Journal of Modern Physics, 2013, 4, 1621 http://dx.doi.org/10.4236/jmp.2013.47A2003 Published Online July 2013 (http://www.scirp.org/journal/jmp) The Determination of Surface Thermodynamic Properties of Nanoparticles by Thermal Analysis George O. Piloyan, Nikolay S. Bortnikov, Natalia M. Boeva Institute of Geology of Ore Deposits, Petrography, Mineralogy and Geochemistry, Russian Academy of Sciences (IGEM RAS), Moscow, Russia Email: bns@igem.ru, boeva@igem.ru Received April 23, 2013; revised May 26, 2013; accepted June 29, 2013 Copyright © 2013 George O. Piloyan et al. This is an open access article distributed under the Creative Commons Attribution Li cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The effect of dispersivity on thermodynamic and kinetic parameters of chemical reactions in nanodispersed systems is theoretically investigated. On the basis of the established theoretical dependences the new method of determination of surface thermodynamic properties of nanoparticles (surface enthalpy, surface entropy and surface energy) by thermal analysis (DTA or DSC) was developed. Three examples of calculation of surface properties of nanoparticles were pre sented to prove the feasibility of this method. Keywords: Surface Enthalpy; Surface Entropy; Surface Energy; DTA; DSC; Nanoparticle 1. Introduction Surface thermodynamic properties of nanoparticles take a distinct effect on thermodynamic and kinetic parame ters of chemical reactions (socalled size effect) in nano dyspersed systems. However, this problem is poorly in vestigated because of complexity of experiment and ab sence of a database on surface thermodynamic properties of solid. The present article attempts to fill an existing gap somewhat. In the article, the new method of determi nation of surface thermodynamic properties of nanopar ticles by thermal analysis is described and its influence on thermodynamic and kinetic parameters of chemical reaction is shown. Owing to its great scientific and practical significance, the influence of solid body dispersibility on the chemical reaction velocity has attracted the attention of researchers for a long time. Three different kinetic models, which take into account substance dispersibility, were proposed for different heterogeneous chemical reactions [1]. At the same time, all the models assume that changes in dis persibility affect only the size of the reactive surface in the solid body. It was shown that the substance dispersi bility growth is accompanied by a shift of the reaction regime from the diffusionkinetic area toward the kinetic one [2]. The processes at the phase interface form the limiting stage of kinetics. The transition to nanoparticles is characterized by size effects. It is established that the thermal effect of the re action Qr for nanoparticles depends on the surface en ergy σ [3]. 2. About Terminology Let’s define some terms which will be used in the fur ther. 2.1. Nanoparticle As this term we shall understand particles of substance which size even in one dimension lay in an interval 10 nm ≤ r 100 nm [4]. In other words, nanoparticles are lar ger clusters, but it is less than microcrystals. Under this definition get also a number of natural substances (some clay minerals, some oxides, etc.). The main characteristic property of nanopartiles is the appreciable contribution of surface energy to their total energy. 2.2. Nanodispersed System We shall understand as this term the system which con sists of nanoparticles. 2.3. Surface Energy, Surface Tension Energetic properties of a surface of solid is usually char acterized value of a surface tension γ (by analogy to liq uids) (J/m2) or surface energy σ (J/m2). The surface tension is caused by unbalanced field of C opyright © 2013 SciRes. JMP
