Pressure and temperature dependence of light fullerenes solubility in n-heptane

Journal of Molecular Liquids 268 (2018) 569–577

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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Pressure and temperature dependence of light fullerenes solubility in n-heptane Konstantin N. Semenov a,⁎, Nikolay A. Charykov b, Enriqueta R. López c, Josefa Fernández c, Juan J. Parajó c, Nikita E. Podolsky a, Igor V. Murin a a b c

Institute of Chemistry, Saint-Petersburg State University, Saint-Petersburg 198504, Universitetskii pr. 26, Russia Saint-Petersburg State Technological Institute (Technical University), Saint-Petersburg 190013, Moskovskii pr. 26, Russia Thermophysical Properties Laboratory, NaFoMat Group, Applied Physics Department, University of Santiago de Compostela, Santiago de Compostela, E-15782, Spain

a r t i c l e

i n f o

Article history: Received 19 February 2018 Received in revised form 22 June 2018 Accepted 23 June 2018 Available online 17 July 2018 Keywords: Light fullerenes C60 C70 Solubility High pressure n-heptane

a b s t r a c t Solubility of individual light fullerenes (C60 and C70) in n-heptane was measured in the ranges of pressure from 0.1 up to 110 MPa and temperature from 298.3 to 358.3 K. At 0.1 MPa, the solubility diagrams of the C60 – nheptane, C70 – n-heptane binary systems consist of only one branch corresponding to crystallization of monosolvated fullerenes – C60·C7H16 and C70·C7H16. The compositions of the solid crystalline solvates were determined by thermogravimetric analysis. Thermodynamic description of the temperature and pressure dependences of solubility was performed using the van-der-Waals differential equation. As a result, the changes of the isothermal compressibility, molar volume, isobaric heat capacity and molar entropy in the process of the light fullerenes dissolution with formation of infinitely diluted solution were calculated. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Light fullerenes have a potential of practical application in various fields of science and technology, in particular in optics [1–5], supercapacitors [4–6], hydrogen storage [4, 7], nanoelectronics [4, 8], photovoltaic solar energy [9, 10], lubricants [11, 12], cosmetic [13] and nanobiomedicine [14–18], but also fullerenes could be used in controlling organic pollution [19]. Solubility of light fullerenes (C60 and C70) is one of the most important properties, which significantly influences the extraction of fullerenes from the fullerene mixture and fullerene black, purification of fullerenes, the subsequent organic functionalization of fullerenes, and development of chromatographic and pre-chromatographic separation methods of industrial fullerene mixtures [5, 20–24]. Analysis of literature reveals the presence of experimental data devoted to investigation of different types of phase equilibria in fullerene-containing systems: (i) solubility diagrams in binary systems (individual light fullerene (C60 or C70) – solvent). Up to this time, isothermal solubility data (at P = 0.1 MPa) for the C60 fullerene were reported in N150 solvents and in the case of C70 fullerene in N50 solvents [22, 23]. Solubility of fullerenes was studied in different classes of solvents (alkanes, halogen-alkanes, aromatic solvents, alcohols, ⁎ Corresponding author. E-mail address: [email protected] (K.N. Semenov).

https://doi.org/10.1016/j.molliq.2018.06.099 0167-7322/© 2018 Elsevier B.V. All rights reserved.

carboxylic acids, inorganic substances) [22–26]; (ii) solubility diagrams in pseudo-binary systems (industrial fullerene mixtures – solvent). Such experimental data are the basis for the development of the industrial fullerene mixtures separation methods based on different solubility of the fullerene components [23, 25]; (ii) solubility diagrams in ternary systems (C60 – C70 – solvent or individual light fullerene (C60 or C70) – solvent 1 – solvent 2) [23, 27–33]. Such experimental data are scarce in number, for example in the case of C60 – C70 – solvent equilibria only few systems are studied: C60 – C70 – o-xylene system over a broad temperature range (293.15–373.15 K), C60 – C70 – styrene at (258.15, 273.15 and 298.15) K and C60 – C70 – 1,2-dichlorobenzene at 423.15 K [23, 29]. The experimental data devoted to the above mentioned types of phase equilibria are presented and summarized in review articles [22, 23]; (iv) solubility diagrams in the multicomponent systems, containing light fullerenes (light fullerenes – natural oils, animal fats, essential oils). The latest data are very important for the creation of the bioactive phases containing fullerenes [22, 23]; (v) extraction equilibria in the fullerene containing systems. These data are very important for elaboration of high-efficient methods of the fullerene mixtures separation [21, 26]; (vi) sorption equilibria in systems containing light fullerenes. In particular, authors of references [34–36] developed a separation method for industrial fullerene mixtures using Norit-Azo carbon and multiwalled carbon nanotubes (MWCNT). Nevertheless, experimental phase equilibria measurements at high pressures are scarce. Thus, the following binary systems were studied in broad

