ISSN 15600904, Polymer Science, Ser. B, 2012, Vol. 54, Nos. 3–4, pp. 197–204. © Pleiades Publishing, Ltd., 2012. Original Russian Text © N.N. Sigaeva, A.K. Friesen, I.I. Nasibullin, N.L. Ermolaev, S.V. Kolesov, 2012, published in Russian in Vysokomolekulyarnye Soedineniya, Ser. B, 2012, Vol. 54, No. 4, pp. 597–604.
CATALYSIS
Initiation of ComplexRadical Polymerization of Methyl Methacrylate in the Presence of Metallocenes1 N. N. Sigaevaa,*, A. K. Friesena, I. I. Nasibullina, N. L. Ermolaevb, and S. V. Kolesova a
Institute of Organic Chemistry, Ufa Research Center, Russian Academy of Sciences, pr. Oktyabrya 71, Ufa, 450054 Bashkortostan, Russia b Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603950 Russia *email:
[email protected] Received August 10, 2011; Revised Manuscript Received December 1, 2011
Abstract—The initiation of radical polymerization of methyl methacrylate in the presence of benzoyl perox ide–metallocene (Cp2Fe, Cp2ZrCl2 , and Cp2TiCl2; (C5Me5)2Fe, (C5Me5)2ZrCl2 , and (AcC5H4)(C5H5)Fe) systems is studied. It is shown that a metallocene affects the rate of initiation and the initial rate of polymer ization. On the basis of quantumchemical calculations, a new mechanism of the initiation reaction may be advanced: Namely, the decomposition of benzoyl peroxide proceeds via the stage of complexation with a metallocene, while the nature of a metallocene determines the probability of complexation and decomposi tion. DOI: 10.1134/S1560090412040057 1
INTRODUCTION
The study of radical polymerization in the presence of initiation systems based on ferrocene (Cp2Fe) and benzoyl peroxide showed that the presence of Cp2Fe in the reaction system leads to a gain in the rate of methyl methacrylate polymerization [1]. In accordance with the literature data, the reaction of diacylperoxides with Fe(II) compounds occurs via the singleelectron transfer Haber–Weiss general scheme [2]. It may be supposed that, in the transient state, peroxide interacts with Fe via the coordination mechanism and, as a result of further electron transfer, the RCOO– anion and the RCOO• radical are formed. With consider ation for these data, it may be assumed that an increase in the rate of polymerization in the presence of fer rocene is due to its catalytic effect on the decomposi tion of benzoyl peroxide via the following scheme:
impossible. Thus, the structure of the intermediate complex remains unclear. It is evident that the rela tionship between the complexation and complex dis sociation (stability) will affect the kinetic features of polymerization. The nature of the used metallocene and the conditions of polymerization can influence both aforementioned factors. In this study, the initiation stage of methyl meth acrylate polymerization was studied in the presence of benzoyl peroxide and the following additives: Cp2Fe and ferrocenes modified with substituents incorpo rated into the cyclopentadienyl ring, such as (C5Me5)2Fe and (АсC5H4)(C5H5)Fe as well as Cp2ZrCl2, (C5Me5)2ZrCl2, and Cp2TiCl2.
Cp2Fe + (PhCOO)2 → [complex] → Cp2Fe•+ + PhCOO– + PhCOO• The ability of ferrocene to form complexes with benzoyl peroxide was shown experimentally [3]. How ever, the Fe atom in a ferrocene molecule is coordi nately saturated; hence, formation of a coordination bond, for example, with the lone pair of oxygen is
O
Fe
Fe
1 This
work was supported by the federal program Scientific and Educational Specialists of Innovative Russia, State Contract no. 02.740.11.0648.
