varied, the system undergoes a transition between a soluble state (i.e., one in which the amount of monomer molecules is appreciably large) and one in which the amount of aggregated states dominates. A typical distribution of these clusters is shown in Figure 6. Here the fraction of molecules Xn in a given aggregation state n (where n is defined as the number of molecules that form a given aggregate) is shown. For a nonaggregated state (crude oil screening factor fasph,m,asph ) 0.32), there is a considerable amount of monomers (n ) 1), in this case more than half of the molecules, and a sharp monotonically decreasing distribution of aggregates. The mean cluster size, N h , or weight average of this distribution, defined as

N h)

∑ nFn ∑ Fn

(14)

where Fn is the asphaltene number density of an aggregate of n particles is N h ) 1.84 (less than two). On the other hand, in a state dominated by aggregation (crude oil screening factor fasph,m,asph ) 0.356), only close to one-third of the molecules are present as monomers. The distribution is not unimodal, presenting a dominant cluster size of average aggregation number of 2.5 and a second, larger cluster size with ap-


Aguilera-Mercado et al.

Table 3. Dielectric Constants, e, Refractive Index in the Visible Spectra, ni, and Hamaker Constants, A, for Some Crude Oils Fractions as Calculated from Eq 25 Using an Estimate of We ) 3.0 × 1015 s-1a

a

ni

A (10-20 J)

substance

e

H TE C TA TK S U B

2.6 2.5 2.2 2.3 2.4 2.4 2.4 2.1

Crude Oils 1.527 1.522 1.487 1.497 1.496 1.498 1.496 1.472

7.7 7.6 6.8 7.0 7.0 7.0 7.0 6.4

H TE C TA TK S U B

3.9 3.8 5.1 4.8 4.5 3.9 3.8 4.7

Resins 1.576 1.587 1.606 1.585 1.608 1.608 1.595 1.606

9.0 9.3 9.8 9.3 9.9 9.9 9.5 9.8

H TE C TA TK S U B

16.2 18.4 15.2 9.2 10.9 10.1 8.2 5.5

Asphaltenes 1.647 1.707 1.638 1.676 1.706 1.668 1.685 1.650

11.0 12.7 10.8 11.8 12.7 11.6 12.0 11.0

Data and nomenclature of crude oils from ref 51.

proximately 16 molecules. The change between the two states is not sharp with any indication of a first-order transition. There is no obvious quantitative measurement of what is considered a dispersed state and when a state can be considered aggregated. Furthermore, it is evident from the above that there is no such definition of a nonaggregated system, even when the solvent is considered ideal. This is a plausible explanation of why it proves to be difficult to measure even the average molecular weight of asphaltenes. Figure 7 shows snapshots of the two state points considered above. In Figure 7a (crude oil

screening factor fasph,m,asph ) 0.32), it can be seen that dimers and higher n-mers coexist with the monomers even in this “soluble” state of the asphaltene system. Figure 7b (crude oil screening factor fasph,m,asph ) 0.356) shows a very different configuration, where some aggregates are clearly seen in the foreground. The reader is reminded that periodic boundary conditions exist in all three Cartesian directions, and thus the clusters near the edges of the simulation box may be larger than apparent, since they may be connected to periodic images of molecules that are in opposite sides of the box. The clusters show a general columnar trend, although it is noted that some lateral aggregation is apparent, and the general shape of the overall clusters tends to be spherical. Figure 7a,b shows two distinct states of the system: one in which only hints of aggregations are apparent, another in which very well-defined large clusters are seen. In line with the above observations, we chose to highlight a region defined by the zone in which the fraction of monomers, X, is between 0.3 and 0.4 as an aggregation threshold. Thus, systems with higher monomer fractions, X > 0.4, are considered herein as those in which the asphaltene is primarily dissociated and those where X < 0.3 are considered as those where aggregation is predominant and in which flocculation and consequent precipitation would be expected. Figure 9 shows how the mean cluster size, N h , changes as function of the free asphaltenes (the monomer fraction), X. The aggregation threshold corresponds to systems in which the mean cluster size is of the order of 5. The description of the thermodynamics of a system in terms of the fraction of nonaggregated molecules has proven adequate in other contexts of modeling associating systems. An unexpected result is that all simulations performed, independent of the temperature, resin concentration, and asphaltene concentration, fell on the same curve, indicating that the aggregation mechanism is unique. In Figure 8 and subsequent plots, this aggregation threshold is plotted as a region stressing the continuous nature of the transition. 4.2. Effect of Temperature and Concentration. Parametric studies of the effect of temperature, asphaltene, and resin concentration were performed around the base case and the

