E-mail: shurzag@materials.kiev.ua Zaginaichenko S.Yu.,
MatysinaZ.A., SchurD.V., Chumak V.A., Pishuk V.K.
Institute for Problems of Materials Science of UAS, Kiev, Ukraine Dnepropetrovsk State University, Dnepropetrovsk, Ukraine Institute of Hydrogen and Solar Energy, Kiev, Ukraine
Theoretical investigation of transition between phases of SCL^BCCL type at fullerit hydrogenation
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ABSTRACT
Theoretical study of structural phase transition of SCL ^ BCCL type in fullerit at its hydrogenation has been carried out. The statistical calculation of crystal free energy has been performed, the equations of phases thermodynamic equilibrium have been found. The concentrational dependence of temperature of phase transformation has been elucidated, the phase diagrams of system have been constructed. The temperature dependence of hydrogen solubility in fullerit has been defined. The results of theoretical calculations have been compared with experimental data.
INTRODUCTION
The fourth allotropic modification of carbon is called fullerenes. They are unique molecules with high reacting abilities [1-4]. Hydrogen easily adds to and separates from fullerenes [5-14] due to relatively weak bonds formed between hydrogen and the carbon atoms of the fullerenes [15, 16]. An investigation into the process of fullerene hydrogenation is of great interest in connection with their various practical applications [3, 17, 18] including systems of hydrogen accumulation and storage [9, 10, 17]. Research into the properties of fullerene hydrides is important because of the large potential they have for use in practical systems. Such research will likely result in the development of both new systems and new materials.
By direct noncatalytic hydrogenation of fullerit in solid phase (solid crystalline material from pure fullerenes or their mixtures) [19-21, 6] the simplest fullerene hydrides can be formed at sufficiently high temperature and hydrogen pressure. These hydrides are mono-, di-, triango-, or tetra- hydrofullerenes $H, $H2, $H3, $H4, [22], where $= C60, C70,
83 % by mass. The remainder of the mass is the high fullerenes, consisting of C60, 15% C70 and 2% of the all rest high fullerenes. Upon hydrogen dissolution, the process sto-ichiometry and the change of crystal lattice type are typical for solid-phase hydrogenation. The pure fullerit (molecular crystal from fullerenes) has a simple cubic lattice (SCL) and the hydrogenous phase has body-centered cubic lattice (BCCL) [10]. In BCCL lattice hydrogen atoms occupy the tet-rahedral interstitial sites of the crystal and their concentration grows with increasing temperature. A type of packing of interstitial solid solution is retained with increasing hydrogen concentration and the crystal lattice parameter rises only slightly [23]. The integrity of the fullerene molecules is also conserved.
The aim of present work is a theoretical study of the phase transformation of SCL BCCL type upon hydrogenation, the estimation of transition the temperature, the construction of the constitution diagram, and the elucidation of the temperature dependence of hydrogen solubility in fullerit. As far as the investigators know, these calculations have not been done.
It is assumed that only C60 and C70 fullerenes enter into the composition of fullerit because the concentration of higher fullerenes is small, having molecular concentrations of c = 0.85 and c2 = 0.15 respectively. We consider that crystal lattice of fullerit of SC phase as well as with hydrogen of BCC phase is geometrically ideal. We don't take into account the small change of crystal lattice parameter with a rise in hydrogen concentration.
To solve these problems, the calculation of the free energy of the crystal is carried out, assuming that it is in a hydrogen atmosphere at constant, high pressure. The calculation is based on the molecular-kinetic concept taking into consideration the interaction of nearest fullerenes $1$1, $2$2,
C C C . These are referred to as fullerenes and make up ®a®2 (®a=C60, ®2=C70) as well as the interaction of fullerenes
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with hydrogen atoms $1H, $2H and also interaction between hydrogen atoms HH. Fullerenes interact with each other in both phases, SC and BCC, but $H and HH interactions are manifested only in the BCC phase.
FREE ENERGY OF SC AND BCC PHASES
At first we calculate the free energy of the SC phase using the starting formula
F1 = E1 - kTlnW,, (1)
where E1 is configurational energy, W1 is thermodynamic probability, k is Boltzmann's constant, and T is temperature. The following notations are entered:
N is the number of sites ($ and $2 molecules) of the lattices;
N, N are the numbers of $, $ molecules;
1 2 1 2 N, N22, N12 are the numbers of nearest pairs of $1$1, $2$2,
$1$2 molecules;
U11(d1), U22(d2), U12(d2) are the interaction
energies, taken with opposite sign, of $1$1, $2$2, $1$2 fullerene
pairs at a distance of SC lattice parameter d1;
C1=N1/N, C2=N2/N are molecular concentrations of
fullerenes,
Ci + C2 = 1, (2)
The E1, W1 values are determined by the following expressions:
■N,
Ei =-N1U11 - N2u22 - N12U12; (3)
In W, = N In N - N lnN, - N2 In N2 ; (4)
The N,, N2, N, N22, N12 numbers for SCL can be expressed in terms of c1? C2 concentrations:
Nn = 3NC2 , N22 = 3NC2 , N12 = 6Nc1c2 (5)
Taking into account equation (5) for numbers of fullerene pairs, we find the free energy of SC phase per
molecule as
F
f = — = - V
f N V°
+
kT (c1lnc1 + c2lnc2 ), (6)
where V0 = 3( Uu + ci; U 22 + 2C1C2 Uu ). (7)
This equation shows the dependence of F free energy on temperature, fullerit composition, and energetic parameters.
