Asphaltene Aggregation and Solubility - Energy & Fuels (ACS

Mar 3, 2015 - The EMS Energy Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, United States. ‡ School of Systems B...
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Asphaltene Aggregation and Solubility Paul Painter,*,† Boris Veytsman,‡ and Jack Youtcheff§ †

The EMS Energy Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, United States School of Systems Biology & Computational Materials Science Center, George Mason University, MS 6A12, Fairfax, Virginia 22030, United States § Turner Fairbanks Highway Research Center, McLean, Virginia 22101, United States ‡

S Supporting Information *

ABSTRACT: An attenuated association model describing the aggregation of asphaltenes in solution is extended to derive an equation for the weight-average degree of association and account for phase behavior. The weight-average molecular weight is calculated to be higher than number average, as it must be for a polydisperse material, but not by enough to explain the very large differences in these quantities reported in the literature. Binodals and spinodals are calculated using expressions derived previously, but modified to account for free volume (thermal expansion) differences. The phase behavior of asphaltene solutions is examined in more detail, particularly in the dilute solution regime. It is shown that the formation of nanoaggregates significantly affects the critical value of the χ interaction parameter. The phase diagram is highly asymmetric and the phase boundary approaches the pure solvent composition limit. This has a number of implications in terms of asphaltene solution characterization and the nature of asphaltene solutions. The results indicate that there are toluene insoluble asphaltene components, but these could exist as microphase-separated clusters stabilized against further aggregation by steric and kinetic factors. This would explain the large difference between observed number and weight-average molecular weights. In addition, because of the shape of the binodal curve at low concentrations, experimental data that have previously been interpreted in terms of a critical cluster or micelle concentration are shown to be consistent with a microphase separation.

■

INTRODUCTION

have often been referred to and considered to be nanocolloidal and colloidal structures, respectively. The existence of a CNAC has been challenged in a number of studies, and it has been proposed that asphaltene molecules associate continuously in a manner similar to linear stepwise polymerization.11−15 Sirota16 and Sirota and Lin17 have pointed out that aggregation does not necessarily imply colloid formation in the sense of the type of structures formed by surfactants (for example). Their view is that asphaltene solutions can undergo a liquid/liquid phase separation to give a solvent rich phase in equilibrium with asphaltene rich droplets (clusters). Because of its proximity to the glass transition temperature, the asphaltene rich phase is viscous or solid-like, and this, in turn, kinetically limits subsequent coalescence. They do point out that the asphaltene rich particles formed upon phase separation can act as colloids but that thermodynamically it is a liquid or glass that at elevated temperature or upon dilution exhibits coarsening and coalescence rather than colloidal aggregation. Fogler and co-workers18−21 have demonstrated the critical nature of kinetics in asphaltene phase separation. They used scattering measurements to show that although there is significant dissociation of nanoaggregates upon dilution to concentrations similar to those reported for the CNAC, the dissociation was gradual and aggregates persisted to the lowest measured concentration.10 Equally important, Hoepfner and

The purpose of this work is to expand on the development and application of an attenuated association model recently applied to a description of the phase behavior of asphaltene solutions.1 The literature concerning the aggregation and solubility of asphaltenes is both rich and deep, and a comprehensive background review is beyond the scope of what we can and should attempt to accomplish here. Accordingly, in this Introduction we will simply focus on the works that appear to us to be most relevant to this particular study. It is now widely accepted that asphaltenes aggregate even in very dilute solutions of good solvents such as toluene, but experimental data describing the nature of the aggregates and the aggregation process remain open to interpretation and are the subject of various disagreements. Mullins2 and Mullins et al.3 point to a wide range of literature that indicates that asphaltene molecules begin to form nanoscale aggregated structures at concentrations well below 100 mg/L. These nanoaggregates are considered to have a limited size as a result of steric inhibitions introduced by the alkyl chains attached to asphaltene aromatic cores. A critical nanoaggregate concentration (CNAC) has been defined as the concentration at which growth of nanoaggregates stops,2,4−6 but also (somewhat confusingly) as the concentration at which nanoaggregates form or start.7−10 At higher concentrations (∼2−10 g/L) these nanoaggregates undergo additional association to form clusters or micelles.2,3 The concentration at which this occurs has been referred to as the critical micelle concentration (cmc) or critical cluster concentration (CCC). These aggregates and clusters © 2015 American Chemical Society

Received: November 6, 2014 Revised: March 2, 2015 Published: March 3, 2015 2120

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Energy & Fuels Fogler10 demonstrated that the solvation shell of asphaltenes varies significantly with solvent. Related to this, Wang et al.22 using atomic force microscopy determined that in toluene there is a long-range steric repulsion between asphaltene surfaces but the interaction becomes attractive in toluene/heptane mixtures as the volume fraction of heptane is increased. This result was interpreted in terms of polymer brush theory. In a good solvent, asphaltenes are expanded and there is a steric repulsion between the surfaces to which they are attached. In a poor solvent, there is a conformational change to a more collapsed structure and van der Waals forces then result in a weak attraction. Taken together, these results would suggest that upon phase separation from solvent asphaltenes could form particles or clusters that depending upon solvent and temperature are both kinetically and sterically inhibited from further aggregation. In discussing asphaltene aggregation, it is important to clearly define what type of aggregates are being considered. In some studies, particles in the size range 5−10 nm are referred to as both aggregates and nanoaggregates,23,24 which of course they are. However, in scattering work a clear distinction is made between an initial, limited association to form nanoaggregates approximately 2−3 nm in size and a subsequent aggregation of these particles to form mass fractal clusters in the size range 5− 10 nm.10,20,25−27 In keeping with this latter literature, we will refer to nanoaggregates as the smaller particles formed initially by the self-association of asphaltene molecules and refer to the larger aggregates of these particles as clusters. We will apply an association model to the calculation of the degree of association of asphaltenes in nanoaggregates. This association has a marked effect on phase behavior, and we will argue that further aggregation to form clusters is a result of a phase separation of nanoaggregates from solution, in keeping with some of the work cited earlier. The seminal work on applying association models to asphaltenes is the linear association model of Agrawala and Yarranton,13 recently modified to model molecular weight measurements.28 Merino-Garcia et al.,15 Shirani et al.,29 and Acevedo et al.30 have also used association models to model aggregation, but the equations for association were left as sums and solved numerically. In the attenuated association model described in our preceding work, closed form expressions were obtained for the degree of association and an expression for the free energy of mixing was derived. In order to address issues such as the very large difference between number-average and weight-average molecular weight measurements, the formation of clusters and the relationship between nanoaggregation and phase behavior, we will first extend the equations describing the stoichiometry of association and then modify the free energy expression obtained previously to account for free volume differences between asphaltenes and solvents.

number of charge carriers present in a given solution, hence conductivity measurements. Association models have been widely used to describe the self-association of hydrogen-bonding materials such as alcohols. The book by Acree31 describes and discusses various models. These usually take the form of a linear, continuous association, and in the model based on Flory’s work an equilibrium constant defined in terms of volume fractions is used,32,33 such that the addition of a “monomer” of A molecules to an i-mer is described by the equilibrium condition: ϕA + ϕA ⇌ ϕA i

