Studies On Nonequilibrium Physico-Chemical Processes In The System "fuel Containing Masses -- Aqueous Solutions Of Radioactive Elements"

The main forms of nuclear fuel remaining inside the \Shelter" object and the main interrelated factors of nuclear and ecological danger are considered. Processes of interaction of fuel containing masses with water are analysed on the basis of experimental data. A statistical model for the description of radioactive elements is proposed. The pair structure distribution functions for ions UO 2+ 2 , Cs + and Sr 2+ in aqueous solutions use this model. Chemical reactions of complex formations with the participation of UO 2+ 2 and PuO 2+ 2 as well as reactions of radiolysis in aqueous solutions of radioactive elements are analysed. Nonuniform equations for the description of UO 2+ 2 , Cs + and Sr 2+ ions diiusion from glassy-like fuel containing masses into water and equations of chemical kinetics of radiolysis processes are obtained as well. The shear viscosity and mutual diiusion coeecients of ions UO 2+ 2 , PuO 2+ 2 , Cs + and Sr 2+ in aqueous solutions are calculated numerically for concentrations which are typical of the \Shelter" object.


Introduction
The investigation of ecological problems of the Chornobyl zone was renewed in 1994 and aimed at active elements migration in soils, ground water near the places of temporary location of active wastes (PTLAW).The main problem was to investigate physical and mathematical models of active elements migration in PTLAW taking into account ion exchange, hydration, mutual di usion mechanisms of interaction with water and ground corrosion including porous, acidic and adsorption properties.The model, coupled with GIS technologies for PTLAW, was expected to predict PTLAW active elements migration and enable one to work out practical ways of solving these problems.
The subject of early studies was unique water solutions containing uranium (the whole isotopic content), plutonium, americium, curium, cesium, strontium and others.One cannot create them in laboratory conditions.c I.R.Yukhnovskii, M.V.Tokarchuk, V.V.Ignatyuk, A.E.Kobryn, I.P.Omelyan, R.I.Zhelem, G.S.Dmytriv, O.L. Ivankiv, 1997 The theoretical description of physical and chemical processes in these speci c objects has not been done yet.Besides, a great interest arises as to aqueous solutions of this kind in view of unpredictable consequences of water interaction with fuel containing masses (FCM) inside the \Shelter" object.
We have made a review 1,2] of the published data on the state of FCM-{ nuclear magma in the \Shelter" object and considered the main interaction processes of FCM with water.The conclusion is as follows: water by means of complex physical and chemical impacts destroys glassy FCM that results in an uncontrolled out ow of uranium, plutonium, cesium, strontium from the lakes at the \Shelter" object.
First of all, a survey of archive materials, reports of the Interbranch Scienti c and Technical Center (ISTC) \Shelter"of the National Academy of Sciences of Ukraine concerning FCM state, their chemical composition, ,and -radiation, neutron sources and currents, as well as water availability in the \Shelter" object has been done.Generalizations and a sophisticated analysis of the published materials, archive documents and reports  have led us to the conclusion that the nuclear, physical and chemical state of fuel in the destroyed nuclear reactor of the 4th power block has remained the main problem at the Chornobyl nuclear power plant during 11 years.The rst and the most important question is the amount of nuclear fuel which was in the destroyed reactor immediately after the accident.There is no clear answer to it so far.The second important question is related to the FCM-water interaction inside the \Shelter" object.These questions arise in almost all the conclusions of "Technical substantiation of the nuclear safety of the \Shelter" object " 3].Investigation on the results of such an interaction and the corresponding processes which are expected in the near future will be the line of further activity of the scienti c group at the Institute of Condensed Matter Physics (ICMP) of the National Academy of Sciences of Ukraine.
Kinds of the nuclear fuel and connected with it factors of nuclear and ecological safety of the \Shelter" object were ascertained 26,27].
By April 26, 1986 nuclear fuel with the uranium content 231.5 tons had been distributed in the 4-th unit of the Chornobyl nuclear power plant while the reactor mine contained 190. 3

tons (215 tons of UO 2 ).
There was almost 700 kilograms of plutonium in the reactor core ( 239 Pu 420 kg, 240 Pu 175 kg, 241 Pu 50 kg, 242 Pu 15 kg).In accordance with o cial MAGATE reports 4-8], the \Shelter" object contains 96% of nuclear fuel.11-year research conducted by the Complex Expedition from the Kurchatov Institute of Atomic Energy of the Russian Academy of Sciences and after 1992 by the ISTC \Shelter" has led to the conclusion that the nuclear reactor mine is, in fact, empty.Nuclear fuel of various kinds is in the central hall (core fragments), in rooms of the steam distributive corridor, underequipment apartments (304/2, 305/2), in the bubbling pond (ground and rst oors).Lava-like masses containing fuel will be called nuclear magma.
Calorimetric measurements show that the nuclear magma contains 135 30 tons of UO 2 [8][9][10], in 1994 additional measurements showed 70 90 tons 11,12].Direct measurements 13, 16] found 23.8 4.5 tons of UO 2 .Thus, summarizing the data in 8-12], [13][14][15][16] and estimating the nuclear fuel amount in the central hall from 15 to 40 tons, we can see that more than 100 tons of nuclear fuel have been out of control for 11 years.Even allowing for the accuracy correction of the both measurement procedures, the problem remains unsolved.
