The Mathematical Simulation for the Photocatalytic Fatigue of Polymer Nanocomposites Using the Monte Carlo Methods

We consider an approach to mathematical modeling of photodegradation of polymer nanocomposites with photoactive additives using the Monte Carlo methods. We principally pay attention to the strength decrease of these materials under solar light action. We propose a new term, “photocatalytic fatigue”, which we apply to the particular case when the mechanical strength decreases only owing to the presence of photocatalytically active components in polymeric nanocomposite material. The propriety of the term is based on a relative similarity of photostimulated mechanical destructive processes in nanocomposites with photoactive additives and mechanical destructive processes typical for metal high-cycle fatigue. Formation of the stress concentrations is one of the major causes of fatigue cracks generation in metals. Photocatalytic active nanoparticles of semiconductors initiate a generation of the stress concentrations under sunlight irradiation. The proposed mathematical model is a Wöhler curve analog for the metal high-cycle fatigue. We assume that equations for high-cycle fatigue curves of samples with stress concentrations could be used in mathematical modeling of polymer nanocomposites photodegradation. In this way, we replace the number of loading cycles with the exposition time in the equations. In the case of polypropylene and polyester samples with photoactive titanium dioxide, the experimental parameters of phenomenological equations for “photocatalytic fatigue” are calculated using one of the Monte Carlo methods based on the random search algorithm. The calculating scheme includes a solution of the extreme task of finding of the minimum of nonnegative transcendent multivariable function, which is a relative average quadratic deviation of calculated values of polymeric nanocomposite stress in comparison with corresponding experimental values. The applicability of the “photocatalytic fatigue” model for polymer nanocomposites with photoactive nanoparticles is confirmed by the example of polypropylene and polyester samples. The approximation error of the experimental strength values for them did not exceed 2%.