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Multiscale Modeling of Mass Transfer and Adsorption in Liquid− Liquid Dispersions. 2. Application to Calcium Naphthenate Precipitation in Oils Containing Mono- and Tetracarboxylic Acids K. Kovalchuk, E. Riccardi,† and B. A. Grimes* Ugelstad Laboratory, Department of Chemical Engineering, Norwegian University of Science and Technology, Sem Sælands vei 4, 7491 Trondheim, Norway ABSTRACT: A dynamic multicomponent mass transport model is constructed and solved to determine the interfacial composition and bulk phase concentrations of surfactant mixtures containing a synthetic tetracarboxylic acid (BP10) and decanoic acid (DA) for water droplets dispersed in oil. The transport model employs a molecular mixed monolayer adsorption model that was parametrized by MD simulation and interfacial tension experiments in part 1 [Kovalchuk, K.; Riccardi, E.; Grimes, B. A. Ind. Eng. Chem. Res. 2014, ]. The model accounts for oil−water partitioning, the pH determined dissociation state of the acids, and micelle formation in the water phase. Since the interfacial concentration of tetracarboxylic acids in crude oil emulsions is hypothesized to influence the extent of fouling by calcium naphthenate precipitation, trends for the amount of calcium naphthenate precipitate formed in the system can be predicted as a function of the water volume fraction for various BP10:DA concentration ratios and drop sizes. The model provides an experimentally testable prediction that could support the hypothesis that calcium naphthenate precipitation is an interfacial reaction and have implications on petrochemical and engineering based inhibition strategies. The modeling framework outlined in parts 1 and 2 of this work is well-suited to studying interfacial phenomena in well-defined model systems employing a library of synthetic and purified indigenous crude oil surfactants.

1. INTRODUCTION An increasing portion of existing global petroleum reserves is comprised of heavy crude oils which present a new set of challenges to the petroleum industry in the field of colloid and interface science.1 One challenge in particular occurs with immature, biodegraded, heavy crude oils that have a high content of naphthenic acids;2,3 examples of such oils can be found across the globe from West Africa and the North Sea to China and South America.2 Naphthenic acid compounds in crude oil tend to become very interfacially active at oil/water interfaces when the pH of the production water increases during the successive pressure drops experienced by the fluid as it is transported from the reservoir to the surface. 1,2 Consequently, this high interfacial activity means naphthenic acids could (a) react with calcium in the water to form insoluble calcium soaps that cause scale to form inside piping and process equipment,1−22 and/or (b) significantly influence the formation of stable emulsions that are difficult to separate, and/or (c) generate discharge waters with a high hydrocarbon and naphthenate content.1−22 In this respect, both stable emulsions and calcium naphthenate scale have considerable environmental and economic impacts on the petroleum transport and processing chain. Therefore, understanding the critical mechanisms governing the interfacial adsorption and mass transport of naphthenic acid compounds, along with various other types of interfacially active compounds, is paramount to understanding the mechanistic behavior of calcium naphthenate fouling on the process level as well as the formation and separation of petroemulsions. Recently, a class of C80−82 isoprenoid tetraacids has been discovered4−7 which is a significant constituent of calcium © 2014 American Chemical Society

naphthenate precipitate. These C80−82 isoprenoid tetraacids have been named ARN tetraacids by Baugh et al.4,5 The ARN tetraacids are very interfacially active, and it is suggested that calcium ions can bridge individual ARN acid molecules to form a cross-linked network of calcium naphthenate scale that deposits on process equipment due to its insolubility in both the oil and water phases.1,2,4−22 These cross-linked networks most likely form at the oil−water interface.8−22 Therefore, based on the hypotheses that calcium naphthenate scale formation is likely to occur when the interfacial concentration of these ARN tetraacids becomes high enough to achieve a percolation threshold that facilitates the cross-linking of ARN tetraacid molecules and calcium ions, the interfacial concentration of these ARN tetraacids should be a key variable that indicates the likelihood of fouling. It has been observed that lowering the pH of the production water can inhibit the formation of calcium naphthenate scales1,2,5,9−22 as this limits dissociation of the carboxylic acid groups and lowers the interfacial activity of the acidic compounds. Additionally, observations11,22 indicate that the volume fractions of oil and water in the emulsion as well as the drop size distribution of the emulsion influences the propensity for scale formation since these parameters control the total interfacial area available in the system. Finally, it has been shown2,11,19,20,22 that calcium naphthenate precipitates can be limited by competition with other interfacially active molecules (such as typical naphthenic Received: Revised: Accepted: Published: 11704

March 27, 2014 June 20, 2014 June 20, 2014 June 20, 2014 dx.doi.org/10.1021/ie501296t | Ind. Eng. Chem. Res. 2014, 53, 11704−11719

Industrial & Engineering Chemistry Research

Article

Nikas et al.25 developed a molecular mixed monolayer adsorption model for nonionic surfactants at air−water interfaces based on a two-dimensional hard disk representation of the monolayer. The model of Nikas et al.25 was later extended by Mulqueen and Blankschtein26,27 to include ionic surfactants26 and oil−water systems.27 A significant advantage of this monolayer model27 is that it allows one to fully parametrize the adsorption isotherm by a combination of (a) molecular simulations of each of the N surface active molecules at an oil−water interface and (b) experimental equilibrium interfacial tension measurements of single surfactant solutions for each of the N molecules studied. In this respect, there is a 1:1 correspondence between the number of experimental equilibrium interfacial tension measurements required to parametrize the adsorption isotherm and the number of surface active molecules present in the system.25−27 Considering the library of synthetic asphaltenes and tetracarboxylic acids developed by Norgård and Sjöblom12 along with commercially available compounds that are similar to interfacially active compounds found in crude oils (e.g., mononaphthenic acids), it becomes practically feasible to study and quantify the interfacial composition of model oil−water dispersion systems with several model surface active molecules that mimic the interfacial behavior of surface active molecules found in real crude oils at various concentrations and compositions due to the fact that a large amount of the experimental work required to parametrize the isotherm is offset by computer simulation. Additionally, the isotherm parameters explicitly account for the molecular structure of the surface active compounds and their interactions through the molecules of the oil and water phases at the interface. In part 1 of this work,28 the molecular dynamics (MD) simulation framework for tetraacid molecules at oil− water interfaces developed by Riccardi et al.29 was employed to parametrize the molecular mixed monolayer adsorption model25−27 for the synthetic tetraacid BP10 and decanoic acid (DA). In this work (part 2), the general mechanistic framework developed by Chatterjee and Wasan23 and the adsorption isotherm developed by Mulqueen and Blankschtein27 for liquid−liquid systems is employed in a continuum interfacial mass transport model for liquid−liquid dispersions. The model is applied to a water-in-oil dispersion containing the synthetic tetraacid BP1012−14 and decanoic acid (DA), whose structure is similar to each of the four acidic arm groups of BP10. The parameters determined in part 1 of this work28 are employed in the model presented here, and the dynamic transport of the acidic compounds from the oil phase to the interface, and then to the water phase, is simulated. The dynamic model accounts for deprotonation of the acids at the interface, oil−water partitioning, and micelle formation in the bulk water phase. The equilibrium value for the interfacial concentration of the tetraacid BP10 after the initial formation of the monolayer is then employed to estimate the behavior of the amount of calcium naphthenate produced per unit process volume as a function of (a) the composition of the mono- and tetraacids in the oil, (b) the water cut, and (c) the radius of the dispersed water droplets. It should be noted that, while the dynamic model has been applied to the problem of calcium naphthenate precipitation in crude oil systems, the dynamic model is developed in general for many types of surfactants. Therefore, the dynamic model developed in this work could be applicable to liquid−liquid dispersions encountered in a wide variety of applications.

monoacids) which can compete for interfacial area and possibly terminate the cross-linking of ARN tetraacid molecules.11 Naturally, other interfacially active compounds such as asphaltenes or basic compounds can compete for interfacial area as well.2,11,22 There have been several attempts to specifically model the formation of calcium naphthenate precipitates in the literature.2,15,18,19 The basic approach common to these models is to employ the oil/water partition coefficients, acid dissociation constants, solubility product constants, and bulk mass balances to determine the mass of calcium naphthenate precipitate in terms of the naphthenic acid concentration in the oil, calcium (and other ion) concentration in the water, pH, oil volume, and water volume. While these models can make quantitative predictions of the calcium naphthenate precipitate mass for single component model systems under certain conditions, there are certain shortcomings for more complex systems. First, since the precipitate mass is determined from solubility product constants defined in terms of bulk water concentrations, the models2,15,18,19 necessarily treat the calcium naphthenate formation as a bulk reaction as opposed to an interfacial reaction, which the authors themselves consider to be the most likely reaction mechanism.2,15,18,19 Furthermore, these models2,15,18,19 do not consider the effects of other interfacially active components such as monoacids or asphaltenes present in real systems that could affect the partitioning behavior of the naphthenic acids. Finally, these models may be difficult to apply to the emulsions encountered in real systems due to the extremely large surface areas encountered in the liquid−liquid dispersions (emulsions) where the calcium naphthenate precipitate forms in real systems. Thus, it becomes apparent that an important tool to study the interfacial behavior of tetraacids would be an interfacial mass transport model for liquid−liquid dispersions that employs a thermodynamically consistent23−27 mixed monolayer model that can relate the depletion of interfacially active components in the bulk oil to not only their bulk water phase concentrations but also, most crucially, to their interfacial concentrations. Chatterjee and Wasan23 developed a model to describe a mixed adsorbed layer for an acidic oil/alkali/ surfactant system that correctly couples the interfacial behavior acidic compounds present in crude oil to bulk solution properties. Additionally, they23 accounted for dissociation of the acid compounds and formation of salt complexes at the oil/ water interface, the formation of mixed micelles in the aqueous phase, and molecular interactions between molecules at the interface. It should also be noted that the model of Chatterjee and Wasan23 was formulated for experimental equilibrium interfacial tension (IFT) systems where the oil phase sits on top of the water phase and did not account for the large interfacial area present in a liquid−liquid dispersion of small water droplets in oil. The model23 described the experimental data very well and provides a general mechanistic framework for describing the interactions and reactions of acidic compounds in oil/water systems. It should be noted that the mixed monolayer and mixed micelle models employed by Chatterjee and Wasan23 are based on regular solution theory.24 In practical terms, this implies that, as the number of interfacially active components increases, the number of experiments required to quantify the interaction parameters of the model will increase in a pyramiding manner.25−27 11705