At present there are the following main kinds of nuclear fuel remaining in the \Shelter" object after the accident at the Chornobyl nuclear power plant: uranium dioxide UO 2 inside the fuel elements and their fragments { the most dangerous kind, presumably located in the central hall (thousands R ontgens per hour).It has not been investigated practically for 11 years because of the complicated radiative situation and technical state.The mathematical modelling 17,18] of an interaction between core fragments and water for a certain volume in the central hall displayed that an e ective neutron multiplication factor was able to exceed unity that is a supercritical state; nuclear dust { hot nuclear particles, nearly 15 tons of uranium content 10]; lava-like nuclear magma { formation that arose after cooling the molten mixture of nuclear fuel and lling materials (dolomite, lead, sand, clay, combinations with boron, etc.): 8,9, [12][13][14][15] { brown ceramics { brown glass-like mass with the average density from 1.6 to 3.15 g/cm 3 and 10 2% nuclear fuel content having burn-up fraction 12.6 0.4 (MW day/kg of U); { black ceramics { black glass-like mass with the average density from 2.0 0.2 to 2.9 g/cm 3 and 5 1% nuclear fuel content having burn-up fraction 12.5 0.5 (MW day/kg of U); { slag and slag-like granular nuclear magma; { pemza-like nuclear magma with the average density 0.14 0.18 g/cm 3 { produced by an interaction between the molten nuclear fuel and water.The data generalization of radiological properties of radionuclides and characteristics of various nuclear magma samples taking into account the information on the percent content of isotopes 234 U, 235 U, 236 U, 238 U is given in 10,11,27].aqueous solutions of uranium salts formed as a result of the interaction of nuclear magma with natural and technical water.About 2000 m 3 is located in the machine hall, but only about 1000 m 3 is under control.There are near 3000 m 3 of water inside the premises of the \Shelter" object.The content of boron and gadolinium salts as neutron absorbers is not under control in the reservoirs.In 1994 the upper limit of enriched uranium masses in the water of the lower apartments of the \Shelter" object reached several kilograms 10].The generalized data on the radionuclide content in the water within the \Shelter" object are presented in many archive materials and reports 10,26 The mentioned above kinds of nuclear fuel in the \Shelter" object ensure its nuclear and ecological safety to a great extent.For this reason, the basic factors of nuclear and ecological danger at the \Shelter" object are: nuclear transformations in the core fragments of lavas at the interaction with water (the central hall was not investigated).The contribution of ( ; n) 20,21] reactions is important because they generate almost 50% of neutrons in the nuclear magma, -transition of 241 Pu into 241 Am is an intensive -particle source 20,21,29,30]; the lava-like solid amorphous state of the nuclear magma transforms into nuclear dust with a gradually decreasing particle size converting into a ne grained fraction 2,10]; the probability of nuclear dust ejections increases as a result of the disruption under the continuing interior construction-borne radioactivity at the \Shelter" object; the mobility of the nuclear fuel ne grained fractions grows essentially due to the interaction with water penetrating inside the \Shelter" object through holes, cracks in the roof, walls and its condensation from the air onto internal walls of the \Shelter" object; the interaction between water and lava-like amorphous medium leads to a gradual leaching of radioactive elements which form hydrated complexes in aqueous solutions and may display colloid and polimeric properties 2,31,32]; the interaction of nuclear magma with water under -, -decays and -radiation results in water radiolysis.Products of the water decomposition (H 2 O 2 ), radicals (OH, HO 2 ) take an active part in the uranium dioxide hydration, while atomic hydrogen H, OH groups a ect the magma fragility and disintegration 2, [30][31][32].The interplay of these processes brings about a rapid increase of the uranium concentration in water: 1991 { 4900 ( g of U)/l, 1992 { 23000 ( g of U)/l, 1993 { 14000 ( g of U)/l, 1994 { 18000 ( g of U)/l, 1995 { 14000 ( g of U)/l; if the nuclear fuel in the lava-like amorphous medium is in a subcritical state, its accumulation in water inside the \Shelter" object due to an interaction with water may result in the formation of polymeric structures and increasing the active elements concentration after the swelling of these structures and sedimentation together with all the fragments of the fuel core.Then the probability of appearing a supercritical mass and a local self-supported chain reaction increases; radioactive elements can penetrate from the \block" water region into the ground waters and travel outside the \Shelter" object.Analysis of the water from the boreholes of the industrial part of the \Shelter" object reveals that the concentration of strontium and cesium increases; tritium oxide in the reservoirs of the \Shelter" object.So, the water factor in the \Shelter" object a ects its safety essentially if the lling materials in the destroyed reactor have promoted an intensive decomposition of fuel particles and damping nuclear ssion reactions.The nuclear magma pulverization and its interaction with water proceed in the opposite direction and may cause a nuclear fuel concentrating.
An essential trouble is caused by the state of core fragments, their isolated elements and conglomeration being observed after the accident near the broken 4-th unit.The core fragments were thrown out by the explosion onto the ventilation tube areas, the roof of the 3-rd unit machine hall, however, an evidently greater part remains in the central hall.When managing with the accident consequences, the core fragments were thrown into the central hall from the roof of the object as well.
In the above context one has to pay a particular attention to the results of the reports of 17,18].In 17] there are numerical calculations of the e ective neutron multiplication factor for the system modelling conglomerations of reactor's active core fragments in various apartments of the \Shelter" object.The calculations have shown that K ef reaches the value of 0.61 for fragment conglomerations in drum separators with the consideration of structural heterogeneity, the value of 0.88 on the scheme \E" and 0.97 for conglemerations in the central hall.In 25], displacements of the building construction elements of the \Shelter" able to vary the geometry of FCM are analysed.Those are shown to be most dangerous that can form nuclear magma of a spherical form.The ruin of the scheme \E" and fall on conglomerations of active core fragments in the central hall and their uni cation with the nuclear magma in apartment 305/2 were observed.In the case of ood with 0.2 m deep water 17] the situation can originate a self-sustaining nuclear chain reaction.Computer simulations 17] point out that the water ood of nuclear magma and active core fragments with the increasing percent content of UO 2 brings about a rise in the nutron multiplication factor K ef up to 1.05 that is a supercritical state again.
Similar studies have been recently done in paper 33].The authors have drawn conclusions about a deep subcriticality for FCM in apartments 304/3 and 305/2 even in the case of water ood.However, this investigation, as well as the previous ones, treat both apartments on the basis of a onedimensional model to be far from real values of multiplication factors and neutron uxes.One can get a more realistic picture with the help of a threedimensional model taking into consideration fuel heterogeneity, physical and chemical processes which do occur in the system \FCM-water".
Summarizing all that, one can a rm that the irreversible process allowing the nuclear fuel out ow from the nuclear magma into the indoor reservoirs is in progress at the \Shelter" object.It is one of the factors of the object nuclear danger.Therefore, the problem of the water interaction with the active core fragments, nuclear magma, lavas and nuclear dust becomes very acute, since: water interaction with nuclear magma can essentially increase e ective neutron multiplication factor K ef 10,18,25] and possibly result in local self-sustaining nuclear chain reactions (SNCR); water destroys nuclear magma lava by means of radiolysis and complex leaching processes, enabling uncontrolled active elements transport indoors and outdoors of the object.Hence, the main kinds of nuclear fuel and the basic factors of nuclear and ecological danger of the \Shelter" caused by it are primarily connected with the availability of water.The water factor discussed during the last four years, shifted the accents of the Technical substantiation of nuclear safety (TSNS -1990 3]) of the \Shelter" object entirely.Moreover, it was shown in 3] that the object is nuclear-dangerous with the presence of 300 g. of 235 U, Ru or their mixture.
The interaction of water with active core fragments (fuel elements, tablets of uranium dioxide), lava-like FCM and nuclear dust should be considered when predicting disruption phenomena and calculating probability of local SNCRs in the system \active core fragments { FCM { water".According to the latest studies, water is an intensive destroyer of FCM, but at the same time it is an e ective neutron moderator.Special features of transuranic elements aqueous solutions are caused to a great extent by hydrolysis, complex formation and also variety of oxidation degrees.The formation of polynuclear structures under these conditions is to be investigated.
For a detailed understanding of the FCM destruction, uranium yield in water and prognostication of these processes, we have carried out an investigation 31,33] of the structural distribution and di usion processes of ions UO 2+ 2 , Sr 2 +, Cs + in the system \glassy-like FCM { water".Chemical reactions between aqueous solutions of radioactive elements and glassy FCM have been analysed, the main mechanisms of surface destruction (hydrogen, water molecules, OH group in uence) and the leaching of UO 2+ 2 , Sr 2 +, Cs + ions from the glassy matrix have been established.Radiolysis processes 30,31] in alkali aqueous solutions of radioactive elements are considered, the role of hydrogen ions, hydrated electrons and OH groups is established.Chemical reactions for the creation of minerals UO 2 CO 3 , UO 4 4H 2 O, Na 4 UO 2 (CO 3 ) 3 in the system \glassy FCM { water" are analysed.
At present the calculations of di usion, thermal conductivity and viscosity coe cients for ions UO 2+ 2 , Sr 2 + , Cr + in glassy FCM and water solutions of active elements are in progress.
