Cryometry and excess thermodynamic functions in water soluble of the fullerenol C60(OH)24 Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

Cryometry and excess thermodynamic functions in water soluble of the fullerenol C60(OH)24

N. A. Charykov1'5, K. N. Semenov1'2'6, V. A. Keskinov1, N. A. Kulenova3, Z. K. Shaimardanov3, B. K. Shaimardanova3, L. V. Gerasimova1, Ayat Kanbar1, D. G. Letenko4

1St. Petersburg State Technological Institute (Technical University), Moskovsky prospect, 26 Saint-Petersburg, 190013, Russia

2 St. Petersburg State University,7/9 Universitetskaya emb., Saint-Petersburg, 199034, Russia

3D. Serikbayev East Kazakhstan state technical university, A. K. Protozanov Street, 69, Ust-Kamenogorsk city, 070004,The Republic of Kazakhstan

4 St. Petersburg State University of Architecture and Civil Engineering (SPSUACE), 2-nd Krasnoarmeiskaya St. 4, 190005 St. Petersburg, Russia

5 St. Petersburg Electrotechnical University "LETI", ul. Professora Popova 5, 197376 St. Petersburg, Russia

6Pavlov First Saint Petersburg State Medical University L'va Tolstogo str. 6-8 Saint Petersburg, 197022, Russia

[email protected]

PACS 61.48.+c DOI 10.17586/2220-8054-2020-11-2-205-213

The temperature of water-ice crystallization initiation decreases (AT) were determined in the binary water solutions of water soluble fullerenol:

C60(OH)24 — H2O at 272.85-273.15 K. Solution concentrations (in molar fractions) vary over a wide range xnano-ciuster = 5.0 • 10-6 ^

1.6 • 10-4 a.un. Liquidus temperatures were determined with the help of Beckman thermometer with a linear resolution of the device scale AT

- ^ 0.01 K/mm (h - height of Hg capillary raising). For the thermodynamic description of the discussed systems, we have elaborated an

Ah

original semi-empirical model, the virial decomposition asymmetric model (VD-AS), with assistance from partial molar functions of nano-clusters (activities and activity coefficients) were calculated. The Gibbs energies for the solutions and miscibility gap concentration regions were calculated. VD-AS model excellently describes pre-delamination or micro-heterogeneous-structure formation in the considered solutions. These calculations are confirmed by the dynamic light scattering data.

Keywords: fullerenol C60(OH)24, water solutions, cryometry, excess functions, delamination. Received: 19 February 2020 Revised: 1 March 2020

1. Introduction

The article is the development of the investigations, devoted to the synthesis, identification and investigation of physico-chemical properties of water soluble derivatives of light fullerene C60 and C70, such as: complexes of eithertwo-based dicarboxylic acids (malonates, oxalates), poly-hydroxylated forms (fullerenols), amino-acid derivatives (arginine, alanine) some other derivatives [1-32]. In these works, in particular, such physical-chemical properties of water solutions of light fullerene derivatives, depending on the solutions concentration, were investigated: poly-thermal solubility and crystal hydrate compositions; volume properties (density, average and partial components molar volumes); refraction indices, specific and molar refractions; conductivity and hydrogen indicator (seeming dissociation degrees and concentration dissociation constants); associates dimensions and electro-kinetic (-potentials, some others.

The investigation of the excess thermodynamic functions in such systems (activities, activity coefficients, excess (or mixing) Gibbs energies (enthalpies, entropies) etc, as authors know, until this time was not provided, except for some original works [33-38]. In these articles, in the binary systems with the help of cryometry investigations, the authors determined the decrease of the temperatures of the beginning of ice crystallization (liquidus temperatures), water activities, water activity coefficients, and then, numerically solving the Gibbs-Duhem equation - activities and activity coefficients of fullerene derivatives. No other data, concerning excess function in considered systems, obtained, for example by isopiestic method, were not found, unfortunately. Meanwhile, such data may be very scientifically interesting, because these water solutions have very specific and rare consistent hierarchical type of association (vide infra).

2. Cryometry data in the binary water solutions of water soluble derivatives of light fullerenes

The temperature of water-ice crystallization initiation decreases (AT) were determined in the binary water solutions of water soluble derivative of light fullerenes: C6o(OH)24-H2 at 272.85-273.15 K. Solution concentrations (in molar fractions) vary in the wide range Xnano-cluster = 5.0 • 10-6 ^ 1.6 • 10-4 rel.un. Liquidus temperatures were

AT

determined with the help of Beckman thermometer with the linear resolution of the device scale —— «0.01 K/mm

Ah

(h - height of Hg capillary raising). Cryometry data AT(xnano-cluster) are depicted in the Fig. 1 and in Table 1.