G. O. PILOYAN ET AL. 17 intermolecular forces, i.e., by definition, it is value iso tropic. The condition is satisfied always for liquids. An alternative pattern is observed for solid. Stress tensors in volume and in surface layer of solid are not isotropic in general so also the surface tension should be not obliga tory isotropic, that contradicts its definition. Thus, the concept of a surface tension for a solid has concrete phy sical sense only within the range of melting temperature of a solid. More universal concept is surface energy. There is a simple thermodynamic relationship between a surface tension and surface energy for liquid [1]: , F (1) where is area of surface of liquid. For pure liquids (onecomponent system) 0,F so numerically r G , r GG F . According to Gibbs, thermodynamic properties of na nopartiles is described the same thermodynamic func tions, as microcrystals. For example, Gibbs free energy of nanoparticle is stated as follows: G (2) where —Gibbs free energy of microcrystal, —the area of a surface of nanoparticle, is surface energy of nanoparticle: , F TS (3) where —surface enthalpy, S T 0, —surface entropy, is temperature. For pure substances (onecomponent systems) 0, FF SH and S do not depend on temperature as a first approximation [1]. is often named surface energy or total surface energy to suppose approximately the equality for nanoparticles at first approximation. 3. The Offered Model Let investigated substance represents ensemble indepen dent nanoparticles, contacting, but not cooperating with each other. We shall admit, that everyone nanoparticle represents pure substance (onecomponent system). The total area of a surface of ensemble nanoparticles we shall characterize the specific area F carried to 1 mole sub stances (m2/mol): , Sc r Sc 3Sc 2Sc2.5Sc (4) where is shape coefficient ( for spherical particles, for lamellar particles, for particles of the complex or uncertain form, etc.), — mole density of a particle, r—the characteristic size of a particle. Let chemical reaction occur in the system: , SD gas C D SB gasC AB where i are stoichiometric coefficients, A1, i.e. all calculations are related to a mole of the initial sub stance. The assumption, that the transformation process in a system can be described by one chemical reaction, is equivalent to an assumption, which this process depends on one independent variable. The number of moles of any reaction component can be selected as such a vari able, but is more convenient to introduce a new variable α which is known as the degree of transformation or frac tional extent of reaction. According to the definition: (5) 00 , iiii nn n (6) is dimensionless quantity, 01. If we assume, that all components of system mutually insoluble, reaction in system should go up to the end 1 , unless one of the components is exhausted. Hence, nanopartile is changed into nanoparticle as a result of reaction (5) according to our model. N 1 n ri i GG 4. Some Features of Thermodynamics of Chemical Reactions For our model Gibbs free energy of reaction ∆G1 is de termined as: i G ,i (7) where i is thermodynamic potential of component i , are stoichiometric coefficients, 0 i 0 for reagents and i 0 r G for the reaction products. In the equilibrium state, we have (8) Let’s consider separate types of transformations, de scribed by Equation (5). Phase transitions 0,1. BD C . rCCAA GGF F With account for (2) and (5), we can write the follow ing equation: ,GHTS (9) Taking into account equality constancy of mass and (4), (8), we get (for spherical particles): 2 3 0 3, C AC A T QTr Q (10) where is heat of phase transition related to the mi crocrystalline state, TT T 00 is equilibrium tem perature difference between temperatures of phase transi tions for the microcrystalline and nanodispersed state of the substance, ρ is the mole density of nanoparticles and Copyright © 2013 SciRes. JMP
