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Structuring elements of zirconium oxyhydrate gels under unbalanced conditions
Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 197 /207 www.elsevier.com/locate/colsurfa
Structuring elements of zirconium oxyhydrate gels under unbalanced conditions Yuri Ivanovich Sukharev a,*, Vladimir Aleksandrovich Potyomkin b,1 a b
Department of Common and Engineering Ecology, Southern Ural State University, 76 Lenin Avenue, 454080 Chelyabinsk, Russia Department of Organic and Inorganic Chemistry, Chelyabinsk State University, 70b Molodogvardeitsev Street, Chelyabinsk, Russia Received 13 June 2002; accepted 27 March 2003
Abstract Hydrated water in zirconium oxyhydrate gel is shown to be located in the inner region of the polymer compound in the gap between the molecules of zirconium oxide and to provide the bond between monomer units. For example, with the hydration of the compound (ZrO2(H2O)3)2 ×/ZrO2(H2O)5, the water molecule cannot enter directly into the inner region; therefore, the hydration of such a chain is not energetically equivalent to the reaction of addition of the hexahydrated form to the dimer. Structural difficulties accompanying the hydration of trimers and other forms of oxyhydrated polymers require the restructurization of the polymer chain to transfer into a more stable state. Such a transformation includes the successive processes of destruction, hydration and further construction of the chain. The activation energy of the destruction process is comparable with the energy of hydrogen bonds; therefore, the destruction rate of the chain is rather high. Moreover, it is to increase with the increase in the number of units in the chain. The pentahydrate is the most stable molecular form in the system under consideration. Therefore, it is the interaction of the pentahydrate form of oxyhydrate with the polymer chain, which is the most probable. The interaction of the pentahydrate with the metastable trimer (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 results in the formation of the metastable tetramer with 16 water molecules, while the stable tetramer should contain 19 water molecules. The fact confirms the theoretical scheme of oxyhydrate gel polymerization reactions described in the literature and in the present work, since the splitting-off of water molecules as a result of the olation-polymerization reaction is accounted for by the formation of this kind of metastable products. The successive growth of chains and their discrete destruction provide the time periodicity of gel properties. The structural peculiarities of the polymer chain formation are of special interest. The location of monomer units in the most beneficial tetramer (ZrO2(H2O)3)2 ×/ZrO2(H2O)6 ×/ZrO2(H2O)7 resembles the origin of the helix coil growth. In this case, the molecules of bond water are in the center of the coil to form some chain providing the structure-formation. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Model; Oxyhydrate gel; Spiral structure; Polymer; Chain-monomer unit; Zirconium; Oxyhydrate gel; Metastable trimer; Tetramer; Structure-formation
* Corresponding author. Tel.: /7-3512-399-022. E-mail address: [email protected] (Y.I. Sukharev). 1 Tel.: /7-3512-429-012. 0927-7757/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0927-7757(03)00136-5
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Fig. 1. Functional dependence of the hydration enthalpy (DHr) on the degree of hydration (n ).
1. Introduction
2. Formulation of the research problem
At present, the molecular mechanism of gel formation at the quantitative level is poorly studied. Studies are usually carried out at the level of the phenomenological description with the detailed analysis of particular properties. The objective difficulties, in this case, are as follows: the low stability and non-reproducibility of gels, the impossibility of direct structure studies, the problems with kinetic studies, since the formation of gel proceeds, as a rule, heterophasically and is known as the macrokinetic process including several kinetic and diffusion stages. The diffusion, in this case, plays an important role due to high viscosities of gel formations. At the same time, it is the understanding of the processes underlying the formation of gel systems which is the key to understand the structural regularities for the formation of the gel phase as well as the distinction of gel systems from liquid, amorphous and crystal states. Therefore, the simultaneous experimental and theoretical study of the regularities of gel phase-formation are of great importance.
In previous works [1,2], certain periodical regularities, both of structural and time nature, were found for oxyhydrate gels of heavy metals. As regards the structural nature, there was found the periodicity of the density variation (mass differentiation of gel) accounted for by the anisotropy of the field of forces of the gel-forming particles, which provides their ordering manifested later on at the macroscopic level. The problem of time periodicity, i.e. the periodical change-in-time of a number of gel properties, seems to be more complicated. This is due to the fact that a gel is a ‘‘living’’ system where some restructurization continuously proceeds, i.e. that of gel polymer chains. Examples of such a restructurization may be either the breakdown of a polymer chain followed by the growth of its fragments, or the breakdown of chains followed by their recombination. This phenomenon is similar to the dynamics of liquids in which the constant formation, destruction and restructuring of associates are observed. In liquids, however, the above processes
Y.I. Sukharev, V.A. Potyomkin / Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 197 /207
proceed very quickly due to the low degree of association and low viscosity. At the same time, high structurization of gel formations makes these systems similar to liquid-crystal ones. The growth of the polymer chain of heavy metals oxyhydrates being the structure elements of gels proceeds by the following scheme [1]:
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where (/R /) stands for a polymer fragment that is equally capable of entering the reactions of olation chain propagation (reactions 1 and 6), degradation (reaction Eq. (3)), and chain termination (polycondensation). The analytical description is complicated by a great number of components taking part in the
