Introduction

The disposal of high-level radioactive waste is the focus and difficulty of nuclear waste treatment, and at present, deep geological disposal is widely accepted and the most feasible high-level radioactive waste disposal method. The construction of the repository takes into account the time scale of tens of thousands of years to prevent the migration of radionuclides out of the repository and leakage into the biosphere. Therefore, the determination of the migration parameters of key radionuclides in high-level radioactive wastes in barrier materials is critical and is a key issue in safety assessment. In China, the Beishan area of Gansu has been identified as the preferred pre-selection area for high-level radioactive waste geological repositories [1]. The surrounding rock in the area is granite, which serves as the outermost barrier to the repository, and its barrier properties are critical, so it is particularly necessary to study the migration behavior of radionuclides in Beishan granite.

The chemical composition and structure of granite are commonly characterized by X-ray diffraction and scanning electron microscopy. Granite is mainly composed of quartz, feldspar, mica, etc. A considerable number of nuclide adsorption studies in granite and mica have shown that mica minerals are the main contributors to the adsorption occurring in the mineral composition of granite components, especially biotite [2,3,4], mainly due to its high ion exchange capacity and large amount of Fe(II) in the octahedral layer [5]. In the study of the migration behavior of Cs in granite porous media [6], biotite is a strong affinity site for Cs+ in Beishan granite. In granite and its main constituent minerals feldspar, quartz and mica, the adsorption order of Th is mica > feldspar > quartz [7].

As fission product, 79Se and 99Tc have long half-lives, low retention capacity, and are key nuclides in the safety evaluation of repositories, which have received a lot of attention and research [8]. To avoid radiological hazards, stable species of selenium, and stable rhenium that have similar diffusion behavior to technetium, are often used as substitutes for 99Tc and 79Se [9]. In recent years, researchers have done many studies on the migration adsorption behavior of nuclides Se(IV) and Re(VII) on Beishan granite, and obtained many migration parameters [10]. Diffusion and adsorption of Se(IV) in four granite matrices drilled at depths of 380–550 m in Beishan area, China, showed that biotite can represent the adsorption of Se(IV) in granite minerals [2]. Because selenium species are sensitive to redox, their migration characteristics cannot be predicted without considering possible speciation. Among them, Se(IV) has a stronger adsorption capacity than Se(VI) [11], the adsorption of Se(IV) on Beishan granite is independent of ionic strength, on the contrary, the adsorption of Se(VI) is strongly dependent on ionic strength, and XPS indicates that the adsorption of Se(IV) by granite is dominated by Fe(II) and Fe(III) in biotite. A surface complexation model of Se(IV) adsorbed on biotite with one strong adsorption site and two weak adsorption sites was established [12, 13], adsorption can be well simulated. In the fracture filling materials (FFMs) [14], the adsorption and migration of Se(IV) is mainly influenced by biotite and albite in granite samples.

The above research results have proved that biotite has a certain adsorption retention ability for Se(IV), when paying attention to the migration and adsorption of Se(IV) in granite, biotite is the focus of discussion. But it is generally considered to be the diffusion of Se(IV) of weak adsorption nuclides and Tc(VII) or Re(VII) of non-adsorbed [9] nuclides on biotite has not been reported, and the measurement of the diffusion behavior or diffusion coefficient of nuclides in mica minerals is quite limited. 79Se(IV) and 99Tc(VII) mainly exist as anions in groundwater, less adsorption in bentonite, and diffuse faster [15], which will threaten the human environment if they are finally migrated out of the surrounding rock. The radionuclide migration behavior of Beishan granite under the influence of biotite in it is an issue that needs to be considered for far-field transport. Clay compaction is one of the important factors affecting the diffusion behavior of radionuclides in clay, and in general, the total pore space and interlaminar pore space decrease with the increase of dry density, slowing down the diffusion of nuclides [9, 16, 17]. On the other hand, ionic strength in groundwater also affects the migration behavior of nuclides [18, 19]. However, the law and mechanism of diffusion of Se(IV) and Tc(VII) in biotite minerals under these different ionic strength and compaction density conditions are not clear. And we used Se(IV), which is more soluble and inclined to diffuse [20] and Re(VII), a common alternative to technetium, Tc(VII), a common form of technetium [9], the diffusion experiment was carried out on biotite powder after compaction, and the influence of reachable porosity on diffusion and the dependence of diffusion on the ionic strength of solution.

