RESEARCH Revista Mexicana de F´ısica Vol. 59, No. 6. Pages 584–593, 2013

NOVEMBER-DECEMBER 2013

Tricritical phenomena in asphaltene/aromatic hydrocarbon systemsJuan Horacio Pacheco-SanchezInstituto Tecnologico de Toluca. Division de Estudios de Posgrado e Investigacion,Av. Tecnologico s/n, 52149 Metepec, Estado de Mexico, Mexico, jpachecos@toluca.tecnm.mx

G.Ali MansooriDepartments of BioEngineering, Chemical Engineering and Physics,University of Illinois at Chicago (M/C 063), Chicago, Illinois 60607-7052, mansoori@uic.edu

Abstract: The calculation of phase behavior performed for asphaltene-micelles systems that assume several forms when they are mixed

with petroleum fluid depending on its aromaticity, and on the relative sizes and polarities of the particles present in such fluid mixtures, is extended to tricritical phenomena. Asphaltene monomers form micelles in the presence of excess amounts of aromatic hydrocarbons and polar solvents, and they are dispersed in a petroleum fluid. The coupling between the micellization and phase separation may, in principle, lead to a near-tricritical coexistence of a monomer phase and a micellar phase; however, this tricritical phenomenon has not been experimentally observed. In this report, such tricritical phenomenon for asphaltene / aromatic systems is predicted.

Keywords: Asphaltene–aromatic systems; phase separation; micellar solutions; tricritical points; second order phase transition.

El calculo de la conducta de fase efectuado para sistemas de micelas de asfaltenos que supone varias formas cuando son mezclados con fluidos petroleros dependientes tanto de su aromaticidad como de los tamanos relativos y de las polaridades de las partıculas presentes en tales mezclas de fluidos, es extendido al fenomeno tricrıtico. Monomeros de asfaltenos forman micelas en la presencia de cantidades de hidrocarburos aromaticos en exceso y solventes polares, y estan dispersados en un fluido petrolero. El acoplamiento entre la micelizacion y la separacion de fase puede, en principio, llevar hacia una coexistencia tricr ıtica de una fase monomero y una fase micela; sin embargo, el fenomeno tricr ıtico no ha sido observado experimentalmente en soluciones de asfaltenos. En este reporte, se predice dicho fenomeno tricrıtico para sistemas asfalteno / aromatico.

Descriptores: Sistemas asfalteno-aromatico; separacion de fase; soluciones micelares; puntos tricrıticos; transiciones de fase de segundo orden.

1. Introduction

Asphaltene is defined as the fraction of carbonaceous fossilenergy sources (petroleum crude oil, coal, tar sand and oilshale). It is insoluble in low-boiling paraffin hydrocarbonsolvents, such as methane, ethane, propane, etc. and solu-ble in aromatic hydrocarbons like benzene, toluene, and xy-lene. Such compounds as asphaltenes and resins are presentin petroleum fluids in minute amounts, but in heavy oils andother fossil energy sources in insignificant quantities. In na-ture, it is hypothesized that asphaltenes are formed as a re-sult of oxidation of natural resins. On the contrary, the hy-drogenation of asphaltenic compounds (resins, asphaltenesand asphaltenic acids) may produce heavy hydrocarbon oilswhich contain polycyclic aromatic or hydroaromatic hydro-carbons. They differ, however, from polycyclic aromatic hy-drocarbons due to presence of oxygen and sulfur in theirstructures in varied amounts [1,2]. On heating above 300-400C, asphaltenes are not melted, but decompose, formingcarbon and volatile products. Asphaltenes react with sulfuricacid forming sulfonic acids, as might be expected on the basisof the polyaromatic structure of these components. The dark-brown color of crude oils and residues is due to the combinedeffect of neutral resins and asphaltenes. The black color ofsome crude oils and residues is related to the presence of as-phaltenes, which are not properly peptized.

Asphaltene after heating [3] are subdivided in: Non-volatile (heterocyclic N and S species), and, volatile (paraf-fin + olefins), benzenes, naphtalenes, phenanthrenes, sev-eral others). Speight [3] also reports a simplified repre-sentation of the separation of petroleum into the followingsix major fractions: volatile saturates, volatile aromatics,nonvolatile saturates, nonvolatile aromatics, resins and as-phaltenes. He also reports arbitrarily defined physical bound-aries for petroleum using carbon-number and boiling point.The molecular structures proposed for the asphaltene (heavyorganic) molecules [4] includes carbon, hydrogen, oxygen,nitrogen, sulfur as well as polar and non-polar groups seeFig. 1.

It can be said that asphaltene particles are known to bepolymeric and polydisperse. They behave lyophobic andsteric, and they posses electrostatic and associative forces.Due to polydispersivity of asphaltenes, entropic effects areimportant in mixtures containing asphaltenes [5].

Asphaltene particles in crude oil assume various formsdepending on the oil aromaticity, and on the other compoundspresent in the oil. Experimental observations have indicatedthat asphaltene monomers form micelles in the presence ofcertain concentrations of aromatic hydrocarbons, and the re-sulting micelles are dispersed in the petroleum fluid [6,7].

The behavior of asphaltenes in presence of excess amountof toluene can be analogous to the behavior of a surfactant in

PACS: 05.70.Fh; 05.90.+m; 83.70.FhISSN: 0035-001X

TRICRITICAL PHENOMENA IN ASPHALTENE/AROMATIC HYDROCARBON SYSTEMS 585

presence of an oil-water mixture. Pacheco and Mansoori [5]proposed a model for the phase behavior of asphaltenes inaromatic hydrocarbons and could predict the conditions whendifferent kinds of micelles may be formed in such mixtures.In the present paper we report our theoretical studies on thecoupling between formation of asphaltene micelles and phaseseparation which may, in principle, lead to a near-tricriticalcoexistence of a monomer asphaltene phase and a micellarphase.