The nonequilibrium transport of particles and energy in lava-like fuel containing materials inside the \Shelter" to a great extent determine their stability and disruption processes.In doing so, the description of ionic or neutral uranium, plutonium, americium and curium in nuclear magma is very important.First of all, it is connected with the fact that the main sources of -activity and spontaneous ssion neutrons are the isotopes 238?240 Pu, 241 Am, 242 Am (spontaneous ssion), 242 Cm, 244 Cm providing 99% and 96% of -activity and neutron out ow from 5 to 10 years after the accident.The interaction of the emitted -particles with the atoms of B, O, Na, Mg, Al and Si in the nuclear magma is accompanied by the reaction ( , n) generating a supplementary neutron ux.Estimations of ( ; n) reactions for some magma samples from under-reactor apartments 305/2 and 304/3 reveal that their contribution to the neutron generation rate in the nuclear magma reaches 50%.It should be noted that the neutron generation rate will increase with time ow due to ( ; n) reactions on light elements.It is caused, rst of all, by the accumulation of americium 241 Am (after -decay of 241 Pu) as an intensive -particle source.
One has to point out that americium 241 Am, among similar to it isotopes, has a large thermal-neutron ssion cross-section, namely 3.13 barns.For the consideration of actinides in the nuclear magma, the curves for ssion fragments of 241 Pu, 241 Am by slow neutrons may be useful.Neutron elds in the nuclear magma in apartments 305/2, 304/3 were studied.In particular, a signi cant divergence between experimental and numerical data was observed at the investigation of ssion densities, cadmium ratios and spectral indices for the nuclear magma in apartment 304/3.Especially great di erence in the magnitude is found for the neutron ux density (the experimental values are almost 20 times as large as the numerical ones).Evidently, it can be explained by the fact that the computational models do not take into account the peculiarities of the content, structure, neutron source strength and radionuclide transport in the nuclear magma.Recent investigations dealt with the estimation of the neutron current for the FCM models in the central hall, which contact with water.But the contribution of ( ; n) reactions, that can amount 50%, has not been considered in the calculations.
In connection with the above mentioned problems, we have obtained 29,30] a system of equations for neutron transfer kinetics taking into consideration the transport processes of particles in FCM.This is one of the versions for the investigation of neutron di usion and nuclear processes using a three-dimentional model of FCM.
The transport of radionuclides by the ground soil and underground waters in the system the \pumping of the nearby zone of the Chornobyl nuclear power plant { the Prypjat' river { the cascade of the Dnieper storage lakes" and the processes occurring in the burial grounds of PTLAW in the 30-km Chornobyl zone are the main problems concerning the consequences of the accident at the Chornobyl plant.The PTLAW (burial places of parts of buildings, radioactive grounds, forests, parts of metallic constructions, etc.) were created in the conditions of a highly radioactive background in a very short period of time without a proper technical execution (the absence of hydroisolated layers) during the desactivation of the territory around the Chornobyl plant.The PTLAW near the \Yaniv" station and the point \Oil Storehouse" 34,35] (investigations for the content of 137 Cs and 239 Pu in the ground waters at their migration from trenches), which are ooded by water, attract special attention.The \red" wood 36,37] has been creating conditions for an uncontrolled penetration of radionuclides, such as Cs, Sr, Pu and Am, into the environment.In particular, in 37] the problem of americium and plutonium in the ground waters of the \red" wood was emphasized.
Another possible source of radionuclides in the ground waters is the \Shelter" object.In the bore holes on its industrial territory one can observe an increase in the concentration of Sr and Cs.It can be connected with the penetration of the \block" water into the ground outside the object 38] or with the interaction of the surrounding wet grounds with the dry radioactive grounds of the \Shelter" object.The di usion processes in this case are determined not only by concentration gradients, but also by electrostatic e ects which have not been taken into account up to now.In our papers 1,30], osmotic processes of the radionuclide transport in the ground waters were discussed.
In this paper we present some results of the investigation of physico-chemical processes (di usion, radiolysis, chemical reactions) in the system \glassy FCM (nuclear magma) { aqueous solutions of radioactive elements (water)".
2. Nonequilibrium physico-chemical processes in the system \glassy fuel containing masses { aqueous solutions"

Statistical Model
We are going to treat an interaction of nuclear magma with water by means of a model two-phase system \nuclear magma{water".The nuclear magma was formed as a result of high temperature melting and gradual solidi cation of the nuclear fuel along with boron, dolomite, lead, sand, clay combinations which were thrown into the reactor to decrease the temperature and terminate active nuclear processes.The global problem of nuclear fuel decomposition was solved.It resulted in a glassy inhomogeneous solid medium in the form of avalanches containing a number of highly active nuclides: U, Pu, Cs, Ce, Am, Cm, Zr, Sb and their isotopes.Reasoning from the chemical content of the samples 11,12,19,39], nuclear magma is argued to be multicomponent glass 40,41] in a structure with the inherent ionic bond -O-Si-O-, -Ca-O-, -O-Al-O-, and belong to sital-like glass: Thus, nuclear magma has a silicate matrix lled with impurities which are di erent in the chemical content 11] including up to 18% UO 2 uranium oxide.Mainly, ionic bonds are available between the impurities and the silicate matrix, thus, those impurities might be regarded as ionic clusters (e.g.uranyl ion UO 2+ 2 ) inside the silicate matrix.In connection with this the nuclear magma can be represented as a system of interacting ionic clusters with polyvalent radicals (SiO 3 ) 2n? n creating a polymeric glass structure.
In view of the aforesaid, we can consider the system \nuclear magma{ water" as a statistical model of interaction between ionic clusters and a water solution.It is clear that advancing the model, one should require it to describe the physical and chemical processes occurring in the \Shelter" object in the most realistic manner.
What basic processes need to be treated when investigating nuclear, physical and chemical transformations in the system \nuclear magma{water"?The category of necessary questions includes the following: 1. Di usion of ionic clusters incorporating active elements or isolated ions in the silicate matrix taking account of coulombic, dipole and resonant kinds of interaction; 2. Di usion of active elements ions from the nuclear magma surface to water.The investigation of an interaction between water and nuclear magma, taking into consideration both radiolysis, because of -,decays, -radiation, and chemical reactions.The di usion of active elements (in an ionic form) from the nuclear magma surface to water is a ected by both the transport of ions or ionic clusters in the very matrices and water radiolysis with chemical reactions occurring in it.It is apparent that the basic processes leading to the di usion from the silicate matrix to water take place, rst of all, in the near-surface layer of the system \nuclear magma { water".The water radiolysis has a speci c e ect on them, in particular, on the products of radiolysis: ions, radicals and molecular products: H 2 O ! e ?aq , H + aq , OH ?aq , OH, H, HO 2 , H 2 , H 2 O 2 ; the index \aq" stands for hydrated ions.To a great extent the above radiolysis products destroy the silicate matrix, increasing the probability of active elements out ow from it into water in an ionic form.
