Mutual solubility–interfacial tension relationship in aqueous binary and ternary hydrocarbon systems

Fluid Phase Equilibria 285 (2009) 24–29

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Mutual solubility–interfacial tension relationship in aqueous binary and ternary hydrocarbon systems Alireza Bahramian ∗ Institute of Petroleum Engineering, University of Tehran, Tehran 11365-4563, Iran

a r t i c l e

i n f o

Article history: Received 16 September 2008 Received in revised form 16 January 2009 Accepted 21 February 2009 Available online 5 March 2009 Keywords: Water–hydrocarbon Liquid–liquid interface Thermodynamics Interfacial tension Regular solution

a b s t r a c t It is shown that the liquid–liquid interfacial tension in binary and ternary aqueous–hydrocarbon systems at any temperature can be determined by knowing the molar compositions of the bulk fluid phases and a system-dependent constant. The system-dependent constant has been found to be related to saturation adsorption. Based on these findings, a new equation has been proposed for reliable prediction of interfacial tension in aqueous–organic systems. The model is simple to use and reasonably describes the interfacial tension of binary water–hydrocarbon mixtures and some ternary aqueous–organic systems. The new equation shows an apparent inconsistency near the critical point which is explained using the underlying mono-molecular layer assumption. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Placing a small amount of oil on the top of water in a capillary tube, Young [1] noticed a smaller capillary rise than with water alone. Liquid–liquid interfaces therefore, according to Young’s observations, similar to liquid–vapour interfaces are in a state of tension. Young suggested that the liquid–liquid interfacial tension was of the same value as the difference of the pure compounds surface tensions [1]. Although some modifications of the Young’s suggestion appeared later in the literature [2,3], none of them received as much attention as the simple modification of Antonow [4,5]. In his suggestion, which is now referred to as Antonow’s rule, he postulated the equality of the liquid–liquid interfacial tension,  ˛ˇ , between two liquid phases ␣ and ␤ and the difference of the two coexisting liquid surface tensions,  ˛ and  ˇ , against their common vapour ␶. That is, Antonow just amended the Young’s suggestion by considering the surface tensions of the equilibrated liquid phases instead of the pure compounds surface tensions. Despite Antonow’s belief [6] that “the validity of Antonow’s rule cannot be disputed and since it is a law of equilibrium, it is bound to be a precise law”, the rule soon received experimental [7,8] and theoretical [9,10] critiques and now its validity has become evident just near the critical point, where two of the three phases are to be fully miscible.

∗ Tel.: +98 21 88632975; fax: +98 21 88632976. E-mail address: [email protected]. 0378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2009.02.013

While most of the investigations on liquid–liquid interfacial tension related its values to the surface tensions of the coexisting liquids, the work of the Donahue and Bartell [11] was the first that suggested a relationship between the interfacial tension of two immiscible liquids and their mutual solubility, using experimental data of water and organic liquids:  OW = a − b log(XO(W ) + XW (O) )

(1)

where  OW is the water–organic liquid interfacial tension, and XO(W) and XW(O) are the mole fractions of the organic liquid in the waterrich phase and the mole fraction of water in the organic-rich phase, respectively. By regression of experimental data at 25 ◦ C, constants a and b were found [12] to be 3.31 and 15.61, respectively. The Donahue–Bartell equation was modified later by Treybal [13] for ternary systems as,

 

OW

= a − b log

XO(W ) + XW (O) +

X3(W ) + X3(O) 2

 (2)

where the compound 3 refers to the organic component that is more soluble in the aqueous phase. While none of the above approaches accounts explicitly for intermolecular forces, the impact of such forces on liquid–liquid interfacial tension first appeared in the work published by Girifalco and Good [14] where similarly to the approach taken by Berthelot [15] in describing cross-molecular interactions, set out an equation to relate the liquid–liquid interfacial tension to the liquid surface tensions through intermolecular forces.