G. O. PILOYAN ET AL. 18 r is the size of nanoparticles. Formula (10) for the first time has been derived by Hill [5]. This formula can be considered as the general ized analogue of known Gibbs—Thomson formula. For heterogeneous endothermic (or exothermic) reac tions the equation is obtained similar to Equation (9): rr GG . CCCAA FF 0 r G (11) In the equilibrium state, we have (12) A little manipulation yields as follows (for spherical particles): 2 3 , CC A A C M M i 0 3A AC AA MT QTr (13) where are the molecular masses of nanoparticles A and i ,C are their densities. Complex MM CC A is known as Pilling—Bedward coefficient in the litera ture [6]. 5. Some Features of Kinetics of Chemical Reactions in the Nanodispersed Systems The effect dispersivity of a solid on the chemical reac tions rate has attracted the attention of researches for a long time, owing to its great scientific and practical sig nificance. The various kinetic models, which take into account dispersivity, were proposed for different chemi cal reactions [2]. However in all models it is supposed, that the change of dispersivity changes only the area of a reactionary surface of a solid. The dispersivity growth has been shown to be accompanied by a shift of the reaction regime from the diffusion—kinetic area toward the kinetic one [7]. A limiting kinetic stage becomes the processes going on surface of the interface. In this case, reaction rate de pends on value of the surface and can be presented Equa tion [2]: d, d t t VkF t V (14) where t and t are accordingly volume and a surface of the particle, not reacted by the moment , is the rate constant. tk By introducing degree of transformation α (5) into Equation (14), we obtain following equation: 01n kr d dt (15) where n is the order of reaction (for particles of the spherical form 23,n for flat 1n2, etc.0 r), is the size of the particle. The rate constant k is usually described by the Ar rhenius equation: expkA ERT (16) where A is a preexponential term, Е is empirical (ap parent) energy of activation, R is a gas constant, Т is temperature in Kelvin. In the theory of the activated complex (one of the ba sic theories of chemical kinetics) [8], the rate constant is defined by the equation: 10 exp ,kA GRT (17) , where 1B kT h k G is Boltzmann’s constant, h is Planck’s constant, 0 is activation free energy, is transmission coefficient. defines a probability that the system to jump activation barrier. It usually is as sumed that 1 . Equation (17) for nanoparticle must be changed by analogy with Equation (9). According to work [9] the rate constant of reaction of the activated complex which has already formed on a surface nanoparticle, should not de pends on dispersivity. According to this assumption Equ ation (9) may be transformed as follows: 00 0rArAAA AA GGG GGFGF G, (18) A where Ar is free energy initial nanoparticle G is Gibbs free energy for microcrystals of an initial com ponent , 0 G is free energy of formation of the ac tivated complex. Thus Formula (18) may be written as follows: 10 expkAGFRT (19) where is surface energy, is the mole area of surface. Formula (19) can be transformed taking into account equality (3): 20 expkAHF RT (20) where 0 is activation enthalpy, 21 0 exp , ASR 0 S 0 is activation entropy. The following equation has been derived in the the ory of the activated complex [7]: ERT (21) where E is activation energy of reaction for microcrys talls. Having substituted (21) in (20) we shall receive: 0expkAEF RT , (22) where 02 eA Substitution in (15) Formulas (20) and (21) gives: 00 dexp 1 d n ArEFRT t (23) From Equation (23) we may deduce that rate of Copyright © 2013 SciRes. JMP