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above kinetic stages of the process, e.g. oxyhydrate forms of metals with various degrees of polymerization, water, protons, hydroxyl anions. Besides kinetic stages, the diffusion processes of carrying the components to the ends of the growing chains as well as those of removing the products of interaction from the reaction zone play an important role in the viscous gel system. The diffusion of chain oxyhydrate units is of special importance, since their mobility is much lower than the diffusion of water molecules, hydrated protons and hydroxyl anions. Thus, to analyze the growth process of polymer chains of heavy metal oxyhydrates, it is reasonable to consider the following kinetic processes of growth and destruction: addition and elimination of a monomer fragment; hydration and dehydration; protonation and deprotonation, addition and splitting out of a hydroxyl anion. Kinetic and thermodynamic peculiarities of the processes under consideration appear to differ depending on the chain length; therefore, the analysis of these stages should be carried out for polymer chains with various degrees of polymerization. 2.1. Calculation techniques To study the mechanism of growth for polymer chains of oxyhydrate gels, the zirconium oxyhydrate system was chosen as being more thoroughly investigated, both theoretically [1,3,4] and experimentally [5,6]. The ways of polymer chain formation are largely determined by the possibility of the location of the next monomer unit at the ends of the chain as well as by the energy of interaction. In this case, the description by the Mayo /Lewis equation which is traditionally applied to determine the composition of organic copolymers is impossible due to the essential difference in the energies of interaction of each of these forms with the growing polymer chain. Besides, this equation makes it possible to determine the polymer composition, but does not enable to establish the mechanism of the chain growth. The calculation of the optimal location of a monomer unit in the vicinity of a growing chain may be performed in the framework of a number of models (e.g. either
quantum-chemically, or by the methods of molecular mechanics [7 /9]). In gels, however, the associates of such a kind are dynamic formations with conformation mobility. Their composition and structure vary and as a matter of fact these associates are the probabilistic values. Therefore, to determine the composition and structure of the associates in gels by the methods of molecular mechanics or quantum chemistry requires the calculation of a great number of possible structures regarding the probability of their existence. The problem in itself is very complicated. Thus, to determine the local structures of the ‘‘polymer chain-monomer unit’’ type, the probabilistic approach described in [10] is used. With this method, the probabilities pij of atom /atom contacts are the basic model representation of the molecular surroundings. The calculations are based on the notion of a polymer chain and a monomer as a set of overlapping atomic spheres whose radii are functionally dependent on the set of valence and non-valence interactions and temperature. This is realized in the framework of DENSON and MERA models [10 /12]. The calculation of probable structure configurations of the ‘‘polymer chain-monomer unit’’ type is performed in the framework of the MERA model [12] (for the case of solutions), in which the calculation of atom /atom probabilities pij of contacts of molecules for all system components is performed. The probabilities pij for the contact of the atom i of a given molecule with the atom j of an adjacent one are determined as follows:
pij
nij xi xj Si exp( Eij =kT) P 1 N k 1 exp( Eik =kT)
(1)
where xi , xj are the mole fractions of the solution components, atoms i and j included, nij the maximum number of atoms j of an adjacent molecule which may be located in the contact with the atom i of a given molecule, si the fraction of the free surface of the atom i, Eij the potential energy of interaction of atoms i and j , k the Boltzmann constant, T the absolute temperature and N the number of atoms in the system (in a single molecule or in a solvate complex).
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The interaction energy of the atom i of the molecule being considered with the atom j of the adjacent one is written using the Lennard-Jones potential regarding the Coulomb interaction between atoms: qq 14kRaij Raij 6 7kRaij Raij 12 Eij i j (2) 4po 0 Rij 48a Rij 48a Rij where qi and qj are the charges on atoms i and j, Rij the minimum distance between atoms i and j calculated as the sum of minimum distances from atoms i and j to the molecule surface and o0 the dielectric permittivity of the vacuum. Since a monomer unit and a polymer chain approach each other for a distance of contact, it is reasonable to determine the atom of the monomer unit and that of the polymer chain, taking into account the maximum probability of their contact. A great number of variants for the relative position of the chain and the monomer unit at the distance of the atoms contact appear to exist, thus, the position corresponding to the energy minimum is chosen. To avoid entering into the local minimum of the potential energy, the MonteCarlo method with the local optimization [13 /16] in the framework of the semiempirical quantumchemical ZINDO/1 method is used. 2.2. Discussion of the calculation results The monomer unit of the unbalanced polymer chain of such a system may be presented by different zirconium oxyhydrate forms having the general formula ZrO2(H2O)n . The detailed analysis of possible monomer units in the framework of the semiempirical quantum-chemical ZINDO/1 method showed that the functional dependence of the hydration enthalpy DHr on the degree of hydration n is of the extreme nature, which is shown in Fig. 1. The functional dependence obtained does not contradict to the experimental data. Zirconium acid ZrO(OH)2 (or ZrO2(H2O)), or its hydrated form ZrO(OH)2 ×/H2O (or ZrO2(H2O)2) are usually considered to be the structural basis for gel polymer formations of zirconium oxyhydrate. In the compounds mentioned water is valence-bound. The addition of the
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next water molecule occurs in the coordination sphere. The inception of the polymer chain may represent the interaction of the pair of monomer units, in which case one should consider three variants of interaction: 2ZrO2 (H2 O) 0 (ZrO2 (H2 O))2 ;
(I:1)
DH 105:89 ZrO2 (H2 O)2 ZrO2 (H2 O)
(I:2)
0 ZrO2 (H2 O)2 ×ZrO2 (H2 O); DH 58:75 2ZrO2 (H2 O)2 0 (ZrO2 (H2 O)2 )2 ;
(I:3)
DH 17:01 ZrO2 (H2 O)2 ZrO2 (H2 O)3
(I:4)
0 ZrO2 (H2 O)2 ×ZrO2 (H2 O)3 ; DH 49:01 2ZrO2 (H2 O)3 0 (ZrO2 (H2 O)3 )2 ;
(I:5)
DH 81:90 ZrO2 (H2 O)3 ZrO2 (H2 O)4
(I:6)
0 ZrO2 (H2 O)3 ×ZrO2 (H2 O)4 ; DH 118:50 2ZrO2 (H2 O)4 0 (ZrO2 (H2 O)4 )2 ;
(I:7)
DH 121:62
Hydration competes with the above processes. The analysis of hydration processes shows that the most beneficial penta- and dehydrated forms of the incipient polymers are bound to have the highest concentration content in the solution. The minimum possible amount of water added
Table 1 Enthalpies of the dimerization process (chain growth DH) and hydration of monomer units (DHH) No. DH (kcal mol 1)
DHH (kcal mol 1)
DH /DHH (kcal mol 1)
(I.1) /105.89 (I.2) /58.75 (I.3) /17.01 (I.4) /49.01 (I.5) /81.90 (I.6) /118.50 (I.7) /121.62
/173.54 /124.26 /75.98 /59.43 /43.88 /121.63 /199.38
67.65 65.51 58.97 10.42 /38.02 3.13 77.76