1

(1)

i+1

where ϕAi is the volume fraction of i-mers (so that ϕAi is the volume fraction of “monomers” in dynamic equilibrium with associated species). An equilibrium constant (KA) describing the association is given by ϕA

i ϕA ϕA i + 1

KA =

i+1

i

(2)

1

This is a dimensionless equilibrium constant that is equal to an equilibrium constant defined in terms of molar concentrations divided by a molar volume per structural unit. As Flory pointed out,32,33 because this latter quantity is a constant, either equilibrium constant definition is acceptable. In addition, although the equilibrium defined in eq 1 is in terms of the addition of a monomer to a an i-mer, as long as KA is independent of chain length, equations of equivalent form can be used to describe the addition of an i-mer to a j-mer. To account for a system where the degree of association is limited by steric or other factors, an equilibrium constant that is continuously attenuated as monomers are added to a nanoaggregate can be defined by modifying eq 2 in such a fashion that the effective value of the equilibrium constant is continuously decreased as the nanoaggregate size (i) increases. This can be accomplished by dividing KA in eq 2 by a factor i + 131 ϕA KA i i+1 = i+1 ϕA ϕA i + 1 i

(3)

1

This allows the use of a single parameter in calculations, rather than the multiple parameters that would be required to describe an association where KA is allowed to vary with nanoaggregate size. By successive substitution using eq 3 (see Supporting Information), it follows that

ϕA = i

KA i − 1ϕ A i 1

(i − 1)!

(4)

The volume fraction of asphaltene molecules present in the mixture (ϕA) is then given by

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STOICHIOMETRY OF ASSOCIATION Equations. The derivation of equations describing the association of asphaltenes in an attenuated association model was described in our preceding work.1 Here we will just reproduce the main results, but for the interested reader a detailed description of the derivations is provided as Supporting Information. The model will then be extended to develop an equation for the weight-average molecular weight and the number of species (monomers plus aggregates) per unit volume. The former can be related to the results of scattering studies while the latter can be related to NMR data and the

∞

ϕA =

∞

∑ ϕA i=1

i

=

∑ i=1

ϕ A (KAϕ A )i − 1 1

1

(i − 1)!

∞

=

∑ i=0

ϕ A (KAϕ A )i 1

1

i! (5)

The lower limit of the summation is changed from 1 to 0 in order to make use of the Taylor series for an exponential function and obtain an expression in a closed form: ∞

ex =

∑ i=0

2121

xi x2 x3 =1+x+ + + ... i! 2! 3!

(6)

DOI: 10.1021/ef5024912 Energy Fuels 2015, 29, 2120−2133

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Energy & Fuels Substituting x = KAϕA1

is proportional to the volume fraction of asphaltenes divided by their number-average degree of association.

ϕA = ϕ A e KAϕA1

(7)

1

∑ NA i

Note that for simplicity in the derivation of eq 7 we have only considered adding a monomer at a time to a nanoparticle. The equations for the more general form of adding an i-mer to a jmer to make an (i+j)-mer are given in the Supporting Information, but the final result is the same. The definition of KA used here also assumes that it does not depend on solvent. This is probably a reasonable assumption for solvents that do not engage in specific interactions with asphaltenes, but nevertheless remains an assumption that must be verified by how well the model fits a range of data. The number-average degree of association, NAn , is given by ∞

NAn =

∑

N A ii NA i

i=1

∑ ϕA

=

V

(8)

i

where NAi is the number of i-mers. In a similar fashion eq 6 can be used (see Supporting Information) to obtain a closed form expression NAn =

(e

ϕAKA KAϕ A 1

− 1)

(9)

Similarly, the weight-average degree of association, NAw , can be derived as follows:

∑

=

∞

N A ii 2

∞

NAw

=

N A ii

i=1

∑i = 1 ϕ A i i

∞

∑i = 1 ϕ A

(10)

i

Using ∞

∞

∑ ϕA i = ∑

iϕ A (KAϕ A )i − 1 1

=

(i − 1)!

i

i=1

∞

1

i=1

∑

iϕ A (KAϕ A )i

i=0

∂(x e x) 2x 3x 2 4x 3 =1+ + + + ... ∂x 1! 2! 3!

1

1

i!

ϕA NAn

(16)

Degree of Association and Scattering Studies. In order to calculate the degree of association, we first need a value for the equilibrium constant, KA. One of the goals of this work is to reproduce the main features of asphaltene association and phase behavior using as few adjustable parameters as possible. In this regard, Lisitza et al.34 calculated the enthalpy and entropy of association of an asphaltene from NMR data and obtained values for the free energy change upon association of the order of 25−26 kJ/mol using Debye’s two state model and 20 kJ/mol using a Flory−Huggins type model. Our association model is different from this two state model (monomers and aggregates), but we will assume a value of ∼23 kJ/mol, a value between these limits, so that KA is 11000 at 25 °C. This is an average of course, as asphaltenes consist of a broad range of molecules with a corresponding range of chemical characteristics. However, as we showed previously, the calculated degree of association does not vary significantly with KA over a broad range of values.1 (Additional calculations using a range of values of KA are presented in the Supporting Information to further illustrate this.) Using KA = 11000 and eqs 9 and 13, the values of the number- and weight-average degrees of association were calculated as a function of composition (asphaltene volume fraction) and the results are plotted in Figure 1. It can be seen that the degree of association increases rapidly and continuously with increasing asphaltene concentration and reaches a maximum value of ∼7−8 at high concentrations, with the weight average being larger than the number average, as it must be for a system that is not monodisperse. This result is

i

∑ (ϕ A / i )

∝

(11)

(12)

together with eq 5, we obtain NAw = 1 + KAϕ A

(13)

1

In the association model used here, the total number of species or i-mers (monomers plus aggregates) per unit volume is given by ∑ NA i V

=

=

1 VA

∞

∑ i=1

ϕi i

=

1 VA

∞

∑

ϕ A (KAϕ A )i − 1 1

i=1

⎡ ∞ (K ϕ )i ⎤ A A1 1 ⎢ ∑ − 1⎥ ⎥⎦ VAKA ⎢⎣ i = 0 i!

1

i!

(14)

where NAi is the number of asphaltene i-mers and VA is the average molar volume of an asphaltene molecule. As above, the summation term is a series expansion of an exponential, so we get the following: ∑ NA i V

=

(e KAϕA1 − 1) VAKA

Figure 1. Number-average and weight-average degree of association plotted as a function of asphaltene concentration (volume fraction): top two curves, number and weight averages calculated at 25 °C; bottom two curves, weight average calculated at 150 and 300 °C.

