Quantitative Modeling of Formation of Asphaltene Nanoaggregates

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Quantitative Modeling of Formation of Asphaltene Nanoaggregates Murray R Gray, and Harvey William Yarranton Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.9b02400 • Publication Date (Web): 28 Aug 2019 Downloaded from pubs.acs.org on August 28, 2019

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Quantitative Modeling of Formation of Asphaltene Nanoaggregates Murray R Gray1 and Harvey W. Yarranton2 1. Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 1H9, Canada. 2. Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, T2N 1N4, Canada. KEYWORDS Asphaltenes, aggregation, supramolecular, association, dimerization



Corresponding Author: Murray R Gray, [email protected]

ACS Paragon Plus Environment

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ABSTRACT

Association of asphaltene molecules to form aggregates is a defining property of this fraction of petroleum, which impacts on all aspects of production and refining. Association has been detected over a very wide range of concentrations, but the quantitative modeling of this aggregation of molecules has mainly been restricted to fitting relatively highasphaltene concentration data from vapor-pressure osmometry. This paper examines the capability of the termination/propagation model for association to represent lowconcentration behavior. The addition of an additional strongly-associating fraction to the model is necessary to obtain dimer formation over the observed range of concentrations. Unfortunately, the data required to fit such a model with confidence are lacking. Data from ultracentrifugation

of

asphaltene

aggregates

can

also

be

fitted

to

a

termination/propagation model. Transient measurements of ultracentrifuge concentration profiles have the potential to enable more comprehensive models of association behavior, especially when applied to subfractions with known aggregation behavior.


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1. Introduction The association of asphaltene molecules into aggregates in crude oil and in solvents such as toluene is one of the most important and enigmatic properties of this solubility fraction of petroleum. Asphaltene association has significant implications for many crude oil applications. For example, asphaltene aggregates are surface active and have been shown to adsorb at the water-oil interface and contribute to emulsion stability1. Asphaltene association impacts phase behavior at minimum by altering the mole fraction and effective molecular weight of the asphaltenes2. The interactions between the asphaltene aggregates suspended in crude oil make a significant contribution to its high viscosity3,

4

and to viscoelastic behavior at low temperature5. During processing, the

formation of aggregates changes the local environment of asphaltene species for transport to heated surfaces and into catalyst pores, and may direct reaction pathways6, 7.

Throughout this paper, we define aggregates as assemblies of two or more molecules formed by molecular association interactions. These aggregates have sizes in the range


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of 2-20 nm

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and are stably suspended in most unmodified crude oils and in aromatic

solvents such as toluene, as distinct from much larger domains of asphaltene material that forms during flocculation or precipitation when it is no longer stable in the crude oil or solvent. Asphaltene self-association has been investigated mainly using asphaltenes extracted from the source oil and dissolved in toluene, an effective solvent for dissolving or dispersing this fraction of crude oil. Many techniques have been applied over a range of five orders of magnitude of asphaltene concentration, from 10-1 to 104 mg/L. The average size of asphaltene molecules is in the order of 2-3 nm 8 while the average aggregate size is in the range of 5-20 nm, depending on the experimental method and the concentration. The average molar mass, which is a more accessible property for modeling of association, is in the range 700-1200 Da for the asphaltenes detected as dissociated species9, 10, and order 104 for aggregates based on vapor-pressure osmometry (VPO)8. Significant evidence suggests that the true molecular weight distribution of the asphaltene molecules that tend to aggregate has not yet been determined11.


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The concentration at which asphaltenes begin to associate is not well established. At infinite dilution, an asphaltene molecule is by definition dissociated. Hence, after equilibration at sufficiently low concentrations, most asphaltenes must be dissociated. Association has been detected by spectroscopic methods at concentrations as low as 0.15 mg/L

12, 13.

At much higher concentrations, where most studies are conducted, the

asphaltene aggregates are expected to predominate. In typical crude oils, where the asphaltene concentration is over 1 wt% (over 8000 mg/L), most of asphaltenes capable of self-association are likely to be aggregated. The aggregation mechanism is still debated with two main proposed concepts: threshold aggregation and step-wise aggregation. Threshold aggregation is characterized by a critical aggregation concentration (CAC) below which only free molecules exist and above which aggregates form, each with a significant number of molecules per aggregate. Hence, there is a step change in aggregate size at the CAC and the aggregate size distribution does not necessarily change as the asphaltene concentration increases. A qualitative thermodynamic model for the aggregation of alkyl-aromatics into colloids, a form of threshold model, was proposed by Rogel14. Linear step-wise aggregation is


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characterized by a gradual change in size from single molecules to dimers to trimers, etc. There is no critical aggregation concentration and the size distribution will tend to broaden as the asphaltene concentration increases. Linear step-wise models for aggregation have been successful in quantitative modeling of VPO data for asphaltenes at concentrations in the range from 103 to 105 mg/L 15-18. Despite the importance of asphaltene self-association, no quantitative model has been proposed to model molecular association to form a distribution of aggregate sizes over the full range of concentration, ranging from highly dilute solutions in pure solvents to heavy crude oil. The objectives of this paper are to critically examine the proposed association concepts and to examine the ability of thermodynamic equilibrium models for association to represent the available data.

1.1 Available Data on Asphaltene Association To assess the two proposed association concepts, consider the available data on asphaltene association. Table 1 lists the experimental evidence that has been attributed to observation of a critical aggregation concentration in asphaltene solutions. Table 2 lists


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the experimental evidence for dimerization and Table 3 lists the experimental evidence on the effect of asphaltene concentration on the aggregate size distribution.

