#### 2.1. Spectral-IQ Method

The wave-particle issue was in the “heart” of quantum mechanics, even in its very principles, Heisenberg one in particular; see [

28] and references therein. Currently assumed as a complementarity reality, it was just recently quantified with the aim of the path integrals’ quantum fluctuation factor (

n) through considering the quantum averages for the Gaussian wave packet to the harmonic one for the particle and wave representations, respectively. The results were finite and apart of consistently explaining the atomic (and thus the matter) stability through particle-wave equivalency at the quantum level; they permit also a general formulation of the particle-to-wave ratio content for an observed event [

28]:

as well as for the free quantum evolution [

28]:

One notes, for instance, that when quantum fluctuations asymptotically increase, the wave contents become infinite and cancel the particle observability, according to the

Equation 1.

Instead, even when a system hypothetically experiences zero quantum fluctuations, the wave nature of the system will be still slightly dominant over its particle side at both observed and free evolutions; see the upper branches of

Equations 1 and

2. These extremes show that the wave nature of matter will never be fully transferred to particle contents and the mesosystems will never be fully characterized by pure particle (or mechanical) features.

However, it is apparent from

Equation 2 that free evolution of a stable system is merely associated with particle dominance, however, without being manifestly observable; in fact, such peculiar particle behavior of the free evolutions of stable matter confirms its inner quantum nature by quantifiable features.

Figure 2 depicts the main tendencies of the particle-to-wave ratios of a stable system in terms of its quantum fluctuation, in free or observed conditions, alongside the present inverse quantum (IQ) index introduced as their competition.

Indeed, the inverse quantum index (

3) showcases the manifestly inverse behavior respecting free quantum evolution while accompanying the observed evolution for the respective quantum fluctuations’ range; therefore, it may constitute a suitable index for accounting the particle information degree in a general quantum evolution, from a free-to-observed one. Moreover, if one considers also the residual inverse quantum information

RQ = 1 −

IQ, one also gets a symmetrical tool with respect to

IQ for treating the free evolution at the quantum level.

Being the quantum fluctuation factor crucial for assessing the free and observed quantum behavior, it should be noted it may discriminate between these two quantum sides of motion, however, based solely on experimental measures of classical and quantum paths, since one considers their squared averages 〈x_{0}^{2}〉_{Exp} and 〈x^{2}〉_{Exp}, respectively, as:

and

It is obvious that for a given experimental set-up and records that the resulting observed evolution associates with higher quantum fluctuation than the corresponding free evolution, this feature being consistent with the (extended) Heisenberg uncertainty principle [

28].

However, when applied to spectroscopic data, they involve three classes of spectra information in terms of wave-numbers, namely:

The maximum absorption line wave-number υ̃_{0} (A_{max}) that relates to the classical path, and the same for squared average measure in the inverse manner as:

The left and right wave-numbers υ̃_{L}, υ̃_{R} of the working absorption band, being arithmetically-to-geometrically averaged to get the average of quantum paths “inside” the band:

The full width of a half maximum (FWHM) wave-number Δ υ̃_{FWHM} of the concerned absorption band that is reciprocally associated with the dispersion of the quantum paths of vibrations within the band:

Now, taken together, the quantum averaged path (

8) and its dispersion (

9) provide the average of the squared quantum paths, according to the general definition [

29]:

Altogether, the classical and quantum paths’ information of

Equations 6–

10 inversely correlate to the specific spectroscopic wave-numbers for a given absorption band and correlate the quantum fluctuations’ indices of

Equations 4 and

5 with the actual spectral-inverse quantum ones, respectively:

and

They will be eventually used to compute the observed, free, inverse and residual inverse quantum indices to in depth characterizing of a given material for its porosity-to-free binding ordering through recognizing the particle vs. wave quantum tendency of the investigated state by spectroscopy in general and by absorption spectra in the present approach. Specific examples and analyses follow.

#### 2.2. Results on Silica Sol-gel-based Mesosystems

Measurement of FT-IR absorption for samples under thermal treatment [

30–

32], e.g., same ionic liquid chain length, Cetyltrimethylammonium bromide (CTAB), respectively with DTAB or with their combination CTAB+DTAB, in different basic environment, are summarized in the

Table 2, and are reported in

Figures 3 and

4 for analysis at 60 °C and 700 °C, respectively (refer also to the Experimental Section).