The diffusion cell method and capillary method have been widely used to explore the diffusion of radionuclides in bentonite and granite rocks. In this study, the effects of ionic strength and compaction density on Se(IV) and Re(VII) diffusion were studied by using the diffusion cell through-diffusion method and capillary in-diffusion method. The purpose of this study is to obtain their diffusion parameters, De, Da, Kd values, etc. in biotite and discuss them, so as to provide a reference for nuclide migration behavior in granite.

Materials and methods

Materials

The biotite used in this work came from Xinlei mineral powder processing plant in Xingtang County, Hebei Province, with a particle density of about 3 g/cm3, and chemical and surface analysis of the biotite was performed using SEM–EDS(Phenom, Pharos G2) and XRF(Bruker, S4 TSTAR).

Se(IV) and Re(VII) used stable nuclides, reagent selenium dioxide and ammonium perrhenate, background electrolyte of 0.01–1 mol/L NaCl, HCl or NaOH to adjust pH, and the reagents used in the experiment were analytically pure. The capillary tube is peek material, the capillary is filled with a stainless steel needle, the concentration of Se and Re in solution was determined by ICP-OES (PerkinElmer, Avio 220 Max).

Experimental methods

In-diffusion of capillary method

The experiment was carried out in an aerobic room temperature environment, and the schematic diagram of the experimental steps is shown in Fig. 1, the capillary tube adopts PEEK tube, the inner diameter of the capillary is 1.016 mm, and the outer diameter is 1.588 mm. Cut the peek tube into a 24 mm section, weigh its mass, fill the biotite powder in, use a stainless steel to compact, weigh its mass after filling, calculate the density, so as to achieve the specified density. Put the filled capillary into 5 ml of NaCl solution with a certain ionic strength and adjusted pH, equilibrate for 7 days after the capillary quality remains unchanged and take out the wiped moisture to weigh (in the equilibrating process, the loss of soil is negligible), close one end of the capillary, and then put it into the source solution containing nuclides for diffusion. The initial concentration of diffusion source solution CSe = 3000 mg/L, CRe = 1000 mg/L, diffusion lasts for two days, after taking out the end of the capillary and wiping dry, every 2 mm cut and pushed each soil block into 3 ml of 2 mol/L nitric acid. Nitric acid extracts it for a certain period of time and analyzes the nuclide concentration in the solution by ICP-OES measurement. According to the concentration of the sample solution at different distances x, the diffusion profile C/C0 = f(x) is obtained, where x corresponds to the distance from the diffusion source.

Fig. 1
figure 1

Experimental procedure of capillary method

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Through-diffusion of diffusion cell method

The entire connected diffusion device is shown in Fig. 2, and the diffusion device and experimental procedure are the same as those described by Van Loon et al. [21] using the device and experimental procedure for the through-diffusion experiment. Weigh the mass m of biotite powder, put it into a diffusion tank, press it with screws, and achieve the required compaction density according to the quality of placement. The diffusion cell is placed between the source liquid bottle and the sampling bottle, and the pipeline connects the diffusion tank with the bottle and the peristaltic pump. The background solution NaCl is added to both bottles, and the peristaltic pump is started to flow the liquid, equilibrate for a period of time(about a month). After equilibration, replace the source liquid bottle on one side of the diffusion cell with the nuclear solution of Se(IV) and Re(VII), add the equilibrium solution to the collection bottle on the other side, and change the collection bottle every 1–5 days, the nuclides diffuse under the action of concentration difference. As the concentration of nuclides to be measured in the diffusion collection bottle increases, the relevant diffusion coefficient can be calculated by analyzing the concentration of Se(IV) and Re(VII) in the collection bottle by ICP-OES.

