ON THE KINETICS OF DISSOLUTION OF PRE-IRRADIATED CRYSTALS

P. A. Savintsev, A. F. Naumov, I. G. Berzina

The formation of defects in a crystal under irradiation leads to an increase in the potential energy of the lattice \[1, 2\] and in the thermodynamic potential of the crystal. The energy stored in this process must change the conditions of thermodynamic equilibrium of the crystal with the solution. An increase in the thermodynamic potential \(\Phi\) under irradiation causes an increase in the concentration \(C_0\) of the equilibrium solution over the irradiated crystal. Thus, irradiation of crystals increases their solubility and, correspondingly, the dissolution rate \(v\).

At very small undersaturations or supersaturations \(z\) of the mother solution and with intensive stirring, the rate of dissolution and growth of crystals becomes proportional to \(z^2\) \[3\], where \(z = (C_0 - C)/C_0\), and \(C\) is the concentration of the dissolved substance in the volume of the solution.

Let irradiated and non-irradiated crystals dissolve under identical conditions in one and the same solution. We shall denote the undersaturation for the non-irradiated crystal by \(z_{\text{nonirr.}}\), and for the irradiated one by \(z_{\text{irr.}}\). The ratio of the dissolution rate of the irradiated crystal \(v_{\text{irr.}}\) to the dissolution rate of the non-irradiated crystal \(v_{\text{nonirr.}}\) will be equal to

\[ \frac{v_{\text{irr.}}}{v_{\text{nonirr.}}} = \frac{z_{\text{irr.}}^{2}}{z_{\text{nonirr.}}^{2}} . \tag{1} \]

Here \(z_{\text{irr.}} \cdot z_{\text{nonirr.}}\) is a constant quantity, not changing when the concentration of the solution is changed.

Equation (1) shows that the ratio \(v_{\text{irr.}} : v_{\text{nonirr.}}\) must increase sharply as the concentration of the solution approaches the saturation concentration relative to the non-irradiated crystal.

The considerations set forth were confirmed experimentally during the dissolution of irradiated and non-irradiated crystals of NaCl, KCl, and KBr in their aqueous solutions. The dissolution of the crystals was carried out as follows. A cylindrical hole was drilled in the crystal, through which a jet of solvent passed. The dissolution regime was varied by changing the rate at which the solution flowed through the hole, and also by changing the length and diameter of the hole. The transfer of substance from the crystal into the solution occurs through a laminar—

the laminar sublayer adjacent to the dissolution surface, whose thickness at the end of the turbulent-flow stabilization region is equal to

\[ h = 64.2 \frac{d}{Re^{0.875}}, \tag{2} \]

where \(d\) is the diameter of the opening, and \(Re\) is the Reynolds number [4].
In nonstationary flow of a jet of solution through an opening in a thin crystal plate, the thickness of the laminar sublayer becomes smaller than that determined by this formula.

Crystals of alkali-halide salts were prepared for the experiments in the form of plates \(0.93 \pm 0.01\) mm thick. The velocity of passage of a jet of aqueous solution of the corresponding salts through a cylindrical opening at the center of the plate, 1.2 mm in diameter, was chosen so that, regardless of the solution concentration, the Reynolds number remained constant and equal to 4000. Under these conditions the velocity of the solution jet proved to be approximately \(4\ \text{m/sec}\).

The linear dissolution rate was found from the formula \(v = \dfrac{d - d_0}{\tau}\), where \(d - d_0\) is the increase in the diameter of the cylindrical opening during dissolution of the crystal, and \(\tau\) is the dissolution time. The concentration of the solution was measured by determining the density of the solution, which was found by weighing 100 ml of solution. The experiments were carried out in a thermostated room at a temperature of \(20^\circ\)C.

For irradiation of the crystals, a URS-70 X-ray apparatus with a BSV-4-Cu tube was used. A voltage of 47.5 kV was applied to the tube at an anode current of 16 mA. The irradiated crystal was located at a distance of 5 cm from the center of the tube and received a dose of 130 r in one minute (the crystal plate was placed perpendicular to the direction of the rays).

In addition to irradiation with X-rays, experiments were carried out to study plates of alkali-halide crystals prepared in the same way and irradiated with neutrons and \(\gamma\)-rays from a Ra–\(\alpha\)—Be source. The samples were placed in a paraffin block that slowed neutrons at a distance of 15 cm from the source. The activity of the source was \(4 \cdot 10^7\) neutrons per second. The irradiation time was from 40 to 70 hours. Dissolution of the crystals was carried out immediately after the end of irradiation.

