A review of the migration of radioactive elements in clay minerals in the context of nuclear waste storage

Abstract

Clay is a widespread natural mineral. The review considers physical and chemical properties of clay minerals which are important in terms of geological high-level radioactive waste disposal (HLRW). The articles under consideration present that the properties of clay as a barrier material for the isolation of radionuclides are influenced by temperature, density (external pressure), ionic strength and pH of the solution, and the presence of cations of stable elements. These conditions are studied by both experimental and calculation methods. However, there are not enough research results in the publications for conditions close to those of HLRW geological disposal. The research status worldwide is introduced in detail, and at the end of the article, the future directions of the research on radionuclide diffusion in HLRW disposal are discussed.

Introduction

Radionuclides are widely used in scientific researches and medicine. Nuclear power uses uranium and plutonium as nuclear fuel. This produces irradiated fuel and other radionuclides. After shutting down a nuclear reactor or reprocessing irradiated fuel, radioactive waste (RAW) is generated and requires disposal. Due to the end of the first nuclear reactors lifespan, the number of RAWs is increasing every year. According to modern requirements, high-level radioactive waste must be isolated from the environment for 1,000–10,000 years. During storage, radionuclides decay, and their residual amount in RAW should not harm people and their environment [1]. Therefore, RAW long-term storage technology is needed.

The following requirements are imposed on the RAW barrier material for disposal: low cost (availability), reliable blocking of radionuclides, and radiation resistance. Clay minerals are the cheapest materials. Clay is available and is an excellent barrier material due to its low hydraulic conductivity.

Finland, Sweden and Canada were the first to propose building underground RAW storage tanks, after some processing, in a layer of clay 500 to 1000 m underground. For this, spent nuclear fuel and high-level radioactive waste are placed in highly durable metal containers (for example, made of copper, steel or titanium). The long-term disposal containers are then placed in a buffer material consisting of compacted clay in nearby rock [2]. In some of these countries clay minerals are part of the geological formation where the disposal facility is planned, but in some countries the clay minerals are considered as a part of a filler around the waste canisters in the engineered setup. The extremely low clay permeability contributes to the long-term isolation of waste containers, ensuring that no radionuclides migrate out of the storage facility even in the case of a major natural disaster (earthquake, tsunami). Moreover, no pollutants will migrate from the storage facility to the Earth's surface after many years of storage. Therefore, studies on the assessment of the rate of radionuclides migration in clay minerals and the current state of such studies are relevant. Figure 1 shows the timeline of introducing RAW underground disposal technology in metal containers into different countries.

Fig. 1

The construction status of in-depth laboratories and high-level waste disposal sites in various countries [3,4,5]

One of the important characteristics of clay minerals is their ability to strongly sorb cations [6], which affects the concentration of ions in the pore solution and their transport through the clay layer. Illite, vermiculite, and other minerals are successfully used for the sorption of ¹³⁷Cs and Pu radionuclides [7]. The radionuclide is directly adsorbed on the areas of the constant charge on the surface of the clay mineral layer due to cation exchange [8]. Most of the research is devoted to the study of alkali and alkaline earth metals. The heavier the alkali metal ion, the higher the sorption energy [9]. Therefore, it is more difficult for the ion to diffuse in the space between the layers of the illite mineral. It is possible to form a series of decreasing sorption strength [9, 10]: Cs⁺ > Rb⁺ > K⁺ > Na⁺ > Li⁺andBa²⁺ > Sr²⁺ > Ca²⁺ > Mg²⁺. At low pH values, H⁺ ions are able to replace other cations. This means that hydrogen ions are more active in acidic solutions.

Due to the variety of processes of different cations interaction with clay minerals, in the development of RAW disposal technologies, a thorough analysis of the processes and experimental studies are required for the given composition of radionuclides. In this case, it is necessary to experimentally determine the diffusion coefficient, active porosity, the ability to exchange cations, the coefficient of adsorption distribution, which determine the mass transfer through the barrier material. In this article, an attempt is made to analyze the results of studies of the parameters of radioactive ions migration in compacted clay minerals obtained in recent years. The research results indicate a trend towards improved technology for using clay minerals as a barrier layer in RAW storage facilities.

Methods for determining the diffusion coefficient of radionuclides in clay

Physicochemical properties of radionuclides contained in high level nuclear wastes

The main factors that counteract the migration of radionuclide cations through clay minerals are their strong surface and volume sorption, high swelling capacity, and low water permeability. The main radioactive elements that determine the RAW hazard class are actinides with long half-lives, such as ²³⁸U (T_1/2 = 4.468·10⁹ y), ²³⁷Np (T_1/2 = 2.144·10⁶ y), ²³⁹Pu (T_1/2 = 2.411·10⁴ y), ²⁴³Am (T_1/2 = 7370 y) and their short-lived fission products ⁹⁰Sr (T_1/2 = 28.90 y), ¹³⁷Cs (T_1/2 = 30.167 y). The composition of RAW determines the necessary method for their disposal which should provide isolation of radionuclides for more than a thousand years.

Actinides are characterized by a wide variety of oxidation states. One of the factors determining the rate of actinides migration in the environment is their oxidation states. In the range of groundwater acidity as shown in Fig. 2, thorium, americium and curium (Th⁴⁺, Am³⁺ and Cm³⁺) exist in only one oxidation state. When uranium, neptunium and plutonium ions are reduced from the state of higher valency to ions in lower oxidation states (An³⁺ and An⁴⁺ are the designations of actinide ions), their solubility decreases and a high tendency to adsorption on the surfaces of minerals is observed.

Fig. 2

Diagram of the main valence states of actinides under various conditions. (-)—unstable; (?)—possible, but not confirmed; (bold/red) are the most common [11]

On the contrary, ions with valences V and VI in an aqueous solution are hydrolyzed to form linear antidioxo cations AnO₂⁺ and AnO₂²⁺, containing actinides [11, 12]. The strength of the actinide complexes (for a given ligand) usually decreases in decreasing order. At the same time, tetravalent actinides form stable aqueous complexes and poorly soluble solid phases, while pentavalent actinides form the least stable complexes and more soluble solid phases. The tendency to hydrolysis of actinides increases with increasing effective charge of the cation [13]: An⁴⁺  > AnO₂²⁺  > An³⁺  > AnO₂⁺.

The only stable oxidation state of americium is + 3. Therefore, the contribution to the migration of the solid and liquid phases, including the metallic form, can be neglected. The sorption of Am(III) by clays increases monotonically with pH value and becomes constant in the absence of CO₂ in the acidity range from neutral to basic [14]. The increased negative charge of the outer surface of clay minerals promotes the sorption of positively charged ions (Am³⁺, AmOH²⁺, Am(OH)₂⁺). However, in an alkaline environment, the presence of carbonates is a very important factor and, as a rule, reduces the absorption of Am(III) due to the formation of a negatively charged Am(III) carbonate compound.

The behavior of Np(V) needs to be studied because of its high solubility, mobility, and the possibility of entering groundwater during spills of RAW storage facilities [13, 15]. Under oxic conditions, the possible oxidation states of neptunium are Np(IV) and Np(V) [13]. [13]. Under slightly oxidizing conditions, the reduction reaction, as a result of corrosion of the steel tank, leads to the predominance of Np(IV). However, the presence of Np(V) should also be taken into account, since the ion in the form of NpO₂⁺ has a higher mobility [16]. To quantitatively characterize the mobility of NpO₂⁺ in the environment, a deep understanding of the adsorption behavior of this cation on artificial and natural barrier materials is required. Np(V) adsorption by montmorillonite is affected by pH. When pH < 5, adsorption is mainly determined by the ion exchange process. When pH > 5, Np(V) adsorption is controlled by surface complexation and increases monotonically with pH [17]. In the presence of carbonates, adsorption tends to decrease above pH = 8 due to the stability of aqueous solutions of NpO₂CO₃ and NpO₂(CO₃)₂³⁻ and their complexes with water [16, 18].

