The Non-Stationary Case of Maxwell-Garnett: Growth of Nanomaterials (2D gold flakes) in Solutions
The solution-based growth mechanism is a common process for the nanomaterials. Maxwell Garnett (for light-matter interactions), describes the solution growth in an effective media, homogenized by a mean electromagnetic fields, which applies when the materials are in a stationary phase. However, charge transitions (inter and intra transitions) during the nano-materials growth, leads to non-stationary phase and associated with a time-dependent permittivity constant transitions (for the nano-materials). Therefore, the time-independence in the standard Maxwell Garnett is lost and we have a time-dependent ɛi(t). This is become important when we need to deconvolute the optical spectra of the solution at the different reaction times as each peak represents a specific charge/energy transfer with a specific permittivity constant. Based on that, we developed a time-resolved deconvolution approach f(t) α ɛi(t) which has led us to identify those transitions (inter and intra transitions) with their dominated growth regimes. Two gold ions peaks are precisely measured (322nm and 367nm) for inter transition, and three different polyaniline oxidation states (PAOS) for intra transitions: A (372 nm), B (680 nm), and C (530 nm). At an initial reaction time regime (0-90min), the permittivity constant of gold is found to be highly dependent on time i.e. fE α ɛi(t) as a charge transfer takes place from the PAOS to gold ions (i.e. inter transition leads to reduction reaction). In the second time regime (90-180min), the permittivity constant of gold changes as the material deform from 3D to 2D (fS α ɛ3D-2D) i.e. intra transition (combined with thermal reduction). Our approach provides a new framework for time-dependent modelling of (an)isotropic solutions of other nanomaterials and their syntheses.