Table 1. Experimental observations attributed to critical concentration for asphaltene aggregate formation in toluene Method

Observation

Reference

Mass spectrometry of

Onset of detection of large

McKenna et al.19

aggregates

aggregates over 2000 Da at 50 mg/L from solution of 50/50 toluene and methanol

Ultracentrifugation

Sedimentation measured

Mostowfi et al.20, Goual et

at concentrations over 50

al.21

mg/L Calorimetry of dissolution

Change in slope of heat

Andersen and Birdi22;

evolved versus

Andersen and

concentration at 3240 to

Christiansen23

4000 mg/L in toluene DC conductivity

Change in slope with

Zheng et al.24; Goual et

concentration at 40-150

al.21

mg/L Ultrasonic velocity

Change in slope below 50

Andreatta et al.25

mg/L


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NMR signal intensity and

Non-linear change at low

spin-diffusion

concentration

Page 8 of 49

Lisitza et al.26

Table 2. Experimental detection of dimerization of asphaltene components at low concentration in toluene Method

Observation

Reference

Refractive index

Shift in refractive index due Evdokimov and Fesan12 to dimer formation at circa 0.15 mg/L

Steady state fluorescence

Shift in spectra due to

Evdokimov and Fesan12;

emission spectroscopy

dimer formation at 0.15

Evdokimov et al.13

mg/L to 0.75 mg/L Synchronous fluorescence

Detection of quenching of

Merino-Garcia and

spectroscopy

fluorescent intensity at 50

Andersen27

mg/L Small-volume isothermal

Association in solution

Merino-Garcia and

titration calorimetry

persists below 34 mg/L

Andersen27

Table 3. Experimental evidence on the effect of asphaltene concentration in toluene on aggregate size. Method

Observation

Reference

Isothermal titration

Continuous decrease in

Merino-Garcia and

calorimetry

heat of mixing on addition

Andersen27

of asphaltenes to anhydrous toluene


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Vapor-pressure

Smooth increase in AMW

Agrawala and Yarranton15;

osmometry (VPO)

with concentration from

Zhang et al.18; Powers et

1000 to 60000 mg/L

al.16

Shift in refractive index

Evdokimov and Fesan12

Refractive index

with concentration above 0.15 mg/L Small-angle x-ray and

Gradual dissociation on

neutron scattering

dilution to 15 mg/L

Hoepfner and Fogler28

(0.00125 vol%) Rayleigh scattering

Smooth dependence of

Morimoto et al.29

aggregate size on concentration from 2020,000 mg/L

Critical Aggregation Concentration: The studies listed in Table 1 presented data that were interpreted by the respective authors as indicating a critical aggregation concentration. The data from mass spectrometry19 give proof of the detection of aggregated species at surprisingly low concentrations circa 50 mg/L, but do not provide calibration to relate signal intensity to mass or number of aggregates. The work on ultracentrifugation20 is a first step in developing more quantitative methods to study the mass concentrations of aggregated species, but the data do not show results below the


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proposed onset concentration. Interpretation of ultracentrifugation data will be discussed in a subsequent section. In addition, the application of these methods to complex mixtures at very low concentrations has not been validated by detection of aggregation of defined components in the presence of a mixture of asphaltene components, or by verification that a mixture of aggregating components will give a detectable transition. Merino-Garcia and Andersen27 suggested that a critical concentration for complex mixtures of asphaltene components was unlikely, by analogy to surfactant mixtures which do not show a distinct critical micelle concentration by interfacial tension30 or ultrasonic velocity31 methods. Asphaltenes are an extremely complex mixture of components as underscored by the rapidly developing literature on the diversity of molecular structure32 and the range of sub-fractions of asphaltenes that display different stabilities in solution and different of molecular architectures33. Hence, it is more likely that the reported critical aggregation concentrations are artifacts created by the detection limits of the experimental methods used in these studies. Dimerization: The smallest possible asphaltene aggregate is a dimer, formed by the association of two molecules. Formation of dimers can only be considered a threshold


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event if higher order aggregates are not possible; in fact, dimers are normally the first step in the sequence of interactions leading to a distribution of aggregate sizes. Hence, the detection of dimers would be more consistent with the step-wise association interpretation. Data from refractive index, fluorescence spectroscopy, and titration calorimetry suggest that dimers are detected at concentrations in the range 0.15-50 mg/L (Table 2). However, the unambiguous detection of dimer formation is complicated by the broad distribution of the molecular weight of asphaltene molecules, mainly between 600 and 1600 Da33, and the complex functionality of these molecules. Does each method detect association of 0.1%, 1%, or 10% of the components in a complex mixture? A small fraction of highly associated aggregates could, in some cases, be interpreted as dimers. Full validation would require detection of association of calibration compounds at lower concentrations against the background of the complex asphaltene mixture. None of the studies reported such positive control experiments, or a detailed analysis of the fraction of the mixture detected as dimer, but the results suggest that models for asphaltene association should be tested down to very low concentrations.


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Effect of Concentration on Aggregate Size: The data from the studies listed in Table 3 show a continuous smooth response of the measured variable over the experimental range of concentration, indicating no sharp change in the size or quantity of the aggregates in solution. In combination, these studies suggest continuous growth in size and number of aggregates over a very wide range of concentration, from 0.15 mg/L to 60,000 mg/L, as shown in Figure 1. The gradual monotonic increase is characteristic of step-wise association as opposed to threshold association where the aggregate size is expected to be nearly constant above the critical association threshold. Therefore, we conclude that a step-wise addition mechanism is the appropriate approach for modeling of asphaltene aggregation.


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Detection of dimerization

Detection of higher aggregates (n>2)

5

Degree of polymerization

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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4

3

VPO

Sedimentation Mass spec

2

1

0 10-3

10-2

10-1

100

101

102

103

104

105

Asphaltene concentration, mg/L Figure 1. Schematic of asphaltene association as a function of concentration. Onset of dimer formation is from UV spectroscopic data 12, critical concentration of asphaltene for formation of large aggregates is from mass spectrometric data polymerization at higher concentrations is from VPO data

16

19,

and degree of

with an assumed molecular

weight of 1100 Da.