Being the main component of fuel for nuclear reactors, U and Pu are also elements that need to be given priority in the processing and disposal of radioactive waste. Plutonium can exist simultaneously in four oxidation states [19]. Under mild oxidation conditions, the dominant oxidation states are Pu(IV) having the lowest mobility and Pu(V) having the highest underground mobility [16]. Adsorption or ion exchange studies of Pu(V) or Pu(IV) are complicated by the simultaneous presence of unoxidized states and solubility limitations imposed by Pu(IV) hydroxide colloids [20]. The aqueous phase morphology of U(VI) can determine the distribution of uranium on mineral surfaces, the reduction of U(VI) to U(IV), and the incorporation of uranium into secondary precipitates such as iron oxides and other ones [21]. Most uranium salts are water soluble and usually exist in aqueous solutions as the linear UO₂²⁺ ion [22]. When pH > 4, uranium hydroxyl complexes compete with other inorganic and organic ligands in a solution, including carbonates, phosphates, sulfates, silicates, and n-carboxylic and humic acids (HA) [23]. Uranyl easily interacts with carbonate ions and organic complexes in a solution, forming various uranyl compounds UO₂²⁺, UO₂CO₃, UO₂(CO₃)₂²⁻, UO₂(CO₃)₃⁴⁻. According to the redox morphology of actinides, in an oxidizing environment (oxic conditions) they exist in a highly oxidized state, with the exception of plutonium, most of which is in the tetravalent state. Under anoxic conditions, An(IV) part is proportional to the total U(IV) concentration. This indicates the formation of hydroxides inherent U(IV) that bind other actinides. On the contrary, under oxidizing conditions, uranium and plutonium are adsorbed by natural colloids (amorphous aqueous oxides of Fe and Mn) [24].

Structure and general properties of clay minerals

Clay minerals (such as illite and mixed illite / montmorillonite) are the main mineral components of clay. Aluminum ions in the octahedral lattice of alumina in the clay mineral are isomorphically replaced by metal ions (such as magnesium, iron) with a lower charge. As a result, the flat (basal) surface of the clay mineral layer acquires a negative charge. Depending on the location of the charged defects of the mineral, the cations to be immobilized and exchanged in the interlayer space of the mineral are hydrated or form complexes, which may contain one or more layers located on a spherical surface around the cation [25]. The parameters of the radionuclide complexes affect the physical properties of the clay such as swelling, plasticity and permeability [26].

Clay minerals can be divided into several types, depending on their structure and isomorphic substitution in different parts of the mineral layers. According to their structure, clay minerals can be divided into three categories: 1:1 (for example, kaolinite), 2:1 (for example, montmorillonite, vermiculite, illite) and 2:1:1 (for example, chlorite). The 1:1 structure is a tetrahedral silicon oxide sheet and an octahedral aluminum oxide sheet, forming layers called TO [27, 28]. In contrast, clay minerals such as montmorillonite and illite, having the same 2:1 structure, in which some of the central cations in octahedral or tetrahedral sheets are isomorphically replaced by lower charged cations (for example, substitution of Al³⁺ with Mg²⁺, Al³⁺ with Fe²⁺, Si⁴⁺ with Al³⁺). Such incomplete substitution results in the formation of a negatively charged basal surface of the clay mineral layers, which is neutralized by the adsorption of counterions on the outer surface of the clay microparticles or between adjacent clay mineral layers. In montmorillonite, the isomeric substitution is concentrated on the octahedral sheet. Therefore, the interaction between clay layers and interlayer ions is weak and hydration of interlayer cations causes the clay to swell. If isomorphic substitution occurs on a tetrahedral sheet, a negative charge is formed on the outer surface of the clay mineral layer and interacts directly with interlayer counterions.

So, when weak hydration of interlayer K⁺ ions [29, 30], illite clay is characterized by a weak swelling property. Interlayer counterions have a fundamental influence on the swelling of clay—their type, nature, charge and even position relative to the structure of the mineral have a strong influence on the adsorption process [31,32,33,34]. In contrast, if the interlayer space is used to neutralize the negative charges on the surface of the clay layers with hydroxide sheets, 2:1:1 clay structure is formed. A mineral of this structure, such as chlorite, is often used as a material for filling cracks in rocks around nuclear waste storage sites [35]. Figure 3 shows the different structures of clay minerals, and Fig. 4 describes various types of clay swelling.

Fig. 3

Possible structures of clay minerals

Fig. 4

Schematic diagram of clay layer expansion [36]

The parameters of the radionuclide complexes affect the physical properties of the clay such as swelling, plasticity and permeability [26]. Due to the swelling of the clay mineral on contact with water, the compacted clay has extremely low hydraulic conductivity [37]. Therefore, diffusion is an important cation transfer process in clay [38,39,40,41,42,43]. In comparison, the diffusion of cations in clay has the highest rate, while the diffusion of neutral substances such as water is slower. Due to the repulsion of anions by negative charges on the surface of a clay particle, the diffusion of anions in the clay layer is the slowest one [44,45,46]. Cations can diffuse through the clay layer by different mechanisms. They can also pass into the intermediate (octahedral) layer of the clay mineral upon collapse of the interlayer space or abrasion of the layer edges [47].

Only two models explaining the mechanism of water absorption by clay are widely recognized: crystalline swelling [48] and osmotic swelling [49]. Due to the formation of one, two, three, or even five layers of hydrated ions in the space between the clay mineral layers, the distance between them can vary in the range of 9–20 Å [50]. For information: the diffusion coefficient of cations in monohydrated montmorillonite (MMT) is about 10^–10 m²∙s⁻¹ at atmospheric pressure and room temperature [51].

Concepts and constants used in clay-water and clay-ions interactions

Diffusion is a fundamental and irreversible process. During diffusion, matter is spontaneously transported from one part of the system to another by the random movement of molecules [52], passing from a high chemical potential to a low one. Diffusion is the migration of atoms only due to thermal vibrations caused by the difference in concentrations at different points in space.

In clay minerals, four different types of diffusion can be distinguished [53,54,55]: Self-diffusion; tracer diffusion; salt diffusion; and counter diffusion or interdiffusion. Initially the salt concentrations in solutions are the same on both sides. The diffusion process, measured by the amount of diffusing substance Na⁺, replaced by its isotope ²²Na⁺, is called self-diffusion. If the isotope is an ion of another element, this is called the diffusion of the absent agent to distinguish it from self-diffusion. On the other hand, the diffusion of salt is the diffusion of the solute into water (cations and anions in the same direction). Counter diffusion or interdiffusion describes a process in which different ions of the same concentration diffuse in opposite directions [54, 56].

Two methods are suitable for laboratory determination of the diffusion coefficients of substances in clays: stationary and transitional [57, 58]. The stationary method involves the diffusion of a single solute in a porous medium, assuming that the concentration gradient in the clay is constant. The diffusion process is described by Fick's first law:

$$ J = - D_{eff} \frac{\partial C}{{\partial x}}, $$

(1)

where J is the diffusion flux of the solute (g/cm²∙s), D_eff is the effective diffusion coefficient (cm²/s), C is the concentration of the diffusing substance in the solution, x is the spatial coordinate.

Fick's first law shows that a concentration gradient always causes diffusion. The "-" sign means that the direction of the diffusion flux is opposite to the direction of the concentration gradient. The diffusion flux per time unit through a unit of cross-sectional area perpendicular to the direction of the diffusion is proportional to the concentration gradient in the considered section of the route.

The steady state method is used to determine the effective diffusion coefficient of the solute without taking into account the value of the delay coefficient R, since, by definition, the solute has no delay in the steady state. By establishing a constant concentration gradient in a clay mineral when a steady state condition is reached, the effective diffusion coefficient D_eff can be determined using formula (2) [55]:

$$ D_{eff} = - \frac{\Delta x}{{\Delta C}} \cdot J_{D} = - \frac{L}{\Delta C} \cdot \frac{\Delta m}{{A\Delta t}} = - \frac{L}{A\Delta C} \cdot \frac{\Delta m}{{\Delta t}}, $$

(2)

where L and A are the width and cross-sectional area of the clay mineral, respectively; ∆C—concentration gradient; ∆m is the change in the mass of a chemical with time ∆t. The L/(A ∆C) value is constant. Therefore, only the change in mass ∆m/∆t needs to be measured.

The apparent diffusion coefficient can be obtained using Eq. (3):

$$ D_{a} = \frac{{l^{2} }}{{6\;t_{e} }}, $$

(3)

where t_e is the delay time of the cation release relative to the time of entry into the clay layer (s); D_a is the apparent diffusion coefficient (m² s⁻¹), l is the thickness of the clay layer.