The experimental investigations of the object's aqueous solutions of radioactive elements, with the average concentration of uranium 10 mg/l and more, show that their pHs are usually in the intervals of 6.5 7 and 9 10.The most interesting are solutions in those places of interaction of the nuclear magma with water, where we observe a substantial yield of uranium in water and a high level of the -, -and -radiation of water.High values of pH are pointing to a certain shift of equilibrium (due to chemical reactions) between groups OH ? and H + , and the fact that ions of hydrogen, which are formed in the process of radiolysis, hydratation and other characteristic reactions, quickly react transforming into molecular hydrogen and molecules of water.Besides, atomic hydrogen can be formed in reduction reactions.This value of pH corresponds to the basic solutions where the concentration of OH ?groups is dominant and there is a de cite of hydrogen ions H + .From this point of view it is necessary to analyze chemical reactions with UO 2+ 2 , PuO 2+ 2 , AmO 2+ 2 in the system \nuclear magma { basic solution" taking into account radiolysis and alkalization processes.
In aqueous solutions U, Pu and its isotopes, strontium Sr, cesium Cs and other radioactive elements produce di erent forms of hydrated ions, molecules, double and mixed complexes, mono-and polynuclear products of hydrolysis and colloid particles 42,43].Let us analyze a set of reactions where uranium is involved.The approximated schematic set of reactions can be seen as follows.In aqueous solutions uranium has +3, +4, +5, +6 states of oxidation.The stability of valency states of uranium in a solution is characterized by the sequence U( 6)>U( 4)>U(3)> U(5) 43].The stabilization of the large positive charge of uranium U(6) takes place because in aqueous solutions uranyl ion UO 2+  Ion complexes, which are formed as a result of an interaction with water molecules, can actively interact with the products of radiolysis, namely, with the corresponding radicals OH ? or molecular compounds H 2 O 2 and HO 2 .These chemical reactions should be taken into account along with those described above.It should also be noted that uranyl can form stable complexes with anions, like NO ? 3 , Cl ?, SO 2? 4 , OH ?, CO 2? 3 .Experimental ultra ltration investigations 43] show that for some de nite concentrations of OH ?ions the six-valent uranium in the solution is in a colloidal state.Taking into account an excess of OH ?groups, we can make an assumption about the existence of the ionic complexes UO 2 (OH) 2 UO 2 ]] 2+ n in such solutions, which points to their polymer structure with hydrogen bonds: 3), it is necessary to draw the conclusion that the excess of OH ?, the yield of reactions (2.2) according to the Le-Chatelle principle, shifts the reactions to the right towards the formation of polymer complexes, particularly for UO 2 (O UO 2 ) 2+ n with the oxygen bridges: magnitude of rate constants of the polymerization process decreases signi cantly with the increase of n, which causes the probability of chain formations with n > 4 to be small.As we can see, uranyl hydrolysis reactions produce hydrogen ions, as one of the sources of atomic and molecular forms of hydrogen during the interaction of water with nuclear magma.The reaction of molization H+H!H 2 is accompanied by the liberation of 0.34 ridbergs per molecule.This energy transforms into a kinetic energy of H 2 .Hydrogen molecules, unlike the molecules of the gases He, Ne, Kr and others, have a large kinetic energy and this causes large pressures in magma cavities.Thus magma is mechanically destroyed.So, we are faced here with two problems, namely, the problems of hydrogen and uranium compounds.Hydrogen causes a mechanical disintegration of the system, which increases contact with water, and concentration of uranium compounds.Uranium compounds in water form complexes which tend to coagulate.
Plutonium Pu in aqueous solutions has ve degrees of oxidation: +3, +4, +5, +6, +7.Pu 4+ has the most interesting properties.It is in the form of hydrated ions Pu(H 2 O) 4+ 6?8 in polymer-acidic solutions.Also for some concentrations of Pu, when pH of the solutions increases, the hydrolysis of Pu 4+ ions leads to the formation of mononuclear hydroxocomplexes Pu(OH) 3+ , Pu(OH) 2+ 2 , which further can exhibit polymeric and colloid properties 43,47].The hydrolysis of Pu( 4) takes place at the concentrations of H + ions less than 0.3 M: Pu 4+ +H 2 O PuOH 3+ +H + .
(2:4) Here we have to distinguish between two types of reaction products: monomeric hydrated ions of Pu(OH) (4?n)+ n type and products of polymerization, which are formed at the same values of acidity.A slow decrease of acidity in the region of H + hydrolysis below 0.3 M helps the formation of polymers.The dissolving of acidic solutions with water causes an instantaneous local decrease of acidity to the values favourable for polymerization.The formation of polymers is also accelerated when moderately acidic solutions of plutonium Pu(4) are heated 47].The process of polymerization is accompanied by polarization of a water molecule by two plutonium ions.
The presence of Fe 3+ and CO 2? 3 ions in aqueous solutions causes alkalization of uranium from nuclear magma into water.As it was mentioned, observations, which had been made since 1990, showed a creation of needlelike crystals of uranium minerals UO 2 CO 3 , Na 4 UO 2 (CO 3 ) 3 .
The formation of the latter mineral can be represented by the following reactions: UO 2 +O+3CO 2?
Three-valent Fe 3+ is known to be a good oxidant: (2:7) This reaction also causes alkalization of uranium from nuclear magma to water.The process of alkalization of uranium from nuclear magma to water can run actively, because UO 2 reacts with OH radicals which are formed in the process of radiolysis and they are in excess in aqueous solutions: UO 2 +2OH !UO 2+ 2 +2OH ? .
(2:8) Obviously, in the process of destruction of nuclear magma, the whole parts of UO 2 pills (fragments of the destroyed fuel elements, which got into the alloy of nuclear fuel in the process of its cooling) can fall into water.Oxidation reactions of UO 2 by ion Fe 3+ and radicals O and OH in water will be accompanied by the yield of uranyl ions to water.Further, uranyl UO 2+ 2 will be hydrolysed.Thus, on the basis of radioactive elements solutions, which are typical of the object, it is necessary to analyze and then calculate chemical reactions of plutonyl and uranyl polymerization, reaction of hydrolysis, reactions with the participation of OH radicals and ions Fe 3+ .Besides, it is very important to calculate chemical reactions which yield a sedimentation of uranium and plutonium and to investigate their transfer into water solutions.
It is necessary to note that all the mentioned above reactions of hydrolysis and alkalization are of the type A j +B j C j +D j , (2:9) and the change in the concentration of the components in a solution versus time, as a result of the di usion and chemical reactions, can be described in the di use approximation by the following equations: d where D l and k lf correspond to the di usion coe cients of the components and constants of reactions between the components in the solution.
When describing the disruption of the silicate matrix, an important question is to elucidate destructive mechanisms, rst of all for magma surface undergoing interaction with water.Interplay of an aqueous solution with a glassy surface is accompanied by a set of interconnected physical and chemical processes: ion-exchange mutual di usion, di usion of ions H + or H 3 O + , hydrolysis of the silicon{oxygen network and its corrosion, formation of pores and cracks.According to 48] they may be brie y described with the aid of chemical rections, namely, by the ion-exchange di usion: L + + SiO ?M + SiO ?L + +M + (2:11) As a result of the reaction (2.11), an equilibrium point is shifted, in fact, entirely to the right creating the hydrated complexes SiOH OH.Leaching and formation of the rst hydrated layer occurs, which interacts afterwards with a glassy surface according to the reaction: SiO ?M + + SiOH SiOH+ SiO ?M + (2:12) In this manner the ion-exchange layer is formed with the inherent mutual di usion of ions H + and M + .The interaction of water molecules with a silicate surface can be described by the chemical reactions SiO ?M + +H 2 O SiOH+OH ?+M + (2:13) which results in the outlet of metal ions into the solution, while hydroxide complexes OH ?take part in further reactions of the depolymerization of the silicon-oxygen network.
