On the Help of Mathematical-Analytical Modeling for Present and Future Advances in Current Engineering

1. Introduction

In last years the ability to handle matter, combined with advances in structures synthesis and assembly at micro- and nano-scale, resulted in considerable goals in scientific and technological areas. In order to capitalize on nano-scale phenomena in devices, a better comprehension of the electronic, magnetic and photonic interactions at this size scale is required, across experiments, phenomenology, theory, and modeling [1,2].

The use of innovative methodologies, based on advanced bottom-up and top-down approaches, is allowing to design and synthesize new organic, organic/inorganic hybrid, and inorganic materials, aimed at a controlled assembly in active (micro-)nano-structured devices with applications in all technological sectors of benefit for human being [3].

Through the study and development of new technologies and methodologies, the functionalities of innovative devices can be explored for an ever-increasing range of applications. All fundamental studies are making possible the exploration of new techniques and methodologies in the most diverse engineering fields.

Present and future challenges concern in particular:

a) Civil engineering: solving complex problems typical of the construction of large infrastructures, in relation to the principles of structural modeling and the most advanced digital tools to support the engineer [4–6];

b) Energy engineering: the energy transition from fossil to synthetic fuels, renewable energy technologies, micro- and macro-conversion (energy harvesting and controlled nuclear fusion); these are scenarios that humanity is called to face in the transition towards a sustainable use of energy resources [7–9];

c) Electronic and biomedical engineering: light technology for bio-medicine and human health, nano-devices between electronics and nano-technologies, the new frontiers of digital twin, “learning by doing”, also through advanced software tools used for modeling and simulation, in parallel with the development of e-noses [10–22];

d) Computer engineering: questions of computer security, advances in programming for mobile devices, visual programming of applications for IoT. We also think about programming of applications for mobile devices, with particular attention to the aspects of creating the user interface [23–25];

e) The soil ecosystem: a network of relationships in the fabric of life. Soil is a very complex living system, extraordinarily rich in biodiversity, in which a wide range of biochemical and biophysical processes occur, including decomposition, a key process for the functioning of ecological systems, that transforms organic fractions into real nutrients for the living community. This leads to the need to monitor the health of the arboreal component of the ecological network, through the collection of parameters related to the physiology of plants, environmental conditions and the structural stability of plant elements inserted in the urban fabric [26–28].

2. Discussion

These challenges have a common denominator, i.e. nano-bio-technologies, which today allow the creation of miniaturized devices on a micro- and nanometric scale that, integrated with electronics, become intelligent nano-devices with multiple capabilities and surprising functionalities.

One of the most important aspects at the nano-scale affects the charge transport, which can be guided by particles dimensions and takes different features with respect to those of bulk. From a theoretical point of view, several techniques can be used for the knowledge of transport phenomena, both numerical and analytical ones, based on transport equations. The numerical approaches are important, but not able to provide an overall vision, instead typical of analytical approaches; these latter are of high mathematical interest and lead to newsworthy results and predictions, implemented via experimental literature data and obtained at any time by experiments [29].

The most used formulations are as follows:

A) The “Tight-Binding” Method (T-BM), semi-empirical method mainly preferred when the computational calculation is considerable [30];

B) The “ab-initio” formulations, that produce accurate results, although dependent by the initial choices;

B₁) The “Density Functional Theory” (DFT), that gives the total energy of the system through the functional of the electrons total density [31];

B₂) Local Density Functional Theory (LDFT).

Other approaches include: 

C) Non-Local Functional Approach (N-LFA);

D) Car-Parrinello Molecular Dynamic Method (C-PMDM) [32];

E) Conjugate-Gradient Method (C-GM) [33];

F) Augmented Plane Wave Method (APWM) [34];

G) Korringa-Kohn-Rostoker Method (K-K-RM) [35];

H) Linearized-Muffin-Thin-Orbital Method (L-M-T-OM) [36];

I) Full  Potential  Linearized Augmented Plane Wave Method (FPLAPWM) [37];

J) “Drude-Lorentz-type” models: they are basically improvements of Drude model [38]. The “Smith model” contains a parameter c_n accounting for the scattering anisotropy upon the first scattering event [39]. In the “Effective Medium Theories” (EMTs) the electromagnetic interactions among pure materials and host matrixes are roughly taken into account. Among EMTs, we remember in particular the “Maxwell-Garnett model” (MG) and the “Bruggeman model” (BR) [40,41].

About new generalizations, the “DS model” fits significantly well with experimental data and provides also new predictions of several peculiarities at nano-level. It is structured on the analytical form of three most important functions describing the charge transport, namely the velocities correlation function  ${〈\overrightarrow{v}(t)\cdot \overrightarrow{v}(0)〉}_T$, the mean square deviation of position $R^2(t)$ and the diffusion coefficient $D(t)$. This approach embraces classical and quantum-relativistic aspects [42–45]; it studies the dynamics of transport from sub-pico- to macro-level, thanks to a gauge factor

It is based on the total Fourier transform of the frequency-dependent complex-valued far-infrared photo-conductivity $\sigma (\omega )$, as deduced by the linear response theory, considering the Cauchy integration on the entire time axis (-∞, +∞), using the residue theorem in the plane of complex numbers.

3. Future Perspectives

\Tuning on every physical-chemical, systemic and model-intrinsic parameter, namely the temperature T, values of relaxation times ${\tau }_i$ and frequencies ${\omega }_i$, the carrier density N, the effective mass $m*$, the chiral vector, the quantum weights of modes, the speed of carriers, we can conduct a fine tuning of ${〈\overrightarrow{v}(t)\cdot \overrightarrow{v}(0)〉}_T$, $R^2(t)$ and $D(t)$, in order to gauge the work of nano-bio-devices, with “a-priori” and “a-posteriori” use of the model [46].

Drude-Lorentz-type models are very promising tools from every point of view, because they involve (“DS model” in particular) quantum and relativistic effects too. The totally analytical form of DS model looks very elegant; it is giving notable confirmations, and new knowledge on the dynamics at the nano-scale, properly tested with experimental time-resolved techniques, as TRTS, Photon-Induced Near-Field Electron Microscopy and Graphene based Plasmonics [47–60].