Figure 1 shows the increase in the ratio \(v_{\text{irr.}}/v_{\text{unirr.}}\) of the linear dissolution rates of rock-salt crystals irradiated and not irradiated with X-rays as the irradiation dose increases, at concentrations of NaCl solution in water equal to 24, 25, and 26% by weight. On the abscissa axis, besides the irradiation dose in roentgens, the time during which the crystal received the corresponding doses is also plotted. As can be seen from Fig. 1, the increase in the dissolution rate of irradiated crystals slows down as the irradiation time increases. This probably indicates that, at a certain irradiation time, the rate of a kind of annealing of defects appearing in the crystal during irradiation becomes comparable with the rate of formation of new defects in the crystal. The ratio

\[ \frac{v_{\text{irr.}}}{v_{\text{unirr.}}} \]

increases with increasing solution concentration (Figs. 1, 2), while the absolute value of the difference between these rates decreases.

Direct measurement of the increase in the equilibrium concentration of the solution \(\Delta C_0\) above the crystal during its irradiation is made difficult

![Graph: dependence of the ratio of dissolution rates of X-ray-irradiated and unirradiated NaCl crystals on irradiation dose. Vertical axis: \(V_{\mathrm{irr.}}/V_{\mathrm{unirr.}}\). Horizontal axis: irradiation time in hours; dose in roentgens. Curves labeled \(C = 26\%\), \(C = 25\%\), and \(C = 24\%\).]

Fig. 1. Dependence of the ratio of the dissolution rates of NaCl crystals irradiated with X-rays and unirradiated crystals on the irradiation dose.

![Graph: dependence of the ratio and difference of dissolution rates of irradiated and unirradiated rock-salt crystals on solution concentration. Left vertical axis: \((V_{\mathrm{irr.}} - V_{\mathrm{unirr.}})\cdot 10^4\ \mathrm{cm/sec}\). Right vertical axis: \(V_{\mathrm{irr.}}/V_{\mathrm{unirr.}}\). Horizontal axis: \(C\%\) by weight. Curves labeled \(V_{\mathrm{irr.}} - V_{\mathrm{unirr.}}\) and \(V_{\mathrm{irr.}}/V_{\mathrm{unirr.}}\).]

Fig. 2. Dependence of the ratio and difference of the dissolution rates of irradiated and unirradiated rock-salt crystals on the concentration of the solution.

because, as a result of particle exchange between the irradiated crystal and the solution, there occurs a change in the layers of the crystal lattice that are in contact with the solution. But an approximate calculation can be carried out from equation (1), using the data of Fig. 2 on the dependence of

\[ \frac{v_{\text{irr.}}}{v_{\text{unirr.}}} \]

on the concentration of the solution. Taking the solution concentration \(C = 25\%\) by weight (\(v_{\text{unirr.}} = 0.053\)), \(\frac{v_{\text{irr.}}}{v_{\text{unirr.}}}=1.12\), we obtain \(\Delta C_0 \simeq 0.08\%\) by weight.

Thus, at a temperature of \(20^\circ\text{C}\), the equilibrium concentration of a solution of NaCl in water will be \(26.40\%\) by weight over an unirradiated crystal and \(26.48\%\) by weight over an irradiated one (irradiation dose with X-rays \(6\cdot 10^4\ r\)).

From these data one can calculate the increase in the thermodynamic potential of an NaCl crystal at the indicated irradiation dose. The quantity \(\Delta \Phi\) will be equal to the difference between the chemical potentials of NaCl in solutions of concentrations 26.48 and \(26.40\%\) by weight, since the thermodynamic potential \(\Phi\) of a crystal, calculated per gram-mole of substance, is equal to the chemical potential of this substance \(\mu\) in a solution of equilibrium concentration. Writing the chemical potential in the usual form for a real substance, \(\mu=\mu_0+RT\ln fC\), where \(R\) is the universal gas constant, \(T\) is the absolute temperature, \(f\) is the activity coefficient, and \(C\) is the concentration of the substance in the volume, we obtain

\[ \Delta \Phi=\mu_1-\mu_2=RT\ln\frac{fC_1}{fC_2}=RT\ln\frac{C_1}{C_2}, \tag{3} \]

where \(\mu_1\) and \(\mu_2\) are the chemical potentials of the dissolved substance in solutions of concentrations \(C_1\) and \(C_2\), respectively.