Si{O{Si +OH ?SiO ?+ SiOH (2.14) 2 SiOH+OH ? and so on (2.15) Reactions of this kind can proceed from a glassy magma surface to an alkali active solution (pH=9 11 for the reservoirs of the \Shelter" object).Apart from reactions 48], water molecules are able to destroy the silicon-oxygen network due to the interaction between oxygen ions of water and Si atoms, and simultaneously between H + ions and oxygen atoms of silicon-oxygen network Si{O{H Si{O{Si +H{OH !Si{O Si +H OH !Si{O{H+ Si{OH; (2:16) A set of reactions (2.13)-(2.16)represents hydrolysis and depolymerization of the silicon-oxygen network.It brings about the formation of hydrated complexes =Si(OH) 2 , {Si(OH) 3 , Si(OH) 4 , Si(OH) 2? 6 which are transferred to the solution.In such a manner the corrosion of the silicon-oxygen network occurs being enhanced by hydrogen ions H + , hydrated complexes OH ?, which arise additionally as a result of radiolysis, hydrolysis and hydration of uranium UO 2 , plutonium PuO 2 and other oxides 26,27,30,31].
When ions of uranyl UO 2+ 2 are present in the near{surface layer, which can be manifested by its yield from nuclear magma, the following reactions with hydrated silicium complexes are possible: UO 2+  For the description of interplay of an aqueous solution with glassy nuclear magma according to reactions (2.11)-(2.16)we need to formulate a physical model to research structural functions, particle di usion from one phase to another and reaction constants.The prelude of any microscopic treatment is structural information in the form of density pro les and higher distribution functions.
Active elements are supposed to be present in water in small amounts.Within this model the active elements particles (UO 2+ 2 , Cs + , Sr 2+ ) are considered as charged hard spheres having an overall charge compensated according to the electroneutrality condition by negative OH ?-groups in a continuous medium with the dielectric constant " = 81 (water).
Active aqueous solutions inside the \Shelter" object contact mostly with nuclear magma, concrete, clay and numerous construction materials.To predict destruction and investigate di usion of active particles, it is necessary to know the structure of a solution near these materials.The problem reduces to the treatment of the solution model near a hard wall outlined above, which, along with a solvent, is a continuous medium with the dielectric constant " = 1 15 (glassy-like medium, clay, concrete).In this approach pro les are a ected by structural ordering caused by their sizes and the presence of surface, as well as electrostatic images most pronounced at small distances.However, their consecutive consideration is a fairly intricate problem.So far we used the rst equation of the Bogolubov-Green-Yvon hierarchy, modi ed to describe image charges.The latter are assumed to be ctitious charged particles inside the wall (glassy medium, clay, concrete, volume V 0 ), which have the charges "p?"c "p+"c Z i e and sizes i , whereas Z i e , i are charges and sizes of particles in the solution (volume V ).Applying the method given in 49], we arrive at the following equation: df a (z 1 ) dz 1 + dU a 1 (z 1 ) dz 1 f a (z 1 ) (2.17) hs (r 12 ) + U ac C (r 12 ))F ac (z 1a ; z 2c ; r 2c ) whereas f a (z 1 ), U a 1 (z 1 ) are the pro le and interaction potential of the ath species particle with the wall; F ac ,U ac hs , U ac C , U ac im are the binary distribution function for the particles of species a, c, hard sphere potentials, Coulomb interaction between the particles and between the particles and the images, respectively; c , 0 c are the densities of species c particles and their images.
It is easily seen that the second integral term takes into account image e ects on equal terms.But at small concentrations the integral terms are minor (the pair correlation is not essential), so the ionic distribution near the surface is determined by the potential U a 1 (z 1 ) that is advantageously used as a screened potential.The problem of the point particles screened potential near a hard wall has a rigorous analytical solution 2,50].In the case of ion-ion interaction the result is as follows: g(r 1 ; r 2 ) = Z 1 Z 2 e2 " p ( e ?r12 r 12 + " p ? " s " p + " s e ?r 0 12 r 0 12 ) ; (2.18) with r 12 { a distance between the particles, r 0 12 { that between the rst particle and the image of the second one.It is evident that the potential consists of the bulk part dependent on r 12 and the surface one.The problem of the bulk screened potential evaluation for an arbitrary amount of ions, having distinct sizes and valences, is solved in 50].We make use of the result for small concentrations in view of the fact that the potential of an inhomogeneous system is expressed through bulk ones, as is seen from (2.18).Thus, U a 1 (z 1 ) = " p ? " c " p + " c (Z a Then the solution of (2.17) under the condition f a (1)=1 is the following: f a (z) = exp(?U a 1 (z)): (2.20)For a pair distribution function the superposition approximation yields: F (z 1 ; z 2 ; r) = f (z 1 )f (z 2 )F (r) = exp ??U 1 (z 1 ) ?U 1 (z 2 ) + g(z 1 ; z 2 ; r) : (2.21) Figure 1 shows qualitatively the two-body functions for ions UO 2+ 2 , Cs + estimated at various concentrations with the aid of (2.21).There are 10 pair distribution functions in the system under consideration.We restrict ourselves to those for identical radioactive particles.It means that we have computed F , =UO 2+ 2 , Cs + .Since F depends, in general, on three parameters, it is convenient to x particle 1, for example, in contact with the magma wall.As the dependence on z 1 cannot be plotted, the gures exhibit the behaviour of functions F 0 = F =f (z 1 ).And nally, the position of the second particle with respect to the xed one is de ned via distance r and angle rather than z 2 , so that z 2 = z 1 + r cos .Such functions F 0 (r; ) are plotted in gure 1.
It can be seen that small concentrations of active elements result in an essential wall e ect even at long distances, while large ones bring about screening due to which the system quickly attains bulk properties.The overall drop of functions for all the elements at angles more than 70 0 is the evidence of a negative adsorbtion occurring in a thin near interface layer of magma.The phenomenon might terminate the penetration of radioactive matter into the grounds.