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results in the formation of hepta-, tetra- and trihydrated forms. Energetically, the dimerization enthalpy of the first process is characterized as DH1 / /105.89 kcal mol 1; nevertheless, the process of monomer units hydration is more beneficial, since 2DHH / /173.54. Table 1 presents enthalpies of the competing processes of chain growth and hydration of the initial units. The data from Table 1 illustrates that the dimerization process is thermodynamically beneficial with dimerization of trihydrated forms of ZrO2(H2O)3. As for the other cases, it would be more beneficial, if zirconium dioxide were present in the solution in the form of hydrated compounds. At the same time, the trihydrated form is one of the least beneficial. Penta- and dihydrated forms of compounds are certain to be more beneficial. Dimerization of these forms, however, is not beneficial, which is quite evident, since it is the unstable oxyhydrate forms which must tend to the dimerization process. Since the energetically beneficial dimerization process is possible only from the forms, whose content in the solution is low, it is dimerization that must constraint the inception and growth of the polymer chain. The further growth of the polymer chain is provided by a number of successive processes: (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)2 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)2 ;
(II:1)
DH 80:99 (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)3 0 (ZrO2 (H2 O)3 )3 ; DH 85:35 (II:2) (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)4 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)4 ;
(II:3)
DH 180:84 (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)5 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)5 ;
(II:4)
DH 123:22 (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)6 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ; DH 108:50
(II:5)
(ZrO2 (H2 O)3 )2 ZrO2 (H2 O)7 (II:6)
0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)7 ; DH 111:70
With these reaction as well as with dimerization, the hydration process may be competitive to that of the trimer growth. Chain hydration follows the equation: (ZrO2 (H2 O)3 )2 H2 O 0 ZrO2 (H2 O)3 ×ZrO2 (H2 O)4 ; DH 58:54 Enthalpies of the monomer hydration were given before. Enthalpies of the competing processes of the dimer chain growth and total hydration of monomers and dimers are presented in Table 2. It is quite obvious that the trimerization process competes with the hydration in interacting of (ZrO2(H2O)3)2 with monomer units, from a trihydrate to hexahydrate, i.e. from ZrO2(H2O)3 to ZrO2(H2O)6. In the other cases, it is more energetically beneficial both for a dimer and a monomer to be present in the solution as individual hydrated compounds. The addition of hexa- and tetrahydrated forms of a monomer to a dimer is considered to be the most energetically beneficial. However, tetrahydrated and hexahydrated oxyhydrate forms are not the most probable formations in the solution. In this case, the addition of the dominating hydrated form, i.e. pentahydrate (reaction (II.4)), is thermodynamically the most probable process, but results in the formation of the metastable Table 2 Enthalpies of trimerization (chain growth DH ) and total hydration of a dimer and monomer units (DHH) No.
DH (kcal mol 1)
DHH (kcal mol 1)
DH /DHH (kcal mol 1)
(II.1) (II.2) (II.3) (II.4) (II.5) (II.6)
/80.99 /85.35 /180.84 /123.22 /108.50 /111.70
/96.03 /80.48 /158.23 /116.62 /79.82 /115.13
15.04 /4.87 /22.61 /6.60 /28.68 3.43
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product (ZrO2(H2O)3)2 ×/ZrO2(H2O)5, i.e. not the most probable. This compound could be assumed to add water with the formation of a more energetically beneficial compound (ZrO2(H2O)3)2 ×/ ZrO2(H2O)6. In fact this is not the case, which is accounted for by the structural peculiarities of and trimers (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 (ZrO2(H2O)3)2 ×/ZrO2(H2O)6. Hydrated water in the above compounds is in the inner region between the molecules of zirconium oxide and provides the bond between monomer units (Fig. 2). With hydration of the compound (ZrO2(H2O)3)2 ×/ZrO2(H2O)5, the water molecule cannot enter directly into the inner region; therefore, hydration of such a chain is not energetically equivalent to the reaction of
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addition of a hexahydrated form to a dimer. Such complicated hydration requires isomerization, including the opening of the inner region, the entering of a water molecule into it, followed by the closing of that region. Since the process is rather complicated, it is unlikely. In addition to the reasons mentioned, this reaction requires the maximum energy to open the inner region with the breaking of Zr /O /Zr bridge bonds. The most probable process is that including the destruction of the trimer (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 with hydration of a dimer and pentahydrated form of a monomer. Such a process requires only 6.60 kcal mol 1 (reaction is reversible to (II.4)). Further, there will take place the addition of the resultant hexahy-
Fig. 2. Structures of hydrated trimers (a) (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 and (b) (ZrO2(H2O)3)2 ×/ZrO2(H2O)6.
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drate to a dimer with the release of 28.68 kcal mol 1 (process (II.5)). In this case, the destruction enthalpy of (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 may be considered as the activation energy for the formation of a more beneficial chain of (ZrO2(H2O)3)2 ×/ ZrO2(H2O)6. Such a value of the energy barrier (6.60 kcal mol 1) is small and is comparable with the energy of diffusion activation. With the further growth of the oxyhydrate chain from the most beneficial trimer (ZrO2(H2O)3)2 ×/ ZrO2(H2O)6, the interaction processes of the trimer with various hydrated forms of monomer units may continue: (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 ;
(III:1)
DH 139:51 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)
(ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)8 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)8 ; DH 153:02 (III:9) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)9 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)9 ; DH 207:72 (III:10) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)10 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)10 ; DH 143:08 (III:11) The competing process of chain hydration proceeds by the following equation: (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 H2 O
0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O); DH 180:34 (III:2) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)2 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)2 ; DH 146:32 (III:3) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)3 0 (ZrO2 (H2 O)3 )3 ×ZrO2 (H2 O)6 ;
0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)7 ; DH 24:48 Enthalpies of monomer hydration were given before. Enthalpies of the competing processes of the dimer chain growth as well as of the total reaction of monomers and dimers hydration are presented in Table 3. It is quite obvious that the process of adding the dominating pentahydrated form to the growing chain, except for the first stage of dimer formation,
DH 168:68 (III:4) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)4 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)4 ; DH 190:66 (III:5) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)5
Table 3 Enthalpies of tetramerization (chain growth DH ) and total hydration of dimers and monomers (DHH) No.