The numerical Spectral-IQ results, as abstracted from

Figures 3 and

4, are presented in

Tables 3 and

4, for the particle-to-wave (P/W) ratio values in observed and free evolutions,

Equations 1 and

2, as based on the quantum fluctuation factors of

Equations 11 and

12, along the inverse quantum ratio of

Equation 3, respectively. Accordingly, one clearly observes the almost particle-to-wave equivalence throughout all samples, although the residual inverse quantum information 1 –

IQ makes the significant difference (in some cases, to adouble extent) in the wave- or free-binding content of samples; see for instance I-60 and V-60 with respect to II-60, IV-60 and VI-60 for samples at investigated at 60 °C, and IV-700

vs. I-700, V-700

vs. III-700 and VI-700

vs. II-700 for samples investigated at 700 °C, respectively.

However, in aiming to establish a hierarchy in binding potency, one should run on the residual IQ of the samples for identifying the decreased potency of free bindings information. Accordingly, for 60 °C, one notices from

Table 3 the main series VI > IV > II followed by III > V > I, indicating two important features:

Nevertheless, these binding potency series are apparently changing with the rising of the samples’ temperature, as results in

Table 4 provide for the 700 °C case; however, agreement with 60 °C is to be researched, while considering specific thermal analysis, as will be exposed later.

#### 2.3. Discussion: Cross-Check by Thermal Analysis

Samples of

Table 2 were considered for thermal decomposition treatment, see

Figure 5 and the Experimental Section below. The resulting thermo-gravimetric data are summarized in

Table 5, as abstracted from

Figure 5, while the corresponding correlation curves

TG[%] =

f(

T[

^{0}C]), either as parabola or lines, depending on the number of temperature change points used, three or two, respectively, are provided for each of the samples of

Table 2.

These regression curves, while recovering at least the mass loss at the ending temperatures on the interpolation domains, served to provide the missing TG

_{700}[%] information,

i.e., for characterizing the percent of mass loss by rising temperature from 60 to 700 °C; for samples of

Table 2, see

Table 6. They are to be compared with the information obtained from the recorded spectra of

Figure 4 and the allied residual inverse quantum information (1 −

IQ) of

Table 4.

However, to this aim, the relative residual IQ index is computed through containing the relative behavior with respect to the 60 °C data of

Table 3; it is thus considered as:

with the corresponding values for the samples of

Table 2 by the residual IQs from

Tables 3 and

4; the results are reported in

Table 6. It is worth noting that, since

Equation 13 models the amount of relative decrease of the residual of the inverse quantum particle-to-wave information, it naturally correlates with the particle and, thus, to the mass losing amount at 700 °C; thus, it may be compared with the mass loss as provided by thermal treatment.

However, warnings should be made, since the thermal decomposition method belongs merely to classical characterization of the structure with respect to the spectroscopic records of quantum motions (IR vibration in this work); therefore, the comparison is only meaningful unless it is not taken as a one-to-one correspondence. In fact, this is the reason for which, in

Table 6, the percent values of

Equation (13) were displayed along the TG extrapolated values at 700 °C upon the interpolation curves of

Figure 5, here aiming for only a semi-quantitative cross-check.

Nevertheless, if one hierarchically arranges the samples’ increasing absolute differences of

Table 6, you will get the series II→VI→IV followed by the succession I→V→III; remarkably, they correspond with the classification provided by (1-IQ) residual analysis of the samples’ spectra at 60°C drying synthesis conditions; see the previous sub-section.

However, both analyses tell us that the most reliable compound for free bindings and, thus, with less potential as a particle carrier, is predicted for the sample VI of

Table 2; it is followed by sample IV at drying condition or by sample II under higher temperature circumstances; instead, the silica films obtained within ammonia catalyst (samples I, V and III) are more reliable for being further engaged in the effectors’ interaction due to higher porosity or particle quantum information contained therein; see their higher IQ in

Table 3 and

4. Interestingly, the interchange in the last series between samples III and IV for IQ porosity in

Table 4 at higher temperature treatment is probably due to the difference in which the shorter chain of DTAB (respecting CTAB) interacts with basic cosolvent at elevated thermal conditions.

All in all, the present spectral-IQ method provides a consistent structural tool in analyzing the spectra for their particle-to-wave quantum content, in the view of establishing the porosity and free binding potential, at various temperatures, respectively, having the silica sol-gel based films as the current working example. Further applications and illustrations of the present method will help in generalizing it towards an in-depth understanding of the quantum influence on the chemical binding potency by physical analysis (spectral, thermal, etc.).