A. Bahramian / Fluid Phase Equilibria 285 (2009) 24–29

25

of Eq. (4), but it must be noted that the above inconsistency is solely a direct consequence of the Shain and Prausnitz assumption. Substitution of Eq. (4) in Eq. (3) and solving the resultant equa-



␣␤

␣␤

Xi = 1, results in Eq. (6) tion for Xi , bearing in mind that which describes the relationship between bulk compositions and the interfacial tension of a c-component in a regular solution system [17], c



␤ 0.5

(Xi␣ Xi )

exp

i=1

a

i

RT

 ␣␤



=1

(6)

The author recently has reported another equation for prediction of interfacial tension of the two immiscible liquid phases, while developing a predictive model for solid–fluid interfacial tension [22]. The equation relates the interfacial tension of a binary system of two immiscible liquid phases to the mutual solubility as; Fig. 1. Relationship between the bulks and interface compositions in a binary immiscible regular solution assuming an equivalent activity coefficient function for all phases.

Using thermodynamic concepts, the liquid–liquid interfacial tension can be related to the compositions of the bulk phases (␣ or ␤) and the interface (␣␤) and the intermolecular forces through activity coefficients () as [16,17];





␣␤

RT = ln ai

␣␤

Xi

␣␤

i



␣␤



=

(3)

Xi␣ i␣

␤

i␣ i

(4)

Considering a binary system which is adequately described by the two-suffix Margules equation, juxtaposing with the Shain and Prausnitz conclusion while making use of Eq. (4), the following system of equations is logically inevitable;



␣␤ 2

␤ 2

␣␤ 2

␤ 2

2(X1 ) = (X1␣ )2 + (X1 )

(5)

2(X2 ) = (X2␣ )2 + (X2 )

␤

That is, a right triangle made by the legs X1␣ and X1 has the ␣␤ 20.5 X1

1 RT 2 a

␤

ln(X1␣ /X1 )

hypotenuse equal to for component 1. This is also the case for compound 2. To illustrate the problem, those triangles are depicted in Fig. 1 as YON and MOU with their hypotenuses, ON and OM, respectively, that indicate the magnitude of the interface compositions, according to Eq. (5). The incongruity arises in the triangle MON, where it seems to have a side, MN, equal to the sum of the other two, OM and ON. Such inconsistency is an inherent consequence of the Shain and Prausnitz assumption on the equality of the interface composition with those of the adjacent layers. Developing this argument, one may however cast doubt on the reliability

␤ 2

(7)

(X2 ) − (X2␣ )2

where a is the average partial molar surface area. The above equation was subsequently reduced to a much simpler form for water–hydrocarbon systems as;  ␣␤ = 0.022

where R is the universal gas constant, T is the absolute temperature, and ai is the partial molar surface area of the ith component. The literature on the modern molecular thermodynamics is rich in activity coefficient models developed based on the intermolecular interactions [18]. Hence, in addition to the compositional effects suggested by Eqs. (1) and (2), Eq. (3) implies the influence of intermolecular forces on interfacial tension. Shain and Prausnitz [16] assumed compositions equal to the compositions at the interface for the layers adjacent to the interface layer, to adhere with the Gibbs–Duhem constraint, and concluded that the expression for activity coefficients of the components at the interface is equivalent to that for the bulk liquid phases. This conclusion, although used extensively by other investigators [19–21], contains a subtle contradiction. For immiscible regular solutions, it has been shown [17] that; i

 ␣␤ =

T ln(XW (O) /XW (W ) ) (XO(W ) )2 − (XO(O) )2

(8)