For quantitative modeling of association as a function of concentration or temperature, the experimental data must give a molar or mass balance on the entire asphaltene mixture. The majority of methods listed in Table 1 and Table 2 rely on detecting a shift or


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inflection in a signal as a function of sample concentration.

Page 14 of 49

For example, the

spectroscopic methods give a qualitative indication of the shift in solution due to the onset of dimer formation, rather than a quantitative measurement of dimer concentration. Two methods stand out as being amenable to more quantitative modeling: 1) VPO in toluene to give the average molecular weight of the components in a sample, calibrated against a known standard compound, and; 2) sedimentation in an ultracentrifuge to give the mass fraction recovered as sediment in the bottom of the tube. Only VPO gives direct data on degree of polymerization (DP) via the experimental determination of average molecular weight. The DP is defined as: 𝑛

𝐷𝑃 =

∑𝑖 = 1𝑥𝑖𝑀𝑊𝑖 𝑀𝑊𝑚𝑜𝑛𝑜

=

𝑀𝑊𝑎𝑣𝑔 𝑀𝑊𝑚𝑜𝑛𝑜

(1)

where the mixture contains n species in solution (aggregates and remaining free molecules). When the asphaltenes are fully dissociated, DP=1. Unlike defined chemical mixtures, the exact value of DP for asphaltenes depends on the average molecular weight assigned to the dissociated asphaltenes (MWmol). In Figure 1, we used 1100 Da as a representative value.


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Both VPO and sedimentation are dominated by the bulk behavior of the mixture and are consequently insensitive to low-concentration species. For example, the average molecular weight from VPO will not differentiate a value of DP of 1.01 from 1.001. These small differences are important for the methods used at low concentration, which must be sensitive to the behavior of small sub-fractions of the complex asphaltene mixture in order to define the onset of dimer formation.

1.2 Models for Step Wise Association in Solution Non-covalent or supramolecular polymerization has been extensively studied, with an excellent comprehensive review provided by De Greef et al. 34. The extensive studies of association of defined chemical compounds show three types of polymerization behavior; isodesmic growth, ring-chain behavior, and cooperative growth. In isodesmic growth, molecules add linearly in a stepwise fashion, giving a high degree of polydispersity. The most important parameter is the association constant of the linking supramolecular units. The DP increases monotonically with concentration. To help account for the tendency for large aggregates to reach a limiting size, an attenuation factor can be added to the


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association step35. In ring-chain behavior, the molecules can either form linear chains or cyclic groups, and the degree of polymerization is determined by the equilibrium between these states. In this case a critical concentration is observed; below the critical concentration the molecules form only rings and DP is low. Above the critical value much larger linear chains are formed and the DP increases with concentration. In the cooperative case, the initial association follows the stepwise isodesmic path until a critical nucleus is achieved, then the association constant increases due to cooperative effects. The DP increases with increasing concentration, then accelerates above the nucleation condition. Unlike supramolecular polymerization, where DP >102 is expected as concentration increases, the average size of asphaltene aggregates in crude oil or in toluene solution never exceeds a value in the range of 10-20 nm, even at high concentrations. The VPO data imply DP < 5, as indicated in Figure 1. The challenge for asphaltenes is to model aggregation that begins at very low concentration but then stops even when the aggregates are small. This behavior is not consistent with the ring-chain or cooperative


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growth models. The isodesmic growth model may apply, but a limit to the aggregation is required. Two approaches have been used to represent this experimental observation in a stepwise association model. The Yarranton group15,

16, 36

assumed the presence of

different species in the asphaltene mixture, each with different properties. The size of the aggregates in this model is determined by the composition of the mixture. In contrast, Zhang et al.18 used a hindrance factor to represent a reduction in association equilibrium constant as a function of the size of the aggregate. Both models are successful in fitting the experimental data, and give parameters that empirically represent changes in the underlying chemistry of the asphaltene components. In this paper, we adopt the model of Powers et al.16 for elaboration because it is formulated to represent the interactions of different types of components in solution.

2. Theory 2.1 Propagator-Terminator (P-T) Model with Three Asphaltene Components


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The propagator-terminator model15, 16, 36 assumes that some components in the crude oil (mainly found in the asphaltene fraction) self-associate in a manner analogous to linear polymerization. Molecules that have multiple active sites, such as heteroatoms and aromatic stacks, can link to other similar molecules to form a linear chain, similar to propagators in polymerization. Molecules with only one site can end a chain, similar to terminators. Molecules with no active sites are neutral and do not participate in the selfassociation. In this analogy, the links between molecules arise from intermolecular forces (dispersion, polar, hydrogen bonding) rather than covalent bonds. The analogy is an oversimplification of the actual association mechanism but is sufficient to model the available data from VPO.