2. Background

Dickie and Yen [8], using X-ray diffraction studies on solidasphaltenes, proposed unitary sheets made up a system ofhighly condensed polinuclear aromatic ring with alkylicalchains tied to the ring structure. They also proposed that theasphaltene association takes place through stacking of aro-matic sheets due toπ − π interactions. These aggregates orparticles were then proposed to associate into what was calledmicelles.

Several more investigators have established the existenceof asphaltene micelles when there is an excess of aromatichydrocarbons present in a crude oil [7,9-12]. Of course, thewell-studied micelles known today are the kind, which areformed between water and oil in the presence of surfactants.Since asphaltene in oil has a large size distribution, investi-gators have concluded that the asphaltene micellar system isconceptually equivalent to an oil/many-surfactant/water sys-tem [13-15].

The existence of asphaltene micelles is confirmed bymeasurements of surface tension [6,15] and viscosity [7].The data were also used to show that below the critical mi-celle concentration (cmc) with small concentrations, the as-

FIGURE 1. Molecular structures of asphaltenes derived from vari-ous sources. The asphaltene structures shown here are separatedfrom asphaltene deposits consisting of asphaltene steric colloidflocs and encapsulated in them other compounds [4]. Benzene ringsare represented with double bonds, and ciclohexane rings with sin-gle bond.

FIGURE 2. Lambda “λ” form of the coexistence curve showingtricritical phenomenon, because lineaCb is first order and linedCis second order.

phaltenes in the solution are in a molecular state, while, abovethecmc, asphaltene micelle formation will occur, in a mannersimilar to that in oil/surfactant/water micellar systems whilesurfactant monomers are much uniform in their structure andless polydisperse. Experimental measurements for the phasediagram of asphaltene micelle formation were performed byRogacheva O. V.et. al. [6] and Priyantoet al. [7].

Blankschteinet. al. [16] developed the thermodynamicmethod of phase separations in micellar solutions. Pachecoand Mansoori [5] used their method to propose the phase be-havior of asphaltene in aromatic systems. Experimental stud-ies by various investigators have indicated that the solute as-phaltene molecule (amphiphile or monomer) forms micellesin the presence of aromatic hydrocarbon solvents [6,7,11,15].

Landau and Lifshitz [17] briefly discussed the criticalpoint for second-order phase transitions in mixtures of twosubstances. It can be shown that the state diagram near sucha point takes theλ-form observed in Fig. 2. As abscissa it isplotted the concentrationX of the mixture, and as ordinate,the temperature. The curvesCa andCb are first-order phasetransition boundaries andCd is a curve for the second-orderphase transition. The dashed areaaCb is the region split intotwo phases of which phase I is the less symmetrical and phaseII the more symmetrical. The pointC is the critical point; thecurvebC joins smoothly ontoCd. One more thing is that atthe critical point the specific heat Cp of the mixture under-goes a finite discontinuity.

According to Uzunov [18] in binary alloys as FeAl and,especially, in ternary and quaternary fluid mixtures (See Grif-fiths [19]) critical points, which have no analogue in simplefluids, can be observed. These are thetricritical points- crit-ical points where three phases co-exist and have identicalcompositions. The experiments on systems which possesstricritical points show that at such points the first-order phasetransition line in the thermodynamics 3D space of fields (T,

Rev. Mex. Fis.59 (2013) 584–593

586 J. H. PACHECO-S ANCHEZ AND G.A. MANSOORI

P,µ) changes to a line of critical points (second-order transi-tion) as shown in Fig. 2 whereCd is a line of critical points.

At a pure-substance critical point, densities of the twocoexisting fluid phases become identical and the meniscusbetween them disappear. In principle, at a tricritical point,densities of the three coexisting phases simultaneously be-come identical. Knobler and Scott [20] performed experi-mental studies for multicritical points in fluid mixtures. Theyconcluded that the disappearance of the meniscus at an or-dinary critical point could be observed in the laboratory byheating (or sometimes cooling) the system in a sealed tube.However, the analogous experiment cannot be performed fora fluid mixture at a tricritical point: for fluid mixtures, thereis no closed-system path along which three visually distinctphases can be observed going into one.

Knobler and Scott [20] also commented that the first ex-perimental discovery of a tricritical point has been attributedto Efremova by her coworkers Krichevskiiet al. Efremovaand Prianikova found upper and lower critical end-points inthe ternary system n-C4H10+ CH3COOH + H2O. In a moredetailed study in collaboration with Krichevskiiet al., theyobserved a separation of only 0.3C between the critical end-points. At both critical end-points (189.2 and 189.5C) andin the intervening three-phase region, all three phases showednear-critical opalescence. The third-order critical point wasestimated to lie at 190C. One year prior to the publicationof the study on the ternary system, Radyshevskayaet al. re-ported experiments on phase behavior in the system water +ammonium sulfate + ethanol + benzene as a function of tem-perature at fixed (atmospheric) pressure. They described a“triple critical point”, which they estimated to exist at 49C.

In case of binary systems, upper and lower critical solu-tion points (UCST, LCSP) have been known since 1800 [21]and it is reported in every solution thermodynamic book [22].In this case, when the components are reasonably similar,e.g.n-hexane + methanol, the system comes first to a liquid-liquidcritical point (UCST=34C). When the components are dis-similar, e.g., n-hexane + water, however, the system comesfirst to a critical point between the gas,i.e., the mixed vapors,and the liquid phase rich in hexane (Tc=210C).

