**1. Introduction**

Models of Brownian motion, both stochastic and deterministic, have been of interest to researchers for a long time [1]. Lucretius considered in 60 BC that the movement of dust particles in the air is caused by the movement of small invisible particles. The chaotic movements of coal particles in alcohol were described by Ingenhousz in 1785. In 1827, the botanist Brown observed the movement of pollen in water under a microscope. A mathematical description of Brown's movement was given in 1880 by Thiele and in 1900 by Bachelier. In 1905, Einstein developed a stochastic theory of Brownian motion [2], which Perrin experimentally proved in 1909. Langevin in 1908 used a stochastic differential equation to describe changes in macroscopic variables. In 1965, Mori described transport, collective motion and Brownian motion within statistical-mechanical theory [3]. Saffman and Delbruck investigated Brownian motion in 1975 in biological membranes. Caldeira and Legget investigated the quantum Brownian motion of 1983 [4].

Fujisaka, Grossmann, Thomae and Geisel from 1982 to 1985 wanted to form a dynamic theory of Brownian motion [5]. In 1998, Gaspard and his associates experimentally proved

**Citation:** Nježi´c, S.; Radulovi´c, J.; Živi´c, F.; Miri´c, A.; Jovanovi´c Peši´c, Ž.; Vaskovi´c Jovanovi´c, M.; Grujovi´c, N. Chaotic Model of Brownian Motion in Relation to Drug Delivery Systems Using Ferromagnetic Particles. *Mathematics* **2022**, *10*, 4791. https://doi.org/10.3390/ math10244791

Academic Editor: Efstratios Tzirtzilakis

Received: 2 November 2022 Accepted: 7 December 2022 Published: 16 December 2022

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microscopic chaos [6]. In 2005, Cecconi attempted to determine the microscopic nature of diffusion by data analysis [7]. Since 2011, Brownian motion in superfluids has been considered. There are various studies of Brownian motion and stochastic and chaotic models are observed, and the nature of this movement is a difficult question and the answer is not determined by the character of the model [8,9].

The first definition of Brownian motion was related to stochastic process [10], in relation to a wide range of different real stochastic processes, and represented by the Wiener process, which describes continuous-time stochastic process with real values [11]. Brownian motion can be observed as stochastic or deterministic in chaos theory, based on the deterministic equations that describe stochastic phenomena [11], but the governing parameters that might provide a full replication of the experiment are difficult to determine or define. Recent computer simulation experiments have shown the possibility to model the chaotic system as a stochastic one, by controlling simulation parameters and initial conditions [11]. Such an approach has enabled research on how to govern the chaotic system (and determine governing parameters) through the study of particle trajectories that have random motion, that is, by using deterministic equations to reproduce random behavior [11].

The generation of deterministic Brownian motion is possible through additional degrees of freedom in the Langevin equation of the phenomenological system of particle mixing and agitation in fluids [12]. Another study [13] replicated Brownian motion by using a fully deterministic set of differential equations and applied it to a real problem of electronic circuit implementation. Their deterministic model showed that some variables within the model can enable modeling of the circuit dynamics as a stochastic Brownian behavior [13]. There are numerous real systems that exhibit Brownian behavior, and modeling such systems by deterministic systems (without random components) is an important area in recent research, including drug delivery systems [13]. An analysis of 126 different combinations of governing parameters is given in [13], and around 10% of those cases involved deterministic Brownian-like motion. They obtained stochastic or deterministic Brownian motion based on the initial setup conditions (assigned initial parameters values for circuit implementation) [13].

Ferrofluids are a suspension of small particles of 10 nm, each of which contains one permanent ferromagnetic domain [14]; thus, each particle is a permanent magnet, which, in the absence of an external magnetic field, rotates randomly under the action of Brown's forces, which are strong due to the small particle size [15]. In ferrofluids, dipoles exist without fields and rotate randomly by Brownian motion [16]. Ferrofluids are interesting magnetic fluids that can be controlled by an external magnetic field [17]. There are several applications for ferrofluids in industrial as well as technological fields such as magnetic memory, inkjet printers, magnetic seals, etc. [18]. They are known for their biomedical applications such as magnetic resonance contrast agents [19], hyperthermia [20], targeted drug delivery to tumor and cancer cells, antibacterial activity, etc. [21,22].

Numerical computational simulations have emerged in the past decade as powerful tools for the analysis and prediction of the material physical behavior at macro/micro and nano scales, with extensive research on applied models and software solutions. Thermal conductivity of fluids is the most influential factor for the fluid behavior, and different approaches to estimate or predict it for novel nanofluids have been studied using mathematical models [23–25] or experimentally based models [26]. For example, models related to the rheological properties of hybrid non-Newtonian nanofluids are important, since it is proven that with the increase in volume fractions of nanoparticles, the effect of temperature increase is more influential, leading to the non-Newtonian behavior of the nanofluid and also having a strong effect on viscosity [26].

A mathematical model was developed, using SigmaPlot software, to study the thermal conductivity of nanofluids, through the study of different volume fractions of ternary hybrid nanofluids and mono and binary hybrid nanofluids [23]. This model correlated volume fractions of different nanofluids and resulting thermal conductivity in order to provide a tool for the estimation of thermal conductivity of a ternary hybrid nanofluid [23]. Thermal properties of DNA structure in water fluid were estimated by using equilibrium and non-equilibrium molecular dynamics approaches and LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) software [24]. Another work on the prediction of the thermal conductivity of hybrid Newtonian nanofluid proposed an algorithm to solve the problem in the Artificial Neural Network (ANN) [25] that considered the volume fraction of nanoparticles and temperature.

This paper studies the possibility to use and influence Brownian motion to produce patterned trajectories of particles in a diffusive motion of the ferrofluid, aiming to assist in more efficient drug delivery nanofluid systems. A chaotic model of Brownian motion was theoretically analyzed and simulated by using Maple software. The chaotic model was mapped with an introduced control parameter, p, which depends on the viscosity coefficient and particle mass and size, in analogy with the Langevin equation. The ferrofluid in the gravitational field without the presence of an external magnetic field in a two-dimensional model was observed.

### **2. Materials and Methods**

### *2.1. Brownian Motion*

Numerous experiments show that there is a constant internal movement in every substance. This internal movement is in fact the movement of the molecules that make up the observed substance. This movement of molecules is unregulated, never stops and depends only on temperature. The phenomenon discovered by Brown directly indicates the stochastic nature of the movement of molecules, where the same initial condition will not replicate the resulting motion (the same trajectory in time). Using a microscope, it was observed that very small particles floating in a liquid are in a state of continuous stochastic motion, and the smaller the particles, the faster they move [27]. This motion, called Brownian motion, never stops, does not depend on any external cause and is a manifestation of the particles' motion due to colliding with surrounding molecules of fluid and internal energy of matter: the potential energy of all the particles and thermal energy of moving particles (kinetic energy), which is correlated to the temperature and number of particles (mass). When they collide with a solid body, liquid molecules, which are constantly moving, are subjected to a certain amount of movement. If the body is in a liquid and has larger dimensions, the number of molecules that come across it from all sides is also very large, and their shocks are compensated at any time and the body remains practically motionless [28].

If the body is small, such compensation may be incomplete: it can accidentally hit one side of the body with a much larger number of molecules than the other, causing the body to move [28]. It is a movement performed by Brown's particles under the action of chaotic blows of molecules. Brown's particles have several billion times the mass of individual molecules and their velocities are very low compared to the speeds of molecules, but their movement can be observed with a microscope. In this way, too, a substance not only has a granular structure—that is, it consists of individual separate parts—but it also consists of particles that are constantly moving.