G. O. PILOYAN ET AL. 19 chemical reaction should increase with growth of disper sivity. Of course, this is an idealized situation, which takes into account neither defects nor the covering degree of active centers on the reactive surface of the substance. A more general example of heterogeneous catalytic re actions is considered in [9], where it is shown that the dispersivity growth under stationary filling of active cen ters of the catalyzing agent 1, 1, the activation energy should decrease, while under T m TT the latter should grow. Below it will be shown, what even the simplified model considered in the article, leads to satisfactory re sults. 6. Some Formulas from the Theory of the Thermal Analysis The thermal analysis experiments are known to run in nonisothermal conditions. According to the theory of the thermal analysis [10], parameters of thermal curves (DTA, DSC, TG, DTG) contains the usefulness information on investigated substance and processes, in it proceeding. For the decision of our problem it is enough to use only one parameter—peak temperature of the thermal ef fect (temperature of the maximal deviation of thermal curves from a base line in an interval of thermal effect). The peak temperature of curve DTA, DTG or DSC is usually assumed to correspond to the temperature of ma ximal rate of chemical reaction. This assumption can be accepted only in the case where inertia of balance for DTG curves or thermal inertia of substance for curves DTA or DSC may neglect. It has theoretically been shown in work [11], that the peak temperature on curves DTG should advance peak temperature on curves DTA for endothermic reactions. However, practice shows, that this difference is usu ally insignificant, so as a first approximation it is possi ble to accept, that for curves DTA or DSC the peak tem perature m of endothermic effect is approximately equal to temperature of maximal rate reaction. Thus, at the equation is accomplished: 2 2 d0 dt (23) More complex picture, especially for fast reactions with greater heat effect, is observed for exothermic reac tions. Let’s consider endothermic reaction of the first order. Twice differentiating (15) with respect to t we shall re ceive: 0 ERT r m TT ^2 expERTdTdt A (24) It is known [10], that for curves DTA or DSC at dd ,Tt b TT (25) where b is heating rate of the furnace or reference sam ple. For curves DTG at m the derivative dd .Ttb For this reason the use of curves DTA or DSC is more preferable, as they contain less unknown parameters than curves DTG. In [12] it has been shown, that at as a first approximation: m TT 1 11 n n . (26) Hence, Equation (24) can be used and for reactions of the norder. If instead of k from (16) to use k from (22) we shall receive: 00 ^2 exp.rbEF RTAEF RT (27) For convenience, Formula (27) transforms more con venient form: 00^2 exp.rART bEFEFRT (28) Denote the term in a square brackets by B. Equation (28) may be rewritten as follows: 0exprBEFRT const.B . (29) At change Т or r term B changes essentially more slowly, than the exponential term. As a first approxima tion the term B may be assumed to be constant: (30) Write down (29) in form more convenient for calcula tion, having substituted instead of and their va lues from Formulas (3) and (4): ln ln , mFm F rBERTScHRrT Sc SRr rE T R (31) where is the size of a particle, is empirical acti vation energy for microcrystals, m is peak temperature of thermal effects on curves DTA or DSC, is a gas constant, Sc —shape factor, is surface enthalpy, ρ is a mole density of substance, S is surface entropy. Thus, the value ln r depends on three independent variables—1,1TrT mm and 1 Coefficients at these variables from Equation (31) is calculated by method of multiple regression. The admissible calculation accuracy can be achieved if to use not less than 6 experimental points. .r 7. The Calculations Let’s consider on the several examples, how much well offered model describes experimental data. Copyright © 2013 SciRes. JMP