The general propagation reaction is given by, 𝐾

𝑃1 + 𝑃𝑛 𝑃𝑛 + 1

(2)

The concentration of the n+1 aggregate is given by,

[𝑃𝑛 + 1] = 𝐾[𝑃1][𝑃𝑛] = 𝐾𝑛[𝑃1]𝑛 + 1

(3)

and the general termination reaction is given by: 𝐾

𝑃𝑛 + 𝑇1 𝑃𝑛𝑇1

(4)

and the general termination equation is given by:


18

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[𝑃𝑛𝑇1] = 𝐾[𝑃𝑛][𝑇1] = 𝐾𝑛[𝑃1]𝑛[𝑇1]

(5)

where T1 is a terminator molecule, P1 is a molecule that propagates, Pn is an aggregate, n is the number of the molecules in the aggregate, and K is the association constant. For simplification, it is assumed that the constant K is equal for all the reactions regardless of molecule type or aggregate size. The reaction scheme is solved from the mass balance equations for propagators and terminators. The equilibrium concentrations of propagators and terminators are given by:

[𝑃1] =

1 + 𝐾(2[𝑃1]0 + [𝑇1]0) ―

(1 + 𝐾(2[𝑃1]0 + [𝑇1]0))2 ― 4𝐾2[𝑃1]0([𝑃1]0 + [𝑇1]0) 2𝐾2([𝑃1]0 + [𝑇]0)

[𝑇1] = [𝑇1]0(1 ― 𝐾[𝑃1])

(6) (7)

where [P1]0 and [T1]0 are the initial concentrations of propagators and terminators, respectively. This model constitutes a simple example of hetero-association, where the growing aggregate is considered to be “capped” by a single terminator. Further discussion of more sophisticated models for hetero-association is provided by Buchelnikov et al.37 .

For solutions of asphaltenes in solvent, the inputs to the model are the ratio of terminator to propagator molecules in the whole solution (T/P)0 and K. The asphaltene mass concentration and the solvent molar volume must also be specified to calculate the initial molar concentration of the terminators ([T1]0) and propagators ([P1]0). In non-aggregated systems with neutrals, the initial mole fraction of terminators and propagators are calculated as follows:

(𝑃𝑇)0 and 𝑥 𝑇0 = 𝑇 (1 ― 𝑥𝑁0) 1 + (𝑃 ) 0

𝑥 𝑃0 = 1 ― 𝑥 𝑇 0 ― 𝑥 𝑁 0

(8)

where xTo, xPo and 𝑥𝑁0are the mole fraction of terminators, propagators and neutrals in the asphaltenes, respectively. The molecular weight of the asphaltene molecules is calculated as: 𝑀𝑊𝑚𝑜𝑙 = 𝑥𝑇0𝑀𝑊𝑇 + (1 ― 𝑥𝑇0 ― 𝑥𝑃)𝑀𝑊𝑃 + 𝑥𝑁0𝑀𝑊𝑁

(9)


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where MWmol is the average molecular weight of the asphaltenes before any association and 𝑀𝑊𝑇, 𝑀𝑊𝑃 and 𝑀𝑊𝑁 are the molecular weights of terminators, propagators and neutrals. The initial mole fractions of propagators, terminators, and neutrals in solution are calculated as follows:

[𝑃1]0 =

1

(1 +

)(1 + ( ) )

𝑀𝑊𝑚𝑜𝑙 𝐶𝐴𝜐𝑆

𝑇 𝑃 0

(1 ― 𝑥𝑁0)

(10)

(𝑃𝑇)0

[𝑇]0 = [𝑃1]0 [𝑁]0 =

(11)

𝑥𝑁0

(1 +

(12)

)

𝑀𝑊𝑚𝑜𝑙 𝐶𝐴𝜐𝑆

The average molecular weight of the aggregated system is given by,

(

𝑛𝑚𝑎𝑥

)

𝑀𝑊𝑎𝑣𝑔 = (1 ― 𝑥𝑁) ∑𝑛 = 1 (𝑥[𝑃𝑛].𝑀𝑊[𝑃𝑛] + 𝑥[𝑃𝑛𝑇].𝑀𝑊[𝑃𝑛𝑇]) + 𝑥𝑁.𝑀𝑊𝑁

(13)

where 𝑥𝑁 is the mole fraction of neutrals in the aggregated system and is different than 𝑥𝑁0. The model is run by adjusting fitting parameters, (T/P)0 and K until the calculated average molecular weight fits the measured experimental molecular weight data. The outputs of the model are the average molecular weight and the mole fraction (or mass fraction) of each aggregate at the specified mass concentration of asphaltenes in the solvent.

2.2 Propagator-Terminator Model with Four Asphaltene Components (T/2P) Given the molecular diversity of the asphaltene components, the mixture is likely to give a distribution of strengths of association, rather than a single value of K. The simplest case is that one fraction of propagators (P1) associates more strongly than the remainder


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(P2), giving more aggregation at lower concentrations. For the strong aggregates, the association reaction is: 𝐾1

𝑃11 + 𝑃1𝑛 𝑃1𝑛 + 1

(14)

The concentration of the n+1 aggregate is given by,

[𝑃1𝑛 + 1] = 𝐾1[𝑃11][𝑃1𝑛] = 𝐾𝑛1[𝑃11]𝑛 + 1

(15)

The addition reaction for a weaker propagator to an aggregate is: 𝐾2

𝑃21 + 𝑃1𝑛 𝑃1𝑛𝑃21

(14)

The concentration of the mixed aggregate is given by,

[𝑃1𝑛𝑃21] = 𝐾2[𝑃21][𝑃1𝑛] = 𝐾2𝐾𝑛1 ― 1[𝑃21][𝑃11]𝑛

(15)

The termination reaction is unchanged, in that addition of a terminator stops the growth of any aggregate as in reaction (4). 𝐾2

𝑃1𝑖 𝑃2𝑗 + 𝑇1 𝑃1𝑖 𝑃2𝑗 𝑇1

(16)

The general termination equation is given by: 𝑗

[𝑃1𝑖 𝑃2𝑗 𝑇1] = [𝑃1𝑖 𝑃2𝑗 ][𝑇1] = 𝐾𝑗2+ 1𝐾𝑖1― 1[𝑃11]𝑖[𝑃21] [𝑇1]

(17)

In this scheme for association of two propagators and a terminator, a given aggregate will have 𝑖 ≥ 0 units of strong propagators, 𝑗 ≥ 0 units of weak propagators, and zero or one terminator. At equilibrium, this model gives a set of non-linear simultaneous equations for the concentrations of all possible aggregates in terms of the free concentrations of the