At steady state, the diffusion coefficient obtained from Fick's first law implies that the difference in solution concentration between the two sides of the clay layer is held so that the concentration in the clay mineral changes linearly. However, the actual concentration gradient across the compacted clay is non-linear (Fig. 5b). Therefore, different assumptions about the concentration gradient across the clay (and inside and outside the clay particle) can give different values for the diffusion coefficients [55, 59, 60].

Fig. 5

Explanation of steady state diffusion. a Diffusion scheme; b Change in concentration of the solution in the clay [55]

The advantage of the stationary approach is evident in the fact that the diffusion coefficient can be obtained directly. However, the disadvantage is that it is necessary to maintain a constant concentration gradient throughout the volume to maintain its stationary conditions, so that Eq. (2) is fulfilled. Undoubtedly, this will take a lot of time and effort and can lead to an increase in the measurement error. However, with the appropriate equipment, the steady-state method can measure very low permeabilities in a short period of time (from 1 to 3 days) due to the low compressibility of clayey rocks, which allows the rapid propagation of pressure waves, as first shown in the work by Boulin et al. [61].

In transient methods, Fick's second law is used to find the diffusion coefficient, which is a combination of Fick's first law with the law of conservation of mass. In accordance with Fick's second law, the equation of transient diffusion of a stable chemical element in a permeable medium can be written as:

$$ \frac{\partial C}{{\partial t}} = D_{a} \frac{{\partial^{2} C}}{{\partial x^{2} }}, $$

(4)

$$ D_{a} = \frac{{D_{p} }}{R} = \frac{{D_{eff} }}{\alpha }, $$

(5)

where D_p is the self-diffusion coefficient of the porous system, R is the Retardation factor, and α is the Capacity factor.

The retention factor R used in Eq. (6) assumes linear, reversible and instant sorption of chemicals. Based on the mass balance of a substance in a porous medium saturated with water, the retention coefficient depends on the ratio of the total mass to the mass of the aqueous phase of a chemical substance in a unit of the total volume of the porous medium:

$$ R = 1 + \frac{\rho }{\varepsilon }K_{d} , $$

(6)

where K_d is the distribution coefficient (L³g⁻¹), ρ is the dry density of the porous medium (clay); ε—porosity of the clay layer.

The relationship between the Capacity factor α and the delay factor R in formula (7) is as follows:

$$ \alpha = \varepsilon + \rho \cdot K_{d} = \varepsilon \cdot R, $$

(7)

If a diffusing chemical substance is radioactive and decays [62], formula (6) is modified to take into account the effect of this decay on the temporal distribution of the components concentration at each point of the propagation path:

$$ \frac{\partial C}{{\partial t}} = D_{a} \frac{{\partial^{2} C}}{{\partial x^{2} }} - \lambda \cdot C, $$

(8)

where λ is the constant of radioactive decay of radionuclides (s⁻¹), λ = 0.693/T_1/2 [58].

If several radionuclides are involved in diffusion, it is necessary to make changes to Eq. (8) [63], taking into account the contribution of each nuclide in the direction of their diffusion:

$$ \frac{{\partial C_{n} }}{\partial t} = D_{a,n} \frac{{\partial^{2} C_{n} }}{{\partial x^{2} }} - \lambda_{n} \cdot C_{n} + \frac{{R_{n - 1} }}{{R_{n} }} \cdot \lambda_{n - 1} \cdot C_{n - 1} . $$

(9)

Three different types of water are distinguished in the clay layer containing an aqueous salt solution [64]. In the process of clay swelling, most of the water penetrates into the interlayer space of the clay mineral. Some of the water molecules are exposed to negative charges on the surface of the edge of the mineral layer, which squeeze out negative ions. Therefore, a double electrostatic diffusion double layer (DDL) is formed. The remaining solution, external water, is electrically neutral, as shown in Fig. 6. A third of the water is in the space between the clay mineral particles (pores in the compacted clay layer). In this case, anions cannot penetrate into the mineral due to the negative charge outside and inside the clay particle. Only cations and neutral molecules can enter the interlayer space of the mineral. Therefore, substances dissolved in water migrate in the clay layer along curved paths characterized by the bending coefficient. The anion has the highest bending coefficient for diffusion in clay. The bending coefficient of the trajectory of the water molecules and neutral particles is less than that of anions. Cations migrate along the least curved trajectories. Moreover, different cations have different bending coefficients [65,66,67].

Fig. 6

Three types of water in the clay layer. ①—interlayer water with exchanged cations, ②—diffuse double layer with excess positive charge, ③—charge balanced external pore water [64, 68]

Since the diffusion coefficient in water DDL is slightly lower than in the free pore water [51, 69], the interlayer water of the clay mineral becomes the dominant medium for the propagation of cations with an increase in the clay density above 1800 kg/m³ [70]. Therefore, the effective diffusion coefficients in the free pore water, DDL water and interlayer water can be separated. It can be seen from the above Fig. 6 that anions are mainly concentrated in the area of the free pore water. Therefore, one can write [71]:

$$ D_{eff} = \varepsilon_{free} \frac{{\delta_{free} }}{{\tau_{free}^{2} }}D_{W} , $$

(10)

where τ is the coefficient of tortuosity of the trajectory ε, a purely geometric coefficient that quantifies the length and complexity of the trajectory of molecules and ions migration through the pores; δ—Constrictivity (used to measure the contraction and expansion of the pore), usually taken δ = 1; and D_w is the diffusion coefficient of water molecules.

For cations and neutral atoms and molecules:

$$ D_{eff} = \left( {\varepsilon_{free} \frac{{\delta_{free} }}{{\tau_{free}^{2} }} + \varepsilon_{DDL} \frac{{\delta_{DDL} }}{{\tau_{DDL}^{2} }} + \varepsilon_{IL} \frac{{\delta_{IL} }}{{\tau_{IL}^{2} }}} \right) \cdot D_{W} , $$

(11)

where subscripts denote: free—free pore water, DDL—diffuse double layer water, IL—interlayer water.

The biggest advantage of the transient method is that it does not require constant maintenance of steady conditions, more accurately reflects the situation than maintaining a constant difference in source concentrations. This method is relatively straightforward. Disadvantages are the following: the saturation of unsaturated clay samples can take a long time and the boundary conditions can be inappropriate to apply the test results.

Simulating with molecular dynamics methods

Currently, many studies are being carried out with the use of various methods and instruments to determine the diffusion coefficient of radioactive elements in clay. However, experimental methods require a lot of resources (time and financial). For example, even a month of experimental research does not allow obtaining scientifically rigorous results. Moreover, it is difficult to experimentally investigate all the structural, mechanical, and kinetic properties of solid clay matrices, as well as liquids and solvents on micro- and mesoscopic scales [72]. Therefore, the understanding of the processes associated with adsorption and diffusion remains mainly at the macroscopic level. Molecular dynamics (MD) methods are often used to study the diffusion behavior of radioactive elements in clays, starting from the atomic level. Simulation makes it possible to study successively the effect of only one parameter on diffusion by stabilizing other influencing factors. On the basis of an ideal and simplified model, which differs from those used in experiments, a basic model is developed for subsequent experimental studies. At the same time, theoretical hypotheses are formulated and then verified by experiments. This can significantly reduce the amount of exploratory empirical research.

Figure 7 shows the spatial arrangement of atoms in the layers forming the illite mineral. Each layer of the mineral contains the central region in which the atoms are located at the nodes of the octahedron. In the peripheral regions of the layer, the atoms are located at the nodes of the tetrahedron. Therefore, the illite mineral layer can be designated as TOT (tetrahedron-octahedron-tetrahedron). The structure of TOT is characteristic of the minerals of many clays. In the octahedral structure of the clay mineral, one third of the oxygen positions are vacant [73]. Therefore, the octahedral structure is divided into two types—cis-vacant (cv) and trans-vacant (tv). In natural clays, octahedral structures cover a wide range of cis–trans-vacant structures. Typically, montmorillonite has cv structure and illite has tv structure [74].

Fig. 7

Schematic diagram of the layered crystal structure of the illite mineral with its two vacancy structures shown on the right

The middle region of each clay mineral layer (octahedron layer) is denser. The electric double layer fields on adjacent mineral surfaces overlap strongly, resulting in specific and improved sorption and stability of the cations contained in the interlayer space of the clay mineral.