DH (kcal mol 1)
DHH (kcal mol 1)
DH /DHH (kcal mol 1)
(III.1) (III.2) (III.3) (III.4) (III.5) (III.6) (III.7) (III.8) (III.9) (III.10) (III.11)
/139.51 /180.34 /146.32 /168.68 /190.66 /134.01 /152.58 /371.28 /153.02 /207.72 /143.08
/76.84 /111.25 /61.97 /46.42 /124.17 /82.56 /45.76 /81.07 /68.06 /94.28 /70.42
/62.67 /69.09 /84.35 /122.26 /66.49 /51.45 /106.82 /290.21 /84.96 /113.44 /72.66
0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)5 ; DH 134:01 (III:6) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)6 0 (ZrO2 (H2 O)3 )2 ×(ZrO2 (H2 O)6 )2 ;
(III:7)
DH 152:58 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)7 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)7 ; DH 371:28 (III:8)
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is thermodynamically beneficial, but results in metastable forms. Structural difficulties accompanying hydration of the latter require the restructurization of the chain to transfer into a more stable state. Such a transformation includes the processes of destruction, hydration, to be followed by the further construction of the chain. The chain destruction requires the activation energy, but the latter is comparable with the energy of hydrogen bonds. Thus, the rate of the chain destruction is quite high and is to increase with the increase in the number of units. The chain growth is thermodynamically beneficial and is determined only by the rate of monomer units addition, the diffusion in the gel phase being rather slow process. At some instant of time, the state of quasi-equilibrium comes (the empirical equilibrium) between the processes of monomer units addition and destruction. Since the destruction is the process of the first order [1,2] and the diffusion is determined by the Fick law, the following equation may be written in the simplified form, regarding the kinetics of destruction up to monomer units: kd aC KD
@ 2 (1 a)C 0 @x2
(3)
where kd is the constant of the destruction rate, C the concentration of monomer units in the solution, a the degree of transformation, KD the constant of the diffusion rate and x the diffusion coordinate. The following expression is the solution for this equation: aC A exp(ivx)
(4)
where v is the wavenumber of monomer concentration vibrations in coordinate x . This expression suggests the structural periodicity of the gel observed by the authors in works [1,2]. Besides, it should be noted that the pentahydrate is the most stable form of particles in the system. Therefore, it is the interaction of the pentahydrate form with the chain which is the most probable. The chain being formed in this case, however, is metastable. So, the collision of the pentahydrate with the dimer (ZrO2(H2O)3)2 is bound to result in the trimer with the total number
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of water molecules being 11. As regards the stable form, it contains 12 water molecules and is thermodynamically more beneficial. The further interaction of the pentahydrate with the metastable trimer (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 results in the formation of the metastable tetramer with 16 water molecules, while the stable tetramer should contain 19 water molecules. In this case, the gain in energy is 223.72 kcal mol 1. Thus, the high kinetic probability of addition of the pentahydrate zirconium oxide results in the formation of thermodynamically metastable polymer chains. With the increase in the chain length its instability increases, which results in the drastic increase in the probability of the metastable state relaxation, i.e. the chain destruction. With a particular length, the polymer chain is bound to break down. Two resultant fragments are more stable and may proceed to grow. In case the solution lacks monomers, the chain growth may proceed at the expense of the isothermic transformation of less stable chains into more stable ones. Thus, the successive growth of chains and their discrete destruction provide time periodicity of gel properties. The structural peculiarities of the polymer chain formation are of special interest as well. The location of monomer units in the most beneficial tetramer (ZrO2(H2O)3)2 ×/ZrO2(H2O)6 ×/ZrO2(H2O)7 presented in Fig. 3 resembles the origin of the helix coil. In this case, the molecules of bond water are in the center of the coil to form some chain providing the structure-formation. The resultant structure is in good agreement with the structural regularities proposed in [1].
3. Conclusions 1) It is shown that hydrated water in zirconium oxyhydrate gel is located in the inner region of the polymer compound in the gap between the molecules of zirconium oxide and provides the bond between monomer units. For example, with hydration of the compound (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 the water molecule cannot enter directly into the inner region; therefore, hydration of such a chain
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Fig. 3. Structure of the hydrated tetramer (ZrO2(H2O)3)2 ×/ZrO2(H2O)6 ×/ZrO2(H2O)7.
is not energetically equivalent to the reaction of addition of the hexahydrated form to the dimer. 2) Structural difficulties accompanying the hydration of trimers and other forms of oxyhydrated polymers require the restructurization of the polymer chain to transfer into a more stable state. Such a transformation includes the successive processes of destruction, hydration and further construction of the chain. The activation energy of the destruction process is comparable with the energy of hydrogen bonds; therefore, the destruction rate of the chain is rather high. Moreover, it is to increase with the increase in the number of units in the chain. 3) The pentahydrate is the most stable molecular form in the system under consideration. Therefore, it is the interaction of the pentahydrate form of oxyhydrate with the polymer chain which is the most probable. 4) The interaction of the pentahydrate with the metastable trimer (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 results in the formation of the metastable tetramer with 16 water molecules, while the stable tetramer should contain 19 water molecules. The fact confirms the theoretical scheme of oxyhydrate gel polymerization reactions described in the literature and in this work, since the splitting-off of water molecules as a result of the olation-polymerization reaction is accounted for by the formation of metastable products of such a kind.
5) The successive growth of chains and their discrete destruction provide the time periodicity of gel properties. The structural peculiarities of the polymer chain formation are of special interest. The location of monomer units in the most beneficial tetramer (ZrO2(H2O)3)2 ×/ZrO2(H2O)6 ×/ZrO2(H2O)7 resembles the onset of the helix coil growth. In this case, the molecules of bond water are in the center of the coil to form some chain providing the structure-formation.
References [1] Yu.I. Sukharev, V.A. Potyomkin, E.Z. Kurmaev, V.A. Markov, I.Yu. Apalikova, I.V. Antonenko, J. Inorg. Chem. 44 (6) (1999) 855 /863. [2] B.A. Markov, Yu.I. Sukharev, V.A. Potyomkin, V.V. Avdin, E.A. Korotkova, Math. Model. 11 (12) (1999) 77 /86. [3] Yu.I. Naberukhin, V.A. Luchnikov, G.G. Malenkov, E.A. Dgelikovskai, J. Struct. Chem. 38 (4) (1997) 723 /732. [4] P.M. Zorky, E.V. Sokolov, G.G. Malenkov, L.V. Lanshina, J. Phys. Chem. 74 (11) (2000) 1951 /1956. [5] I.V. Antonenko, Candidate thesis ‘‘Periodical properties of zirconium oxyhydrate gels’’, Chelyabinsk, 1999. [6] Yu.I. Sukharev, V.A. Potyomkin, B.A. Markov, Colloid. Surf. A 194 (2001) 75. [7] N.L. Allinger, F. Li, L. Yan, J. Comput. Chem. 11 (1990) 848. [8] N.L. Allinger, F. Li, L. Yan, J.C. Tai, J. Comput. Chem. 11 (1990) 868. [9] J.-H. Lii, N.L. Allinger, J. Comput. Chem. 19 (1998) 1001. [10] V.A. Potyomkin, E.V. Bartashevich, A.V. Belik, J. Phys. Chem. 70 (3) (1996) 448.