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Page 22 of 49

two propagators and the terminator, as in Equation (17), subject to the overall material balances: 𝑗

𝑗

[𝑃11]0 = ∑𝑚𝑗= 0∑𝑛𝑖= 1𝑖(𝐾𝑗2+ 1𝐾𝑖1― 1[𝑃11]𝑖[𝑃21] [𝑇1] + 𝐾𝑗2𝐾𝑖1― 1[𝑃11]𝑖[𝑃21] ) 𝑛 𝑚 ∑𝑖 = 1∑𝑗 = 1𝑗

(

𝑗

𝑖

𝐾𝑗2+ 1𝐾𝑖1― 1[𝑃11] [𝑃21] [𝑇1]

)

(18)

+ 𝐾2[𝑃21][𝑇1] + [𝑃21]

(19)

𝑖 [𝑇1]0 = ∑𝑛𝑖= 1∑𝑚 𝐾𝑗 + 1𝐾𝑖1― 1[𝑃11] [𝑃21] [𝑇1]) + 𝐾2[𝑃21][𝑇1] + [𝑇1] 𝑗 = 0( 2

(20)

[𝑃21]0 =

𝑖

+ 𝐾𝑗2𝐾𝑖1― 1[𝑃11] [𝑃21]

𝑗

𝑗

The initial concentrations of total propagator, terminator, and neutral are given by Equations (8), (10), (11) and (12). The average asphaltene molecular weight is given by Equation (9), and the average molecular weight for the aggregated system is given by the sum of the molecular weights of the aggregated and free species as in Equation (13):

(

𝑛𝑚𝑎𝑥

𝑛𝑚𝑎𝑥

1

)

𝑀𝑊𝑎𝑣𝑔 = (1 ― 𝑥𝑁) ∑𝑖 = 0 ∑𝑗 = 0 ∑𝑘 = 0𝑥[𝑃1𝑖 𝑃2𝑗 𝑇𝑘](𝑖 ∙ 𝑀𝑊[𝑃11] + 𝑗 ∙ 𝑀𝑊[𝑃21] + 𝑘 ∙ 𝑀𝑊[𝑇1]) + 𝑥𝑁. (21)

𝑀𝑊𝑁

No analytical solution of the simultaneous equations is possible, so the values for [𝑃11],

[𝑃21],

and [𝑇1] in the simultaneous Equations (18) through (20) were solved by a MATLAB

program using the fsolv function, for specified values of [𝑃11]0, [𝑃21]0, [𝑇1]0, K1 and K2. Aggregate concentrations were calculated for up to 40 propagators in an aggregate, each


22

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Energy & Fuels

of molecular weight of 1200 g/mol, to represent the full potential distribution of aggregate molecular weights.

3. Results 3.1. Step Wise Association Model of Example VPO Data Data for molar mass from VPO are illustrated in Figure 2 for two sets of C7-asphaltenes from Western Canadian bitumens, and their subfractions. The whole asphaltenes from unreacted bitumen vacuum residue (Panel (a) of Figure 2) give an almost three-fold increase of average molar mass, from 1800 g/mol at 1000 mg/L to 5300 g/mol at 60000 mg/L. The heavier, less stable fractions precipitated from heptane/toluene mixtures (H61H and H80H) gave more association, with 4- to 8-fold increases in molar mass with concentration. In contrast, the samples after hydroconversion (Panel (b) of Figure 2) gave asphaltenes with much less tendency to associate at any of the concentrations tested. The whole sample and the two heavy fractions (HT50 and HT80) all gave a two-fold increase in molar mass when the concentration was increased from 1000 to 60000 mg/L. Cracking of attached rings and chains, partial hydrogenation of the larger aromatic cores,


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Page 24 of 49

and significant sulfur removal all contributed to a mixture with much less tendency to associate than in the unprocessed bitumens. The continuous curves in Figure 2 are from fitting of the T/P model using the parameters listed in Table 4. The data from VPO are not sufficient to uniquely fit all the parameters used in the model. The minimum molar mass at low concentration is determined by the weighted sum of the three component, terminators, propagators and neutrals. The increase in average molecular weight with concentration is determined by the mass fraction of neutrals, wN, the ratio T/P, and the association constant K. The molar mass of the subfractions of the two asphaltenes as a function of concentration were fitted by adjusting only wN and T/P (Figure 2 and Table 4). The trends in these parameters from the whole samples through the subfractions are reasonable; the more easily precipitated materials (H61H, H80H, HT50H and HT80H) have lower values of both wN and T/P than the whole sample. This result is consistent with retention of more of the neutrals and terminators in heptane-toluene solutions, and propagators being more abundant in the precipitates.


24

Page 25 of 49

16000

a) Bitumen Vacuum Residue

14000

Whole H61H H80H

12000 10000

Molecular weight, g/mol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

8000 6000 4000 2000 0

b) Hydroconverted Bitumen

6000

Whole HT50H HT80H

4000 2000 0 0

10000

20000

30000

40000

50000

60000

Concentration, mg/L

Figure 2. Molar mass of C7 asphaltenes from unprocessed bitumen vacuum residue (a), and from a bitumen at 77% conversion of vacuum residue by catalytic hydroconversion (b). Data for each sample show whole C7 asphaltenes and heavy subfractions precipitated from heptane-toluene mixtures of 61% and 80% heptane in (a) and 50% and


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Page 26 of 49

80% heptane in (b). Data and curves showing fits to the propagation-termination model are from Powers38 and Powers et al.16 with parameters from Table 4.


