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In the safety assessment of a potential site for high-level radioactive wastes (HLW) disposal, the investigation on the geochemical behaviors of key radionuclides with the possibility for releasing from the potential repository is an important aspect. Due to the high mobility of technetium under most repository conditions, lots of research works were performed to investigate the diffusion of technetium in different potential rocks. In spite of these studies, there remains a lack of data addressing temperature effects. In this paper, the diffusion of ^{99}Tc in Beishan granite at temperatures from 25℃ to 55℃ was studied with laboratory small scale diffusion devices. The experimental data were fitted with a finite difference scheme to get the effective diffusion coefficient (De) of . The results indicated that the relationship of De with temperatures could be described as the modified Stokes-Einstein equation, and the formation factor of Beishan granite was constant in the temperature range of 25℃ - 55℃ with the value of (3.91 ± 1.77) × 10^{-4}.

Safety disposal of high-level radioactive wastes (HLW) is a worldwide challenge of nuclear industry. Deep geological disposal is the most acceptable way in isolating the HLW from biosphere. To demonstrate that the wastes can be safely isolated from the biosphere in a long time scale which is needed for the HLW radioactivity decays to natural level, the safety assessment of the geologic repository is an essential task. In the safety assessment of a potential site for HLW disposal, investigation on the geochemical behaviors of key radionuclides with the possibility for releasing from the potential repository is an important aspect. It is well understood that after the placement of HLW into the repository, the wastes will release considerable amount of heat due to the decay of radionucludes in a period of 100 years. The temperature of the HLW container will be rised up to 90˚C [1-3] or even higher. Naturally, the temperatures of the surrounding backfilling materials and the host rock in the near field will also be increased. The rising temperature increases the solubility and diffusion coefficient of contaminants (radionuclides) in most cases, while lowering the viscosity, density of the groundwater, and the sorption coefficient of radionuclides [4,5]. So that the migration of radionuclides in the near field will be enhanced [6,7]. The effects of temperature on fluid density and viscosity are well understood, and equations describing those effects are readily available [8,9]. In general, temperature increase leads to decreasing in both water density and viscosity, which induces buoyant groundwater flow, and faster mass transport [

In China, Beishan granite area in Gansu province has been pre-selected as one of the most potential research site for HLW repository [^{125}I^{−} as a surrogate for ^{129}I^{−} in Beishan granite obtained at 300-m depth by laboratory diffusion experiments, and the results showed that the effective diffusion coefficient ranged from 2.44 × 10^{−12} to 2.72 × 10^{−12 m2}/s [^{125}I^{−} diffusion in 600-m deep Beishan granite, and found that the relationship between the effective diffusion coefficient and temperature could be described by the modified Nernst equation at 27˚C - 50˚C [^{125}I, ^{99}Tc is one of the most important fission products in the spent fuels from nuclear power plants, due to its high mobility and long half-life [17-25]. Its diffusion and migration data are important to the data base for performance assessment of the potential geologic repository, while the temperature effect on the diffusion of ^{99}Tc in host rocks has not yet been reported. In this work, the temperature effect on its diffusion in Beishan granite was investigated by through-diffusion method.

A Beishan granite core sample was drilled out in the depth of about 600 m at Beishan area, in Northwest China’s Gansu province, China. The core sample was cut into slices which were 64 mm in diameter and 5 mm in thickness. Another small fraction of the same granite core sample was crushed and sieved by 200 mesh sieves to get granite powder sample, which was used for characterization. The mineralogical composition of the granite sample was determined by the XRD method, and the data is given in ^{99}Tc was in the form of NH_{4}TcO_{4} (Eckert & Ziegler Isotope). All other chemicals were in AR grade and bought from Sinopharm Chemical Reagent Co.