Finally, in the asphaltene case, Tran [23] measured heatcapacity of athabasca bitumen asphaltenes using differentialscanning calorimetry (DSC) and temperature modulated dif-ferential scanning calorimetry (TMDSC), founding a secondorder phase transition. We must stress that Pacheco-Sanchez[24] predicted a second order phase transition in asphaltenes-micellar solutions. Now, we will explain tricritical pointsin this work, as part of second order phase transition ofasphaltene-micellar solutions.

3. Asphaltene Micellization Theory

We will theoretically consider asphaltene micellization (mi-celle formation) in this section. In a theory of phase separa-tion in asphaltenes-micellar solutions, Pacheco-Sanchez [24]developed an advance of the behavior of phase separation for

asphaltene-micelles in petroleum fluids proposed by Pachecoand Mansoori [5]. In here, it is now extended to the predic-tion of tricritical points.

We consider that micelles can exchange asphaltene am-phiphile molecules. This exchange is represented as a multi-ple chemical equilibrium among the members of the micelledistribution as follows:

µn = nµ1 n = 1, 2, 3, . . . (1)

where the chemical potential per asphaltene amphiphilemolecule must be the same in all the micelles.

In the thermodynamics of micelle formation is postu-lated that the interactions between asphaltene amphiphilemolecules which result in micelle formation are mainly dueto the following three effects: a)Kinetic effectdue to colli-sions between molecules into micellar solutions, which causesynergetic phenomena, molecular interactions are present. b)Polar effectsin the form of electrostatic interactions presentbetween asphaltene amphiphiles, which are responsible forthe polar heads in the asphaltene amphiphiles. All of thisis reflected in the entropy of mixing of the formed micelles.c) Asphaltene amphiphile effectsdue to association presentin micellar solutions of asphaltene amphiphiles and aromaticsolvents known as micellar interactions.

Based on the above three postulates it is possible to writethe equation for the total Gibbs free energy G = Ga + Gb +Gc for a system ofNar aromatic solvent molecules andNas

asphaltene amphiphile molecules at temperature T and pres-sure P. Asphaltene amphiphile micellization in this systemproduces a distributionNn, whereNn is the number ofmicelles having n asphaltene amphiphile in their structure.Then, the most important equations in this theory are:

i The chemical potential of aromatic solvent in a micel-lar solution:

µar = µ0ar + kT [ln(Xar) + Xas − ΣnXn]

+ CγX2as/2[1 + (γ − 1)Xas]2 (2)

whereµ0ar(T,P) is the pure state chemical potential of an aro-

matic solvent molecule, k is the Boltzmann constant, T is theabsolute temperature,

Xas =Nas

(Nas + Nar)

and

Xar =Nar

(Nar + Nas)

are themole fraction of asphaltene amphiphile molecules andaromatic solvent molecules in the solution, respectively, suchthat

Xar + Xas = 1 Xn =Nn

(Nas + Nar)

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TRICRITICAL PHENOMENA IN ASPHALTENE/AROMATIC HYDROCARBON SYSTEMS 587

is the mole fraction of micelles (each consisting ofnmolecules of asphaltene amphiphiles),ΣnXn is definedaccording to the theory of moments:Mα(Xas, T, P )= ΣnnαXn, C = U(T, P )/Ωas where U(T,P) is the magni-tude of the potential energy, andγ = Ωas/Ωar whereΩas isthe volume of an individual asphaltene amphiphile moleculeandΩar is the volume of an aromatic solvent molecule.

ii By using the Gibbs-Duhem equation (Nardµar +ΣnXndµn = 0) it is possible to obtain an expressionfor the chemical potential of micelles in the solution

µn = µ∗0n + kT [ln(Xn) + n(Xas − ΣnXn − 1)]

+nC

2

(1−Xas)2

[1 + (γ − 1)Xas]2− 1

(3)

whereµ∗0n ≡ µ0n + kT andµ0

n is the hypothetical pure mi-celle chemical potential at the same temperature and pressureas the mixture.

iii The distribution of micelles is given by

Xn = Xn1 exp[−β(µ∗0n − nµ∗01 )]

× exp[(βU(T, P )/Ω)ΣjfnjNj ], (4)

whereβ = 1/kT , Xn is the distribution of micelles con-taining n asphaltene-amphiphile-monomer,Xn

1 representsthe likelihood that n molecules will be localized in the sameregion of space,exp[−β(µ∗0n − nµ∗01 )] is the Boltzmann fac-tor which represents the enhancement of the micellar config-uration due to the chemical potential difference (µ∗0n −nµ∗01 )arising upon assembling ofn dissolved molecules into a sin-gle micelle. In fact, the sequence of the chemical potentialdifferences can be represented as a spectrum of energy lev-els [5], the last termexp[(βU(T, P )/Ω)ΣjfnjNj ] is equal tounity. The conservation of asphaltene amphiphile mass im-plies the following normalization condition:

Xas = ΣnnXn (5)

The separation of the micellar solution into two coexist-ing phases having different total concentrationsYas andZas

of asphaltene amphiphile will be developed in the followingsection.

4. Phase Coexistence

In general, in the absence of any external field, the thermo-dynamic state of a one-component system of fixed size isdetermined by specifying two variables as the independentvariables. Two-component systems require the specificationof a third independent variable. Three-component systemsrequire the specification of four independent variables, and

so on. For practical purposes for a c-component system theusual choice for the independent variables are temperature,pressure, andc− 1 mole-fractions (or other forms of compo-sition variables).

