Multicomponent diffusion in sodium borosilicate glasses

Journal of Non-Crystalline Solids 478 (2017) 29–40

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Multicomponent diffusion in sodium borosilicate glasses a,⁎

b

c

d

e

MARK f

H. Pablo , S. Schuller , M.J. Toplis , E. Gouillart , S. Mostefaoui , T. Charpentier , M. Roskosz

e

a

CEA, DEN, DE2D, SEVT, LDMC, F-30207 Bagnols-sur-Cèze, France CEA, DEN, DE2D, SEVT, LDPV, F-30207 Bagnols-sur-Cèze, France IRAP, Université de Toulouse, CNRS, UPS, Toulouse, France d Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 93303 Aubervilliers, France e Muséum National d'Histoire Naturelle, F-75231 Paris, France f NIMBE, CEA, CNRS, Université Paris-Saclay, CEA Saclay, F-91191 Gif-sur-Yvette, France b c

A R T I C L E I N F O

A B S T R A C T

Keywords: Borosilicate glasses Multicomponent diffusion Highly supercooled liquids Viscosity

Multicomponent chemical diffusion has been investigated in the SiO2-Na2O-B2O3 system for melts with an average composition (mol.%) of 68SiO2-18B2O3-14Na2O. Three diffusion couples were studied at 5 different temperatures between 700 °C and 1100 °C. The extended form of Fick's second law was used to fit the data, derive the diffusion matrix at each temperature, and quantify eigenvectors and eigenvalues. The results reveal diffusive mechanisms in chemical space (i.e. eigenvectors) that appear constant over the temperature range studied. The principal eigenvector is characterized by an exchange between silicon and sodium, while the secondary eigenvector corresponds to an exchange between silicon and boron made possible by sodium. Eigenvalues vary considerably with temperature, but do not follow an Arrhenian law. This behavior has been attributed to the structural changes of the borosilicate network with temperature. The diffusion data are then compared with viscosity and ionic conductivity measurements, revealing close links between viscosity and chemical diffusion.

1. Introduction In France, High Level radioactive Waste (HLW) from spent nuclear fuel reprocessing is incorporated in a multicomponent sodium borosilicate glass. This matrix was selected for its good chemical durability, its low glass synthesis temperature (around 1100 °C) and its ability to incorporate a wide range of radionuclides and fission products [1–3]. The next challenges in the field of nuclear glass research involve finding new glass compositions or optimizing the current formulations, in order to incorporate higher amounts of waste while maintaining chemical durability and attainable synthesis conditions [4]. In this context, a thorough knowledge of the physical and chemical processes that occur in the melt during glass making is essential. In particular, chemical diffusion is of considerable importance given that in the supercooled liquid state, under appropriate thermodynamic conditions, this process can lead to the appearance of crystals that are a source of heterogeneities and may adversely affect long-term stability. As far as we know, no data on chemical diffusion are currently available for nuclear glasses in the supercooled liquid state. A few studies related to the measurement of self-diffusion coefficients have

been performed, but those experiments were carried out below the glass transition temperature [5–8]. There is therefore a lack of knowledge on chemical diffusion at high temperatures for compositions of interest in the field of nuclear waste management. One reason for this lack of data is that nuclear glasses are multicomponent systems, which makes their study complex. In such systems, Fick's second law in its classical form is insufficient and a more generalized matrix notation must be used to describe diffusion [9–11]. On the other hand, deriving the diffusion matrix is of interest as the eigenvectors and eigenvalues of that matrix contain information that is relevant for describing and understanding diffusive phenomena. In the literature, the studies on multicomponent chemical diffusion were conducted in metallurgical systems [12–16] and silicate liquids of geological [17–26] and industrial interest [27–35]. Silicate systems were mainly alkaline [18,19,25], alkaline earth [22–24,26,36] or both alkaline and alkaline earth aluminosilicates [17,27]. To the authors' knowledge, only two studies have focused on borosilicates but the diffusion matrices in these systems were not determined [37,38]. In the present contribution, we focus on multicomponent diffusion of simplified borosilicate liquids (in the SiO2-Na2O-B2O3 ternary) with a target

⁎

Corresponding author. E-mail addresses: [email protected] (H. Pablo), [email protected] (S. Schuller), [email protected] (M.J. Toplis), [email protected] (E. Gouillart), [email protected] (S. Mostefaoui), [email protected] (T. Charpentier), [email protected] (M. Roskosz). http://dx.doi.org/10.1016/j.jnoncrysol.2017.10.001 Received 8 August 2017; Received in revised form 30 September 2017; Accepted 3 October 2017 Available online 10 October 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.

Journal of Non-Crystalline Solids 478 (2017) 29–40

H. Pablo et al.

two glasses in each couple in order to color the glass, providing a convenient means of locating the interface after heat treatment. For a given composition, the powder mixture was stirred in a mechanical mixer for 15 min and poured into a PteRh10 platinum-rhodium crucible before being placed into a muffle furnace. The temperature was increased from room temperature to 800 °C at a rate of 400 °C/h. It was then kept constant for 30 min to allow Na2CO3 decarbonation. After this step, the furnace was heated to 1400 °C at a rate of 400 °C/h. The melt was homogenized for 3 h at this temperature, a time span chosen to ensure bubble-free and crystal-free glass. It was subsequently poured into a preheated carbon crucible with a 2.5 cm2 section, and annealed for 1.5 h at 620 °C in another muffle furnace. Finally, samples were cooled to 300 °C at 30 °C/h before the furnace was turned off. The homogeneity of the glasses was systematically checked by scanning electron microscopy (SEM) and a chip of each glass was also analyzed by electron microprobe, to be sure that its composition was close to that of the nominal composition despite the high volatility of boron and sodium oxides (Table 1).

composition of interest to the nuclear industry. 2. Material and methods 2.1. Choice and synthesis of glasses 2.1.1. Experimental strategy Diffusion-couple experiments were conducted around a central composition of 67.73SiO2–18.04B2O3-14.23Na2O corresponding to a simplified version of the French R7T7 waste-disposal glass (the three oxides are kept in the same molar ratio as R7T7 [39]). According to Trial and Spera [40], only two diffusion experiments are necessary to determine the diffusion matrix in a ternary system. However, some authors [17,27,40,41] have shown that adding more experiments is crucial to derive the diffusion matrix more precisely. Therefore, one more couple was added to this work. The directions of the diffusion couples in concentration space were chosen carefully in order to ensure that no information was missing in the diffusion matrix. Ideally, couple directions should be collinear to eigenvectors in composition space [40]. However, as these vectors are unknown a priori, diffusion directions were chosen in such a way as to be as far as possible from each other. In a ternary representation, this involves having a 60° angle between the three directions (Fig. 1). The difference in composition between the two starting glasses of a given couple was optimized to maintain an acceptable signal to noise ratio during chemical analyses, while keeping chemical variations to a minimum [18] (in case diffusion coefficients vary as a function of composition). With these constraints in mind, glasses were synthesized with a variation of ± 4 mol% in two oxides for each couple. The glasses were labeled SBNx-y, where x is the SiO2 molar concentration in the glass and y is the B2O3 molar concentration (Fig. 1).

