Mathematical Modeling of Photolysis Reactions of Heavy Metal Azides

Yu. V. Gavrishenko, G. G. Savel’ev

(Presented by the scientific seminar of the Department of Radiation Chemistry)

The results of studies of the kinetics of photolysis and photoelectric phenomena in the azides of lead, silver, and thallium make it possible to suppose that electronic stages play the determining role in decomposition \([1–4]\).

However, when interpreting the results of investigation of the initial stages of decomposition, significant difficulties are encountered, since the system of formal-kinetic differential equations describing the decomposition process is usually nonlinear. The method of stationary concentrations is not applicable in this case, since the nonstationary portion of the kinetic decomposition curve is of greatest interest.

Such a system can be modeled on an analog computing machine; moreover, the possibility of varying analogs of the constants of intermediate reactions makes it possible to alter the contribution of one or another proposed elementary stage to the photolysis process. Comparison of the character of changes in the concentrations of intermediate and final products, calculated on the analog machine for different contributions of the proposed stages of decomposition, with the character of changes in the corresponding concentrations obtained experimentally makes it possible to judge the mechanism of decomposition.

From experimental studies \([5]\) it is known that the rate of photochemical decomposition of heavy metal azides is proportional to the intensity of the incident light,

and the photocurrent is proportional to the square root of the intensity. Moreover, these dependences were observed in the steady-state regime of the processes.

The scheme of the photolysis reaction of heavy-metal azides, taking into account all known stages of decomposition, has the following form. The primary stage of decomposition is either the formation of an electron–hole pair or the formation of an exciton, which may subsequently dissociate or annihilate with emission of a light quantum:

\[ N_3^- + h\nu_1 \xrightleftharpoons[k_2]{k_1} N_3^0 + e \tag{1} \]

\[ N_3^- + h\nu_2 \xrightleftharpoons[k_5]{k_3} (N_3^-)^{*} \tag{2} \]

\[ (N_3^-)^{*} + q \xrightleftharpoons[k_6]{k_4} N_3^0 + e \tag{3} \]

where \(q\) is thermal energy.

However, in addition to the capture of electrons and holes by ordinary recombination with formation of an anion (1) or an exciton (3), an electron may also be captured by various types of traps, which, in particular, may be interstitial metal ions:

\[ e + T \xrightarrow{k_8} Te . \tag{4} \]

Here \(T\) is the concentration of electron traps,
\(Te\) is the concentration of traps that have captured electrons.

The process of nitrogen formation may proceed by various paths:

\[ N_3^0 + N_3^0 \xrightarrow{k_2} 3N_2 + 2\ \text{vacancies} \tag{5} \]

\[ N_3^0 + N_3^- \xrightarrow{k_{10}} 3N_2 + Te + 2\ \text{vacancies} \tag{6} \]

\[ (N_3^-)^{*} + N_3^- \xrightarrow{k_7} 3N_2 + 2Te + 2\ \text{vacancies} \tag{7} \]

Processes of this type are described by the following system of differential equations:

\[ \frac{d(N_3^-)^{*}}{dt} = k_3 JN_3^- - k_5 (N_3^-)^{*} - k_4 (N_3^-)^{*} + k_6 N_3^0 e - k_9 (N_3^-)^{*} \]

\[ \frac{dN_3^0}{dt} = k_4 (N_3^-)^{*} - k_6 N_3^0 e + k_1 JN_3^- - k_2 N_3^0 e - k_7 (N_3^0)^2 - k_{10}N_3^0 \]

\[ \frac{de}{dt} = k_4 (N_3^-)^{*} - k_6 N_3^0 e + k_1 JN_3^- - k_2 N_3^0 e - k_8 e^T \]

\[ \frac{dN_2}{dt} = k_7 (N_3^0)^2 + k_9 (N_3^-)^{*} + k_{10}N_3^0 \tag{8} \]

If one takes into account that at room temperature, in the absence of irradiation, decomposition practically does not occur, and that the conductivity of the samples under irradiation by light changes by approximately three orders of magnitude [4], then the dark concentration of charge carriers may be neglected; i.e., as the initial conditions for solving the system of equations (8) we may use the following:

\[ \left. \begin{aligned} (N_3^-)^{*}\big|_{t=0} &= 0,\\ N_3^0\big|_{t=0} &= 0,\\ e\big|_{t=0} &= 0,\\ N_2\big|_{t=0} &= 0. \end{aligned} \right\} \tag{9} \]

Introduce the notation: \(JN_3^- = I\); \((N_3^-)^{*} = X_1\); \(N_3^0 = X_2\); \(e = X_3\); \(N_2 = X_4\); \(k_8T = k_8\).