Spatially inhomogeneous di usion model
To describe the di usion of complex particles from nuclear magma into water realistically, we are going to complicate the system by considering a model comprising M species with N a particles of each species, which combine X sites (each site carries the charge ez a ) interacting between themselves and water molecules.The subsystem \water" with the dielectric constant " v is regarded as a total combination of water molecules H 2 O and ionic radicals, in particular, hydrogen ions H + and complex OH ?, and labelled as particles of s species formed by X charged sites in volume V 2 .It is clear that the aqueous solution can harbour other ions or molecular complexes interacting with nuclear magma particles.At a certain stage of studies they might be taken into account.The change in the charged sites density, which should be incorporated to create an active element (e.g.those in ionic form UO 2+ 2 , PuO 2+ 2 ) in the silicate matrix, can be described by generalized di usion equations for the two-phase system \nuclear magma { water".Similar equations were obtained in our papers 51,52].In the case of a quasiequilibrium di usion this equation system is as follows: Studies on aqueous solutions of radioactive elements 79 a) the phase \nuclear magma", volume V 1 : @ @t n a 1 (r 1 ; t) = dt 0 e "(t 0 ?t)@ @r 1 D a; b 11 (r 1 ; r 0 1 ; t; t 0 ) @ @r 0 1 n b 1 (r 0 1 ; t 0 ) ?
dt 0 e "(t 0 ?t)@ @r 2 D a; s 22 (r 2 ; r 0 2 ; t; t 0 ) @ @r 0 2 n s 2 (r 0 2 ; t 0 ) The last two terms in these equations describe a radionuclide density variation in time because of nuclear transformations promoted by neutrons and spontaneous decay.The rst one describes the formation of a radionuclide in the site a from all the other nuclei in sites b, as a result of (n; ), -, -decays, as well as (n; f) reactions if the relevant a-site radionuclide belongs to ssion products.The second term describes the decay of an a-site radionuclide under the action of neutrons and natural radioactive decay.Functions A (r; t), A (r; t) are rates of the reactions represented as follows: A a b (r; t) = Here J(r; E; t) is the spectrum of neutron density uxes in point r at time t; a; b (E) is the microscopic section of a-site radionuclide formation at catching neutron with energy E by b-site nucleus, a (E) is the microscopic section of catching a neutron with energy E by a-site nucleus, L a; b is the probability for the creation of an a-site radionuclide at radioactive decay of b-site nucleus, a and b are the decay constants of a-, bsite nuclei.The time dependence of reaction rates A a; b (r; t), A a (r; t) is determined by the spectrum of neutron density uxes J(r; E; t) which have been calculated for FCM of the \Shelter" object in papers 18,20,21].The neutron spatial energy distribution in its turn depends on the spatial distribution and concentration of radionuclides in the system, therefore, actually the neutron eld and time variation of density of actinides or ssion products are closely related.
In equations (2.22)-(2.24)n a (r ; t) = h n a (r )i t ; (2.25) n a (r ) = n a (r ) ? hn a (r )i 0 ; (2.26) (r j ?r ): (2.27) It should be remembered that the site densities n a (r ; t) are not irrespective of each other as they compose complex particles and, thus, n a (r ; t) X = n a (r ; t) X = n a (r ; t) in which n a is related to f a from (2.17) through the density factor n a = N a V f a Equation (2.27) de nes the density for the th site carried by the a th species particle in the phase = 1; 2; h: : :i t is the averaging with the total nonequilibrium distribution function for particles of the whole system, h: : :i 0 is the averaging with the total equilibrium function 0 .Respectively, n s (r ) = Ns X l=1 (r l ?r ) (2.28) is the microscopic density of water molecules or dissociated ions (H + , OH ? ) and radicals.We denote D a; b f (r ; r 0 f ; t; dr 00 f 0 h(1 ?P M ) Ĵ a (r ) T(t; t 0 )(1 ?P M ) Ĵ c f 0 (r 00 f 0)i 0 F ?1 (r 00 ; r 0 ) is the momentum density for charged sites in the respective phase , is the Mori projection operator, T 0 (t) = expf(1 ?P M )iL N tg is the time evolution operator, iL N is the Liouville operator appropriate to the system's Hamiltonian ' (1) a (z j ); where ab (r ij ) is the potential of interaction between charged sites, as (r jl ) that for charged sites and water molecules (or ions H + , OH ?, or radicals a (z j ) is the potential at the interface \nuclear magma{ water".The functions ( F ?1 (r; r 0 )) a; b f make up the matrix F ?1 (r; r 0 ) inverse to F (r; r 0 ).This latter consists of the pair equilibrium distribution functions for charged sites F a; b f (r; r 0 ) = hn a (r)n b f (r 0 )i 0 : ( introducing the corresponding constants of chemical reactions.It is important to research on the hydration of ions UO 2+ 2 ; PuO 2+ 2 at their outlet from nuclear magma into water, because uranyl and plutonyl reveal a polymeric structure if uranium or plutonium concentration in water increases 10].The dynamical behaviour of those combinations is indetermined in the case of enhancing uranium or plutonium concentration.But equations (2.22)-(2.24)can yield information by calculating the generalized di usion coe cients (2.29) for the system \nuclear magma{water" and taking account of the peculiarities of interaction between the medium and the water solution.A remarkable point in the studies of di usion coe cients D a; b f ; D a; s f in the subsystem \water" is the consideration of radiolysis in it, induced by -, -decays and -radiation.

Processes of aqueous solutions radiolysis and chemical reactions
From the point of view of statistical theory, the aqueous solutions of radioactive elements are complex ionic-molecular systems with a long-range dipole and short-range interactions leading to the creation of chemical bonds between ions and molecular products.In such solutions it is necessary to take into account the phenomena of ions solvatation, hydratation, appearing and altering the polymer structure.These aqueous solutions are under the permanent in uence of internal and external -, -and -radiations from FCM.That is why the state of these solutions is determined by the character of radiolysis processes which permanently change the nature of interaction between solutions particles due to the emerging of ionized tracks (domains of high concentration of e ?aq , H 2 O + , H + aq , OH ?aq ions).
At the physical stage, during the time interval of 10 ?16 sec after the ionizing particle had passed through the excited H 2 O and superexcited H 2 O , water molecules and ions H 2 O + were created H 2 O !H 2 O + + e ?aq ; (2.32) where the asterisk denotes an excited molecule.The excited electrons during the time interval of about 10 ?15 sec have been hydrated and interacted with water molecules e ?aq + H 2 O !H + OH ?: (2.33) H 2 O + ions also interact with water molecules: (2.34)After 10 ?14 sec the dissociation of excited water molecules occurs: Studies on aqueous solutions of radioactive elements 83 In the case of a large irradiation dose (the excitation energy of water molecules is about 10 17 eV) the hydrogen and oxygen radicals could be created 10 ?14 sec after the process had started: (2.36) In reactions (2.32), (2.35), (2.36) the excitation energy of a water molecule is di erent, hence we use various denotations.
Analyzing reactions (2.32){(2.36) at the end of the physico-chemical stage (up to the time 10 ?13 10 ?12 sec from the beginning), one can see that there are hydrated electrons e ?aq , ions H + , OH ?, H 3 O + (H + aq ) and radical products O, H in aqueous solutions.Initially they are concentrated in tracks and characterized by a strongly inhomogeneous space distribution.Hence, those particles should di use in the bulk of a solution quickly reacting with each other and with the dissolved compounds.A high density of chemical radicals in tracks makes the reactions of their capturing by dissolved compounds ine ective in comparison with the recombination.As a result, the di usive-recombination kinetics of chemical reactions appears.The main point consists in a competition between the mutual di usion of chemically active particles and reactions occurring inside the tracks.At the chemical stage, the following reactions are possible in the tracks: In our paper we have tried to avoid this duality, as it leads to di erent values of the di usion coe cients and kinetic properties of particles H 2 O ? and water molecules which transfer electron (H 2 O) ?n due to their di erent masses and dimensions.This should be applied to a hydrated proton as well: H + or H 3 O + .Any dualities here can be avoided by introducing a time limitation.It does mean that there appears a pure proton H + immediately after the ionized particle has passed (10 ?14 sec).The proton is hydrated at once into the hydroxonium ion H 3 O + , where it is connected with a water molecule by a persistent covalent donor-acceptor bond.
