**3. Results**

### *3.1. Dynamic Model*

We analyzed the theoretical model as previously presented and used Maple software to perform the computational experiment. The ferrofluid in the gravitational field without the presence of an external magnetic field in the xy plane was observed and the initial condition on the x and y axes for the ferrofluid is given. It performs 400 collisions with fluid molecules. For the values of the control parameter *p* that depend on the viscosity coefficient and particle mass and size (in analogy with β coefficient in Equation (26), as shown in the following equations), we obtained a path on the basis of which the movement can be characterized. The character of the movement itself depends on the value of the control parameter. The numerical model can conduct a large number of different computer simulations in a short time. We started with simple low-dimensional deterministic systems that can exhibit diffusive behavior. Chaotic behavior is possible to be associated with diffusion in simple low-dimensional models, supporting the idea that chaos was at the very origin of diffusion [47].

Deterministic diffusion is a phenomenon also present in chaotic maps on the line. Many researchers are dealing with this phenomenon [5]. In 1908, Langevin used a stochastic differential equation to describe slow changes in macroscopic variables. A Stokes viscous force and a fluctuating random force with a Gaussian distribution act on the particle. Einstein views Brownian motion as diffusion. The Langevin equations are defined as stochastic equations [48]. Fujisaka and Grossmann worked on the dynamical theory of Brownian motion [5]. A one-dimensional discrete-time dynamical system example can be given by Equation (25):

$$x(t+1) = [x(t)] + F(x(t) - [x(t)]),\tag{25}$$

where *x (t)* is the position of the particle *x* as a function of time *t* that is performed by the diffusion in the real axis. The brackets [ ] denote an integer number of arguments. *F(u)* is a map defined at interval [0, 1]. Based on the Langevin equation, we can observe the Brownian motion of a particle of mass *m* in a two-dimensional model as follows:

$$\frac{d^2\xi}{dt^2} = -\beta \frac{d\tilde{\xi}}{dt} + f\_{\tilde{\xi}'} \tag{26}$$

where:

$$
\beta = \frac{6\pi\eta r}{m}, f\_{\overline{\xi}} = \frac{F\_{\overline{\xi}}}{m}, \overline{\xi} = \text{x}, y,\tag{27}
$$

where *η* is fluid viscosity, *r* is particle radius and *m* is particle mass. In Equation (26), the Stokes viscous force and a fluctuating random force with a Gaussian distribution act on the particle [12].

### *3.2. Chaotic Model*

Considering the equations given in a previous section, the chaotic model can be mapped as given in Equation (28). Equation (28) is derived from Equation (25) and describes the Brownian motion of particles in a two-dimensional chaotic model, where variable *t* is substituted by *ξ = x, y*.

$$\mathbb{E}(\mathbf{t}+1) = [\mathbb{E}(\mathbf{t})] + \mathbb{F}(\mathbb{E}(\mathbf{t}) - [\mathbb{E}, (\mathbf{t})]); \mathbb{E} = \mathbf{x}\_{\prime} \text{ y}\_{\prime} \tag{28}$$

where *[ξ]* is the integer part of *ξ* while

$$F(u) = \begin{cases} 2(1+q)u, 0 \le u \le \frac{1}{2} \\ 2(1+q)(u-1) + 1, \frac{1}{2} \le u \le 1 \end{cases} \tag{29}$$

where *F(u)* is a map defined on the interval [0, 1] that fulfills the following properties:


If the theoretical model presented in our study is mapped by the chaotic model, it can be stated as in Equation (30). With the introduction of the previous *F(u)* sinusoidal function, Equation (28) can be accordingly stated as Equation (30).

$$
\zeta^\sharp(t+1) = \zeta^\sharp(t) + p \sin(2\,\pi\,\zeta^\sharp(t)),
\tag{30}
$$

where *ξ = x, y* is a time-dependent coordinate *t* and *p* is a control parameter that depends on the viscosity coefficient of the fluid.

When a series of computer experiments are performed where the parameter *p* changes, it is observed that the Brownian particle can be in a stochastic or chaotic motion. Legitimacy can be derived from the following. The particle motion has deterministic patterns for the following values of the parameter *p*:

$$p = N + \frac{1}{2}, \ N = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2} \dots \tag{31}$$

Computer experiments showed that the ferrofluid can exhibit different modes of deterministic dynamics within the two-dimensional model, depending on the initial value of the parameter *p*, as shown in Figures 3–8. Figure 3 shows the linear trajectory of the ferrofluid. Figures 4 and 5 are classic examples of the chaotic motion of a particle. Figures 6–8 show the transition from a deterministic to chaotic state of the system. This is demonstrated by the computer experiment where for the same initial condition (input value of the parameter *p*), the same patterns were obtained each time the computer simulation was repeated.