This choice of independent variables is by no meansunique and separates the variables in the sense that temper-ature and pressure are “field variables” while the composi-tion variables represent “densities”. A ”field variable” isa quantity, sometimes called a “potential variable”, whichhas the same value in all the coexisting phases at equi-librium, while a “density variable” is a quantity that canhave, and normally does have, different values in each ofthe coexisting phases. For each generalized density vari-able, there is a corresponding conjugate field variable, as canbe seen by writing any general thermodynamic equation, asfor example, the generalized Gibbs-Duhem equation: SmdT-VmdP+x1dµ1+x2dµ2+... where the temperature is the fieldconjugate to molar entropy, the pressure to molar volume,and the chemical potential of a component to its correspond-ing mole fraction.

In the system used in this report, the components are as-phaltene and toluene, and the variables are temperature, pres-sure and mole-fraction in each phase. For all the calculationsreported here we assume normal pressure.

According to the Gibbs phase rule [20], a system ofc-components andp-phases has F = C + 2-p degrees of free-dom. At a critical point however, the requirement that twophases become just identical introduces a further constraintand reduces F by one. At annth-order critical point, wheren of thep phases become identical, there aren − 1 such ad-ditional constraints and the Gibbs phase rule must be writtenas

F = c + 2− p− (n− 1) = c + 3− p− n (6)

SinceF ≥ 0 andp ≥ n, it follows that annth-ordercritical point requires a minimum of2n− 3 components.

FIGURE 3. Schematic representation about the phase behavior, ina closed binary system near the tricritical point. While solid lineindicates separation of two different phases, dashed line indicatesthat the phase separation is vanishing, consequently two phases be-come identical.

Rev. Mex. Fis.59 (2013) 584–593

588 J. H. PACHECO-S ANCHEZ AND G.A. MANSOORI

Thus, one component suffices for an ordinary critical point(in this sense, second order orn = 2), three componentsare necessary for a tricritical (n= 3) point, and five com-ponents for a tetracritical point (n= 4) point, etc. Thereforec+3−p−3 = c−p ≥ 0 orc ≥ p. For a two-component threephases in equilibrium Fig. 3 represents the general schematictransition of the phases near a tricritical point [19]. As thetemperature is raised the system goes from three phases (α+ α′ + α′′) to two phases (α + α′ ≡ α′′) to two phases(α ≡ α′ + α′′) to one phase (α ≡ α′ ≡ α′′). The latteris the tricritical point in which three phases become identical.The dashed line in Fig. 3 is the critical interface [20].

In this particular case, asphaltene-monomers dissolvedin an aromatic solvent self-associate. This resulting asso-ciation is called micelle which will constitute a new phase.Therefore, this system can be considered as a two-componentthree-phase (α, α′ andα′′) system, where:

α is the monomer phase.

α′ is the micellar phase

α is the liquid phase due to the solvent.

As it was mentioned beforec ≥ p, so, it is not possible toget more phases than components in a system. Nevertheless,starting on the case in which there are three phases and threecomponents in equilibrium

µ`α = µ`′

α = µ`′′α

µ`α′ = µ`′

α′ = µ`′′α′ (7)

µ`α′′ = µ`′

α′′ = µ`′′α′′

where`, `′, `′′, are asphaltene-monomer, asphaltene micel-lar and aromatic components respectively. Specifically:µ`

α

is the asphaltene chemical potential in a monomer phase,µ`′α

is the micelle chemical potential in a monomer phase, andµ`′′

α is the solvent chemical potential in a monomer phase,µ`

α′ is the asphaltene chemical potential in a micellar phase,and so on. The equilibrium system represented by Eqs. 7 hasan algebraic system of six equations, in spite of that, thereare only two significant equations in view of the equilibriumcondition of Stigter [25] -Tanford [26]:µn = nµ1. On theother hand, it is well known that [27]

µi = µ0i + RT ln ai (8)

whereai is the activity of the componenti, andµ0i is the stan-

dard chemical potential of the componenti. Equations 7 canthen be referred to as the system of equality of activities

a`α = a`′

α = a`′′α

a`α′ = a`′

α′ = a`′′α′ (9)

a`α′′ = a`′

α′′ = a`′′α′′ .

When a componenti is absent in a phase, its activity isunity, i.e.

ai = 1

In the tricritical case the equilibrium conditionµn = nµ1

makes the activities in the monomer phasea`α = a`′

α =a`′′

α = 1 and also activities corresponding to the asphaltene-monomer componenta`

α′ = a`′α′′ = 1. Therefore, this system

of Eqs. (9) reduces to the following system of equations

1 = 1 = 1

1 = a`′α′ = a`′′

α′ (10)

1 = a`′α′′ = a`′′

α′′

This system is equivalent to the one obtained for phaseseparation in asphaltene/toluene solutions [5], because it canbe reduced to:

1 = 1 = 1

1 = µn (Yas, T, P ) = µn (Zas, T, P ) (11)

1 = µar (Yas, T, P ) = µar (Zas, T, P )

In general, when a triple point is reached as the temper-ature is raised, the compositions of the three phases becomemore nearly identical, and theα/α′ andα′/α′′ interfaces maybe observed to vanish simultaneously as the tricritical temper-ature is attained. However, when a tricritical point is reachedone of the interfacesα/α′ or α′/α′′ may never vanish, pro-vided that the overall composition is adjusted so that at leastone interface could remain present. i) In an excess of aro-matic there are no monomer phase. ii) In a scarcity of aro-matic there is no micellar phase.