(15)

VA is a constant, and substituting from eq 9, it can be seen that the number of i-mers per unit volume (concentration of i-mers) 2122

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Energy & Fuels consistent with the work of Tanaka et al.,35 who estimated a degree of association of about 8 from an X-ray diffraction study of asphaltene powders. They also found that the degree of association decreased to ∼6−7 at 150 °C (depending on the sample source) and to near 5 at 300 °C. Based on a free energy of association of about 23 kJ/mol at 25 °C, the association constant KA has values of 700 and 125 at temperatures of 150 and 300 °C, respectively. The weight-average degree of association calculated at these higher temperatures is also shown in Figure 1, and it can be seen that for asphaltene powders values near 6 and 4−5 are calculated at these temperatures, in reasonable agreement with the diffraction results. Good agreement can be obtained by “tweaking” the value of KA, as one would expect that the (average) strength (free energy) of association would vary somewhat from sample to sample. However, our purpose here is to show the model is generally consistent with a broad range of experimental data rather than to fit the model to results from specific samples. Various experimental results, particularly measurements of number-average molecular weight using vapor pressure osmometry (VPO),28,36 have shown that association persists to extremely dilute solutions. Figure 2 shows the calculated

at asphaltene concentrations of 0.00125 vol % (0.0000125 volume fraction). However, the weight-average molecular weight for the samples used in their study was determined to be of the order of 1 × 104 to 1 × 105 g/mol (depending on the sample) in this concentration range, considerably higher than predicted by the association model presented here. If we assume an average asphaltene molecular weight in the range 750−850, we calculate a weight-average value near 1000 (weight-average degree of association NAw ∼ 1.15 at an asphaltene volume fraction of 0.0000125). Figure 1 is expanded in the dilute solution concentration range in the Supporting Information for the interested reader. This brings up the general discrepancy observed between asphaltene molecular weights determined by techniques such as VPO and those by scattering measurements. The results of many scattering experiments give weight-average molecular weights of the order of 1 × 105 or more, even in dilute solutions of good solvents such as toluene.10,20,26,36−38 One might expect some differences according to sample source, but Yarranton et al.36 reported a multilaboratory study of a single source asphaltene characterized by a broad range of techniques. VPO measurements indicated monomer asphaltene molecular weights of the order of 850 g/mol, with nanoaggregates having a number-average molecular weight less than 10,000. However, small-angle X-ray scattering (SAXS) data indicated molecular weights 20 times higher than this. Although weight-average molecular weights are always higher than number-average molecular weights, the results shown in Figure 1 suggest that this cannot be the source of such a large difference. The authors discussed a number of possible reasons for this result, including the formation of flocs of nanoaggregates, which would not be detected by VPO measurements but would significantly affect SAXS (and permeation) measurements. Similarly, Hoepfner et al.10,20 interpreted high molecular weight values determined by scattering studies in terms of a clustering or aggregation of nanoaggregates, as did Eyssautier et al.25−27 in their studies. This would suggest that even in good solvents there is a microphase separation of some asphaltene nanoaggregates into clusters and/or small flocs. We will return to this later when we consider solution phase behavior Conductivity and NMR Data. Zeng et al.8 and Goual et 5 al. reported conductivity measurements on asphaltene solutions in toluene. They observed a break in plots of conductivity vs concentration that they interpreted in terms of a CNAC. These breaks between two linear regions occurred at very low asphaltene concentrations, ranging from 69 to 150 mg/L, depending on the asphaltene. Additional breaks were observed at higher concentrations, in the range 1−3 g/L in these and a later work.7 The breaks in the plots at higher concentrations were interpreted in terms of cluster formation. Direct current (DC) conductivity depends on the number of charge carriers, which is given by

Figure 2. Number fraction of i-mers at 25 °C plotted as a function of asphaltene volume fraction. The inset shows the number fraction of all i-mers and monomers plotted on a log volume fraction scale.

C ion =

fraction of i-mers (ϕAi/ϕA, using eq 5) at asphaltene volume fractions less than 0.002. An insert in this figure shows the fraction of all monomers and i-mers (i > 1) plotted on a logarithmic concentration (volume fraction) scale. Even at asphaltene volume fractions of 0.00001, about 10% of the molecules are present as some form of aggregate. This is also at first sight consistent with the results of Hoepfner and Fogler,10 who could not detect complete dissociation of aggregates even

CasphNAx M

(17)

where Cion is the number of diffusing ions/m3, NA is Avogadro’s number, x is the fraction of asphaltenes that are ionic, and M is the molecular weight of the asphaltene. The quantity x is of the order of 10−5, which suggests that on average there is not more than one charged species per nanoaggregate. To a first approximation, the fraction of charged species should then be 2123

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Energy & Fuels proportional to the concentration of i-mers as expressed in eq 16. Figure 3 shows a plot of the concentration of i-mers (in effect multiplied by the average molar volume of the asphaltene under

calculation presented here suggests that it would be hard to determine this unambiguously using conductivity data. However, Goual et al.7 also performed centrifugation studies on their samples, showing significant aggregation to form clusters that commenced at asphaltene concentrations above 0.3 g/L. The fractional accumulation of asphaltenes increased systematically and continuously, leveling off at values near 0.9 at asphaltene concentrations above 2 g/L. We will discuss this later when we consider phase behavior. Turning to NMR data, Lisitza et al.34 observed that above a critical concentration the spin−echo signal in the spectrum of asphaltene solutions is substantially reduced as a result of fast spin relaxations within aggregates. The concentration dependence of the asphaltene signal should then also depend on the concentration of i-mers, as plotted in Figure 3. The curve reported by these authors actually looks remarkably like that presented in Figure 3, and it would be possible to draw a continuous curve through their data points. Our point in examining these data is that in very dilute solutions these experimental results can also be interpreted in terms of an attenuated association, rather than the formation of nanoaggregates at a critical concentration. We now turn to examination of how the formation of these nanoaggregates affects the free energy and phase behavior.

■

THERMODYNAMICS OF ASSOCIATED ASPHALTENE SOLUTIONS Free Energy of Associated Asphaltene Solutions. As derived in our previous study1 (and reproduced with added detail in the Supporting Information), the free energy of mixing asphaltenes with nonassociating solvents is superficially in the form of a Flory−Huggins equation with additional terms to account for the association of asphaltene molecules to form nanoaggregates. The free energy per mole of lattice sites (ΔG′) is given by

Figure 3. Calculated concentration of i-mers (multiplied by the average molar volume of asphaltenes and proportional to the fraction of charge carriers) plotted as a function of asphaltene concentration. The lines in the large plot are added as a guide to the eye.