Fig. 2
figure 2

Schematic diagram of the diffusion cell used in through-diffusion

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Data processing

In-diffusion of capillaries

In capillaries, the density of biotite after compaction is calculated as follows:

$$\rho = \frac{m2 - m1}{V}$$
(1)

where m1 is the peek capillary weight of 24 mm, m2 is the mass after filling, V is the capillary volume of 24 mm, and the total porosity is calculated as follows:

$$\varepsilon = 1 - \frac{\rho }{{\rho_{0}}}$$
(2)

ρ is the compaction density and ρ0 is the biotite particle density.

In capillary analysis, compacted biotite is considered a homogeneous porous medium. And the diffusion process is in infinite one-dimensional space, then its diffusion process in it can be described by Fick's second law:

$$\frac{\partial C}{{\partial t}} = D_{a} \frac{{\partial^{2} C}}{{\partial x^{2} }}$$
(3)

In capillaries, according to the one-dimensional semi-infinite diffusion assumption, the numerical solution of the diffusion equation in the medium is the formula:

$$\frac{C}{{C_{0} }} = {\text{erfc}} \frac{x}{{2\sqrt {D_{a} t} }}$$
(4)

By fitting C and t, Da and C0 are obtained, where C is the concentration of diffused substances in the pore water at the moment of t(s) at the interface of the capillary opening end, mg; C0 is the amount of diffusion material at the interface between the capillary opening end and the diffusion source solution, mg; Da is the apparent diffusion coefficient, m2/s, erfc is a residual error function and is often defined as the following equation:

$${\text{erfc}} (z) = 1 - \frac{2}{\sqrt \pi }\int_{0}^{z} {\exp } (1 - x)dx$$
(5)

In capillary experiments, the total distribution coefficient can be replaced by the distribution ratio of the nuclide in the first slice. The Kd value is derived from the amount of nuclides C0 and Cslice at the interface between biotite and the diffusion source solution, and after estimation Kd, it can be expressed by the following two formulas, which are expressed by Eq. (7) in this work:

$$K_{d} (slice1) = \frac{{C_{s} }}{{C_{L} }} = \frac{{\left( {C_{{\text{slice }}} - \varepsilon V_{{\text{slice }}} C_{L} } \right)/\rho V_{{\text{slice }}} }}{{C_{L} }}$$
(6)
$$K_{d} ({\text{ calc }}) = \frac{{C_{s} }}{{C_{L} }} = \frac{{\left( {C_{0} - \varepsilon V_{{\text{slice }}} C_{L} } \right)/\left( {\rho V_{{\text{slice }}} } \right)}}{{C_{L} }}$$
(7)

where, ε is porosity; ρ is the compaction density of biotite powder, kg/m3; Vslice is the volume of first sliced, mL; CL is the concentration of nuclides in the diffusion stock solution, mg/L; Cs is the concentration of nuclides in the solid phase, mg/g; Cslice is the amount of nuclides in the first slice at the interface between clay and solution, mg.

The apparent diffusion coefficient is calculated by:

$$D_{{\text{a}}} = \frac{{D_{e} }}{{\varepsilon + K_{d} \rho }}$$
(8)

Da and C0 are obtained by fitting C with distance t by ORIGIN software, and then De, Kd, and C0 are calculated.

Through-diffusion of diffusion cell

The diffusion cell adopts the through-diffusion method, and for the one-dimensional diffusion process, Fick's second law is applied. The analytical solutions and boundary conditions describing the cumulative amount of diffused species with time is given by:

$$A_{{\text{cum }}} = S \cdot L \cdot C_{0} \left( {\frac{{D_{e} \cdot t}}{{L^{2} }} - \frac{\alpha }{6} - \frac{2 \cdot \alpha }{{\pi^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{( - 1)^{n} }}{{n^{2} }}} \cdot \exp \left\{ { - \frac{{D_{e} \cdot n^{2} \cdot \pi^{2} \cdot t}}{{L^{2} \cdot \alpha }}} \right\}} \right)$$
(9)
$$\begin{gathered} c(x,t) = 0\quad {\mkern 1mu} t = 0;\quad \forall x \hfill \\ c(0,t) = C_{0} \quad {\mkern 1mu} x = 0;\quad t > 0 \hfill \\ c(L,t) = 0\quad {\mkern 1mu} x = L;\quad t > 0 \hfill \\ \end{gathered}$$
$$D_{{\text{a}}} = \frac{{D_{e} }}{\alpha }$$
(10)

where α is the rock capacity factor, defined as:

$$\alpha =\varepsilon +\rho \cdot {K}_{d}$$
(11)

The diffusion flux of the collection bottle J is calculated as follows:

$$J(L,t) = \frac{Acum}{{S \cdot\Delta t}}$$
(12)

Results and discussion

Properties and characterization of biotite

Biotite belongs to the layered silicate mineral, its crystal structure is monoclinic crystal system, the crystal structure of biotite is composed of layers of silicate tetrahedron and oxygen octahedron, biotite has a TOT-layer structure [22, 23]. The basic structural unit is two layers of silicon-oxygen tetrahedral sheet (T layer) sandwiched by one aluminum (iron) oxygen octahedral sheet layer (O layer), referred to as T-O-T structural unit, and the layers are connected by K atoms [23]. Due to the phenomenon of charge transfer and electron exchange in its crystal structure, biotite has a negative charge as a whole, so it is more affinity for cations, and has only a weak adsorption effect or no adsorption effect on nuclides in the form of anions. Its general chemical formula is K(Mg, Fe)3AlSi3O10(F, OH)2 [24]. The elemental composition of biotite used in this work was determined by XRF, with Table 1 showing the calculated chemical composition, and the calculated chemical formula is (KMg2.4Fe1.5Ti0.3) (Al0.9Si1.3O10)(OH)2.

Table 1 Chemical composition (wt.%) of biotite determined by XRF
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The biotite used in this work is scanned by SEM as shown in Fig. 3, and its size distribution is relatively uniform, within 200 µm, showing a layered structure, the surface is smoother, the edge is sharp, the pores are less, but there are strip-like traces, and it is stained with fine nanoscale small particles. After EDS analysis of the fixed part of the blue coil, the energy spectrum and main components are shown in Fig. 3D.

Fig. 3
figure 3

SEM and EDS diagrams of biotite; A 600× magnifications; B 5000× magnifications; C 50,000× magnifications; D EDS mapping

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Effect of biotite dry density on Se(IV) and Re(VII) diffusion

The effect of compaction density was studied by capillary in-diffusion, and Da and C0 were fitted by Eq. (4). In Fig. 4A, B, taking C/C0 as the ordinate and the distance X of each 2 mm clay block from the contact surface between biotite and the diffusion source solution as the abscissa, the concentration of C/C0 gradually decreased with the increase of distance. Table 2 shows the relevant diffusion parameters, when the density increases from 1600 kg/cm3 to 1850 kg/m3, the apparent diffusion coefficient Da of Se(IV) decreases from 2.08 × 10–10 m2/s to 3.25 × 10–11 m2/s, while the apparent diffusion coefficient of Re(VII) decreases from 2.28 × 10–10 m2/s to 2.15 × 10–11 m2/s, and the diffusion coefficient of the two is an order of magnitude. It shows that Se(IV) and Re(VII) have a considerable level of diffusion coefficient in biotite and under this experimental condition. Under the experimental environment of 0.1 M ion strength and about 23 °C, the distribution coefficient Kd of Se(IV) and Re(VII) at all different densities of biotite was close to 0 and not change much.—means that the calculated Kd values are <0, which are the result of negative adsorption caused by anion exclusion effects. In short, the effect of adsorption during diffusion is negligible at this time. The change of diffusion coefficient is mainly due to the change of diffusion space layer caused by the change of compaction density. Se(IV) and Re(VII) are mainly in the form of anions, which diffuse mainly through free water space [16]. The increase of compaction density makes the clay granular layer denser, the pores between the particles become smaller, and the decrease of pores directly leads to the reduction of the free water layer, narrows the diffusion space, limits the diffusion path of nuclides, and reduces the diffusion coefficient accordingly. Figure 5A, B show the relationship between Da and dry density, and compare the Da values obtained in this experiment with those of other researchers [9, 25,26,27]. They have a similar trend: the apparent diffusion coefficient decreases with increasing compaction density. At all densities, the Da values of Se(IV) obtained by this work are roughly between GMZ bentonite and Tamusu clay rocks, larger than GMZ bentonite and smaller than Tamusu clay rocks, while the Da values of Re(VII) are at the highest value among these minerals.