5. Tricritical Phenomena

The separation of the micellar solution into two coexistingphases having different total concentrationsYas andZas ofasphaltene amphiphile requires the thermodynamic equilib-rium condition between the phases given by Eqs. (11):

µn(Yas, T, P ) = µn(Zas, T, P )

µar(Yas, T, P ) = µar(Zas, T, P )

Using the later conditions, and after some mathematicalmanipulations, the coexistence condition between the phasesin equilibrium reduces to the following two coupled equa-tions:

Rev. Mex. Fis.59 (2013) 584–593

TRICRITICAL PHENOMENA IN ASPHALTENE/AROMATIC HYDROCARBON SYSTEMS 589

√Zas −

√Yas√

K= (Zas − Yas) + ln

(1− Zas

1− Yas

)+

12βγC

[Z2

as

(1 + (γ − 1) Zas)2 −

Y 2as

(1 + (γ − 1) Yas)2

], (12)

1√K

(1√Zas

− 1√Yas

)+√

Zas −√

Yas√K

= Zas − Yas +12βC

[(1− Zas)

2

(1 + (γ − 1) Zas)2 −

(1− Yas)2

(1 + (γ − 1) Yas)2

]. (13)

The above two equations depend on the interactionβCand on the growthK parameters, the concentrationsYas andZas correspond to the monomer phase and micellar phase,respectively. ParametersβC andK are arbitrary positive val-ues which depend on temperature and pressure, however, ex-plicit βC(P) or K(T,P) functions have not been proposed untilnow, which means the proposition of new equations of stateis required. On the other hand, parameterβC of interactionenergy can be expressed as

βC(T ) =Tc

γT(14)

A wider explanation can be found in the Ref. 24 wherethe corresponding spinodal line is calculated. In that refer-ence,βC values of four different fractions of crude oil werealso calculated, where wasγ = Ωas/Ωar obtained directlyfrom the properties and characteristics of the crude oil, den-sity and molecular weight basically. In practice, to solve thedependence on the critical temperature (Tc), we consideredthat this corresponds to every point incmc line experimen-tally measured. The later is allowed by the knowledge thatcmc is the critical line. The growth parameter K can be ex-pressed as

k(T, P ) ≡ exp[β(µ∗0n0− n0µ

∗0)] (15)

wheren0 is the minimum number of amphiphiles in the mi-celle. Assuming a spherocylinder form of the asphaltene-amphiphile micelle,µ∗0n0

is the reference chemical potentialper asphaltene amphiphile monomer associated to the endregion, andµ∗0 is the reference chemical potential per as-phaltene amphiphile monomer associated to the cylindricalregion. The corresponding sequence of chemical potentiallevels isµ∗0n = µ∗0n0

+ (n − n0)µ∗0, for n ≥ n0. In general,parameter K is responsible for the form of each micelle. As-suming spherocylinder form for micelles, K can assume thefollowing values:

0<K<1: If µ∗0n0< µ∗0 then asphalteneamphiphiles are dis-

tributed in the cylindrical region of the micelles beingformed, i.e., asphaltene amphiphile micelles form isdisk-like. This is the normal flat form of asphaltenesand graphite.

K=1: If µ∗0n0= n0µ

∗0 thenasphaltene amphiphiles are pro-portionally distributed in both cylindrical and end re-gions of the micelles being formed,i.e., asphaltene am-phiphile micelles form is sphere-like. This is the formof fullerenes.

K>1: If µ∗0n0> µ∗0 thenasphaltene amphiphiles are dis-

tributed in the end region of the micelles in formation,i.e., asphaltene amphiphile micelles form is cylinder-like. This is the form of nanotubes.

It has to be mentioned that Raveyet al. [28] foundthese three geometrical forms of aggregated asphaltenes byusing Small Angle X-Ray Diffraction, and more recently,fullerenes were found by Camacho-Bragadoet al. [29] us-ing both High Resolution Transmision Electron Microscopyand Energy Dispersion Spectrometry in a study of asphal-tene separated from resins. During their observation inthe microscope they saw fullerene formation in structuressuch as Onions and C240@C60. Furthermore, using as-phaltenes Wanget al. [30] synthesized carbon microspheresusing Chemical Vapor Deposition, and characterizing by X-Ray Difraction, Scanning Electron Microscopy, High Res-olution Transmision Electron Microscopy and Raman Spec-troscopy. Their results indicate that the monodisperse spheresare graphitic curved structures with diameters: 300 - 400 nm.Asphaltene used consisted in metastable structures at highertemperatures. The aromatic carbon rings promoted the for-mation of graphitic structures of closed cage. Finally, Velascoet al. [31] found nanotubes in crude oil naturally processed.

The system of Eqs. (12) and (13) can be solved simul-taneously for calculating Yas and Zas numerically by theNewton-Raphson iterative method, which for a general con-centration is expressed as:

Xn+1 = Xn − f (Xn)f ′ (Xn)

. (16)

To accomplish this calculation we assume Eq. (12) issolved by the Newton-Raphson method for one of the vari-ables (either Yas or Zas) assigning an initial value to theother variable. Then Eq. (13) is also solved by the Newton-Raphson method using the solution to the Eq. (12) as theinitial value. This iterative procedure converges for

ε = |Xi −Xi−1| << 1. (17)

This iterative solution will provide us Yas and Zas whichare used for producing the theoretical coexistence curve fora micellar solution of asphaltene in toluene. To speed upthe convergence in the Newton-Raphson iterations we use

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590 J. H. PACHECO-S ANCHEZ AND G.A. MANSOORI

the following functionality f((KX)1/2

) ∼= f(K) and

f(K1/2X−1/2

) ∼= f (1) which means that when K<0 thesolution goes to imaginary numbers; however, when K>0 aconvergent solution in real numbers is obtained. Then K isalways positive.

Three asphaltene fractions of Kotur-Tepinsk crude oiland one asphaltene fraction of Arlan crude oil were usedto test different values ofβC and K of this modelby Pacheco-Sanchez [24]. For tar asphaltene fractionof crude oil Kotur-Tepinsk, the following cases wereanalyzed. One case was forβC=314.45/(γT) consid-ered, in which K=0.5,0.7071,1,10,100 in one graph andK=0.1,0.5,0.7,1,10,100,1000 in the other graph, where thecoexistence curves show lambda form; and in the other caseβC=380/(γT) for K=0.003,0.01,0.1,1,50,280 in two graphs,where the coexistence curves were closed without showingany lambda form as expected becauseβC value is abovecmc.In principle, this model will take us to get tricritical points bysolving the set of Eqs. (12) and (13) for particular values ofparametersβC andK as we will see in the results, over allwhenβC is around room temperature and K is lower than 1as expected for asphaltene.

