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\begin{center}
{\large \bf A MATHEMATICAL MODEL OF METAL FILMS DEPOSITION FROM PHOTOACTIVE COMPOUND SOLUTIONS ON SOLID-LIQUID INTERFACE}

\vspace{1cm}

{\large \underline{S.A. Kukushkin}, S.V. N\'emenat}

\vspace{1cm}

Institute of Problems of Mechanical Engineering, RAS, V.O. Bolshoj pr. 61, 199178, St-Petersburg, Russia

e-mail ksa@math.ipme.ru

\end{center}

\vspace{2cm}

\begin{abstract}
The kinetics of the growth of thin metal films from solutions of photoactive chemical coordination compounds irradiated by light at certain wavelength has been theoretically investigated. In the framework of our mathematical model the time dependencies of the basic characteristics of film growth process (size distribution function, density of nucleation, concentration of the metal on the surface) have been computed. They proved to be in accordance with the experimental data. It is shown that the flux of metal (substance) from the solution onto the surface of the substrate can be derived from the stationary diffusion equation with the added term taking into account the influence of an photochemical substance source. It is concluded that the variation of the light intensity, in large range, keeps the kinetics of the process of film deposition out of drastic changes
\end{abstract}

\newpage

\begin{center}
{\bf Introduction}
\end{center}

The processes of metal films formation and growth on solid substrates are of great importance from both theoretical and practical viewpoints \cit{�����}. In the papers \gcite{1}{4}, process of obtaining $\rm{Ag}$, $\rm{Au}$, $\rm{Cu}$ films on flat quartz substrates in solutions of photoactive chemical coordination compounds was experimentally investigated. The essence of this method is the following. Supposing that it is considered a solution of coordination compound $[M^{z+}L_n]X_m$, where $M^{z+}$ is the central metal atom ($z+$ is the oxidation level); $L$ denotes innersphere ligands; $X$ denotes outersphere counterions. If ligand to metal charge transfer ($L\rightarrow M$) occurs in this compound, then irradiation of the solution by light at wavelength $\lambda = \lambda _{ L\rightarrow M }$ results in the reduction of the metal atom. By carrying the irradiation on, the central metal atom can be reducted to  zerovalence state. Thus, we can write the scheme of the entire process as
$$
[M^{z+}L_n]X_m \stackrel{h\nu}{\longrightarrow} M^0 + L^+ + X^-
$$
One should put the substrate to be deposited on the bottom of the vessel filled ith the solution. If we direct a light beam from above, then according to the Lambert-Buger-Behr law its intensity decreases (Fig. 1).
%\newpage
%\epsffile[180, 260, 500, 490]{figure1.eps}

Zerovalence metal atoms generated by light will diffuse toward the substrate and be adsorbed on it. When a critical concenteration of metal on the substrate surface is reached, the growth of new crystal metal phase starts.

It was noted in \gcite{1}{4}, that the essential feature of films obtained in this way is the narrow size distribution of grains in the film. In those papers, on the basis of the Volmer-Weber theory \cit{���쬥�} the qualitative explanation of the results was proposed. The process of formation and growth of a new phase on the surface is the complicated multistage one \cit{���}, and the Volmer-Weber theory does not allow any exact quantitative description.

By our opinion, the method described above is very constructive and interesting for problems of films growing. In this paper, we are theoretically investigating the time dependencies of the basic characteristics of the growing film, on the basis of modern film growth ideas, presented in the publications \cit{�����, ���}.

