PROCEEDINGS
OF THE TOMSK ORDER OF THE OCTOBER REVOLUTION
AND ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV

Volume 277　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1977

STUDY OF A ONE-LEVEL SCHEME OF PHOTOPROCESSES IN INORGANIC AZIDES BY THE METHOD OF MATHEMATICAL MODELING

G. B. AGUTOVA, B. I. BALYSHEV, Yu. V. GAVRISHCHENKO,
A. I. RUBAN

(Presented by the scientific seminar of the Department of Computer Engineering)

The study of photoprocesses in solids (photodecomposition, the internal photoelectric effect) makes it possible to establish the mechanism of chemical reactions proceeding under the action of light radiation. Mathematical modeling of the processes provides substantial assistance in establishing the reaction mechanism, significantly reducing the volume of experimental work.

Attempts had already been made earlier to study a one-level decomposition scheme [4]. However, the methods of trial and error and successive enumeration that were used made it possible only to interpret qualitatively the results obtained.

In the present work an analysis of a similar scheme was carried out by the simplex method [3]. The investigation was performed on an electronic linear modeling installation of the MPT-9-3 type.

The system of differential equations describing the kinetics of photoprocesses in solid inorganic azides has the following form:

\[ \tag{1} \left\{ \begin{aligned} \frac{d(\mathrm{N}_3^-)^{*}}{dt} &= a_3 I \mathrm{N}_3^- - a_5(\mathrm{N}_3^-)^{*} - a_4'(\mathrm{N}_3^-)^{*} + a_6 \mathrm{N}_3^0 e - a_9(\mathrm{N}_3^-)^{*},\\[4pt] \frac{d\mathrm{N}_3^0}{dt} &= a_4(\mathrm{N}_3^-)^{*} - a_6 \mathrm{N}_3^0 e + a_1 I \mathrm{N}_3^- - a_2 \mathrm{N}_3^0 e - a_7(\mathrm{N}_3^0)^2 - a_{10}\mathrm{N}_3^0,\\[4pt] \frac{de}{dt} &= a_4(\mathrm{N}_3^-)^{*} - a_6 \mathrm{N}_3^0 e + a_1 I \mathrm{N}_3^- - a_2 \mathrm{N}_3^0 e - a_8' T e,\\[4pt] \frac{d\mathrm{N}_2}{dt} &= a_7(\mathrm{N}_3^0)^2 + a_9(\mathrm{N}_3^-)^{*} + a_{10}\mathrm{N}_3^0 . \end{aligned} \right. \]

Initial conditions:

\[ (\mathrm{N}_3^-)^{*}\big|_{t=0}=0; \]

\[ \mathrm{N}_3^0\big|_{t=0}=0; \]

\[ e\big|_{t=0}=0; \]

\[ \mathrm{N}_2\big|_{t=0}=0. \]

Introduce the notation:

\[ IN_3^- = I; \qquad (N_3^-)^* = x_1; \]

\[ N_3^0 = x_2; \qquad e = x_3; \]

\[ N_2 = x_4; \qquad \alpha_8' T = \alpha_8; \]

then

\[ \tag{2} \left\{ \begin{aligned} \frac{dx_1}{dt} &= a_3 I - a_5 x_1 - a_4 x_1 + a_6 x_2 x_3 - a_9 x_1,\\ \frac{dx_2}{dt} &= a_4 x_1 - a_6 x_2 x_3 + a_1 I - a_2 x_2 x_3 - a_7 x_2^2 - a_{10} x_2,\\ \frac{dx_3}{dt} &= a_4 x_1 - a_6 x_2 x_3 + a_1 I - a_2 x_2 x_3 - a_8 x_3,\\ \frac{dx_4}{dt} &= a_7 x_2^2 + a_9 x_1 + a_{10} x_2. \end{aligned} \right. \]

Proceeding from the structural diagram (Fig. 1), the machine-simulating equation is written in the following form:

Fig. 1. Structural diagram.

\[ \tag{3} \left\{ \begin{aligned} \frac{dU_1}{d\tau} &= k_1U_0-k_5U_1-k_4U_1+k_6U_2U_3-k_9U_1,\\ \frac{dU_2}{d\tau} &= k_1U_1-k_6U_2U_3+k_1U_0-k_2U_2U_3-k_7U_2^2-k_{10}U_2,\\ \frac{dU_3}{d\tau} &= k_4U_1-k_6U_2U_3+k_1U_0-k_2U_2U_3-k_8U_3,\\ \frac{dU_4}{d\tau} &= k_7U_2^2+k_9U_1+k_{10}U_2; \end{aligned} \right. \]

where \(k_1 \div k_{10}\) are machine coefficients for the machine variables \(U_1 \div U_{10}\), which vary from 0 to 10; \(U_0\) varies from 0 to 100 V.