The through-diffusion technique was used to investigate the diffusion of ^{99}Tc in Beishan granite. The diffusion device was described in detail in previous work and is schematically shown in ^{2}; thickness 0.5 cm) was mounted between the two cells. The cells were filled with a 0.1 M NaClO_{4} solution. The hydraulic heads in the two cells were kept the same during the experiment. A small amount of ^{99}Tc (ca. 0.02 mCi) was added as to the source cell. After mixing for 24 hours, one mL solution was taken from each cell and assayed for ^{99}Tc by liquid scintillation counting (TriCarb 3110 TR, Perkin Elmer) using 10 mL of scintillation cocktail (Optiphase Hisafe 2, Perkin Elmer). The counting efficiency of ^{99}Tc was 54.9%. The specific activity of Tc in the source cell was (5.0 ± 0.1) ´ 10^{2} Bq/L. The one milliliter sample removed from both cells was

^{a}Loss of ignition.

replaced by one milliliter of 0.1 M NaClO_{4} after each sampling.

Before the start of the experiments, 0.1 mol/L NaClO_{4} solution was added into both the source and the sample cell. The hydraulic heads in the two cells were kept the same by monitoring the water levels in the cells and fresh 0.1 mol/L NaClO_{4} solution was added when needed. Then a dose (~0.02 mCi) of ^{99}Tc solution was added into the source cell. After stirring for 24 hours, 1.0 mL solution was taken from each cell and mixed with 10.0 mL scintillation cocktail (Optiphase Hisafe 2, Perkin Elmer) for radioactivity measurement with a Liquid Scintillation Counter (Tri-Carb 3110TR, Perkin Elmer). The counting efficiency of ^{99}Tc was 54.9%. The specific activity of ^{99}Tc in the source cell was measured as (5.0 ± 0.1) ´ 10^{2} into each cell. The activity of ^{99}Tc in the sample cell was measured in the same way at a 2-day interval in the first 10 days and in a 4-day interval in the rest of the 81-day experiment.

As all of our experiments were performed under atmospheric condition, ^{99}Tc could not be reduced into lower oxidation state and stayed as during the experiment [26,27]. Since the concentration of ^{99}Tc in the sample cell was diluted after each sampling due to the replacement of 1.0 mL water sample with 1.0 mL NaClO_{4} solution, the experimental data were corrected with the method used in our previous work [15,16]. The radioactive decay of ^{99}Tc can be ignored, due to its long half life of 2.1 × 10^{5} y.

According to the set up of our experiment, the diffusion of radionuclide in the diffusion cell could be described by the decreasing inlet concentration-increasing outlet concentration through-diffusion model (DC-IC) or approximately by the constant inlet concentration-increasing outlet concentration through-diffusion model (CC-IC), and the diffusion of ^{99}Tc in the granite can be described as [28,29]:

where [cpm/mL] is the pore water concentration of ^{99}Tc in the granite at the time t [s] and at position x along the diffusion direction; D_{e} [m^{2}/s] is the effective diffusion coefficient of ^{99}Tc in the granite; ε, ρ [kg/dm^{3}] are the porosity and dry bulk density of the granite sample.

According to the experimental set up, Equation (1) subjects to the following initial and boundary conditions [

I.C. (2)

B.C. (3)

where L [m], and A [m^{2}] are the thickness and effective diffusion area of the granite sample, respectively; V_{u} [m^{3}] and V_{d} [m^{3}] are the volume of the source and sample cell, respectively; C_{0} [cpm/mL] is the initial concentration of ^{99}Tc in the source cell. [cpm/mL] and [cpm/mL] are the concentration of ^{99}Tc in the source and sample cell when the experiment lasts for time t [s], equals to and in value, respectively. Since , the decrease of C_{u} can be ignored.

In order to obtain the effective diffusion coefficient D_{e}, the experimental data were fitted with Equation (1) by a finite difference scheme. We assumed:

Then Equation (1) could be transformed into:

When Δt and Δx are chosen to satisfy the following condition, Equation (8) will have a stable numerical solution:

For each experiment, we got the D_{e} and K_{d} by solving the equation set of Equation (2)-(4), (8) when L, A, V_{u}, V_{d}, C_{0} and at different temperatures are given.