2.2. Analytical methods 2.2.1. Microprobe Electron microprobe analyses were conducted for diffusion couples obtained at 900 °C, 1000 °C and 1100 °C. The standard chosen for measuring the SiO2, Na2O, and B2O3 content was the International Simple Glass (ISG), whose composition is given in Table 2. Two different microprobes were used in this work: a Cameca SX50 (CEA Saclay) and a Cameca SX100 (Lille University). In both cases, the probes were defocused to limit sodium migration under the electron beam, with a 10 μm beam used each time. Operating conditions implemented an accelerating voltage of 15 kV and a beam current of 10 nA. The sodium and silicon contents were measured on a TAP crystal, while a multilayer PC3 crystal was used for boron quantification. Oxygen was calculated by stoichiometry, with counting times of 30, 20 and 10 s for boron, silicon and sodium respectively. Sodium was measured first, in order to improve the accuracy of its measurement. For the analyses performed at Lille University, sodium measurements were made by counting for 2 s, five times in a row. These measurements were regressed to determine the initial sodium content.

2.1.2. Glass synthesis For each starting glass, a powder mixture corresponding to a decarbonated weight of 100 g was prepared by using reagent grade sodium carbonate, silica and boric acid. The quantities of Na2CO3 and H3BO3 used were slightly higher than the nominal composition in order to compensate for the effect of boron and sodium volatilization during synthesis. A few ppm of cobalt (CoO powder) were added to one of the

Fig. 1. Compositions and names of the glasses used in this study, presented in the ternary SiO2-Na2O-B2O3 diagram.

Na2O (mol.%) 10 30

15 25

20 20 SBN66-18

25

SBN68-16

SBN68-18 SBN66-20

(mol.%)

15

SBN70-18

SBN68-20

B2O3

SBN70-16

30 60

65

70

75

80

30

10 SiO2

(mol.%)

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H. Pablo et al.

Table 1 Nominal and analyzed compositions of the glasses investigated in the study. The last column gives the mean composition for each diffusion couple. Nominal composition (mol.%)

SBN68-18 SBN70-16 SBN66-20 SBN70-18 SBN66-18 SBN68-20 SBN68-16

Analyzed composition (mol.%)

SiO2

B2O3

Na2O

SiO2

67.73 69.73 65.73 69.73 65.73 67.73 67.73

18.04 16.04 20.04 18.04 18.04 20.04 16.04

14.23 14.23 14.23 12.23 16.23 12.23 16.23

65.26 69.00 65.76 70.88 65.42 67.18 67.90

B2O3 ( ± 0.47) ( ± 0.58) ( ± 0.45) ( ± 0.79) ( ± 0.32) ( ± 0.43) ( ± 0.44)

20.98 16.40 19.54 17.69 18.98 19.67 15.86

Na2O ( ± 0.44) ( ± 0.59) ( ± 0.50) ( ± 0.75) ( ± 0.29) ( ± 0.52) ( ± 0.60)

13.76 14.59 14.70 11.42 15.60 13.15 16.23

( ± 0.16) ( ± 0.49) ( ± 0.47) ( ± 0.51) ( ± 0.24) ( ± 0.41) ( ± 0.44)

SiO2

B2O3

Na2O

– 67.38

– 17.97

– 14.64

68.15

17.55

13.51

67.54

17.77

14.69

steps.

Table 2 Nominal composition of ISG glass (mol.%) [42]. SiO2

B2O3

Na2O

Al2O3

CaO

ZrO2

60.1

16.0

12.7

3.8

5.7

1.7

2.2.4. Electrical conductivity The electrical conductivity of SBN68-18 glass was measured in the temperature range 300 °C–1200 °C with a 4-electrode impedance spectrometer. More information about this apparatus may be found elsewhere [44]. Prior to the measurements, the cell was calibrated by measuring a KCl solution whose conductivity is known at room temperature. The platinum-rhodium electrodes were then placed in the melt at 1200 °C and impedance measurements at frequencies ranging from 10 Hz to 1 MHz were performed every 5 °C, during cooling at a rate of 2 °C/min.

2.2.2. NanoSIMS NanoSIMS was used in order to improve the spatial resolution of short diffusion profiles for the experiments heat-treated at 700 °C and 800 °C. Chemical quantification in this case was performed with the Cameca NanoSIMS N50 at the Muséum National d'Histoire Naturelle (National Natural History Museum) in Paris. A 16 keV Cs+ primary ion beam with a spot diameter of smaller than 0.3 μm and an intensity of ~ 3.5 pA was applied through a D1-2 diaphragm to sputter negative secondary ions of 11B−, 18O−, 23Na− and 28Si−. These elements were detected simultaneously in multicollection mode, using four electron multipliers (EMs) whose sensitivities had been adjusted to provide the best possible signal to noise ratio. The mass resolving power was set at about 7000, which was sufficient to resolve the measured secondary ions from potential isobaric interferences. The measurements were made in spot mode with an acquisition time of ~11 min per spot. To quantify the chemical profiles, straight lines of 20 to 100 spots were acquired across sample interfaces at intervals from 1 to 10 μm. In order to reach a steady-state regime and to remove the coating (a 20 nm thickness of gold) as well as initial surface contamination, sample presputtering was performed for ~ 3 min under conditions similar to those used for the analyses. Because the samples were insulating, a normal incidence electron gun was necessary in order to compensate for positive primary beam charge deposition on the sample surface during analysis.

