# Quantitative estimation of chemical microheterogeneity through the determination of fuzzy entropy

^{1}Department of Chemistry, Biology, and Biotechnology, Università Degli Studi di Perugia, Perugia, Italy^{2}Department of Earth and Planetary Sciences and Origins of Life Initiative, Harvard University, Cambridge, MA, United States^{3}Santa Fe Institute, Santa Fe, NM, United States

Chemical micro-heterogeneity is an attribute of all living systems and most of the soft and crystalline materials. Its characterization requires a plethora of techniques. This work proposes a strategy for quantifying the degree of chemical micro-heterogeneity. First of all, our approach needs the collection of time-evolving signals that can be fitted through poly-exponential functions. The best fit is determined through the Maximum Entropy Method. The pre-exponential terms of the poly-exponential fitting function are used to estimate Fuzzy Entropy. Related to the possibility of implementing Fuzzy sets through the micro-heterogeneity of chemical systems. Fuzzy Entropy becomes a quantitative estimation of the Fuzzy Information that can be processed through micro-heterogeneous chemical systems. We conclude that our definition of Fuzzy Entropy can be extended to other kinds of data, such as morphological and structural distributions, spectroscopic bands and chromatographic peaks. The chemical implementation of Fuzzy sets and Fuzzy logic will promote the development of Chemical Artificial Intelligence.

## 1 Introduction

Microheterogeneity refers to systems that are heterogeneous at the microscopic level (*Kalyanasundaram, 2012*). There are numerous examples of micro-heterogeneous systems in both soft and crystalline materials (*Chen, 2022*). They might be broadly classified into five major sets (I) molecular aggregates composed of surfactants, lipids or other compounds, (II) natural and synthetic polymeric systems (III) nanomaterials and colloidal dispersions, (IV) adsorbed and intercalated guest-host solid materials (V) the multi-compartments and multiphase assemblies of living beings and their mimicries (*Cheng and Perez-Mercader, 2020*). The heterogeneity can be at the level of single particles (i.e., intra-entities) and/or inter-entities (*Chen, 2022*) (*Rabanel et al., 2019*).

No single technique can unveil all the details of these micro-heterogeneous systems. Techniques such as electron, fluorescence, Raman and atomic force microscopies, diffraction of X-rays and neutrons allow taking two- and three-dimensional snapshots of these micro-heterogeneous systems at intra- and inter-entities levels (*Rabanel et al., 2019*). Other relevant data on the collective features of micro-heterogeneous samples can be collected by techniques such as NMR and ESR. The Kirkwood-Buff theory (*Kirkwood and Buff, 1951*) describes solution mixtures containing any number of components; and it has been proven a solid framework for providing expressions of macroscopic thermodynamic features for any stable solution mixture as a function of its composition (*Pierce et al., 2008*) (*Newman, 1994*). In time-resolved spectroscopies or other techniques, transient signals monitor the dynamics and kinetics of molecular events occurring in the micro-heterogenous systems, thus providing a picture of their time evolution, including during non-equilibrium stages. The time-evolving signal

In many cases, decay kinetics from complex systems and fractal structures have been fitted by stretched exponential functions (*Berberan-Santos et al., 2005*) of the type:

or the less-known compressed hyperbola (or Becquerel) function (*Menezes et al., 2013*) of the type:

However, it has been shown (*Hirayama et al., 1990*) that stretched exponential functions can be substituted by the fitting function appearing in Eq. 1. In certain conditions, a sum of compressed hyperbolas can be replaced by a sum of exponential functions (*Menezes et al., 2013*). In Eq. 1, the determination of the “image *Berberan-Santos et al., 2005*)) is the inverse Laplace transform of the measured time-resolved profile

Although deconvolution is well conditioned, inverting the Laplace transform is ill-conditioned (*McWhirter and Pike, 1978*). This implies that minor errors in the data can lead to considerable uncertainty in the reconstruction of *Jaynes, 1988*) (*Livesey and Brochon, 1987*) (*Brochon, 1994*) (*Steinbach et al., 2002*).

In the following paragraphs, the basic principles of the MEM are briefly recalled. Then, a discretization of the “image *Gentili, 2021*).