dx.doi.org/10.1021/ie501296t | Ind. Eng. Chem. Res. 2014, 53, 11704−11719


Article

2. MODEL FORMULATION A liquid−liquid dispersion of water droplets in oil with a uniform radius Rd is considered where, in general, N interfacially active molecules (surfactants) are initially present in the oil phase at known, dilute concentrations, C0o,i (i = 1, 2, ..., N). The water droplets are considered to be suspended in the bulk oil and the bulk oil is well mixed such that the concentrations of the surfactants in the bulk oil at any given time, t, can be taken to be uniform throughout the oil phase except for in a thin film of stagnant oil surrounding each water droplet. Consequently, the surfactant molecules will experience a mass transfer resistance by diffusion through the stagnant oil film surrounding the water droplets as they migrate to the oil− water phase interface and adsorb. The N surfactants can be considered to be sufficiently dilute so that the heat of adsorption does not significantly change the temperature of the system (isothermal system). In the adsorbed phase, the ionic compounds (acids and bases) can become charged and the water-soluble surfactants can desorb and partition into the bulk water phase; in the current model formulation, the water phase is considered to be buffered so that the pH remains constant over time. However, the possibility for the watersoluble amphiphilic surfactants to form micelles is considered. A further assumption is that the formation of the monolayer is considered to occur quite fast relative to the average coalescence rate; therefore, at this level of analysis, the changing surface-to-volume ratio of the liquid−liquid dispersion is considered to remain constant during the time to reach the initial equilibrium of the monolayer. It should be noted here that detailed water chemistry that could account for bulk phase reactions and a dynamic pH are neglected at this point to study the effect of the interfacial area on the competitive adsorption and phase partitioning of a mixed tetraand monoacid system. Based on this system description and assumptions, the differential mole balance for a surface active species, i, in the bulk oil containing N dilute surface active species is given by the following expression:30,31 dCo, i dt

=−

ϕ 3 K fo, i(Co, i − Ci*), 1 − ϕ Rd

obtained from the molecular mixed monolayer equation of state developed by Mulqueen and Blankschtein27 for liquid−liquid systems containing ionic and nonionic surfactants is employed in this work as the constitutive equation that relates Γi to Ci* for each surfactant molecule i. In general, the adsorption isotherm of the molecular mixed monolayer model25−27 can be expressed as follows in terms of molar concentrations: ⎛ ⎞ ⎛ C* ⎞ Δμî °, σ /b Nav Γi ⎟ + ln⎜⎜ (1 + |zi|) ln⎜ i ⎟ = ⎟ N kBT ⎝ C Tb ⎠ ⎝ 1 + Nav ∑k = 1 Γkak ⎠ N

+

N

(1 + Nav ∑k = 1 Γkak )2

for i = 1, 2, ..., N

+

⎛ λ ⎞2 ⎫ ⎪ d ε 1 + ⎜ zi Γi⎟ ⎬ + s zi Γi 2ψ εs ⎝ 4ψ ⎠ ⎪ ⎭ (3a)

where

λ=

ψ=

εεokBT 2e 2NavI

(3b)

εεokBT 2e 2Nav

(3c)

In eqs 3a−3c, zi denotes the charge number of component i, CTb is the total molar concentration of the solvent (oil or water depending on which side the isotherm is parametrized; see below), Δμ̂i°,σ/b represents the modified standard state chemical potential difference of molecule i between the interface and the specified bulk phase (see Nikas et al.25 and Mulqueen and Blankschtein26,27 for the definition of the standard state), kB is the Boltzmann constant, T denotes the absolute temperature, Nav is Avogadro’s number, Γi represents the interfacial molar concentration of component i, ri and ai denote the hard disk radius and cross-sectional area of molecule i in the adsorbed phase, respectively, ds is the thickness of the Stern layer, ε denotes the dielectric constant in the bulk aqueous phase, εs is the dielectric constant of the Stern layer, εo represents the permittivity of free space, e denotes the elementary charge, and I represents the ionic strength of the aqueous phase. It should be noted that, for neutral surface active compounds, the last two terms in eq 3a vanish and the leading term on the left-hand side of eq 3a is unity. It is important to note here that the effect of the curved interface of the water droplets is neglected based on the arguments given by Slattery et al.32 The model developed here considers that all the surface active compounds are initially present only in the bulk oil phase. At times t > 0, the surfactant compounds then migrate to the oil−water interface and, possibly, partition into the bulk water phase. Any ionic surfactant compound in the oil will arrive at the oil−water interface in the neutral (undissociated) state. At the moment an ionic surfactant compound enters the interfacial phase, they could become charged under the right conditions, but it is important to stress that the ionic surfactants on the interface in the charged state all came from the oil phase. Therefore, the value of Ci* defined on the oil side of the phase

for i = 1, 2, ..., N

for i = 1, 2, ..., N

πai(Nav ∑k = 1 Γkrk)2

⎧ ⎪ λ 2zi ln⎨ zi Γi + ⎪ 4ψ ⎩

In eq 1, Co,i denotes the concentration of the surface active molecule i in the bulk oil phase, ϕ represents the volume fraction of water droplets in the system (equivalent to the “water cut” for water-in-oil dispersions), Rd represents the radius of the water droplets, Kfo,i is the film mass transfer coefficient of molecule i in the stagnant oil film surrounding the droplets, and Ci* denotes the concentration of component i in the liquid layer adjacent to the oil−water interface. The initial condition for eq 1 is obtained from the known initial concentrations of the surface active molecules in the bulk oil phase and is given by eq 2. Co, i = Co,0 i ,

N

1 + Nav ∑k = 1 Γkak N

+

(1)

at t = 0,

N

Nav(ai ∑k = 1 Γk + 2πri ∑k = 1 Γkrk)

(2)

C0o,i

In eq 2, (i = 1, 2, ..., N) denotes the initial concentration of surface active molecule i in the bulk oil phase. The concentration Ci* (i = 1, 2, ..., N), of component i in the liquid layer adjacent to the phase interface is considered to be the concentration that drives the adsorption process and is assumed to be in quasi-equilibrium with the adsorbed phase concentration, Γi (i = 1, 2, ..., N). The adsorption isotherm 11706



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interface must account for the surfactant in the uncharged and charged states in order to obtain the appropriate molar flux from the oil phase. One complication that arises is that the interfacial molecular areas of the charged surface active molecules, ach,i (i = 1, 2, ..., N), could change23,26,27 relative to the uncharged form, aun,i (i = 1, 2, ..., N), which indicates that eq 3a will be different for the uncharged and charged states of a surfactant ion. In order to address this consideration, we first assume that the ionization reactions are in quasi-equilibrium with respect to the mass transport rate such that the interfacial concentration, Γi, of surfactant i can be expressed as a linear combination of the interfacial concentration Γch,i of the charged state of molecule i and Γun,i of the neutral (uncharged) state molecule i.

fch, i =

fch, i = 0,

In eq 4, fch,i denotes the fraction of surfactant molecule i in the charged state and should be specifically defined for particular types of ionic surfactant compounds (acid or base). Depending on the pH of the aqueous phase, acidic or basic surfactant molecules could become charged at the phase interface according to their acid dissociation equilibrium relationship. In general, if mp,i is the number of protons donated or accepted by the acid (A) or base (B), respectively, we can write the following stepwise dissociation reactions and equilibrium relationships (note that the value of mp,i should always be a positive integer): K a1, i

K a2, i

1 + 10np,i(pH − pK̅ a,i)

(6a)

when i = neutral

N

p, i , i

N

Nav(a un, i ∑k = 1 Γk + 2πrun, i ∑k = 1 [(1 − fch, k )run, k + fch, k rch, k ]Γk) N

1 + Nav ∑k = 1 [(1 − fch, k )a un, k + fch, k ach, k ]Γk N

(5a)

+

where

πa un, i(Nav ∑k = 1 [(1 − fch, k )run, k + fch, k rch, k ]Γk)2 N

(1 + Nav ∑k = 1 [(1 − fch, k )a un, k + fch, k ach, k ]Γk)2

(K̅ a, i)mp,i = [∏ K aj , i] = j=1

(mp, i) −

[H ] [A [H mp,iA]

]

,

In eq 7, C*un,i represents the concentration of surfactant molecule i in the oil layer directly adjacent to the phase interface that is in equilibrium with the surfactants adsorbed on the interface in the uncharged state, CTo is the total molar concentration of the oil phase, aun,i and run,i denote the crosssectional area and corresponding radius of the hard disk representation of molecule i in the uncharged state on the interface, respectively, and ach,i and rch,i represent the hard disk cross-sectional area and radius of molecule i in the charged state on the interface, respectively. Also, in eq 7 the parameter Δμ̂ °i,σ/o represents the modified standard state chemical potential difference of molecule i in the uncharged state between the interfacial phase and the bulk oil phase. The parameter Δμ̂ °i,σ/o is referred to as “modified” because it accounts for bulk phase partitioning of the surfactant27 molecule and is related to the actual standard state chemical potential difference, Δμi,σ/o ° , of molecule i between the interface and bulk oil phase as follows:27

for i = acid (5b)

and K a1, i

K am

K a2, i

p, i , i

BH(mp,i) + [oooZ H+ + BH(mp,i − 1) + [oooZ ... [ooooooZ H+ + B, (5c)

for i = base

where mp, i

(K̅ a, i)mp,i = [∏ K aj , i] = j=1

[H+]mp,i [B] [BH(mp,i) +]