197
Cl
Cl Zr
Fe
Zr
Cl
Cl Ti
Cl
Cl
198
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Table 1. Initiation rate of methyl methacrylate polymerization at various temperatures in the presence of metallocenes ([benzoyl peroxide] = [MC] = 1 × 10–3 mol/L) wi × 106, mol/(L min)
T, K 323 333 343
benzoyl peroxide
Cp2Fe
(C5Me5)2Fe
(AcC5H4)(C5H5)Fe
Cp2TiCl2
Cp2ZrCl2
1.8 5.6 11.3
15.6 30.5 36.9
2.6 4.7 7.7
6.8 8.3 27.8
– 1.2 –
– 0.4 –
EXPERIMENTAL The stabilizer hydroquinone was removed from methyl methacrylate (MMA) via shaking with a 10% KOH solution. Then, the monomer was washed with water until neutrality of the wash water, dried over CaCl2, and distilled twice under vacuum. Benzoyl per oxide was repeatedly recrystallized from methanol and dried at room temperature under vacuum to a constant weight; Тm = 381 K. The metallocenes Cp2Fe, Cp2ZrCl2, and Cp2TiCl2 (all from Aldrich) and (C5Me5)2Fe, (C5Me5)2ZrCl2, and (АсC5H4)(C5H5)Fe (synthesized at the Institute of Applied Physics, Rus sian Academy of Sciences (Nizhni Novgorod) as described in [4–6]) were used without further purifi cation. The kinetics of bulk polymerization of MMA was studied through the dilatometric method in the temper ature range 313–353 K. The concentrations of benzoyl peroxide and metallocene were 1.0 × 10–3 mol/L. The reaction mixture in a dilatometer was evacuated to a residual pressure below 1.33 Pa [7]. The initiation rate was estimated from the time of inhibition determined after addition of the stable nitroxide radical 4phenyl 2,2,5,5tetramethyl3imidazolin1yloxyl to the polymerization system. Quantumchemical calculations were carried out with the PRIRODA06 program [8, 9] that uses Gaus siantype basis sets for solution of the Kohn–Sham equation and expansion of the electron density in the auxiliary basis sets to calculate the Coulomb and exchangecorrelation energies. We used the density
functional PBE [10] and the orbital basis sets of con tracted Gaussiantype functions of size: (5s1p)/[3s1p] for H, (11s6p2d)/[6s3p2d] for C and O, (15s11p2d)/[10s6p2d] for Cl, (17s13p8d)/[12s9p4d] for Fe and Ti, and (20s16p11d)/[14s11p7d) for Zr. The aux iliary densityfitting basis sets⎯the uncontracted Gaussiantype functions of size (5s2p) for H, (10s3p3d1f) for C and O, (14s3p3d1f1g) for Cl, (18s6p6d5f5g) for Fe and Ti, and (22s5p5d4f4g) for Zr. The method allows a good reproduction of the geo metric parameters for the studied compounds, as was shown in [11, 12]. The geometric parameters of the structures were optimized without symmetric limita tions. The type of stationary point on the potential energy surface was determined via calculation of the 298 Hessian. Absolute enthalpies H abs for triplet and excited singlet states of metallocene–benzoyl perox ide complexes were calculated for T = 298 K. The heat effects of reactions were estimated from the difference in total energies of reaction products and initial com ponents at Т = 0 K. RESULTS AND DISCUSSION The Effect of Ferrocenes on the Initiation Rate and Activation Energy of Polymerization The electrochemical potentials of the used fer rocenes are 0.41, 0.68 and –0.12 V for Cp2Fe, (АсC5H4)(C5H5)Fe, and (C5Me5)2Fe, respectively [13, 14]. Therefore, these metallocenes can have a cat alytic effect on the decomposition of benzoyl peroxide
Table 2. Initial rate of methyl methacrylate polymerization at various temperatures in the presence of metallocenes (MCs); [benzoyl peroxide] = [MC] = 1 × 10–3 mol/L wp × 103, mol/(L min)
T, K 313
benzoyl peroxide
Cp2Fe
(C5Me5)2Fe
AcCpFeCp
Cp2TiCl2
Cp2ZrCl2
(C5Me5)2ZrCl2
1.4
15.0
2.5
5.8
2.5
1.0
1.4
323
1.8
25.5
4.0
9.6
2.7
2.2
2.5
328
3.5
38.8
–
–
–
–
–
333
5.6
41.8
6.3
16.2
7.4
5.0
5.6
343
11.3
51.0
11.7
34.8
13.1
12.0
11.2
353
26.8
85.4
20.4
40.1
25.9
26.1
22.9