Fig. 4
figure 4

Diffusion in capillaries at different densities/ionic intensities of biotite; A and C: Se(IV); B and D: Re(VII)

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Table 2 Da and Kd values of Se(IV) and Re(VII) in biotite soils with different compaction densities, capillary method, ionic strength: 0.1 M, pH = 7.00 ± 0.05, T = 23 ± 2 °C
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Fig. 5
figure 5

Comparison of apparent diffusion coefficient and literature data; A, B Da of Se(IV) and Re(VII)(Tc(VII)) at different densities, C, D Da of Se(IV) and Re(VII)(Tc(VII)) at different ionic strengths; a, b: calculated by combining data from the literature using Eq. (10) and Eq. (11)

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Effect of ionic strength

Effect of ionic strength on Se(IV) diffusion

The influence of ion strength on diffusion was explored by through-diffusion diffusion and capillary diffusion. Figure 6 showed the relationship between total diffusion and time under different ion intensities of the diffusion cell, and Fig. 4C, D showed the variation of C/C0 with diffusion distance. Under the condition of compacting biotite at pH 7, the diffusion coefficient and adsorption distribution coefficient of Se(IV) and Re(VII) in NaCl solutions with different ionic strengths are shown in Table 3 under the two experimental methods.

Fig. 6
figure 6

Effect of ionic strength on the total diffusion of Se(IV) (A) and Re(VII) (B) over time in compacted biotite with a bulk dry density of 1700 kg/m3; pH = 7.0, T = 23 ± 2 °C

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Table 3 Da and Kd values of Se(IV) and Re(VII) in biotite with different ionic strengths,pH = 7.00 ± 0.05,T = 23 ± 2℃,Kd is calculated by C0
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In Table 3, the Da values of Se(IV) increase with the increase of ionic strength, in the diffusion cell method, De value increased from 5.42 × 10−13m2/s to 7.11 × 10−13m2/s as the ionic strength changes from 0.01 to 0.1 M. And in the capillary method, when the ionic strength increases from 0.01 to 0.5 M, Da rapidly increases from 1.34 × 10–11 m2/s to 7.22 × 10–11 m2/s, while the ionic strength increases from 0.5 to 1 M, and the diffusion coefficient increases smallly, from 7.22 × 10–11 m2/s to 7.40 × 10–11 m2/s. The Kd values decrease with the increase of ionic strength, except for an ionic strength of 0.01 M, the values of other ion intensities are very close to 0 or less than 0, indicating that Se(IV) only has small adsorption on biotite at 0.01 M. Therefore, the minimum Da value under 0.01 M ionic strength may be partially affected by adsorption, Se(IV) will form inner-sphere complexes between the edge surface of biotite/pore surface and Fe(III) [28], however, ionic strength has less effect on this mechanism [11]. In all cases, the Kd values are small, and it can be assumed that the large change in Da values with ion strength are not directly caused by the difference in the adsorption of Se(IV) on biotite. The pH of this experiment was 7, and the species was mainly HSeO3−, and there was basically no change in the case of weak biotite adsorption [29, 30], and therefore it is not caused by species changes. Rather, it is the results of a change in ionic strength that is related to the diffusion pore space. And changes in the external diffusion environment will be discussed in subsequent chapters.