,

,

(7)

for i = 1, 2, ..., N + mp, i

(6b)

⎛ ⎞ Nav(1 − fch, i )Γi ⎟ + ln⎜ N ⎜ 1 + N ∑ [(1 − f )a + fch, k ach, k ]Γk ⎟⎠ av ⎝ k=1 ch, k un, k

H mp,iA [oooZ H+ + H mp,i − 1A− [oooZ ... [ooooooZ H+ + A(mp,i) −,

mp, i

,

⎛ C* ⎞ Δμî °, σ /o un, i ⎟⎟ = ln⎜⎜ C kBT ⎝ To ⎠

+

for i = acid

10np,i(pH − pK̅ a,i)

In eqs 6a and 6b, np,i represents the number of protons donated by molecule i whose value is positive for acids, negative for bases, and zero for neutral compounds, and pK̅ a,i is the logarithmic value of the acid dissociation constant of molecule i. It should be noted that considering the speciation of molecules at the interface in this manner means that the mole balance equations in this model formulation represent the mole balance on the amphiphilic part of the surface active molecules. Note that in eqs 5a−5d, 6a, and 6b the notation for concentration common to chemistry is employed (i.e., [H+] is the same as CH+). It should be noted here that the model formulation is readily extendable to surfactant salts with the appropriate counterion concentration and dissociation constant. For the surfactant molecules in the uncharged state, the adsorption isotherm in eqs 3a−3c can be formulated from the oil side of the phase interface as follows:

(4)

K am

[H np,iA] + [A(np,i) −]

=

when i = acid or base

Γi = Γch, i + Γun, i = fch, i Γi + (1 − fch, i )Γi, for i = 1, 2, ..., N

[A(np,i) −]

for i = base (5d)

In eqs 5a−5d, mp,i is the number of protons donated or accepted by the acid or base, Kaj,i denotes the acid dissociation constant for dissociation step j of molecule i, and K̅ a,i is an effective acid dissociation constant which is the geometric mean of the Kaj,i (K̅ a,i = (Ka1,iKa2,i...Kamp,i,i)1/mp,i). If the number of protons donated by molecule i is denoted as np,i (where np,i > 0 for acids, np,i < 0 for bases, and np,i = 0 for neutral compounds), then based on the acid equilibria relation presented in eqs 5a−5d, the fraction, fch,i, of ionic surfactant molecules of type i in the charged state can be expressed as follows:

Δμî °, σ /o kBT

=

Δμi°, σ /o kBT

⎛ ⎞ ϕ ⎟, + ln⎜⎜1 + (1 − ϕ)K p, i ⎟⎠ ⎝

for i = 1, 2, ..., N

(8)

In eq 8, Δμ°i,σ/o is the actual standard state chemical potential difference of molecule i between the interfacial phase and the bulk oil phase; Kp,i represents the bulk oil−water partition 11707



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⎡ ⎛ C + ⎞np,i ⎤ * i + ⎢ ϕ K p, i⎜⎜ H ⎟⎟ ⎥Cch, * i, Ci* = Cun, ⎢⎣ 1 − ϕ ⎝ K̅ a, i ⎠ ⎥⎦

coefficient of molecule i in the uncharged state and is defined as follows: K p, i

n p, i 1 − ϕ Co, i 1 − ϕ Co, i ⎛ K̅ a, i ⎞ = = ⎜ ⎟ ϕ Cw(un), i ϕ Cw, i ⎝ C H+ ⎠

for i = 1, 2, ..., N (9)

In eq 12, Cun,i * can be determined from eq 7 and Cch,i * can be determined from eq 10. Now that the driving force concentration Ci* is defined on the oil side of the interface, the differential mole balance for a surfactant molecule i in the interfacial phase can be formulated. Due to the fact that there is a quasi-equilibrium between the concentration, C*i , of a given molecule i in the liquid layer adjacent to the oil−water interface and its concentration, Γi, in the adsorbed phase, the accumulation of molecule i in the adsorbed phase is equivalent to the sum of the molar flux to/ from the oil phase and the molar flux to/from the water phase.

In eq 9, Cw(un),i represents the concentration of surfactant molecule i in the uncharged state in the bulk water phase if molecule i is an acid or base, and Cw,i denotes the concentration of component i in the charged state in the bulk water phase or the concentration of a neutral surfactant in the bulk water phase. For the surfactant molecules in the charged state, the adsorption isotherm in eq 3a can be formulated from the water side of the phase interface as follows: ⎛ C* ⎞ Δμi°, σ /w ch, i ⎟⎟ = (1 + |zi|) ln⎜⎜ C kBT ⎝ Tw ⎠

dΓi = K fo, i(Co, i − Ci*) dt ⎛⎛ C + ⎞np,i ⎞ 1 − ϕ Ci* ⎟ + (1 − fch, i )K fw, i⎜⎜⎜⎜ H ⎟⎟ Cw, i − ϕ K p, i ⎟⎠ ⎝⎝ K̅ a, i ⎠

⎛ ⎞ Navfch, i Γi ⎟ + ln⎜ ⎜ 1 + N ∑N [(1 − f )a ⎟ + Γ f a ] un, ch, av k k k ⎝ ⎠ k=1 ch, k ch, k N

+

N

Nav(ach, i ∑k = 1 Γk + 2πrch, i ∑k = 1 [(1 − fch, k )run, k + fch, k rch, k ]Γk)

n p, i ⎛ 1 − ϕ ⎛ K̅ a, i ⎞ Ci* ⎞ ⎟, + fch, i K fw, i⎜⎜Cw, i − ⎜ ⎟ ϕ ⎝ C H+ ⎠ K p, i ⎟⎠ ⎝

N

1 + Nav ∑k = 1 [(1 − fch, k )a un, k + fch, k ach, k ]Γk N

+

πach, i(Nav ∑k = 1 [(1 − fch, k )run, k + fch, k rch, k ]Γk)2 N

(1 + Nav ∑k = 1 [(1 − fch, k )a un, k + fch, k ach, k ]Γk)2

⎧ ⎪ λ 2zi ln⎨ zifch, i Γi + ⎪ 4ψ ⎩

⎛ λ ⎞ 1 + ⎜ zifch, i Γi⎟ ⎝ 4ψ ⎠

for i = 1, 2, ..., N

+

for i = 1, 2, ..., N

2⎫

⎪ d ε ⎬ + s zifch, i Γi, ⎪ 2ψ εs ⎭

⎛ C + ⎞np,i ϕ * i, K p, i⎜⎜ H ⎟⎟ Cch, 1−ϕ ⎝ K̅ a, i ⎠

(13)

In eq 13, Kfw,i represents the film mass transfer coefficient of molecule i in the water phase, and Cw,i denotes the concentration of molecule i in the charged state in the bulk water phase. It should be noted here that eq 13 is general for neutral surfactants as well as acidic and basic surfactants based on the fact that np,i is positive for acids, negative for bases, and zero for neutral compounds. Also, since the thickness of the stagnant liquid film is at least 2 orders of magnitude smaller than the drop radius for the systems studied, it should be acceptable to employ the film mass transfer coefficient, Kfw,i, to represent the transport coefficient for the molar flux from the interface to the internal space of the drops. Initially, the oil− water interface is assumed to be entirely free of surfactant molecules and, thus, the initial condition for eq 13 can be expressed as follows:

(10)

In eq 10, Cch,i * represents the concentration of surfactant molecule i in the charged state in the water layer directly adjacent to the phase interface that is in equilibrium with the surfactant molecules adsorbed on the interface in the charged state, CTw is the total molar concentration of the water phase, and Δμ°i,σ/w represents the actual standard state chemical potential difference of molecule i in the charged state between the interfacial phase and the bulk water phase. It is important to note that we do not consider the surfactant molecules to exist in the oil phase in the charged state and, thus, the modified standard state chemical potential difference Δμ̂°i,σ/w of molecule i in the charged state between the interfacial phase and the bulk water phase is essentially equal to the actual value of Δμi,σ/w ° . * , of surfactant molecules The effective concentration, Cch(o),i of type i in the oil phase that is in quasi-equilibrium with Γch,i can be obtained from C*ch,i and the acid-dissociation equilibria relationships given in eqs 5a and 5b, the partition coefficient definition given in eq 9, and the droplet volume fraction, ϕ, as follows: * i= Cch(o),

(12)

at t = 0,

Γi = 0,

for i = 1, 2, ..., N

(14)

In the bulk water phase, the surfactant molecules could form micelles which would have an effect on the interfacial composition due to the fact that the concentration of the surfactant in the bulk water phase would remain close to the cmc while additional surfactant molecules that partition to the water phase would go into the micellar phase. At this level of analysis, only the formation of pure micelles of individual surfactant molecules is considered for reasons discussed below. Thus, the accumulation of the surfactants in the water phase can be obtained by an overall mole balance on the system with the condition that the concentration of a surfactant molecule i in the bulk water phase remain constant once its critical micelle concentration, cmci, is reached. Considering this argument, the accumulation of the surfactant molecules in the bulk water phase can be expressed as follows:

for i = 1, 2, ..., N (11)