Table 1. Cryometry data and excess thermodynamic functions in the binary system bis-adduct of light fullerene C70 and amino-acid - lysine V at 272.99-273.15 K

Molar fraction of Ce 0(OH)24 in the solution XC60(OH )24 (rel.un.) Temperature of the beginning of ice crystallization decrease AT (K) lnaH2O ln(water activity) (rel.un.) ln YH2O ln(water activity coefficient) (rel.un.) ln YCeo(OH)24 ln(activity coefficient of C6 0 (OH)24 (rel.un.)

0.000 0.000 0.000 0.000 0.000

1.083 • 10-e 0.005 -5.86 • 10-5 -5.75 • 10-5 3.38

2.166 • 10-e 0.013 -1.44 • 10-4 -1.42 • 10-4 6.65

4.332 • 10-e 0.020 -2.22 • 10-4 -2.18 • 10-4 12.9

8.66 • 10-e 0.030 -3.33 • 10-4 -3.25 • 10-4 24.3

1.73 • 10-5 0.050 -5.49 • 10-4 -5.32 • 10-4 42.2

3.46 • 10-5 0.075 -8.22 • 10-4 -7.87 • 10-4 66.0

6.92 • 10-5 0.11 -0.00122 -0.00116 72.5

1.38 • 10-4 0.16 -0.00178 -0.00164 73.1

One can see, that all dependencies AT(xnano-ciuster) are sharply nonlinear, which prove very high positive deviations of the solutions from ideality for all solutions, even when very diluted. In the Fig. 1 for the comparison by the arrow is represented the value ATid for the ideal non-electrolyte solution. We can see the experimental AT exceeds ATid one-two orders (for comparable concentrated and diluted solutions, correspondingly). So, one should expect probably gigantic positive deviations of the solution from the ideality in thermodynamic sense.

3. Calculation of water excess functions

For the calculation of water activity we have used well-known equation, got from the equality of the chemical H2O potentials in pure solid ice and non-ideal liquid solution [34-38]:

-AHf AT - ACp AT2

W =ln O.H2O, (1)

R (Tf - AT) Tf

where AhW = 5990 J/mole, ACP = -38.893 J/moleK, t0 = 273.15 K heat, temperature of ice fusion and change of heat capacity in the process of ice fusion, correspondingly. Equation (1) was displayed in the symmetrical normalization scale for thermodynamic functions for both solution components:

aH2O (XH2O = 1) = YH2O (XH2O = 1) = 1, (2)

anano-cluster (xnano-cluster 1) Yfnano-cluster (xnano-cluster 1) 1 (3)

where xi and ai, - molar fraction, activity and activity coefficient of i-th component. Calculated data ln[aH2O (xnano -cluster )] are represented in Fig. 2 and Table 1.

Fig. 1. Temperature of water-ice crystallization beginning (liquidus temperature) decrease (AT) against molar fraction concentration in the binary solutions: C60(OH)24-H2O at 272.85-273.15 K. Arrow shows AT-function for the ideal non-electrolyte solution

Further authors [33-38] calculated concentration dependencies ln yh20, derivatives

d ln YH2O

dx

cally). Then, authors [33-38] (especially [36]) calculated the dependencies

d ln y

nano—cluster

(numeri-

nano-cluster

, (according to clas-

nano-cluster

sical differential Gibbs-Duhem differential equation) also numerically and at the end by numerical integration the dependencies ln jnano-cluster (xnano-cluster) were calculated. As a result, as was expected earlier, significant positive deviations of the solution from the ideality for the functions ln jnano-cluster were obtained ln jnano-cluster « n(10°^101) (see Table 1 and lower - Fig. 4). Naturally, in all likelihood, no currently existing thermodynamic model can describe such non-trivial behavior of nano-cluster thermodynamic functions.

Fig. 2. Logarithm of water activity (lnaH2O) against molar fraction concentration in the binary solutions: C6o(OH)24-H2O at 272.85-273.15 K

4. Virial Decomposition Asymmetric Model - VD-AS

For the description of such nontrivial thermodynamic behavior of considered water solutions, we have elaborated semi-empirical model VD-AS (Virial Decomposition Asymmetric Model), based on the virial decomposition of excess molar Gibbs energy on the solution for the components molar fractions. This technique has often been used for the thermodynamic description of binary and multicomponent systems with the different physical-chemical nature, such as: electrolyte solutions [39-42], non-electrolyte (semiconductor) melts [43-46], isovalent substitution solid solutions [47-49]. If one takes only one term in the decomposition for the binary system (corresponding to the invariant second virial coefficient), - see later equation (4) well-known strictly regular solutions model - RSM, is realized. Under the assumption of the temperature dependence for the only virial coefficient, the quasi-regular solution model - QRSM is realized. If one uses third virial coefficients for the decomposition, the sub-regular solution model -SRSM, is realized. Finally, when one additionally takes into account the contribution of the electrostatic non-specific interactions (according to Debye-Huckel theory) Pitzer's model in different variants is realized.