Figure 5C compares the Da values of Se(IV) in different minerals at different ionic strengths [19, 27, 29], in the capillary method of this work, Se(IV) Da values in biotite are closer to GMZ bentonite at all ionic strengths, between 10–11 m2/s and 10–10 m2/s. The large difference in the De values between the diffusion cell method and the capillary method caused in this work may be due to the huge scale difference between the materials for the through-diffusion experiment and the capillary diffusion experiment. In addition, although biotite is an important adsorbed mineral in granite, the apparent diffusion coefficient of Se(IV) in it is greater than in Beishan granite, which may be due to the anion exclusion effect of biotite. And in this work, Se(IV) already had smaller Kd values, while Wang et al. had smaller Kd values [29], which may be due to the difference in experimental methods.

Effect of ionic strength on Re(VII) diffusion

The main diffusion’s species of Re(VII) is ReO4−, which has a similar effective ion radius compared to HSeO3− [31], and has a similar diffusion principle in the diffuse double layer formed by biotite. As the ion strength increases, the diffuse double layer is compressed, the free water space increases, and the diffusion is faster. In Table 3, the ionic intensity in the capillary method increased from 0.01 to 1 M, and the apparent diffusion coefficient increased from 1.04 × 10–11 m2/s to 9.29 × 10–11 m2/s. The apparent diffusion coefficient is still increasing substantially in 0.5 M to 1 M ionic strength, suggesting that increasing the ionic strength of the diffusion solution significantly favors the diffusion of Re(VII) in biotite, which may be more dependent on ionic strength than Se(IV). In addition, the ionic strength in the diffusion cell method ranged from 0.01 to 0.1 M, and the apparent diffusion coefficient Da ranged from 3.97 × 10−13m2/s to 7.11 × 10–13 m2/s. On the other hand, in the capillary method Re(VII) had a large Kd value at 0.01 M ion strength, according to the results of Li et al. [32], they had a large Kd value of Tc(VII) at 0.02 M ion strength in granite. And it is speculated that Re(VII) here will forms outer-sphere complexes with biotite at low ionic strength [33], this mechanism is greatly affected by ionic strength, and the adsorption of Re(VII) is stronger at this ionic strength, which also weakens the diffusion and causes a lower diffusion coefficient. However, the Kd values at other ion strengths are small, so adsorption is not the main reason for the change of diffusion coefficient of other ion strengths. The Kd values of Tc(VII) in Beishan granite decrease with the increase of ionic strength [32], generalizing this trend to our biotite, and our Kd values are in line with their trend. In this study, Re(VII) only had a large Kd value at the smallest ionic strength of 0.01 M, which were in line with the weak adsorption or non-adsorption characteristics of Re(VII) in clay [9, 34].

In the comparison in Fig. 5D, except for the diffusion cell method, the Da values of Re(VII) in biotite at all ionic intensities are between GMZ bentonite and Beishan granite (assuming Tc is equivalent to Re), close to GMZ bentonite, between 10–11 m2/s and 10–9 m2/s. In addition, assuming that Re and Tc are considered the same nuclide, the contrast with Se(IV) is similar, where Re(VII) has an order of magnitude larger apparent diffusion coefficient than that of Tc(VII) in Beishan granite, and the Kd values of Re(VII) are around 0 under the conditions of this work.

Possible diffusion mechanisms

Relationship between accessible porosity and dry density

In Chapter 4, section “Effect of biotite dry density on Se(IV) and Re(VII) diffusion”, the compaction density change significantly changes the diffusion coefficient because the dry density directly affects the accessible porosity, according to Archie's law [16]:

$$D{\text{e}} = D{\text{w}}\varepsilon^{{\text{n }}}$$
(13)

where n is an adjustable parameter and varies according to the type of nuclide and the type of porous media material, which is widely used in diffusion calculations of clays [16, 35].