In eq 11, CH+ represents the concentration of hydronium ions in the water phase. At this point, the concentration, C*i , that will provide the appropriate molar flux of the surfactants from the oil phase can be obtained from a linear combination of Cun,i * and Cch(o),i * as shown in eq 12. 11708



dCw, i dt

dt

decanoic acid (DA). The first major assumption is that the surfactants are dilute. Since tetraacids are normally only present in real crude oils at concentrations up to 100 ppm,1−5 and the monoacid concentrations are also relatively dilute in real crude oils, the assumption of dilute surfactants is applicable to the system studied. The second major assumption is the quasiequilibrium of the adsorption and micelle formation processes as well as the speciation (dissociation) reactions. This assumption may not be entirely appropriate in all cases, especially with respect to the adsorption process of the tetraacids as Riccardi et al.29 have demonstrated that tetraacid molecules (especially BP10 with its rigid core structure) undergo significant conformation changes to orient the four acidic “arm” moieties toward the interface. This reorientation is related to an adsorption energy barrier and suggests that a kinetic (nonequilibrium) adsorption model may be more appropriate to model the adsorption of the tetraacids. However, in this work the equilibrium concentrations in the oil, water, and interfacial phases after the initial formation of the monolayer are the most relevant to modeling the extent of tetraacid precipitation as will be discussed below. In fact, the model was only solved in the dynamic form due to the fact that it facilitates an easier numerical solution with the nonlinear adsorption isotherm employed in this work as discussed below. It should be noted here that mixed micelles of BP10 and DA are not considered because recent studies with neutron scattering experiments33 have indicated that BP10 and DA do not form mixed micelles; thus the assumption of pure micelles is appropriate for the system studied in this work. Finally, the assumption of no coalescence during the formation of the monolayer and the uniform drop size distribution is the most precarious assumption in this model. In real crude oil systems, oil−water emulsions are generally thought to be formed in the intense turbulent flow field around the wellhead or subsequent choke valves early in the processing chain. Generally, the initial drop sizes are quite small, which means the coalescence rate is quite fast at these initial times and the surface-to-volume ratio of the emulsion should be expected to change quite quickly,34−37 especially before the formation of the initial monolayer. Therefore, the assumption of no coalescence during the formation of the initial monolayer is weak. However, the purpose of this work is to develop the framework for a multiphase transport model that accounts for the effect of the surface-to-volume ratio on the interfacial concentration of tetraacids in mixed surfactant (multicomponent) systems. In future work, the framework of the model could be incorporated into a population balance formulation34−37 that accounts for a nonuniform drop size distribution as well as drop−drop coalescence and drop breakage. This approach would tie the dynamic surfactant mass balance and its effect on the interfacial tension to the changing surface-to-volume ratio of an emulsion.38 However, the simplified model as presented here should still provide insight into the qualitative trends one could expect to observe for the interfacial concentration of tetraacids in liquid−liquid dispersions based on the surface-to-volume ratio of the dispersion expressed in terms of the drop size and water cut and can be compared to trends observed11,22 in real crude oil systems. Equations 1, 2, 7, 10, and 12−17b represent a mathematical model that can be used to describe the dynamic behavior of surfactant molecules as they move from an oil phase to a water phase dispersed in the oil when the surface-to-volume ratio of the water phase is large. The set of ordinary differential−

⎧ ⎛ ⎞ ⎛ ⎞np,i ⎪− 1 − ϕ 3 f K ⎜C − 1 − ϕ ⎜ C H+ ⎟ Ci* ⎟ ; ⎜ ⎟ ch, i fw, i ⎜ w, i ⎪ ϕ Rd ϕ ⎝ K̅ a, i ⎠ K p, i ⎟⎠ ⎪ ⎝ =⎨ ⎪ if Cw, i < cmci ⎪ ⎪ 0; if C ≥ cmc w, i i ⎩ (15a)

when i = acid or base

dCw, i

Article

⎧ ⎛ *⎞ ⎪− 1 − ϕ 3 (1 − f )K ⎜C − 1 − ϕ Ci ⎟ ; fw, i w, i ⎜ ch, i ⎪ ϕ Rd ϕ K p, i ⎟⎠ ⎪ ⎝ =⎨ ⎪ if Cw, i < cmci ⎪ ⎪ 0; if C ≥ cmc w, i i ⎩ (15b)

when i = neutral

No surfactants are considered to be in the bulk water phase at the initial time and, thus, the initial condition of eqs 15a and 15b can be expressed as follows: at t = 0,

Cw, i = 0,

for i = 1, 2, ..., N

(16)

It should be pointed out here that eqs 15a and 15b are valid for acids and bases where Cw,i represents the concentration of the conjugate base or acid, respectively, in the water phase. For neutral surfactants, Cw,i simply represents the concentration of the neutral surfactant in the water phase where np,i = 0 for nonionic surfactants. Also, eq 15a is readily extendable to surfactant salts where the term corresponding to the ionization equilibrium would have the appropriate counterion concentration and equilibrium dissociation constant. The number of moles, nM,i, of surfactant molecule i in the micellar phase per unit volume of water (the molar density of surfactant molecule i in the micellar phase) can be obtained from an overall mass balance given by the following expression: nM, i = (1 − ϕ)(Co,0 i − Co, i) − ϕ ⎛ ⎛ C + ⎞np,i ⎞ ⎜ − ϕ⎜1 + ⎜⎜ H ⎟⎟ ⎟⎟Cw, i , ⎝ K̅ a, i ⎠ ⎠ ⎝

3 Γi Rd

when i = acid or base (17a)

nM, i = (1 − ϕ)(Co,0 i − Co, i) − ϕ when i = neutral

3 Γi − ϕCw, i , Rd (17b)

It is important to note here that, in the numerical solution of the model, the accumulation term, dCw,i/dt, for a given surfactant molecule i in the bulk water phase is first calculated according to the top expressions in eqs 15a and 15b at every time step. Then the current concentration, Cw,i, of surfactant i is compared to its cmci. The value of dCw,i/dt will only be set to zero under the condition that Cw,i ≥ cmci and the calculated dCw,i/dt > 0. This way, if the interfacial concentrations adjust so that molecules of type i would be required to return back from the water phase, the mole balance on the bulk water phase will respond accordingly. The final form of the model is given by eqs 1, 2, 7, 10, and 12−17b. Many assumptions have been made in the course of the model derivation. At this point, the model assumptions will be discussed relative to the system studied below. Specifically, the model formulated above will be applied to mixed surfactant solutions containing the model tetraacid compound BP10 and 11709



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Figure 1. Dynamic behavior of (a) bulk oil concentration, Co,i, (b) bulk water concentration, Cw,i, (c) interfacial concentration, Γi, and (d) number of moles in the micellar phase per unit volume, nM,i, at three different initial oil phase concentrations, C0o,DA, of decanoic acid (DA). The value of C0o,BP10 is fixed at 0.01 mM, while the value of C0o,DA is varied as indicated in the legend.

Table 1. Base Parameter Set Used in the Calculations system

component: BP10 (i = 1)

component: DA (i = 2)

parameter (unit)

value

parameter (unit)

value

parameter (unit)

value

T (K) ϕ Rd (μm) pH ρo (kg/m3) ρw (kg/m3) μo (mPa·s) μw (mPa·s) CTo (mol/m3) CTw (mol/m3) I (mol/m3) ε εs ds (Å)

298 0.2 10 8.0 916.7 997.0 0.6 0.9 8529 55 409 600 80.1 42.0 2.32

C0o,BP10 (mol/m3) Dfo,BP10 (m2/s) Dfw,BP10 (m2/s) Kfo,BP10 (m/s) Kfw,BP10 (m/s) Kp,BP10 K̅ a,BP10 (mol/L) np,BP10 zBP10 Δμ̂B°P10,σ/o/kbT Δμ̂°BP10,σ/w/kbT aun,BP10 (Å2) ach,BP10 (Å2) cmcBP10 (mol/m3)

0.01 5.36 × 10−10 2.12 × 10−10 7.59 × 10−5 3.17 × 10−5 2000 1.0 × 10−6 4 −4 −66.31 −184.7 268 721 6.0 × 10−4

C0o,DA (mol/m3) Dfo,DA (m2/s) Dfw,DA (m2/s) Kfo,DA (m/s) Kfw,DA (m/s) Kp,DA K̅ a,DA (mol/L) np,DA zDA Δμ̂ D° A,σ/o/kbT Δμ̂ °DA,σ/w/kbT aun,DA (Å2) ach,DA (Å2) cmcDA (mol/m3)

1.0 1.43 × 10−9 5.64 × 10−10 1.86 × 10−4 7.67 × 10−5 4.1 1.0 × 10−5 1 −1 −50.07 −75.79 78 137 50.0

3. RESULTS AND DISCUSSION The theoretical results presented in Figures 1−6 were obtained from the numerical solution of the mathematical model presented above in order to elucidate the behavior of the interfacial concentration of a tetraacid molecule in a liquid− liquid dispersion as a function of (a) the water cut, ϕ, (b) the drop radius, Rd, and (c) the initial concentration of an interfacially active monoacid that adsorbs competitively on the interface. Therefore, the competitive transport and adsorption of two acidic surfactant molecules are considered here: the model tetraacid compound BP10 (component 1) and decanoic acid (DA; component 2). It should be noted again that decanoic acid was chosen to represent a monoacid compound

algebraic equations are nonlinear and coupled due to the adsorption isotherm given by eqs 7, 8, 10, and 12, and they were solved simultaneously with Gear’s BDF method.39 It should be noted that the time step was limited to 0.1Rd/Kfo,1 in order to ensure that the values of the adsorbed phase concentrations, Γi (i = 1, 2, ..., N), converged to their positive real roots. In fact, the advantage of solving the dynamic model with a limited step size to obtain the equilibrium values of the system variables is that the solution reliably converges to the positive real roots of the adsorption isotherm at each time step over a wide range of parameter values. 11710