In order for equations (2, 3) to be identical, it is necessary to introduce scale coefficients:

\[ \tag{4} \left\{ \begin{aligned} m_1 &= \frac{x_1}{U_1}; \qquad m_2 = \frac{x_2}{U_2}; \qquad m_3 = \frac{x_3}{U_3};\\ m_t &= \frac{t}{\tau}; \qquad m_l = \frac{l}{U_0}; \qquad m_4 = \frac{x_4}{U_4}. \end{aligned} \right. \]

Let us replace the machine variables in the machine equation, using the scale coefficients; then system (3) will be written in the following form:

\[ \tag{5} \left\{ \begin{aligned} \frac{dx_1}{dt} &= \frac{m_1}{m_t m_l} k_3 l_0 -\frac{1}{m_t}k_9x_1 -\frac{1}{m_t}k_4x_1 -\frac{1}{m_t}k_5x_1 +\frac{0{,}01\,m_1}{m_2m_3m_t}k_6x_2x_3,\\ \frac{dx_2}{dt} &= \frac{m_2}{m_1m_t}k_1x_1 -\frac{1}{m_2m_t}0{,}01\,k_6x_2x_3 -\frac{m_3}{m_lm_t}k_1l_0\\ &\quad -\frac{1}{m_2m_t}0{,}01\,k_7x_2^2 -\frac{1}{m_t}k_{10}x_3,\\ \frac{dx_3}{dt} &= \frac{m_3}{m_1m_t}k_1x_1 -\frac{1}{m_2m_t}0{,}01\,k_6x_2x_3 -\frac{m_3}{m_lm_t}k_1l_0\\ &\quad -\frac{1}{m_2m_t}0{,}01\,k_2x_2x_3 -\frac{1}{m_t}k_8x_3,\\ \frac{dx_4}{dt} &= \frac{m_4}{m_2^2m_t}k_7\,0{,}01\,x_2^2 +\frac{m_4}{m_1m_t}k_9x_2 +\frac{m_4}{m_2m_t}k_{10}x_2. \end{aligned} \right. \]

Consequently, the criterion equations will be equal to:

\[ \tag{6} \left\{ \begin{aligned} \frac{m_2}{m_1m_t}k_1 &= \frac{m_3}{m_1m_t}k_1 = \frac{m_2}{m_1m_t}k_1 = a_1,\\ \frac{0{,}01}{m_2m_t}k_2 &= \frac{0{,}01}{m_3m_t}k_2 = a_2,\\ \frac{m_1}{m_lm_t}k_3 &= a_3,\\ \frac{m_3}{m_1m_t}k_4 &= \frac{1}{m_t}k_4 = a_4. \end{aligned} \right. \]

\[ \left. \begin{aligned} &\frac{1}{m_t}k_5=\alpha_5,\\ &\frac{m_1}{m_2m_3m_t}\,0.01\,k_6=\frac{1}{m_3m_t}\,0.01\,k_6=\frac{1}{m_2m_t}\,0.01\,k_6=\alpha_6,\\ &\frac{1}{m_2m_t}\,0.61\,k_7=\frac{m_4}{m_2^{\,2}m_t}\,0.01\,k_7=\alpha_7,\\ &\frac{1}{m_t}k_8=\alpha_8,\\ &\frac{1}{m_t}k_9=\frac{m_4}{m_1m_t}k_9=\alpha_9,\\ &\frac{1}{m_t}k_{10}=\frac{m_4}{m_2m_t}k_{10}=\alpha_{10}. \end{aligned} \right\} \tag{6} \]

From expression (6) it follows: \(m_1=m_2=m_3=m_4=m_t=m\).
Then we obtain

\[ \left. \begin{aligned} &\frac{k_1}{m_t}=\alpha_1,\\ &\frac{k_2\,0.01}{m\,m_t}=\alpha_2,\\ &\frac{k_3}{m_t}=\alpha_3,\\ &\frac{k_4}{m_t}=\alpha_4,\\ &\frac{k_5}{m_t}=\alpha_5,\\ &\frac{0.01\,k_6}{m\cdot m_t}=\alpha_6,\\ &\frac{0.01\,k_7}{m\,m_t}=\alpha_7,\\ &\frac{k_8}{m_t}=\alpha_8,\\ &\frac{k_9}{m_t}=\alpha_9,\\ &\frac{k_{10}}{m_t}=\alpha_{10}. \end{aligned} \right\} \tag{7} \]