In compacted clay, only part of the pores are available for diffusion due to some disconnected pores or the repulsive effect of anions. Accessible porosity εacc is used to describe the transport of anions. When Kd < 0, it is equal to α. In this experiment, Kd are at small values or less than 0, and we take the De values when the Kd values are less than 0 and its corresponding α, that is, εacc. Since the Dw value of HSeO3− has not been found in the literature, we assume that the Dw of HSeO3− is equal to HSO3−, and use this Archie's law equation to fit the relationship between De and accessible porosity. It also includes data from other researchers [9, 34], the fitting parameters are shown in Table 4, and the n values of biotite obtained by our work are close to GMZ bentonite, as can be seen in Fig. 7. The De of HSeO3− and ReO4− increases with the increase of accessible porosity, and the two nuclides have differences.

Table 4 Parameters fitted by Archie's law
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Fig. 7
figure 7

Fitting of De and accessible porosity of HSeO3− and ReO4− in biotite

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Effect of ionic strength on accessible porosity

Biotite has a negative charge as a whole due to the phenomenon of charge transfer and electron exchange in its crystal structure, and it has been reported to have a zero charge point(pHpzc) of around 4 [38, 39],and there is also a pHpzc between 6–7 [24], but both are below 7. When the pH is less than pHpzc, the surface is positively charged, when the pH is greater than pHpzc, the surface is negatively charged. The pH of this working experimental solution is around 7, so the surface of biotite minerals is negatively charged at this time. According to the diffuse double layer theory [16], due to the presence of negative charge, a diffuse double layer structure is formed on the surface of biotite, and there is a TOT structure similar to montmorillonite between the biotite layers, and the pore structure may be similar to montmorillonite [16, 22], Its interlaminar structure and possible pore structure are shown in Fig. 8, and the pore space includes interlayer space and interparticle pore spaces [40, 41], the interparticle pore spaces can be divided into a free water space and a diffuse double layer(DDL). For the interlayer space solution, that is, between the biotite TOT layers, assuming that it is also similar to montmorillonite, the width of the interlayer space is very narrow and the electrostatic potential is quite high, then the anions cannot enter the interlaminar solution at all. In the diffuse double layer, anions are also discharged in large quantities from the double layer due to the action of electrostatic potential and charge exclusion. Therefore, in the three parts of the hypothetical pore solution, only the free water space can be used by anions as a diffusion pathway, while the thickness of the diffuse double layer changes with the change of ion charge number and solution ionic strength, and with the increase of ionic strength, the diffuse double layer is inhibited and narrowed, and the free water space increases. In this working diffusion process, the main cation of the pore solution is Na+, and the anions are Cl− and HSeO3− and ReO4−, both of them are monovalent, so the thickness of the diffuse double layer here is only determined by the ionic strength.

Fig. 8
figure 8

Influence of diffuse double layer structure and ionic strength

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The thickness of the diffuse double layer, i.e., the Debye length, is inversely proportional to the square root of the ionic strength of the electrolyte solution [42]. As the ionic strength of the solution increases, the charge on the pore surface is better shielded and compressed, and the exclusion of the anion on the charge surface is weakened, resulting in a decrease in the repulsive force, the diffuse double layer narrows. And when the diffuse double layer thins, the proportion of free water in the interparticle porous solution increases, the porosity increases, the effective diffusion path of anions increases, and the diffusion coefficient increases. The relationship between accessible porosity and ionic strength is described by the following equation [16]:

$$\varepsilon_{{{\text{acc}}}} = \varepsilon_{\max } - {\text{A}} \cdot e^{{ - {\text{B}} \cdot {\text{I}}}}$$
(14)

where εmax is the maximum value of ε, I is the ionic strength, and A and B are the fitting constants that vary with bulk dry density. In this work, Se(IV) and Re(VII) obtain Kd close to 0, we treat the Kd values at different ionic intensities as 0, and the relationship between Eq. (10) and (13), Da and ε can be simplified to:

$$D_{{\text{a}}} = \frac{{D_{e} }}{\varepsilon }{ = }\frac{{D_{{\text{w}}} \varepsilon^{{\text{n}}} }}{\varepsilon }{ = }D_{{\text{w}}} \varepsilon^{{{\text{n}} - {1}}}$$
(15)

n substitutes the 2.8 and 3.1 fitted earlier in this section to calculate the accessible porosity εacc. According to Eq. (14), the fitting curves of ionic strength and accessible porosity are shown in Fig. 9, which also includes the results of other researchers. In this work, the accessible porosity of the two nuclides was overall between 125I and Re(VII), and Table 5 lists the respective fitting parameters, including Se, Re, I, Cl [19, 40, 43]. With the increase of ionic strength in different media, the accessible porosity of these nuclide species gradually increases and finally reaches a stable value.

Fig. 9
figure 9

Effect of ionic strength Se(IV), Re(VII),125I and 36Cl on the accessible porosity

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Table 5 Parameters fitted by Eq. (14)
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In the curve of this operation in Fig. 9, when the ionic strength increases from 0.01 M to 0.1 M, εacc increases rapidly, and when the ionic strength reaches a certain value of 0.5 M, εacc will be in a relatively stable range, at which time the diffuse double layer is almost completely suppressed. Therefore, in this work, in the ionic strength range of 0.01–0.5 M, the increase of ionic strength makes the diffuse double layer compressed, and the diffusion coefficient of Se(IV) increases rapidly. But when the ionic strength reaches a certain level, that is, above 0.5 M, the diffuse double layer is compressed to a certain extent, this compression inhibition effect weakens with the increase of ionic strength, and the effect is no longer significant when it reaches more than 0.5 M. Therefore, when the ionic strength increases to 1 M, the diffusion coefficient increases very little compared with 0.5 M. For Re(VII), according to the data of this work, between the ionic strength of 0.5-1 M, the εacc of Re(VII) is still increasing greatly, and it seems that the threshold max has not yet been reached, and the stability has not been completely reached, so the apparent diffusion coefficient of Re(VII) is still increasing significantly at this stage.

In Beishan granite, the diffusion coefficients of Se(IV) and Tc(VII) also increase with the increase of ionic strength [29, 32]. Biotite has the same tendency as it has, and the diffusion coefficient is greater, biotite may just enhance the tendency of granite, which also proves that it is necessary to study the migration characteristics of nuclides in biotite.

Conclusion

In this work, the diffusion of Se(IV) and Re(VII) on the adsorbed host mineral biotite of Beishan granite was studied by capillary method and diffusion cell method, and the effects of compaction density and ionic strength on nuclide diffusion were explored, and the results showed that nuclides increased with the decrease of compaction density or the increase of ionic strength. In the capillary method, when the compaction density increased from 1600 to 1850 kg/m3, the apparent diffusion coefficients of Se(IV) and Re(VII) decreased from 2.08–10 m2/s to 3.25–11 m2/s and 2.28–10 m2/s to 2.15–11 m2/s, respectively. This is due to the decrease in total porosity that reduces the proximity porosity, which narrows the diffusion channel. On the other hand, at a density of 1850 kg/m3, with the increase of ion strength from 0.01 to 1 M, the effective diffusion coefficients of Se(IV) and Re(VII) increased from 1.34–11 m2/s to 7.40–11 m2/s and 1.04–11 m2/s to 9.29–11 m2/s, respectively, which was interpreted as the increase of ion strength narrowed the diffuse double layer and the free water space became larger, and finally εacc becomes larger. Many studies have shown that biotite of Beishan granite has a certain adsorption of Se(IV) and is the adsorption subject therein, and the distribution coefficient Kd values obtained by the capillary diffusion method using biotite alone as a diffusion medium are basically around 0, which may be that the biotite scale in the capillary is small. This article also cites other researchers related data parameters for comparison, the Da values of Se(IV) and Re(VII) in biotite are greater than those of other researchers in granite and closer to those in GMZ bentonite. In addition to mineral differences, this may also be a difference in experimental methods and design, as well as experimental environments.