For a hydrated electron everything looks more complicated, because the connection mechanism of the electron to a water molecule is not de nite.From the chemical point of view, it is problematical to interpret the hydrated electron as an H 2 O ? ion.In our kinetic calculations, the hydrated electron means an electron surrounded by water molecules (H 2 O) ?n and it is denoted by e ?aq in chemical transformations.These water molecules transfer electrons only and do not participate in any chemical transformations.In the case of an interaction between the hydrated electron and water molecules, the latter one should be written separately to avoid expressions like e ?aq + e ?aq + !H 2 + 2OH ?: (2.58) Thus, as a result of the ionizing irradiation action on aqueous solutions, the particles with oxidation (OH radicals) and reduction (hydrated electrons and H atoms) properties are generated simultaneously.
In particular, the hydrogenium peroxide, created during radiolysis, participates in reactions with uranyl ions, forming uranium peroxide UO For the calculation of the radiolysis products yield and for its comparison with the experimental data, the equations of chemical kinetics are used [53][54][55][56][57][58], which could be obtained on the basis of nonequilibrium statistical thermodynamics 59].Not taking into account the processes dealing with energy and momentum uctuations and considering the chemical processes in aqueous solutions to have a reaction-di usion character, one can write down an equation for the reagents concentration in the general form: @ @t a (r; t) = G a J + is the sum of reactions when a-species particles disappear, F a describes the a-particles transfer due to molecular di usion and water ow with velocity v(r; t): F a = ?div(v(r;t) a ) + div(D a grad a ): (2.64) D a means the di usion coe cient of a-particles, G a denotes the yield of a-species particles during radiolysis.Units of measuring G are particles/ 100eV, { mol/10 ?3 m 3 , radiation capacity RC is measured in eV per (litre sec).(2.67) where N A is the Avogadro number.Hence, the change in the concentration of particles interacting according to (2.32)-(2.54)will be investigated on the basis of kinetic equations (2.55).
where k 0 (r ab ) denotes the proper reactional ability, g ab (r ab ; t) is the nonequilibrium radial distribution function of particles in solutions.In a steady state, g ab (r ab ) means the equilibrium radial distribution function and the rate constant K ab does not depend on time: K ab = 4 Z dr ab k 0 (r ab )g ab (r ab ): (2.73) The proper reacting ability k 0 (r ab ) of molecules depends on the mechanism of elementary reactions and, in principle, can be calculated either by classical or quantum mechanics methods.The simplest form has the Smoluchowsky proper reacting ability constant k 0 (r ab ) = k 0 (r ab ?R ab )=4 r 2 ab ; (2.74) then K ab = k 0 g ab (R ab ); In view of the statistical theory of interacting particles, the aqueous solutions of radioactive elements, in which radiolysis reactions take place, can be considered as a model electron-ion-atomic-molecular (plasma-molecular) system.The peculiarity of such a system consists in a long-range character of Coulomb and dipole forces between ions and electrons (H + , OH ?, e ?aq , H ? , H 3 O + ), as well as between molecules (H 2 , H 2 O, H 2 O 2 ).Besides, there exist hydrogen atoms H and hydroxyl OH groups in the system.They actively participate in transport processes and chemical reactions in the solution.The multiple dynamical processes, which are marked by both shortrange and long-range correlations, lead to the phenomena of solvatation and complex formation in solutions.The investigation of structure, thermodynamical and kinetic properties of such plasma-molecular solutions is actual.Besides, the nonequilibrium distribution functions g AB (r; t), which are connected with chemical reactions constants (2.64), have not yet been studied appropriately.A considerable achievement in the investigation of plasma-molecular systems was made in 62].On the other hand, the constants of quasiequilibrium states of chemical reactions could be de ned via pair distribution functions (2.65) of electrons, ions, atoms and molecules.Methods of the investigation of equilibrium distribution functions of particles of electrolytes (in the ion and ion-molecular approaches) are described in 62,63].
For qualitative estimations, the rate constants can be calculated on the basis of an ion-dipole model of aqueous solutions 18].Within the framework of this model the particles interact as hard spheres on short ranges with the equilibrium distribution g ab ( ab ) ( ab = 1 2 ( a + b ), where at a ; b are hard spheres diameters), whereas at large distances they interact through the ion, ion-dipole and dipole-dipole potentials which are characterized by electrostatic screening.For an ion-dipole system the equilibrium pair distribution function can be presented in the approximation of the second virial coe cient g ab (r) = g ab ( ab ) expfG ab (r)g; (2.76) where G ab is the screened potential of electrostatic interactions.For an ion-dipole system the screened potentials of electrostatic interactions can be obtained from 62,63]: G ab (r) = ?Z a Z b e 2 "r exp ?p " r G as (r) = Z a ed s 1 "r V Z 2 a e 2 ) 1=2 means the value which is inverse to the Debye screening radius, " denotes dielectric penetreability, = 1=k B T, k B is the Boltzmann constant, T is the temperature, Z a e denotes the a-kind ion charge, d s is the value of dipole moment for molecules, Q s and ' s are orientation angles.In particular, a hydrated electron can be considered as an ion with the charge Z e ?aq = ?1 and the e ective radius e ?aq = 2:5 3:0 A and with the di usion coe cient 4.9 0.25 10 ?5 cm 2 /sec.
The calculation of radiolysis rate constants on the basis of formulae (2.32)-(2.54)for a dipole model of water will be carried out in our further papers.Investigation of the solutions di usion coe cients is important from the point of view of the kinetic calculation of chemical reactions (2.55).One approach of an approximate calculation of transport coe cients for ions UO 2+ 2 , Sr 2+ , Cs + in aqueous solutions and FCM is presented below.

Coe cients of mutual di usion and viscosities for ions UO 2+ 2 , PuO 2+ 2
Sr 2+ , Cs + in aqueous solutions of radioactive elements and in FCM An important stage of the investigation of nuclear physico-chemical processes in the system \FCM { aqueous solutions" is, undoubtedly, the study of di usion and thermal di usion processes and viscosity of radioactive elements (UO 2+ 2 , PuO 2+ 2 , Sr 2+ , Cs + ) both in FCM and in aqueous solutions.Speci cally, the calculation of di usion coe cients for ions, radicals and molecules in aqueous solutions is important for studying chemical reactions of hydrolysis which were considered in subsections 2.1, 2.3.On the other hand, the investigation of di usion coe cients for ions UO 2+ 2 , PuO 2+ 2 in glassy FCM is important from the point of view of nuclear transformations and calculations of neutron density ow spectra.The problem of the calculation of such transport coe cients as di usion and viscosity for ions UO 2+ 2 , PuO 2+ 2 , Sr 2+ , Cs + in glassy-like FCM and in aqueous solutions can be solved on the basis of the generalized Enskog-Landau kinetic equation for a multicomponent system of charged hard spheres [65][66][67][68].We will use a model where the silicium matrix is treated as an environment with the xed dielectric constant " = 1 15 and in which ions of UO 2+ 2 , PuO 2+ 2 , Sr 2+ , Cs + run in an e ective compensating eld correspondingly to the electroneutrality condition.Similarly, aqueous solutions will be treated on the basis of an ion approach, when the solvent is modelled by a molecular subsystem with the dielectric constant " w = 81 and ions UO 2+ 2 , PuO 2+ 2 , Sr 2+ , Cs + run inside it in an e ective compensating eld correspondingly to the electroneutrality condition as well.To calculate the viscosity and di usion coe cients we use the results of our previous works 67,68].It will be, of course, rather an approximate description, as formulae 67,68] were obtained for a two-component system of charged hard spheres.They limit our consideration to two systems, namely \UO 2+ 2 | PuO 2+  Z values of ions charges in the positron charge units: Z 2 Z.