2.2.5. 11B nuclear magnetic resonance spectroscopy 11 B Magic Angle Spinning (MAS) Nuclear Magnetic Resonance (NMR) analyses were performed at CEA Saclay in the Laboratoire de Structure et Dynamique par Résonance Magnétique (LSDRM). Data were collected on a Bruker Avance II 500WB spectrometer operating at a magnetic field of 11.72 T using a Bruker 4 mm (outer diameter of the ZrO2 rotors) HX CPMAS at a spinning rate of 12.5 kHz. Spectra were acquired using a short single radiofrequency pulse of 1 μs (tip angle of about π/8 to ensure quantitativeness) and a recycle delay of 2 s. Data was processed and fitted using an in-house code, as described in [45,46]. 3. Experimental 3.1. Diffusion couple experiments Cylinders were drilled out of the annealed starting glasses, then cut and mirror polished to produce smaller cylinders (5 mm in diameter and 4 mm long). These dimensions were chosen in order to prevent convection and to minimize potential thermal gradients across the samples. The length of the cylinders was set at 4 mm in order to satisfy the approximation that diffusion takes place in a semi-infinite medium and their diameter was chosen to limit edge effects during diffusion. Two disks of contrasting compositions in the SiO2-Na2O-B2O3 ternary diagram were then placed into a 1 cm high, 5 mm wide platinum crucible. The denser glass, which was colored by the few ppm of cobalt, was put at the bottom to avoid convection. Annealing took place in a tube furnace preheated to the required temperature (between 700 °C and 1100 °C). Prior to these experiments, the furnace was calibrated with a TCK thermocouple and temperature uncertainties were found to be lower than 10 °C. Once the target temperature was reached, the diffusion couples were placed in the furnace and isothermally annealed. The run durations were optimized to avoid crystallization or convection, and to observe diffusion over a distance large enough to be accurately analyzed by EMPA or NanoSIMS (Table 3). These durations were also chosen so that the time for the samples to thermalize was negligible. After the experiments, the diffusion couples were annealed for 1.5 h at 610 °C in another preheated tube furnace to limit the buildup of mechanical stress. The glass assemblies were then cut

2.2.3. Viscosity Viscosity measurements were performed on SBN68-18 glass with a stress imposed rheometer (Rheometrics Scientific SR5000) mounted above a vertical tube furnace in Couette geometry [43]. The glass, contained in a platinum-rhodium cylindrical crucible (Ø = 27 mm), was first heated to 1350 °C in the furnace chamber. Then a cylindrical platinum spindle (Ø = 9 mm) suspended from the rheometer was submerged and rotated in the melt with a shear rate, γ̇, ranging from 1 s− 1 to 100 s− 1. As glasses are generally Newtonian, they follow the formula:

η=

Average composition (mol.%)

τ K C = τ γ̇ K γ̇ θ ̇

(1) −1

−1

−1

Where τ (Pa) is the shear stress, Kτ (Pa·N ·m ) and K γ̇ (rad ) are geometric constants and C (N·m) is the torque necessary for the cylinder to rotate at the imposed angular speed, θ ̇ (rad·s− 1). Prior to the measurements, the system was calibrated with a glass of known viscosity (BCL B3959) for determining Kτ and K γ̇ . η was then deduced from the measurements of C with Formula (1). Data acquisition was performed with the Rheomatic-P software between 1350 °C and 950 °C, with 50 °C 31

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Similarly, this equation implies that only N − 1 fluxes are independent in our nominally barycentric system. The flux of component N can be deduced from the other fluxes. However, considering N − 1 fluxes requires the choice of a component N which will be referred to as the dependent component. Eq. (3) can be written in a more condensed way using matrix notation (in what follows, bold face font is used for matrices and vectors):

Table 3 Diffusion test experimental conditions. Annealing temperature

Annealing time

700 °C 800 °C 900 °C 1000 °C 1100 °C

4h 1 h 30 min, 3 h 2h 30 min 20 min,1 h 20 min

J = −D N

3.2. Diffusion profiles

−

Depending on the analytical method used for analyzing the diffusion profiles, 1 (NanoSIMS) or 6 (microprobe) sets of diffusion profiles were measured at the center of the crucibles. Multiplying the number of measurements enabled more accurate results when fitting average profiles. Because the error function used for fitting the profiles is 0centered, the interface position had to be found and set at x = 0. To do this, the infinite couple solution for binary systems was applied:

Ci, −∞ + Ci, +∞ Ci, +∞ − Ci, −∞ x − x0 ⎞ erf ⎛⎜ + ⎟ 2 2 ⎝ 2 Di t ⎠

k=1

N

where the columns of P are the eigenvectors of D and its eigenvalues are the coefficients of the diagonal matrix λ. After replacing DN in Eq. (8) and assuming that DN is constant within the compositional range considered, then Eq. (8) can be written as:

∂P −1C ∂2P −1C =λ ∂t ∂x 2

∂U ∂ 2U =λ 2 ∂t ∂x

(10)

This new formula is equivalent to Eq. (7) but can be broken down into a system of N − 1 independent equations. Each of them has the form of the classical Fick's second law, in which concentration variations with time are correlated to concentration gradient derivatives through the diffusion coefficient λi. 2

⎧ ∂U1 = λ1 ∂ U21 ∂x ⎪ ∂t ⋮ ⎨ 2 ⎪ ∂UN − 1 = λN − 1 ∂ UN2− 1 ∂x ⎩ ∂t

(11)

If the experimental conditions are consistent with the assumption of infinite media, the solutions to these equations are known [48] and, in matrix notation, are expressed by:

U (x ) =

U−  + U+  U − U−  + F (x) +  2 2

(12)

where U(x) is the vector of concentrations at position x and U−∞ ,U+∞ are the vectors of initial concentrations in the first and the second endmembers respectively. All these vectors are expressed on an eigenvector basis. F(x) is a diagonal matrix which depends on x and contains information about the eigenvalues λi. It must obey the Equation [49]:

(3)

∂Fii ∂ 2F = λi 2ii ∂t ∂x

(13)

Hence:

x ⎞ Fii = erf ⎜⎛ ⎟ 2 ⎝ λi t ⎠

N i=1

(9)

If the vector of concentrations expressed in the eigenvector basis is defined as U = P− 1C, Eq. (9) becomes:

When compared to the typical expression of Fick's first law, this equation does not take into account just one single diffusion coefficient, but rather interdiffusion coefficients, Di, kN, which express the diffusion of species i in a gradient of species j. Therefore this new approach takes diffusive couplings into account when describing diffusion. Here k varies from 1 to N − 1 because one concentration variable is not dependent, given that the sum of concentrations is 1:

∑ Ci = 1

(8)

D N = PλP −1 (2)

During the 1940s, Darken [10,11] and Onsager [9] pointed out that Fick's second law in its simplest form is not well adapted to describing diffusion in multicomponent systems. In particular, analytical solutions expressed as error functions are not able to reproduce so-called ‘uphill diffusion’ phenomena. This corresponds to the case where a component diffuses against its own concentration gradient, due to diffusive couplings with another component [19]. Onsager [9] put forward a new definition of Fick's first law for Ncomponent systems. He expressed the flux of one component as a linear combination of the concentration gradients of other components:

∂Ck ∂x

(7)

Because the diffusion matrix is diagonalizable [47], an invertible matrix P can be found that satisfies:

4.1. Theoretical background

∑ DiN,k

(6)

∂C ∂ 2C = DN 2 ∂t ∂x

4. Theory/calculation

Ji = −

∂Jx ∂ ⎛ N ∂C ⎞ = D ∂x ∂x ⎝ ∂x ⎠

Assuming that the diffusion matrix is constant, it may be appreciated that:

where Ci(x) (mol.%) is the concentration at position x, x0 (m) is the position of the interface, t (s) is the annealing time and Di (m2/s) are diffusion coefficients, known as Effective Binary Diffusion Coefficients (EBDCs). These diffusion coefficients can only be calculated for monotonic profiles [17] and are equivalent to an inter-diffusion coefficient between component i and all the other components, which are considered as a solvent. Because they are very sensitive to the bulk composition and to compositional gradients [17], EBDCs are not very helpful in a study such as this, as they change depending on the direction of the diffusion couples in component space. For this reason, they were only used to locate the interface position, x0.

N −1

(5)

in which J is the vector of flux density, C is the vector of concentrations and DN is a diffusion matrix of dimension N − 1. Generalizing Fick's first and second laws leads to:

longitudinally and one half of each sample was prepared for EMPA or NanoSIMS measurements.

Ci (x ) =

∂C ∂x

(4)

(14)

With this equation in hand, it is then possible to return to 32

Journal of Non-Crystalline Solids 478 (2017) 29–40

H. Pablo et al.

concentration space by multiplying the left-hand side of Eq. (12) by P and by replacing C by PU:

700

600

σT

2

10

Log( ) (Pa.s)

C−  + C+  C − C−  + PF (x) P −1 +  2 2

(16)

Where C−∞, C+∞ are vectors that contain initial concentrations of both end members. An equation similar to Eq. (16) was used to adjust the fits to the experimental concentration profiles, with the difference that this study focused (see below) on the difference between the concentration at position x and the mean composition of the profile (noted C∗):

C (x ) = C (x ) −

C−  + C+  2

8

0

6

η 4

2

-4

(17)

0.7

0.8

C+  − C−  2

0.9

1.0

1.1

1.2

1000/T (K-1)

The analytical equation solved to adjust the fit to the data was then:

C (x ) = PF (x) P −1

-2

.cm-1.K)

(15)

and:

C (x ) =

800

12

-1

U−  + U+  U − U−  + PF (x) P −1P +  2 2

900

log( T (

PU (x ) = P

T (°C) 12001100 1000

Fig. 2. Evolution of log(η) and log(σT) with 1000/T.

(18) profiles collected at 800 and 700 °C, analytical errors are implicitly accounted for in the uncertainties provided here. For the low temperature experiments, the limited dataset makes our data and their fitting less accurate and uncertainties must be considered as indicatives.

4.2. Mathematical method 4.2.1. Fitting procedure All diffusion profiles collected on samples annealed at the same temperature were processed together by an iterative computer program written in Python language. All the functions used were grouped in the multidiff package. The overall fitting procedure consisted in minimizing the difference between experimental and theoretical profiles. Input parameters were a set of eigenvectors and eigenvalues that generated theoretical concentration profiles, with the use of Eq. (18). The deviations between these profiles and the experimental profiles were then calculated and minimized using a function from the scipy library which implements the least-squares method developed in the LevenbergMarquardt algorithm. The iteration output was a new set of eigenvectors and eigenvalues, which were implemented as initialization parameters for the next iteration. Once the final eigenvectors and eigenvalues were determined, it was possible to deduce the diffusion matrix from Eq. (8).

5. Results 5.1. Viscosity and conductivity The viscosity and conductivity of SBN68-18 glass are presented in an Arrhenian diagram (Fig. 2). These two properties were fitted with an empirical Vogel-Fulcher-Tamman law:

B ⎞ Φ = Φ0 exp ⎛ − T T0 ⎠ ⎝ ⎜

⎟

(19)

Where Φ is η or σT, T is the absolute temperature and Φ0, B, T0 are fitting parameters. For viscosity, the fit was extrapolated to the glass transition temperature, considering that viscosity at this temperature was 1012 Pa·s. Fig. 2 shows that the evolutions of viscosity and conductivity are clearly not linear, which means that the activation energies for these transport properties change with temperature. In both cases, the activation energies decrease with increasing temperature, as expected.

4.2.2. Uncertainty assessment For the diffusion couples heat treated at 900 °C, 1000 °C and 1100 °C, a Student's t-test was used in order to assess the uncertainty on the fits. At each temperature, there were 6 sets of concentration profiles per experiment. Sets of diffusion profiles were fitted three by three by taking a set from each experiment each time. This procedure was performed 6 times to use all concentration profiles. The Student's t-test was then performed on eigenvalues and eigenvectors retrieved from the fits. The uncertainty calculated from this ensured a confidence interval of 95%. This procedure was not applicable to samples annealed at 700 °C and 800 °C, for which only one set of diffusion profiles was measured. In this case, the procedure used to estimate eigenvalues and their uncertainty was the following. We began by estimating the experimental noise on the diffusion profiles by calculating the standard deviation of concentration at each end of the profiles for the last 5 to 10 points. Theoretical diffusion profiles were then generated for the three exchanges using the same eigenvectors as measured at higher temperature. Finally, the program was run to determine a set of fitted eigenvalues. The same method was performed 6 times. A loop was added to the program to enable different diffusion profiles to be obtained from each run, while keeping the same signal to noise ratio. We note here that the procedures described above mainly evaluate the uncertainty on the fit to the data. However, the fitting procedure developed by Claireaux et al. [27] considers the analytical error during the fitting procedure. So with the notable exceptions of diffusion

5.2.