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due to occurrence of the redox reaction proceeding via scheme (1). Among these ferrocenes, (C5Me5)2Fe is the strongest reducing agent. However, as follows from Table 1, the highest value of initiation rate wi in the polymerization of MMA is observed in the presence of ferrocene; the second highest value, in the presence of (АсC5H4)(C5H5)Fe; and the third highest, in the pres ence of (C5Me5)2Fe. For all ferrocenes, the rates of initiation are higher than the corresponding values in the absence of metallocene additives. The results on the influence of metallocenes on the initiation rate agree well with the data on the effect of metallocene on the initial polymerization rate –logwp (Table 2). This circumstance suggests that the used metallocenes affect the decomposition of benzoyl peroxide. On the basis of the linear dependences of logwp on 1/T, the total activation energies of polymerization Еа were calculated. The activation energies of benzoyl peroxideiniti ated polymerization conducted in the presence of metallocenes are given below: Еа = 81.6 kJ/mol for benzoyl peroxide [15] and 37.5, 47.5, 48.0, 57.5, 65.0, and 76.1 kJ/mol for benzoyl peroxide systems with Cp2Fe, (AcC5H4)(C5H5)Fe, (C5Me5)2Fe, Cp2TiCl2, (C5Me5)2ZrCl2, and Cp2ZrCl2, respectively. (The measurement error for Еа is no more than 5%.) As can be seen, the minimum activation energy is characteristic of the benzoyl peroxide–ferrocene sys tem. Somewhat higher but close values are observed for benzoyl peroxide–(АсC5H4)(C5H5)Fe and ben zoyl peroxide–(C5Me5)2Fe systems. The Complexes of the Ferrocenes with Benzoyl Peroxide The quantumchemical calculations revealed the origins of divergence between the experimental data and the electrochemical potentials of ferrocenes. The study of different routes for the interaction of fer rocene with benzoyl peroxide made it possible to localize the structures of chargetransfer complexes that form via the simultaneous interaction of carbonyl groups of benzoyl peroxide (a, b) with hydrogen atoms of cyclopentadienyl rings in the case of (a) Cp2Fe and (b) (АсC5H4)(C5H5)Fe and (c) with hydrogen atoms of methyl groups of (C5Me5)2Fe. The structures of the chargetransfer complexes (a) Cp2Fe–benzoyl perox ide and (c) (C5Me5)2Fe–benzoyl peroxide in triplet state T0 and excited singlet state S* are shown below. (Interatomic distances are given in Å, and the dihedral angles are given in degrees.) The acetylferrocene– benzoyl peroxide complex is similar to Cp2Fe–ben zoyl peroxide; therefore, it is omitted. In accordance with calculations, triplet T0 and excited singlet S* states of the aforementioned com plexes have almost identical structures and energy characteristics (Table 3). With allowance for the spinconservation law, the scheme of chargetransfercomplex formation is as follows (with the example of ferrocene and benzoyl POLYMER SCIENCE
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peroxide). When two molecules (Ср2Fe and benzoyl peroxide) interact in ground singlet states S0, forma tion of triplet structure T0 must be preceded by forma tion of excited singlet S*. The latter then relaxes into the triplet, and an energy of about 1 kJ/mol is released. H
H
H
H Fe H H
H
O C
+ O O C
H
H S0 H
Ph
O
O
O S0 ΔHо= 13.9 kJ/mol
H
H
H
H
O
H
Ph
Fe O
S*(↑ ↓)
H
H
H
H H
ΔHo= –0.9 kJ/mol
Ph
O
O
H
H
H
H
O
H Fe
Ph O
T0(↑ ↑)
H
H
H
H H
Two unpaired electrons are distributed in the Cp2Fe–benzoyl peroxide complex as follows. The spin density on ferrocene is almost completely concen trated on the Fe atom and comprises 0.8 au. The spin density on each benzoyloxyl fragment is 0.6 au and is mainly localized on oxygen atoms of the peroxide group. 298
Table 3. Absolute enthalpies H abs of metallocene–benzoyl peroxide complexes 298
H abs , kJ/mol
Complex state complex a
2012
complex b
complex c
S*
–6536409.08 –6936773.53 –7566851.29
T0
–6536409.94 –6936774.14 –7566852.11
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SIGAEVA et al. (a)
O3
O2
C1
O1
C2 H3
O4 H4
H1
H5 H2
Fe
H6
(b)
C1
O3
O2
C2 O1
H3
H4 H5
H1 H2
O4 H6
Fe
Geometric parameters О1–Н1 О1–Н2 О1–Н3 О1–Н4 О2–Н2 О2–Н3 О2–Н4 О3–Н3 О3–Н4 О3–Н5 О4–Н3 О4–Н4 О4–Н5 О4–Н6 О2–О3 С1–О2–О3–С2 C1–O1 C1–O2 C2–O3 C2–O4
T0
S*
T0
complex a 3.845 2.599 3.891 2.548 4.116 2.345 2.980 3.459 2.314 4.876 2.464 2.942 2.847 3.133 2.264 148.7 1.244 1.293 1.288 1.251
S*
Individual molecule of benzoyl peroxide
2.330 2.607 2.166 3.340 2.956 3.326 2.932 2.934 3.320 2.955 3.346 2.168 2.595 2.334 2.293 123.4 1.250 1.285 1.284 1.250