Y.I. Sukharev, V.A. Potyomkin / Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 197 /207 [11] A.V. Belik, V.A. Potyomkin, J. Phys. Chem. 66 (1) (1992) 140. [12] V.A. Potyomkin, E.V. Bartashevich, A.V. Belik, J. Phys. Chem. 72 (4) (1998) 650. [13] N. Metropolis, A. Rosenbluth, A. Teller, J. Chem. Phys. 21 (1953) 1087.
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[14] S. Kirkpatrick, C.D. Gelatt, M.P. Vecci, Science 220 (1983) 671. [15] Z. Li, H.A. Scheraga, Proc. Nat. Acad. Sci. USA 84 (1987) 6611. [16] D.J. Wales, M.P. Hodges, Chem. Phys. Lett. 286 (1998) 65.
Structuring elements of zirconium oxyhydrate gels under unbalanced conditions Yuri Ivanovich Sukharev a,*, Vladimir Aleksandrovich Potyomkin b,1 a b
Department of Common and Engineering Ecology, Southern Ural State University, 76 Lenin Avenue, 454080 Chelyabinsk, Russia Department of Organic and Inorganic Chemistry, Chelyabinsk State University, 70b Molodogvardeitsev Street, Chelyabinsk, Russia Received 13 June 2002; accepted 27 March 2003
Abstract Hydrated water in zirconium oxyhydrate gel is shown to be located in the inner region of the polymer compound in the gap between the molecules of zirconium oxide and to provide the bond between monomer units. For example, with the hydration of the compound (ZrO2(H2O)3)2 ×/ZrO2(H2O)5, the water molecule cannot enter directly into the inner region; therefore, the hydration of such a chain is not energetically equivalent to the reaction of addition of the hexahydrated form to the dimer. Structural difficulties accompanying the hydration of trimers and other forms of oxyhydrated polymers require the restructurization of the polymer chain to transfer into a more stable state. Such a transformation includes the successive processes of destruction, hydration and further construction of the chain. The activation energy of the destruction process is comparable with the energy of hydrogen bonds; therefore, the destruction rate of the chain is rather high. Moreover, it is to increase with the increase in the number of units in the chain. The pentahydrate is the most stable molecular form in the system under consideration. Therefore, it is the interaction of the pentahydrate form of oxyhydrate with the polymer chain, which is the most probable. The interaction of the pentahydrate with the metastable trimer (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 results in the formation of the metastable tetramer with 16 water molecules, while the stable tetramer should contain 19 water molecules. The fact confirms the theoretical scheme of oxyhydrate gel polymerization reactions described in the literature and in the present work, since the splitting-off of water molecules as a result of the olation-polymerization reaction is accounted for by the formation of this kind of metastable products. The successive growth of chains and their discrete destruction provide the time periodicity of gel properties. The structural peculiarities of the polymer chain formation are of special interest. The location of monomer units in the most beneficial tetramer (ZrO2(H2O)3)2 ×/ZrO2(H2O)6 ×/ZrO2(H2O)7 resembles the origin of the helix coil growth. In this case, the molecules of bond water are in the center of the coil to form some chain providing the structure-formation. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Model; Oxyhydrate gel; Spiral structure; Polymer; Chain-monomer unit; Zirconium; Oxyhydrate gel; Metastable trimer; Tetramer; Structure-formation
* Corresponding author. Tel.: /7-3512-399-022. E-mail address: [email protected] (Y.I. Sukharev). 1 Tel.: /7-3512-429-012. 0927-7757/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0927-7757(03)00136-5
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Fig. 1. Functional dependence of the hydration enthalpy (DHr) on the degree of hydration (n ).
1. Introduction
2. Formulation of the research problem
At present, the molecular mechanism of gel formation at the quantitative level is poorly studied. Studies are usually carried out at the level of the phenomenological description with the detailed analysis of particular properties. The objective difficulties, in this case, are as follows: the low stability and non-reproducibility of gels, the impossibility of direct structure studies, the problems with kinetic studies, since the formation of gel proceeds, as a rule, heterophasically and is known as the macrokinetic process including several kinetic and diffusion stages. The diffusion, in this case, plays an important role due to high viscosities of gel formations. At the same time, it is the understanding of the processes underlying the formation of gel systems which is the key to understand the structural regularities for the formation of the gel phase as well as the distinction of gel systems from liquid, amorphous and crystal states. Therefore, the simultaneous experimental and theoretical study of the regularities of gel phase-formation are of great importance.