11

B MAS NMR

The proportions of BO4− and BO3 units in the samples were quantified by 11B MAS NMR. The spectra collected show two distinct contributions: a symmetrical band centered at around 0 ppm corresponding to four-fold coordinated boron, and an asymmetrical band centered at around 10 ppm corresponding to three-fold coordinated boron [45].The quantification of the relative proportions of these two units was obtained after data deconvolution (Table 4). Furthermore, in sodium borosilicate glasses, sodium can be both a charge-balancing and a network-modifying cation. Sodium first charge-balances the negative charge of BO4− units. Excess sodium plays a network-modifying role, creating non-bridging oxygens around silica tetrahedrons. Assessing the proportion of BO4− therefore helps to quantify the number of nonbridging oxygens present in the glass, according to the equation:

(

2 × 100 × CNa2 O − CB2 O3 × %NBO =

% BO4− 100

CNa2 O + 3 × CB2 O3 + 2 × CSiO2

) (20)

Where CNa2O, CB2O3and CSiO2are molar concentrations (mol.%). In the glasses investigated here, a large fraction of boron atoms were 33

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Table 4 Comparison between %BO4− given by the Dell and Bray model and %BO4− from MAS NMR results. %NBO was calculated from NMR results.

11

Glass

%BO4−_Dell and Bray [50]

%BO4−_NMR

%NBO

SBN68-18 SBN70-16 SBN66-20 SBN70-18 SBN66-18 SBN68-20 SBN68-16

65.6 76.3 71.0 64.6 71.5 66.9 76.8

63.9 73.7 65.7 62.8 67.7 63.2 76.7

0.3 2.5 1.8 0.0 2.7 0.0 4.08

were also found to be symmetric with respect to the interface at all temperatures, demonstrating that diffusion can be considered as constant in this compositional range (Fig. 4). Qualitatively, all the profiles for a given couple exhibit the same shape and features at all temperatures. This suggests that the nature of diffusive exchanges that define diffusion are not significantly dependent on temperature (as discussed below). Another common feature of these diffusion profiles, and which is characteristic of chemical diffusion in multicomponent systems, is the occurrence of uphill diffusion. This points to diffusive couplings between cations, where a given cation can be led to diffuse against its own concentration gradient. Such behavior can also be rationalized in terms of the composition dependence of the activity coefficients of the different species present in the melts, as described by the formalism developed by Onsager [9]. In the borosilicate system investigated here uphill diffusion of sodium and silicon was observed, depending on the direction of diffusion (Fig. 4).

B

four-fold coordinated. Sodium thus essentially has a charge-balancing role. Consequently, the glasses contain very limited to no NBOs. The %BO4− measured were then compared to the Dell and Bray model [50] (Table 4). Despite some limitations [51–54], this model realistically predicts changes in the coordination chemistry of boron depending on the bulk composition of the glass. The data reported here are fully consistent yet systematically slightly lower than model predictions. Such a systematic difference may be related to the higher fictive temperature of our glasses compared to the samples analyzed by Dell and Bray [50].

5.4. Results from the fit The complete data sets were fitted simultaneously following the procedure described above. All the eigenvalues and eigenvectors and their relative uncertainties are listed in Table 5. The quality of the fits is excellent (Fig. 4). It reproduces even subtle uphill diffusion phenomena, like that of sodium when boron and silicon diffuse against each other. Furthermore, the uncertainties on eigenvalues and eigenvectors are small. At 700 °C, however, data points from the diffusion profiles associated to the exchange between boron and silicon are very dispersed. An explanation for this could be phase separation induced by the cesium ion beam, a phenomenon observed for a similar glass under electron irradiation [55]. As summarized in Table 5, eigenvectors are not very sensitive to temperature over the studied range. This negligible temperature dependence has also been observed in previous works [19,21,25]. Conversely, eigenvalues vary drastically, by about 4 orders of magnitude between 700 °C and 1100 °C. Because eigenvectors are almost the same at any temperature considered, hereafter only one set of eigenvectors will be considered when describing the chemical diffusion mechanisms.

5.3. Diffusion profiles Concentration profiles were plotted against x / t , the so-called ‘normalized distance’. In the absence of convection and if the diffusion matrix is independent of time, concentration profiles obtained with the same experimental configuration but conducted for different annealing durations should superimpose when plotted together as a function of the normalized distance. This is indeed observed for the samples, especially at the highest temperatures where convection could have been a problem (Fig. 3). The exchange between Na2O and B2O3 at 1100 °C is not presented here because the samples were found to be homogeneous after 90 min. These results indicate that no convection occurred during the experiments and that the terms of the diffusion matrix are independent of run duration. The concentration profiles

SiO2-Na2O exchange

Fig. 3. Concentration profiles from diffusion experiments performed at 1100 °C for 20 min and 90 min.

SiO2-B2O3 exchange

18 16

20 min 90 min

20

18

16

20 min 90 min

18

16

Na2O (mol. %)

Na2O (mol. %)

14 18

14 12 20 min 90 min

10 8 72

SiO2 (mol. %)

B2O3 (mol.%)

20

16 14 12 20 min 90 min

10 8 72

20 min 90 min

SiO2 (mol. %)

B2O3 (mol. %)

22

70 68 66 64

20 min 90 min

70 68 66 64

-10 -8

-6

-4

-2

0

2

xt-1/2(μm.s-1/2)

4

6

8

10

-10 -8

-6

-4

-2 -1/2

xt

34

0

2 -1/2

(μm.s

4

)

6

8

10

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SiO2-Na2O exchange

SiO 2-B2O3 exchange

Na2O-B 2O3 exchange

700°C

700 °C

700 °C

800 °C

800 °C

800 °C

900 °C

900 °C

900 °C

1000 °C

1000°C

1000 °C

1100 °C

1100 °C

1100 °C

Fig. 4. Diffusion profiles between 700 °C and 1100 °C for the 3 diffusion couples.

dependent component is very important, because it implies that the system must be regarded in a reference frame in which the flux of k is considered to be zero [18]. The relationships necessary to switch the dependent components have been expressed by Kirkaldy and coworkers [56]. In order to have a more precise view of diffusion phenomena, all three diffusion matrices were calculated here (Table 6). Diffusion coefficients, Di, jk, provide information about the intensity

The eigenvectors are those calculated for diffusion experiments performed at 1100 °C, as their uncertainties are the smallest. 5.5. Diffusion matrices For the calculation of diffusion matrices, Dk, in our system, SiO2, Na2O or B2O3 are potential dependent components, k. The choice of the 35

