Matrix-graph model of the polymer materials destruction process

A A Khvostov,A V Karmanov,A A Zhuravlev,I A Khaustov,S G Tikhomirov

doi:10.20914/2310-1202-2018-3-50-55

Abstract

The paper deals with the problem of mathematical modeling of the process of thermochemical destruction using the theory of graphs. To synthesize a mathematical model, the Markov chain is used. For the formalization of the model a matrix-graph method of coding is used. It is proposed to consider the process of destruction as a random process, under which the state of the system changes, characterized by the proportion of macromolecules in each fraction of the molecular mass distribution. The intensities of transitions from state to state characterize the corresponding rates of destruction processes for each fraction of the molecular weight distribution (MWD). The processes of crosslinking and polymerization in this work have been neglected, and it is accepted that there is a probability of transition from any state with a lower order index (corresponding to fractions with higher molecular weights) to any state with a higher index (corresponding fractions with lower molecular weights). A computational formula is presented for estimating the number of arcs and model parameters from a given number of fractions of the molecular weight distribution of the polymer. An example of coding in a matrix form of a graph model of the process of degradation of polybutadiene in solution for the case of six fractions of the molecular weight distribution is shown. As the simulation environment, the interactive graphical simulation environment of MathWorks Simulink is used. To evaluate the parameters of the mathematical model, experimental studies of the degradation of polybutadiene in solution were carried out. The chromatography of the polybutadiene solution was used as the initial data for the estimation of the MWD polymer. The considered matrix-graph representation of the structure of the mathematical model of the polymer destruction process makes it possible to simplify the compilation of the model and its software implementation in the case of a large number of vertices of the graph describing the process of destruction

Highlights

The paper deals with the problem of mathematical modeling
of the process of thermochemical destruction using the theory of graphs
It is proposed to consider the process of destruction as a random process

Summary

Introduction

Определяемого числом фракций и описываемых функцией ММР, а также общности процессов деструкции полимеров, актуальна задача разработки общей матрично-графовой модели, описывающей основные закономерности процесса. Первой задачей для синтеза матричнографовой модели процесса деструкции является нахождение числа дуг. Для этого сначала рассчитаем число дуг K по заданному числу вершин и структуре графа в виде Г. Рассчитаем число дуг для 2-x узлов (рисунок 2).

Results

Conclusion