The set of experimental curves on the nonlinearity blocks (NB) was carried out by the method of piecewise-linear approximation. The problem was first solved by trial and error in the periodic computation mode. This method consisted in successively bringing the solution curve of the equation on the machine closer to the experimental curves. In this process, as a quantitative standard characterizing the degree of similarity of the curves, the quality function \(\Phi\), obtained from the scheme shown in Fig. 2, was taken. As a result, the following value-

...of the coefficients and of the disturbing action for which the curves from the model come as close as possible to the experimental ones.

Fig. 2. Circuit implementing the quality function with respect to two parameters.

1. For the region of strong absorption

Table 1

2. For the region of weak absorption

Table 2

Further approximation of the curves was carried out by the simplex method. For this purpose the matrix of the initial simplex was constructed:

\[ \left[ \begin{array}{cccccc} k_1 & k_2 & \cdots & \cdots & k_{n-1} & k_n\\ -R_1 & k_2 & \cdots & \cdots & k_{n-1} & k_n\\ 0 & -R_2 & \cdots & \cdots & k_{n-1} & k_n\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & \cdots & k_{n-1} & k_n\\ 0 & 0 & \cdots & \cdots & R_{n-1} & k_n\\ 0 & 0 & \cdots & \cdots & 0 & -R_n \end{array} \right], \tag{8} \]

where

\[ R_i=\sqrt{\frac{i}{2(i+1)}};\qquad \]

\[ k_i=\sqrt{\frac{1}{2i(i+1)}};\qquad \]

\[ i=1,2,\ldots,n. \]

For the present problem the initial numerical matrix has the form (8*). The matrix of the required simplex of natural variables is found from the formulas:

\[ \left\{ \begin{aligned} x_1&=\frac{\alpha-l_1}{\Delta_1};\\ x_2&=\frac{\beta-l_2}{\Delta_2}; \end{aligned} \right. \tag{9} \]

where

\(\alpha, \beta\) are the factors;
\(x_1, x_2\) are the values from the initial numerical matrix;
\(l_1, l_2\) are the basic level of the factors;
\(\Delta_1, \Delta_2\) are the intervals of variation.

In our problem, the coefficients \(k_1 \div k_{10}\) and the perturbing action \(U_0\) serve as the basic level of the factors. The basic levels of the factors are given in Tables 1, 2, and the intervals of variation in Tables 3 and 4.

1. For the region of strong absorption

Table 3

2. For the region of weak absorption

Table 4

Analysis of the results obtained showed that the worst experiment in the series is experiment No. 8 \((\Phi=16\ \text{V})\). “Reflection” of the worst point (experiment) relative to the center of the opposite face of the simplex gives the conditions for carrying out a new experiment in place of the excluded one.

The coordinates of the center of the face were determined by the formula

\[ x_i^c=\frac{\sum_{k=1}^{n+1}x_i^k}{n};\quad k\ne j. \tag{10} \]

The conditions for carrying out the experiment at the reflected point:

\[ x_i^{n+2}=2x_i^c-x_i^j;\quad i=1,2,\ldots,n, \tag{11} \]

\(x_i^c\) is the \(i\)-th coordinate of the center of the opposite face;
\(x_i^j\) is the \(i\)-th coordinate of the worst point;
\(x_i^{n+2}\) is the \(i\)-th coordinate of the new point obtained as a result of reflection.

Table 5

As is seen from Table 5, experiment No. 8 is the worst. In accordance with the method described above, we shall replace point No. 8 by a new point No. 13. The coordinates of this point and the coordinates of the center of the face formed by the points numbered 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12 have the following values:

\[ k_1^c=\frac{0{,}654+0{,}6361+0{,}645\cdot9}{11}=0{,}6451, \]

\[ k_2^c=\frac{2\cdot8{,}51+5{,}5+7{,}5\cdot8}{11}=7{,}48, \]

\[ \ldots \]

and so on,

and the coordinates of the thirteenth point are expressed as:

\[ x_1^{(13)}=20{,}6451-0{,}645=0{,}5451, \]

\[ k_2^{(13)}=2\cdot7{,}5-7{,}5=7{,}5, \]

\[ \ldots \]

and so on.