\begin{center}
{\bf Flux of metal atoms onto the substrate}
\end{center}

In this paper, the initial stage of the film condensation on the solid substrate is investigated. The substrate is considered to be a homogeneous continuous surface. Thermal effects of crystallization and adsorption are neglected. Passing through the solution, light interacts with molecules of coordination compounds. That results in the formation of zerovalence metal atoms. The power of metal atoms source is given by
\begin{equation}
g(x) = I_0\varphi \alpha exp(-\alpha x),
\label{LBB}
\end{equation}
where $I_0$ is the incident light intensity; $\alpha $ is the absorbtion coefficient; $\varphi$ is the quantum yield. The absorption coefficient $\alpha $ can be represented by $\alpha = 2.3\varepsilon c_{complex}$, where $\varepsilon $ is the extinction; $c_{complex}$ is the concentration of absorbing centres in the solution. Here, we suppose that $c_{complex}$ is constant. If the height of solution $l$ (Fig. 1) is small, the probability of the formation of metal clusters in the bulk of solution (homogeneous nucleation) vanishes. In this case, the flux of metal onto the substrate can be derived from 1D stationary diffusion equation, written by
\begin{equation}
D\Delta c(x) + g(x) = 0,
\label{diffusion}
\end{equation}
where $\Delta $ denotes the Laplacian; $D$ is the diffusion coefficient; the power of source is given by the equation (\ref{LBB}). We make use of stationary diffusion equation for the following reasons:
\begin{enumerate}
\item the source of metal atoms is stationary
\item we suppose that the characteristic time of photochemical reaction is much less than that one of diffusion and,  i.e. the photochemical reaction is not the ratelimiting stage.
\end{enumerate}
The boundary conditions to the equation (\ref{diffusion}) are
\begin{equation}
\frac{dc}{dx}\Biggr| _{x=0} = 0
\label{bcond0}
\end{equation}
\begin{equation}
c(l) = c_s^{\ast}
\label{bcondl}
\end{equation}
It means that the flux of metal into the system at the upper boundary of solution ($x = 0$) misses and concentration of metal on the surface ($x = l$) is constant and equal to $c_s^{\ast}$.

In fact, some mean time-dependent concentration of adsorbed atoms on the substrate exceeding $c_s^{\ast}$ exists. It is the reduced difference of these concentrations referring to as supersaturation
\begin{equation}
\xi (t) = \frac{c_s(t)-c_s^{\ast}}{c_s^{\ast}},
\end{equation}
that is the driving force of phase transformation on surface. However, due to the growth is carried out at room temperature and the solubility of metal is very small value, the equilibrium concentration of metal $c_s^{\ast}$ is small also. This results in formation of surface metal islands at small supersaturations. Therefore, following \cit{�����, ���, ���}, we assume that application of the boundary condition (\ref{bcondl}) is enough satisfactory. By differentiation of solution of the equation (\ref{diffusion}) with the boundary conditions (\ref{bcond0}) and (\ref{bcondl}), we can derive the form of the flux of metal to the surface
\begin{equation}
J_s = I_0\varphi \biggl( 1 - exp(-\alpha l)\biggr).
\label{flux}
\end{equation}

\begin{center}
{\bf Processes of metal islands formation}
\end{center}

According to the paper \cit{���}, a process of new phase formation goes through three basic stages. The first stage is the fluctuation formation of new phase nuclei. It is described by the Zeldovich theory, modified for formation on a surface \cit{���}. During this stage, the stationary flux of nuclei, which are spherical segments, has the form \cit{���}:
\begin{equation}
I(\xi ) = K(\xi +1)ln(\xi +1)exp\Biggr(-\frac{b}{ln^2(\xi +1)}\Biggl),
\label{I}
\end{equation}
where $\xi $ is the supersaturation; $b = \Bigl(\sigma /(kT)\Bigr)^34\pi v^2$, $\sigma $ is the surface tension; $v$ is the volume per atom in nucleus; $K$ is  proportionality coefficient depending on duffusion coefficient of adatoms. The time lag is estimated by:
\begin{equation}
t_s = \frac{1}{-F''(n_{cr})W(n_{cr})},
\label{t_s}
\end{equation}
where $n_{cr}$ is the number of atoms in critical nucleus; $F(n)$ is the Helmholtz energy of nucleus taking form of a spherical segment; $W(n)$ is the diffusion coefficient in size space taking into account the probability of a random eleminating or joining atoms to nucleus (fluctuations of nucleus in size space) \cit{������}.
\begin{equation}
F(n_{cr}) = \frac{b}{ln(\xi +1)}
\label{F(n_{cr})}
\end{equation}

At $t > t_s$ the second stage starts. This is the process of formation of new phase islands in the system. This stage is described by the system of equations:
\begin{equation}
\frac{\partial f\Bigr(n;t\Bigl)}{\partial t} + \frac{\partial \Bigr(v(n;t)f(n;t)\Bigl)}{\partial n} = 0,
\label{Fokker-Planck}
\end{equation}
where $v(n;t) \equiv \frac{dn}{dt}$ is the linear rate of nucleus growth, written as
\begin{equation}
v(n;t) = a\xi (t)n^{-m},
\label{rate of growth}
\end{equation}
where $a$ and $m$ are coefficients depending on the mechanism of nuclei growth,
\begin{equation}
\frac{\xi _0}{\xi (t)}-1 = \int\limits_{0}^{t}J_sdt - k\int\limits_{0}^{t}\int\limits_{n_{cr}}^{\infty}f(n;t)ndndt,
\label{xi/xi}
\end{equation}
where $k$ is coefficient depending on the metal under investigation. The equation above represents the matter conservation. The equation (\ref{Fokker-Planck}) should be complemented by initial and boundary conditions \cit{���, english}
\begin{equation}
f(n;0) = 0,
\qquad
f(n_{cr};t) = \frac{I\Bigr( \xi (t)\Bigl) }{v(n;t)}\Biggr|_{n=n_{cr}}
\label{b.c. for Fokker-Planck}
\end{equation}