In the MD methods, the basic structure of the clay mineral is set first, then the parameters of its model are set (for example, the size of the supercell, the magnetic field, the bond angle, the atomic charges). For structural optimization, the energy of the computational cell is minimized by the MD methods. In molecular simulating, some state variables are external parameters, while others need to be calculated.

Changes in any system are associated with the minimization of energy described by the laws of thermodynamics. Thermodynamic systems have state variables that describe such macroscopic states as the number of particles (N), volume (V), temperature (T), pressure (P), and total energy (E). The desired thermodynamic system for clay is combined according to the simulated situation at the RAW disposal site. The following thermodynamic states are commonly used for modeling by MD methods.

• 1. Microregular system (NVE), in which N, V and E are guaranteed to be constant, and the kinetic energy of atoms is continuously changing at a certain rate under the action of a force field. This changes the corresponding system temperature. It is an isolated system in which energy is stored inside the computational cell.

• 2. In the NVT system, the values of E and P parameters can fluctuate around a certain average value. Using an integrated NVT system requires a thermostat (e.g. Nosé-Hoover Thermostat, Langevin Thermostat, and the Andersen thermostats) [43, 75] to maintain a constant temperature by adding or removing energy to the system.

• 3. In the isobaric-isothermal (NPT) ensemble, not only temperature but also pressure is monitored. The NPT ensemble assumes a varying cell size.

For MD simulating, there are a number of force fields designed specifically to reproduce the properties of crystals and minerals, such as the BKS potential (van Beest, Kramer, van Santen), the pcff-INTERFACE force field [76] or the CLAYFF force field [38]. The Morse-charge equilibration force field (MS-Q FF) is designed to simulate processes typical for oil production [77]. Many force fields are interconnected using only combinatorial rules for the parameters of the unbound potential, for example, the Lorentz-Bertlow rule for these force fields. This relationship can lead to the fact that the charge on the two atomic sites of the bond will not be determined by means of a force field or will not correspond to the required chemical environment [78]. Therefore, the choice of an appropriate force field is also a key issue in MD modeling.

In MD, the spatial displacement of the cation is determined by calculating the mean square displacement (MSD) parallel to the basal surface of the clay mineral layer. MSD determines the diffusion rate. The diffusion coefficient is an important parameter for describing the movement of particles and can serve as an estimate of how quickly radionuclides migrate. For example, this parameter provides important parametric support for RAW geological disposal safety assessment methodologies.

In the interlayer space of clay minerals, movements are possible only in two orthogonal directions. Possible displacements in the direction perpendicular to the basal surface of the mineral layers do not exceed 20 Å, which does not affect the conditions for the macroscopic displacement of the cation [79]. The slope of the MSD plot (R(t)-R(0))²versus time t is used to estimate the mean value of the diffusion coefficient. To calculate the self-diffusion coefficients of interlayer cations and water molecules, the relation is used [79,80,81]:

$$ D = \frac{1}{2n} \cdot \frac{d}{dt}\left( {R\left( t \right) - R\left( 0 \right)} \right)^{2} , $$

(12)

where R(0) is the initial position of the cation, R(t) is the position of the ion in time t, and n is the dimension of space (the number of independent directions of ion diffusion).

When calculating the diffusion coefficient of cations in a clay mineral, n = 2 is used. The advantages of MD modeling are very clear. When simulating, you can specify conditions that are difficult to achieve in experiments (for example, high temperature and pressure). Simulation ensures no radiation hazard, and no material loss. Simulation allows for strict control of variables, describing atomic details and thermodynamic properties intuitively, starting from the molecular level. However, the disadvantages of MD modeling cannot be ignored. The calculated results obtained deviate from their possible values due to the use of a small calculation cell of the clay mineral, inaccurate setting of the distribution of charges in the layers. A real mineral is replaced by a structure obtained by periodically stitching the boundary conditions of individual calculated cells [82]. In a compacted clay layer, the mineral particles are oriented randomly, which makes it difficult to use the MD method.

Factors affecting the rate of diffusion cations in clay minerals

The ability of clay to absorb water depends on its minerals lattice, localization of charges on the surface of clay mineral layers, parameters of the surrounding solution and the pressure of the environment. All of these factors affect the mechanical stability of the clay [83]. Diffusion of ions and water molecules in the interlayer region of clay minerals is an important process which is determined by its ion exchange and sorption properties. Diffusion is also critical for the safe operation of high water discharge treatment tanks [84].

Influence of interlayer ions composition in clay minerals

The swelling of clay minerals as the beginning of the exchange of cations between the mineral interlayer space and an external aqueous solution is of great importance for RAW disposal. This is due to the fact that the pressure arising from the swelling of clay can change the range of local stresses and form the probability of shear-type failure [85]. Therefore, it is necessary to understand the effect of hydration of interlayer cations on the size of the interlayer space of clay minerals.

Many authors have studied the effect of different types of interlayer ions of alkali and alkaline earth metals on crystal swelling [86, 87]. The results of these studies show that the degree of crystal swelling depends on the intensity of cation hydration in the order Na-MMT > K-MMT > Cs-MMT > Mg-MMT > Ca-MMT. However, some authors have a different opinion [88, 89]. When comparing the swelling of Ca-MMT and Na-MMT, from the simulation results, it was found that the effect of hydration of Ca²⁺ on swelling was more significant than that of Na⁺. This is probably due to the use of different parameters for the simulation. Clay swelling depends mainly on the valence of the cation, rather than on its radius [31, 90]. Moreover, lower charges on the basal surface of the layer lead to greater swelling. This is also confirmed in [91], where it was shown that the order of hydration of interlayer cations changes in the following order: Mg > Ca > Na > K. If the hydration shell of the cation is larger and more stable, it has fewer opportunities for diffusion, compared to less hydrated complexes [75, 92].

The hydratability has a significant effect on the sorption of the cation on the clay mineral. For example, Cs⁺ is a weakly hydrated cation [93] and can easily detach from its hydration shell and form an “Inner-sphere complex” with oxygen atoms from the illite tetrahedral layer. These centers cannot absorb any highly hydrated monovalent and divalent cations (for example, Na⁺ and Ca²⁺). Therefore, illite selectively adsorbs Cs⁺ from a solution containing Na⁺, Ca²⁺, Cs⁺ [8, 94]. Indeed, it was found that regardless of the concentration of K⁺, Na⁺, Rb⁺ or NH₄⁺ cations used for the desorption of Cs⁺ from vermiculite and illite, Cs⁺ ions adsorbed on the surface of the mineral layers can only partially exchange [95, 96].

It was shown that the position of charges on the basal surface of the mineral layer also affects the transport of cations through the clay layer [82, 90]. The specific localization of the isomeric substitution of Cs-MMT in the TOT layer of the mineral does not change the solvation of the clay minerals, but can only slightly affect the thermodynamic, structural, and transport properties of the system. The higher the surface charge density of MMT, the higher the sorption energy between it and water molecules, the stronger the interaction. However, with an increase in the surface charge, the amount of absorbed exchangeable ions also increases.

It can be assumed that the influence of the radius value of interlayer cations may be associated with the effect on:

• 1. Residence time on the clay mineral layer. Cations with large radii are more likely to be immobilized by being captured in hexagonal rings on the basal surface of the clay mineral layer and remain in this position for some time [97];

• 2. Pressure of repulsion (Disintegrate) of the clay mineral layers. With a minimum interlayer space without water, the interaction force of the Cs-MMT layers is greater than that of Na-MMT and Sr-MMT due to the larger ion radius [98].

The effect of interlayer ions on clay mineral parameters is often studied using MD simulation. In some works, only the presence of one ion type in a mineral is simulated [31,32,33,34]. In many studies, data for single ions in the interlayer space of a clay mineral are compared with data for several types of ions. For example, we studied the swelling of clay containing simultaneously two exchangeable ions (Na/Cs-MMT, Na/Mg-MMT) and one interlayer ion (Cs-, Mg-MMT). It was found that the distance between the layers of the mineral after swelling is smaller for the clay containing also the Na ion in the interlayer space of the mineral [99]. This may be the result of the smaller interlayer Na⁺ cation taking some of the water molecules away from the clay mineral layer and forming a more compact complex. An interesting fact is that the self-diffusion of Na⁺ significantly decreases with an increase in the amount of interlayer Ca²⁺ ions, and the self-diffusion coefficient of Ca²⁺ remains practically unchanged and equal to 8·10⁻¹² m²/s. It was shown that Ca²⁺cations significantly affect the diffusion of Na⁺ and Ca²⁺ during the interlayer ion exchange of swollen clay particles [100].