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Table 4. Yield of subfractions and parameters for fitting the T/P model to the VPO data of Figure 2 Sample

Yield, wt% of

Mass

MWT,

MWP,

C7

fraction of

g/mol

g/mol

asphaltenes

neutrals, wN

T/P

K

Bitumen Vacuum Residue C7 Asphaltenes Whole

100

0.08

800

1200

0.08

38000

H61H fraction

48

0.0

800

1200

0.02

38000

H80H fraction

76

0.02

800

1200

0.06

38000

Hydroconverted Bitumen C7 Asphaltenes, at 77% conversion of vacuum residue Whole

100

0.19

450

650

0.17

7500

HT50H fraction

47

0

450

650

0.04

7500

HT80H fraction

68

0.05

450

650

0.14

7500

The H61H and the HT50H fractions contain the material that precipitates at much lower heptane concentration than the whole asphaltenes, and these sub-fractions also aggregate more readily at the molecular level in toluene than the respective whole C7asphaltenes, giving higher molecular weights (Figure 2). Because the T/P model parameters are not unique, the data for the subfractions can also be fitted by holding T/P constant and adjusting K. For example, the HT61H fraction is fitted equally well by T/P


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Page 28 of 49

values ranging from 0.01 to 0.04, with a compensating range of K from 31000 to 49000. There are no independent analytical data to define the concentrations of the components used in the model; neutrals, terminators, and propagators, therefore, fitting either T/P or

K for solubility subfractions is a valid approach. Zhang et al.17 used a comparable stepwise-association model to fit VPO data from a series of supercritical-extraction subfractions of vacuum residue, and determined a range of association constants from 2.5 for the lightest fraction to 1304 for the C5-asphaltene rich end cut. These model parameters are not directly comparable to the association constants of Table 4, but the results emphasize that the asphaltene fraction of petroleum is expected to contain components with a wide range of association behavior.

3.2 Extrapolation to Low Concentration Range The data of Figure 3 and Figure 4 show the results from the extrapolation of the P-T model of Powers et al.

16

down to 0.1 mg/L. This extrapolation goes far below the

concentration range where the model was been fitted to experimental VPO data. The extrapolated results indicate that dimerization should begin to affect the DP at circa 10


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Energy & Fuels

mg/L, a much higher concentration than indicated by spectroscopic studies12. As expected, the model indicates a monotonic decrease in DP with decreasing concentration, with no critical concentration range where DP drops more rapidly. In order to obtain a rapid change in DP as a function of concentration, a stepwise model would require that the association constant be a function of aggregate size, with a maximum at some most favorable value. As with any stepwise model for association of molecules, the distribution of molecular weight at low concentration is dominated by free molecules (Figure 4). As concentration increases, the dimers and trimers become more important, and the mode of the distribution moves to dimers, trimers, and higher-order aggregates with increasing concentration in solution. A stepwise addition model will always give a distribution of molecular weight with a single mode14, 18, 38.


29

Energy & Fuels

5 Model of Powers et al. 10% strong aggregators with K1=100*K2

4

3

2

Detection of higher aggregates (n>2)

Detection of dimerization

Degree of polymerization

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 49

Range of VPO data

1 10-2

10-1

100

101

102

103

104

105

Asphaltene concentration, mg/L Figure 3. Propagation-termination (P-T) model of Powers et al.16 extrapolated to low concentrations, with representative parameters (Molecular weight of terminators 800, neutrals 800, and propagators 1200 with molar ratio of terminators to propagators of 0.08 and mol fraction of neutrals 0.12. Association constant, K, was 38000 (gmol/L)-1). The dashed line shows the results for a four-component model (T/2P) tuned to the same DP at the maximum concentration (K = 62000 (gmol/L)-1).


30

Page 31 of 49

1.0 Concentration, mg/L 0.5 50 250 400

0.8

Mass fraction

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

0.6 0.4 0.2 0.0 0

2

4

6

8

10

12

Number of molecules per aggregate Figure 4. Size distribution as number of molecules per aggregate from propagationtermination model of Powers et al.16 at low concentrations. Parameters for the model are the same as in Figure 3.

3.3 Extension of P-T Model to a Distribution of Association Strengths The addition of more strongly-associating species can enable a step-wise association model to predict or fit association behavior over a wider range of concentrations. Given the complexity of the asphaltene fraction and the continuum of molecular structures, we expect a priori that a range of association constants would be present in such a mixture.


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For the whole vacuum residue, this principle is illustrated by Zhang et al.

Page 32 of 49

18

who fitted

VPO data for extracted fractions of vacuum residue with association constants that increased with the strength of the extraction solvent. Similarly, the definition of neutrals in the T/P model of Powers et al.16 effectively defines a fraction with an association constant so much smaller than K for the majority of species in the mixture that its contribution to aggregation is insignificant. From supramolecular chemistry, the strength of association of molecules with more than one point of interaction will increase with the number of favorable interactions between any pair of molecules, or between a molecule and the surface of an aggregate39. The T/2P model from Equations (14) through (21) is the simplest extension of the P-T to represent a mixture of strong propagators with large association constants, propagators, terminators to suppress aggregate growth, and neutrals that do not associate significantly; i.e., extending the model from three asphaltene sub-species to four. The data in Figure 3 illustrate the effect of dividing the “propagator” fraction of asphaltenes in the model of Powers et al.16 into strong and moderate aggregators, with a 100-fold increase in K for the strong propagators. Association of the asphaltenes, defined