So, let the following numeration: 1 - is the number of the dissolved component (nano-cluster in our case), 2 - is the number of the solvent (H2O). Let us postulate the following expression:

Gex ^^ • • y^- 1 y, • 1 nin2 * Ajo

^ = (rn + «2) EEXX * Aij = ^'=« +=1 j j, (4)

RT i=1 j = i («1 + «2) j

where Gex - full molar excess solution Gibbs energy, R = 8.31 J/K, T - temperature (K), « and X - molar number and molar fraction of i-th component, correspondingly, Aj-ij-th virial coefficient in the decomposition of Gex/RT on the component molar numbers. In other words, Aj are naturally identified as divided by RT specific energy of interaction of i particles of the 1-st component and j particles of the 2-nd component. If one takes into account huge (two orders of magnitude) differences in components molar masses (1-st are heavy) and it's linear dimensions (almost one order - nanoclusters in our case are hollow), it is quite clear, that if upper limit of the summation according to the 1-st component is small natural number (not more than 4-6), that if upper limit of the summation according to the 2-st component may be more on one-two order.

Let us calculate excess thermodynamic functions of the components ln y

In 71 = d (GJn/RT) = E E [i - (i + j - 1) X1] X^X* * Aj,

1 i=1 j=1

(5)

lnY2 = d (G!en/Rr) = EE [j - (i + j - 1) X2] X1X2-1 * Aij. (6)

2 1=1j=1 In our case molar fractions f the components are incomparable:

X2 >>X1;X2 > 0.999 « 1X1 << 1. (7)

So, equations (5), (6) may be simplified:

ln 72 « E (1 - i) Xi E Aij = £ (1 - i) X1 E Aij. (8)

1=1 j=1 1=2 j=1

From the systems (5), (6), (8) one can see that, they are agreed in thermodynamic sense, i.e. Gibbs-Duhem equation is valid at T,P = const:

X1d ln 71 + X2d ln y2 =0, (9)

or, for example from the equations (7) and (8), we can get identity:

X1 E i (i - 1) X1-2A1jdX1 + E (1 - i) iX1-1A1jdX1 = 0. (10)

i=1 i=1 Let us transfer system (7), (8), denoting as A1 summary virial coefficients:

E A1j = A1 (T), (11)

j=1

So, in our conditions of consideration:

ln Y1 iA1 X1-1 (12)

i=1

ln Y2 « E (1 - i) A1X1 = E (1 - i)A1X1. (13)

i=1 i=2

One can see, that, if Ai = 0, then logarithms of the limit activity coefficients - ln y0 = limXi^0 ln yí, correspond to symmetrical method or normalization of the excess thermodynamic functions, namely:

ln Y? = Ai =0, Yi(Xi ^ 0)= Y? = 1, (14)

ln y0 = 0, Y2(Xi ^ 0) = 1. (15)

In our case incomparable more conveniently is to use asymmetrical normalization scale, (for this one should only demand the performing of the single condition A1 = 0):

ln y0 = 0, Yi (Xi ^ 0)= y0 = 1. (16)

ln y0 = 0, Y2 (Xi ^ 0) = 1, (17)

ln Yiss « EiAiXi-i, (18)

i=2

ln Y?ss « E (1 - i)A¿X i. (19)

i=2

This normalization scale will be used by us always default.

-mix

Let us introduce the function (divided by RT molar Gibbs energy of mixing), the second isothermal -

RT

isobaric concentration derivative of which is equal to the full solution molar Gibbs energy:

Qmix

-—- = X i ln Xi + X2 ln X2 + Xi ln Y i + X2 ln Y2, (20)

RT

d = 1 + 1 - (ln X2 + 1) + E i2AiXi- i + E (1 - i) Ai [i (i - 1) Xi-2 - (i +1) X i- i] (21)

dXi Xi X2 . 1

i=

32« e i (i _ 1)AiX i

2

dx2 X1 + ^i(i - 1)AjX 1 • (22)

Absolutely similarly we can calculate (divided by RT) first isothermal - isobaric concentration derivative of the chemical potential or logarithm of the activity of the 1-st component - ^ 1 and ln a 1, correspondingly:

1 = dln a i 1 (i Xi-2 (23)

íTdX = "dX" « X + i(i - 1)AiXi . (23)

RT dX 1 dX1 X 1

i=2

So, the equation of the diffusional (spinodal) stability loss will be the following:

—mix-ass = d2 [-"/RT] = d ln a^ 1 ^ i - =0 (24)

=-dX-= "dX" « Xi+ 2^(i- 1)AiXi =0. (24)

i i=2

5. Application of VD-AS model to the description water soluble derivatives of light fullerenes water solutions. Calculation of nanoclusters excess functions. Miscibility gaps