K.N. Semenov et al. / Journal of Molecular Liquids 268 (2018) 569–577

2. Experimental 2.1. Materials Table 1 summarizes the characteristics of the samples. Individual light fullerenes (C60 and C70) were purchased from NeoTechProduct, St. Petersburg. The n-heptane sample, purchased from Sigma Aldrich, is anhydrous (N99% wt.). The samples were used without further purification. 2.2. Solubility measurements techniques at atmospheric pressure Solubility measurements of C60 or C70 fullerene in n-heptane were carried out in the temperature range 298.3–358.3 K by the method of isothermal saturation during 8 h using a LAUDA ET 20 thermostatic shaker at a shaking frequency of 80 Hz. Temperature was measured with an uncertainty of 0.1 K (at the 95% confidence level). The required volume (about 5 cm3) of the heterogeneous system was sampled at atmospheric pressure for carrying out the spectrophotometric analysis. Then, the liquid phase was separated by filtration on a Schott filter (porosity factor 10) under vacuum, and the aliquot of fullerene solution was diluted with n-heptane for carrying out spectrophotometric analysis (the reference solution was n-heptane). The concentration of C60 and C70 fullerenes in the saturated solution was determined using a Specord M40 double-beam spectrophotometer (Karl Zeiss, Germany) at characteristic wavelengths of (335 and 472) nm and at 298.15 K. For this purpose, we dilute with n-heptane a known amount of the liquid phase. The photometric and wavelength accuracies are respectively 0.005 and 0.5 nm. The thickness of the absorption layer was 1 cm. This experimental technique was previously used to study the light fullerenes solubility in 1-hexanol [37] and n-nonane [42]. The expanded relative uncertainty of the solubility values is estimated to be 10% (at the 95% confidence level). The experimental measurements of solubility at atmospheric pressure were carried out two times. Relative air humidity was 40–50%. The solvent content in the solid crystal solutes was determined by dynamic thermogravimetric analysis under air atmosphere on a Shimadzu DTG-60H, from 323 to 700 K, at a heating rate of

Table 1 Provenance and mass fraction purity of the samples studied in this work. Name

Supplier

Mass fraction puritya

Analysis method

n-heptane C60 C70

Aldrich NeoTechProduct NeoTechProduct

N0.99 0.999 0.999

Gas chromatography Liquid chromatography Liquid chromatography

a

The purity analysis was performed by the supplier.

10 K·min−1. The expanded estimated uncertainty (at the 95% confidence level) of the solid solvate concentration in the mixture is 5%. 2.3. Solubility measurement technique at high pressures The solubility of light fullerenes in n-heptane at high pressures has been experimentally determined in a cylindrical stainless steel variable-volume view cell. It consists of a horizontal cylinder of 2 cm internal diameter with a movable piston at one end, which permits to change the pressure [48]. Description of experimental procedure and device has been presented previously [37, 42]. A pressure transducer (Kulite, model HEM375) with a typical uncertainty equal to 0.03 MPa was used to measure the pressure. A Pt100 probe with an expanded uncertainty of 0.02 K was employed to measure the temperature. For carrying out the high pressure experiments we have used the following procedure: the cell was charged initially with a known amount of solution of individual fullerene (C60 or C70) in n-heptane. The mass of the solution was measured with a Sartorius MC210P balance and the fullerene concentration was determined by the spectrophotometric method. Then a weighted portion of fullerene was added to the solution in the cell and the heterogeneous system of known composition was compressed, under isothermal conditions, to achieve a single phase under stirring. The pressure at which the disappearance of a solid phase occurs was determined visually by slowly increasing pressure. This procedure was repeated several times being the lowest value taken as the experimental equilibrium pressure. The estimated expanded uncertainty (at the 95% confidence level) of this pressure is 0.1 MPa. Subsequently a new temperature was selected. When solubility at all the desired temperatures was determined, a small weighted quantity of fullerene was added. The expanded overall uncertainty of the fullerene weight fraction of the saturated solution was estimated to be 10% (at the 95% confidence level). 3. Results and discussion 3.1. Experimental data on solubility of C60 or C70 in n-heptane Fig. 1 and Table 2 present the experimental values of the solubility of light fullerenes (C60, C70) in n-heptane at 0.1 MPa. In the whole experimental temperature range, the solid phases in equilibrium with the saturated solution are monosolvated fullerenes: C60·C7H16 and C70·C7H16. This type of temperature dependences of the solubility (with formation of crystal solvates) often takes place in the fullerene-containing binary systems: for example in binary systems containing n-alkanoic carboxylic acids (C60 – Сn-1H2n-1СOOH (n = 5–9), C70 – Сn-1H2n-1СOOH (n = 6, 7, 8)), n-alcohols (C60 – СnH2n+1OH (n = 5–11)), aromatic

0.020

0.016

60,70

pressure ranges: (C60 or C70) with 1-hexanol [37], C60 – toluene [22, 38, 39], C60 – n-hexane [22, 26, 40], C60–water [22, 41], (C60 or C70) with nnonane [22, 42]. We can also mention a series of studies devoted to investigation of phase equilibria and non-equilibria phase transitions in one-component systems containing individual light fullerenes under high-pressures at broad temperature ranges [43, 44]. In this work we have studied the light fullerenes (C60 and C70) solubility in n-heptane in the temperature range from 298.3 to 358.3 K and pressures up to 110 MPa. Additionally, we applied the Van-der-Waals equation [42, 45–47] to describe the phase equilibria in binary twophase systems for calculation of the change of isothermal compressibility, molar volume, isobaric heat capacity and molar entropy in the process of the light fullerenes dissolution with formation of infinitely diluted solution. The choice of n-heptane as a solvent is mainly due to the visual method used to measure the solubility at different temperatures and pressures, which can be applied only for sufficiently transparent solutions. In addition, one of the assumption of the thermodynamic description is only valid for the solvents, as n-heptane, characterized by low solubility values of fullerenes in this solvent.

wC / %

570

0.012

0.008

0.004 300

310

320

330

T/ K

340

350

360

Fig. 1. Temperature dependence of the individual light fullerene solubility (C60 (●) and C70 (○)) in n-heptane at 0.1 MPa. w is the mass fraction of fullerene in saturated solution.