Although limited to regular solutions, the above equation is the latest published development in the above approach, capable of describing the liquid–liquid interfacial tension of multi-component hydrocarbon–water mixtures with respect to mutual solubility [22]. The applicability of Eq. (6) in predicting liquid–liquid interfacial tension and its superiority over some of the available models [11,13,22–24] have been discussed elsewhere [17,22,24]. Although limited to regular solutions, the equation has been shown to be able to predict of water–hydrocarbon interfacial tensions with reasonable accuracy even at high pressure and high temperature conditions [24]. The equation has also been successful in predicting the water–hydrocarbon liquid–vapour interfacial tension at high pressures, where the vapour phase is a liquid like condensed phase [24]. However, it does not explicitly relate the interfacial tension to mutual solubility, while Donahue–Bartell equation (Eq. (1)) and also Treybal equation (Eq. (2)) suggest a universal correlation between interfacial tension and mutual solubility. Using measured data at different temperatures and pressures, the universality of such correlations has been disputed recently [24,25]. Evaluating the correlation of Donahue–Bartell for several organic–organic and water–organic systems at different temperatures, Chavepeyer et al. [25] found very poor correlation between interfacial tension and mutual solubility and concluded that; such correlation “does not exist and other parameters should be taken into account, namely molecular arrangement parameters of two bulk phases (bulk entropies?) and for the interface.” Most recently, Ayirala and Rao [26] have explored the relationship between solubility and miscibility with the measured values of interfacial tension of ternary water–ethanol–benzene systems. They produced a correlation that strongly related the solubility to interfacial tension in the form of; interfacial tension = C/solubility, where C is a system-dependent constant and, consistent with the conclusion of Chavepeyer et al., may simply imply a certain value of the bulk and the interface entropies. The main aim of this paper is to obtain an explicit relationship between the interfacial tension and mutual solubility, starting from the fundamental equation express by Eq. (3).

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A. Bahramian / Fluid Phase Equilibria 285 (2009) 24–29

Table 1 The deviation of the experimental interfacial tensions from the suggested linear trends of Eq. (13). System

T (K)

R-squared value

a (Å2 /molecule)

Water–benzene-1,4-dioxane Water–trichloromethane–acetone Water–1,1,2-trichloroethane–acetone Water–benzene–acetic acid Water–acetic acid isopropyl ester–acetic acid Water–2-ethylhexanol–acetic acid Water–n-hexane–acetone Water–hexane–methyl ethyl ketone Water–benzene–methanol Water–benzene–isopropanol Water–methyl isobutyl ketone–acetone Water–ethyl propionate–methanol Water–ethyl butyrate–methanol Water–butyl acetate–methanol Water–1,1,2-trichloroethylene–acetone Water–1,1,2-trichloroethylene–methyl ethyl ketone Water–benzene–acetone Water–carbontetrachloride–propanol Perflluro-n-hexane–benzene–hexane Perflluro-n-hexane–carbon disulfide–hexane

298.15 298.15 298.15 298.15 297.60 297.60 298.15 298.15 298.15 298.15 298.15 303.15 303.15 303.15 298.2 298.2 303.2 293.2 293.2 293.2

0.9112 0.9291 0.9279 0.9042 0.9524 0.9113 0.9866 0.9618 0.9784 0.9851 0.9699 0.9334 0.9655 0.9531 0.9598 0.9457 0.9358 0.9294 0.8988 0.9217

48.5 34.4 35.3 58.0 43.8 36.0 135.4 70.5 67.4 107.1 39.3 45.3 36.7 44.0 32.4 34.4 42.2 65.5 66.4 57.5

5 5 7 5 5 5 8 10 8 8 11 5 5 9 8 7 6 4 5 4

Total

The mole fractions of components at the interface, according to Eq. (3), cannot be negative or zero, while in the simplest case of adsorption from a binary solution, the Gibbs’ convention defines a zero mole fraction for the solvent. Eq. (3) therefore can only be applied to a surface phase, where all the components have a positive adsorption value. The sum of all adsorptions, or the total adsorption value, can be reasonably assumed constant and equal with the maximum adsorption value of  ∞ , as has been discussed in more details elsewhere [27]. Therefore, i =  ∞ = const.

(9)

i

and,









∂i

=

␣␤

∂nj

i

␣␤ i= / j

n

∂ ∞ ␣␤

∂nj

 =0

(10)

␣␤

␣␤

␣␤

−

A + dA

nj

A

+

i= / j



␣␤

ni

A + dA

␣␤

−

ni

A



= 0.

