The non-stationary case of the Maxwell-Garnett theory: growth of nanomaterials (2D gold flakes) in solution

The solution-based growth mechanism is a common process for nanomaterials. The Maxwell-Garnett theory (for light-matter interactions) describes the solution growth in an effective medium, homogenized by a mean electromagnetic field, which applies when materials are in a stationary phase. However, the charge transitions (inter- and intra-transitions) during the growth of nanomaterials lead to a non-stationary phase and are associated with time-dependent permittivity constant transitions (for nanomaterials). Therefore, time-independence in the standard Maxwell-Garnett theory is lost, resulting in time dependence, i ( t ). This becomes important when the optical spectrum of a solution needs to be deconvoluted at different reaction times since each peak represents a specific charge/energy transfer with a specific permittivity constant. Based on this, we developed a time-resolved deconvolution approach, f ( t ) ∝ i ( t ), which led us to identify the transitions (inter- and intra-transitions) with their dominated growth regimes. Two gold ion peaks were precisely measured (322 nm and 367 nm) for the inter-transition, and three different polyaniline oxidation states (PAOS) for the intra-transition, including A (372 nm), B (680 nm), and C (530 nm). In the initial reaction time regime (0-90 min), the permittivity constant of gold was found to be highly dependent on time, i.e. f E ∝ i ( t ), since charge transfer takes place from the PAOS to gold ions ( i.e. inter-transition leads to a reduction reaction). In the second time regime (90-180 min), the permittivity constant of gold changes as the material deforms from 3D to 2D ( f S ∝ 3D-2D ), i.e. intra-transition (combined with thermal reduction). Our approach provides a new framework for the time-dependent modelling of (an)isotropic solutions of other nanomaterials and their syntheses. We followed the growth of the gold flakes by the optical spectra. The optical spectra were deconvoluted based on non-stationary case of Maxwell-Garnett. Two regimes were resolved: 0-90 min which driven by the oxidation process and from 90-180 min which driven by thermal process.