K.N. Semenov et al. / Journal of Molecular Liquids 268 (2018) 569–577 Table 2 Solubility of light fullerenes in n-heptane at 0.1 MPa. w is the mass fraction of fullerene in the saturated solution in weight percentage, T – temperature. T/K

w/%

Solid phase

C60 – n-heptane 298.3 303.3 308.3 313.3 318.3 323.3 328.3 333.3 338.3 342.3 348.3 353.3 358.3

0.0040 0.0041 0.0043 0.0044 0.0046 0.0049 0.0050 0.0053 0.0055 0.0058 0.0061 0.0065 0.0068

C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16 C60·n-C7H16

C70 – n-heptane 298.3 303.3 308.3 313.3 318.3 323.3 328.3 333.3 338.3 343.3 348.3 353.3 358.3

0.0057 0.0060 0.0062 0.0063 0.0065 0.0075 0.0102 0.0131 0.0141 0.0154 0.0172 0.0181 0.0201

C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16 C70·n-C7H16

Expanded uncertainties (at the 95% confidence level) are U(T) = 0.1 K, Ur(P) = 0.5% and Ur(w) = 10%.

solvents (C60, C70 – 1,2-dimethylbenzene, C60, C70 – styrene, C60, C70 – bromobenzene, C60, C70 – 1,3,5-trimethylbenzene), etc. [22–24, 44]. The results of thermogravimetric analysis expressed as relative mass loss as a function of temperature, i.e., the TG curve, and its derivative, the DTG curve, are presented in Fig. 2 for C60·C7H16 and C70·C7H16 crystal-solvates. Both DTG curves consist of one peak corresponding to dissociation of the light fullerenes crystal-solvates according to the schemes: C 60
C 7 H 16 ðsolidÞ→C 60 ðsolidÞ þ C 7 H 16 ðvapor Þ;

ð1Þ

C 70
C 7 H 16 ðsolidÞ→C 70 ðsolidÞ þ C 7 H 16 ðvapor Þ:

571

ð2Þ

Thus, according to the above schemes the desolvation of C60·C7H16 (solid) and C70·C7H16 (solid) and the heptane vaporization correspond to the relative mass losses of 12.21% and 10.65%, respectively. These values agree with the total mass losses of 12% and 10% respectively (Fig. 2). Experimental P-T-w data in the individual light fullerenes (C60 and C70) – n-heptane binary systems in the range of pressures 0.1–110 MPa and in the range of temperatures (298.3–358.3) K are presented in Table 3 and in Fig. 3. This temperature interval was chosen due to vaporization of n-heptane at higher temperatures. Nevertheless, the range of temperatures and pressures is sufficient to reveal the regularities of dissolution of fullerenes in this solvent. In the whole range of temperatures and pressures in both binary systems the only one tentative solid phase in the equilibrium with saturated liquid solutions is the corresponding crystal monosolvated fullerene: C60·C7H16 or C70·C7H16. 3.2. Thermodynamic description of the solid-liquid equilibrium in the C60 or C70 fullerene – n-heptane binary systems Similar to a previous paper [42], in this work we will consider the two-phase equilibrium of saturated fullerene solution (phase l) – fullerene solvates (C60·C7H16 or C70·C7H16) (phase s). Subscript 1 will be used to n-heptane, and subscript 2 to individual light fullerenes (C60 or C70). For the two-phase solid (s) – liquid (l) equilibrium in binary systems, van-der-Waals equation (Eq. (3)) can be written as [42, 45–47]:
 ðl Þ ðlÞ x2 ðsÞ −x2 ðlÞ G22 dx2 ¼ Sðs→lÞ dT−V ðs→lÞ dP

ð3Þ

(l) where x(s) 2 and x2 are the mole fraction of the individual fullerenes in the fullerene solvates and in the saturated solution respectively; S(s→l) and V(s→l) are the changes of entropy and volume functions in the process of formation of 1 mol of phase l from an infinitely big mass of the phase s at constant T and P given by Eqs. (4), (5) [42, 45–47]:

" #
 ∂S ðlÞ ð sÞ ðl Þ ; Sðs→lÞ ¼ SðlÞ −SðsÞ þ x2 −x2 ∂x2

ð4Þ

" #
 ∂V ðlÞ ðsÞ ðlÞ V ðs→lÞ ¼ V ðlÞ −V ðsÞ þ x2 −x2 ; ∂x2

ð5Þ

Fig. 2. TG and DTG curves for the C60·C7H16 (grey lines) and C70·C7H16 (black lines) crystal solvates. Δm/m – mass loss (solid lines), derivative of mass with respect to temperature (dashed line).

572

K.N. Semenov et al. / Journal of Molecular Liquids 268 (2018) 569–577

Table 3 Experimental (P-T-w) data for the C60 – n-heptane and C70 – n-heptane binary systems. w is a mass percentage of fullerene in the saturated solution, T – temperature, P – pressure. The tentative solid phases equilibrium with saturated liquid solutions are monosolvated C70 (C70·C7H16) in the case of C70 – n-heptane binary system and monosolvated C60 (C60·C7H16).