In previous works [1,2], certain periodical regularities, both of structural and time nature, were found for oxyhydrate gels of heavy metals. As regards the structural nature, there was found the periodicity of the density variation (mass differentiation of gel) accounted for by the anisotropy of the field of forces of the gel-forming particles, which provides their ordering manifested later on at the macroscopic level. The problem of time periodicity, i.e. the periodical change-in-time of a number of gel properties, seems to be more complicated. This is due to the fact that a gel is a ‘‘living’’ system where some restructurization continuously proceeds, i.e. that of gel polymer chains. Examples of such a restructurization may be either the breakdown of a polymer chain followed by the growth of its fragments, or the breakdown of chains followed by their recombination. This phenomenon is similar to the dynamics of liquids in which the constant formation, destruction and restructuring of associates are observed. In liquids, however, the above processes
Y.I. Sukharev, V.A. Potyomkin / Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 197 /207
proceed very quickly due to the low degree of association and low viscosity. At the same time, high structurization of gel formations makes these systems similar to liquid-crystal ones. The growth of the polymer chain of heavy metals oxyhydrates being the structure elements of gels proceeds by the following scheme [1]:
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where (/R /) stands for a polymer fragment that is equally capable of entering the reactions of olation chain propagation (reactions 1 and 6), degradation (reaction Eq. (3)), and chain termination (polycondensation). The analytical description is complicated by a great number of components taking part in the
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above kinetic stages of the process, e.g. oxyhydrate forms of metals with various degrees of polymerization, water, protons, hydroxyl anions. Besides kinetic stages, the diffusion processes of carrying the components to the ends of the growing chains as well as those of removing the products of interaction from the reaction zone play an important role in the viscous gel system. The diffusion of chain oxyhydrate units is of special importance, since their mobility is much lower than the diffusion of water molecules, hydrated protons and hydroxyl anions. Thus, to analyze the growth process of polymer chains of heavy metal oxyhydrates, it is reasonable to consider the following kinetic processes of growth and destruction: addition and elimination of a monomer fragment; hydration and dehydration; protonation and deprotonation, addition and splitting out of a hydroxyl anion. Kinetic and thermodynamic peculiarities of the processes under consideration appear to differ depending on the chain length; therefore, the analysis of these stages should be carried out for polymer chains with various degrees of polymerization. 2.1. Calculation techniques To study the mechanism of growth for polymer chains of oxyhydrate gels, the zirconium oxyhydrate system was chosen as being more thoroughly investigated, both theoretically [1,3,4] and experimentally [5,6]. The ways of polymer chain formation are largely determined by the possibility of the location of the next monomer unit at the ends of the chain as well as by the energy of interaction. In this case, the description by the Mayo /Lewis equation which is traditionally applied to determine the composition of organic copolymers is impossible due to the essential difference in the energies of interaction of each of these forms with the growing polymer chain. Besides, this equation makes it possible to determine the polymer composition, but does not enable to establish the mechanism of the chain growth. The calculation of the optimal location of a monomer unit in the vicinity of a growing chain may be performed in the framework of a number of models (e.g. either
quantum-chemically, or by the methods of molecular mechanics [7 /9]). In gels, however, the associates of such a kind are dynamic formations with conformation mobility. Their composition and structure vary and as a matter of fact these associates are the probabilistic values. Therefore, to determine the composition and structure of the associates in gels by the methods of molecular mechanics or quantum chemistry requires the calculation of a great number of possible structures regarding the probability of their existence. The problem in itself is very complicated. Thus, to determine the local structures of the ‘‘polymer chain-monomer unit’’ type, the probabilistic approach described in [10] is used. With this method, the probabilities pij of atom /atom contacts are the basic model representation of the molecular surroundings. The calculations are based on the notion of a polymer chain and a monomer as a set of overlapping atomic spheres whose radii are functionally dependent on the set of valence and non-valence interactions and temperature. This is realized in the framework of DENSON and MERA models [10 /12]. The calculation of probable structure configurations of the ‘‘polymer chain-monomer unit’’ type is performed in the framework of the MERA model [12] (for the case of solutions), in which the calculation of atom /atom probabilities pij of contacts of molecules for all system components is performed. The probabilities pij for the contact of the atom i of a given molecule with the atom j of an adjacent one are determined as follows:
pij
nij xi xj Si exp( Eij =kT) P 1 N k 1 exp( Eik =kT)
(1)
where xi , xj are the mole fractions of the solution components, atoms i and j included, nij the maximum number of atoms j of an adjacent molecule which may be located in the contact with the atom i of a given molecule, si the fraction of the free surface of the atom i, Eij the potential energy of interaction of atoms i and j , k the Boltzmann constant, T the absolute temperature and N the number of atoms in the system (in a single molecule or in a solvate complex).
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The interaction energy of the atom i of the molecule being considered with the atom j of the adjacent one is written using the Lennard-Jones potential regarding the Coulomb interaction between atoms: qq 14kRaij Raij 6 7kRaij Raij 12 Eij i j (2) 4po 0 Rij 48a Rij 48a Rij where qi and qj are the charges on atoms i and j, Rij the minimum distance between atoms i and j calculated as the sum of minimum distances from atoms i and j to the molecule surface and o0 the dielectric permittivity of the vacuum. Since a monomer unit and a polymer chain approach each other for a distance of contact, it is reasonable to determine the atom of the monomer unit and that of the polymer chain, taking into account the maximum probability of their contact. A great number of variants for the relative position of the chain and the monomer unit at the distance of the atoms contact appear to exist, thus, the position corresponding to the energy minimum is chosen. To avoid entering into the local minimum of the potential energy, the MonteCarlo method with the local optimization [13 /16] in the framework of the semiempirical quantumchemical ZINDO/1 method is used. 2.2. Discussion of the calculation results The monomer unit of the unbalanced polymer chain of such a system may be presented by different zirconium oxyhydrate forms having the general formula ZrO2(H2O)n . The detailed analysis of possible monomer units in the framework of the semiempirical quantum-chemical ZINDO/1 method showed that the functional dependence of the hydration enthalpy DHr on the degree of hydration n is of the extreme nature, which is shown in Fig. 1. The functional dependence obtained does not contradict to the experimental data. Zirconium acid ZrO(OH)2 (or ZrO2(H2O)), or its hydrated form ZrO(OH)2 ×/H2O (or ZrO2(H2O)2) are usually considered to be the structural basis for gel polymer formations of zirconium oxyhydrate. In the compounds mentioned water is valence-bound. The addition of the
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next water molecule occurs in the coordination sphere. The inception of the polymer chain may represent the interaction of the pair of monomer units, in which case one should consider three variants of interaction: 2ZrO2 (H2 O) 0 (ZrO2 (H2 O))2 ;
(I:1)
DH 105:89 ZrO2 (H2 O)2 ZrO2 (H2 O)
(I:2)
0 ZrO2 (H2 O)2 ×ZrO2 (H2 O); DH 58:75 2ZrO2 (H2 O)2 0 (ZrO2 (H2 O)2 )2 ;
(I:3)