In the paper \cit{english}, the method of solution of this equations system was developed. The solution of the system allows us to derive the time dependencies of the basic characteristics of film growth process:
\begin{enumerate}
\item the supersaturation $\xi (t)$ is as
\begin{equation}
\xi (t) = \xi _0\frac{1}{1+\frac{1}{\Gamma }T^k(t)\phi _k\bigl(T(t)\bigl)}
\label{xi(t)}
\end{equation}
\item the frequency of nuclei formation $I(t)$ is as
\begin{equation}
I(t) = I(\xi _0)\frac{exp\Bigl(-T^k(t)\phi _k\bigl(T(t)\bigr)\Bigr)}{1+\frac{1}{\Gamma }T^k(t)\phi _k\bigl(T(t)\bigl)}
\label{I(t)}
\end{equation}
\item the size distribution function $F(N;t)$ is as
\begin{equation}
f(n;t) = \frac{1}{k+1}n^{-\frac{k}{k+1}}\frac{I(\xi _0)}{\alpha \xi _0}exp\Biggl(-\Bigl(T(t)-\frac{\rho (n)}{\alpha t_k\xi _0}\Bigr)^k\phi _k\Bigl(T(t)-\frac{\rho (n)}{\alpha t_k\xi _0}\Bigr)\Biggr),
\label{distribution}
\end{equation}
where the auxillary function is $\rho (n) = (n^{m+1}-n_{cr}^{m+1})/(m+1)$
\end{enumerate}
The formulae (\ref{xi(t)})-(\ref{distribution}) contain the parameter $\Gamma$ -- quantity, which is inverse with respect to expansion small parameter. This parameter has the very important property: $\Gamma \approx n_{cr}$. All other symbols in these formulae correspond to ones in the origin paper \cit{english}.

\begin{center}
{\bf Results of calculations and conclusions}
\end{center}

Assuming that the growing film is determined by the following constants: $I_0 = 10^{20} quantum \cdot m^{-1}\cdot s^{-1}$, $\varphi = 0.8$, $\alpha = 50 m^{-1}$, $l = 2\cdot 10^{-4} m$ \gcite{1}{4}, $D = 10^{-9} m^2\cdot s^{-1}$, $v = 10^{-30} m^3$, $c^{\ast}_s = 10^{20} atoms\cdot m^{-2}$, $\sigma = 1 J\cdot m^{-2}$, $k = 1$, $a = 10^{14}$, $K = 10^{32}$, $\tau _r = 10^{-2} s$ \cit{�ᨯ��}, $S_{surf} = 10^{-4} m^2$, $T = 298 K$.

Thus the value of the steady flux (equation (\ref{flux})) is $2\cdot 10^{18} atoms\cdot m^{-2}\cdot s^{-1}$. $n_{cr} \approx \Gamma = 53$. That criterium proves that the solution obtained by small parameter method describes sufficiently the film growth the kinetics. The time lag value is $t_s = 1.8\cdot 10^{-6} s$.

The results of calculations are presented in Figs. 2-5.

%\newpage
%\epsffile[170, 230, 500, 490]{figure2.eps}
%\epsffile[140, 230, 500, 590]{figure3.eps}
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%\epsffile[200, 230, 500, 580]{figure5.eps}
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It is seen from Fig. 4, that size distribution takes the form of a bell. It is the form that is predicted by the theory \cit{�����, ���}.

We have calculated the time dependencies of the size distribution function, the frequency of nucleation, and the supersaturation on the surface at three different values of the incident light intensity being $10^{20}$, $10^{19}$ and $10^{18} quantum \cdot m^{-2}\cdot s^{-1}$. The variation of light intensity, in large range, keeps the kinetics of the process of film deposition out of drastic changes.

\begin{center}
{\bf Acknowledgments}
\end{center}

This work was supported by Russian Found for Basic Research (grant $\rm N^{\underline{\circ}}$ 99-03-32768). We would like to thank Prof. I.P. Kalinkin for helpful discussions.