G. O. PILOYAN ET AL. 20 The published experimental data of different authors (see Table 1) have been used as initial data. The first example is related to dehydration of boehmite. (α—AlOOH). This reaction is known to belong to the class of topochemical reactions; i.e., their limiting stage is represented by processes at the phase interface. The rate reaction is described by Equation (23). The authors [14] synthesized boehmite nanoparticles 1 to 26 nm across and described their thermal curves ob tained on the differential scanning calorimeter (DSC). The size of particles was defined by Xray method. Their experimental data are presented in Table 1. Using the boehmite data from Table 1, the multiple regression equa tion was calculated: ln18.675112366.1 1 0.0049445 1 rT r 3.37732 1 mm rT The coefficient R2 = 99.68%. Using the coefficients from Equation (31), we obtain (see Table 2 that the activation energy of the boehmite dehydration reaction is E = 102.8 kJ/mole, the surface enthalpy is 0.58 J/m2, surface entropy is 0.000844 J/ (m2·K), surface energy is calculated at 298 K σ298 = 0.329 J/m2, using Formula (3). At calculation shape coef ficient has been put (Sc) = 2.5 as synthesized boehmite nanoparticles have the uncertain forms. In the literature there are data on the surface enthalpy of boehmite, ob tained by a method hightemperature calorimetry [15]: НF = 0.52 J/m2. Our data are fairly consistent with liter ary data. The second example is related to the oxidation of mag netite. Table 1. Initial experimental data. Al2O3 Magnetite Fe3O4 Boehmite αAlOOH Sample r Tm r Tm r Tm 2.65 1463 9.5 358 1.13 653 2.69 1471 16 378 1.56 676 3 1476 30 388 2.04 686 3.3 1479 44 398 2.42 701 4.5 1522 48 418 6.9 744 6.2 1562 60 433 14.2 781 6.6 1563 80 433 26.3 801   95 438   Phase tansformation Al2O3 αAl2O3 Oxidation, the formation Fe2O3 Dehydration, the formation of Al2O3 The reaction [13] [14] [13] References The note: Tm is taken in К, r is taken in nm. Table 2. The results of calculation of surface thermody namic properties of investigated substances. Sample Е, kJ/mol HF, J/m2 SF, J/m2K σ298, J/m2 ρ, g/cm3(Sc) Boehmite102.80.58 0.0008437 0.329 3.082.5 Magnetite16.1 1.85 0.00567 0.16 5.2. 3 γAl2O3 95.1 1.90.00144 1.474 3.6 3 The note: Е is activation energy of chemical reaction for microcrystals, is surface enthalpy, is surface entropy, σ298—surface energy at Т = 298 К, ρ—density of substance, (Sc)—factor of the form. F HF S Several kinetic models of oxidation reaction were proposed for many classes of solid substances [6,15]. All of them describe, however, processes for microcrystal line substances. The changeover to nanoparticles alters the picture of reaction. The limiting stage of reaction, as well as in case of with boehmite, becomes the process going on at the phase interface. Reaction rate is described by Equation (22). The authors of [14] synthesized mag netite nanoparticles 9.5 to 95 nm across and investi gated oxidation reaction of magnetite and formation γ Fe2O3 by DSC. The size of particles was determined by Xray method. Their experimental data are listed in Ta ble 1. Using a calculation procedure similar to that in the situation with boehmite, we obtain the following multiple regression equation: ln9.392711937.36 129.7373 1 0.091178 1 mm rTrT r 0 0.01 , Fy The coefficient R2 = 99.53%. Using coefficients from Equation (31), we obtain that the activation energy of the magnetite oxidation reaction is E = 16.1 kJ/mole, the surface enthalpy is HF = 1.85 J/m2, surface entropy is SF = 0.00567 J/(m2·K). The sur face energy under 298 K calculated in line with Equation (3) is σ = 0.16 J/m2 (Table 2). No data on magnetite surface properties are available in the literature. Let us use for the theoretical assessment of the surface enthalpy the Orovan equation [16], which allows the surface enthalpy for metals and some oxides to be calculated at first approximation: Ea (32) where E is Young module and a0 is the parameter of the lattice. Using the available published data Ey = 231.3 gPа [17], a0 = 0.8394 nm [18], we obtain НF = 1.941 J/m2. Quite satisfactory agreement to our result is ob served. The third example is related to the phase transforma tion γAl2O3 αAl2O3. In work [13] γAl2O3 has been synthesized at dehydra tion of boehmite nanoparticles. The size of particles was determined by Xray method. Thermal curves DSC have Copyright © 2013 SciRes. JMP