Since montmorillonite is the most common highly swellable clay, the diffusion coefficients of various interlayer ions are compared using MMT as an example. These data are shown in Table 1.

Table 1 Diffusion coefficients of different interlayer ions and water molecules under different hydration

There is relatively little research on the effect of clay mineral layers charge on diffusion, especially on charge distribution over the surface of the layer. Also, relatively few types of ions have been studied compared to those that may be present in the disposal of radioactive waste. Further studies of diffusion and sorption of multiple ions or radionuclides and comparison of their differences are required. It should be noted that during the disposal of RAW, it is necessary to provide additional barriers for radionuclide ions present in impurity quantities.

Influence of solutes composition in the solution on diffusion

Remnants of nuclear fuel (containing uranium and plutonium) and its fission products (Cs, I) are simultaneously located in RAW disposals which use clay as a barrier material. Therefore, it is important to study the migration of radionuclide cations and the effect of other cations on their migration. Knowledge of the migration mechanism of these cations through the barrier material is a prerequisite for long-term RAW disposal.

Since Al in the octahedral layer of the clay mineral is replaced by other elements with a lower valence (Mg, Fe, Li, etc.), a negative charge is formed, which will facilitate better diffusion of cations to the outer surface of the clay particle and between the layers of the mineral. Under these conditions, anions will repel the clay mineral. The process of sorption of most ions on clay is influenced by cation exchange. It is known that the adsorption force of metal cations to the outer surface of an MMT particle decreases in the order K⁺ > Na⁺ > Ca²⁺ > Cs⁺ > Ba²⁺ [106]. Comparison of the data for the elements of alkali and alkaline earth metals located in the same period of the table of elements shows that the adsorption force of alkali metals is greater. Moreover, the smaller the radius is, the stronger the adsorption.

When simulating the adsorption of Ca²⁺, Na⁺, Cl⁻, Br⁻ ions in clay by MD methods, it was established [79] that cations are attracted by a clay particle when they are in close proximity to its surface. Cl⁻ ions are repelled by the clay particle due to the interaction of negative electric charges. The presence of Cl⁻ ions in the interlayer space is excluded and also prohibited in some small nanopores between clay particles [107]. It should be borne in mind that Cl⁻ ions have a strong attraction to water molecules located between the layers of the clay mineral [108]. In experimental studies, salts such as NH₄Cl, AlCl₃, MgCl₂ and FeCl₃ were added to Na-MMT. It was observed that these inorganic salts inhibited the expansion of the clay layer by increasing the fluidity of the ions between the layers of the clay mineral and decreasing the hydration shell of the ions. The greatest effect of suppressing clay swelling was observed for NH₄Cl at high temperature and pressure [108]. These data show that there is competition in adsorption between ions in solution. Therefore, it is possible to use a large amount of a salt solution with stable isotopes to reduce the adsorption of radionuclides and counteract the migration of radionuclides.

For example, the migration of Eu³⁺ is significantly affected by the presence of carbonate in the solution, as shown in Fig. 8a. At pH > 5, CO₂ from the air dissolves in the solution and reacts with hydroxyl radicals. This leads to the formation of Eu(OH)CO₃. Coexistence of Eu(OH)CO₃, Eu(OH)₂CO₃⁻, EuOH(CO₃)₂²⁻ ions in the solution is observed, which reduces the concentration of ions from one element (Eu³⁺). Therefore, the presence of carbonate helps to delay the migration of Eu³⁺ ions [109]. Experimental studies Tachi et al. [18] showed that after the addition of 0.01 M sodium bicarbonate to the NaCl solution, the effective diffusion coefficient of Np⁵⁺ decreases by up to 30 times from the initial value.

Fig. 8

The distribution of aqueous species of a) Eu(III) in 0.001 mol/L NaCl with 0.001 mol/L CO₃²⁻[109], b) solutions as a function of pH at the atmospheric partial pressure of CO₂(g) (pCO₂ = 10.^−3.45 atm) [110]

Since uranium is the main component of nuclear fuel, UO₂²⁺ is also the most studied actinide ion. Uranyl ions can be found in different forms [21, 111]. The presence of carbonates in solution inhibits the production of U(VI) precipitates in solution [110], forming multiple carbonates (see Fig. 8b). The series in the order of the decrease in the diffusion coefficient is associated with an increase in the mass of the ion UO₂²⁺ > UO₂CO₃ > UO₂(CO₃)₂²⁻ > UO₂(CO₃)₃⁴⁻. Therefore, it is obvious that the appearance of carbonate in the solution slows down the diffusion of uranium. This may also be associated with a decrease in clay porosity [112]. From the above experimental data, it can be assumed that carbonate can effectively slow down the migration of radionuclides. This creates prospects for using the acidity of clay to control the rate of the diffusion of radionuclides. For example, by adding a mixture of hydrocarbons to clay, a reliable RAW storage facility can be created when disposing of nuclear waste. This is because when CO₂ is added, a solid rock is formed from the clay used as a barrier material.

Likewise, the presence of sulphate will also have some effect on the migration of cations in the clay. Although SO₄²⁻ is a divalent anion, its repulsive effect is similar to that of a monovalent anion such as Cl⁻. The sulphate content in the pore water of opal can be increased by microorganisms, the colony of which is located in the immediate vicinity of the RAW storage [113]. Therefore, sulphate can be considered as a consequence of chemical or biological processes. The diffusion coefficient of SO₄²⁻ in opal is D_e = 0.2–0.6 × 10⁻¹² m² s⁻¹. When sulfate ion interacts with opal, the available porosity decreases by almost 10 times relative to the total porosity [113]. This leads to a decrease in the diffusion coefficient. In addition, sulfate-reducing microorganisms can decrease excess hydrogen generated by oxygen-free corrosion of steel through reduction reactions and improve the safety factor of geological disposal of nuclear waste [114].

The influence of different ions on diffusion turns out to be very significant not only in terms of cations from one element, but also in terms of complex cations (NH₄⁺, UO₂²⁺) and anions. A wide variety of chemical elements are observed in RAW disposal, which requires further study of joint diffusion of two or more radionuclides in clay. For example, the clay barrier layer may come in contact with regions containing acidic salts composed of other radioactive elements such as Eu, Np, U, or acidic residues such as SO₃²⁺ and NO₃⁺.

Influence of the ionic strength of the solution on diffusion

The concentration gradient is known to induce diffusion. However, an increase in thesalt concentration decreases the degree of sorption of cations in clay and decreases the concentration gradient of cations in the pores of a clay layer. Therefore, it can be assumed [46] that the effective diffusion coefficient in clay depends on the ionic strength of the solution. There are two main factors in this process [115,116,117]:

• 1. Different complexing ability of ions containing different heavy metals.

• 2. Competition between metal ions and their positively charged compounds during sorption in the interlayer space of a clay mineral [115, 118].

These two factors act with different efficiencies in accordance with different types and amounts of metal ions. The solutes are expected to behave differently with increasing salt concentration.

Kozaki et al. [119] found that an increase in the salt concentration from 0 to 0.1 M decreases the distance between the swollen Na-MMT layers from 18.8 Å to 15.6 Å, which means a decrease in the number of single-molecular water layers. In this case, the diffusion coefficient of ²²Na changes insignificantly. With an increase in the salt concentration to 0.5 M, the diffusion coefficient slightly increases.

Increasing the concentration of substances used to pretreat clay with a background solution also accelerates the migration of solutes (Cs and Sr) in solution [120, 121]. This is due to the increased competition among Cs⁺, Sr²⁺ and Na⁺ ions in the area of the clay mineral containing silicon oxide. Therefore, an increase in the ionic strength of the background solution from 0.1 mM to 2.0 mM leads to a change in the mobility ratio and an excess of the migration rate of Cs ions relative to Sr ions. However, according to the results [122], the sorption of cesium with the formation of the inner-sphere complex should not be so sensitive to changes in ionic strength than for strontium. Therefore, other control factors (e.g. water content, ion concentration) must also be considered in the effect of clay pretreatment on diffusion. It is known that Cs makes a greater contribution to sorption even when the Cs concentration is only one tenth of the Sr concentration [121].