Then the system of equations (8) will have the form:

\[ \left. \begin{aligned} \frac{dX_1}{dt} &= k_3 I - k_5X_1 - k_4X_1 + k_6X_2X_3 - k_9X_1, \\[4pt] \frac{dX_2}{dt} &= k_4X_1 - k_6X_2X_3 + k_1I - k_2X_2X_3 - k_7X_2^2 - k_{10}X_2, \\[4pt] \frac{dX_3}{dt} &= k_4X_1 - k_6X_2X_3 + k_1I - k_2X_2X_3 - k_8X_3. \end{aligned} \right\} \tag{10} \]

\[ \frac{dX_4}{dt} = k_7X_2^2 + k_9X_1 + k_{10}X_2 \]

It is known from experimental data that the establishment of stationarity of the process lasts several seconds. For this reason, in solving the problem it is convenient to use a real time scale, i.e., \(t_{\text{real}} = t\) machine. The structural scheme of the set of solving elements for solving the system of differential equations (10) is shown in Fig. 1. Since all equations of the system are of first order, the set of systems was implemented in the same form. The variables \(X_1, X_2, X_3, X_4\) in the computer are represented by voltages and vary from 0 to 100 volts. In analog computers the variables vary from \(-100\) to \(+100\) volts; however, the physical analog of our variables is either the concentration of excited states and charge carriers, or the amount of reaction products evolved; for this reason we consider positive values of the variables.

The studies were carried out as follows. Since, in machine form, the contribution of one or another term of the equation is determined only by a coefficient, we modeled the process taking into account the stages needed for our investigation by changing the coefficients. Then, by varying the coefficient determining the intensity of the incident light, we studied the dependence of the concentrations of electrons and holes, as well as the rate of gas evolution, on the irradiation intensity. Owing to the enormous number of combinations of coefficients, we performed calculations only for three extreme values of each coefficient: 0, 10, and 30. The dependence on intensity was determined at four points: 2.5, 5, 10, and 20; such changes in intensity values are close to the real changes produced in the experiment. Typical curves of the variation with time of the real and computed values of the variables \(X_1 - X_4\) are shown in Figs. 2 and 3. By varying the coefficients, we attempted to obtain kinetic curves similar to the kinetic curves of photoconductivity and of the rate of photolysis of lead, silver, and thallium azides; however, as the studies showed, we were unable to obtain an initial peak-like maximum on the kinetic curves \(N_3^\circ\) and \(\dfrac{dN_2}{dt}\).

Fig. 1. Block diagram of a set of computing elements for analyzing a system of differential equations.

Fig. 2. Form of the calculated kinetic curves when the process proceeds through the stage of direct ionization. Dashed lines indicate the form of the experimental curves for lead azide.

Fig. 3. Form of the calculated kinetic curves when the process proceeds through the stages of exciton formation and direct ionization. Dashed lines indicate the form of the experimental curves for silver and thallium azides.

Fig. 4. Dependences of the variable \(x_2\) (an analogue of the concentration of holes \(N_3'\)) on \(J\) (an analogue of the intensity of the incident light) for different sets of coefficients. (The number of the curve corresponds to the number of the set of coefficients).

Fig. 5. Dependence of the gas-evolution rate \(\frac{dN_2}{dt}\) on the intensity of the incident light.

analogues of photoconductivity and photolysis rate for \(PbN_6\). On this basis we made the assumption that equations taking into account ordinary recombination processes are inapplicable for describing the initial sections of kinetic curves of this type.

The study was carried out as follows. Since the photocarriers in lead, silver, and thallium azides are holes [1], from the set of kinetic curves obtained by varying the coefficients we selected only those for which the hole concentration was proportional to the square root of the intensity of the incident light (Fig. 4), i.e., curves corresponding to the experimental results. We obtained 12 such curves.

Then, from the set of curves obtained in this way, we selected those for which the gas-evolution rate was proportional to the intensity of the incident light. There proved to be only 4 such curves; these are coefficient sets I; 4; II; 12 (Fig. 5).

The listed sets of coefficients correspond to cases in which the process of formation of gaseous nitrogen proceeds mainly through the stage

\[ 2N_3^0 \longrightarrow 3N_2 \]

We considered a case in which the change in the concentration of electron traps with time is not taken into account.

The absence of a peak-shaped maximum in the calculated kinetic curves indicates that this scheme is inapplicable for describing the initial stages of photolysis of lead azide.

REFERENCES

1. G. G. Savel’ev, O. V. Gavrishenko, O. A. Zakharov, Izv. VUZov, “Fizika”, No. 7, 71 (1968).

2. O. V. Gavrishenko, G. G. Savel’ev, O. A. Zakharov, Izv. TPI, Izd. Tomskogo universiteta, 199, 112 (1969).

3. G. G. Savel’ev, O. V. Gavrishenko, V. I. Mertvinskii, S. I. Rukoleev, ibid. 199, 116 (1969).

4. Z. J. Dedman, J. C. Lewis, Trans. Farad. Soc. 62, 881 (1966).

5. O. V. Gavrishenko, Candidate’s dissertation, Tomsk, 1969.