Model description of the H–bonded transuranium complexes

Figure 1. The schematic structure of polymer complex of uranile. The first part of the work intends to propose a simple and reliable quantum statistical model for describing the actinide hydroxocomplex formation in aqueous solutions investigated in the range of experiments (see e.g. [1]–[4]), to make semiempirical estimation of complex configurational energies based on the experimental data and to investigate an effect of saturation in the process of complex formation for the solution with high concentration of metal ions. The second part of the work is devoted to an investigation of polymer complexes of uranile UO2[(OH)2UO2] 2+ n with sheet–like double bridges of ionic groups OH (Figure 1) [5]. Objects of such a type are formed under the influence of hydrolysis and radiolysis of electrolytic aqueous solutions. Our main purpose is to describe the thermodynamics of the polymerization process from the microscopic point of view allowing for the quantum–statistical character of distribution of protons on hydrogen bonds.


Introduction
Figure 1.The schematic structure of polymer complex of uranile.
The first part of the work intends to propose a simple and reliable quantum statistical model for describing the actinide hydroxocomplex formation in aqueous solutions investigated in the range of experiments (see e.g.[1]-[4]), to make semiempirical estimation of complex configurational energies based on the experimental data and to investigate an effect of saturation in the process of complex formation for the solution with high concentration of metal ions.
The second part of the work is devoted to an investigation of polymer complexes of uranile UO 2 [(OH) 2 UO 2 ] 2+ n with sheet-like double bridges of ionic groups OH − (Figure 1) [5].Objects of such a type are formed under the influence of hydrolysis and radiolysis of electrolytic aqueous solutions.Our main purpose is to describe the thermodynamics of the polymerization process from the microscopic point of view allowing for the quantum-statistical character of distribution of protons on hydrogen bonds.

Configurational model of metal hydroxocomplexes
Main thermodynamic properties of the subsystem of metal ion interacting with ligand cations can be obtained based on a simple Hamiltonian Ĥ where p=0, . . ., p max , i=1, . . ., N M , X pp i is the projection operator for the complex i onto the configuration p with the energy λ p and n p ligands bounded to metal ion, µ is the chemical potential of ligands, p max is the maximal number of possible configurations for the complex and N M is the full number of metal ions.The average values of projection operators (i.e.complex formation probabilities) are equal to where ∆ p =λ 0 −λ p .If there is only one state with the given number of ligands n p =p, the mean value X pp correspond to the partial mole fraction of complexes with p ligands.Average number of ligands per complex n L can be expressed as n L = p n p X pp .Thus the number of ligands bounded to metal ions is N L =n L N M .Corresponding concentrations are to be defined as C M =N M /N, C L =N Lsol /N and C Ltot =N Ltot /N, where N=N M +N L +N solv is the total number of particles and N solv is the number of solvent molecules.
For small concentration of metal ions the expression for chemical potential µ could include only ideal gas and electrostatic interaction (in Debye-Hückel approximation) terms: βµ=βψ+ ln C L , where the variable ψ depends on: the mass of the ligand particle, pressure (here the atmospheric one), the ionic strength and the dielectric constant of the solution.Partial mole fraction is equal to corresponding complex formation probability and comparing their definitions one can see that ∆ p =β −1 ln B p −pψ, where B p is the formation constant of the complex ML p .By means of this expression the following values of energy parameters ∆ p have been calculated here based on the data of forming the constant measurement originated from the papers cited below: for Pu 4+ they are ∆ [1], while data from [2] give the values 9600, 19000, 27300 and 34200 cm −1 correspondingly; for Th 4+ they are 8860, 17300, 25700 and 34000 cm −1 [2]; for Am 3+ -7000, 14000 and 21000 cm −1 [3]; for UO 2+ 2 -7300, 14000 and 21000 cm −1 [2]; for PuO 2+ 2 -8200, 16000 and 21000 cm −1 [2]; for NpO 2+  2 -8000, 15000 and 20000 cm −1 [4].One can consider two main cases for such a model: 1.Fixed concentration of ligands in solution (fixed pH value).Regions of pH values, where particular forms of hydroxocomplexes exist, depend significantly on values of the ∆ p set (compare Figure 2a and 2b).Chemical experiments are usually done at so called normal conditions at temperature T=25 • C.But as one can see in Figure 2c variation of temperature has a similar effect as variation of ∆ p .

Saturation effect (fixed N Ltot value).
At fixed total number of ligands concentrations should satisfy the equation C Ltot =n L C M +C L , which can be solved numerically.Saturation effect manifests itself by sharp decrease of the formation probability value of highly occupied forms of the hydroxocomplex and increase of the bare metal ion fraction with increase of concentration C M (Figure 3a).When the concentration of metal ions C M increases to the threshold value C ′ M =p max C Ltot the concentration of ligands C L rapidly decreases from its saturation value C Ltot (Figure 3b).The threshold concentration C ′ M depends on the total concentration of ligands C Ltot (Figure 3c).
For fixed concentration C Ltot at the concentration C M below the threshold value lowering of temperature leads to a decrease of bare metal ion fraction from 1 to 0, as well as to the appearance of intermediate complex configurations and, finally, to the domination of fully occupied complexes at T→0 (Figure 4a).The concentration of ligands in solution C L equals to C Ltot at T→∞ and C Ltot −p max C M at T→0 (Figure 4d) and the corresponding values of n L are 0 and p max (Figure 4e).For C M slightly above the threshold, behaviour of partial mole fractions changes significantly (Figure 4b).In this case at T→0 the concentration C L →0 and the average number of ligands is n L =C Ltot /C M (Figure 4d and 4e).For large C M fully occupied complexes are not practically present and bare ions dominate in solution (Figure 4c).

Statistical model of double-bounded polymer chain
The model of a molecular complex, where pairs of hydrogen bonds join in series of ionic groups, is presented in Figure 1.A case is considered, when intermediate ionic groups and groups (A) and (B) at the ends of a polymer are the same regarding their properties.
describes the short-range configurational interaction, and is an electrostatic dipole-dipole long-range part (here m is the number of intermediate ionic groups).The components of {B pq } matrix are expressed in terms of the energies of proton configurations in a potential minima near ionic groups (Figure 5).By the terms ĤA 1 = p α Ap Xpp 1 and ĤB m+1 = p α Bp Xpp m+1 the boundary effects are described; the α A,B p coefficients take certain values of configuration energies from the set of a possible one (Figure 5) depending on the number of protons joined outside to the boundary ionic groups; Ze is the effective charge of proton, a is the distance between potential minima on the hydrogen bond and c is the distance between centres of ionic groups.

Summary
The configurational model is proposed for describing the H-bonded transuranium complexes in aqueous solutions, which gives a correct dependence of partial mole fractions of actinide hydroxocomplexes on the solution pH, temperature, etc.At high metal ion concentration a saturation effect can take place, which leads to the rapid change of the ligand concentration in the solution.If the metal ion concentration crosses the threshold value, suppression of complexes with high number of ligands at low temperature takes place due to exhaustion of ligands in solution.
For the proposed model of polymer chain, the dependence of free energy per one ionic group on the number m ′ of groups is investigated.In some cases the nonmonotonous character of dependencies f (m ′ ) on m ′ is discovered (see, for example, bends (Figure 6b) or minimum (Figure 6c) of f (m ′ ) function).In almost all boundary conditions we observe the decrease of the free energy per one ionic group with the increasing number of ionic group m ′ .This gives an evidence that the formation of a polymer chain structure is thermodynamically advantageous.
This work was supported by the INTAS-Ukraine-95-0133 project.The authors are pleased to express their gratitude to the INTAS.