**Figure 3.** The linear trajectory of the particle in a two-dimensional chaotic model for the value of parameter p = 1.0.

**Figure 4.** The trajectory of the particle in a two-dimensional chaotic model for the value of parameter, *p* = 0.9.

**Figure 5.** Trajectory of the particle in a two-dimensional chaotic model for the value of parameter, *p* = 0.8.

**Figure 6.** Trajectory of the particle in a two-dimensional chaotic model for the value of parameter, *p* = 0.7.

**Figure 7.** Trajectory of the particle in a two-dimensional chaotic model for the value of parameter, *p* = 0.6.

**Figure 8.** The trajectory of the particle in a two-dimensional chaotic model for the value of parameter, *p* = 0.5.

The condition of the system is described by the vector *ξ(t)* of *d* dimension—Equation (30). The trajectory is discretized in time where the discretization step is *τ* and the vector of dimension *d* is introduced in Equation (32) with the associated string (of *m* length) given in Equation (33).

$$
\Xi^{m}(t) = \left(\xi(t), \xi(t+\tau), \dots, \xi(t+m\tau-\tau)\right) \tag{32}
$$

$$\mathcal{W}^{\rm m}(\varepsilon, t) = (i(\varepsilon, t), i(\varepsilon, t + \tau), \dots, i(\varepsilon, t + m\tau - \tau)) \tag{33}$$

where *i(<sup>ε</sup>,t + jτ)* denotes the cell of *ξ(t + jτ)*, with a length of *ε*.

The value of 0.3060 for Kolmogorov–Sinai entropy (equal to the sum of positive Lyapunov exponents) was obtained, calculated according to the following equations:

$$h\_{KS} = \lim\_{\varepsilon \to 0} h(\varepsilon, \tau) \tag{34}$$

$$h(\varepsilon, \tau) = \frac{1}{\pi} \lim\_{m \to \infty} \frac{1}{m} H\_m(\varepsilon, \tau) \tag{35}$$

$$H\_{\mathfrak{m}}(\varepsilon, \mathfrak{r}) = -\sum\_{\mathcal{W}^{\mathfrak{m}}(\varepsilon)} P(\mathcal{W}^{\mathfrak{m}}(\varepsilon)) \ln P(\mathcal{W}^{\mathfrak{m}}(\varepsilon)) \tag{36}$$

Variable *hKS* has a value between zero and infinity, thus proving that the system is chaotic.

### **4. Discussion**

The behavior of a particle (ferrofluid) moving in a fluid under the influence of a gravitational field without the presence of an external magnetic field is observed. Likewise, the delivery of drugs to the body could be possible without the presence of any electric or magnetic field, but only under the influence of the gravitational field. Accordingly, Brownian motion is studied, which, under the influence of the gravitational field, can be stochastic, deterministic or chaotic. Different models of the aforementioned movement have been observed, showing stochastic and chaotic movement [10–13]. Based on the model in our study, it can be observed that the particle moves randomly for certain values of the control parameter *p* and exhibits linearity in motion for other values of the parameter. The control parameter affects the movement of the particle. Linear motion of the particles was observed for certain values of the parameter *p* (as shown in Equation (31) and Figure 3). For other values of the parameter *p*, the particles move randomly without any rule (Figures 4 and 5). It can be noticed from Figures 4–8 that even the chaotic motion can exhibit patterns of a deterministic movement for certain material properties of the particles (and the surrounding fluid, as well as their interrelated properties) that will result in the desired nanofluid behavior.

The control parameter, p, is related to the friction constant and viscosity coefficient. Friction constant in a fluid motion has a direct relation with Reynolds number that further determines whether the laminar or turbulent flow of fluid will occur. Parameter *p* can be further correlated to the Peclet number in a microfluidic setup, thus indicating advectively dominated distribution or diffuse fluid flow. Changes in the parameter *p* are associated with changes in the viscosity coefficient and particle mass and size. The rheological behavior of nanofluids is complex because the increase in volume fractions of nanoparticles in a fluid may result in non-Newtonian nanofluid, with more pronounced temperature effects on viscosity changes [26].