6. ResultsThis part of the analysis will be done by choosingγ=10.4038, βC = 314.45/(γT) and different positive valuesof the parameterK, which reflects the tendency of asphaltenemonomers to their aggregation in the form of spherocylindermicelles. The Rogachevaet al. [6] experimental points ofcmc for tar asphaltene fraction of crude oil Kotur-Tepinskwere assigned as the initial values in Newton-Raphson it-erative calculations. A graph for a superposition of coexis-tence curves whenK = 100, 10, 1, 0.7071 and 0.5 was calcu-lated by Pacheco-Sanchez [24]. Using these values ofK wesolve the system of Eqs. (12) and (13). WhenK=100 andthe temperature is greater than 320 K all the experimentalpoints are almost reproduced without region of two coexist-ing phases. This means that the two-phase region is quitethin. WhenK = 10 the two-phase region is wider than forK=100. The two-phase region is more and more open asthe value ofK is decreasing. The widest open two-phasesregion is shown forK = 0.5. The later K-value is in thedisk-like form of the micelle, which corresponds to planarasphaltenes. It has to be mentioned that asphaltenes werefound in both planar and curved forms using molecular dy-namics simulations [32]. Curved forms of asphaltenes areconsistent with several facts: i) reversed micelles [33], ii) ex-istence of nanotubes and fullerenes [28-31], iii) activated car-bons containing pentagonal to heptagonal rings, which adoptcurved forms [34]. To these activated carbons we could callthem asphaltene-blankets whether the asphaltenes grew ex-tensively.

Fixing the parameterβC = 314.45/(γT) and allowing thevariation of parameter K, Pacheco-Sanchez [24] shows thatthere are some differences when we use as initial values ei-

FIGURE 4. Phase behavior (of asphaltene-toluene systems forγ=10.4038, K = 500 andβC = 314.45/γT) in λ-form as in theLandau theory of tricritical points as reported in Fig. 2. Rogachevaet al. [6] cmcpoints were used as initial values in the numericalsolution. Horizontal coordinate is the asphaltene mole fraction.

thercmcor arbitrary points, in the Newton-Raphson numer-ical solution of Eqs. 12 and 13. He exhibits two generalgraphs on which lambda form of second order phase transi-tion and tricritical points of asphaltene micelar solutions canbe inferred. Here we clearly exhibit this idea.

Thecmcpoints are used as initial values in the numericalsolution on Fig. 4. Such Figure shows a behavior similar toa lambda in a continuous line calculated by this theoreticalmodel. This line approximately coincides within fourcmcpoints, however the otherscmcpoints remained inside of thetwo phases region. This is due to the critical temperatureused forβC value, which produces a very thin (or, vanishedin fact) two phase region for temperatures greater than crit-ical temperature. In principle,cmcpoints are considered ascritical points in this way, from which we can take criticaltemperature. Then, this is not a first order phase transition asthat obtained for water (just one component) at a triple pointsolid liquid and vapor. This is a second order phase transi-tion as that proposed by Landau on certain variation in thesymmetry of the body due to a change in its pressure or itstemperature or its composition as in the Fig. 2.

The tricritical point in Fig. 4 is such that the three linesare joined as in Fig. 2. Then, below tricritical point it is ob-served a region of two phases, the monomer phase and themicellar phase of asphaltenes, this might be considered asthe flocculation region. Above tricritical point it is just ob-served separation between monomers and micelles as it wasobserved by Yudinet al. [12] because they detected asphal-tene “molecules” (monomers) belowcmcand micelles abovecmc by measurements of dynamic light scattering (photon

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TRICRITICAL PHENOMENA IN ASPHALTENE/AROMATIC HYDROCARBON SYSTEMS 591

FIGURE 5. The horizontal coordinate corresponds to the mole frac-tion of asphaltene in toluene. Allcmcsquare points of Rogachevaet al. [6] were reproduced usingγ = 10.4038,βC = 293/γT andK = 500). A small improvement ofβC or K will provide just onecalculated curve.

FIGURE 6. Phase behavior (of asphaltene+toluene system forγ=10.4038, K = 500 andβC = 293/γT) in λ-form as in the Lan-dau theory of tricritical points. Arbitrary initial values to solveequations (18) and (20) were used in our numerical solutions. Hor-izontal coordinate is the asphaltene mole fraction.

correlation spectroscopy) on asphaltenes in toluene + heptanemixtures.

Threecmcpoints were exactly reproduced in Fig. 4 with-out region of two phases; however, in Fig. 5 allcmcpointsof Rogachevaet al. [6] were reproduced using the follow-ing values of the parameters:γ = 10.4038,βC = 293/γT andK = 500. This K-value is in the cylinder-like form ofthe micelle, which corresponds to nanotubes. Thecmcpointsused as initial values in this numerical solution are still show-ing a very tiny lambda form. When arbitrary initial points areused to solve the system of equations by Newton-Raphsonmethod, as the important difference respect to the last one,Fig. 6 was obtained. Two things are observed in this Figure,

one is its lambda form and the other is the line above tricriti-cal point is one order of magnitude parallel to the left of thecmcpoints.

The γ value changes when one of the othercmccurvesin the same graph of Rogacheva’s experimental data is used,however, the general information can hardly increase if it isanalyzed another one. Then, it is believed that the analysesdone until now are enough to describe tricritical phenomenain asphaltene/aromatic solutions.