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Table 5 Eigenvectors and eigenvalues derived from the fits. At 700 °C, eigenvectors determined at 1100 °C were imposed during the fit in order to have less uncertainty on eigenvalues. Temperature (°C)

Major eigenvector (× 10− 8 cm2/s)

700

B2O3 Na2O SiO2

800

B2O3 Na2O SiO2

900

B2O3 Na2O SiO2

1000

B2O3 Na2O SiO2

1100

B2O3 Na2O SiO2

0.38 −1 0.62 ± 0.18 0.06 −1 0.94 ± 0.03 0.02 −0.98 1 ± 0.03 0.09 −1 0.91 ± 0.01 0.11 −1 0.89 ± 0.01

Major eigenvalue (× 10− 8 cm2/s)

Minor eigenvector (× 10− 8 cm2/s)

Minor eigenvalue (× 10− 8 cm2/s)

1.82 × 10− 3 ± 0.99 × 10− 3

B2O3 Na2O SiO2

1.40 × 10− 4 ± 0.14 × 10− 4

8.57 × 10− 2 ± 2.58 × 10− 2

B2O3 Na2O SiO2

6.78 × 10− 1 ± 0.31 × 10− 1

B2O3 Na2O SiO2

5.44 ± 0.32

B2O3 Na2O SiO2

1.01 × 101 ± 0.01 × 101

B2O3 Na2O SiO2

Table 6 Diffusion matrices expressed with different dependent components and for diffusion experiments performed at 1100 °C. DSiO2(× 10− 8cm2/s)

B2O3 Na2O

B2O3 0.81 − 2.83

Na2O − 1.04 9.99

DNa2O(× 10− 8cm2/s)

B2O3 SiO2

B2O3 1.85 10.97

SiO2 1.04 8.95

Na2O SiO2

2.60 × 10− 3 ± 0.28 × 10− 3

3.60 × 10− 2 ± 0.54 × 10− 2

1.55 × 10− 1 ± 0.32 × 10− 1

5.07 × 10− 1 ± 0.22 × 10− 1

Table 7 Diffusive exchanges with their corresponding eigenvalue at 1100 °C.

DB2O3(× 10− 8cm2/s) Na2O 12.82 − 10.97

0.73 0.27 −1 ± 0.04 0.73 0.27 −1 ± 0.02 0.77 0.23 −1 ± 0.01 0.75 0.25 −1 ± 0.02 0.76 0.24 −1 ± 0.01

Major Minor

SiO2 2.83 − 2.02

Diffusive exchange

Eigenvalue (× 10− 8 cm2/s)

0.10 B2O3 + 0.90 SiO2 ↔ 1 Na2O 0.75 B2O3 + 0.25 Na2O ↔ 1 SiO2

10.1 ± 0.1 0.51 ± 0.02

used for their calculation is described elsewhere [19,27]. Table 7 summarizes both the eigenvectors and their eigenvalues that can be related to the principal mechanisms of diffusive exchange. In the system studied here, the dominant diffusion mechanism (i.e. that with the largest eigenvalue) involves exchange between ~1 SiO2 and ~1 Na2O. The secondary mechanism involves an exchange between boron and silicon with minor participation of sodium. This exchange is more surprising because boron and silicon are both strongly bonded to the network, making their exchange less intuitive. This exchange may be made possible by the presence of Na, as discussed below. Turning to the eigenvalues of diffusive exchange, at 1100 °C the eigenvalue of the principal exchange mechanism is 20 times higher than that of the secondary mechanism, a ratio that is remarkably constant at all the temperatures studied.

of the couplings between components. In this study, the off-diagonal terms of the diffusion matrices are of the same order of magnitude as the diagonal terms. Couplings between components are thus not negligible and must be taken into account when describing diffusion. For the matrix expressed with B2O3 as the dependent component,| DSiO2, Na2OB2O3 | ≫ | DSiO2, SiO2B2O3 |, meaning that a strong coupling exists between silicon and sodium. The negative sign of DSiO2, Na2OB2O3 means that transport of silicon is accelerated by the presence of a concentration gradient of sodium. This result is consistent with the observations made on diffusion profiles where sodium exchanged with silicon. Another strong coupling between silicon and boron is highlighted through the diffusion coefficient | DSiO2, B2O3Na2O |,which is higher than | DSiO2, SiO2Na2O |. This time, DSiO2, B2O3Na2O > 0 so silicon is slowed down when it diffuses in a concentration gradient of boron. A final coupling that participates, to a lesser extent, is apparent in the diffusion coefficient DB2O3, Na2OSiO2,which reveals that diffusion of boron is accelerated in the presence of a sodium concentration gradient. Such behavior confirms the observations made on diffusion profiles where boron diffused faster in the B2O3-Na2O exchange than in the B2O3-SiO2 exchange. These results demonstrate that the magnitude of Di, j coefficients is not the only factor controlling the extent of coupling, but that this also depends on the choice of dependent component and reference frame. It's also important to note that a same diffusion matrix can result in uphill diffusion or not depending on the direction in compositional space where diffusion occurs. This point has been made by Chakraborty et al. [18,19] but is still misunderstood.

5.7. Diffusion paths In a ternary system, the plot of diffusion profiles in compositional space (called the diffusion path) is another practical way to quickly obtain information about the direction of eigenvectors and diffusive couplings. For the sake of clarity, diffusion paths were calculated using the theoretical starting compositions as well as eigenvalues and eigenvectors derived from the fit (rather than using individual data points), as illustrated in Fig. 6. In this representation, the trend close to the starting compositions follows the direction of the principal eigenvector. Near the average composition, both diffusive exchanges occur but the minor diffusive exchange dominates, so that the diffusion path locally follows the direction of the secondary eigenvector [19]. The results show that diffusion paths have very different shapes depending on the directions chosen for the diffusion couples (Fig. 6). They are Sshaped when uphill diffusion occurs in the sample [19,27] (e.g. for the diffusion path of the Na2O-B2O3 exchange). The uphill diffusion of Na2O during the SiO2-B2O3 exchange is also significant though less pronounced. Diffusion profiles that are associated with Na2O-SiO2 and

5.6. Diffusive exchanges The eigenvectors of the diffusion matrix are another way to quantify the couplings involved during diffusive exchange. The methodology 36

B2O3-SiO2 exchanges are much flatter, because their orientation in the composition space nearly parallels those of eigenvectors.