– – – – – – – – – – – – – – 1.432 87.6 1.203 1.389 1.389 1.203
complex c 3.958 2.713 3.912 2.497 4.208 2.388 3.013 3.472 2.397 4.930 2.433 3.004 2.903 3.240 2.173 144.4 1.241 1.294 1.291 1.245
2.321 2.643 2.174 3.321 2.987 3.344 2.923 2.935 3.335 3.003 3.321 2.173 2.615 2.334 2.283 123.3 1.250 1.285 1.285 1.250 POLYMER SCIENCE
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Table 4. Averaged charges (au) on atoms of the initial metallocene and benzoyl peroxide molecules and on the formed complexes Individual molecules Atoms Cp2Fe Fe
(АсC5H4)СрFe
Complex a
Complex b
Complex c
(C5Me5)2Fe
1.0
0.6
0.7
0.8
0.8
0.9
C(Ср)
–0.4
–0.3
–0.1
–0.3
–0.3
–0.1
H(Ср)
0.3
0.3
–
0.3
0.3
–
С(С=О)
–
0.6
–
–
0.6
–
О(С=О)
–
–0.6
–
–
–0.5
–
С(Ме)
–
–0.8
–0.7
–
–0.8
–0.7
Н(Ме)
–
0.3
0.2
–
0.3
0.3
Total charge on molecule
0
0
0
0.93
0.63
0.91
Complex a
Complex b
Complex c
Atoms
Benzoyl peroxide
С(Ph)
–0.2
–0.2
–0.2
–0.2
C(С=О)
0.8
0.8
0.8
0.8
O(С=О)
–0.6
–0.7
–0.7
–0.7
O(О О)
–0.3
–0.5
–0.5
–0.5
H(Ph)
0.2
0.2
0.2
0.2
Total charge on molecule
0
–0.93
–0.63
–0.91
Charge distributions for initial molecules and com plexes are shown in Table 4. It is evident that, in com plexes a and c, ferrocene bears a positive charge close to unity, while benzoyl peroxide carries an equal (in magnitude) negative charge. In the case of acetylfer rocene, the charge transfer proceeds to a lesser extent. The structure and distribution of electron density in the complex is in agreement with the literature data [16]: The complex is paramagnetic with a significant charge transfer from the metallocene to benzoyl per oxide. Calculations show that, during complexation of ferrocenes with benzoyl peroxide, the peroxide bond becomes longer to a great extent; for example, its length in the Cp2Fe–benzoyl peroxide complex is on the order of 0.35. In a free molecule of benzoyl perox ide, the order of the peroxide bond is 0.84. Therefore, formation of the aforementioned complex may facili tate the decomposition of benzoyl peroxide and affect the rate of initiation. Heat effects calculated for reac tions of complex formation are listed in Table 5. The values of heat effects are in logical agreement with a wellknown fact: The incorporation of electron acceptor substituents into a ferrocene molecule decreases its ability to donate an electron and to form complexes via the donor–acceptor mechanism, whereas, in the case of electrondonor groups, this POLYMER SCIENCE
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ability is increased. It follows from the heateffect val ues that the formation of the (C5Me5)2Fe–benzoyl peroxide complex is an exothermic reaction, while the formation of the Cp2Fe–benzoyl peroxide complex and the formation of the (AcC5H4)(C5H5)Fe–benzoyl peroxide complex are endothermic reactions. Thus, the (C5Me5)2Fe–benzoyl peroxide complex is more stable than the complexes of the two other ferrocenes. An increase in temperature should shift the equilib rium of the reaction between (C5Me5)2Fe and benzoyl peroxide to the left; for reactions of Cp2Fe and (AcC5H4)(C5H5)Fe with benzoyl peroxide, the equi libria should be shifted to the right. (C5Me5)2Fe, which forms the most stable complex with benzoyl peroxide, has the smallest effect on the rate of initiator decom position. During examination of the effect of ferrocenes on the rate of initiator decomposition, not only the stage of chargetransfercomplex formation but also further transformations of this complex should be considered. To verify the previous assumption about the occur rence of reaction (1), which yields the ferrocenium cation, benzoyloxyl radical, and benzoate anion, its heat effect was calculated. Because reactions leading to ionic products are strongly affected by the polarity of a medium, the effect of a polar solvent was taken into account in calculations. (Individual calculations 2012
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Table 5. Energy changes (kJ/mol) during formation and dissociation of the metallocene–benzoyl peroxide complexes Energy change, kJ/mol Reaction Cp2Fe МC + (PhCOO)2
[МC···(PhCOO)2]
[МC···(PhCOO)2]
[МC···PhCOO ] + PhCO O
.