7. Discussion

For asphaltene dissolved in toluene, at a certain concentrationand temperature asphaltene micelles will be formed. Thenself-assembly (coacervation) of asphaltene micelles may takeplace at higher concentrations and/or lower temperature andaround one of the transition points. Of course, micelle coac-ervates may deposit if the coacervate size exceeds a certainlimits, while micelles change to become coacervates due totheir growth depending on the concentration of the solution.In these reversible processes asphaltene micelle-coacervatesmay swell in the presence of a proper solvent. The micelliza-tion point starts on the critical micelle concentration (cmc),and ends on self-assembly (coacervation) of asphaltene mi-celles.

In a closed binary system, made up of monomers andmicelles in liquid state abovecmc (below cmc micelles donot exist), micelles can coexist in several forms and differ-ent (polydisperse) sizes depending on the nature of asphal-tene molecule and temperature of the solution. Two phasesare expected for micelles: monomer and micellar phases.Monomer phase is due to low aromaticity, and micellar phaseis due to high aromaticity of the solution.

In this model the fact experimentally found by Yen [35],about self-association in stacking of two to four asphaltenesis taken as a natural way of the micelle formation. Such as-phaltene stacking is emphasized by considerations proposedby Rogel [36,37] using the free energy of Gibbs also forasphaltene micellar considerations. The stacking of 2-4 as-phaltenes is a self-associating model which has a strong ten-dency to be considered as oligomer [38]. Recently, accord-ing to the thermodynamic theory of phase separation in selfassociating micellar solution Huanget al. [39] open the pos-sibility that instead of oligomers of asphaltene stacking theymight be unimers (subunits of a surfactant micelle, as op-posed to monomer, the subunit of a polymer) which is a natu-ral starting for micelle growing in different geometric forms,for which the intermolecular forces allow self associating inmicellar phase for planar and curved asphaltene as follows:

i) Disk-like can be considered as planar asphaltene mi-cellar growth toward graphite (which is also stackingof huge planar pentagonal and hexagonal carbon rings)

ii) Sphere-like are considered as curved asphaltene mi-cellar growth to cumulus of spheres aggregates. Each

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592 J. H. PACHECO-S ANCHEZ AND G.A. MANSOORI

sphere is made up pentagonal and hexagonal carbonrings. These spheres might be fullerenes.

iii) Cylinder-like are also considered as curved asphaltenemicellar growth to cylinder aggregates. Each cylin-der is built of pentagonal and hexagonal carbon rings.They might be nanotubes.

Until now, asphaltenes are considered only as planarmolecules, according to the Yenet al. model [35]; how-ever, asphaltene stacking for planar and curved asphaltenescan be observed in a previous work [32], which reports struc-ture factors agree with Yenet al. work [35], and it is nearer tothe asphaltene habitat conditions than that proposed by Yenet al. [35]. Furthermore, the existence of activated carbonsas those proposed by Harriset al. [34] gives us confidenceabout the existence of curved asphaltenes, that might arriveto be nanotubes, or fullerenes according to the conditions aswe observed in this work.

In general, the m-phase region (m = 1,2,3,. . . ) is thefocus of most SANS investigations since it is the region ofmicelle formation and micelles are of nanometer size. Them-phase region is rich in mesophases (with various mor-phologies). It contains spherical, cylindrical and lamellarmicelles depending on the temperature range. Structures forthese mesophases correspond to cubic (spherical micelles),hexagonal (cylindrical micelles) and lamellar symmetry re-spectively. Moreover, oil-in-water micelles are obtained atlow temperature and “reverse” (water-in-oil) micelles are ob-tained at high temperatures.

This method can be used as a model of two parameters tobe adjusted. The model is good for: i) aqueous solutions, ii)asphaltene in aromatic micellar solutions, iii) asphaltene inchromatographic fractions. In this work, the model has beentested for temperatures among 0 – 80C and concentrationslower than 10−2 (mole fraction). The model is still in devel-opment to be tested for temperatures 10-400C and higherconcentrations of asphaltenes among 0.01-1 (mole fraction)where the aggregated stability of petroleum dispersed sys-tems is a central problem.

Rogachevaet al. [6] extended their work in order to findthe phase state of asphaltenes in petroleum dispersed systems

in the temperature range 10-400C [40], and concentrations0-40 mass %. Their results are very far of the CMC; how-ever, they affirm that the new measurements are consistentwith those previously accomplished by themselves. Theyused optical density, calorimetry, and surface tension at leastfor their measurements. A preliminary calculation using ourmodel needs more revision, because at those temperaturesthey report two concentrations for one temperature value, andwe still need to validate the calculation accomplished. Ourmodel works nearcmcvalues, the most probable idea is thatnew considerations have to be taken into account in order toget a successful model.

8. Conclusions

All the experimentalcmcpoints were reproduced in Fig. 5for specific values of the parametersβC andK tuned bycmcexperimental points. In principle, the model can give an ideaof the phase diagram in case onecmc point is given, suchpoint will be considered as the tricritical point by this model.

If the cmcpoint were in the top of the phase diagram itwill hardly be found a second order phase transition, how-ever, it is expected a limited miscible fluid, this means thatasphaltene can stay dispersed in the solution. If thecmcpointwere in the middle of the phase diagram a second order phasetransition can be obtained as in the Fig. 4. If thecmcpointwere in the bottom of the phase diagram a second order phasetransition can be found as in Fig. 6.