## 2 The basics of maximum entropy method

The MEM’s roots reside in Bayesian probability (*Jaynes, 1957*) (*Brochon, 1994*). It provides “the least biased estimate possible on the given information; i.e., it is maximally non-committal with regard to missing information”. According to Bayes’ theorem, the probability of the hypothesis

The term

In Eq. 5,

with

The term

The term *Gull and Daniell, 1978*):

where

By introducing all the terms in Eq. 4, the explicit definition of the “posterior probability”

where

## 3 The discretization of the “image $\mathit{w}\left(\mathit{\tau}\right)$ ”

A possible shape of *Cover and Thomas, 2006*), there exists a value

The quantized variable

Then, the probability that *Friar et al., 2016*)

As

If

## 4 Definition of fuzzy entropy

The information entropy of the continuous probability distribution function

Introducing the quantized variable

Since

But as *Cover and Thomas, 2006*). Hence, we get that

The fitting procedure of the time-resolved signals through the MEM allows determining the weights

The variable

The distribution of lifetimes, obtained by fitting a specific time-resolved signal, *Gentili, 2018*). A Fuzzy set is different from a classical Boolean set (*Zadeh, 1965*) because an element belongs to a Fuzzy set with a degree of membership *i*th lifetime

Based on this second property, it is reasonable to propose the normalized version of the Fuzzy Entropy that becomes independent of the number of exponential terms used in the fitting procedure:

The

β_{bis})

Finally, both

γ) The value of Fuzzy Entropy depends on the physicochemical context of the chemical system: the more significant its micro-heterogeneity, the larger its Fuzzy Entropy.

Some experimental proofs of this third property are reported in the next paragraph. This paragraph is concluded by asserting that among the different definitions of Fuzzy Entropy that have been proposed (*Al-sharhan et al., 2001*), only that presented in Eqs. 20c and 21 is valuable for our case. The Fuzzy Entropy, appearing in Eqs. 20c and 21, is appropriate for characterizing the micro-heterogeneity of a chemical sample, based on the information retrieved by fitting any exponential time-resolved signal using the MEM.

### 5 Determination of fuzzy entropy for some chemical systems

The γ property of normalized Fuzzy Entropy, defined in Eq. 21, implies that *Comez et al., 2021*) (*Chakraborty et al., 2018*), microemulsions (*Penconi et al., 2014*), nanomaterials (*Bellacanzone et al., 2020*), dyes (*Gentili et al., 2010*) in different micro-environments, and membranes (*Krishnamoorthy and Ira, 2001*; *Haldar, 2022*)^{1}.

**FIGURE 2**. In **(A)**, the fluorescence lifetimes’ distributions of Pyr in water (black points), SDS micelles (pink points), pOoBSK micelles (red points), pDoAO hydrogel (cyan points), and Gemini hydrogel (blue points) are shown. In **(B)**, the fluorescence lifetimes’ distributions of PyI in PhAA/dMdDNCl after adding 0% (black points), 5% (blue points), 10% (green points), and 15% (red points) of water are reported. In **(C)**, the photochromism of PP (see the molecular structures of both colored and uncolored forms and the spectra recorded upon UV irradiation) has been investigated in acetonitrile and in PMMA. The plots containing the black points refer to the distributions of lifetimes for the coloration (on the left) and bleaching (on the right) kinetics in acetonitrile. The plots below, with red dots, refer to the same kinetics collected in PMMA.

The first example refers to the fluorescent salt 2-[(1-pyrenyl)-ethenyl]-1-methylpyridinium (Pyr). The Pyr’s fluorescent lifetimes distribution is susceptible to its micro-environment (*Cesaretti et al., 2016*). When Pyr is dissolved in pure water, the distribution is rather sharp (see Figure 2A): there are two principal components (

The second example refers to another fluorescent N-methylpyridinium iodide, i.e., 2–4-(diphenylamino)phenyl-ethenyl-1-methylpyridinium iodide (PyI) used to probe the effect of water into the microheterogeneity of a Deep Eutectic Solvent (DES), which is made of phenylacetic acid (PhAA) and N,N,N,N-dimethyldidodecylammonium chloride (dMdDNCl) mixed in 2:1 M ratio (*Tiecco et al., 2021*). Figure 2B reports the fluorescent lifetimes’ distribution for PyI dissolved in PhAA/dMdDNCl and determined after addition of different amounts of water. The distribution in black (see graph on top of Figure 2B) has been determined for the DES having only its hydration water: the weighted averaged lifetime *Ma et al., 2018*).

