Non-Ideality Factor in Multifractal and Entropy-Based Analysis of Self-Organized Structures of Plant Polymers (Lignins)

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Non-Ideality Factor in Multifractal and Entropy-Based Analysis of Self-Organized Structures of Plant Polymers (Lignins)

Journal Title: Lesnoy Zhurnal (Russian Forestry Journal) - Year 2021, Vol 18, Issue 2

Abstract

An attempt has been made to introduce the generalized non-ideality factor of systems g (GNF) into information entropy equations that describe self-organized structures of essentially nonequilibrium systems with the use of studying the topological properties of high molecular weight compounds in solutions using wood lignins as an example. The factor g as a relative thermodynamic characteristic connects the ideal and real models of systems in which two competitive (opposite in sign and action) processes can be distinguished: order (−) ↔ chaos (+); attraction (−) ↔ repulsion (+); compression (−) ↔ extension (+); clustering (−) ↔ decay (+), etc.g = 1 + 〈– βord + αnord〉 = 1 + 〈– pi (β) + pi (α)〉, где – βord ≡ 1/nΣinβi и αnopd ≡ 1/nΣniαi are relative average characteristics (pi – probabilities) of oppositely occurring processes. The factor g varies in the interval 0≤ g ≥1 and depends on which of the competitive processes prevails. For αnord = 0 g → 0, for βord = 0 g→2, for g = 1 the behavior of the elements of the system will be ideal. The factor g is introduced into any classical equations suitable for studying ideal systems with the aim of using them to describe real systems (for example, the equations of Henry, Raoult, Van’t Hoff, general gas, etc.). Strictly mathematically, the factor g is defined through the values M – measure, ε – size (scale), and d – dimension as a ratio of logarithms of measures of real (М*) and ideal (М0) states of the object: gth = lnМ*/lnМ0 = d/D, where M* and M0 can be the number of elements in the structure of the fractal real (for example, cluster) or mathematical object (for example, Sierpiński triangle) Nd and the number of elements in the structure of the object in the perfect condition, having the property of multi-scale and self-similarity, ND, where d and D are the fractal and Euclidean dimensions. As a thermodynamic characteristic gth is defined by the ratio of thermodynamic functions, functionals, for example, ΔGi*/ΔGi, where ΔGi* = –RTlnаi is real and, ΔGi = –RTlnNi is ideal state; the number of moles of n* − real state of matter to n − ideal state of matter; relative entropies of the system ΔSreal/ΔSid (ΔSid − Boltzmann entropy). New expressions of the information and thermodynamic entropies with a fractional (0:1) moment of order and with the entropic gS and gth non-ideality factors are obtained for the analysis of self-organized quasi-equilibrium structures in the Renyi formalism SgSM–Rn(p) = R/(1 – gS)lnΣNipgSi; SgthM–Rn = R/(gth)ln(ΣNi=1pigth – 1); in the Tsallis formalism SgSM–TS (p) = R(1 – ΣiN(ε)pigS)/(gS – 1); SgthM–TS (p) = R(1 – ΣiN(ε)pi1–gth)/gth with an application for studying the topological properties of high-molecular compounds by hydrodynamic methods, as well as the thermodynamics of polymer solutions. For citation: Makarevich N.A. Non-Ideality Factor in Multifractal and Entropy-Based Analysis of Self-Organized Structures of Plant Polymers (Lignins). Lesnoy Zhurnal [Russian Forestry Journal], 2021, no. 2, pp. 194–212. DOI: 10.37482/0536-1036-2021-2-194-212

Authors and Affiliations

N. A. Makarevich

Keywords

non-ideality factor of systems fractal structures fractal dimensionality information and thermodynamic entropy polymer solutions thermodynamics of polymer solutions lignins

EP ID EP693569
DOI https://doi.org/10.37482/0536-1036-2021-2-194-212
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