#### 4.1. Hydrolytic Durability

Hydrolytic stability of aluminum–phosphate glass matrices was reported in References [

11,

44,

45,

46,

47]. However, very few publications are devoted to the effect of self-irradiation of the glasses on structure and properties [

11]. Therefore, analysis of the effect of irradiation on the leaching of glass matrices is the most important part of this work. To assess the chemical (hydrolytic) stability of glasses of different compositions, leaching experiments were performed. The leaching of glass samples (NAPas0, NAPas1, NAPcm0, NAPcm1, NAP1 and NAP2) was carried out in the water in autoclaves at 90°C; the solution was changed after 1, 3, 7, 10, 14, 21, and 28 days. Besides, similar leaching tests were carried out for samples NAPcm0 and NAPcm1 at 25°C. Due to the small amount of material and difficulties in the preparation of monolithic cubic or parallelepiped specimen, the samples were crushed. The particle size fraction in each sample was selected using sieves with mesh sizes of 0.16 mm and 0.071 mm. From average particles size, the surface area of the powdered glass (S) in runs was estimated to be 200 сm

^{2}.

Analysis of solutions after experiments on interaction with glass was carried out by inductively coupled plasma mass spectrometry on an X-Series instrument at the GEOKHI RAS (Vernadsky Institute of Geochemistry and Analytical Chemistry of Russian Academу of Sciences).

Let us denote the time at which the water was replaced (and the aliquot of the solution for analysis was taken) as {t

_{i}, i = 1, ..., 7}; t

_{0} = 0. Then, for the time interval between t

_{i−1} and t

_{i}, the differential dissolution rate of the glass, normalized by one of its main structural elements, is determined as

where C

_{El} is the concentration of the glass element, which determines the normalized dissolution rate; V is the volume of the autoclave; S is the total surface area of the particles, m

_{El} is the mass fraction of the element El in the glass. Since J

_{El} is the average value of the normalized dissolution rate in the time interval between t

_{i−1} and t

_{i}, when determining the function J

_{El}(t), it is approximately attributed to the middle of this interval, i.e., at time τ

_{i} = (t

_{i−1} + t

_{i})/2, since the smooth function is assumed.

Data on integral dissolution rates of the glasses investigated are presented in

Appendix B. A typical dependence of the normalized dissolution rate of glasses on time is shown in

Figure 7.

In general, all values of J

_{El} decrease with time, which is consistent with the simplest diffusion models for the advancement of the leach front. However, at τ

_{i} = 8.5 days, a local maximum occurs. Such jumps in J

_{El} values were noted in many studies on the leaching of borosilicate glasses [

26,

48,

49,

50,

51,

52]; they are termed as leaching resumption stage. The presence of these events is typically explained by redeposition of sparingly soluble matrix components as secondary phases. The formation of these phases circumvents the protective effect of the altered glass layer, which acts as a barrier against leaching [

53,

54]. However, this explanation cannot be considered exhaustive, since in many works, the effect of leaching recovery was not observed, although secondary phases were also deposited on the glass surface. An alternative explanation is the formation of micro-cracks on glass surface [

26]. In the case of investigated NAP glasses, the presence of the local maximum is neither an artifact nor a random deviation, as proved by maxima of J

_{El}(t) in experiments with other glasses (

Figure 8).

In all the samples studied the mass fraction of phosphorus is approximately the same. In addition to the presence of modifiers in samples NAPcm, NAPcm0, NAP1 and NAP2, significant differences of these four types of glasses from samples NAPas0 and NAPas1 relate only to the ratio of mass fractions of sodium and aluminum. Analysis of values of J

_{El} (t) in these two groups of glasses shows that glasses with a higher mass content of aluminum dissolve more slowly in water, i.e., have smaller J

_{El}(t) (

Figure 8).