After carrying out the experiment at point No. 13, the worst point of the simplex \((1, 2, 3, 4, 5, 6, 7, 13, 9, 10, 11, 12)\) turned out to be point No. 7. Its determination relative to the face \((1, 2, 3, 4, 5, 6, 7, 13, 9, 10, 11, 12)\) also gives the worst point No. 4 in the simplex. The simplex \((1, 2, 3, 5, 6, 9, 10, 11, 12, 13, 14, 15)\) corresponds to an almost stationary region. The coefficients of this simplex are given in Table 6.

Table 6

Using system (7) and the values of the scale coefficients, we find the values of the coefficients for system (2):

\[ \begin{gathered} \alpha_1=\frac{k_1}{m_t}=\frac{0,649}{1}=0,649,\\[4pt] \alpha_2=\frac{0,01\,k_2}{m\,m_t}=\frac{0,01\cdot 7,5}{1\cdot 0,125\cdot 10^9}=0,6\cdot 10^{-9},\\[4pt] \alpha_3=\frac{k_3}{m_t}=1,\\[4pt] \alpha_4=\frac{k_4}{m_t}=10,\\[4pt] \alpha_5=\frac{k_5}{m_t}=0,62,\\[4pt] \alpha_6=\frac{0,01\,k_6}{m\cdot m_t}=0,548\cdot 10^{-9},\\[4pt] \alpha_7=\frac{0,01\,k_7}{m\cdot m_t}=0,409\cdot 10^{-9},\\[4pt] \alpha_8=\frac{k_8}{m_t}=0,078,\\[4pt] \alpha_9=\frac{k_9}{m_t}=0,144,\\[4pt] \alpha_{10}=\frac{k_{10}}{m_t}=0,391. \end{gathered} \]

System (2) can be written as

\[ \left\{ \begin{aligned} \frac{dx_1}{dt}&=I-0,62\,x_1-10\,x_1+0,548\cdot 10^{-9}x_2x_3-0,144\,x_1,\\[4pt] \frac{dx_2}{dt}&=0,649\,x_1-0,548\cdot 10^{-9}x_2x_3+0,649\,I-0,6\cdot 10^{-9}x_2x_3-\\ &\quad -0,409\cdot 10^{-9}x_2^{\,2}-0,391\,x_2,\\[4pt] \frac{dx_3}{dt}&=10\,x_1-0,548\cdot 10^{-9}x_2x_3+0,649\,I-0,6\cdot 10^{-9}x_2x_3-\\ &\quad -0,078\,x_3,\\[4pt] \frac{dx_4}{dt}&=0,409\cdot 10^{-9}x_2^{\,2}+0,144\,x_1+0,391\,x_2. \end{aligned} \right. \]

The curves corresponding to experiment No. 15 for the weak-absorption region are presented in Figs. 3, 4; the numerical data are in Tables 6 and 7. Studies of the photoprocesses for the strong-absorption region were carried out analogously. For the strong-absorption region the following results were obtained: the graphs in Figs. 5, 6; the numerical data in Tables 8, 9; and the values of the coefficients and of the perturbing action corresponding to them in Table 10.

Fig. 3. Curve of the gas-evolution rate in the region of weak absorption.
— — — experimental curve, ——— curve from the model

Fig. 4. Concentration of holes \(N_3^0\) under illumination with light
(weak absorption). — — — experimental curve,
——— curve from the model.

Table 6a

Table 7

Table 8

Fig. 5. Curve of the gas-evolution rate in the region of strong absorption.
- - - - - experimental curve, ——— curve from the model.

Fig. 6. Curve of the concentration of holes \(N_3^0\) \((\times 10^{10})\) under irradiation by light (strong absorption). - - - - - experimental curve, ——— curve from the model.

Table 9

Table 10

In view of the fact that the measurement error of the experimental curves is \(10\%\), the results obtained are a good approximation.

References

1. L. Levin. Methods for Solving Technical Problems Using Analog Computers. “Mir,” 1966.

2. Theory of Analog and Combined Computing Machines. Methods of Mathematical Modeling. “Nauka,” 1969.

3. V. G. Gorskii, V. E. Brodskii. The simplex method for planning extremal experiments. “Zavodskaia laboratoriia,” 1965, No. 7.

4. Yu. V. Gavrishchenko, G. G. Savel’ev. Izv. TPI, 251, 255, (1970).