Table 2 shows the knowledge of the diffusion coefficients of Sr at various ionic strengths studied by various authors. The table shows that the effective diffusion coefficient D_eff of strontium decreases significantly as the concentration of the MMT treatment solution increases. Experimental data [46] show that there is a decrease in the diffusion coefficients by up to 4 times. This does not contradict the data of Ning et al. [123], according to which the sorption of Sr²⁺ changes by almost 10 times with a tenfold change in the concentration of the background solution. Due to the expansion behavior of clays, the sorption capacity for Sr²⁺ions decrease in the order allophane > vermiculite >  > illite > kaolinite. These data show that ion concentration does have a significant effect on the diffusion of dissolved radionuclides.

Table 2 Comparison of the diffusion coefficient and sorption data for Sr

If the ionic strength of the solution affects the adsorption process by weakening the surface electric field, an increase in the ionic strength leads to a decrease in the sorption of metal cations and promotes the sorption of anions [127]. Due to the electrostatic repulsion of negative charges by the layers of the clay mineral, the anion cannot get into the interlayer space of the mineral or even is displaced from free pore water. However, a high salt concentration reduces the size of the DDL region by decreasing the volume per charge, which is expressed by the Debye formula:

$$ k^{ - 1} = \frac{0.304}{{\sqrt I }}, $$

(13)

where k⁻¹ is the thickness of the diffusion double electric layer; I is the ionic strength of the electrolyte.

The ionic strength of the solution affects the diffusion of the solutes in compacted clay by changing the thickness of the electric double layer on the charged surface of the clay particles [128]. Therefore, with increasing salt concentration, the layer of free water between clay particles expands, increasing the effective diffusion coefficient [129, 130]. However, upon the sorption of several solutes from the solution, as the ionic strength increases, the sorption capacity of both anions and cations will decrease [109, 127, 131]. It should be taken into account that in MD, according to the classical model of the pore water diffusion, only geometric effects affect the effective diffusion coefficient of a solute in pore water and it does not depend on the coefficient of interphase distribution of ions [132].

The sorption of Cd²⁺ and Pb²⁺ ions in Turkish illitic Clay (TIC) was measured under various conditions [133]. As the salt concentration increased, the sorption capacity of TIC for ions decreased [134]. The diffusion of Re⁷⁺ and Se⁴⁺ in compacted bentonite forms such as GMZ and MMT was compared. The change in ionic strength had little effect on the sorption of Se⁴⁺ into bentonite [129]. However, at pH < 8.0, the effect of ionic strength on the sorption of Eu³⁺ and Sr²⁺ is greater than at pH > 8.0 [109, 135]. Therefore, it is necessary to consider other external factors as well. When studying the effect of ionic strength on the diffusion of a solute, factors such as the type and structure of the clay, as well as the pH of the solution, should also be taken into account.

Characteristics of the surface charge were investigated on colloidal bentonite at varying ionic strength [136, 137]. Bentonite colloids are in a stable state when the ionic strength of the NaClO₄ solution is below 0.01 M. With an increase in the concentration of NaClO₄ solution to 0.1 M, colloidal particles of bentonite begin to show instability. This instability demonstrates itself as a tendency for the colloidal particles to coagulate. In ionic strength studies, three solutions of salts NaCl, NaClO₄, CaCl₂ were used, mainly MMT and vermiculite [138]. The research results show that the properties of different clays differ significantly at different values of the ionic strength of the solution.

Effect ofsolution acidity (pH value) on diffusion

It is known that MMT and vermiculite can sorb heavy metal ions through two mechanisms [139]:

• Cation exchange at a point with minimum potential energy located in the interlayer space, the probability of which depends on the magnitude of electrostatic interaction with a constant negative charge of the basal surface of the mineral layer.

• Sorption with the formation of a complex on the basal surface with groups of mineral atoms, in the center of which are the Si or Al atoms of the octahedral layer.

Both mechanisms are affected by the pH value, due to the fact that most of the groups of silyl and aluminum alcohols on the basal surface of the mineral layer are protonated at a pH value of less than 4 [140].

The surface functional atomic groups of clay can be classified into two types: ≡LH and ≡XOH. The first ≡LH type is only deprotonated functional groups of Si–O atoms. The second type includes amphoteric surface Al–OH functional groups. The behavior of both centers during protonation can be described by schemes [127]:

$$ \equiv {\text{LH}} \leftrightarrow {\text{H}}^{ + } + \equiv {\text{L}}^{ - } , $$

(14)

with process constant (K_a) $K_{a} = \frac{{\left[ { \equiv L^{ - } } \right] \cdot \alpha_{H} }}{{\left[ { \equiv LH} \right]}}$,

$$ \equiv {\text{XOH}} \leftrightarrow {\text{H}}^{ + } + \equiv {\text{XO}}^{ - } , $$

(15)

with process constant (K_a-)$K_{{a^{ - } }} = \frac{{\left[ { \equiv XO^{ - } } \right] \cdot \alpha_{H} }}{{\left[ { \equiv XOH} \right]}}$,

$$ \equiv {\text{XOH}} + {\text{H}}^{ + } \leftrightarrow \equiv {\text{XOH}}_{{2}}^{ + } , $$

(16)

with process constant (K_a+)$K_{{a^{ + } }} = \frac{{\left[ { \equiv XOH_{2}^{ + } } \right]}}{{\left[ { \equiv XOH} \right] \cdot \alpha_{H} }}$,where square brackets indicate the concentration of the surface functional group, α_H⁺ is the activity of protons in the solution.

The sorption of metal ions by clay decreases with decreasing pH, since the protonation of the groups of aluminol and silicol increases. This effect can be used to retain a small number of metal ions under study. For example, in the article [141] it was shown that at pH higher than 5.5, the sorption capacity of MMT for Cd²⁺, Cu²⁺, Ni²⁺, Mn²⁺, Pb²⁺ and Zn²⁺ ions is close to 100%. Also, vermiculite at pH = 8.0 sorbs almost 100% of Cu²⁺, Pb²⁺ and Zn²⁺ ions. The sorption capacity in vermiculite decreases in the following order [142]: Mn²⁺ > Ni²⁺ > Zn²⁺ > Cd²⁺ > Cu²⁺ > Pb²⁺. This order can be related to the concentration and atomic weight of free metal ions in the solution, but does not depend on the ionic radius, ionic potential, and enthalpy of hydration. The sorption of Cd²⁺ and Cu²⁺ ions on Ghassoul clay yielded the same result [143]. At low pH, the K_d value increases with decreasing ionic strength [144], but at pH > 5, the K_d value does not depend on acidity. This behavior can be interpreted as an increase in the contribution of the cation associated with a flat surface at low pH. At the same time, in the marginal region of the interlayer space of the mineral, preference is given to protons [145].

According to a study [138], the negative charge of the minerals of three clays: kaolinite, MMT, and illite increased monotonically with an increase in the pH value from 3 to 9. The highest charge is observed in MMT. This may be due to the fact that the clay was pretreated with an acid. In this case, the surface ions of Al and Si are partially dissolved, which increases the surface area of the clay particles and the contribution of the clay structure. Of course, pH will have some effect on the properties of the clay. For example, under low pH conditions, the process of illite sorption is the capture of Na⁺ or H⁺ ions by the basal surface through the hydration shell (complex of the outer sphere). At high pH, complexes of the inner sphere are the main ones, when the ion interacts directly with the mineral. As a result, when pH is less than 4.5, the sorption capacity of Th⁴⁺ on illite increases with increasing pH or temperature, but decreases with increasing ionic strength [146].

Similarly, diffusion of Se⁴⁺ in clay from the Chinese Tamusu region decreases with increasing pH under acidic conditions (pH = 4–7) while under alkaline conditions the diffusion coefficient increases with increasing pH [130]. This behavior can be interpreted as follows: at pH below 7, the anion in the solution binds with hydrogen and oxygen ions to form an acidic salt (see Fig. 9).

Fig. 9

Types of Se salts at different pH values [62]

The interaction of these anions with a positive charge at the edge of the mineral layer leads to an increase in the sorption rate, and therefore to a decrease in the diffusion rate. When the pH is higher than 7, the anion in the solution is repelled by the negative charge on the surface of the clay particles, which accelerates the diffusion [130, 147]. Whether the result is a simple action on the Se ion, or whether all salts of the acid show the same effect, further research is required.