Analysis of J

_{El}(t) over the entire range of time values shows that for all t, the ratio J

_{El}(t) for sample NAPas1 and samples NAPcm, NAPcm0, NAP1, and NAP2 for all elements for which the normalized dissolution rate was calculated, has close values apart from the unreliable value of J

_{El}(t) for NAPcm0. The dependence of J

_{El}(t) on the ratio of sodium and aluminum in phosphate glasses has the form

where

${\mathrm{J}}_{\mathrm{E}\mathrm{l}}^{0}\left(\mathrm{t}\right)$ is a function that does not depend on the ratio m

_{Na}, m

_{Al}, e.g., mass fractions of sodium and aluminum in glasses.

Dependences J

_{El}(t) are satisfactorily approximated by power functions of the form (see

Figure 7):

where A is a coefficient depending on a certain element by which the normalized dissolution rate of the glass is calculated, and on the ratio of the mass fractions of sodium and aluminum in the glass composition. Values of normalized dissolution rates calculated for different elements satisfy the inequalities:

This seems to be well justified: Sodium passes into solution more easily than phosphorus, and that, in turn, is easier leached compared to aluminum. If values of J

_{Na}(t) are used in the estimate of the safety of repositories to calculate leaching rates, then this estimate will be conservative (i.e., pessimistic). In this regard, it seems appropriate to find such a form of the analytic function

${\mathrm{J}}_{\mathrm{N}\mathrm{a}}^{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}}\left(\mathrm{t}\right)$ that approximates with satisfactory accuracy (with the mean-squared error of about 30%) the dependences obtained from experiments in the entire range of ratios m

_{Na} and m

_{Al} studied. Accounting for Equation (2), we seek this analytical approximating dependence in the form

where a, b are unknown parameters that we will determine by the least-squares method [

55] from the condition

The procedure for finding the minimum of the function Ф was carried out by the gradient method with the control of convergence as follows [

53]. The initial approximation was set:

${\mathrm{a}}_{0}$= 9 · 10

^{−5},

${\mathrm{b}}_{0}$ = 0.8 Then, with the known n-th approximation—a

_{n}, b

_{n}—the (n+1)-th approximation was determined by the formulae

Here we have

where m is the minimum positive integer under which the condition for convergence of the method is satisfied:

The minimum function Φ (a,b) found in this way from the data on the leaching rate of Al is reached at

$\mathrm{a}=1.183\xb7{10}^{-4},\mathrm{b}=0.75$. Therefore, if we substitute these values of the parameters into dependence (5), it will have the form

where [J

_{Na}] = g/cm

^{2} · day, [t] = day,

${\mathrm{F}}_{\mathrm{N}\mathrm{a},\mathrm{A}\mathrm{l}}=\mathrm{exp}\left\{0.75{\mathrm{m}}_{\mathrm{N}\mathrm{a}}/{\mathrm{m}}_{\mathrm{A}\mathrm{l}}-1.875\right\}$. The calculated dependence (9) is compared with the experimental data on the leaching of glasses (

Figure 9) and shows a satisfactory accuracy of the approximation of experimental data by power functions.

The temperature has a significant effect on the leaching of sodium–aluminum–phosphate glasses [

54]. A comparison of the J

_{El}(t) dependences obtained on the NAPcm0 sample in experiments at 90 °C and 25 °C for different structural elements of glass is shown in

Figure 10.

The dependence of leaching rates on temperature, as well as the intensity of many chemical reactions, can be described by the Arrhenius formula:

where E

_{a} is the activation energy, and R is the gas constant (R = 8.3 J/(mol · K)).

As noted above the final portion of J

_{El}(t) dependences of NAPcm0 can be ignored. Thus, for calculating the J

_{El}(t) ratios at different temperatures, only two time points remain, e.g., 5 and 8.5 days. The values of the relations J

_{El}(t) at different temperatures T at these points are given in

Table 7.