considerationsee eqs 13 and 14) plotted as a function of asphaltene concentration in grams per liter, calculated from the corresponding volume fraction by assuming an asphaltene density of 1.2 g/cm3. The main plot is in the concentration range 0.0−0.6 g/L, while an inset shows the concentration range extended to 8 g/L. In any association model, this type of plot will be continuous and this is obvious in the plot inset shown over a broad concentration range. However, the plot has a low degree of curvature, and in the more dilute concentration range (0.0−0.6 g/L) where there are fewer calculated data points it can appear that there are two linear regimes separated by a break, as indicated by the lines connecting data points (drawn as a guide to the eye). The apparent change in slope is not very large, and if fewer data points are plotted and if the effect of experimental errors are considered, it is possible that a change in slope may not be evident at all, as in the data reported by Lesaint et al.9 These authors suggested that conductivity measurements are not appropriate for the detection of a CNAC. Similar arguments apply to conductivity measurements in the 1−8 g/L concentration range. The curve shown as an insert in Figure 3 is continuous. It appears similar in form to the conductivity data reported by Goual et al.7 (Figure 9 in that paper). However, only data at concentrations above the CNAC were used to obtain a linear fit by these authors. Again, if we consider a limited number of data points, it is possible to obtain linear fits to the calculated number of i-mers at the upper and lower ends of this range. If there is a phase separation into clusters, there could indeed be a break in the plot, but the

ΔG′ ΔG Vref = RT RT V ⎧ ϕA ⎫ ϕ = ⎨ ln(ϕ A /ϕ A0 ) + B ln ϕB⎬ 1 1 rB ⎩ rA ⎭ ⎧ ϕ (e KAϕA01 − 1) ⎫ ⎪ (e KAϕA1 − 1) +⎨ A − + ϕAϕBχeff ⎬ ⎪ ⎪ KArA KArA ⎭ ⎩ ⎪

(18)

V is the total volume of the mixtures, and an arbitrary reference volume, Vref, equal to the molar volume of a lattice site, has been defined. R and T have their usual meaning. VA and VB are the molar volumes of an asphaltene molecule and the second component of the mixture (in this study, solvent), respectively. Also rA = VA/Vref, rB = VB/Vref, and r = rA /rB. For asphaltene solutions we will put rB = 1. The quantities ϕA1 and ϕ0A1 are the equilibrium volume fractions of (unassociated) asphaltene monomer molecules in the mixture and pure state, respectively, while ϕB is the volume fraction of the second component of the mixture. The χeff term accounts for interactions between the components of the mixture, in this case associated asphaltene nanoaggregates and solvent, as discussed later. It excludes those interactions that lead to association, which are accounted for by the other terms in eq 18. The form of this equation results from applying Prigogine’s observation that in an associated mixture in equilibrium the 2124

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Energy & Fuels

volume effects also result in composition dependent excess terms. We will also neglect these and use an approximate form of the contribution of free volume to χeff, χfv. Graessley45 and Milner et al.46 have discussed and applied this simplified form of the contribution to χ from free volume effects based on the work of Patterson47 and Flory et al.48−50 This is expressed in terms of a mismatch of polymer and solvent thermal expansion coefficients, α, as

chemical potential of a stoichiometric component is equal to the chemical potential of the monomer.39 For comparative purposes, the terms have been collected into two sets of parentheses. The first set contains terms that have the appearance of the Flory−Huggins combinatorial entropy of mixing, but with the asphaltene log term expressed in terms of the monomer concentration at equilibrium. It is therefore not just an entropy term but also depends on the enthalpy of association. The first two terms in the second set of parentheses also contain terms associated with free energy changes due to association in the mixture relative to the pure state. These arise from the change in the degree of association on going from the pure state to the mixture and do not appear in the Flory− Huggins equation for polymer solutions, because in the latter the (covalent) polymer chain length does not change. Finally, χeff is an “effective” interaction term that accounts for nonspecific interactions between the asphaltene i-mers and the second component of the mixture (in this study, solvent) together with excess free energy terms that were not considered in our initial work.1 Flory χ Parameter−Free Volume Effects. In eq 18 we labeled the Flory χ parameter effective, χeff, because it has long been recognized that χ has both entropic and enthalpic components and is composition dependent. Blanks and Prausnitz,40 for example, used solubility parameters to estimate the enthalpic component of χ for polymer solutions and found empirically that the entropic component (χS) had an average value of 0.34. Earlier, Small41 had estimated the value of χS to be of the order of 0.3. This results in a widely used “rule of thumb” estimate that, for a polymer to dissolve in a solvent, the solubility parameter difference between the two should be less than ±1 (cal/cm3)0.5. This is a result of putting the critical value of χ = 0.5 and using χeff

V = χS + χH = 0.34 + s (δp − δs)2 RT

χ

V α P* ⎛ Tp* − Ts* ⎞ ⎟ = s s s ⎜⎜ 2R ⎝ Ts* ⎟⎠

fv

2

(20)

where P*s is related to the cohesive energy density of the solvent (in turn, related to δs2) and Tp* and Ts* are characteristic temperatures given by 3 (1 + 4αT /3)4 α (1 + αT )3

T* =

(21)

χ has a slight temperature dependence and is not purely entropic. Milner et al.46 showed that, for polymer solutions where there are large differences between the thermal expansion coefficients of polymers and solvents χfv ∼ 0.3, close to empirically determined values of χS. Chemical Potentials and the Spinodal. The chemical potentials are given by fv

μA − μA0 RT

=

μ A − μ A0 1

1

RT

= ln(ϕ A /ϕ A0 ) − 1

1

1 KAϕ A K ϕ0 2 1 − e A A1) − rϕ + r ϕ χ ′ (e A B eff B KA

(22) μB −

μB0

RT

(19)

= ln ϕB −

1 KAϕ A 2 1 − 1) + ϕ + r ϕ χ ′ (e B A eff A rKA

(23)

In this study we have made approximations that result in a composition independent χ, so that

The quantity Vs is the molar volume of the solvent and defines the cell size in this lattice model, while the solubility parameters δp and δs refer to the polymer and solvent, respectively. The composition dependence of χ also means that experimentally determined values will depend on the technique used to measure them, some related to the first derivative of the free energy of mixing with respect to composition (e.g., osmotic pressure measurements) while others depend on the second derivative of the free energy (scattering measurements). These quantities (χeff ′ and χeff ″ ) can be related to one another and a “bare” χ parameter, χ0.42 Entropic contributions and the composition dependence of χ can be related to various factors, not least free volume (thermal expansion) differences between the components of the mixture. In addition, it has long been recognized that the Flory lattice approximation (lattice coordination number z → ∞) also plays a role. For simplicity, Flory assumed that the probability of a polymer segment being adjacent to a chosen solvent molecule is given by ϕp, the volume fraction of polymer in the mixture (i.e., a random distribution of segments). Essentially, this neglects chain connectivity and treats polymer/solvent contacts as if the polymer segments were disconnected and randomly dispersed with solvent molecules. We have shown in previous work that connectivity effects result in composition dependent terms,43,44 but inclusion of this factor here would lead to unnecessary complications in terms of the general points we wish to make. Similarly, various approaches to modeling free

′ = χeff = χ fv + χ0 χeff

(24)

The spinodal is ∂ 2(ΔG′/RT ) 1 ∂ϕ A1 1 ″ =0 + − 2χeff = 2 r ∂ r ϕ ϕ ∂ϕA A A1 BϕB A

(25)

where ∂ϕ A

1

∂ϕA

=

⎤ ϕA ⎡ 1 1⎢ ⎥ ϕA ⎢⎣ (1 + KAϕ A ) ⎥⎦ 1

(26)

and again ″ = χeff = χ fv + χ0 χeff

(27)