Preliminary calculation shows that the 3-term approximation in VD-AS - model (i.e. i = 2, 3,4) is sufficient for more or less successful for the excess thermodynamic functions calculation with the satisfactory accuracy:

ln 7r « 2A2X 1 + 3A3X 2 + 4A4X 3, (25)

ln y^8 « -A2X 2 - 2A3X 3 - 3A4X 4. (26) So, the equation of the diffusional (spinodal) stability loss will be the following:

12A4X 3 + 6A3X2 + 2A2X1 + 1 = 0, (27)

or is elementary explicitly solved relatively X 1 (according to Cardano formula) cubic equation. All calculated parameters of VD-AS - model is represented in Table 2.

In all considered binary systems we calculated concentration dependencies ln yh20 (see Table 1):

ln Y2 = lnH2O = ln aH2O - ln XH20 = ln aH2O - ln (1 - X1), (28)

and from these functions ln y2(X 1) according to equation (26) we determined parameters of VD-AS model: (A2, A3, A4) (see Table 2).

Formally, the huge values of summary virial coefficients Aj, in general, are not surprising, if we remember that, according to the physical sense, the last ones are the sums of the rows, consist of probably thousands terms, responsible for the energies of interactions of few nano-clusters with very large number of water molecules.

Table 2. Parameters of model VD-AS (A2, A3, A4) and concentration boards of diffusion instability regions (Xdiff-instab) in the binary system: C60(OH)24- H2O 272.85-273.15 K

VD-AS model parameters (rel.un.)

A2 (aun.) A3 (aun.) A4 (aun.) ~Xdiff-instab (a.un.)

2.10 • 106 -1.58 • 10 10 3.99 • 10 13 6.0 • 10-5

Then, according to the equation (24) we calculated concentration dependencies ln Yass(X^ (See Table 1 and Fig. 3). From Fig. 3 one can see that some dependencies ln yjss(Xi) will cross through the maximum at the values: X1 « 1.5 • 10-4 rel.un. (in our case, it lays slightly beyond the investigated concentration range). Maximum state may be determined easily, form the equation (24), solving square equation:

1 d ln Yf

2 dX1

In our opinion, it interesting to compare cryometry data in these system C60(OH)24 - H2O (pure derivative without any traces on Na-substituted fulerenol forms, such as C60(OH)23ONa) and in the system fullerenol-d - H2O (mix of C60(OH)22, C60(OH)23, C60(OH)24 etc with the traces of sodium mono-substituted forms - nearly 10-3 mol. %). If Fig. 3 one can observe very good agreement between our data and data from [36], despite on some inconsistency in the composition.

= A2 + 3A3X + 6A4X2 = 0.

(29)

Fig. 3. Logarithm of light fullerene water soluble derivative activity C60(OH)24 - H2O at 272.85273.15 K (open points - our calculation; solid points symbolize experimental cryometry data from [36] in the related system fullerenol-d - H2O)

Then, according to the equation (27), we determined the boards of diffusional stability loss - Xdiff (see Table 2 and Fig. 4), solving cube equation:

F = 12A4X3 + 6A3X2 +2 A2X1 + 1=0. (30)

As one can see from Table 2, all cubic equations (27) for our considered systems have real positive roots in real concentration ranges, which correspond to the existence of liquid solutions of nano-clusters in water. From this fact, one can conclude, in some concentration range: xfff « 6.0 • 10-5 rel.un. system begins to flake or may be pre-flake (see later). In the Fig. 4 the concentration dependencies of the diffusional instability functions Fdiff-mstab = 12A4X3 + 6A3X2 +2 A2X1 + 1=0 is represented. From Fig. 4, we see that almost all functions Fdif f -instab(x 1) have the initial intersection with the abscissa axis (red point), which corresponds to the diffusional stability loss.

Fig. 4. Instability board function F against molar fraction concentration in the binary solutions: C6o(OH)24-H2O at 272.85-273.15 K (solid point corresponds to the border of solutions instability region, dash curve corresponds to the border of solutions instability region from [36])

As expected, even such minor amounts of Na-substituted forms considerably stabilized phase instable nano-cluster solutions.

6. Conclusions

Thus, the water soluble fullerenol C6o(OH)24 demonstrated very high positive deviations from the ideality, which in turn caused formation of micro-heterogeneous system or pre-delamination in the liquid phase. VD-AS semi-empirical thermodynamic model provided a sufficiently accurate description of the concentration dependence of excess thermodynamic functions in this system.

Acknowledgements

This work was supported by Russian Foundation for Basic Research (RFBR) (Projects Nos. 18-08-00143 A, 19-015-00469 A, and 19-016-00003 A).

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