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(dashed lines), and 1000 (dashed−dotted lines). It should be noted that the value of C0o,BP10 was fixed at 0.01 mol/m3 (see Table 1) and the value of C0o,DA was varied from 0.1, 1.0, and 10 mol/m3 to achieve the desired value of the initial oil concentration ratio C0o,DA/C0o,BP10. It should also be pointed out here that the interfacial concentrations, Γi, in Figure 1c have been normalized by a term denoted as Γi,max; the normalization term Γi,max is defined for convenience and defined in terms of the hard disk interfacial area, ach,i, of the charged acids (Γi,max = 1/(ach,iNav)) and is used consistently for the remainder of the results presented. The results in Figure 1a indicate that, for all values of the initial oil concentration ratio C0o,DA/C0o,BP10, the tetraacid BP10 is essentially completely removed from the bulk oil phase while around 90−95% of the decanoic acid leaves the bulk oil phase. This result is primarily due to oil−water partitioning requirements in order to ensure that eq 9 is satisfied. Also, according to Figure 1a, the effect of C0o,DA/C0o,BP10 on the rate of removal of BP10 from the oil is negligible while a slight deviation is observed for DA when C0o,DA/C0o,BP10 = 1000. In Figure 1b, it can be observed that the decanoic acid accumulates significantly in the water phase but there is a significant drop in Cw,DA when C0o,DA/C0o,BP10 = 1000. On the other hand, it does not appear that the BP10 partitions much into the water phase except when the value of ratio C0o,DA/ C0o,BP10 is 1000; in this case, the BP10 partitions to the water phase where it quickly reaches its cmc value and forms micelles as shown in Figure 1d. In Figure 1c, the dynamic behavior of the interfacial concentration of BP10 indicates that the tetraacid dominates the interfacial behavior. This can be inferred from the fact that the value of the ratio C0o,DA/C0o,BP10 has little effect on ΓBP10 except for when C0o,DA/C0o,BP10 = 1000. In this case, ΓBP10 begins to decrease slightly after approaching the equilibrium value of ΓBP10 obtained for the other two lower values of the ratio C0o,DA/C0o,BP10; the slight decrease in the value of ΓBP10 corresponds to its bulk water phase partitioning and formation of BP10 micelles. The results seem to suggest that, ° /kbT and ΔμBP10,σ/w ° /kbT for since the values of both ΔμBP10,σ/o BP10 are significantly greater than the corresponding values for DA, BP10 will dominate the adsorption behavior. Additionally, the requirement of eq 9 on the oil−water partitioning behavior has an effect of limiting the amount of DA that can be adsorbed since a significant amount of DA will be required to partition into the bulk water phase. However, when the amount of monoacid relative to tetraacid becomes large enough, the increased driving force for adsorption of the monoacids can drive the tetraacid into the water phase, where it starts to form micelles due to its low cmc. The effect of partitioning to the water phase and forming micelles leads to a slightly lower interfacial concentration for the tetraacid when the monoacid is in a large excess in the oil phase relative to the tetraacid. On the other hand, as the tetraacid partitions to the water phase and forms micelles when the value of the ratio C0o,DA/C0o,BP10 = 1000, the interfacial concentration of the monoacid increases, while both the bulk oil and water phase concentrations of the monoacid decrease to accommodate the higher interfacial concentration of the monoacid; thus, the interfacial and bulk oil and water concentrations of the monoacid are linked to the water phase partitioning behavior of the tetraacid. Additionally, Figure 1c indicates that increasing the initial concentration of the monoacid in the bulk oil phase can lead to faster dynamics of adsorption and larger transient adsorbed phase concentrations for the monoacid due to the larger driving force in the film mass transfer flux expression.

due to its chemical similarity to the four-arm groups of BP10. The base set of parameters used in the simulation of the model is summarized in Table 1; these values are used for all simulations unless otherwise indicated. The nonpolar (oil) phase was taken to be 9:1 v/v p-xylene/chloroform in order to be consistent with the calculations and experiments presented in part 1 of this work28 as well as with previous experimental work.12−14 The pH of the water phase was set equal to 8 in order to ensure that both acid molecules are, for all practical purposes, fully dissociated.19,40 The ionic strength of the water phase was taken to be 600 mol/m3 in order to be consistent with previous experimental work with BP10 12−14 that employed 600 mM aqueous NaCl solutions to mimic seawater as well as with the equilibrium IFT experiments used to parametrize the adsorption isotherm in part 1 of this work.28 The value for the dielectric constant of the Stern layer, εs, was obtained from Mulqueen and Blankschtein,26 noting the argument that the predicted surface pressure is not particularly sensitive to this value. Additionally, the Stern layer thickness, ds, was also simply taken to be the ionic diameter of a sodium ion as the sensitivity of the model to ds should be similar to that of εs. The initial concentration, C0o,BP10, of BP10 in the bulk oil phase was taken to be 0.01 mol/m3, which corresponds to a concentration of about 10 ppm (by mass) and represents a reasonable tetraacid concentration for crude oils.8 The free molecular diffusion coefficients for BP10 and DA in the bulk oil and water phases, Dfo,i and Dfw,i (i = BP10, DA), respectively, were obtained from the Wilke−Chang equation41 and were employed to calculate the film mass transfer coefficients for BP10 and DA in the oil and water phases, Kfo,i and Kfw,i (i = BP10, DA), respectively, from the expression developed by Geankoplis41 for mass transport in suspensions of small spherical particles. The partition coefficient, Kp,BP10, of protonated BP10 between the oil and water phases was taken from Simon et al.19 and lies in the range determined by Nordgård et al.40 The value of the partition coefficient, Kp,DA, of protonated decanoic acid (DA) was taken from the value reported for an octanol/water system since no experimental information is available about the partition coefficient of DA between water and the 9:1 (v/v) xylene−CHCl3 oil phase considered here. The effective acid dissociation constant, K̅ a,BP10, for BP10 was assigned considering the discussion given by Simon et al.19 based on the work of Sundman et al.,16 while the effective acid dissociation constant, K̅ a,DA, for decanoic acid was assigned considering the arguments of Hurtevent et al.2 The cmc of BP10 in water at an ionic strength of 600 mM was determined by Nordgård et al.13 The value of the cmc of DA was taken from Namani and Walde42 as an estimate; it should be noted that in the results presented in Figures 1−6 the concentration of ionized decanoic acid in the bulk water phase never approaches any of the cmc values reported by Namani and Walde42 and, thus, the value taken here does not have any significant effect on the system behavior. Finally, the values of the adsorption isotherm parameters, Δμi,σ/o ° /kbT, Δμi,σ/w ° /kbT, aun,i, and ach,i (i = BP10, DA), were obtained from the calculations and experiments presented in part 1 of this work.28 In Figure 1, the dynamic behaviors of the bulk oil concentrations, Co,i (Figure 1a), the bulk water concentrations, Cw,i (Figure 1b), the interfacial concentrations, Γi (Figure 1c), and the molar density in the micellar phase, nM,i (Figure 1d), are presented for BP10 (blue lines) and decanoic acid (black lines), when the initial oil concentration ratio of the two components, C0o,DA/C0o,BP10, is equal to 10 (solid lines), 100 11711



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Figure 2. Dynamic behavior of (a) bulk oil concentration, Co,i, (b) bulk water concentration, Cw,i, (c) interfacial concentration, Γi, and (d) number of moles in the micellar phase per unit volume, nM,i, at three different values of the drop radius, Rd, when C0o,DA/C0o,BP10 = 100.

the oil−water interface since no water phase partitioning or micelle formation appears to occur for BP10 at the small drop size. This result is likely due to the fact that the total surface area for adsorption is very large when Rd = 5 μm relative to the larger drop sizes while the total amount of moles of BP10 in the overall system remains unchanged. Consequently, the large surface-to-volume ratio of the smaller drop radius system drives down the tetraacid interfacial concentration simply because smaller drops provide a larger area to distribute a fixed amount of tetraacid molecules. These results can be further elucidated by comparing the plot of the interfacial concentration of BP10 in Figure 1c when the ratio C0o,DA/C0o,BP10 = 100 where the drop radius is 10 μm and the value of ΓBP10 as the system approaches equilibrium falls between the value of ΓBP10 obtained when the drop radius is 5 and 20 μm. This observation seems to suggest that the interfacial concentration of tetraacid compounds is strongly controlled by the amount of accessible surface area for adsorption up to the point where the total available surface area becomes small enough such that total amount of tetraacid in the system can fully saturate the interface. Naturally since the tetraacids are found in such small concentrations in the bulk oil phase,8 vary large values of the drop radius might be required to provide a small enough surface area for tetraacids to reach their largest possible interfacial concentrations that would be dictated by the adsorption isotherm. It should also be noted that Figure 2d indicates that the amount of moles of BP10 in the micellar phase increases as the drop radius increases. Again, this result could be simply due to the fact that there is less overall interfacial area for adsorption when the drop radius increases. Specifically, since the BP10 interfacial concentrations are similar in the cases for Rd = 20 and 50 μm, then lowering the total amount of interfacial area would require more of the