The “bare” χ, χ0, is related to solubility parameters in the usual way: χ0 =

Vs V (δA − δs)2 = s (Δδ)2 RT RT

(28)

The solubility parameters δA and δs refer to the asphaltene and solvent, respectively. As above, the derivation of these equations is described in the Supporting Information. Parameters for Asphaltene Solution Phase Behavior Calculations. As in out calculations of the degree of 2125

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Energy & Fuels association, we will use a value of KA = 11000 at 25 °C to calculate the contributions of association to the free energy. Essentially, this just involves calculating the volume fraction of monomers present at a given concentration using the stoichiometric relationship given in eq 5. A simple way of doing this is to start with values of ϕA1 in a spread sheet and calculate values of ϕA. All other quantities are then calculated at these volume fraction values. (Note, values of ϕA1 are very smallsee Figure 2.) Equation 28 is used to calculate χ0 using solubility parameter values from the literature for the solvent and is an equation we will discuss below for the solubility parameters of asphaltenes. In order to calculate χfv, we require values of the (liquid state) volumetric thermal expansion coefficient (α) of asphaltene. Experimental data do not seem to be available, but values have been estimated using simulations. Diallo et al.51 calculated a value of 1.49 × 10−4 K−1, which is smaller than the experimental value for phenanthrene (2.56 × 10−4 K−1). However, assuming an increment of ∼5 × 10−4 K−1 at the Tg, this would give values of α in the liquid state of ∼(6−7) × 10−4 K−1, about the same as many high molecular weight polymers.45 Li and Greenfield52 also calculated values in this range but in addition cited density data on a single asphaltene that indicated a value of about 6.5 × 10−4 K−1. This seems reasonable, and we will assume that the thermal expansion coefficient of asphaltenes is of the order of 6.5 × 10−4 K−1. This gives a value for the characteristic temperature for asphaltenes, TA*, of ∼6764. Graessely45 lists the thermal expansion coefficients of a number of solvents, and these were used to calculate χfv. Solvent solubility parameters were taken from the literature.53 Table 1 lists the data used here. It is interesting that pentane has a significantly larger value of χfv than heptane.

solubility parameters. Accordingly, solubility parameters for solvents such as methanol are not listed in Table 1, as hydrogen bonding provides a major contribution to their cohesive energy density. We will treat the effect of self-associating solvents (e.g., methanol) in a separate publication.57 In order to use eq 26, we need a method for calculating the solubility parameter of an “average” asphaltene molecule and, more subtly, this method should be based on data that do not include heats of vaporization of molecules such as methanol that self-associate, as an assumption of the model is that solubility parameters are being used to account for dispersion and weak polar interactions only. An atomic group contributions method based on this premise was constructed a number of years ago58 and is an extension of earlier work by van Krevelen.59 The solubility parameter of an “average” asphaltene molecule is given by (N + S) H O + 106( C ) + 51.8 C ( C) δA = 2.045 (N + S) H O − 10.9 + 12fa + 13.9( C ) + 5.5( C ) − 2.8 C

7.0 + 63.5fa + 63.5

(29)

The application of this equation requires that elemental analysis data (to give atomic ratios H/C, O/C, and so on) and fraction aromaticity (fa) data for the asphaltenes of interest are available. Taking data from various sources,60−68 values of asphaltene solubility parameters were calculated and are plotted against their atomic H/C ratios in Figure 4. Included in this plot are

Table 1. Solvent Parameters solvent

molar volume (cm3/mol)

solubility parameter (MPa0.5)

χfv

toluene xylene pyridine cyclohexane pentane heptane octane carbon tetrachloride dichloromethane trichloroethylene THF acetone nitrobenzene

106.9 123 80.5 108.1 116.1 148 164 97.1 64.1 90 81.9 73.4 102

18.2 18.2 21.7 16.8 14.5 15.1 15.5 17.6 19.8 18.7 18.0 18.5 22.5

0.12 0.05 0.14 0.13 0.28 0.19 0.16 0.17 0.22 0.15 0.16 0.25 0.11

Figure 4. Solubility parameters, δ (units MPa0.5), for asphalt and bitumen fractions calculated using eq 29 and plotted as a function of the atomic H/C ratio of the material.

As originally used by Hildebrand and described in the book by Hildebrand and Scott,54 solubility parameters were designed to deal with the mixing of molecules with relatively weak interactions. Many attempts have been made to extend their use to systems that involve strong, specific interactions, such as hydrogen bonds, by breaking down contributions to molar attraction constants into various components. In association models, however, the strong associative forces (not to mention nonrandom contacts) are accounted for in terms that describe association,55,56 and it is only weaker interactions between associated species and other molecules that are described using

data from oxidized bitumen samples and asphalts together with data from maltenes and SARA fractions. These were included because taken as a whole it can be seen that there is an interesting, continuous variation of the calculated solubility parameter, δ, with H/C ratio. This is partly because the values calculated for bitumen fractions using eq 29 depend largely on carbon and hydrogen contents (which predominate over heteroatom content in these materials) and aromaticity. Most of the asphaltenes from bitumen and crude oils are calculated to 2126

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solutions.32,33 It has the well-recognized limitations of all lattice models of this type, but also the strength of simplicity and the ability to provide insight. Furthermore, the calculated critical value of χ for high molecular weight polymer solutions (∼0.5) has turned out to be remarkably close to experimentally determined values. The critical values of χeff calculated here for molecules and aggregates that are smaller than macromolecules are consistent with this limiting value for polymer solutions. Accordingly, this result supports the view of the authors cited in the Introduction, that association to form nanoaggregates does not imply colloid formation. These aggregates can form solutions in the thermodynamic sense, but they also phase separate more easily than nonassociating molecules of similar chemical character. There are a number of implications related to the calculated critical value of χ and the asymmetry of the phase diagram. First, it has been observed that asphaltenes have a component that is non-self-associating. In the sample studied by Yarranton et al.,36 this constituted about 10% of the whole asphaltene fraction. We will label this fraction As-NA and the associating fraction As-A. For an As-NA fraction with a molecular weight in the same range as As-A monomers, the critical value of χeff is about 0.92, as noted previously. If we assume that, in a solution where χfv ∼ 0.2, the critical value of the solubility parameter difference calculated from the bare value of χ, χ0 (∼0.72), would be about 4.2 MPa0.5, relative to a value of about 3.3 MPa0.5 for the As-A component. Oil or bitumen As-NA components with somewhat higher aromaticities and lower H/ C ratios than As-A components would therefore precipitate with the latter. On the basis of mass spectrometry measurements, Yarranton et al.36 suggest that the As-NA components do indeed have higher aromaticities and lower H/C ratios than the As-A component. The effect of χfv is also apparent in attempts to determine the solubility parameter of asphaltenes on the basis of their solubility/insolubility in a range of solvents. For example, in older work solubility parameter spectra were determined on the basis of the solubility of asphaltenes in a range of solvents on the assumption that dissolution is a maximum when Δδ ∼ 0.69,70 Lian et al.70 determined that an asphaltene from AAM-1 asphalt (SHRP asphalt sample) became completely soluble in a solvent with a solubility parameter of 17.6 MPa0.5 at a concentration of 5% (w/v). On the basis of the model presented here, that value would not correspond to the solubility parameter of the asphaltene. At a volume fraction of 0.05 the value of χeff defining the phase boundary is ∼0.66. If we again assume χfv ∼ 0.2, then

have average solubility parameters in the range 20−24 MPa0.5. We emphasize here that the values calculated for asphaltenes from various sources are an average over what is no doubt a distribution of values within each sample.