The free energy F2 of BCC phase is calculated with the formula
E2 - kTlnW2 - kTNH lnX ,
(8)
where NH is the number of hydrogen atoms and l is their activity.
The configurational energy E2 and thermodynamic probability W2 for fullerit with hydrogen are defined by the formulas (9) and (10) below
(9)
E2 = -N'U - - N2 ln N'2 -
_ N V - N V - N V - N' U
H1 H1 H 2 H 2 HH HH HH HH J
lnW2 = NlnN - NJn^ - N2lnN2 +
+6Nln6N - NHlnNH - (6N-NH)ln(6N - NH), (10)
where indicated numbers of pairs are at a distance of d2 V2 for fullerenes pairs, at a distance of d2 "s/5/4 for $H pairs and at distances of d2 /2^[2 and d2 V2 for HH pairs respectively (for UHH and vHH energies), d2 is the parameter of BCC lattice. The interaction energies in formu-
la (9) are determined at the distances
u11 = ^(/72), U2 = U22(/V2), uj2 = (/72), (11)
UH1 ^(V5/4), UH2 = UH2(d2^/4), UHH = Uhh((il2^2), uHH = uHH(dJ2)
The pair number are expressed in terms of concentrations for BCC phase by formulas
N1, = 4Nc2 , N22 = 4Nc2 , N12 = 8Nc,c2
NHH = ^NcH
N HH = '^NcH .
(12)
The number of tetrahedral interstitial sites in the BCC lattice is equal to 6N. They are partially occupied by hydrogen atoms and the part of them are vacant. The hydrogen concentration in relation to the fullerenes numbers CH = NH/N can be changed in the range
0 < CH < 6 (13)
Using the expressions for fullerenes pairs, hydrogen atoms and considered pairs (12) in term of concentration, we obtain the free energy f2 for BCC phase per fulleren as
f2 = F" = -Vi(c! )-V№ (ci )cH - VH cH + kT(cilnci + c2lnc2 ) +
+ kT^cHlncH + (6-cH)ln6j-kTcH lnX ,
where
K = 4(cX + C¡v'22 + 2c1c2v;2 ), = 4(uhi + C2VH2 ),
V„
if 1 '
VH ~ 3 I UHH + 2 UHH
(14)
(15)
The derived formula (14) allows determination of the dependence of free energy of the BCC phase on temperature, fullerene concentrations, hydrogen content, and its activity and energetic constants. An investigation of f2(CH) function for the presence of extremum shows that this function can have minimum at the condition
- 3|u
+1 u'
hh 2 hh
+ kT-
i(6 - ch )
> 0
(16)
As an example the plots of free energies f1, f2 as a function of hydrogen concentration are presented in Fig. 1 (a) and (b) for different temperatures kT = 0; 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9; 1.0 eV, fullerene concentrations C1 = 0.85, C2 = 0.15, activity X = 1 (a), 10 (b) and energetic parameters equal to
V$(0.85) = —2 eV, v;(0.85) = -4 eV , (17)
Vm(0.85) = 0.1 eV, Vh = 0.05 eV , (17a)
Vh (0.85) = —0.1 eV, VH=—0.1 eV . (17b)
The interactions of $H and HH pairs correspond to the attraction type (a) and the repulsion type (b).
With increase in temperature the f2(CH) function assume minimum in accordance with condition (16).
The free energy, f1 is independent of CH concentration. Because of this, in Fig. 1 it is presented by a straight line parallel to the abscissa and for studied temperatures, the f1 energy will fall into the range of 1.58 < f1 < 2eV. Only extreme plots of the f1 function for kT = 0 and kT = 1eV are shown in Fig. 1. The intersection points of f1, f2(CH) curves are marked off by solid circles. For each temperature the phase with smaller value of free energy is realized. Thus at kT = 0 the SC phase is realized. But with increasing temperature the BCC phase come to realization and at kT = 1eV it is realized over all interval of hydrogen concentrations CH.
Fig. 2 shows the plots of temperature dependence of f1(T), f2(T) functions for the same values of energetic param-
Figure 1. The plots of free energies fl5 f2 of SC and BCC phases respectively as a function of hydrogen concentration CJor different temperatures kT= 0; 0.1; 0.2;... 1.0 eV (curves 0; 1; 2;... 10 for f2function), activity X = l(a), 10(b), fullerenes concentrations C} = 0.85, Cz = 0.15 and energetic constants (17) which correspond to interaction of $H and EE pairs of attraction type (a) and repulsion type (b). Only extreme plots for temperatures kT= 0 and kT = leV are presented for f2 function. The intersection points of fff2(Cg) curves are marked off by solid circles. The dotted line connects these points. The positions of f2( CJ curves minimums are marked off by open circles, these points correspond to the equilibrium state of BCC phase. The open circles are connected by dot-and-dash line.
Fig. 2. Temperature
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