(11)

Considering Eq. (9), the above equation can be simply reduced to dA ␣␤

dnj

=

A n␣␤

=



RT =− ln a

c



␤ 0.5 (Xi␣ Xi )

(13)

i=1

Thus, at any temperature, the interfacial tension between two liquid phases relates simply to the mole fractions of the bulk phases through a system-dependent constant, a. The next section comprises the corollary tests for verification of the above equation making use of the assumptions that the molar surface area of the interface, a, is constant and does not vary with temperature and composition. If Eq. (13) holds and these assumptions are valid, then plotting the interfacial tension of a system versus the temperaturecomposition function (right hand side term in Eq. (13)) must result in a straight line with zero intercept. 3. Results

(= ˙ni ai ) is the total surface area. Making use of the definition of adsorption in Eq. (10) yields; + dnj



␣␤

␣␤ i= / j

␣␤

␣␤

component k, ak . That is, ai = ak = B. However, not necessarily, A is equal to B, and the partial molar surface area of component i in the first system may be totally different with that in the second one. The above conclusion simplifies Eq. (6) to,

n

where  i (= ni /A) stands for adsorption of component i and A

nj

Ref. [12] [12] [12] [12] [12] [12] [29] [29] [29] [29] [29] [30] [31] [32] [33] [33] [33] [33] [33] [33]

130

2. Theory



No. data points

1 ∞

(12)

That is to say, the partial molar surface area of each component is equal to the molar surface area of the interface (a1 = a2 = . . . = a = 1/ ∞ ). It must be emphasised that the molar surface area of the interface, a, is a parameter related to the position of the dividing surface and has no geometrical meaning. However, the above result deserves more clarification using an example. According to the above discussion, the partial molar surface area of component i, ai , is identical to the partial molar surface area of component j, aj , in binary mixtures of i–j system (ai = aj = A), and similarly in a binary system of i–k the partial molar surface area of compound i would be equal to the partial molar surface area of

Table 1 reveals the results of such corollary test for 20 ternary systems, presenting the R2 (the goodness of linear fit) of the fitted lines to the experimental data and the resulted slopes in terms of a values. The number of data points and the experimental data references are also included in Table 1. The mutual measured compositions of each system can be found in the cited references and their reliabilities are verified by making use of the Othmer–Tobias plots [28]. Some of the results also are graphically demonstrated in Figs. 2–4. Fig. 5 depicts the results for water–n-hexane system at temperatures from 10 to 50 ◦ C. The measured interfacial tensions for this system and for temperatures between 10 and 50 ◦ C can be found in Zeppieri et al. [34]. The mutual solubility values reported by Tsonopoulos and Wilson [35] have been employed for this calculation. As it is evident from these figures and also Table 1, the linear trend which is proposed by Eq. (13) is reasonably the case. It is worth noticing that Eq. (13) is developed based on the regular solution assumption, while many of the tested systems (almost all of them) are far from such assumption. Moreover, Eq. (13) basically has the assumption of considering the interface as mono-molecular layer. This assumption, as will be shown later, poses unreliable results near the critical consolute point. Nevertheless, Eq. (13) can be used

A. Bahramian / Fluid Phase Equilibria 285 (2009) 24–29

Fig. 2. Linear relationship between experimental [29] liquid–liquid interfacial tension values and the bulk phase compositions (Eq. (13)) for two ternary systems at 298 K.

27

Fig. 5. Linear relationship between experimental [35] liquid–liquid interfacial tension values of water-n-hexane mixtures and the mutual solubility [36] at different temperatures.

Fig. 3. Linear relationship between experimental [29,31] liquid–liquid interfacial tension values and the bulk phase compositions (Eq. (13)) for two ternary systems at 298 K (open circles) and 303.2 K (solid circles). Fig. 6. Linear relationship between experimental liquid–liquid interfacial tension values [23,35,37] of water–hydrocarbon mixtures and the temperaturecompositions function proposed by Eq. (13).

Fig. 4. Linear relationship between experimental [29,32] liquid–liquid interfacial tension values and the bulk phase compositions (Eq. (13)) for two ternary systems at 298 K (open circles) and 303.2 K (solid circles).

to develop a general correlation (such as Donahue–Bartell equation) for water–hydrocarbon systems. An interface, in a simple view, can be assumed to be comprised of some lattices or interface sites, where two molecules from both phases meet in each site. At a water–hydrocarbon interface, the interface site is an area where a molecule of water meets a molecule of hydrocarbon. It is reasonable to assume that all these sites have the same area for the hydrocarbons, regardless of their carbon numbers, that orient themselves near the interface and approach the water molecules by one segment of their chains. Therefore plotting the data of different water–hydrocarbon system must result in a single straight line. Fig. 6 shows the variation of the values of a number of experimentally measured [23,34,36] interfacial tensions of a large number of binary systems with the composition-temperature function in Eq. (13). According to this figure, the linear relationship of the interfacial tension with the developed temperature-composition function in Eq. (13) is adequately evident. Moreover, the corresponding