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PHASE BEHAVIOR OF ASPHALTENE SOLUTIONS In our preceding study1 spinodal and binodal curves for hypothetical asphaltene solutions were calculated in terms of χ (χeff here) as a function of composition. The calculation of the spinodal is straightforward using eq 25, but as always the binodal is more difficult. In our previous publication we used a relatively simple method based on Maxwell’s construction, but care has to be taken in how the chemical potentials are used in the calculation. We discuss this and give an example of the calculation of binodal points in the Supporting Information. The calculated spinodal and binodal phase boundaries are presented in Figure 5. A value of KA = 11000 at 25 °C was

Figure 5. Calculated spinodal for a non-self-associating hypothetical asphaltene molecule plotted as χef f vs the volume fraction of asphaltene using the Flory−Huggins model (ratio of molar volumes, r = 8) compared to the calculated spinodal and binodal for asphaltenes that associate to form nanoaggregates.

assumed together with an asphaltene molar volume eight times that of the solvent (i.e., an asphaltene molar volume ∼ 800 cm3/mol). Also shown in this figure is the Flory−Huggins spinodal calculated for an asphaltene of the same size, but in the absence of association. In their multilaboratory study of a single source asphaltene, Yarranton et al.36 noted that it is unclear how self-association relates to asphaltene precipitation. This phase diagram suggests one answer. The formation of nanoaggregates reduces the value of the critical value of χeff significantly, from about 0.92 in a nonassociating material where the molar volume of the solute and solvent differ by a factor of 8, to about 0.64 in a solution where the solute associates to a limiting value of about eight of these units. One can question these values and the validity of the model described here, but it is essentially a Flory−Huggins model that accounts for association with modifications that can also be traced back to Flory’s original work on polydisperse polymer

χ0 =

Vs (Δδ)2 = χeff − χ fv = 0.46 RT

(30)

Assuming Vs ∼ 100 cm3/mol, then Δδ ∼ 3.4 at 25 °C. This would suggest an asphaltene solubility parameter value of ∼(17.6 + 3.4) = 21 MPa0.5. The H/C molar ratio of the AAM-1 asphaltene is 1.15, and according to the correlation shown in Figure 4 this material has an average solubility parameter of ∼21 MPa0.5. Because many studies of asphaltene aggregation are conducted in very dilute solutions, the marked asymmetry of the phase diagram has a number of ramifications. Figure 6 displays the binodal and spinodal calculated in terms of the bare value of χ, χ0, for an asphaltene/toluene dilute solution (0−20 g/L) obtained by subtracting χfv (see Table 1) from χeff. This, in turn, was used to calculate a corresponding solubility parameter 2127

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average solubility parameters for various asphaltenes are somewhat larger than what we calculate for the same samples, but the important point is that broad, often bimodal, distributions of values were calculated, most lying between 20.0 and 25.0 MPa0.5. The heterogeneous nature of asphaltenes is well-established, and various studies have shown that asphaltenes can be fractionated using various solvents into components with different chemical characteristics (e.g., H/C ratios).72−74 Rogel et al.74 separated asphaltenes into easy to dissolve and difficult to dissolve fractions, with the former having H/C ratios ∼ 1.2−1.3 and the latter 0.9−1.1. On the basis of the correlation shown in Figure 4, the latter would have average solubility parameters in the range 22.0−24.0 MPa0.5. (These fractions are no doubt also heterogeneous). Acevedo et al.73 separated their asphaltene into two fractions with H/C ratios ∼ 1.0 and 1.1, suggesting average solubility parameters in the range 22.0−23.0 MPa0.5. This indicates that there are components of many if not most asphaltenes that are in the thermodynamic sense insoluble in toluene. Taking the calculated values of Δδ for asphaltene/ toluene solutions shown in Figure 6 and adding them to the solubility parameter of toluene (δsolv) gives the top curve in Figure 7 and shows the phase boundary in terms of a Δδ + δsolv

Figure 6. Binodal and spinodal (insert) calculated using the attenuated association model (as in Figure 5) adjusted to account for χfvfor asphaltene/toluene solutions and plotted as χ0 (left-hand y-axis) vs the concentration of asphaltene for dilute solutions. The right-hand y-axis shows values of the solubility parameter difference squared, Δδ2, corresponding to these values of χ0.

difference, Δδ, from eq 28 and Δδ2 (linearly related to χ0) is included in the plot as the right-hand y-scale. Note that the spinodal line is outside the range of χ0 values used for the binodal plot and is shown as an insert. The two phase region in the main plot shown in Figure 6 is therefore metastable, and phase separation would occur by a process of nucleation and growth. Depending on the glass transition temperature of the asphaltene rich phase, this may be a very slow process. We will discuss this in more detail later. It can be seen from Figure 6 that at an asphaltene concentration of 12 g/L (calculated from the volume fraction, 0.01, by assuming a density of 1.2 g/cm3), the model predicts that only those asphaltenes or asphaltene components that differ in solubility parameter from toluene by a value of less than about 3.9 MPa0.5 will form a single phase solution of nanoaggregates dispersed in solvent. If stock solutions at this concentration are prepared, centrifuged to remove any insoluble material and diluted to obtain data at lower concentrations, then all such solutions should be single phase. However, solutions prepared directly at higher concentrations would have slightly different soluble asphaltene compositions. In addition, this assumes that the initially “dissolved” material forms a solution in the thermodynamic sense and does not contain microphase-separated clusters or loosely flocculated material that is difficult to remove by centrifugation and which is stabilized against further aggregation by the steric and kinetic factors discussed in the Introduction. The operational definition of an asphaltene is in terms of its solubility in toluene, but as Lesueur71 points out, this generally means not producing a precipitate rather than molecular solubility. Rogel64 estimated the solubility parameter distribution of asphaltenes using an additive group method applied to a large number of hypothetical asphaltene molecules constructed to reflect known structural parameters. Rogels’s calculated

Figure 7. Plots of Δδ + δsolv vs asphaltene concentration for various solvents and solvent mixtures using values of Δδ calculated using χ0 = χeff − χfv and values of χfv and δsolv from Table 1.