C60 – n-heptane 298.3 K 33.4 0.0048 68.3 0.0055 96.9 0.0062 313.3 K 5.01 0.0048 44.4 0.0055 79.9 0.0061

0.0074 0.0085 0.0093 0.0109 0.0127 0.0145 0.0161

348.3 K 2.07 4.57 7.5 18.1 35.4 60.9 95.4

0.0048 0.0055 0.0062 0.0055 0.0062 0.0068 0.0074 0.0085 0.0064 0.0068 0.0074 0.0083 0.0091 0.0109 0.0127

0.0074 0.0085 0.0091 0.0109 0.0127 0.0145 0.0161

P/MPa 308.3 K 23.4 55.9 90.2 323.3 K 7.9 16.3 28.4 44.4 82.2 338.3 K 2.19 4.23 7.2 14.8 22.2 44.5 68.2 98.6 353.3 K 3.43 4.66 9.9 25.3 46.1 74.7 98.3

w/%

0.0048 0.0055 0.0062 0.0055 0.0062 0.0068 0.0074 0.0085 0.0062 0.0068 0.0074 0.0085 0.0091 0.0109 0.0127 0.0145

C70 – n-heptane 298.3 K 30.0 0.0065 57.5 0.0074 107.5 0.0087

w/%

P/MPa

w/%

and G(l) 22 is the second derivative of the molar Gibbs energy of the phase l respect to the fullerene mole fraction x2 (Eq. (6)): ðlÞ

G22 ¼

!ðlÞ 2 ∂ G : ∂x22

ð6Þ

T;P

Taking into account that fullerene solutions in n-alkanes are very di−5 –10−6, thus x(l) luted (x(l) 2 takes values in the range 10 1 ≈ 1), it is can be assumed: γ1 ðlÞ ≈ const ≈ 1; γ 2 ðlÞ ≈ const≠1;

ð7Þ

is the activity coefficient of i-th component in the liquid where γ(l) i phase. Eq. (7) reproduces the assumption of pseudo-ideality of fullerene

0.0085 0.0091 0.0109 0.0127 0.0145 0.0161 0.0170

0.0085 0.0091 0.0109 0.0127 0.0145 0.0161 0.0175

P/MPa

Expanded uncertainties (at the 95% confidence level) are U(T) = 0.02 K, U(P) = 0.1 MPa and Ur(w) = 10%.

0.018

(a)

0.016 0.014

60

343.3 K 3.82 8.41 12.9 22.3 45.9 70.7 106.5 358.3 K 1.01 2.31 7.11 17.9 35.1 62.1 95.3

0.0062 0.0065 0.0074 0.0085 0.0091 0.0109

303.3 K 28.1 61.1 93.9 318.3 K 12.3 29.2 45.4 60.3 96.0 333.3 K 4.32 6.7 12.5 26.9 40.2 68.3 99.6

w/%

0.0264 0.0279

0.012 0.010 0.008 0.006

313.3 K 2.12 8.7 23.8 60.1 96.5

0.0065 0.0074 0.0087 0.0104 0.0115

303.3 K 6.2 21.5 46.4 98.6 318.3 K 4.13 12.1 34.6 45.5 89.1

328.3 K 11.2 18.5 26.8 54.6 99.2

0.0129 0.0137 0.0146 0.0164 0.0187

333.3 K 10.9 26.7 49.9 64.6 97.9

0.0164 0.0195 0.0210 0.0217 0.0231

343.3 K 11.1 20.5 26.6 50.5 71.3 97.5

0.0195 0.0210 0.0217 0.0231 0.0239 0.0248

348.3 K 4.58 11.9 17.2 34.4 47.5 65.9 98.2

0.0195 0.0210 0.0217 0.0231 0.0239 0.0248 0.0264

358.3 K 1.43 3.47 10.6 17.7 32.5

0.0210 0.0217 0.0231 0.0239 0.0248

0.0065 0.0074 0.0087 0.0104 0.0074 0.0087 0.0104 0.0111 0.0129

308.3 K 4.3 15.1 35.4 87.5 323.3 K 4.36 14.5 22.2 45.7 72.9 94.8 338.3 K 7.07 18.5 30.5 43.4 76.7 99.0 353.3 K 1.23 5.77 10.0 20.5 30.7 48.1 81.2

0.0065 0.0074 0.0087 0.0104

0.004 0

0.0087 0.0104 0.0111 0.0129 0.0137 0.0146 0.0165 0.0195 0.0210 0.0217 0.0231 0.02387 0.0195 0.0210 0.0217 0.0231 0.0239 0.0248 0.0264

20

40

60

P / MPa

80

100

(b)

0.028 0.024 0.020

70

328.3 K 6.3 10.6 24.6 45.0 56.3 98.1

P/MPa

w/%

63.4 94.3

wC / %

w/%

P/MPa

wC / %

P/MPa

Table 3 (continued)

0.016 0.012 0.008 0.004 0

20

40

60

P / MPa

80

100

Fig. 3. P-T-w experimental values for binary systems C60 – n-heptane (a) and C70 – nheptane (b) in the pressure range 0.1–100 MPa. (○) 298 K, (●) 303 K, (Δ) 308 K, (▲) 313 K, (∇) 318 K, (▼) 323 K, (□) 328 K, (■) 333 K, (◊) 338 K, (►) 343, (⊲) 348 K, (◄) 353 K, (⊳) 358 K.