Figure 6. Histogram of the fraction Xn of asphaltene molecules forming clusters of size n. Gray bars correspond to a condition for which the asphaltenes are considered essentially soluble (crude oil screening factor fasph,m,asph ) 0.32). Black bars correspond to a condition for which the asphaltenes are considered aggregated (crude oil screening factor fasph,m,asph ) 0.356) (base case).


Figure 7. Snapshots of equilibrium configurations for the state points mentioned in Figure 6. (a) System with low aggregation (crude oil screening factor fasph,m,asph ) 0.32). (b) System with significant aggregation (crude oil screening factor fasph,m,asph ) 0.356). Resin molecules are not shown to aid in the visualization.

asphaltene monomer fraction monitored as a function of the Hamaker constant of the crude. In all cases, the screening factor was varied from fasph,m,asph ) 0.275 to 0.386, corresponding to the range in which precipitation is likely to occur. Figure 9 shows the effect of changing the temperature of the base case. The higher temperature curves allow the asphaltenes to remain dispersed at higher screening factors (corresponding roughly to the lower molecular weight alkanes), while as the temperature is lowered, the precipitation is facilitated. Figure 9 shows the approximate screening factors of pure solvents: “good” ones such as toluene and benzene are seen to disperse the asphaltenes regardless of their condition, while “poor” ones as n-pentane are seen to facilitate aggregation, again regardless of other conditions. Figure 10 is a similar plot where the concentration of the asphaltenes in increased and decreased 2-fold and 4-fold. The effect on the aggregation is more pronounced than in the case


of the other variables. In fact, at the higher concentrations, close to 10% (w/w) of asphaltenes, it is seen that the system aggregates to a certain degree in almost any solvent, with the exception of the “good” solvents. The trends are those expected and seen experimentally: if the concentration of asphaltene molecules is increased, they will aggregate even under “appropriate” conditions, while removing asphaltenes from a system precludes precipitation. These results correspond to the changing of composition of a giVen type of asphaltene. It is well-known that concentration alone is not an indicator of precipitation, as examples are abundant in which a crude oil with a high asphaltene concentration is stable with respect to precipitation, while another with a much lower concentrations has severe separation problems. This is not in contradiction with our results. The role of the resin molecules, which in the preceding analysis were not mentioned, seems to be marginal. This is caused by the fact that the energy gain associated with the asphaltene-asphaltene interaction is much larger than that of the asphaltene resins (cf. Figure 3). Figure 11 shows a snapshot of a system with almost complete aggregation, corresponding to the base case in a crude oil with a screening factor of fasph,m,asph ) 0.392. With the exception of one monomer, a dimer, and a trimer, the rest of the asphaltene molecules form a tight cluster (note that periodic boundary conditions are enforced and that the dominating cluster is thus seen partially among two opposing simulation cell walls). Resin molecules do not seem to solvate the cluster, as is commonly argued. On the contrary, they appear homogeneously dispersed along the cell. There is, however, a group of resins that are trapped within the asphaltene structure, filling in the interstices left in the structure. This trend was seen in almost all cases studied, the system being very insensible to the overall resin composition. This behavior conforms to experimental observations.41 Resins themselves aggregate in a media with fasph,m,asph ) 0.463 (similar to propane), confirming that the overall model behaves in accordance to phenomenological expectations.10 It is illustrating to point out that the effect of the resins is similar in nature to that of an asphaltene molecule with a smaller Hamaker constant (e.g., a lighter asphaltene). In that sense, the model reinforces the idea that crude oils are composed of complex and continuous distributions of molecular species, which increase in complexity as the average molecular weight increases. 4.3. Linear Aggregation Model. The asphaltene molecule model shares some resemblance to self-assembling lyotropic discotic liquid crystals,42,43 of which probably the most studied is the family of triphenylenes with six symmetrical peripheral oxygenated alkane branches (β-side chains).44 As an example, the 2,3,6,7,10,11-hexa-(1,4,7-trioxaoctyl)triphenylene (TP6EO2M) molecule is a prototype of a discotic amphiphile. When placed in an appropriate solvent (in this case water, since the outer alkane chains are hydrophilic), it self-assembles into columnar aggregates. The discotic nature of some asphaltene fractions seems in line with the recent work of Hurt and Hu45 on carbon pitch, where a thermodynamic model was presented that considered (41) Liao, Z.; Zhou, H.; Graciaa, A.; Chrostowska, A.; Creux, P.; Geng, A. Energy Fuels 2005, 19, 180. (42) De Gennes, P. G.; Prost, J. The Physics of Liquid Crystals; Clarendon Press: Oxford, 1993. (43) Bushby, R. J.; Lozman O. R. Curr. Opin. Colloid Interface Sci. 2002, 7, 343. (44) Kumar S. Liq. Cryst. 2004, 31, 1037. (45) Hurt, R. H.; Hu, Y. Carbon 1999, 37, 281.