DH 17:01 ZrO2 (H2 O)2 ZrO2 (H2 O)3
(I:4)
0 ZrO2 (H2 O)2 ×ZrO2 (H2 O)3 ; DH 49:01 2ZrO2 (H2 O)3 0 (ZrO2 (H2 O)3 )2 ;
(I:5)
DH 81:90 ZrO2 (H2 O)3 ZrO2 (H2 O)4
(I:6)
0 ZrO2 (H2 O)3 ×ZrO2 (H2 O)4 ; DH 118:50 2ZrO2 (H2 O)4 0 (ZrO2 (H2 O)4 )2 ;
(I:7)
DH 121:62
Hydration competes with the above processes. The analysis of hydration processes shows that the most beneficial penta- and dehydrated forms of the incipient polymers are bound to have the highest concentration content in the solution. The minimum possible amount of water added
Table 1 Enthalpies of the dimerization process (chain growth DH) and hydration of monomer units (DHH) No. DH (kcal mol 1)
DHH (kcal mol 1)
DH /DHH (kcal mol 1)
(I.1) /105.89 (I.2) /58.75 (I.3) /17.01 (I.4) /49.01 (I.5) /81.90 (I.6) /118.50 (I.7) /121.62
/173.54 /124.26 /75.98 /59.43 /43.88 /121.63 /199.38
67.65 65.51 58.97 10.42 /38.02 3.13 77.76
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results in the formation of hepta-, tetra- and trihydrated forms. Energetically, the dimerization enthalpy of the first process is characterized as DH1 / /105.89 kcal mol 1; nevertheless, the process of monomer units hydration is more beneficial, since 2DHH / /173.54. Table 1 presents enthalpies of the competing processes of chain growth and hydration of the initial units. The data from Table 1 illustrates that the dimerization process is thermodynamically beneficial with dimerization of trihydrated forms of ZrO2(H2O)3. As for the other cases, it would be more beneficial, if zirconium dioxide were present in the solution in the form of hydrated compounds. At the same time, the trihydrated form is one of the least beneficial. Penta- and dihydrated forms of compounds are certain to be more beneficial. Dimerization of these forms, however, is not beneficial, which is quite evident, since it is the unstable oxyhydrate forms which must tend to the dimerization process. Since the energetically beneficial dimerization process is possible only from the forms, whose content in the solution is low, it is dimerization that must constraint the inception and growth of the polymer chain. The further growth of the polymer chain is provided by a number of successive processes: (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)2 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)2 ;
(II:1)
DH 80:99 (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)3 0 (ZrO2 (H2 O)3 )3 ; DH 85:35 (II:2) (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)4 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)4 ;
(II:3)
DH 180:84 (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)5 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)5 ;
(II:4)
DH 123:22 (ZrO2 (H2 O)3 )2 ZrO2 (H2 O)6 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ; DH 108:50
(II:5)
(ZrO2 (H2 O)3 )2 ZrO2 (H2 O)7 (II:6)
0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)7 ; DH 111:70
With these reaction as well as with dimerization, the hydration process may be competitive to that of the trimer growth. Chain hydration follows the equation: (ZrO2 (H2 O)3 )2 H2 O 0 ZrO2 (H2 O)3 ×ZrO2 (H2 O)4 ; DH 58:54 Enthalpies of the monomer hydration were given before. Enthalpies of the competing processes of the dimer chain growth and total hydration of monomers and dimers are presented in Table 2. It is quite obvious that the trimerization process competes with the hydration in interacting of (ZrO2(H2O)3)2 with monomer units, from a trihydrate to hexahydrate, i.e. from ZrO2(H2O)3 to ZrO2(H2O)6. In the other cases, it is more energetically beneficial both for a dimer and a monomer to be present in the solution as individual hydrated compounds. The addition of hexa- and tetrahydrated forms of a monomer to a dimer is considered to be the most energetically beneficial. However, tetrahydrated and hexahydrated oxyhydrate forms are not the most probable formations in the solution. In this case, the addition of the dominating hydrated form, i.e. pentahydrate (reaction (II.4)), is thermodynamically the most probable process, but results in the formation of the metastable Table 2 Enthalpies of trimerization (chain growth DH ) and total hydration of a dimer and monomer units (DHH) No.
DH (kcal mol 1)
DHH (kcal mol 1)
DH /DHH (kcal mol 1)
(II.1) (II.2) (II.3) (II.4) (II.5) (II.6)
/80.99 /85.35 /180.84 /123.22 /108.50 /111.70
/96.03 /80.48 /158.23 /116.62 /79.82 /115.13
15.04 /4.87 /22.61 /6.60 /28.68 3.43
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product (ZrO2(H2O)3)2 ×/ZrO2(H2O)5, i.e. not the most probable. This compound could be assumed to add water with the formation of a more energetically beneficial compound (ZrO2(H2O)3)2 ×/ ZrO2(H2O)6. In fact this is not the case, which is accounted for by the structural peculiarities of and trimers (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 (ZrO2(H2O)3)2 ×/ZrO2(H2O)6. Hydrated water in the above compounds is in the inner region between the molecules of zirconium oxide and provides the bond between monomer units (Fig. 2). With hydration of the compound (ZrO2(H2O)3)2 ×/ZrO2(H2O)5, the water molecule cannot enter directly into the inner region; therefore, hydration of such a chain is not energetically equivalent to the reaction of
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addition of a hexahydrated form to a dimer. Such complicated hydration requires isomerization, including the opening of the inner region, the entering of a water molecule into it, followed by the closing of that region. Since the process is rather complicated, it is unlikely. In addition to the reasons mentioned, this reaction requires the maximum energy to open the inner region with the breaking of Zr /O /Zr bridge bonds. The most probable process is that including the destruction of the trimer (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 with hydration of a dimer and pentahydrated form of a monomer. Such a process requires only 6.60 kcal mol 1 (reaction is reversible to (II.4)). Further, there will take place the addition of the resultant hexahy-
Fig. 2. Structures of hydrated trimers (a) (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 and (b) (ZrO2(H2O)3)2 ×/ZrO2(H2O)6.
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drate to a dimer with the release of 28.68 kcal mol 1 (process (II.5)). In this case, the destruction enthalpy of (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 may be considered as the activation energy for the formation of a more beneficial chain of (ZrO2(H2O)3)2 ×/ ZrO2(H2O)6. Such a value of the energy barrier (6.60 kcal mol 1) is small and is comparable with the energy of diffusion activation. With the further growth of the oxyhydrate chain from the most beneficial trimer (ZrO2(H2O)3)2 ×/ ZrO2(H2O)6, the interaction processes of the trimer with various hydrated forms of monomer units may continue: (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 ;
(III:1)
DH 139:51 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)
(ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)8 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)8 ; DH 153:02 (III:9) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)9 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)9 ; DH 207:72 (III:10) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)10 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)10 ; DH 143:08 (III:11) The competing process of chain hydration proceeds by the following equation: (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 H2 O
0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O); DH 180:34 (III:2) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)2 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)2 ; DH 146:32 (III:3) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)3 0 (ZrO2 (H2 O)3 )3 ×ZrO2 (H2 O)6 ;
0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)7 ; DH 24:48 Enthalpies of monomer hydration were given before. Enthalpies of the competing processes of the dimer chain growth as well as of the total reaction of monomers and dimers hydration are presented in Table 3. It is quite obvious that the process of adding the dominating pentahydrated form to the growing chain, except for the first stage of dimer formation,
DH 168:68 (III:4) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)4 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)4 ; DH 190:66 (III:5) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)5
Table 3 Enthalpies of tetramerization (chain growth DH ) and total hydration of dimers and monomers (DHH) No.