K.N. Semenov et al. / Journal of Molecular Liquids 268 (2018) 569–577

solutions. This means that the activity coefficient of the solute does not depend on the mole fraction of fullerene and temperature over the whole range of concentrations and temperatures. This approach is traditional for description of the fullerenes solutions [5, 23]. Additionally, due to the extremely low solubility of fullerenes in n-heptane we can equate the chemical potential of the solvent in the fullerene solution to the chemical potential of the pure solvent. We have previously [42] derived qualitative analogues of some wellknown thermodynamic laws (Clausius–Clapeyron relation, GibbsKonovalov laws) and semi-quantitative equations for calculation of sol the molar volume (ΔVsol 2 ) and entropy of dissolution (ΔS2 ) changing in the process of dissolution of fullerenes with formation of very diluted solution (in the extreme case, an infinitely diluted solution). These properties can be determined according to Eqs. (8), (9) [42]:
 ðlÞ ∂ ln x2 =∂P RT ≈ −ΔV sol 2 ;

ð8Þ


 ðl Þ ∂RTlnx2 =∂T ≈ ΔSsol 2 :

ð9Þ

T

P

3.2.1. Thermodynamic calculations for the individual light fullerenes – nheptane binary systems under isothermal conditions For the determination of the molar volume change (ΔVsol 2 ) in the process of the light fullerenes (C60 or C70) dissolution with formation of infinitely diluted solution, we have assumed as in reference [42] that for two systems containing C60 or C70 the values of (ln x(l) 2 )TRT at a constant temperature T follow a quadratic function of pressure in the temperature range 298.3 K ≤ T ≤ 358.3 K:


ðlÞ

ln x2

 T

RT ¼ AðT Þ þ BðT ÞP þ C ðT ÞP 2 ;

ð10Þ

where x(l) 2 is the mole fraction of fullerenes in the saturated solution. We have correlated the isothermal values of (ln x(l) 2 )TRT with pressures for the different temperatures to Eq. (10) for determining the temperature dependences of the coefficients A(T), B(T) and C(T). As a result, we obtained Eqs. (11)–(13) for the C60 – n-heptane binary system and Eqs. (14)–(16) for the C70 – n-heptane system: AðT Þ ¼ 1816−206
T þ 0:277
T 2 ;

ð11Þ

BðT Þ ¼ 8:55−0:8352
T þ 0:0028
T 2 ;

ð12Þ

C ðT Þ ¼ −3:9623 þ 0:03
T−5:82
10−5
T 2 ;

ð13Þ

AðT Þ ¼ 12666−278:4
T þ 0:32825
T 2 ;

ð14Þ

BðT Þ ¼ −2485 þ 15:4
T−0:0235
T 2 ;

ð15Þ

C ðT Þ ¼ 18:73−0:11501
T þ 1:74
10−4
T 2 :

ð16Þ

Table 4 and Fig. 4 present the results of calculation of molar isothersol mal compressibility (Δβ,sol T, 2) and differential molar volume effect (ΔV2 ) in the process of light fullerenes dissolution with formation of infinitely diluted solution at constant pressure (P = 0.1 MPa) or at constant temperature (Т = 298 K). It is necessary to point out that thermodynamic functions referred to components (or phases of constant composition) are positively defined (μi N 0, V N 0, βT N 0, S N 0, Cp N 0). At the same time the sign of partial molar functions of the solution component as well as the sign of the difference of the thermodynamic functions in the process of dissolution can be arbitrary, moreover they can change the sign, pass through an extremum or have a discontinuity of the first kind. For example the ΔVsol 2 function for the C70 – n-heptane system changes the sign in the range of pressures (0.1–100) MPa (Fig. 4a), thus we have a point ,sol where ΔVsol 2 = 0 and in which the ΔβT, 2 function has a discontinuity of the first kind according to Eq. (18) (Δβ,sol T, 2 = ± ∞) (see Fig. 4b). 3.2.2. Thermodynamic calculations in the individual light fullerenes – nheptane binary systems under isobaric conditions Temperature dependences of solubility for the C60 – n-heptane and C70 – n-heptane binary systems at various pressures are presented in Fig. 5. Isobaric data were obtained by interpolation according to the obtained P-T-w experimental data. Temperature dependences of solubility expressed as mole fraction of fullerenes in saturated solution (x(l) 2 ) in the temperature range (313.3 ≤ T ≤ 353.3 K) were fitted using the Eq. (19):


ðlÞ

ln x2

 P

RT ¼ F ðP Þ þ DðP ÞT þ EðP ÞT 2 :

ΔV sol 2 ≈ −BðT Þ−2C ðT Þ
P:

ð17Þ

On the basis of isothermal data we can easily calculate the change of the isothermal compressibility in the process of fullerene dissolution (Δβsol T ) with formation of infinitely diluted solution according to Eq. (18): ! ! ! 1 ∂ΔV sol ∂ ln ΔV sol sol 2 2 ¼− Δβ T;2 ¼ − ∂P ∂P ΔV sol 2 T T −2C ðT Þ ¼ : ð18Þ ½BðT Þ þ 2C ðT ÞP 

ð19Þ

The temperature dependences of the quadratic function coefficients can be expressed by Eqs. (20)–(22) for the C60 – n-heptane binary

Table 4 sol Changes of molar isothermal compressibility (Δβsol T, 2) and of molar volume effect (ΔV2 ) in the process of individual light fullerenes (C60 and C70) dissolution with formation of infinitely diluted solution at constant pressure (P = 0.1 MPa) or at constant temperature (Т = 298 K). T/K

3 −1 ΔVsol 2 /cm ∙mol

−1 Δβsol T, 2/MPa

P/MPa

3 −1 ΔVsol 2 /cm ∙mol

−1 Δβsol T, 2/MPa

Т = 298 K

P = 0.1 MPa

According to Eq. (8), the ΔVsol 2 (P,T) can be approximated as Eq. (17):