Figure 8. Mean cluster size N h as a function of the asphaltene monomer fraction X for a system of 100 asphaltene molecules. Multiple symbols correspond to simulations at different parameters studied. Line is a guide to the eye. Shaded section corresponds to the estimated aggregation threshold.

Figure 9. Effect of temperature on the asphaltene monomer fraction X as a function of the crude oil screening factor fasph,m,asph. Aggregation threshold is marked by the gray region. Aggregated states lie below this region, and dispersed states lie above it. Open circles are the base case (25 °C). Diamonds, triangles, and squares are results at 10, 40, and 60 °C, respectively. The values corresponding to some pure solvents are marked as a reference. Lines are guide to the eye.

Figure 10. Effect of concentration on the asphaltene monomer fraction X as a function of the crude oil screening factor fasph,m,asph. Open circles are the base case. Diamonds, triangles, squares, and crosses are results at 0.25, 0.5, 2, and 4 times the base case concentration of asphaltenes, respectively. Other conditions and comments as in Figure 9.

equilibrium set of partially condensed discotic entities in an otherwise isotropic fluid. The conditions here are different, in the sense that the asphaltenes may be surrounded by a fluid whose average properties may be akin only to some parts of the molecules, while not so to others. The discotic geometry of the mesoscopic asphaltene model seems to suggest that a linear aggregation chemical-like theory46 used in the context of describing these amphiphiles could be used to describe the physics underlying our model.47 A linear

aggregation model inspired by polymerization reaction has been proposed by Agrawala and Yarranton.48 (46) Edwards, R. G.; Henderson, J. R.; Pinning, R. L. Mol. Phys. 1995, 86, 567. (47) A much more elegant derivation of the linear aggregation model based on statistical mechanics has been published: Attard, P. Mol. Phys. 1996, 89, 691. (48) Agrawala, M.; Yarranton, H. W. Ind. Eng. Chem. Res. 2001, 40, 4664.



Xn )

Fn ) nYn-1(1 - Y)2 Ftotal

(21)

and the monomer (or free asphaltene) fraction X

X1 ) X )

F1 ) (1 - Y)2 Ftotal

(22)

are derived. The implied distribution function in terms of clusters size n may be expressed as

P(n) )

Figure 11. Snapshot of an equilibrium configuration of the base case at an aggregated state corresponding to a fasph,m,asph ) 0.392.