28

A. Bahramian / Fluid Phase Equilibria 285 (2009) 24–29

area per molecule (30.5 Å2 ), found by the least squares method (slope = 0.045), is very much in line with the values obtained from the adsorption data of fatty acids at water surface [37] (about 30 Å2 ) and the reported 32 Å2 by Rideal [38]. Therefore, from this figure a general correlation can be deduced for water–hydrocarbon systems as;



c



␤ 0.5 (Xi␣ Xi )

␤

ln(X1␣ /X1 ) TC = T 2

(23)

indicates the proximity to the critical point and it will be small at near critical conditions. Considering Eq. (21), it is clear that;

(14)

⎧ ⎨ X1␣ = 1 (1 + )

This equation can be used for prediction of water–hydrocarbon interfacial tension at any temperature and pressure provided that the mutual solubility of components are known. It is worth noticing that Eq. (14) for binary water–hydrocarbon systems at 25 ◦ C can be reduced to the empirical equation of Donahue–Bartell, Eq. (1). Since the mutual solubility of water and hydrocarbon is very small, Eq. (14) will be approximated by the following equation in binary systems,

⎩ X ␤ = 1 (1 − ) 1



␣␤

= −0.045T ln

i=1

 OW = −0.045T ln[(XW (O) )0.5 + (XO(W ) )0.5 ]

2

(15)

which is identical to,  OW = −

 0.045 T ln(XW (O) + XO(W ) + 2 XW (O) XO(W ) ) 2

(16)

XW (O) XO(W ) ∼ = XW (O) + XO(W )

2

(17)

Inserting Eq. (17) in Eq. (16) yields, 

OW

0.045 0.045 =− T ln(2) − T ln(XW (O) + XO(W ) ) 2 2

(18)

Writing Eq. (18) at 298 K and changing the ln function to log function, will result in,  OW = −4.65 − 15.44 log(XW (O) + XO(W ) )

(19)

which has exactly the same form as the Donahue–Bartel empirical correlation, Eq. (1). Moreover, the theoretical intercept and slope are close to those of regression results (−3.31 and 15.61, respectively). However, the consistency of Eq. (13) can be questioned for systems very close to their critical points, where the predicted values may show a trend incompatible with the measured data. A statistical analysis of the results obtained by Eq. (13) for the tested ternary systems also shows a mean average deviation of 0.9 mN/m and a standard deviation of about 1.2, implying poor prediction by the model near critical point. Such inconsistency, as mentioned earlier, stems from the underlying assumption of considering the interface as a mono-molecular layer, and can be resolved by considering a multi layer interface. This problem is discussed in next section.

(24)

2

and therefore,



ln

X1␣





= −ln

␤

X1



1− 1+

 = 2 tanh−1 ( ) = 2

+

3

3

 + ··· (25)

Substituting Eq. (25) in Eq. (23) and neglecting the orders higher than three for at near critical conditions, yield, 2

Since the mole fractions in the above equation are very small, their geometric and arithmetic means are almost equal (this is reasonably true for regular solutions). Thus,



␤

= X1␣ − X1 results in,

Defining a parameter

T − T C

=3

(26)

T

Meanwhile, substitution of Eq. (24) in Eq. (13) gives the following relation between the interfacial tension and as,  ␣␤ = −

1 RT ln(1 − 2 a

2

)

(27)

Near the critical point, 2 is very small and therefore the above equation can be simplified to,  ␣␤ =

1 RT 2 a

2

(28)

Substitution of Eq. (26) into the above equation and then making differentiation of the interfacial tension with respect to temperature results in,



d ␣␤ dT



=− T =Tc

3R = / 0 2a

(29)