The D_{e} and K_{d} values of ^{99}Tc^{−} in Beishan granite at different temperatures are listed in _{e} value obtained at room temperature is comparable to the data reported in literatures (_{d} values do not significantly change with temperature, but that the D_{e} values do. In our previous work, it was found that the relationship between the effective diffusion coefficients of ^{125}I^{−} in the Beishan granite with temperature could be described by modified Nernst equation [^{99}Tc, the limit ionic equivalent conductivity of at different temperatures was needed. Since there was no such data in the literature, the relation between the D_{e} and the temperature of ^{99}Tc should be discussed in a new way.

Since the diffusion of a solute in a porous media is related to the tortuosity (τ) and contrictivity (δ) of the pores, the porosity (ε) of the media and the self-diffusion coefficient of the solute (D_{w}) which describes its diffusion behavior in free water [_{e}) of the solute is described as:

As δ and τ is hardly measured individually with experiments, the formation factor F_{f} is carried out to describe the relation between D_{e} and D_{w}. F_{f} equls to and only depends on the inner structure of the porous media [_{w} depends on temperature [

where k [J/K] is the Boltzmann constant, ζ [kg/s] is the friction constant. For a spherical solute particle larger than the particles of the solvent, ζ is taken approximately as the friction term in Stokes’ law, ζ = 6πηr_{i}, where η [kg∙m^{−1}∙s^{−1}] is the viscosity of the solvent and r_{i} [m] is the radius of the particle of the solute. Then Equation (11) is

_{e} and K_{d} of in Beishan granite in comparison with literature data.

transformed into Stokes-Einstein equation:

In this work, ^{99}Tc was in the form of, which could be taken as a spherical particle (close to the size of I^{−}, r = 2.16Å) and definitely larger than the water molecular (r = 1.4Å). Therefore the Stokes-Einstein equation can be used to describe the diffusion of in water. Substituting Equation (12) into Equation (10), the following relation between D_{e} and temperature is obtained:

In Equation (13), the viscosity of water η is changed with temperature. The data of η in the temperature range in this study are taken from Lange’s handbook of chemistry, 13th edition and listed in _{e} and T/η was constant (see _{f} of the Beishan granite was constant in the temperature range. H. Sato et al. measured the D_{w} value of at 25˚C as (1.95 ± 0.56) × 10^{−}^{9} m^{2}/s [_{w} value of at the corresponding temperature was calculated. So F_{f} of Beishan granite could be derived by dividing D_{e} with D_{w}. The results were listed in ^{125}I^{−} [_{f} values we obtained in this work are comparable to the literature data (_{f} may due to the difference in D_{e} values of obtained in this work (7.4 × 10^{−}^{13} - 1.4 × 10^{−}^{12} m^{2}/s) to ^{125}I^{−} obtained in our previous work (3.4 × 10^{−}^{13} - 5.3 × 10^{−}^{13} m^{2}/s), which was caused by the characteristics of the granite slices and the different diffusion behaviors of and ^{125}I^{−} in the granite.

^{a}Calculated from the viscosity values at other temperatures by fitting the viscosity-temperature curve.

The constant formation factor of Beishan granite indicated that the inner structure of the granite did not change in the temperature range of 25˚C - 55˚C. This is very important for the granite as a host rock for the potential geologic repository. The enhanced diffusion of in Beishan granite at higher temperatures were come from the increasing of D_{w} of.

The effect of temperature on diffusion of in Beishan granite was studied using the through-diffusion technique. It was indicated that the effective diffusion coefficient of in Beishan granite was linearly related with T/η in the temperature range of 25˚C - 55˚C. The formation factor of Beishan granite was constant in the temperatures range, which suggested that the inner structure of Beishan granite did not change in the temperature range. The result of this work was in good agreement with our previous work except that the value of F_{f} is larger. The difference of F_{f} could be considered as experiment uncertainty. The result indicated Beishan granite was stable below 55˚C and appropriate to be a host rock.