[МC···PhCOO ]
.
28
–15
64
77
13
8
17
–3
–8
70
62
95
37
28
–87
–92
–84
–104
–108
–31
–39
–5
–64
–72
.
.
.
.
[МC···PhCOO ] +
Cp2ZrCl2
14
МC + PhCO O
[МC···(PhCOO)2] + ММА
(АсC5H4)СрFe (C5Me5)2Fe Cp2TiCl2
PhCOO ММ A
.
[МC···PhCOO ] + ММА
.
МC + PhCOO ММ A
of the energy of reactants in ethanol were performed within the polarizedcontinuum model.) According to calculations, even under such conditions, the reaction proceeds with a consumption of energy of 159 kJ/mol, and it may be suggested that the reaction does not pro ceed during polymerization. In contrast, as can be seen from Table 5, in the case of ferrocenes, the reactions of radical decomposition of metallocene–benzoyl peroxide complexes pro ceeding without participation of the monomer and resulting in the formation of one free benzoyloxyl rad ical [MC···PhCOO]• and one radical complexed with a metallocene, [МC•..(PhCOO)2] . → [МC•..PhCOO] + PhCOO•, are characterized by low positive heat effects. Note that, in [MC···PhCOO]• intermediates, an unpaired electron is localized at the Fe atom; moreover, a signif icant positive charge (~0.5 au) is localized on fer rocene, while a negative charge (equal in magnitude) is localized on the PhCOO fragment. The release of the residual benzoyloxy radical from the [MC···PhCOO]• intermediate, [MC···PhCOO]• → MC + PhCOO• requires a much higher energy consumption. As is known, chargetransfer complexes can easily dissociate in the presence of a third component, for example, a monomer molecule. In fact, as follows from Table 5, the reactions of dissociation of metal locene–benzoyl peroxide complexes, in which MMA molecules participate, are exothermic. Thus, on the basis of quantumchemical calculations, it may be suggested that, in the case of ferrocene–benzoyl per oxide systems, there is formation of chargetransfer complexes, which easily dissociate in the presence of the monomer and form free radicals and regenerate metallocene. Therefore, the same metallocene mole cule can repeatedly participate in complexation with
benzoyl peroxide. The activation energy of the ther mal decomposition of benzoyl peroxide is ~120 kJ/mol [2]. The formation of metallocene–ben zoyl peroxide complexes is accompanied by a much lower energy consumption (and their dissociation pro ceeds with energy release); therefore, it may be con cluded that an increase in the initiation rate of MMA polymerization in the presence of metallocenes is related to formation and further dissociation of metal locene–initiator complexes. Effect of Titanocene Dichloride and Zirconocene Dichloride on the Initiation Rate and Activation Energy of Polymerization Metallocenes (Cp2ZrCl2, (C5Me5)2ZrCl2, and Cp2TiCl2) containing metals with the highest oxida tion number have an insignificant effect on the initia tion rate and the initial rate and activation energy of MMA polymerization (Tables 1, 2). A certain increase in the initial rate of polymerization is observed in the presence of Cp2TiCl2 (Table 2). The electrochemical potentials of Cp2TiCl2 and (C5Me5)2ZrCl2 are unknown; for Cp2ZrCl2, the electrochemical potential is +1.86, i.e., far in excess of the values known for fer rocenes [17]. It is assumed that the potentials of Cp2TiCl2 and (C5Me5)2ZrCl2 will likewise be above +1. Complexes of Titanocene Dichloride and Zirconocene Dichloride with Benzoyl Peroxide As calculations showed, titanocene dichloride and zirconocene dichloride form triplet complexes with benzoyl peroxide and their structures are similar to those of ferrocene–benzoyl peroxide complexes. In these complexes, the carbonyl groups of peroxide interact with the hydrogen atoms of Cp rings on the opposite side of chlorine atoms. The structure of the
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Cp2TiCl2–benzoyl peroxide triplet complex is shown below.