Log( ) x10-8 cm2/s

6. Discussion 6.1. Mechanisms of diffusive exchange The eigenvectors of the diffusion matrix provide information about the stoichiometry of exchange between different species. Some authors have tried to understand how they are produced [19,27] but this task requires additional information to be taken into account, especially concerning the glass structure. The main diffusive exchange found in this study is produced as a more or less binary exchange between ~ 1 mol of Na2O and ~1 mol of SiO2. Stebbins and his collaborators [57,58] have already suggested an atomic displacement mechanism to describe such behavior in SiO2-X2O glass systems (where X are alkalis). This mechanism has the particularity that it requires an intermediate step with five-fold coordinated silicon to occur. We believe that it is sufficient for describing the diffusion mechanism which dominates in the system investigated here even if silicon with coordination five has never been directly found in sodium borosilicate systems. It is implicit in this observation that this diffusion mechanism does not bring boron into play, and that the sodium-silicon exchange takes place without disturbing the boron-related network. Concerning the secondary diffusive exchange, a strong coupling between silicon and boron can be inferred, responsible for the uphill diffusion of SiO2 visible on the concentration profiles of the exchange between Na2O and B2O3 (Fig. 4). A simple exchange between B2O3 and SiO2 seems difficult to imagine, as boron and silicon are immobile species connected to the network by strong bonds. However, the importance of sodium in the eigenvector in Table 6 is of potential importance. For example, if sodium is added to both sides of the equation, it can be re-written in the form:

1.5 B2 O3 + 1.5 Na2 O ↔ 2 SiO2 + 1 Na2 O

-1

1

-2

0

-3

-1

-4

-2

-5

-3

-6

-4

-7 -8

-5 max

-6

-9

min

-7

-10

-8

-11 -12

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1000/T (K-1) Fig. 5. Comparison of the evolution with temperature of eigenvalues (λ), viscosity (η) and conductivity (σ). The squares correspond to major eigenvalues and the circles are minor eigenvalues.

polymerize the vitreous network, breaking the SieOeSi or SieOeB bonds [39,60]. This change in the structural role of sodium leads to the conversion of BO4− units to BO3 units as follows:

BO4− + Qn ↔ BO3 + Qn − 1 n

(23)

Where Q is a silicon tetrahedron surrounded by n bridging oxygens. For example, for glass compositions similar to those studied here, it has been shown that the %BO4− can drop by up to 18% between the glass transition temperature and 800 °C [61], highlighting the potential importance of Eq. 23. However, it should be appreciated that the constant nature of eigenvectors with temperature indicates that the fundamental exchange at work (i.e. Eq. 22) remains the same between 700 °C and 1100 °C. A change in the abundance of the different species present thus does not change the exchange mechanism, but may be related to the observation that eigenvalues evolve non-linearly as a function of temperature (because the number of BO4−, BO3, Qn and Qn1 units available control the likelihood of encounter and thus exchange). The structural changes in the glasses used here are potentially drastic because, at low temperature our base composition contains few or no NBOs. Consequently, with increasing temperature melt structure becomes significantly de-polymerized. This effect may be responsible for the strong non-Arrhenian behavior of viscosity (and thus diffusivity) in this system, as discussed below. In order to have more insight into diffusive mechanisms, two other transport properties were investigated (i.e. viscosity and ionic conductivity). A comparison of the changes in these properties and eigenvalues as a function of temperature (Fig. 5) shows that ionic conductivity seems disconnected from chemical diffusion over the full temperature range investigated, but that viscosity follows the same non-Arrhenian trend as that of the eigenvalues, within experimental errors. In the ‘high temperature’ liquid (i.e. between 900 °C and 1100 °C), all three properties seem to converge. This observation can be related to the fact that at high temperature, the relaxation times of the processes leading to transport properties tend to converge [62–64]. In detail, chemical diffusion has an activation energy (184 kJ/mol ± 22 kJ/mol) which is very close to that of viscosity (186 kJ/mol ± 16 kJ/ mol) but higher than that of ionic conductivity (79 kJ/mol ± 8 kJ/ mol). Thus, the transport property which seems the best correlated with diffusion is clearly viscosity (Fig. 5). Several studies have already shown that viscosity could be linked to self- or tracer- diffusion of network forming cations [65–68] and attempts have been made to develop models that relate chemical diffusion to tracer diffusivities (e.g. [69]). Although these models fail in some cases [20], our study clearly establishes a link between chemical diffusion and viscosity. Exploring this idea in more detail, this link may be rationalized if it is considered that the kinetics of diffusive exchange are limited by the frequency of

(21)

Assuming that the species which diffuse are ions, the exchange (21) can also be written as:

3(B3 + + Na+) ↔ 2(Si4 + + Na+)

2

-Log( ) (Pa.s) Log( ) ( -1.cm-1)

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(22)

In this form, it would seem that the minor diffusive exchange could be described as an exchange between the Q3 species associated with Si, and BIV tetrahedra associated with Na. This mechanism appears to be consistent with the work reported by Du and Stebbins [59], who used 11 B NMR MQMAS to show considerable mixing between boron and silicate units in SBN68-18 glass (≈ 60% of SieOeSi bonds, ≈ 35% of SieOeB bonds and ≈5% of BeOeB bonds). These observations are also in agreement with the Dell and Bray model [50,54] which predicts medium-range order in the glass compositions studied here marked by the presence of reedmergnerite- and danburite-type structural units. The minor diffusive exchange could thus be associated with a diffusion mechanism requiring the formation/dissociation of these units in the glass. 6.2. Kinetic of diffusive exchanges The eigenvalues of the diffusion matrix (λmin and λmax) can be considered as diffusion coefficients. They provide the rate of exchange reactions represented by eigenvectors [19]. Their variation as a function of temperature can be assessed in an Arrhenian diagram (Fig. 5). This shows that for the temperature range from 700 °C to 1100 °C, the two eigenvalues have a very similar temperature dependence, but their behavior deviates from an Arrhenius law. This latter observation may be the result of structural changes of the borosilicate network with temperature. Indeed, Raman and NMR MAS spectroscopy show that when heating a glass to the liquid state, sodium ions which are initially charge-compensators for BO4− units leave their position and de37

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a)