Substituting into equation (3) the numerical data of our example, we obtain an increase in the thermodynamic potential of the crystal approximately equal to \(2\ \text{cal/mole}\).

Fig. 3. Dependence of the ratio of the dissolution rates of NaCl crystals irradiated from a Ra-γ-Be source and unirradiated on the storage time of the specimen at room temperature.

Similar results were obtained in the dissolution of single crystals of KCl and KBr, grown artificially by the Kyropoulos method.

Irradiation with X-rays causes in these crystals an increase in the dissolution rate that is the stronger, the lower the energy of the crystal lattice of the substance.

Figure 3 shows the dependence of the ratio \(\dfrac{v_{\text{irr.}}}{v_{\text{unirr.}}}\) for unannealed rock-salt crystals irradiated with neutrons and \(\gamma\)-rays from a Ra–\(\gamma\)–Be source, according to the method described above, on the time of annealing of the sample at room temperature, equal to \(20^\circ\)C, after irradiation. The concentration of the solution was 25.85 wt.% NaCl in water. With annealing times up to 8 hours, \(v_{\text{irr.}} > v_{\text{unirr.}}\). With further annealing, \(v_{\text{irr.}} < v_{\text{unirr.}}\).

The dependence of the difference in dissolution times of unirradiated and irradiated polycrystalline indium in a liquid indium–tin melt on the annealing time of indium at \(20^\circ\)C has the same character. The indium crystals were irradiated from a Ra–\(\gamma\)–Be source under the conditions described above. The experimental procedure is described in [5]. Figure 4 shows the dependence of the difference in dissolution times

Fig. 4. Dependence of the difference in dissolution times of unirradiated and irradiated indium crystals from a Ra–\(\gamma\)–Be source on storage time at room temperature: 1—previously quenched sample; 2—previously annealed sample.

Fig. 5. Dependence of the difference in melting times of unirradiated and irradiated indium crystals from a Ra–\(\gamma\)–Be source on storage time at room temperature for different annealing temperatures.

of unirradiated and irradiated indium, \(\Delta\tau = \tau_{\text{irr.}}-\tau_{0}\), on annealing time at room temperature for quenched (1) and annealed (2) indium crystals kept at room temperature for one month before irradiation. The temperature of the experiment was \(121.5^\circ\)C. The irradiation time was 40 hours. As can be seen from the graph, the difference in melting times of unirradiated and irradiated crystals, and consequently the number of induced defects under identical irradiation conditions, is significantly greater in quenched indium crystals than in annealed ones. The motion of defects in crystals and their blocking of dislocations over time [6] leads to strengthening of NaCl crystals (Fig. 3) and indium crystals (Fig. 4), although the mechanism of defect formation in these two cases is apparently different.

Figure 5 shows the dependence of \(\Delta\tau\) on annealing time at \(20^\circ\)C for different experimental temperatures for irradiated and unirradiated indium crystals. As can be seen from the figure, as the experimental temperature approaches the temperature of the onset of contact melting [7], the quantity \(\Delta\tau = \tau_{\text{irr.}}-\tau_{0}\) increases in absolute value with time, and the process taking place reaches equilibrium after longer intervals of time.

Thus, radiation defects in crystals lead to an increase in the dissolution rate of solids in liquids. This phenomenon—

…relation can be used to determine $\Delta \varphi$ in irradiated crystals. For a rock-salt crystal irradiated with X-rays ($6 \cdot 10^4$ roentgens), $\Delta \varphi \simeq 2$ kcal/mole.

References

1. L. Dienes, G. H. Vineyard, Radiation Effects in Solids, IL, Moscow, 1960.
2. I. L. Newgard, J. Appl. Phys., 30, No. 9, 1449–1451, 1959.
3. A. A. Chernov, Dokl. Akad. Nauk SSSR, 133, No. 6, 1960.
4. G. N. Voronin, Fundamentals of Thermodynamics and Heat Transfer. Oborongiz, p. 290, 1958.
5. I. G. Berzina, P. A. Savintsev, “The action of radiation on the contact melting of crystals” (article included in this volume).
6. F. Seitz, Rev. Mod. Phys., 26, 7, 1954.
7. I. G. Berzina, A. F. Naumov, P. A. Savintsev, Kristallografiya, 6, 466, 1961.