Influence of the clay layer porosity on diffusion

The degree of compaction of a clay layer depends on the mutual geometric arrangement of its particles and the distribution of water molecules between its free pores and mineral layers [148]. When pressed under high pressure, clay forms a porous homogeneous medium with low permeability. As the compaction of the clay increases, the dry density increases and the porosity and pore size decrease [149]. Compactness mainly affects the total porosity, which is composed of pores of medium and micro sizes. It is shown that the pore size affects the adsorption of cations in nanopores between clay particles (on the outer surface of the particles). At the same time, a change in the pore width affects the degree of adsorption, structure, dynamics, and stability of metal ions representing complexes of the inner and outer spheres [32].

Figure 10 shows the relationship between the dry density of bentonite clay and its porosity. The Figure shows that the dry density of bentonite clay has an inverse relationship with its porosity, which decreases linearly with increasing density. Then it follows from Eq. (7) that ρ/ε is constant, and the delay coefficient R depends only on the distribution coefficient K_d.

Fig. 10

Dry density of bentonite as a function of porosity [150]

Wu et al. [151] and Muurinen [150] similarly concluded that an increase in dry density leads to a corresponding decrease in permeability, thereby reducing the diffusion coefficient of Re²⁺ ions. The difference between these data and the data of other authors led them [150, 151] to propose the concept of accessible porosity ε_acc, which in the case of radionuclide anions is equal to the free porosity ε_free. The specific migration can see Fig. 11. The relationship between different types of porosity is as follows:

$$ \varepsilon_{acc} = \varepsilon_{free} = \varepsilon - \varepsilon_{IL} - \varepsilon_{DDL} , $$

(17)

which ε – which ε is the total porosity between clay microparticles and between mineral layers, ε_DDL – porosity of diffuse double layer, ε_IL – porosity of interlayer.

Fig. 11

Diagram showing different pores in bentonite and potential diffusion trajectories [151]. 1- interlayer, 2—diffusion double layer, 3—free layer, 4—interparticle space

It was shown that the porosity between clay mineral layers depends on the MMT type and the dry density of bentonite [152, 153]. However, when clay is compacted, the distance between clay particles decreases, while the interlayer space of the mineral changes to a lesser extent by mechanical forces [151, 154]. The change in the size of the interlayer space with an increase in the degree of compaction is associated with the amount of interlayer water. An increase in the density of dry matter from 1300 to 1800 kg/m³ reduces the amount of interlayer water of bentonite from 3 to 2 layers, and when the density increases above 1900 kg/m³, only 1 layer of water remains [155]. This means that when the density of dry bentonite (or its main component, MMT) is more than 1900 kg/m³, the total porosity of the clay is minimal.

Bentonite clays with a dry density above 1900 kg/m³ have a low total porosity, which can significantly limit the diffusion of salts in solution. However, most researchers are studying various aspects of using clay with a dry density of less than 1500 kg/m³ [66, 124, 126, 154].

Kozaki et al. [156]showed that the activation energies for the diffusion of Na⁺, Cl⁻ and HTO change depending on the dry density of bentonite. For example, with a dry density of less than 900 kg/m³, the activation energies of Na⁺ and Cl⁻ are less than in free pore water. On the contrary, if the dry density is greater than 1000 kg/m³, then the activation energies of Na⁺ and Cl⁻ increase significantly. In general, the activation energy in dry clay is higher than in free pore water, which indicates the breaking of chemical bonds between the cation and the mineral [153]. The higher the density of bentonite and the higher the activation energy, the more significant is the diffusion of a water molecule between layers of clay mineral, in comparison with free pore water.

It is known that as the force exerted to compact the clay increases, the porosity and cation permeability of the clay decreases [157]. With the same porosity, there are also differences in the mobility of water in different pore volumes. In comparison, the porosity between clay particles has a significant effect on the water diffusion process, while a small number of water molecules is sorbed between the clay mineral layers [158]. In MMT, the mobility of water molecules between the layers of the clay mineral is up to 3 times lower than in pores between clay particles [159]. Pore fluids can be adsorbed in an open pore after the pore volume is reduced. It is also possible to change the direction of movement due to a change in the orientation angle of the surface of the clay particles relative to the flow [160]. This changes the length and tortuosity of the cation diffusion trajectory. An important role in the diffusion of a solute in water is played by the mobility of water molecules. The diffusion coefficients of the corresponding ions can be obtained using formulae (11) and (12).

Effect of temperature on the diffusion of cations in clay

Temperature has a great influence on the diffusion of a solute in clay. Therefore, changes in soil temperature affect the transport of pollutants in clay. The thermal effect of the solute diffusion in compacted clay caused by a temperature gradient (Soret effect) shows that the effect of temperature on the solute diffusion coefficient (Ds) depends not only on the viscosity of pore water, but also closely correlates with cation mobility [161].

The construction of a RAW disposal reservoir using clay is based on the assumption that the conductivity of the aqueous solution in clay is very low (less than 10^–7 cm²/s). Moreover, the diffusion of water-soluble forms of radionuclides has an important effect on the transport of radioactive substances in clay [162]. It was found that the diffusion coefficient of water molecules and ions in a clay layer increases with increasing temperature [59, 163]. An increase in temperature causes a decrease in the hydration shell of cations, increasing the number of free water molecules between layers in the mineral and increasing the diffusion of the interlayer substance [97]. In addition, a comparison of the diffusion coefficients of water molecules and cations shows that temperature has a greater effect on the diffusion of water molecules than cations—the diffusion of water molecules is more intense at high temperatures [21, 51, 164]. Figure 12 shows the graphs of the dependence of the diffusion coefficient of water molecules on temperature. The observed patterns are associated with the fact that heating changes the structure of clay particles. For example, with increasing temperature, both the total volume and the porosity of warmer clay particles increase [165].

Fig. 12

Water diffusion coefficients as a function of T in the bulk and in mono- and bihydrated homoionic montmorillonites [51]

The diffusion of water is also strongly influenced by the degree of hydration of clay. With a small interlayer distance, low hydration leads to the attraction of the cation by negatively charged layers of clay minerals. Due to the Coulomb force, the distance between particles of low-hydrated clay increases slightly with increasing temperature, which does not lead to an increase in the size of the space for the diffusion of cations. Therefore, the diffusion coefficient increases only slightly. With increasing hydration, the Coulomb force is weakened, which increases the effect of temperature on the expansion of the clay mineral. In this case, the cation can diffuse in a larger interlayer space, and the diffusion coefficient increases [166]. It is known that the smaller the cation radius, the larger the size of the hydration shell. As the size of the hydrate decreases with heating, the number of free water molecules increases. Therefore, the diffusion movement increases significantly with increasing temperature [81].

This may also explain why in the experiments carried out in 1974 [53] the diffusion rate of anions was higher than that of cations, and this result was the opposite of the data presented by other authors [44,45,46]. This is possible because the measured clay hydrates were less in size and strongly influenced by Coulomb forces. This led to the dependence of the diffusion coefficients on the ion radius.

On the other hand, the effect of temperature on the diffusion of different types of cations in clay is significantly different. The effect of temperature on the sorption of Pb in bentonite was studied. It was found that the sorption of lead decreases with increasing temperature [167]. A higher temperature of bentonite containing sodium promotes copper sorption [168], and the zero charge point (pH_pzc) of bentonite is observed at lower pH as the temperature rises. This enhances the copper sorption effect at higher temperatures. The influence of temperature on the diffusion and sorption of heavy metals is a very important factor. The effective diffusion coefficient of metal ions (Zn and Cd) when heated from 15 °C to 55 °C increases up to 10 times, depending on the retention mechanism in the clay and the metal retention factor [59]. The migration of several common radioactive metal cations (Np, Se, Cs, Sr, Eu, U) in various clays at different temperatures is presented in Table 3. In granular clay with a dry density of 1.2–2.6 g/cm³ in the temperature range of 25–100 °C, the effective diffusion coefficient of uranium is 0.05·10^–11-2.32·10^–11 m²/s. Under the same conditions, due to a change in the distribution of pores by size as a result of mechanical action or heating, the apparent diffusion coefficient of uranium changes to 0.17·10^–11–4.20·10^–11 m²/s [165].