Finally, the third example is shown in Figure 2C. It regards the photochromic spiroxazine PP. When PP_{Un} is irradiated by UV, the spiro C-O bond of the oxazine is broken, and a merocyanine (PP_{Col}) is produced. PP_{Col} also absorbs in the visible region. The spectral modifications that are recorded upon UV irradiation are shown in the graph below the PP’s molecular structures. Merocyanine is metastable. If UV irradiation is discontinued, spontaneous thermal bleaching of the color can be observed. The PP’s coloration and bleaching kinetics have been collected in two very different micro-environments: PP dissolved in a homogeneous solvent, such as the acetonitrile, and PP encapsulated in a micro-heterogeneous and viscous environment, such as a film of poly(methyl methacrylate) (PMMA) (*di Nunzio et al., 2010*). Both the coloration and bleaching kinetics have been fitted by poly-exponential functions through MEM. The outputs are shown in Figure 2C. They reveal that PMMA slows down the PP’s photochromism and that the polymer significantly broadens the distributions of lifetimes. The black traces, which are relatively sharp, refer to acetonitrile, whereas the red traces refer to PMMA. The values of Fuzzy Entropy quantitatively remark the differences in the lifetimes’ distributions. _{Col} that exist under many conformers (*Gentili, 2014*). On the other hand,

## 6 Discussion

This work proposes an approach for quantitatively determining the degree of micro-heterogeneity of any chemical sample. Our approach requires, at first, the acquisition of a time-resolved signal that can be fitted by a poly-exponential function. Then, the least number of exponential terms and their relative weights are determined through the MEM. The relative weights are then used to calculate the normalized Fuzzy Entropy *Steinbach, 2002*).

It is interesting to note that the definition of

Any compound will exhibit different *Gentili et al. (2016)* after granulating the UV and visible regions through the absorption bands of properly chosen photochromic compounds. This approach allows encoding a chemical language in ways alternative to those already proposed by *Dueñas-Díez and Perez-Mercader (2019)*.

## 7 Methods

In this paper, we point out that there exists a way for quantifying micro-heterogeneity. The approach we propose is synthetically the following one.

First, we collect a transient signal generated by our sample in its environment (for instance, a luminescence decay after photo-excitation).

Second, we fit the transient signal using the Maximum Entropy Method (MEM). Then, MEM gives us the least number of exponential terms needed to describe the experimental signal (please, see paragraph 2). A weight

Finally, we propose the Normalized Fuzzy Entropy, expressed through Eq. 21, to quantitatively determine the micro-heterogeneity which is then based on the weight

## Data availability statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

## Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Funding

PG thanks Università degli Studi di Perugia (Italy) for their funding of this work within the program “Ricerca di Base 2021”. JP-M thanks Repsol, S.A. (Spain) for their funding of this work and their continued support. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Footnotes

^{1}We remind the reader that De Luca and Termini (*1972*) were the first to propose a definition of the entropy *Singh and Sharma, 2019*; *Prakash et al., 2008*; *Bathia et al., 2013*). These definitions have been utilized in many applications (*Bathia et al., 2015*): for instance, in the field of machine learning for features selections in pattern recognition problems. They assure that the Fuzzy Entropies of a Fuzzy set and its complement are equivalent. This condition does not have any physicochemical meaning because the chemical implementation of a Fuzzy set based on the states of a single compound cannot simultaneously include the above and the potential states associated with the complement. Therefore, the contribution of the complement is excluded from the definition proposed in Eq. 21 for quantitatively characterizing chemical micro-heterogeneity. This is akin to when in chemical kinetics concentrations are always greater than zero.

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Keywords: micro-heterogeneous chemical systems, time-resolved signals, maximum entropy method (MEM), molecular information, molecular computing, fuzzy sets, chemical artificial intelligence

Citation: Gentili PL and Perez-Mercader J (2022) Quantitative estimation of chemical microheterogeneity through the determination of fuzzy entropy. *Front. Chem.* 10:950769. doi: 10.3389/fchem.2022.950769

Received: 23 May 2022; Accepted: 26 July 2022;

Published: 30 August 2022.

Edited by:

Djalal Trache, Ecole Militaire Polytechnique, AlgeriaReviewed by:

Dilip H. Dagade, Shivaji University, IndiaSurender Singh, Shri Mata Vaishno Devi University, India

Copyright © 2022 Gentili and Perez-Mercader. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Pier Luigi Gentili, pierluigi.gentili@unipg.it