vs concentration plot. It can be seen that at volume fractions of 0.01 (∼12 g/L) only those asphaltene components with solubility parameters less than 22.2 MPa0.5 are calculated to be soluble in this solvent. The binodal in this concentration range appears fairly flat until concentrations of less than about 0.5 g/L are reached, but even at this concentration only those asphaltene components with a solubility parameter less than 23.0 MPa0.5 are predicted to form single phase solutions. However, the high solubility parameter components may form stabilized microphase-separated clusters where macroscopic 2128

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that asphaltene components with a solubility parameter difference (asphaltene/toluene) of less than about 4.5 MPa0.5 are initially dissolved, but at higher concentrations at least some of these will phase separate from solution. This again assumes that the initial solutions are truly single phase in the thermodynamic sense. Microphase-separated clusters may be initially dispersed in toluene but become susceptible to further aggregation at higher concentrations under the large gravitational forces imposed by centrifugation for extended periods of time. As Goual et al.7 pointed out, centrifugation is not a precise method for measuring what they interpreted as a CCC. However, near-infrared spectroscopy has been used to measure the onset of asphaltene precipitation, and this technique is presumably detecting molecular level events.77 Oh et al.78,79 titrated asphaltenes dissolved in various solvents with heptane, and break points in plots of the onset of precipitation vs asphaltene concentration were interpreted in terms of a critical aggregation concentration. We interpret the onset of precipitation in terms of a phase separation. If we take the concentration of (for example) toluene/heptane mixtures at the onset of precipitation from the figures reported by Oh et al.,79 we can calculate a volume fraction average of their solubility parameters and χfv using the parameters in Table 1. If we then assume a value for an asphaltene component solubility parameter, the data of Oh et al.79 can be replotted in terms of Δδ. For example, for asphaltene components with solubility parameters of 21.28 and 21.03 MPa0.5, we calculate the plots shown in Figure 8, where we have drawn lines through the data points simply as a guide to the eye. There appears to be a clear break in the plots at a volume fraction of asphaltene near 0.0014. These plots are superimposed upon the calculated binodal in this dilute concentration regime. For an asphaltene with a solubility parameter of 21.28 MPa0.5, all of the data

demixing is prevented by steric and kinetic factors. This is consistent with the results of scattering studies discussed earlier, where material with a molecular weight well in excess of that determined by VPO measurements is detected, even at very low concentrations. In particular, in order to account for X-ray scattering results obtained on nanofiltered material, Eyssautier et al.26 proposed a model where there are two characteristic asphaltene aggregation levels, the first to form nanoaggregates, while the second is a fractal aggregation of these nanoaggregates to form clusters of 5−16 nanoaggregates. A phase separation into a phase rich in nanoaggregates and one rich in clusters is also consistent with DOSY NMR work by Durand et al.75 and some older scattering work by Espinat et al.76 Durand et al.75 observed that an asphaltene from a Buzurgan feedstock displayed behavior characteristic of nanoaggregates at low asphaltene concentrations, less than about 0.25 wt %, but two families of aggregates above this concentration, one diffusing more quickly than the other. They interpreted this in terms of the formation of clusters at higher concentrations, which they referred to as macroaggregates. The asphaltene they studied had an average H/C atomic ratio of ∼1.07, suggesting according to Figure 4 an average solubility parameter of ∼22 MPa0.5. Referring back to Figure 7, this is consistent with a phase separation, which is predicted to occur at an asphaltene concentration near 25 g/L, or about 0.25 wt %. This close agreement is to some degree no doubt fortuitous given the assumptions of the model, but the important point is that the phase-separated domains are calculated to consist of a dilute asphaltene solution consisting of nanoaggregates and a solution that is more concentrated containing clusters. These would have very different diffusion rates. Similarly, in work that preceded this Espinat et al.76 also detected heterogeneities in toluene solution that are consistent with a phase separation into an asphaltene poor phase and a more concentrated phase that showed gel-like behavior. We will have a little more to say about gels later. Using χfv values tabulated in Table 1, plots of Δδ + δsolv vs asphaltene concentration were calculated for heptane and pentane and these are also shown in Figure 7, together with phase boundaries for 50/50 by volume mixtures of these solvents with toluene. These were calculated on the assumption of a volume fraction additivity of χfv and δsolv. It can be seen that these mixed solvents would dissolve some asphaltene components with solubility parameters between 20.0 and 21 MPa0.5, with the heptane/toluene mixtures dissolving some asphaltene components with higher solubility parameters than pentane/toluene mixtures. Looking at this from the point of view of precipitation, pentane and heptane would precipitate asphaltene components with different chemical characteristics. However, microphase separation into clusters would probably occur well before precipitation, as Wang et al.22 have shown that there is still some steric repulsion between asphaltene surfaces at toluene volume fractions greater than 0.2. Asphaltene “solutions” are frustratingly complex and not easily described. A microphase separation of nanoaggregates into small clusters is also consistent with the centrifugation data of Goual et al.7 mentioned earlier in this work. This aggregation appeared to commence at concentrations of about 0.3 g/L and achieved a maximum value of more than 90% of asphaltenes in aggregates at a concentration of about 1.5 g/L. These authors prepared initial stock solutions having an asphaltene concentration of 0.5 g/L. On the basis of Figure 6, this would suggest

Figure 8. Plots of the onset precipitation determined by Oh et al.79 as a function of composition. Solvent composition was used to calculate solubility parameters and χfv, which in turn were used to calculate values of χeff for assumed values of the asphaltene solubility parameters. Two such plots are shown. These plots are superimposed upon the calculated binodal. 2129

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in turn, kinetically limits subsequent coalescence. This can be explored in more detail using the calculated binodals and knowledge of asphaltene and solvent Tgs. However, a complex multicomponent material such as asphaltene would not be expected to have an easily measured or well-defined Tgand it does not. Nevertheless, Bazyleva et al.80 and Fulem et al.81 have shown that there is a broad solid to liquid transition that occurs between temperatures of 330 and 530 K in the materials they studied. Using a value of 117 K for the Tg of toluene82 and Tg values of 330 and 530 K to define a range for asphaltenes, we used the Fox equation83 to calculate (approximate) values for the Tgs of hypothetically single phase asphaltene/toluene mixtures as a function of composition. The Fox equation is w w 1 = 1 + 2 Tg Tg Tg (31)

points lie in the immiscible region of the phase diagram and precipitation would occur at all concentrations. However, it can be seen that the first data point (volume fraction asphaltene ∼ 0.0004) essentially lies on the binodal curve and would define the onset of precipitation. By adjusting the assumed solubility parameter of the asphaltene, the experimental data can be shifted to coincide with the calculated binodal. For an asphaltene with a solubility parameter of 21.03 MPa0.5, for example, the point of incipient precipitation occurs at a volume fraction of asphaltenes near 0.0046. In this fashion, the data of Oh et al.79 for three different asphaltene solutions (toluene, pyridine, and trichloroethylene) were used to calculate the solubility parameter of the asphaltene components that first precipitated from solution at various concentrations. These values are plotted in Figure 9, together with the calculated

1

2

where w is the weight fraction and the subscripts 1 and 2 refer to the components of the mixture. The results are shown in Figure 10, where we have used the component densities to