However, once BP10 arrives at the interface, the decanoic acid is displaced and partitions to the water phase. By evaluating the BP10 concentrations in all four subplots of Figure 1, it is clear that, as the system approaches equilibrium, almost all of the BP10 mass is located at the oil−water interface in the adsorbed phase except when the monoacid concentration becomes large enough to force some tetraacid to partition to the water phase. On the other hand, while a significant fraction of the total mass of monoacid is located on the interface, a very significant amount of monoacid is also located in the bulk water phase in order to satisfy the partitioning relationship in eq 9. In Figure 2, the dynamic behaviors of Co,i (Figure 2a), Cw,i (Figure 2b), Γi (Figure 2c), and nM,i (Figure 2d) are again plotted for BP10 (blue lines) and decanoic acid (black lines). However, now the value of the drop radius, Rd, is varied: Rd = 5 μm (solid lines), Rd = 20 μm (dashed lines), and Rd = 50 μm (dashed−dotted lines). It is clear from Figure 2a that the dynamic behaviors for the bulk oil concentrations of both BP10 and DA are similar to those shown in Figure 1a. In effect, there is very little influence of the drop radius on the rate and extent that the acidic molecules leave the bulk oil phase as this is dictated by the partitioning relationship in eq 9 as discussed above. Figure 2b,c indicates that the concentration, Cw,DA, of decanoic acid (DA) in the bulk water phase increases starting when the interfacial concentration, ΓDA, of DA begins to approach its maximum value when it begins to be displaced by BP10. These results suggest that the adsorption of the tetraacid tends to force the monoacid preferentially into the water phase. Figure 2b,c also indicates that the interfacial concentration, ΓBP10, of BP10 at the smallest drop radius, Rd = 5 μm, is low relative to the values of ΓBP10 at the large drop radii despite the fact that all the BP10 in the system appears to be adsorbed on 11712



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Figure 3. Effect of initial decanoic acid (DA) concentration, C0o,DA, and water cut, ϕ, on (a) interfacial concentration of BP10, Γ∞ BP10, at equilibrium, (b) concentration, C∞ w,BP10, of BP10 in the bulk water phase at equilibrium, (c) fraction of the total BP10 mass on the interface at equilibrium, and (d) number of moles, n∞ M,BP10, of BP10 in the micellar phase at equilibrium.

BP10 molecules to go to the water phase. It is also worth noting that in Figure 1 no micelle formation was observed for BP10 when the drop radius was 10 μm and the initial oil phase concentration ratio was equal to 100, but in Figure 2, BP10 micelle formation was observed when the drop radius was 20 μm and the initial oil phase concentration ratio was equal to 100. Thus, it appears that the partitioning behvior and micelle formation behavior are also strongly influenced by the amount of interfacial area available in the system. Figures 1 and 2 seem to indicate that the interfacial concentration, ΓBP10, of the tetraacid BP10 adsorbing in the presence of decanoic acid is strongly controlled by the total amount of surface area available for adsorption. Additionally, limiting the amount of surface area for adsorption seems to drive more of the acid compounds into the aqueous phase at the slightly basic pH studied here. The two parameters in the system that control the amount of accessible surface area for adsorption are the water cut, ϕ, and the drop radius, Rd; increasing ϕ increases the available surface area since the total number of drops in the system increases when ϕ increases, while decreasing the drop radius, Rd, increases the surface-tovolume ratio of the dispersion. The composition of the oil phase (namely, the initial monoacid concentration relative to the initial tetraacid concentration) has a weaker effect on the interfacial concentration of the tetraacid for this specific system of BP10 and DA mainly due to restrictions imposed by the partitioning behavior as well as the significant differences in the chemical potential differences between the interface and the bulk oil and water phases. In other words, at basic conditions and for a given amount of interfacial area in the system, increasing the amount of DA (monoacid) relative to BP10 (tetraacid) in the oil phase can push the tetraacid toward the

water phase and slightly decrease the tetraacid interfacial concentration. Since the extent of calcium naphthenate precipitation is hypothesized to be related to the interfacial concentration of tetraacid,8−22 it would be advantageous to examine the effect of the water cut, ϕ, initial oil phase monoacid to tetraacid concentration ratio, C0o,DA/C0o,BP10, and the drop radius, Rd, on the equilibrium values of the interfacial and bulk water phase ∞ tetraacid concentrations, Γ∞ BP10and Cw,BP10, respectively, as well as the equilibrium value of the number of moles of tetraacid in the micellar phase, n∞ M,BP10 (the superscript “∞” denotes an equilibrium value). Since the monolayer establishes an equilibrium rather quickly (on the order of about 10 s) and the reaction with calcium to form precipitates also occurs rather quickly,9,12−14,19,20 the equilibrium interfacial concentration of tetraacid after the initial formation of the monolayer could potentially be a useful point of reference when evaluating the possibility and extent of calcium naphthenate formation based on the water cut, (average) drop radius, and monoacid to tetraacid concentration ratio in the oil. In Figure 3, the equilibrium values of the interfacial BP10 ∞ (Figure 3a), the bulk water BP10 concentration, ΓBP10 ∞ concentration, Cw,BP10 (Figure 3b), the fraction of the total BP10 mass on the interface in the adsorbed phase, Minterface BP10 / Mtotal BP10 (Figure 3c), and the molar density of tetraacid in the micellar phase, n∞ M,BP10 (Figure 3d) are presented as a function of the water cut, ϕ, for six different values of the initial oil phase monoacid to tetraacid concentration ratio, C0o,DA/C0o,BP10. The results in Figure 3 indicate that the equilibrium interfacial concentration of the BP10 (tetraacid) seems to be linked directly with the water phase partitioning and formation of micelles as the results in Figure 1 suggested as well as with the available surface area for adsorption as Figure 2 suggested. 11713



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Figure 4. Effects of initial decanoic acid (DA) concentration, C0o,DA, and water cut, ϕ, on estimated mass of calcium naphthenate precipitated on the interface per unit volume.

There appears to be a lower boundary line for Γ∞ BP10 when micelles are present and an upper boundary line for Γ∞ BP10 when micelles are not present. When comparing Figures 3a, 3b, and 3c, the equilibrium interfacial tetraacid (BP10) concentration, Γ∞ BP10, seems to transition between these lower and upper boundary lines at the point when micelles disappear from the bulk water phase (departure from the lower boundary line) and at the point where the tetraacid no longer significantly partitions into the bulk water phase (joining to the upper boundary line). This behavior is reminiscent of a tie-line on a triangular phase diagram. Also, Figure 3c indicates that as the interfacial concentration transitions from the lower boundary line to the upper boundary line, the fraction of the total mass of the tetraacid (BP10) goes to unity meaning that all of the tetraacid is on the interface. Furthermore, when the initial oil phase monoacid to tetraacid concentration ratio, C0o,DA/C0o,BP10, is low, the value of Γ∞ BP10 increases slightly with increasing water cut, ϕ, until the point where ϕ becomes large enough (in this case at ϕ ∼ 0.15) such that the total amount of tetraacid in the system is inadequate to the cover the total interfacial area available. From this point onward, any additional increases in the water cut, ϕ (which amounts to additional increases the interfacial area) will act to drive down the interfacial concentration. As noted for the results in Figure 1, increasing the ratio C0o,DA/C0o,BP10 can result in an increasing tendency to drive the tetraacid (BP10) toward the bulk water phase. Thus, as the results in Figure 3 indicate, the point at which the value of Γ∞ BP10 transitions from the lower boundary line to the upper boundary line occurs at increasingly higher water cuts as the value of C0o,DA/C0o,BP10 increases. Based on the knowledge of the interfacial concentration of BP10 after the initial formation of a monolayer, the mass of Ca2BP10 precipitate per unit volume of the dispersion that could form could be estimated from the following expression: MCa 2BPP10 = WBP10ϕ

3 ∞ Γ BP10f (Γ∞ BP10) Rd

In eq 18, MCa2BP10 represents the estimated mass of Ca2BP10 precipitate per unit volume of the dispersion, WBP10 denotes the molecular weight of BP10, and f(Γ∞ BP10) represents the fraction of the tetraacid (BP10) that cross-links at a given interfacial concentration Γ∞ BP10. At this point, the precise functional form for f(Γ∞ BP10) is not known although one would expect a ∞ sigmoidal function that approaches unity when Γ BP10 approaches ΓBP10,max. Preliminary experiments have been performed to establish f(Γ∞ BP10) which seem to indicate a sigmoidal functional form that transitions from 0 to 1 at about a 50−60% surface coverage (S. Simon, private communication, 2013). Therefore, at a first level of analysis, the fraction, f(Γ∞ BP10), of the tetraacid (BP10) that cross-links to form Ca2BP10 precipitate at the oil−water interface could be expressed as follows: f (Γ ∞ BP10)

⎧ ⎡ (Γ ∞ ⎤⎫ BP10/ ΓBP10,max ) − f1/2 ⎪ 1⎪ ⎥⎬ ⎨ 1 + erf⎢ = ⎢ ⎥⎦⎪ 2⎪ 2 (1 f )/4 − ⎣ ⎭ ⎩ 1/2 (19)

Γ∞ BP10/ΓBP10,max

In eq 19, f1/2 represents the value of where 50% of the BP10 molecules on the interface are cross-linked with calcium. In this work, the value of f1/2 has been taken to be equal to 0.5 ( f1/2 = 0.5). In Figure 4, the estimated mass of Ca2BP10 precipitate that could form per unit volume of the dispersion is plotted, based on the equilibrium values of the interfacial BP10 concentration, Γ∞ BP10, that are presented in Figure 3, as a function of the water cut, ϕ, for six different values of the initial oil phase monoacid to tetraacid concentration ratio, C0o,DA/C0o,BP10. The results in Figure 4 indicate that the Ca2BP10 precipitation behavior goes through a maximum at the value of the water cut that provides the largest tetraacid (BP10) interfacial concentration before the increasing interfacial area of the dispersion begins to drive down the value of Γ∞ BP10. Beyond this water cut that provides the maxima in MCa2BP10, the mass of Ca2BP10 precipitate that

(18) 11714



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∞ Figure 5. Effect of drop radius, Rd, and water cut, ϕ, on (a) interfacial concentration of BP10, Γ∞ BP10, at equilibrium, (b) concentration, Cw,BP10, of BP10 in the bulk water phase at equilibrium, (c) fraction of the total BP10 mass on the interface at equilibrium, and (d) number of moles, n∞ M,BP10, of BP10 in the micellar phase at equilibrium.