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as the formation of dimers and higher aggregates, occurs over a wider range of concentrations. Obviously, the addition of even stronger associations between propagators would extend association to lower and lower concentrations, and lead to the formation of higher-order aggregates (trimers and tetramers) at low concentration. Both curves in Figure 3 show that an average property for the mixture, like DP or average molecular weight, gives poor discrimination of very low concentrations of dimers, and by extension will not indicate association of a small sub-fraction of a complex mixture. We can more clearly observe the effect of model parameters at low concentration by plotting the fraction of the asphaltenes in aggregates of n≥2, as illustrated in Figure 5. When the association constant was increased by a factor of 100, the concentration required for 1% of the total asphaltenes to forming at least dimers dropped from 1.7 mg/L to 0.18 mg/L. Evdokimov and Fesan12 suggested that the onset of dimerization was 0.5 mg/L, because at lower concentrations the shape of UV fluorescence spectra were no longer concentration dependent. This observation does not indicate the precision of detecting dimerization, but we assume that dimerization of over 1 mol% of the asphaltenes at 0.5 mg/L would be detected by this method. On this basis, the data of


33

Energy & Fuels

Figure 5 for 10×K and 100×K suggest an upper limit for the association constant of 17 times the base value of K. For all values of K for the strongly-associating fraction, the molecular weight distribution on a molar basis was nearly exponential, similar to Figure 4, and no bimodal distribution was observed.

1

Fraction in aggregates

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 49

0.1

0.01

Base case 10% strong propagators, 10 x K 10% strong propagators, 100 x K Onset of dimerization

0.001

0.0001 0.1

1

10

100

1000

Asphaltene concentration, mg/L Figure 5. Mol fraction of total asphaltenes in aggregates from a propagation-termination model with the addition of strongly aggregating species (T/2P model). Parameters are as in Figure 3 for the base case (Molecular weight of terminators 800, neutrals 800, and propagators 1200 with molar ratio of terminators to propagators of 0.08 and mol fraction


34

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Energy & Fuels

of neutrals 0.12. Association constant, K, was 38000 (gmol/L)-1). The two comparison cases set 10% of the asphaltenes as strong aggregators with 10×K and 100×K.

The model results in Figure 3 and Figure 5 for the T/2P model illustrate the expected result that addition of stronger associations of components in the asphaltene mixture gives an extension of association behavior to lower concentrations. The same approach could be used to represent multiple sub-fractions of propagators, each with a characteristic association constant. The practical limit is the lack of data to verify the molecular weight distribution aggregates over a range of concentrations. Data from mass spectrometry can suggest the range of molecular weights, but data for the average molecular weight as a function of concentration from VPO does not provide enough discrimination of behavior to extend the P-T model to include more components. The lack of availability of quantitative data for aggregate size distribution is the main limitation to such any extension of the P-T model, or indeed to any stepwise association model. The potential for use of sedimentation data from ultracentrifugation is considered in the next section.


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Page 36 of 49

3.4 Sedimentation by Ultracentrifugation The use of ultracentrifugation for determining aggregate size of asphaltenes has been proposed by Mostowfi et al. 20 and Goual et al.21. The concentration of molecules or other species under centrifugal force is governed by the balance between the body force due to the centrifugal field, which drives sedimentation, and the diffusion or Brownian motion of the species, following the Lamm equation40: ∂𝑐𝑘 ∂𝑡

1∂

[(

)]

∂𝑐𝑘

= 𝑟∂𝑟 𝑟 𝐷𝑘 ∂𝑟 ― 𝑠𝑘𝜔2𝑟𝑐𝑘 + 𝑄𝑘

(22)

where ck is the concentration of the kth species in a mixture with diffusion coefficient Dk and sedimentation coefficient sk at radius r and angular velocity . The term Qk allows for reaction or association of the species k. Analytical centrifugation most commonly uses the transient data for concentration as a function of radial position in order to determine the diffusion and sedimentation coefficients, which in turn provides the molecular weight

MWk from the Svedberg equation: 𝑠𝑘 =

𝐷𝑘𝑀𝑊𝑘(1 ― 𝑣𝑘𝜌) 𝑅𝑇

(23)


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Energy & Fuels

where 𝑣𝑘 is the partial specific volume of component k,  is the solution density, R is the gas constant and T is temperature. At long times, the concentration profile will reach an equilibrium value where the diffusion and sedimentation are balanced40:

[

𝑐𝑘(𝑟) = 𝑐𝑘(𝑟0)𝑒𝑥𝑝

𝑀𝑊𝑘(1 ― 𝑣𝑘𝜌)𝜔2 2𝑅𝑇

(𝑟2 ― 𝑟20)]

(24)

This equation gives the concentration of species ck(r) at radial position r with reference to the concentration at the liquid interface ck(r0) at position r0. Integration of the concentration from the top to the bottom of the liquid column will give the average concentration in solution, 𝑐𝑘. The experimental data of Mostowfi et al.20 are given in Figure 6, showing the amount of sediment at the bottom of the tube at long centrifugation times. The centrifugation times in these experiments were long enough that the amount of sediment did not change with more spinning time. At low concentrations the aggregate-aggregate interactions are minimal, therefore, the observed changes in sediment mass as a function of concentration in Figure 6 are due to association behavior. The higher the initial concentration, the higher the concentration of aggregates of any given size, and the greater the amount of


37

Energy & Fuels

sediment. The very low amount of sedimentation at the lowest concentration of 33 mg/L suggests that the asphaltene molecules and aggregates are small enough to be largely stable in suspension at the experimental conditions.

60

Mass sedimented, wt%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 49

50 40 30 20 10 0 0

100

200

300

400

500

Initial asphaltene concentration, mg/L Figure 6. Sedimentation equilibrium data from centrifugation of asphaltene solutions, indicating the fraction of asphaltene removed by centrifugation for 7.8 d at a mean relative centrifugal force (RCF) of 25000 g

20

The line is calculated from the molecular weight


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distributions from the P-T model of Powers et al.16 based on no sedimentation of molecules, 70% for dimers, and complete sedimentation of trimers and higher aggregate species.