One might suggest that clusters form following a quasichemical reversible equilibrium, defined by an equilibrium constant K

[A1] + [An-1] ) [An]

(15)

where [An] is the concentration of aggregates of n individual asphaltene molecules. The mass action requires that

Kn )

F˜ n F˜ n-1F˜ 1

for all n g 2

N h)

Kn ) K for n ) 2,3...

(17)

It is convenient46 to express the results in terms of a dimensionless quantity Y, defined as

F˜ n F˜ n-1 ) ) ... F˜ n-1 F˜ n-2

(18)

from which

F˜ n ) F˜ n-1Y ) F˜ n-2Y2 ) ... ) F˜ 1Yn-1

(19)

and the total number density of asphaltene molecules, Ftotal, in the system is

)

(23)

where nmax ) -(ln Y)-1 can be understood as the most probable cluster size. The series summation in eq 20 is only convergent if Y < 1. Furthermore, only for values of Y < 0.5 corresponding to values of X > 0.25 can one expect the distribution to be a monotonically decreasing function. At this point, one can notice significant differences with the results obtained by simulation. Figure 6 exemplifies two distributions of the base case. Only at the conditions of low aggregation (e.g., at crude screening factors of fasph,m,asph ) 0.32 corresponding to a monomer fraction of X ) 0.58) do we notice a monotonic decrease in the distribution function. At higher aggregation (e.g., at fasph,m,asph ) 0.386) the distribution is bimodal. We conclude form our observations that the linear aggregation model will only be appropriate for very low aggregations. To quantify furthermore its range of application, one can calculate the weight average of the distribution according to the linear aggregation model as

(16)

where the ∼ denotes the number density of the aggregates (rather than the molecules). The asphaltene number density in the above terms is found as Fn ) nF˜ n where a peculiar case is F1 ) F˜ 1, which corresponds to the “monomer” or free asphaltene number density. Due to the lack of further information, one is forced to make the assumption (discussed further later on) that the equilibrium constants are all equal and independent of each other (isodesmic model). It is notable that the isodesmic models preclude the existence of a critical micelle concentration.49

Y ≡ KF1 )

(

Fn (1 - Y)2 n (1 - Y)2 n ) nY ) n exp Ftotal Y Y nmax

∑ nFn ) (1 + Y) ) 2 - 1 ∑ Fn (1 - Y) xX

(24)

A plot of the weight average as a function of the monomer fraction (Figure 12) shows that the linear aggregation model is only useful in the very soluble region, roughly for X > 0.75 (1/xX < 1.15). In this region, the asphaltene is very much solubilized and the aggregates are sparse and of small n. In a fashion similar to other attempts of fitting a linear aggregation model to real discotic amphiphiles,50 the results show that the equilibrium “constant” K is actually concentrationdependent or, in other words, that the assumption made in eq 17 is not applicable. Furthermore, it is also clear from the analysis of the clusters formed that, within this model, a single asphaltene molecule has the ability to cluster with more than two other ones, since the outer spheres are attractive, although in a weaker fashion than the central ones. Thus, even for a relatively modest concentration of asphaltene molecules, the clusters, while having an overall columnar shape, show other 3D clustering patterns. This makes the system deviate significantly from the ideal linear aggregation model. Figure 11 shows a system with a significant aggregation where this point is made clear. 5. Conclusions

(20)

Although the molecular model for the asphaltene has a total of seven geometric and energetic parameters (size σ and energy of the spheres, distance L), the behavior of the model is

From where the definitions of the fraction of molecules in a given aggregation state Xn

(49) Evans, D. F.; Wennestrom, H. The Colloidal Domain, 2nd ed.; Wiley-VCH: New York, 1999. (50) Henderson J. R. J. Chem. Phys. 2000, 113, 5965.