Figure 6. Effect of drop radius, Rd, and water cut, ϕ, on the estimated mass of calcium naphthenate precipitated on the interface per unit volume.

could be formed begins to steadily decrease with increasing ϕ, where MCa2BP10 decreases rapidly in the range 0.2 ≤ ϕ ≤ 0.3 and becomes negligible when ϕ > 0.4. It should be stressed here that this prediction for the range of ϕ > 0.2 is consistent with observations of the calcium naphthenate precipitation behavior as a function of the water cut in real crude oil transport processes.11,22 Additionally, the effect of the tetraacid water

phase partitioning and micelle formation seems to reduce the amount of Ca2BP10 precipitate formation until the interfacial area of the dispersion increases to the point where the BP10 (tetraacid) micelles disappear and water phase partitioning is reduced; at this point the estimated amount of Ca2BP10 precipitate increases since the interfacial concentration, Γ∞ BP10, of BP10 increases. It should be mentioned at this point that the 11715



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dissociated (charged) state. The model employs a molecular mixed monolayer isotherm which is able to predict interfacial tension of mixed surfactant solutions of BP10 and DA as shown in part 1 of this work.28 The molecular mixed monolayer isotherm was parametrized by molecular dynamics (MD) simulation as well as experimental interfacial tension measurements, and in this respect the model presented in this work could be described as a multiscale model. The dynamic and equilibrium behaviors of the bulk oil, bulk water, and interfacial concentrations of BP10 and DA were described in terms of the surfactant composition of the oil phase, water cut, and droplet size. It was shown that, due to the high interfacial activity of the tetraacid BP10 relative to the monoacid DA as well as bulk oil−water partitioning requirements, BP10 dominates the interfacial composition when the initial concentration of DA in the bulk oil phase is up to approximately 2−2.5 orders of magnitude (for Rd = 10 μm) greater than the initial concentration of BP10 in the bulk oil phase. When the initial concentration of DA in the bulk oil phase was about 2−2.5 orders of magnitude larger than the initial concentration of BP10 in the bulk oil phase, the DA began to displace a portion of the BP10 from the interface causing the BP10 to partition to the water phase. Upon displacement from the interface and partitioning to the water phase, the BP10 rapidly reached its low cmc (cmcBP10 = 6 × 10−4 mol/m3) and began to form micelles which stopped the displacement of BP10 from the interface and stabilized the interfacial composition in the system. This phenomenon plays an important role in terms of affecting the interfacial composition at low values of the water cut and diminishes in effect as the water cut increases. Additionally, the results indicate that the interfacial composition of BP10 and DA is strongly controlled by the surface-to-volume ratio of the dispersion. As the drop radius decreases below about 20 μm (Rd < 20 μm), the increasing surface area of the dispersion drives down the interfacial concentration of the tetraacid. This result is due to the fact that, at these conditions, almost all of the BP10 initially present in the system is in the adsorbed phase, and thus, the increasing surface area leads to lower interfacial concentrations of the tetraacid. In general, a similar effect can be observed by increasing the water cut. As the water cut increases, the interfacial area of the dispersion increases which drives down the interfacial tetraacid concentration. However, due to the competitive adsorption and partitioning behavior, there is a transitional subdomain of the water cut region where the interfacial tetraacid concentration increases with the water cut. This transitional water cut subdomain begins at the point where the water volume becomes large enough that the tetraacid concentration in the bulk water phase falls below its cmc and ends at the point where the tetraacid concentration in the bulk water phase becomes negligible. The equilibrium interfacial tetraacid (BP10) concentration described by the model was also employed to estimate the mass, MCa2BP10, of Ca2BP10 precipitate per unit volume of the dispersion that could form in the system by the hypothesized interfacial cross-linking reaction.1,2,4−22 The results indicated that MCa2BP10 goes through a maximum value as a function of the water cut, ϕ. The value of the water cut where MCa2BP10 reaches its maximum value is dictated by the surface-to-volume ratio of the dispersion. The water cut at which the maximum value of MCa2BP10 occurs increases as the surface-to-volume ratio of the dispersion decreases (Rd increases). The results also

results in Figure 4 represent an experimentally testable prediction to confirm the hypothesis that the bulk of the calcium naphthenate precipitate formed in crude oil systems containing tetranaphthenic acids occurs due to an interfacial cross-linking reaction between the tetraacids and the Ca2+ ions. Of course, it is important to point out that the model predictions for the estimated amount of Ca2BP10 precipitate are not expected to be quantitative at this point due to the fact that several model assumptions would be violated in real systems (e.g., the neglecting of coalescence, constant pH) and real naphthenic monoacids and the indigenous ARN tetraacids found in crude oils would have different adsorption, partitioning, and micellization behaviors (lower cmc’s for the monoacids and formation of mixed micelles). However, the general trend in the precipitate formation with respect to the water cut should be expected to be similar as this is mainly a consequence of the available interfacial area for adsorption. In effect, if calcium naphthenate precipitation occurs in real systems shortly after the oil−water emulsion is first formed and the initial monolayer is established, then the qualitative trends with respect to the available interfacial area for adsorption predicted by the model should be reasonable. In Figure 5, the equilibrium values of the interfacial BP10 ∞ concentration, ΓBP10 (Figure 5a), the bulk water BP10 concentration, C∞ w,BP10 (Figure 5b), the fraction of the total BP10 mass on the interface in the adsorbed phase, Minterface BP10 / Mtotal BP10 (Figure 5c), and the number of moles of tetraacid in the micellar phase, n∞ M,BP10 (Figure 5d) are presented as a function of the water cut, ϕ, for six different values of the drop radius, Rd. In these plots, the value of the initial oil phase monoacid to tetraacid concentration ratio, C0o,DA/C0o,BP10 = 100, is high enough such that the delineation between the interfacial tetraacid (BP10) concentration with water phase partitioning and micelle formation is significant in all systems studied. Also, the effect of the available surface area for adsorption is clear. In particular, as the drop size, Rd, increases, the total available interfacial area for adsorption decreases. Consequently, as the drop size, Rd, increases, it takes larger values of the water cut, ϕ, to provide enough interfacial area for adsorption such that the interfacial concentration of the tetraacid is driven down. It is clear from Figure 5 that the value of the water cut, ϕ, where (a) ∞ the value of Γ∞ BP10 decreases, (b) the value of Cw,BP10 decreases from the cmcBP10 toward zero, (c) the fraction of the total BP10 mass on the interface reaches unity, and (d) the BP10 micelles vanish, increases when the radius, Rd, of the drops increases. In Figure 6, this phenomenon causes the peak value of MCa2BP10 to shift to larger values of ϕ as the drop radius, Rd, increases while the magnitude of MCa2BP10 decreases slightly as the value of Rd increases. It is also clear in Figure 6 that the range of ϕ over which the mass of Ca2BP10 precipitates form increases significantly as the value of Rd increases.

4. CONCLUSIONS AND REMARKS In this work, a multicomponent mass transfer model was constructed and solved to describe the adsorption and phase partitioning of a model tetracarboxylic acid (BP10) and decanoic acid (DA) in a liquid−liquid dispersion consisting of water droplets dispersed in a nonpolar (oil) phase. The model takes into account partitioning of components between two phases, acid dissociation due to pH, and micelle formation in the water phase; the simulations were performed at basic conditions (pH 8) where most of the acid molecules are in the 11716



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Technologies, Clariant, Conoco Phillips, ENI, Petrobras, R.E.P., Statoil, Shell Global Solutions, Talisman, and Total.

indicated that the amount of Ca2BP10 can be suppressed by increasing the amount of monoacid in the system relative to the amount of the tetraacid system up to the point where the water cut is small enough such that the BP10 partitions into the water phase and forms micelles. The behavior of MCa2BP10 as a function of the water cut is an experimentally testable prediction that could support the hypothesis of the interfacial cross-linking reaction1,2,4−22 believed to be the primary mechanism responsible for the formation of calcium naphthenate precipitates in crude oil systems. The model prediction is a reasonable estimation of the parametric trends for MCa2BP10 if the precipitation reaction occurs quickly upon the establishment of the monolayer immediately after formation of a crude oil emulsion (e.g., at the wellhead) without too much drop growth due to coalescence occurring during the monolayer formation. Due to the assumptions of the model, it is important to stress that the predicted precipitation behavior should be interpreted as a qualitative trend to be expected for different values of the water cut and drop radius. However, the general trend with respect to the fact that the amount of precipitate falls off as the water cut increases is consistent with field observations11,22 in real crude oil systems. Additionally, the effect of a higher initial monoacid concentration in the bulk oil suppressing the value of MCa2BP10 could also be an explanation of why the calcium naphthenate precipitation problem can be more significant for crude oils with a lower total acid number (TAN) than for crude oils with a higher TAN number.2 The model presented in this work for the interfacial mass transfer and estimation of the mass of Ca2BP10 precipitate formed per unit volume of the dispersion is the first model to explicitly consider the interfacial cross-linking reaction that is hypothesized1,2,4−22 to be responsible for the major portion of calcium naphthenate precipitates that form in crude oil systems. Considering the library of synthetic asphaltene and tetraacid molecules developed by Nordgård et al.,12−14 the modeling approach presented in part 1 of this work28 could be employed to formulate model crude oil systems with pseudocompounds representing interfacially active resins and asphaltenes with bulk oil concentrations and compositions linked to a real crude oil. The model presented in part 2 of this work could then be employed to evaluate the risk of calcium naphthenate precipitation as a function of water cut, drop size, and perturbations on the composition of the oil as well as developing chemical inhibitor strategies based on competitive adsorption.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +47 735 90 338. Present Address †