The data of Table 5 give the predicted concentration ratios in the centrifuge tubes (assuming no sedimentation) for the experimental conditions of Mostowfi et al.20, calculated as a function of MWk using Equation (24). For a molecular weight from 600 to 1200 Da, the ratio between the mean concentration and the concentration at the liquid surface (𝑐/c0) will range from 1.5 to 2.6. In contrast, higher molecular weights would give much larger gradients in concentration (Table 5); for example, a tetramer gives a 𝑐/c0 ratio of 110.9. In reality, such large concentration gradients would not be sustained at the initial solution concentration, so that sedimentation of the higher-mass aggregates would occur. At equilibrium, the tetramer in solution at the bottom of the tube would still be over 800 times higher than at the liquid surface, but sedimentation would remove mass and reduce the liquid-phase concentration to very low levels. In contrast, the gradient of the free molecules at equilibrium would be more stable at the initial concentration.


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Page 40 of 49

Table 5. Calculation of concentration ratios in ultracentrifugation as a function of molecular weight. The average molecular weight (Mn) is the same as in Figure 4. Parameters for equation (24) are from the experimental conditions of Mostowfi et al.20 (r0=62 mm, rmax=104 mm, =1885 s-1) and from Nikooyeh and Shaw41 (𝑣=0.837 cm3/g) at 20°C. Number of

Mn

Concentration ratio

Average

molecules in

(bottom of tube/top

concentration/top of

aggregate

of meniscus,

meniscus concentration

crMax/cr0)

(𝑐/c0)

1 (free

1131

5.4

2.4

2 (dimer)

2262

29.0

7.5

3 (trimer)

3393

155.9

27.2

4 (tetramer)

4524

838.9

110.9

molecule)

In the absence of data for the concentration gradients and the molecular weight distribution of molecules and aggregates in solution, direct application of Equation (24) to the data of Figure 6 is not possible. The results show that at low concentration, the suspended asphaltenes become small enough that they are quite stable in solution, with only 9% sediment. Trimers and tetramers will generate large concentration gradients in solution, and are unlikely to remain at long centrifugation times. The data of Table 5


40

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Energy & Fuels

suggest that dimers may partition, with some remaining in suspension and some appearing in the sediment. The simplest method of fitting the data of Figure 6, therefore, is to assume than molecules are stable, trimers and tetramers are sedimented, and that dimers are partitioned between the solution and the sediment. The molecular weight distributions of Figure 4 from the P-T model were used to determine the molecular weight distributions as a function of concentration. The curve in Figure 6 was calculated using sedimentation of 70% of the dimers present at each concentration. Clearly a single adjustable parameter was sufficient to represent the trend. Ultracentrifugation has the potential to provide a richer data set for analysis. At short times, the transient curves can be fitted using an integrated form of Equation (22) using well established methods40, and a model such as T/P or T/2P fitted to the distribution of

MWk as a function of initial concentration if the association kinetics are rapid. At equilibrium conditions, measurement of the MW of the remaining components in solution as a function of initial concentration would enable a more thorough definition of the relationship between the initial molecular weight distribution, the amount of sediment, and the distribution of molecular weight remaining in solution.


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Page 42 of 49

3.4. Further Improvements to Models for Asphaltene Association In principle, the form of Equations (14) through (21) can be extended to give a T/nP or mT/nP model with a distribution of association constants. Such a model would fit the expectation that a wide range of association constants would be expected based on the diverse molecular architectures revealed by FT-ICR MS of separated fractions33. The challenge is how to obtain data to fit the parameters of a more complex model, given that not enough data are available even to fit a T/2P model. Strong association would dominate the observable interactions at low concentrations; therefore, the essential requirement to extend association models is to obtain accurate data on the dimer concentration at low concentration. Spectroscopic techniques such as fluorescence are potentially sensitive enough, but they cannot give accurate quantitative measurements in a complex mixture of dissociated species. Experiments on fractions that are more carefully prepared to concentrate the most strongly aggregated material would be more effective. Using strongly associated species in ultracentrifuge experiments with transient in situ measurements of concentration gradients as well as equilibrium


42

Energy & Fuels

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conditions would generate a wealth of data under more carefully controlled conditions in the low concentration range. The fractionation techniques revealed by Chacón-Patiño et

al.33 would enable studies of aggregation from weakly aggregated to strongly aggregated components, all within the asphaltene fraction. Combining ultracentrifugation and VPO data would give a powerful test of the validity of models over a wide range of concentration. Such a dataset would justify the derivation and testing of more sophisticated hetero-association models37, ultimately formulated to represent the vacuum residue as a mixture of multiple species with defined interactions. A combination of molecular analysis of the fractions to define the chemical basis for strong association, and use of selected model compounds, could enable a systematic link between the chemical structures and their association parameters. Further desirable extensions include kinetics of disaggregation and temperature dependence of association constants and kinetic behavior, to enable more effective analysis of asphaltene behavior under refinery process conditions.


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Conclusions 1. Data for average molecular weight of asphaltenes from vapor-pressure osmometry, as a function of concentration, can be represented by linear-association models.

2. Extrapolation of these models to much lower concentrations does not give results that are consistent with spectroscopic studies of molecular dimer formation.

3. Addition of more components with stronger association constants enables dimer formation over a wider range of concentration, but the data to enable fitting of the parameters for such multicomponent models are lacking.

4. Data from centrifugation of asphaltenes can be represented using the results of simple linear association models. Dynamic ultracentrifuge experiments, and measurements on strongly and weakly aggregated subfractions, may provide more data to enable fitting of more capable multicomponent models.

AUTHOR INFORMATION


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Energy & Fuels

Author Contributions The manuscript was written through contributions of both authors, and both authors have given approval to the final version of the manuscript.


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Quantitative Modeling of Formation of Asphaltene Nanoaggregates

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