DH (kcal mol 1)
DHH (kcal mol 1)
DH /DHH (kcal mol 1)
(III.1) (III.2) (III.3) (III.4) (III.5) (III.6) (III.7) (III.8) (III.9) (III.10) (III.11)
/139.51 /180.34 /146.32 /168.68 /190.66 /134.01 /152.58 /371.28 /153.02 /207.72 /143.08
/76.84 /111.25 /61.97 /46.42 /124.17 /82.56 /45.76 /81.07 /68.06 /94.28 /70.42
/62.67 /69.09 /84.35 /122.26 /66.49 /51.45 /106.82 /290.21 /84.96 /113.44 /72.66
0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)5 ; DH 134:01 (III:6) (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)6 0 (ZrO2 (H2 O)3 )2 ×(ZrO2 (H2 O)6 )2 ;
(III:7)
DH 152:58 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ZrO2 (H2 O)7 0 (ZrO2 (H2 O)3 )2 ×ZrO2 (H2 O)6 ×ZrO2 (H2 O)7 ; DH 371:28 (III:8)
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is thermodynamically beneficial, but results in metastable forms. Structural difficulties accompanying hydration of the latter require the restructurization of the chain to transfer into a more stable state. Such a transformation includes the processes of destruction, hydration, to be followed by the further construction of the chain. The chain destruction requires the activation energy, but the latter is comparable with the energy of hydrogen bonds. Thus, the rate of the chain destruction is quite high and is to increase with the increase in the number of units. The chain growth is thermodynamically beneficial and is determined only by the rate of monomer units addition, the diffusion in the gel phase being rather slow process. At some instant of time, the state of quasi-equilibrium comes (the empirical equilibrium) between the processes of monomer units addition and destruction. Since the destruction is the process of the first order [1,2] and the diffusion is determined by the Fick law, the following equation may be written in the simplified form, regarding the kinetics of destruction up to monomer units: kd aC KD
@ 2 (1 a)C 0 @x2
(3)
where kd is the constant of the destruction rate, C the concentration of monomer units in the solution, a the degree of transformation, KD the constant of the diffusion rate and x the diffusion coordinate. The following expression is the solution for this equation: aC A exp(ivx)
(4)
where v is the wavenumber of monomer concentration vibrations in coordinate x . This expression suggests the structural periodicity of the gel observed by the authors in works [1,2]. Besides, it should be noted that the pentahydrate is the most stable form of particles in the system. Therefore, it is the interaction of the pentahydrate form with the chain which is the most probable. The chain being formed in this case, however, is metastable. So, the collision of the pentahydrate with the dimer (ZrO2(H2O)3)2 is bound to result in the trimer with the total number
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of water molecules being 11. As regards the stable form, it contains 12 water molecules and is thermodynamically more beneficial. The further interaction of the pentahydrate with the metastable trimer (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 results in the formation of the metastable tetramer with 16 water molecules, while the stable tetramer should contain 19 water molecules. In this case, the gain in energy is 223.72 kcal mol 1. Thus, the high kinetic probability of addition of the pentahydrate zirconium oxide results in the formation of thermodynamically metastable polymer chains. With the increase in the chain length its instability increases, which results in the drastic increase in the probability of the metastable state relaxation, i.e. the chain destruction. With a particular length, the polymer chain is bound to break down. Two resultant fragments are more stable and may proceed to grow. In case the solution lacks monomers, the chain growth may proceed at the expense of the isothermic transformation of less stable chains into more stable ones. Thus, the successive growth of chains and their discrete destruction provide time periodicity of gel properties. The structural peculiarities of the polymer chain formation are of special interest as well. The location of monomer units in the most beneficial tetramer (ZrO2(H2O)3)2 ×/ZrO2(H2O)6 ×/ZrO2(H2O)7 presented in Fig. 3 resembles the origin of the helix coil. In this case, the molecules of bond water are in the center of the coil to form some chain providing the structure-formation. The resultant structure is in good agreement with the structural regularities proposed in [1].
3. Conclusions 1) It is shown that hydrated water in zirconium oxyhydrate gel is located in the inner region of the polymer compound in the gap between the molecules of zirconium oxide and provides the bond between monomer units. For example, with hydration of the compound (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 the water molecule cannot enter directly into the inner region; therefore, hydration of such a chain
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Fig. 3. Structure of the hydrated tetramer (ZrO2(H2O)3)2 ×/ZrO2(H2O)6 ×/ZrO2(H2O)7.
is not energetically equivalent to the reaction of addition of the hexahydrated form to the dimer. 2) Structural difficulties accompanying the hydration of trimers and other forms of oxyhydrated polymers require the restructurization of the polymer chain to transfer into a more stable state. Such a transformation includes the successive processes of destruction, hydration and further construction of the chain. The activation energy of the destruction process is comparable with the energy of hydrogen bonds; therefore, the destruction rate of the chain is rather high. Moreover, it is to increase with the increase in the number of units in the chain. 3) The pentahydrate is the most stable molecular form in the system under consideration. Therefore, it is the interaction of the pentahydrate form of oxyhydrate with the polymer chain which is the most probable. 4) The interaction of the pentahydrate with the metastable trimer (ZrO2(H2O)3)2 ×/ZrO2(H2O)5 results in the formation of the metastable tetramer with 16 water molecules, while the stable tetramer should contain 19 water molecules. The fact confirms the theoretical scheme of oxyhydrate gel polymerization reactions described in the literature and in this work, since the splitting-off of water molecules as a result of the olation-polymerization reaction is accounted for by the formation of metastable products of such a kind.
5) The successive growth of chains and their discrete destruction provide the time periodicity of gel properties. The structural peculiarities of the polymer chain formation are of special interest. The location of monomer units in the most beneficial tetramer (ZrO2(H2O)3)2 ×/ZrO2(H2O)6 ×/ZrO2(H2O)7 resembles the onset of the helix coil growth. In this case, the molecules of bond water are in the center of the coil to form some chain providing the structure-formation.
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