573

C60 – n-heptane 298 −8.273 303 −12.51 308 −16.88 313 −21.39 318 −26.04 323 −30.83 328 −35.76 333 −40.83 338 −46.04 343 −51.39 348 −56.88 353 −62.50 358 −68.27

0.0461 0.0345 0.0288 0.0256 0.0236 0.0223 0.0215 0.0209 0.0205 0.0202 0.0201 0.0200 0.0200

0.1 10 20 30 40 50 60 70 80 90 100 110 120

−8.273 −4.498 −0.684 3.130 6.944 10.758 14.572 18.385 22.199 26.013 29.827 33.641 37.455

0.0461 0.0848 0.5577 −0.1218 −0.0549 −0.0355 −0.0262 −0.0207 −0.0172 −0.0147 −0.0128 −0.0113 −0.0102

C70 – n-heptane 298 −17.29 303 −23.66 308 −28.86 313 −32.88 318 −35.74 323 −37.42 328 −37.92 333 −37.25 338 −35.41 343 −32.40 348 −28.21 353 −22.85 358 −16.32

0.0105 0.0121 0.0129 0.0135 0.0139 0.0142 0.0144 0.0147 0.0150 0.0153 0.0157 0.0163 0.0175

0.1 10 20 30 40 50 60 70 80 90 100 110 120

−17.29 −15.48 −13.66 −11.84 −10.02 −8.198 −6.376 −4.554 −2.733 −0.911 0.911 2.732 4.554

0.0105 0.0118 0.0133 0.0154 0.0182 0.0222 0.0286 0.0400 0.0667 0.2000 −0.2000 −0.0667 −0.0400

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K.N. Semenov et al. / Journal of Molecular Liquids 268 (2018) 569–577

10

0.018

(a)

wC / %

V2 sol/cm3 mol-1

0

-20

0.004

0

50

100

300

P / MPa

T/ K

300

350

0.028

(b)

0.15

0.024

0.10

0.020

wC / %

/ MPa-1

0.010

0.006

-40

70

0.05

sol

0.012

0.008

-30

T,2

0.014

60

-10

0.20

(a)

0.016

0.00

-0.05

310

320

310

320

330

340

350

330

340

350

T/ K

360

(b)

0.016 0.012 0.008

-0.10 -0.15

0.004 300

-0.20 0

50

P / MPa

100

300

T/ K

350

Fig. 4. Temperature and pressure dependences of the molar volume change (ΔVsol 2 ) (a) and of the isothermal compressibility (Δβsol T, 2) (b) in the process of C70 dissolution with formation of infinitely diluted solution at constant pressure (P = 0.1 MPa) or temperature (Т = 298 K).

Fig. 5. P-T-w data for binary systems C60 – n-heptane (a) and C70 – n-heptane (b). (○) 0.1 MPa, (●) 20 MPa, (Δ) 40 MPa, (▲) 60 MPa, (∇) 80 MPa, (▼) 100 MPa.

Additionally, it is possible to calculate the change of the isobaric heat capacity in the process of fullerenes dissolution (ΔCsol P, 2) with formation of infinitely diluted solution using Eq. (27):

system and by Eqs. (23)–(25) for the C70 – n-heptane system: 2

F ðP Þ ¼ 10255−119:5
P−2:37
P ;

ð20Þ

DðP Þ ¼ −151:77−0:0767
P þ 0:02108
P 2 ;

ð21Þ

EðP Þ ¼ 0:184 þ 0:00169
P−4:38
10−5
P 2 ;

ð22Þ

F ðP Þ ¼ 8583−2183
P þ 15:85
P 2 ;

ð23Þ

DðP Þ ¼ −249:6 þ 13:45
P−0:0969
P 2 ;

ð24Þ

EðP Þ ¼ 0:27861 þ 0:0204
P þ 1:46
10−4
P 2 :

ð25Þ

According to Eq. (9) ΔSsol 2 (P, T) can be approximated by Eq. (26) (see Table 5): 1 0
ðl Þ RTlnx2 @∂ A ≈ ΔSsol ≈ DðP Þ þ 2EðP Þ∙T: 2 ∂T P

360

T/ K

ð26Þ

ΔC sol P;2 ¼

∂ΔSsol 2 ∂T

! T ¼ 2EðP ÞT:

ð27Þ

P

Table 5 presents the results of calculation of molar entropy change sol (ΔSsol 2 ) and molar isobaric heat capacity change (ΔCP, 2) in the process of fullerenes dissolution with formation of infinitely diluted solutions at constant pressure (P = 0.1 MPa) or at constant temperature (Т = 298 К). As an example, Fig. 6 presents temperature and pressure dependences of the change of molar entropy (ΔSsol 2 ) (Fig. 6a) and of molar isobaric heat capacity (ΔCsol P, 2) (Fig. 6b) for the C70 – n-heptane system. The estimated accuracy of the thermodynamic function calculation is the following: (i) for the first derivatives of Gibbs energy change in the process of fullerene dissolution (ΔGsol) with formation of infinitely diluted sol solutions (ΔVsol 2 , ΔS2 ) the accuracy is equal to 5%; (ii) and for the second sol derivatives of ΔGsol (Δβsol P, 2, ΔCP, 2) the accuracy is equal to 10%. 3.2.3. Verification of self-consistency of approximating forms for isothermal and isobaric solubility data It is possible to check the feasibility of the approach of both ΔSsol 2 (T, P) and ΔVsol 2 (T, P) functions by the parity of second, third and fourth