Na2O

Na2O

0

16

100

19 17

20

18

80

18 17

SiO2-B2O3 exchange

40

SiO2-Na2O exchange

19 16

60

Na2O-B2O3 exchange

20

Zhabrev et al. Major eigenvector direction Minor eigenvector direction

15 21

60

80

22

3

13 12 66

67

68

69

70

71

72

20

2

23

65

40

1

14

B2O3

Fig. 6. (a) Diffusion paths from this study plotted with eigenvectors in the SiO2-Na2O-B2O3 ternary diagram. (b) Comparison between the diffusion paths found here and the diffusion paths from Zhabrev et al. [37]. The ternary diagrams are in mol.%.

b)

SiO2

100

B2O3

0

20

40

60

80

100

0 SiO 2

SiO2-Al2O3-Na2O-K2O-B2O3 system. It was shown that when boron is exchanged with silicon, diffusion occurs as it would in a binary system, meaning that this diffusion direction is along an eigenvector (all the other components have constant concentrations along the diffusion couple). These observations suggest that in sodium borosilicate systems, couplings are conserved whatever the composition, but that the stoichiometry of diffusive exchanges varies slightly for different compositions. Furthermore, when other components are added to the system, the exchange between SiO2 and Na2O still persists. This reinforces the view that diffusive exchanges are maintained when considering chemical diffusion in more complex compositions [25,27].

SieO and BeO bond breaking. In this case, the same temperature dependence of the two eigenvalues and viscosity may be simply explained by the fact they are all controlled by the time-scale of the same limiting microscopic change. This reasoning also explains why ionic conductivity is uncorrelated with chemical diffusion in our system. This is because the former is associated with sodium self-diffusion, which is fast and independent of viscous flow in the supercooled liquid domain [65].

6.3. Comparison with other systems from the literature 6.3.1. Borosilicate systems To the authors' knowledge, multicomponent chemical diffusion experiments in simple sodium borosilicate glasses has only been performed by Zhabrev and co-workers [37] (at 800 °C). The diffusion couples they chose had very different mean compositions, and the concentration gradients between their glasses were more than 10 mol %. For these reasons, no attempt was made to derive the diffusion matrix from their results, but their diffusion paths are compared with our new data in Fig. 6. The starting compositions of their diffusion paths 2 and 3 evolve following a direction almost collinear with this study's major eigenvector, which suggests a strong coupling between silicon and sodium. Diffusion path 2 is also rather flat and almost collinear with our minor eigenvector. It is therefore reasonable to assume that both diffusive exchanges derived here are good approximations for the experiment of [37]. The same conclusion can be drawn for diffusion path 3. Its S-shape means that an uphill diffusion of silicon appears when boron is exchanged with sodium, revealing strong couplings between silicon and boron. Diffusion path 1 is the only one that suggests an exchange stoichiometry slightly different from that obtained in this study. It is flat, meaning that either both eigenvalues are the same or that the direction of diffusion is along an eigenvector [25]. In the light of the results obtained here, it can be assumed that the second option is correct, in which case the diffusive exchange between silicon and boron does not require sodium. This would explain why no uphill diffusion of sodium was observed by Zhabrev and co-workers [37] when boron was exchanged with silicon. A similar kind of coupling was also identified by Chakraborty and co-workers [38] in a more complex glass from the

6.3.2. Other systems The data obtained here can also be compared with multicomponent diffusion experiments carried out in other ternary systems. This comparison reveals that the stoichiometries found for diffusive exchanges are very close to those of peraluminous compositions (KASD3,Table 8) from the SiO2-K2O-Al2O3 system [19]. However, for peralkaline compositions, aluminum is more strongly coupled to silicon, even though the major diffusive exchange remains approximately the same [19] (KASD17a, Table 8). Results from Liang and co-workers for the SiO2Al2O3-CaO system enable comparison of the role of alkaline and alkaline-earth cations during chemical diffusion [23]. In alkaline-earth systems, the major diffusive exchange still relates SiO2 to the fastest diffusive species (CaO in that case), but couplings between the components are stronger and result in different stoichiometries for the exchanges. In turn, elements from the same columns of the periodic table (i.e. B and Al, Na and K) seem to play very similar roles during chemical diffusion. However, alkaline-earth and alkaline diffusive mechanisms are not comparable, no doubt due to the fact that alkaline earths are more strongly bonded to the network. 7. Conclusions Multicomponent chemical diffusion was studied at between 700 °C and 1100 °C in the SiO2-Na2O-B2O3 ternary system for glasses with an

Table 8 Diffusive exchanges from Chakraborty et al. [19] and Liang et al. [23]. Glass (mol.%) Name KASD3 KASD17a D

SiO2 85.3 84.9 64.1

Diffusive exchange Al2O3 9.3 5.2 10.0

K2O 5.4 9.9 –

CaO – – 25.9

Major 0.14 Al2O3 + 0.86 SiO2 ↔ 1 K2O 0.11 Al2O3 + 0.89 SiO2 ↔ 1 K2O 0.28 Al2O3 + 0.72 SiO2 ↔ 1 CaO

38

Minor 0.24 K2O + 0.76 Al2O3 ↔ 1 SiO2 0.15 K2O + 0.85 Al2O3 ↔ 1 SiO2 0.52 CaO + 0.48 Al2O3 ↔ 1 SiO2

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average composition (mol.%) of 68SiO2–18B2O3-14Na2O. Diffusion couples were obtained and concentration profiles were fitted using the matrix formalism of multicomponent diffusion. Eigenvectors were found to be more or less identical between 700 °C and 1100 °C, so diffusive mechanisms seem to be unaffected by temperature. They can be modeled in terms of two exchange mechanisms. The dominant exchange mechanism reveals strong couplings between silicon and sodium. It is suggested that this exchange occurs through a diffusive mechanism which requires an intermediate state with five coordinated silicon. The secondary diffusive mechanism involves exchange between silicon and boron which is made possible through the involvement of sodium. Both eigenvalues were found to vary dramatically with temperature and follow the same non-Arrhenian behavior. It was proposed that structural modifications of the borosilicate network with temperature were responsible for the deviation from the Arrhenian law. The evolution of eigenvalues with temperature was compared to that of viscosity and an excellent correlation was found. Their similar behavior was assigned to the fact that both viscosity and diffusion are controlled by the time-scale of SieO and BeO bond-breaking/formation, which limits the global dynamics of the system. Finally, these results were compared with other multicomponent chemical diffusion studies. The complete data set raises hopes that forthcoming studies will make it possible to build a predictive model to describe chemical diffusion in wide compositional domains of nuclear and industrial interest.

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