The coupling between the micellization and phase sepa-ration may, in principle, lead to a near-tricritical coexistenceof a monomer phase and a micellar phase, however, this tri-critical phenomenon has not been experimentally observed,nevertheless Fig. 6 is very indicative that this can be a goodpossibility. Rowlinson gives another good indication of thispossibility because he reported tricritical points at 273 K forbinary systems, and it can be considered that tricritical pointis at 293 K in Fig. 6, which is considered for a binary asphal-tene + toluene system. Fig. 5 is also indicative of tricriticalpoints or second order phase transition, because it has a su-perposition of differentK values showing how can be closedthat open region in a lambda form as Landau reported.

1. T.F. Yen, and G.V. Chillingarian, (Editors),Asphaltenes and as-phalts, Vol. 1, (Elsevier Science. New York 1994).

2. T.F. Yen, and G.V. Chillingarian, (Editors) 2000,Asphaltenesand asphalts, Vol. 2, (Elsevier Science. New York 2000).

3. J.G. Speight, in the bookAsphaltenes and Asphalts, 1, Devel-opments in Petroleum Science, 40 edited by Yen T. F. and G.V. Chilingarian, (Elsevier Science, New York 1994). Chapter:Chemical and physical studies of petroleum asphaltenes,

4. G.A. Mansoori,Int. J. Oil, Gas and Coal Technology2 (2009)141.

5. J.H. Pacheco-Sanchez, and G.A.Mansoori,Petroleum Scienceand Technology16 (1998) 377.

6. O.V. Rogacheva, R. N. Rimaev, Y. Z. Gubaidullin, and D.K.Khakimov,Colloid J.USSR, translated (1980) 490.

7. S. Priyanto, G.A. Mansoori, and A. Suwono,Chem. Eng. Sci-ence56 (2001) 6933.

8. J.P. Dikie, and T.F. Yen,Anal. Chem.39 (1967) 1847.

9. J.P. Pfeiffer, and R.N. Saal,J. Phys. Chem. 44 (1940) 139.

10. V.E. Galtsev, L.M Ametov, and O.Y. Grinberg,Fuel 74 (1995)670.

Rev. Mex. Fis.59 (2013) 584–593

TRICRITICAL PHENOMENA IN ASPHALTENE/AROMATIC HYDROCARBON SYSTEMS 593

11. S.I. Andersen, and K.S. Birdi,Journal of Colloid and InterfaceScience142(1991) 497.

12. I. K. Yudin et al., Petroleum Science and Technology16 (1998)395.

13. E.Y. Sheu, M.M. de Tar, D.A. Storm and S.J. DeCanio,Fuel71(1992) 299.

14. E.Y. Sheu, and D.A. Storm. in the bookAsphaltenes Funda-mentals and Applications, edited by Sheu E. Y. and Mullins O.C., (Plenum Press, New York 1995). Chapter I

15. E.Y. Sheu,J. Phys.: Condens. Matter8 (1996) A125.

16. D. Blankschtein, G.M. Thurston, and G.B. Benedek,PhysicalReview Letters54 (1985) 955.

17. L.D. Landau and E.M. Lifshitz,Statistical Physics. (PergamonPress, London, 1958).

18. D.I. Uzunov,Theory of Critical Phenomena. (World Scientific,London 1993).

19. R.B. Griffiths,The Journal of Chemical Physics60 (1974) 195.

20. C.M. Knobler, and R.L. Scott,Phase Transitions and Criti-cal Phenomena. Vol. 9, Edited by Domb C. and Lebowitz J.L.,(Academic Press, London 1984).

21. Schreinemakers,Zeitschrift fur physikalische Chemie29(1899)597.

22. J. S. Rowlinson, and F. L. Swinton,Liquids and liquid Mixtures,(Butterworth Scientific, London 1982).

23. K. Q. Tran, MSc Thesis.Reversing and Non-reversing PhaseTransitions in Athabasca Bitumen Asphaltenes. (University ofAlberta, Canada 2009).

24. J.H. Pacheco-Sanchez,Revista Mexicana de Fısica 47 (2001)324.

25. D. Stigter,J. Colloid Interface Science, 47 (1974) 473.

26. C. Tanford,Science200(1978) 1012.

27. E. A. Guggenheim,Thermodynamics. (North-Holland Publish-ing Company, Amsterdam 1957).

28. J. C. Ravey, G. Ducouret and D. Espinat,Fuel67 (1988) 1560.

29. G.A. Camacho-Bragadoet al., Carbon40 (2002) 2761-2766.

30. X. Wang, J. Guo, X. Yang, and B. XuMaterials Chemistry andPhysics113(2009) 821-823

31. C. Velasco-Santos, A. L. Martinez-Hernandez, A. Cosultchi, R.Rodriguez, V.M. Castano,Chem. Phys. Lett.373 (2003) 272-276.

32. J. H. Pacheco-Sanchez, F. Alvarez, and J.M. Martınez,Energy& Fuels18 (2004) 1676.

33. S. I. Andersen, J. M. del Rio, D. Khvostitchenko, S. Shakir, andC. Lira-Galeana,Langmuir17 (2001) 307-313

34. P. J. F. Harris, L. Zheng and S. Kazu,J. Phys.: Condens. Matter20 (2008) 362201.

35. T. F. Yen, G. J. Erdman, and S. S. Pollack,Analytical Chemistry33 (1961) 1587.

36. E. Rogel,Langmuir18 (2002) 1928

37. E. Rogel,Langmuir20 (2004) 1003

38. H.W. Yarranton,Journal of Dispersion Science and Technology26 (2005) 5

39. Y. Huang, H. Cheng, C.C. Han,Macromolecules44 (2011)5020.

40. O.V. Rogacheva, Y. Z. Gubaidullin, R. N. Rimaev, and T.D.Danilyan,Colloid J.USSR, translated (1984) 715.

Rev. Mex. Fis.59 (2013) 584–593