K.N. Semenov et al. / Journal of Molecular Liquids 268 (2018) 569–577

-35

Table 5 sol Changes of molar entropy (ΔSsol 2 ) and of molar isobaric heat capacity (ΔCP, 2) in the process of individual light fullerenes (C60 and C70) dissolution with formation of infinitely diluted solution at constant pressure (P = 0.1 MPa) or at constant temperature (Т = 298 K). −1 ΔSsol 2 /J∙K

−1 ΔCsol P, 2/J∙K

P/MPa

−1 ΔSsol 2 /J∙K

-45

−1 ΔCsol P, 2/J∙K

-50

Т = 298 K

-55

C60 – n-heptane 298 −42.01 303 −40.17 308 −38.33 313 −36.49 318 −34.65 323 −32.80 328 −30.96 333 −29.12 338 −27.28 343 −25.44 348 −23.60 353 −21.75 358 −19.91

109.8 111.6 113.4 115.3 117.1 119.0 120.8 122.7 124.5 126.3 128.2 130.0 131.9

0.1 10 20 30 40 50 60 70 80 90 100 110 120

−42.01 −33.30 −25.51 −18.71 −12.92 −8.141 −4.363 −1.590 0.178 0.942 0.700 −0.547 −2.798

109.8 117.1 119.4 116.4 108.2 94.76 76.12 52.26 23.17 −11.13 −50.66 −95.41 −145.4

C70 – n-heptane 298 −83.42 303 −80.65 308 −77.89 313 −75.12 318 −72.36 323 −69.59 328 −66.83 333 −64.06 338 −61.29 343 −58.53 348 −55.76 353 −53.00 358 −50.23

164.8 167.6 170.4 173.1 175.9 178.7 181.4 184.2 187.0 189.7 192.5 195.3 198.0

0.1 10 20 30 40 50 60 70 80 90 100 110 120

−83.42 −71.62 −61.67 −53.70 −47.70 −43.68 −41.63 −41.57 −43.48 −47.36 −53.23 −61.07 −70.89

164.8 53.17 −42.31 −120.4 −181.1 −224.3 −250.2 −258.7 −249.7 −223.4 −179.6 −118.5 −39.93

S2 sol/ J K-1

P = 0.1 MPa

(a)

-40

-60 -65 -70 -75 -80 -85 -90

0

50

P / MPa

100

300

T/ K

350

250

(b)

200 150 100

Cp,2sol / J K-1

T/K

575

50 0 -50 -100 -150

mixed derivatives (see Eqs. (28)–(30)).

-200 2

2

Δμ sol 2

∂ ∂ ¼ ; ∂P∂T ∂T∂P 3

Δμ sol 2

Δμ sol 2

3

Δμ sol 2

3

-300

∂ ∂ ¼ ; ∂T∂P∂P ∂P∂T∂P 3

Δμ sol 2

Δμ sol 2

ð29:2Þ

ð30Þ

100

300

T/ K

350

Fig. 6. Temperature and pressure dependences of the change of molar entropy (ΔSsol 2 ) (a) and of molar isobaric heat capacity (ΔCsol P, 2) (b) in the process of C70 dissolution with formation of infinitely diluted solution at constant pressure (P = 0.1 MPa) or at constant temperature (Т = 298 К).

∂P 2 ðlÞ

ð sÞ

Δμ sol 2 ¼ μ 2 −μ 2 ;

ð31Þ

is the change of chemical potential in the process of dissolution of individual fullerenes (C60 and C70) in n-heptane with formation of infinitely diluted liquid solutions. On account of our approximation (see Eqs. (10), (19)) we can easily determine the mixed derivatives according to Eqs. (32)–(34):         ∂D ∂E ∂B ∂C þ2 T¼ þ2 P; ∂P T ∂P T ∂T P ∂T P 2

∂ D

!

2

þ2

2 T

2

∂ B

2

∂ E

where:

∂T

50

P / MPa

4

∂ Δμ sol ∂ Δμ sol 2 2 ¼ ; ∂T∂P∂P∂T ∂P∂T∂P∂T

∂P

0

ð29:1Þ

∂ ∂ ¼ ; ∂T∂P∂T ∂P∂T∂T 4

-250

ð28Þ

! P

∂P

∂ C ∂T

ð33:1Þ

ð33:2Þ

T

  ∂E P¼2 ; ∂P T

2

T

∂ C ∂T 2

! :

ð34Þ

P

Table 6 summarizes the boundary data of second, third and fourth mixed derivatives (see Eqs. (32)–(34)). One can see a satisfactory agreement between these functions taking into account that successive derivatives lead to high uncertainties. At the same time, it is more correct to compare the fourth mixed derivatives because in the latest case we deal with invariants. 4. Conclusions

T

!

2

¼

ð32Þ

  ∂C T¼2 ; ∂T P

2

2

þ2

2

∂ E

!

!