Table 3 Migration of different cations at different temperatures

Exposure to high temperatures leads to the destruction of the mineral interlayer space and the fixation of ions in the lattice of clay, which makes it impossible for the continuation of diffusion. This process is shown schematically in Fig. 13. Smaller ions such as Cu ions can be effectively incorporated into the lattice of a clay mineral at relatively low temperatures (200 °C) and cannot diffuse further. Ions with a larger radius, such as Cd ions, require a higher temperature heat treatment (above 500 °C) for incorporation into the mineral, leading to a complete cessation of diffusion [180]. The fixation of cations is due to the fact that at high temperatures the structure of the clay changes irrevocably.

Fig. 13

Scheme of isolation of heavy metal cations in MMT by high-temperature heating [180]

The core temperature of a nuclear waste repository can reach 70 °C [181], and the temperature of the entire repository rises to 30–50 °C [182]. At the same time, the temperature of heated radioactive waste stored in the container is in the range of 100–150 °C [183,184,185]. However, at present, the diffusion of radioactive cations in clay at temperatures of ~ 150 °C is insufficiently studied.

Effect of clay colloids on the migration of radioactive metal ions

Various materials, including mineral and organic phases, can form colloidal particles. However, Aluminosilicate clay minerals are one of the most common types of colloids found in groundwater [186]. Clay colloids are abundant in groundwater and can act as carriers for many organic and inorganic pollutants. For this reason, the stability and migration of clay colloids has a significant impact on the transport of pollutants and the quality of groundwater. Upon contact with low ionic strength groundwater, bentonite barriers can release stable colloidal montmorillonite particles [187]. Most importantly, the instability of clay colloids can significantly slow down the transport of contaminants adsorbed on the surface of the clay. Due to this, the instability of clay colloids has a noticeable effect on the migration of pollutants [186].

The behavior of colloids in the environment is mainly determined by the chemical nature of surface functional groups, microparticle size and zeta potential. Different types of colloids can interact through electrostatic interactions, ligand exchange reactions and weak interactions (for example, van der Waals interactions) both among themselves and with the environment [188], thereby controlling the ecological behavior of colloids. The nature of the colloids formed by various clays is related to the properties of the clay mineral, in particular: the content of montmorillonite, the composition of interlayer cations, the total charge of the layer, and the charge distribution between tetrahedral and octahedral layers [189].

It was shown that inorganic clay colloids resulting from the degradation of engineering barrier materials affect the migration of radionuclides [190]. Experimental studies at the Grimsel test site showed that the presence of bentonite colloids significantly affects the migration abilities of Am and Pu. It was found that the amount of Am and Pu passing through the granite layer is 20–30% in the absence of bentonite and 60–80% in the presence of bentonite colloids [191]. The migration of Sr in the presence of colloidal particles is slower than in the dissolved form. This is due to the retention of Sr²⁺ ions by the bentonite colloid, 70% of which remains in the faults of the granite layer of the soil [192].

However, there are also opposing opinions that the creation of colloids can lead to the degradation of engineering barriers. The subsequent joint migration of radionuclides adsorbed on colloids may lead to a decrease in the effectiveness of the natural barriers of clay deposits [193]. Radionuclides form complexes with colloidal particles that diffuse into the rock matrix and can facilitate the migration of previously immobile contaminants. The combination of radioactive Pu with bentonite colloids forms complexes that are weakly retained in cracks, even at relatively low water flow rates. This may be a key factor in the increased mobility of Pu [194]. In contrast, the effect of bentonite colloids on Eu migration is limited due to the reversibility of the bond between radionuclide and colloid.

Due to their small size, clay colloids (1.0 nm to 1.0 μm) can continuously migrate along with the water flow [136]. This provides the radioactive metal ions that are attached to the clay colloids with an additional migration route. The colloids allow even ions that otherwise could not move to migrate with the clay colloid. This means that it is necessary to take into account not only the migration of radioactive metal cations in the clay colloid, but also the migration of the clay colloid in aqueous systems.

Important factors influencing the migration of clay colloids are ionic strength and acidity/alkalinity mentioned above. Bentonite colloids are very stable at low ionic strength (⩽1 × 10^–3 M) and in alkaline water. They were found to remain stable in the Grimsel groundwater for several months [195]. At the same time, a number of scientists note that the presence of humic acid (HA) substance has a strong effect on clay colloids [187]196. Experiments on the diffusion of lignin sulfonate nanoparticles (average size 80 nm) and humic acid (average size < 10 nm) through MX-80 bentonite (dry density from 600 to 1800 kg/cm³) showed that the diffusion rate of these organic nanoparticles is in the same range as negatively charged ions such as Cl⁻ and I⁻ [187]. Bovine serum albumin (BSA) is particularly effective in preventing colloidal aggregation at high ionic strengths and improving the colloidal stability and transport [196]. The increased energy of attractive interactions between kaolinite particles and quartz sand makes the transport and retention of kaolinite highly dependent on ionic strength in the absence of BSA [196, 197]. Due to electrostatic attraction, gibbsite colloids and HA are linked and migrate simultaneously, regardless of the presence of U(VI). In addition to the influence of pH and ionic strength, the ability of gibbsite colloids to transfer U(VI) depends on the concentration of HA [198]. Studies by Chen et al. showed that adsorption of HA-U(VI) on kaolinite colloids forms ternary complexes that increase the mobility of uranium in heterogeneous media [199].

Conclusion

To this date, many studies have been carried out to improve the external and internal properties of clays, aimed at improving the parameters and increasing the service life of barrier materials for RAW geological disposal sites. Although many studies on the diffusion of cations of various radionuclides in clays have been conducted, the models for predicting the migration of radionuclides from RAW repositories are still not developed. Many articles ignore the influence of time and uncontrollable factors on the barrier properties of clay. This article summarizes some of the research findings.

• 1. Considerable attention is paid to simulating various aspects of clay by methods of molecular dynamics. Simulations have both advantages and disadvantages. Various authors, when simulating, considered significantly fewer types of ions between clay layers and limited boundary conditions, far from reality. These limitations can lead to significant errors in the results of calculations at the molecular level. However, an increase in the size of the computational cell, bringing the simulation conditions closer to real ones, leads to a significant increase in the computation time.

• 2. Exchangeable ions in clay mineral layers play a decisive role in clay swelling. However, at present, researches are rather monotonous and most of them study the finding of one type of ions in the interlayer space of the mineral, which does not correspond to the conditions in natural clays. Therefore, it is necessary to investigate the simultaneous effect of several interlayer ions on the properties of clays. In addition, the distribution of charges and exchanging ions over various surfaces of the clay mineral has also been little studied.

• 3. An increase in a dry density of clay over 1900 kg/m³ reduces the total porosity, which can significantly limit the diffusion of salts in solution. At present, most studies have been carried out for clays with a dry density of less than 1500 kg/m³ solution [66, 124, 126, 154]. However, even at a density of 1500 kg/m³ clay has a significant impact on the rate of cations diffusion.

• 4. Competition for a constant positive potential located on the basal surface in the interlayer space of the clay mineral leads to the displacement of more mobile Na⁺ and K⁺ ions. Therefore, at high concentrations of Na⁺ and K⁺ salts during pre-treatment, the migration of radionuclides in clay is largely suppressed.

• 5. The pH of the solution has the greatest influence on diffusion, since it can promote the combination of hydrogen and oxygen with radionuclides when at a certain pH value, the formation of new anions or cations occurs. At the same time, some acidic residues can also react with radionuclides to form acidic salts in a certain pH range. All these processes complicate the radionuclide diffusion trajectory, effectively slowing down the diffusion.

• 6. Colloidal transport of radionuclides is possible, explained due to sorption of forms of radionuclides insoluble in water. Thus, it is necessary to consider not only the migration of radioactive metal cations in the clay colloid, but also the migration of the clay colloid in aqueous systems.

• 7. The study of radionuclides has so far been limited to Sr²⁺, Cs⁺, I⁻, UO₂²⁺ and other isotopes with a long half-life. At the same time, no less important radionuclides such as Np and Pu have been studied insufficiently.

• 8. Pressure and temperature increase with the depth of the underground well. However, the number of the studies simulating such RAW geological disposal conditions is limited. The temperatures of most of the studies carried out at this stage do not reach the actual temperatures of disposal sites. Therefore, research results cannot be used to predict ion mobility at the actual temperatures (70–150 °C) at RAW disposal sites. Also, the possibility of changing the structure of clay at these temperatures is not taken into account.