E.R.: Department of Chemistry, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge support for this work by the Norwegian Research Council and the members of the Ugelstad Laboratory JIP-2 industrial consortium: Champion 11717

LIST OF SYMBOLS aun,i = hard disk area of surfactant molecule i in the uncharged state at the interface (m2) ach,i = hard disk area of surfactant molecule i in the charged state at the interface (m2) Co,i = concentration of surfactant molecule i in the bulk oil phase (mol/m3) CTo = total molar concentration of the oil phase (mol/m3) CTw = total molar concentration of the water phase (mol/ m3) Cw,i = concentration of surfactant molecule i in the charged state in the bulk water phase (mol/m3) Cw(un),i = concentration of surfactant molecule i in the uncharged state in the bulk water phase (mol/m3) Ci* = concentration of component i in the oil layer adjacent to the interface (mol/m3) C*ch,i = concentration of surfactant molecule i in the charged state in the oil layer adjacent to the interface (mol/m3) Cun,i * = concentration of surfactant molecule i in the uncharged state in the oil layer adjacent to the interface (mol/m3) Dfo,i = free molecular diffusion coefficient of surfactant molecule i in the bulk oil phase (m2/s) Dfw,i = free molecular diffusion coefficient of surfactant molecule i in the bulk water phase (m2/s) e = elementary charge (1.602 × 10−19 C) fch,i = fraction of surfactant molecule i in the charged state (defined in eqs 6a and 6b) I = ionic strength of the aqueous phase (mol/m3) Kp,i = oil−water partition coefficient of molecule i (defined in eq 9) Kfo,i = film mass transfer coefficient of surfactant molecule i in the oil phase (m/s) Kfw,i = film mass transfer coefficient of surfactant molecule i in the water phase (m/s) Kaj,i = acid dissociation constant for dissociation step j of molecule i (mol/L) K̅ a,i = effective acid dissociation constant (mol/L) MCa2BP10 = mass of Ca2BP10 precipitate per unit volume of the dispersion (g/m3) mp,i = number of protons donated or accepted by surfactant molecule i (mp,i = |np,i|) Nav = Avogadro’s number (6.022 × 1023 mol−1) nM,i = number of moles of surfactant molecule i in the micellar phase per unit volume of water (mol/m3) np,i = number of protons donated by surfactant molecule i (np,i > 0 for acids, np,i < 0 for bases, np,i = 0 for neutral surfactants) pK̅ a,i = logarithmic value of the effective acid dissociation constant of molecule i Rd = radius of water droplets (m) R = ideal gas constant (8.314 J/mol/K) ri = hard disk radius of surfactant molecule i (m) rch,i = hard disk radius of surfactant molecule i in the charged state (m) run,i = hard disk radius of surfactant molecule i in the uncharged state (m) T = absolute temperature (K) t = time (s) Wi = molecular weight of component i (g/mol) dx.doi.org/10.1021/ie501296t | Ind. Eng. Chem. Res. 2014, 53, 11704−11719


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(11) Brocart, B.; Hurtevent, C. Flow Assurance Issues and Control with Naphthenic Oils. J. Dispersion Sci. Technol. 2008, 29, 1496−1504. (12) Nordgård, E. L.; Sjöblom, J. Model Compounds for Asphaltenes and C80 Isoprenoid Tetra-Acids. Part I: Synthesis and Interfacial Activities. J. Dispersion Sci. Technol. 2008, 29, 1114−1122. (13) Nordgård, E. L.; Magnusson, H.; Hanneseth, A.-M. D.; Sjöblom, J. Model Compounds for C80 Isoprenoid Tetra-acids: Part II. Interfacial Reactions, Physicochemical Properties and Comparison with Indigenous Tetra-acids. Colloids Surf., A 2009, 340, 99−108. (14) Nordgård, E. L. Model Compounds for Heavy Crude Oil Components and Tetrameric Acids. Ph.D. Dissertation, Norwegian University of Science and Technology, Trondheim, Norway, 2009. (15) Mohammed, M. A.; Sorbie, K. S.; Shepherd, A. G. Thermodynamic Modeling of Naphthenate Formation and Related pH Change Experiments. SPE Prod. Oper. 2009, 24, 466−472. (16) Sundman, O.; Nordgård, E. L.; Grimes, B. A.; Sjöblom, J. Potentiometric Titrations of Five Synthetic Tetra-acids as Models for Indigenous C80 Tetra-acids. Langmuir 2010, 26, 1619−1629. (17) Sundman, O.; Simon, S.; Nordgård, E. L.; Sjöblom, J. Study of the Aqueous Chemical Interactions Between a Synthetic Tetra-Acid and Divalent Cations as a Model for the Formation of Metal Naphthenate Deposits. Energy Fuels 2010, 24, 6054−6060. (18) Mohammed, M. A.; Sorbie, K. S. Thermodynamic Modeling of Calcium Naphthenate Formation: Model Predictions and Experimental Results. Colloids Surf., A 2010, 369, 1−10. (19) Simon, S.; Reisen, C.; Bersås, A.; Sjöblom, J. Reaction between Tetrameric Acids and Ca2+ in Oil/Water System. Ind. Eng. Chem. Res. 2012, 51, 5669−5676. (20) Nordgård, E. L.; Simon, S.; Sjöblom, J. Interfacial Shear Rheology of Calcium Naphthenate at the Oil/Water Interface and the Influence of pH, Calcium, and in Presence of a Model Monoacid. J. Dispersion Sci. Technol. 2012, 33, 1083−1092. (21) Ge, L.; Vernon, M.; Simon, S.; Maham, Y.; Sjöblom, S.; Xu, Z. Interactions of Divalent Cations with Tetrameric Acid Aggregates in Aqueous Solution. Colloids Surf., A 2012, 396, 238−245. (22) Baugh, T. Oil Field Fouling: Tales of the Unexpected. Presented at the Thirteenth International Conference on Petroleum Phase Behavior and Fouling, St. Petersburg Beach, FL, June 10−14, 2012; lecture K-3. (23) Chatterjee, J.; Wasan, D. T. An interfacial Tension Model for Mixed Adsorbed Layer for a Ternary System: Application to an Acidic Oil/Alkali/Surfactant System. Colloids Surf., A 1998, 132, 107−125. (24) Chang, C.-H.; Franses, E. I. Adsorption Dynamics of Surfactants at the Air/Water Interface: A Critical Review of Mathematical Models, Data, Mechanisms. Colloids Surf., A 1995, 100, 1−45. (25) Nikas, Y. J.; Puvvada, S.; Blankschtein, D. Surface Tensions of Aqueous Nonionic Surfactant Mixtures. Langmuir 1992, 8, 2680− 2689. (26) Mulqueen, M.; Blankschtein, D. Prediction of Equilibrium Surface Tension and Surface Adsorption of Aqueous Surfactant Mixtures Containing Ionic Surfactants. Langmuir 1999, 15, 8832− 8848. (27) Mulqueen, M.; Blankschtein, D. Theoretical and Experimental Investigation of the Equilibrium Oil-Water Interfacial Tension of Solutions Containing Surfactant Mixtures. Langmuir 2002, 18, 365− 376. (28) Kovalchuk, K.; Riccardi, E.; Grimes, B. A. Multiscale Modeling of Mass Transfer and Adsorption in Liquid−Liquid Dispersions. 1. Molecular Dynamics Simulations and Interfacial Tension Prediction for a Mixed Monolayer of Mono- and Tetracarboxylic Acids. Ind. Eng. Chem. Res. 2014, DOI: 10.1021/ie501295k. (29) Riccardi, E.; Kovalchuk, K.; Mehandzhiyski, A. Y.; Grimes, B. A. Structure and Orientation of Tetra Carboxylic Acids at Oil-Water Interfaces. J. Dispersion Sci. Technol. 2014, 35, 1018−1030. (30) McCoy, M. A.; Liapis, A. I. Evaluation of Kinetic Models for Biospecific Adsorption and its Implications for Finite Bath and Column Performance. J. Chromatogr. 1991, 548, 25−60. (31) Grimes, B. A.; Liapis, A. I. The Interplay of Diffusional and Electrophoretic Mass Transport Mechanisms of Charged Solutes in

zi = charge number of molecule i Greek Symbols

Γi = interfacial molar concentration of surfactant molecule i (mol/m2) Γch,i = interfacial molar concentration of surfactant molecule i in the charged state (mol/m2) Γun,i = interfacial molar concentration of surfactant molecule i in the uncharged state Γi,max = maximum interfacial molar concentration of surfactant molecule i in the charged state (Γi,max = 1/ach,i/ Nav; mol/m2) λ = Debye−Hückel screening length (m) Δμ̂ i,σ/b ° = modified standard state chemical potential difference of surfactant molecule i between the bulk (oil or water) and the interface (J) Δμ°i,σ/b = standard state chemical potential difference of surfactant molecule i between the bulk (oil or water) and the interface (J) ε = dielectric constant in the bulk aqueous phase εo = permittivity of free space (8.854 × 10−12 C2/J/m) εs = dielectric constant of the Stern layer ϕ = droplet volume fraction ψ = constant term defined in eq 3c (mol/m)

Superscripts

0 = initial value ∞ = equilibrium value

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REFERENCES

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