Solubility of individual light fullerenes (C60 and C70) in n-heptane was investigated in the pressure range of 0.1–100 MPa and in the temperature range from 298.3 K to 358.3 K. At atmospheric pressure, the solubility diagrams consist of only one branch corresponding to crystallization of mono-solvated fullerenes – C60·C7H16, C70·C7H16. The composition of solid crystalline-solvate was determined by thermogravimetric method. Along isotherms, solubility increases monotonously with increasing pressure. Using van-der-Waals differential

576

K.N. Semenov et al. / Journal of Molecular Liquids 268 (2018) 569–577

Table 6 i

Boundary data of second, third and fourth mixed derivatives (

∂ Δμ sol 2

∂T j ∂P k

T/K

P/MPa

2 −1 sol ∙MPa−1 ∂2Δμsol 2 /∂T∂P = ∂ Δμ2 /∂P∂T/J∙K

) for the С60 – n-С7Н16 and С70 – n-С7Н16 binary systems. 2 −1 ∂3Δμsol ∙MPa−2 2 /∂T∂P /J∙K 2

2

2(∂C/∂T)P

(∂ B/∂T )P+2(∂ C/∂T )TP

2(∂E/∂P)T

(∂2C/∂T2)T

(∂2E/∂P2)T

C60 – n-heptane 298 0.1 0.93 298 100 −0.07 358 0.1 1.13 358 100 −0.92

0.83 −0.10 1.17 −1.16

−0.01 −0.01 −0.02 −0.02

−0.01 −0.01 −0.02 −0.02

0.006 −0.018 0.006 −0.018

0.003 −0.014 0.003 −0.014

−2∙10−4 −2∙10−4 −2∙10−4 −2∙10−4

−2∙10−4 −2∙10−4 −2∙10−4 −2∙10−4

C70 – n-heptane 298 0.1 1.29 298 100 −0.69 358 0.1 −1.15 358 100 0.37

1.39 −0.87 −1.42 0.49

−0.02 −0.02 0.02 0.02

−0.02 −0.02 0.02 0.02

−0.047 0.023 −0.047 0.023

−0.041 0.018 −0.041 0.018

7∙10−4 7∙10−4 7∙10−4 7∙10−4

6∙10−4 6∙10−4 6∙10−4 6∙10−4

Acknowledgements The work was supported by the interuniversity exchange program between Saint-Petersburg State University (Russia) and University of Santiago de Compostela (Spain), Russian Science Foundation (grant № 17-73-20060). Part of this research was performed by using the equipment of the Thermogravimetric and Calorimetric Research Centre and Chemical Analysis and Materials Research Centre of Research park of St. Petersburg State University. JF and ERL acknowledge the support of both the Spanish Ministry of Economy, Industry and Competitiveness and the European Regional Development Fund through ENE201455489-C2-1-R and ENE2017-86425-C2-2-R projects and that of Xunta de Galicia (AGRUP2015/11 and GRC ED431C 2016/001). References [1] G.F. Malgas, D.E. Motaung, C.J. Arendse, Temperature-dependence on the optical properties and the phase separation of polymer-fullerene thin films, J. Mater. Sci. 47 (2012) 4282–4289. [2] G. Brusatin, R. Signorini, Linear and nonlinear optical properties of fullerenes in solid state materials, J. Mater. Chem. 12 (2002) 1964–1977. [3] D.M. Guldi, N. Martin, Fullerenes: From Synthesis to Optoelectronic Properties, Springer Netherlands, 2002. [4] L.N. Sidorov, M.A. Yurovskaya, Fullerenes, Ekzamen, Moscow, 2005. [5] N.O. McHedlov-Petrossyan, Fullerenes in liquid media: an unsettling intrusion into the solution chemistry, Chem. Rev. 113 (2013) 5149–5193. [6] P. Bairi, R.G. Shrestha, J.P. Hill, T. Nishimura, K. Ariga, L.K. Shrestha, Mesoporous graphitic carbon microtubes derived from fullerene C70 tubes as a high performance electrode material for advanced supercapacitors, J. Mater. Chem. A 4 (2016) 13899–13906. [7] H. Ren, C. Cui, X. Li, Y. Liu, A DFT study of the hydrogen storage potentials and properties of Na-and Li-doped fullerenes, Int. J. Hydrog. Energy 42 (2017) 312–321. [8] R. Majidi, M. Ghorbani, Structural and electronic properties of C and BN nanotubes based on periodic fullerenes: a density functional theory study, Fullerenes, Nanotubes, Carbon Nanostruct. 25 (2017) 265–268. [9] A. Pivrikas, N.S. Sariciftci, G. Juška, R. Österbacka, A review of charge transport and recombination in polymer/fullerene organic solar cells, Prog. Photovolt. Res. Appl. 15 (2007) 677–696. [10] N. Kaur, M. Singh, D. Pathak, T. Wagner, J.M. Nunzi, Organic materials for photovoltaic applications: review and mechanism, Synth. Met. 190 (2014) 20–26. [11] E. Ettefaghi, A. Rashidi, H. Ahmadi, S.S. Mohtasebi, M. Pourkhalil, Thermal and rheological properties of oil-based nanofluids from different carbon nanostructures, Int. Commun. Heat Mass Transfer 48 (2013) 178–182. [12] M. Xing, R. Wang, J. Yu, Application of fullerene C60 nano-oil for performance enhancement of domestic refrigerator compressors, Int. J. Refrig. 40 (2014) 398–403. [13] S.Z. Mousavi, S. Nafisia, H.I. Maibach, Fullerene nanoparticle in dermatological and cosmetic applications, Nanomedicine 13 (2017) 1071–1087. [14] F. Moussa, [60]Fullerene and derivatives for biomedical applications, Nanobiomaterials (2018) 113–136.

2

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2 2 −2 ∂4Δμsol ∙MPa−2 2 /∂T ∂P /J∙K

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2

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(∂D/∂P)T+2(∂E/∂P)TT

2

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