\newpage

\begin{thebibliography}{99}

\bibitem{�����}
S.A. Kukushkin, V.V. Slyozov, Disperse Systems on the Surface of Solids (Evolution Approach): Mechanisms of Thin Films Formation (in russian) (Nauka, St.-Petersburg, 1996)

\bibitem{1}
T.B. Boitsova, A.V. Loguinov, V.V. Gorbunova // Zhurnal Prikladnoi Khimii 70 (1997) 1585

\bibitem{2}
A.V. Loguinov, L.V. Alexeeva, V.V. Gorbunova, G.A. Shaguisultanova, T.B. Boitsova // Zhurnal Prikladnoi Khimii 67 (1994) 803

\bibitem{3}
T.B. Boitsova, V.V. Gorbunova, A.V. Loguinov // Zhurnal Obshchei Khimii 67 (1997) 1741

\bibitem{4}
A.V. Loginov, T.B. Boitsova, V.V. Gorbunova. // Macromolecular Symposia 136 (1998) 103 

\bibitem{���쬥�}
M. Volmer, Kinetik der Phasenbildung (Steinkopf, Dresden, 1939)

\bibitem{���}
S.A. Kukushkin, A.V. Osipov // Physics-Uspekhi 41 (1998) 983

\bibitem{���}
S.A. Kukushkin, A.V. Osipov // Technical Physics 40 (1995) 615

\bibitem{english}
S.A. Kukushkin, A.V. Osipov // Journal of Chemical Physics 107 (1997) 3247

\bibitem{������}
V.V. Slyozov, S.A. Kukushkin // Physics of the Solid State 38 (1996) 239

\bibitem{�ᨯ��}
A.V. Osipov // Physics of the Solid State 36 (1994) 664

\end{thebibliography}

\newpage

\begin{center}
{\bf Figure Captions}
\end{center}

Listing of the figure captions to the paper of S.A. Kukushkin
and S.V. N\'emenat "A MATHEMATICAL MODEL OF METAL FILMS DEPOSITION FROM PHOTOACTIVE COMPOUND SOLUTIONS ON SOLID-LIQUID INTERFACE":

\vspace*{0.6cm}

Figure 1. The scheme of the process of photochemical film deposition; $l$ - thickness of the solution layer, $I_0$ - the incident light intensity

\vspace*{0.6cm}

Figure 2. The time dependence of supersaturation on the surface at different values of incident light intensity: curve 1 - $10^{20} quantum \cdot m^{-2} \cdot s^{-1}$; curve 2 - $10^{19} quantum \cdot m^{-2} \cdot s^{-1}$; $10^{18} quantum \cdot m^{-2} \cdot s^{-1}$; $t$ - time ($s$), $\xi (t)$ - supersaturation (non-dimensional quantity)

\vspace*{0.6cm}

Figure 3. The time dependence of nucleation frequency on the surface at different values of incident light intensity: curve 1 - $10^{20} quantum \cdot m^{-2} \cdot s^{-1}$; curve 2 - $10^{19} quantum \cdot m^{-2} \cdot s^{-1}$; $10^{18} quantum \cdot m^{-2} \cdot s^{-1}$; t - time ($s$), $I(t)$ - nucleation frequency ($s^{-1}$)

\vspace*{0.6cm}

Figure 4. The size distribution function at incident light intensity $I_0 = 10^{20} quantum \cdot m^{-2} \cdot s^{-1}$ and irradiation time: curve 1 - 5 min; curve 2 - 10 min; curve 3 - 40 min; $n$ - the number of nuclei (non-dimensional quantity), $f(n)$ - the size distribution function (non-dimensional quantity)

\vspace*{0.6cm}

Figure 5. The size distribution function at irradiation time 40 min ans incident light intensity: curve 1 - $10^{20} quantum \cdot m^{-2} \cdot s^{-1}$; curve 2 - $10^{19} quantum \cdot m^{-2} \cdot s^{-1}$; $10^{18} quantum \cdot m^{-2} \cdot s^{-1}$; $n$ - the number of nuclei (non-dimensional quantity), $f(n)$ - the size distribution function (non-dimensional quantity)



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