The average value over

Table 7 is 3.62. Then we get from expression (11)

$\mathrm{ln}\text{}3.62=\frac{{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}\left(-\frac{1}{363}+\frac{1}{298}\right)$, therefore for glass NAPcm0 we obtain

${\mathrm{E}}_{\mathrm{a}}=$ 17.7 кJ/(mol · К). In a similar manner, we consider the J

_{El}(t) ratios at temperatures of 90 °C and 25 °C of NAPcm1 glass. These data are given in

Table 8. The average over

Table 8 value of the ratio of leaching rates at 90 °C and 25 °C, in this case, is 3.92. Hence, the activation energy of dissolution of glass NAPcm1 is E

_{a} = 18.9 kJ/(mol · K). This value is close to the previously calculated value of 17.7 kJ/(mol · K) for NAPcm0, which supports the correctness of the results. Note, however, that these activation energies are significantly lower compared with those in borosilicate nuclear waste glasses, which are more durable in water [

56].

#### 4.2. Effect of γ-Radiation on Dissolution of Glass

Effects of irradiation of glasses on their dissolution behavior remain insufficiently studied, but recent works on International Simple (borosilicate) Glass showed that it is indeed necessary to take into account radiation damage effects in the prediction of water-glass interaction in HLW repository [

57]. The absolute majority of irradiation experiments employ much higher dose rates than is expected in real HLW. Whereas, the dose-rate effect is difficult to estimate consistently, one can reasonably expect that higher rates may alter the mechanism of radiation damage (e.g., cascade overlap vs. defects accumulation) and influence recovery kinetics. Presumably, in most cases, the influence of high dose rates on dissolution in irradiation experiments will be larger than in real HLW glasses.

Only very few studies addressed the effects of irradiation on properties of aluminum–phosphate glasses. In Reference [

58] it was shown that during electron irradiation of aluminum–phosphate glass gas bubbles were formed even at low doses (0.8 · 10

^{23} electrons/m

^{2}). As the absorbed dose increases, the bubbles grow, and they move towards the glass surface. At a dose of 2.2 · 10

^{26} electrons/m

^{2} (6.6 · 10

^{11} Gy) all bubbles leave the glass sample. With further exposure of the glass to a dose of 4.5 · 10

^{26} electrons/m

^{2}, areas enriched in Al and P appeared. Sodium–aluminum–phosphate glasses were irradiated to a dose of 10

^{8} Gy (electrons and γ-photons), and up to 2 · 10

^{18} α-decays/cm

^{3} [

11]. These values are close to the radiation dose that real vitrified HLW will receive in 10,000 years. Study of the irradiated glass samples by x-ray diffraction analysis, nuclear magnetic resonance and electron paramagnetic resonance, showed absence of changes in the matrix structure [

11]. It was also found that in experiments lasting one day, the rate of Na leaching at room temperature increases with the rise of radiation dose up to 10

^{8} Gy, but its value remains constant or slightly decreases after 30 days of interaction, amounting to 4 · 10

^{−7} g/(cm

^{2}·day). The damage during gamma-irradiation is mostly due to Compton electrons, thus, the studies employing electrons mentioned above are fully relevant.

A comparison of data on the dissolution of NAPas0 and NAPas1 glass samples is shown in

Figure 11. On a logarithmic scale, the dependency graphs J

_{El}(t) for the same structural elements obtained on samples NAPas0 and NAPasa1 run approximately parallel to each other. This suggests that over the entire time range, the ratio J

_{El}(t) obtained for the initial and irradiated glasses remains roughly the same. The ratios of values of J

_{El}(t) obtained on samples of the pristine NAPas0 and irradiated NAPas1 glasses at 90 °C are given in

Table 9. The average value over the whole table is 2.03; the data of

Table 8 and

Table 9 compared in

Figure 11.

Figure 12 shows that the J

_{El}/J

_{El0} ratios obtained on samples NAPas0 and NAPas1 remain close to 2 through the entire studied range of duration of runs.

Thus, as a result of γ-irradiation, the leaching rate of the sodium–aluminum–phosphate glass has averagely decreased by about 2 times. This can be caused by undetectable structural changes such as radiation-induced annealing that occurs in some other type of glasses [

22]. Much higher effect on the deterioration of properties of the aluminum–phosphate vitreous waste form, including their solubility in hot water, is observed due to glass crystallization in temporary storage and after ultimate disposal in deep underground repository [

11,

14,

15,

36,

37,

38,

39,

40,

43,

44,

45].