G. O. PILOYAN ET AL. Copyright © 2013 SciRes. JMP 21 REFERENCES been written down. Necessary for the further calculations the initial data are placed in Table 1. Using already de scribed procedure of calculation we has been obtained the following multiple regression equation: [1] Yu. G. Frolov, “Course of Colloid Chemistry,” Khimiya, Moscow, 1982. ln9.899711442.90 119 – 0.0147631 rT r .468 1 mm rT [2] Yu. D. Tret’yakov, “Solid Phase Reactions,” Khimiya, Moscow, 1978. [3] G. O. Piloyan and N. S. Bortnikov, Dokl. Akad. Nauk, Vol. 416, 2007, pp. 247249. The coefficient R2 = 99.98%. [4] A. I. Gusev and A. A. Rempel, “Nanocrystal Materials,” Fizmatlit, Moscow, 2001. Using coefficients from Equation (31), we obtain that the activation energy of the phase transformation E is 95.1 kJ/mole, the surface enthalpy γAl2O3 is = 1.90 J/m2, surface entropy γAl2O3 is SF = 0.00144 J/(m2·K). The surface energy under 298 K calculated in line with Equa tion (3) is σ = 1.47 J/m2 (Table 2). [5] T. L. Hill, “Thermodynamics of Small Systems,” Benja min, New York, 1963. [6] P. Barret, “Reaction Kinetics in Heterogeneous Chemical Systems,” Elsevier, New York, 1975. [7] D. A. Frank Kamenetskii, “Diffusion and Heat Transfer in Chemical Kinetics,” Nauka, Moscow, 1967. In the literature there are data on the surface enthalpy of γAl2O3, obtained by the method hightemperature calorimetry [19]: НF = 1.7 J/m2. Our data are fairly con sistent with literary data. [8] N. M. Emanuel and D. G. Knorre, “Chemical Kinetics,” Vyshaya Shkola, Moscow, 1969. [9] V. N. Parmon, Dokl. Akad. Nauk, Vol. 413, 2007, pp. 5359. 8. Conclusions [10] G. O. Piloyan, “Introduction to the Theory of Thermal Ana lysis,” Nauka, Moscow, 1964. The basic equation of the method has been received: [11] R. L. Reed, L. Weber and B. S. Gottfried, Industrial & En gineering Chemistry Fundamentals, Vol. 4, 1965, pp. 38 46. doi:10.1021/i160013a006 ln lnm F rBERT Sc SRr Fm ScHRrT [12] H. E. Kissenger, Analytical Chemistry, Vol. 29, 1957, pp. 17021706. where r is the size of a particle, Е is empirical activation energy for microcrystals, Tm is peak temperature on curves DTA or DSC, R is a gas constant, (Sc) is shape coefficient, HF is surface enthalpy, ρ is mole density of substance, S is surface entropy. F 2 0 B RT [13] X. Bokhimi, J. A. T. Antonio, M. L. GuzmanCastillo, et al., Journal of Solid State Chemistry, Vol. 161, 2001, pp. 319326. doi:10.1006/jssc.2001.9320 [14] C. Sarda, F. Mathieu, A. Vajpei and A. Rousset, Journal of Thermal Analysis, Vol. 32, 1987, pp. 865873. doi:10.1007/BF01913772 b FE , b—heating rate of the refer ence substance or the furnace, ln(B) ≈ const. The equation involves one independent variable ln(r) and three dependent variables 1/Tm, 1/rTm and 1/r. Coef ficients in the equations are calculated by the multiple regression method. Surface energy is calculated by the known equation F [15] J. Majlan, A. Navrotsky and W. H. Casey, Clays Clay Minerals, Vol. 48, 2000, pp. 699707. [16] M. I. Gol’dshtein, V. S. Litvinov and B. M. Bronfin, “Me tallophysics of High Strength Alloys,” Metallurgiya, Mos cow, 1986. TS . On an example of three reactions (dehydration of boehmite (αAlOOH), oxida tions of magnetite (Fe3O4 γFe2O3) and phase transi tion (γAl2O3 αAl2O3) methods of calculation of sur face properties were shown. The obtained results were compared with literature data. Our results quite well co incide with known literary data. [17] I. N. Frantsevich, F. F. Voronov and S. A. Bakuta, “Hand book on Elastic Constants and Moduli of Elasticity for Metals and Nonmetals,” Naukova Dumka, Kiev, 1982. [18] S. Clark Jr., “Handbook on Physical Constants,” Geolo gical Society of America, New York, 1966 (Mir, Mos cow, 1969). [19] A. Navrotsky, Proceedings of National Academy of Sci ence of the USA, Vol. 101, 2004, pp. 1209612101. doi:10.1073/pnas.0404778101 Three examples of calculation of surface properties of nanoparticles for three types of reactions show, what even for such simplified model which is accepted in pre sent paper, it is possible to receive quite satisfactory re sults by thermal analysis.
