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, ϵ_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) ∝ ϵ_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 ∝ ϵ_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 ∝ ϵ_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.