If we compare Figures 3–8, it can be seen that trajectory shapes were significantly changed for slight changes in parameter *p*: a value of 1 resulted in a fully linear trajectory, while a value of 0.9 produced a fully random path. A further decrease in p to the lowest value, 0.5, again introduced patterns of linearity within chaotic motion. Since parameter *p* is related to the viscosity and particle radius and mass, it could be assumed that such transitional behavior with changes in p can be attributed to a complex phenomenon underlining dependencies between viscosity and volume fractions of particles in nanofluids, consistent with [26]. The rheological behavior of nanofluids is dramatically different for Newtonian and non-Newtonian nanofluids and dynamic transitions among these two modes, as influenced by the changes in viscosity are still not fully clarified. Figure 3 (with the highest p value) and Figure 8 (with the lowest p value) exhibit certain similarities in trajectory pattern, since both of these have fully linear parts of trajectories, in accordance with values in Equation (31). For values of p in between these numbers (Equation (31)),

fully random trajectories were generated by simulation. However, for both of these regimes (random and linear trajectories), a decrease in p value produced a closer path (denser total trajectory), in accordance with the fact that viscosity decreases with the decrease in parameter *p*.

Accordingly, we could tailor the trajectory path of the particle in the liquid, regardless of the exogenous power propulsion strategy (e.g., external magnetic field), by tailoring the values of the parameter *p* which is related to viscosity and volume fractions of nanoparticles in a fluid. This means that it could be possible to realize targeted drug delivery by designing the system of nanoparticles in a fluid media at certain temperatures, consistent with recent research articles [49]. Research showed that there is dependence between the motility of particles and the density of neighbors, which has been a foundation for designing self-organizing nanofluids for drug delivery by tailoring the active Brownian motion of the particle [50]. The density of trajectories in our simulation significantly changed with changes in parameter *p* (Figures 3–8), in accordance with research [50] that showed a different size, density and shape of nano-cluster aggregates due to changes in Brownian motion.

Fine tailoring of the Brownian motion can produce different desired effects, including tailoring of the time and amount of the drug release [51]. On the other hand, drug delivery systems based on micro/nanomotors have been designed to overcome the influence of the Brownian motion through the control of nanoparticles' motion by some exogenous force (like external magnetic field) [49]. If the immobilization of nanoparticles increases, heating efficiency decreases [51]. How is this related to the confinement of the space within which the particles' trajectories can appear (as in the case of path shown at Figure 8) has not been the study yet, even though there are some studies related to the nanoparticle motion in a cylindrical tube and associated effects of the boundaries, curvature, size and density of the particle, including the influence of the Brownian dynamics [52] and transport phenomena in confined flows of nanoparticles [53]. Tuning of polymer amphiphilicity can increase the efficiency of drug delivery systems [54]. Amphiphilicity has direct influence on the particle collision modes, thus indicating that chaotic models of Brownian motion might exhibit patterns in particle trajectories for certain conditions.

Néel relaxation of magnetic nanoparticles has been studied, but the study on the correlation of Brownian motion to another magnetic relaxation mechanism is recent, showing the influence of Brownian relaxation on nanocage size [55]. There are complex interactions in the coupling of Brownian and Néel relaxation processes [56], which produces a highly nonlinear field-dependent magnetization response, including the pronounced influence of the size of nanoparticles clusters [57].

There is a correlation between the magnetization curve of the ferromagnetic particles system and Langevin curve [58]. If we observe single-domain ferromagnetic particles, their magnetic behavior at elevated temperatures can be correlated to the atomistic Langevin paramagnetism [59]. On the other hand, changes in temperature result in viscosity changes; thus, it is reasonable to expect that we could apply our model to a colloidal suspension of single-domain magnetic particles—ferrofluid, as described in previous chapters. The magnitude of the uniform magnetization vector for a single-domain ferromagnetic particle is proven to be constant with the direction of fluctuation based on a random motion of particles due to the heat changes (thermal agitation) [60]. Accordingly, the deterministic stochastic processes might be representative of such a process, meaning that the Langevin equation is relevant [60]. In the case of our model, we assume that parameter *p* has a correlation to the viscosity coefficient, particle radius and its mass, which further influences the degree of deterministic behavior of the chaotic system, as shown in Figures 3–8. However, further study is needed in relation to additional parameters that describe the magnetic behavior of ferrofluids.

Based on the above results, it can be concluded that the delivery of drugs could be executed without the presence of an external magnetic or electric field. Patterns of deterministic trajectories can be designed by predefined values of the parameter *p* in a

computer simulation, which can further lead to the design of the nanoparticle system for targeted drug delivery without an exogenous power propulsion strategy. However, complex relations between different influential factors need further study, including further development of the theoretical model for the motion of nanoparticles in an external field and fluid environment. The significance of such a dynamic model for the development of drug delivery systems is related to the possibility to control the motion of the drugcontaining nanoparticles, through the design of the inherent material properties of the particles and surrounding media. The possibility to use and influence Brownian motion to produce patterned particles' trajectories in diffusive motion of the ferrofluid aiming to assist in more efficient drug delivery systems of ferromagnetic nanofluids would support significant advancements in medical treatments.