O3
O2
C1
C2 H2
O1
O4 H3
H1 H4
Ti
C1
Geometric parameters
C1
Complex
Individual molecules
2.446 2.200 2.788 2.803 2.200 2.436 150.6 1.256 1.282 2.243 1.282 1.255 2.333
– – – – – – 87.6 1.203 1.389 1.432 1.389 1.203 2.352
О1–Н1 О1–Н2 О2–Н3 О3–Н2 О4–Н3 О4–Н4 С1–О2–О3–С2 C1–O1 C1–O2 О2–О3 C2–O3 C2–O4 Ti–Cl
However, here, unlike in the case of ferrocene– benzoyl peroxide complexes, no significant delocal ization of spin density to the metal atom occurs. The total charge transferred from the Cp2TiCl2 molecule to benzoyl peroxide is about –0.5 au. It follows from Table 5 that the formation of such complexes requires much more energy than that in the case of ferrocenes. This means that, for the abovementioned metal locenes, the equilibrium in complexation with benzoyl peroxide will be significantly shifted to the initial com ponents. Nevertheless, the energy consumed for for mation of this complex is lower that necessary for the decomposition of benzoyl peroxide. Moreover, as is clear from Table 5, the mentioned complexes, like fer rocenecontaining complexes, easily dissociate to form radicals and regenerate metallocene. A certain retardation of the process observed experimentally in the case of Cp2ZrCl2 is related to the formation of products that are inactive in polymeriza tion [18]. POLYMER SCIENCE
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CONCLUSIONS The specific feature of initiation of methyl meth acrylate polymerization in the presence of benzoyl peroxide–metallocene systems is that an initiator– metallocene chargetransfer complex is formed. Rad ical species are not formed as a result of the redox pro cess, as was generally agreed in [1]: (C5H5)2Fe + (PhCOO)2 → (C5H5)2Fe+• + PhCOO– + PhCOO• (C5H5)2Fe+• + PhCOO• → Fe2+ + PhCOO– + 2C5H5 Fe2+ + (PhCOO)2 → Fe3+ + PhCOO– + PhCOO• According to quantumchemical calculations, the acceleration of polymerization in the presence of met allocenes occurs owing to radical dissociation of com plexes with the participation of monomer molecules. The probability of complex formation and dissoci ation depends on the nature of the used metallocene. The decomposition of peroxide is accelerated most efficiently in the presence of ferrocene and acetylfer rocene. This phenomenon may be associated with a low positive heat effect of the formation of corre sponding complexes and, in contrast, with a highly exothermic reaction of their dissociation. When titanocene dichloride and zirconocene dichloride were used, the formation of complexes with the initia tor requires significant energy consumption; there fore, they have a minor effect on the decomposition of peroxide. REFERENCES 1. Yu. I. Puzin, R. Kh. Yumagulova, V. A. Kraikin, I. A. Ionova, and Yu. A. Prochukhan, Polymer Science, Ser. B 42, 90 (2000) [Vysokomol. Soedin., Ser. B 42, 691 (2000)]. 2. V. L. Antonovskii, Organic Peroxide Initiators (Khimiya, Moscow, 1972) [in Russian] 3. L. Rixin, Z. Xiaohong, and W. Shikang, Gaofenzi Xue bao, No. 3, 374 (1994). 4. J. M. Manriquez, D. R. McAlister, E. Rosenberg, A. M. Shiller, K. L. Williamson, S. I. Chan, and J. E. Bercaw, J. Am. Chem. Soc. 100, 3078 (1978). 5. Yu. T. Struchkov, V. G. Andrianov, T. N. Sal’nikova, I. R. Lyatifov, and R. B. Materikova, J. Organomet. Chem. 145, 213 (1978). 6. R. B. King and M. B. Bisnette, J. Organomet. Chem. 8, 287 (1967). 7. G. P. Gladyshev and K. M. Gibov, Polymerization at High Conversions and Methods of Its Investigation (Nauka, AlmaAta, 1968) [in Russian]. 8. D. N. Laikov, Candidate’s Dissertation in Mathematics and Physics (Moscow State Univ., Moscow, 2000). 9. D. N. Laikov and Yu. A. Ustynyuk, Izv. Akad. Nauk, Ser. Khim. 54, 804 (2005). 10. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 2012
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