Relaxation of Physicochemical Processes during the Chemical Synthesis of Silver Nanoparticles in Reverse Micellar Solutions

Abstract

The kinetics of optical properties is studied experimentally on an OT Aerosol (dioctyl sulfosuccinate sodium salt)/isooctane/Qr (quercetin)/Ag⁺) reverse micellar system (RMS) during the chemical synthesis of Ag nanoparticles. The kinetic light optical absorption data acquired for an RMS at a wavelength λ = 432 nm with a chronometric resolution of 0.14 and 1 s are used to calculate the boundaries of the stages of Ag NP formation (at a confidence factor of at least 0.99). The determination of relaxation times (almost-periods) in an RMS allows evaluation of the kinetics parameters at certain stages of the chemical synthesis of NP metals.

Appendices

APPENDIX 1

The general kinetics of a process is usually represented by nonlinear fluctuations with a trend. Analysis of periodic components associated with deviation from the trend requires the transformation of measured data according to the algorithm of trend exclusion based on the theory of proportions [5]. Thus derived fluctuations with near-zero mean values, are then used for determining the periodical components closest to a period.

The following definition of the almost-periodic function is being suggested: number τ is regarded as ε-almost-period (ε-shift) of a the function f(t) (−∞ < t < ∞), if for all t the following inequality is valid:

$$\left| {f(t + \tau ) - f(t)} \right| < \varepsilon .$$

If f(t) is a periodic function and τ is the period of this function (i.e., f(t + τ) = f(t)), then τ is also an almost-period for any ε > 0, just as for any number nτ (n = ±1, ±2, …).

For a discrete case where n is the total number of readings for function $f({{t}_{i}})$ determined by experimental values, we have the following metrics to determine almost-periods:

$$a({{\tau }_{k}}) = \frac{1}{{n - k}}\sum\limits_{t = 1}^{n - k} {\left| {f({{t}_{i}} + {{\tau }_{k}}) - f({{t}_{i}})} \right|} .$$

This function, referred to as Alter-Johnson shear function [18], is applied to discrete time row within duration interval T (kinetic data). The discretization interval is $T{\text{/}}(n - 1)$ (where n is the total number of function readings) and determines the temporal accuracy of physico-chemical process kinetics’ measurements. Here ${{\tau }_{k}} = kT{\text{/(}}n - 1{\text{)}}$ is the test period and k is being determined in accordance with a number of assumptions [5]. The system of almost-periods τ of function f(t) can be determined as the set of the local minima of the shear function.

APPENDIX 2

Figure 8 shows a theoretical curve plotted using the Gompertz model to describe (a) the processes of limited rise and (b) its rate of change. The coordinates of the singular points for this curve are (the relevant moments in time are given in parentheses) (1) $y{\text{/}}{{y}_{\infty }} = {{e}^{{ - {{\Phi }^{2}}}}}({{t}_{1}})$, the inflection point in the curve of kinetic parameter ($D{\text{*}} = y$) to the left of the point of maximum velocity; (2) $y{\text{/}}{{y}_{\infty }} = {{e}^{{ - 1}}}({{t}_{*}})$, the inflection point in the integrated curve or the point of maximum velocity; (3) $\frac{y}{{{{y}_{\infty }}}} = {{e}^{{ - 1{\text{/}}{{\Phi }^{2}}}}}({{t}_{2}})$, the inflection point in the curve of the rate of change in the intensity of absorption to the right of the point of the maximum. Here, $e \approx \;2.718$ is Neper’s number and $\Phi \; \approx \;1.618$ is Fidium’s number (the golden proportion).

Fig. 8.

(a) Curve within the Gompertz model and (b) relevant velocity curve.

Note that the characteristic time corresponds to the time intervals between the singular points of the Gompertz model: ${{t}_{*}} - {{t}_{1}} = {{t}_{2}} - {{t}_{*}} = 1{\text{/}}k$, where k is the model parameter (see Fig. 4).

APPENDIX 3

As is well known [11], the time dynamics of the solution of a wave equation is represented by harmonic fluctuations of constant amplitude. The diffusion equation in turn determines the exponential change in the time component of its solution. Their combination is described by the equation

$$\frac{{{{\partial }^{2}}U}}{{\partial {{t}^{2}}}} + u\frac{{\partial U}}{{\partial t}} = ({{a}^{2}} + {{b}^{2}})\frac{{{{\partial }^{2}}U}}{{\partial {{x}^{2}}}}.$$

Seeking a solution in the form $U = T(t)X(x)$ results in $T(t) = A{{e}^{{ - \frac{u}{2}t}}}\sin (\xi t + \varphi )$. Here, u is the parameter characterizing the contribution from the diffusion mechanism to the parameters of the chemical synthesis of Ag nanoparticles in an RMS. In our case, u > 0 in the range of t₁ to ${{t}_{*}}$, u < 0 from ${{t}_{*}}$ to around 400 s, and u = 0 when t > 400 s (see Fig. 1b).

Figure 9 shows modulus χ of the amplitude of fluctuation versus the trend (Fig. 1b), for which $u\; = \frac{2}{{0.0131}}\; \approx \;153\;$ s.

Fig. 9.

Logarithm of module χ of fluctuations relative to zero, according to the deviation from the trend (see Fig. 1b).