The coe cients B 0 appear at the expansion on Sonine-Laguerre polynomials and are calculated via the so-called -integrals 71].For the system under consideration their evident structures were found in papers 67,68].

Conclusions
In this paper an analysis of the interaction of FCM with water, as well as an investigation of radiolysis of aqueous solutions of radioactive elements in the \Shelter" object has been carried out.Considering the interaction between the radiolysis products and FCM we have shown that the way of the formation of uranium peroxide tetrahydrate UO 4 4 H 2 O (its availability was con rmed in the \Shelter" object experimentally) is in accordance with the reaction UO 2+  whereas the uranyl ion UO 2 in its turn is obtained from FCM after the reaction UO 2 + 2OH !UO 2+ 2 + 2OH ?: In both the reactions the products of water radiolysis take part (hydrogen peroxide and radicals OH, whose concentration is large at pH=9 12 in aqueous solutions, are typical of the \Shelter" object).The same refers to the formation of other uranium minerals found on the FCM surface.It obviously follows that one needs a profound analysis of radiolysis kinetics, the calculation of coe cients for the reaction rates of the formation of such  products as hydrogen peroxide and others.At the same time, there is a problem of the formation of atomic and molecular hydrogen taking an active part in the glassy FCM disintegration.Except for the problem of radiolysis, we have discussed the mechanisms of the glassy FCM disintegration at the interaction with pH=9 11 aqueous solutions.We have computed the structural distribution functions for ions UO 2+ 2 , Sr 2+ , Cs + in these systems, which enables one to calculate the rates for the corresponding chemical reactions and their dynamics in further research.The mutual di usion and viscosity coe cients are studied for ions Cs + , Sr 2+ , Pu O 2+ 2 , UO 2+ 2 in aqueous solutions depending on concentration and temperature.The evaluations are qualitative, since the solvent molecules have not been treated microscopically.In view of the transport prediction in such systems, it is important to investigate the coe cients of di usion, thermodi usion and thermal conductivity for ions Cs + , Sr 2+ , Pu O 2+ 2 , UO 2+ 2 .We intend to ful l it in our further papers.
Nuclear, physical and chemical processes occurring at the FCM interaction with aqueous solutions in the \Shelter" object, are tightly related.Our research on radiolysis, hydration, chemical reactions with the participation of UO 2+ 2 , Pu O 2+ 2 , Cs + , Sr 2+ , their structural distribution functions and transport coe cients is only an element of the physical and mathematical model for the behaviour of FCM in the \Shelter" object.The model is designated to predict nuclear, physical and chemical processes and provide the object stabilization, gradual extraction of the nuclear fuel and its further processing.
The problem of water in the \Sarcophagus" should be considered in view of nuclear material extraction from the object.Water is an important factor for FCM disintegration and the egress of radioactive elements into the basins.The controlled pumping out of water into assigned containers will enable one to gradually remove the nuclear fuel present in the dust (sprinkling the dust) and glassy states (intensive disruption of glassy lavas occur).The pumped radioactive water can be stored in the containers to prepare it for further processing at a radiochemical plant.
; E; t) a; b (E)dE + L a; b b ; ; E; t) a (E)dE + a : 29) as generalized di usion coe cients for charged sites, in particular, D a; b 11 corresponds to nuclear magma, D a; b 12 governs the di usion from nuclear magma into water; D a; b 22 concerns water.Here Ĵ a (r) = Na X j=1p j (r j ?r )(2.30) ; n ; b denote the concentrations of a, m, n, bspecies particles; K mn and K ab denote the rate constants of the creation and annihilation of a-specie particles in the reactions between m; n; b-species particles, corover all the reactions leading to the creation of a-species particles, P b J = RC=(100N A ): (2.65)Initial yield G a during ionization (before reactions inside tracks) is determined by formula 58]: ) 1=2 , = m a m b =(m a + m b ) is the reduced mass of particles a and b.So the problem of the calculation of rate constants for chemical reactions is reduced to the evaluation of di usion coe cients and structural distribution functions of reagents.
2 " and \Cs + | Sr 2+ ".The results of the calculation at di erent concentrations are shown in gures 2 and 3.The mutual di usion D and shear viscosity coe cients read 67,68] The following conventional designations are used in those formulae: inverse local temperature analogue; D screening radius borrowed from 69]; g 2 two-particle correlation function borrowed from 70]; bulk viscosity borrowed from 67,68]; m reduced mass; m partial masses of particles; n total density of particles number; n partial densities of particles numbers; averaged value of hard spheres diameter;
relatively high content of 234 U is revealed, furthermore 235 U is 1.6 times as large as a normal ratio of 235 U/ 238 U for RBMK-1000.Therefore, a conclusion may be drawn that because of complicated physical and chemical processes with the participation of water, uranium is concentrated in these formations.Since new combinations are well soluble in water, we have an active way of uranium out ow from nuclear magma into the \Shelter" reservoirs in a concentrated state; chemical deposits containing uranium, plutonium, curium, americium, strontium and cesium in aqueous solutions in the \Shelter" object.In general, there are such dangerous nuclear ssion materials in the \Shelter" object: UO 2 , UO 2 UO 3 , UO 2 CO 3 , PuO 2 , (ZrU)O x , (ZrU)SiO 4 , UO 3 2H 2 O, UO 4 4H 2 O, Na 4 UO 2 (CO 3 ) 3 with a large amount of neutron moderators SiO 2 , Al 2 (Si 2 O 3 )(OH) 4 , Na 3 PO 4 , C x H y O z , CaMg(CO 3 ) 2 , CaCO 3 , K 2 O, CaO,MgO, Fe 2 O 3 , ZnO 2 , H 2 O, C. A 2 is produced 44].The hydrolysis of uranyl ions is governed by the reactions 43,45]: , OH?and radicals H 2 , H 2 O 2 , HO 2 .In the subsystem \water" it is necessary to investigate the mutual di usion coe cients for charged sites of UO 2+ 2 , Pu O 2+ 2 and other ions and water molecules, ions H + , OH ? and radicals{D a; b + An important result of the presented reactions is the creation of molecular hydrogen H 2 , hydrogen peroxide H 2 O 2 and water H 2 O. Molecular products H 2 , H 2 O 2 can be annihilated by the reactions e ?aq + H 2 O 2 !OH + OH ? ; The uranium peroxide (studtite) falls into the sediment in the form of UO 4 4H 2 O when its solvability limit is exceeded.It can decay thermally according to the reactions UO 4 + 2H + !UO 2+ 2 + H 2 O 2 ;