Figure 9. Calculated asphaltene component solubility parameters determined using the procedure described in Figure 8, together with the calculated solubility parameters of the solvent/heptane mixtures.

solubility parameters of the solvent/heptane mixtures. In this concentration range the calculated solubility parameters of the asphaltenes that precipitate first lie in a band of values ∼21.2 ± 0.4 MPa0.5. The scatter in the plot can be reasonably attributed to errors in calculated values of solubility parameters and χfv together with any uncertainties in the experimental data. The calculated values of the asphaltene component solubility parameters appear to vary continuously with concentration and mirror the composition dependence of the solubility parameters of the solvents. We interpret the apparent break in the onset of precipitation experimental data plots as being due to the shape of the binodal curve, which in this concentration range descends steeply at low concentrations and then flattens out. So far we have largely focused on dilute asphaltenes solutions, but it is interesting to examine the asphaltene rich phase of a phase-separated asphaltene solution. As mentioned earlier, Sirota and Lin17 proposed that asphaltene solutions undergo a liquid/liquid phase separation. They argued that because of its proximity to the glass transition temperature (Tg), the asphaltene rich phase is viscous or solid-like and this,

Figure 10. Plots of the calculated Tgs of asphaltene/toluene mixtures plotted as a function of asphaltene concentration. Also shown are calculated asphaltene/toluene binodals calculated using the values of solubility parameter differences shown on the plot.

convert to a volume fraction concentration scale. Also shown in Figure 10 are binodals calculated as temperature/composition plots. We have found that in the temperature range used here the variation of KA results in only minor changes to the critical value of χ, because the asphaltenes still maintain a significant degree of association at the high end of this temperature range. For example, if KA is 11000 at 25 °C, then at 125 °C its value is 1000 (assuming the free energy of association is ∼23 kJ/mol). However, this large decrease in KA results in only a small increase in the critical value of of χeff of 0.03, from 0.65 to 0.68. (Plots showing the variation of χeff with KA are given in the Supporting Information.) Similarly, we neglect small changes in χfv in this temperature range. As a result, the temperature dependence of phase behavior depends largely on χ0, so we can use eq 28 to calculate approximate temperature/composition 2130

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Energy & Fuels binodals for selected values of the asphaltene/toluene solubility parameter difference, Δδ. It can be seen from Figure 10 that, for asphaltene components with the highest Tgs, there is an intersection of the Tg/composition line with the binodal at temperatures close to ambient for asphaltene solutions with values of Δδ between 5 and 6 MPa0.5, corresponding to asphaltenes with a solubility parameter ∼23.7 MPa0.5 for toluene solutions. Asphaltenes with smaller (average) solubility parameter values would give an intersection in toluene/heptane or toluene/pentane mixed solvents. As mentioned previously, there is a lot of evidence that there are hard to dissolve asphaltene fractions, which would presumably have higher solubility parameters than more easily dissolved fractions. It would therefore be expected that an asphaltene rich phase would have components that would be a highly viscous gel-like material or even glassy. This would to some degree arrest or limit the extent of phase separation. Another interesting aspect of the concentrated asphaltene/ solvent regime is shown in Figure 11, where the binodals

It is well-known that asphaltenes swell in various solvents.84,85 For example, Nikooyeh et al.85 recently showed that the projected area of a pentane asphaltene observed under a microscope increased ∼17% in octane. Assuming a value of δsolv = 15.5 MPa0.5 and referring to Figure 11, this would suggest that asphaltene components with solubility parameters ∼21 MPa0.5 will show a degree of swelling consistent with this value (i.e., volume fraction asphaltene ∼ 0.83). Other components with larger solubility parameters will also swell, but to a lesser extent. However, this prediction is simply on the basis of the calculated composition of phase-separated domains and says nothing about local structure. Would such miscible solutions be essentially a mixture of compact nanoaggregates and solvent, or would the nanoaggregates themselves swell? In this regard, we suggest that one has to distinguish between nanoaggregates and clusters. Gray et al.86 have cited a wealth of literature that indicate that asphaltenes form supramolecular structures with association taking various forms to give a range of architectures. These structures are porous enough to exchange solvent molecules with surrounding media and occlude and trap material. On the other hand, Eyssautier et al.25 proposed a core/shell model for nanoaggregates and used neutron contrast variation to show that the chemical composition of the core is not modified by solvent diffusion through this aromatic region. We suggest that these models are consistent with one another if the supramolecular structures proposed by Gray et al.86 are phase-separated fractal clusters made up of the core/shell nanoaggregates described by Eyssautier et al.25

■

CONCLUSION It has been shown that the attenuated association model can be used to obtain closed form expressions for the number and weight-average degree of association and the number of aggregated species. Using these equations, data reported in the literature that have been interpreted in terms of a CNAC are reexamined and are shown to be also consistent with a continuous process that results in the formation of nanoaggregates, as previously proposed by other workers.11−15 In addition, the equation for the free energy given in previous work1 is modified to account in an approximate way for free volume effects. Spinodals and binodals were then calculated, and it is shown that nanoaggregate formation does not necessarily imply colloid formation, but does reduce the critical value of χ relative to nonassociating molecules of the same size. The calculations suggest that asphaltene components with a high solubility parameter are insoluble in toluene but on the basis of various studies reported in the literature17−22 we suggest that at least in toluene there is only a microphase separation of these components into clusters that are sterically and kinetically inhibited from further aggregation.

Figure 11. Calculated binodals plotted as a solubility parameter difference between an asphaltene and the solvents toluene, heptane, and octane. Also shown on the plots are asphaltene solubility parameters defining the phase boundary that would be expected for heptane (δ = 15.1 MPa0.5) and octane (δ = 15.5 MPa0.5).

calculated for toluene, octane, and heptane are shown as a plot of Δδ vs composition (at 25 °C). These binodals were calculated from values of χeff by subtracting values of χfv listed in Table 1. It can be seen that, at volume fractions of asphaltenes greater than about 0.75, the calculations indicate that a single phase solution is formed with these solvents if the solubility parameter difference is between about 4.9 and 5.1 MPa0.5. Less concentrated asphaltene solutions would separate to form a phase with this concentration (and a phase that is almost pure solvent). This means that in heptane (δ ∼ 15.1 MPa0.5), asphaltene components with a solubility parameter close to 20 MPa0.5 would be miscible at this concentration. This material would be a 75/25 (asphaltene/heptol by volume) swollen asphaltene gel.5

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ASSOCIATED CONTENT

S Supporting Information *

Text describing the attenuated association models for stoichiometry and thermodynamics, sensitivity, calculation of binodals, variation of χeff with temperature, and additional references and figures showing degree of association of the asphaltenes, number fractions of i-mers as a function of asphaltene volume fraction, chemical potentials of an asphaltene and solvent, replacement chemical potential of an asphaltene and solvent, and spinodal calculated as χeff vs volume 2131

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Energy & Fuels

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fraction of asphaltenes. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the support of the Federal Highway Administration under Subcontract SES No. 12-01. REFERENCES

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