Ftotal )

F1

∑ nF˜ n ) F1∑ nYn-1 ) (1 - Y)2



Figure 12. Average cluster size N h as a function of the inverse square root of the asphaltene monomer fraction X. The solid line is the linear aggregation model prediction (eq 24).

governed very strongly by the Hamaker constant of the asphaltene (i.e., the model parameters might be used in a general fashion as “representative” of a class of fluids leaving the Hamaker constant of the asphaltene as a sole adjustable parameter). To model a given crude oil system with this approach, one will need general information on the general composition of the system, for example, via a SARA, or preferably, a more complete chemical analysis. This would allow the estimation of the model asphaltene and resin compositions. Chemical information of the light fraction enables one to estimate its Hamaker constant (refraction index, density, etc.) For similar chemical families (e.g., saturated hydrocarbons), the dielectric constant is rather constant, regardless of the molecular weight (cf. Table 2). Therefore, the behavior of the system is influenced primarily by the change in the Hamaker constant of the media. Hamaker constants of a particular component (or fraction), interacting across a vacuum, may be estimated from experimental information by use of the approximation34

( )

3hVe (ni2 - 1)2 3 ei - 1 2 Ai,i ) kT + 3/2 2 4 ei + 1 16x2 (ni + 1)

25 relates the Hamaker constant of the media with experimentally available parameters. We are aware only of a single study51 that has published refractive indexes and dielectric constants of asphaltenes, resins, and their corresponding crude. By using an average value of Ve, we have estimated the Hamaker constants in Table 3 for these crude fractions. The values of these experimental Hamaker constants of asphaltenes coincide quantitatively with that of our model, which came out naturally from the fitting process described in section 3, Aasphaltene ) 17.15 × 10-20 J. The Hamaker constant of our model resin appears to be somewhat larger than expected, probably indicating that a less “asphaltene-philic” character should be assigned to it, although these are only rough guidelines. The model presented in this work seems general enough and at the same time has a solid physical ground, allowing it to become a sensible tool for the study and prediction of asphaltene aggregation in crude oils. The fact that the model may be related to a unique parameter (upon making the caveat that the same “family” of solvents is considered) is consistent with the experimental results of Buckley et al.,52 where they present the thesis that the onset of asphaltene precipitation is directly correlated to the refractive index of the oil. The parallelism between this phenomenological observation and the model proposed here is very significant. A change in the thermodynamic state of the crude conditions, be it via a change in volumetric conditions (depressurization, change in temperature) or be it via a change in composition (by dilution, by blowdown of dissolved gases, etc.), will have a significant impact on the Hamaker constant of the crude and may be calculated through the expected change in the refractive index.52 Flocculation rates in related systems have been directly linked53 to the corresponding Hamaker constants. It would be expected that the aggregation process seen here could continue into a full precipitation of given quantities of asphaltenes. However, here, hydrodynamic variables come into play and are important in the complete picture. Further work is in progress to evaluate them. EF050272T

(25)

where k and h are Boltzmann’s and Planck’s constants, Ve is the main electronic absorption frequency in the UV region (which in the cases studied may be estimated as Ve ) 3 × 1015 s-1), ei is the dielectric constant, and ni is the refractive index of the medium in the visible spectra. Some calculated values for relevant solvents are presented in Table 2. Essentially, eq

(51) Goual, L.; Firoozabadi, A. AIChE J. 2002, 48, 2646. (52) (a) Buckley, J. S. Fuel Sci. Technol. Int. 1996, 14, 55. See also: (b) Buckley, J. S.; Hirasaki G. J.; Liu, Y.; Von Drasek, S.; Wang, J. X.; Gil, B. S. Pet. Sci. Technol. 1998, 16, 251. (c) Christiansen, S.; Andersen, S. I.; Buckley, J. S. Pet. Sci. Technol. 2004, 22, 719. (53) Urbina-Villalba, G.; Toro-Mendoza, J.; Lozsa´n, A.; Garcı´a-Sucre, M. J. Phys. Chem. B 2004, 108, 5416. (54) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 85th ed.; CRC Press: Cleveland, OH, 2004. (55) Speight J. G. In Asphaltenes and Asphalts. I. DeVelopments in Petroleum Science; Yen, T. F., Chilingarian, G. V., Eds.; Elsevier: Amsterdam, 1994.

Mesoscopic Simulation of Aggregation of Asphaltene and Resin

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