Therefore, as it is evident from the above equation, the discussed model gives a non-zero gradient for interfacial tension at the critical point. The negative slope in Eq. (29), although shows correctly the decreasing trend of the interfacial tension with increasing temperature, the result of non-zero gradient at the critical point flies in the face of both experimental and theoretical evidences [39]. This incongruity, however, roots in the monolayer assumption and can be removed simply by assuming the interface as series of layers. Assuming the interface as an n-layer model, with the first and the nth layer adjacent to the ␣ and ␤ phase, respectively, the interfacial tension in kth layer is given by,

⎛

␣␤

⎞

(K)

Xi RT ⎠ = ln ⎝  a (K+1) (K−1) Xi Xi

3.1. Mono-layer assumption

(K)

For a binary immiscible regular solution, it can be shown from the equality of chemical potentials that;

The total interfacial tension, therefore, is the sum of all the tensions,

˛12 = RT

␤ ln(X1␣ /X1 ) ␤ X1␣ − X1

(20)

␤

, i = 1, 2

RT ln = 2a

(1)

Xi Xin



(31)

␤

Xi␣ Xi

Near the critical point, the sigmoid form of the composition variation at the interface [40] is reasonably close to a linear trend, (21) Xi␣ − Xi

(1)

where ˛12 is the interaction coefficient in the two-suffix Margules activity coefficient model and relates to the critical temperature as, ˛12 = 2RTC



␣␤ (K)

K=1

and, Xi = 1 − Xi␣

 ␣␤ =

n

(30)

(22)

(n)

= Xi

␤

− Xi =

(32)

n+1 ␤

Making use of the approximate value of 0.25 for Xi␣ Xi near the critical point and remembering that 2 is very small in that region,

A. Bahramian / Fluid Phase Equilibria 285 (2009) 24–29

the following equation can be deduced from Eqs. ((31) and (32)) and Eq. (26).  ␣␤ ∼ =

6n

R (T − T ) 2 a C (n + 1)

(33)

Therefore,



d ␣␤ dT



=− T =Tc

6n

R (n + 1) a 2

(34)

To obtain a zero gradient at the critical point, the above equation implies that the interface must be formed by infinite layers. This result corresponds well with the results of the diffuse interface theory of Cahn and Hilliard [40], where it has been shown that “the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature”. It is interesting to note that an expression somehow similar to Eq. (34) has been found by Rayleigh [41,42] a long time ago through a different approach. The multilayer treatment of the interface, although removes the nonzero gradient and, as discussed elsewhere [39,43], has an important impact on relative adsorptions for liquid–vapour interfaces, has practically an insignificant effect on the magnitude of the calculated surface and interfacial tension, especially away from the critical point. 4. Conclusion Some available models on liquid–liquid interfacial tension are briefly reviewed and their features are discussed. It is shown that the approach of Shain and Prausnitz [16] in using a similar activity coefficient model for both the interface and the bulk phase may yield a paradoxical result while relating the activity coefficient of the interface to that of bulk is a reasonable and practical approach. The Gibbs’ convention of dividing surface is then used to show that all the partial molar surface areas are related to the maximum adsorption value. Employing this finding along with the geometrical mean average rule [17], a new equation has been developed and it is shown that at any temperature the liquid–liquid interfacial tension can be calculated using the molar compositions of the bulk liquid phases and a system dependant constant (=maximum adsorption value). Orientation of hydrocarbon molecules at the interface is discussed and a new equation has been developed for prediction of water–hydrocarbon mixtures. The equation simply reduces to the well-known Donahue–Bartel empirical correlation at 25 ◦ C for immiscible binary water–hydrocarbon systems. Evaluation of the new model against a large number of ternary aqueous–organic systems also showed a remarkable consistency between the theoretical and experimental trends, so that it can be used as a consistency test for measured data. The model, however, gives a questionable trend near the critical point, a non-zero gradient for interfacial tension at the critical point, which is explained to be stemmed from the underlying mono-molecular layer assumption.

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Acknowledgments This paper is dedicated to the memory of the late Professor Ali Danesh. The author has enjoyed the invaluable benefit of working with him, first as his PhD student at the Heriot-Watt University of Scotland and then as a colleague at the University of Tehran, Iran. The author is also extremely grateful to Dr. Mehran Sohrabi from Heriot-Watt University for his considerable help, reviewing the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

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