# Observing Dynamical Systems Using Magneto-Controlled Diffraction

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Discussion

#### 3.1. Light Polarization in Thin Film and Magnetic Contour Patterns

^{2}+ (x′-x)

^{2}+ (y′ − y)

^{2}]

^{1/2}and cos θ = d/r. Without the ferrofluid, the two functions will become identical, but in the presence of the diffraction grating formed by the ferrofluid subjected to the magnetic field, there is a constant factor γ. In this case

_{p}and H

_{q}:

_{0}is the maximal intensity of the magnetic field, H

_{m}(x) and H

_{n}(y) are the spatial derivatives of the magnetic field functions in the x-axis and y-axis respectively. I(x,y,z) is the light intensity directly observed from the Ferrolens with two crossed polarizers. These solutions are represented in the simulations of Figure 5b, Figure 6b and Figure 7c.

_{m}, the magnets can be modeled as two magnetic charges M

_{c}producing a magnetic dipole with [3]:

_{n}and r

_{s}is the position of poles north and south, respectively, and μ is the magnetic constant. The nanoparticles create a structure very similar to a diffracting grating from the influence of the orientation of a magnetic field. The light diffracted by the ferrofluids grating seemed to follow isopotential lines of this scalar field, having the light source as the origin of the light line, because each diffraction line is perpendicular to the scatterer. Starting with the case of the polar magnet configuration by considering (r

_{s}$\to $ infinity), which reduces the magnetic potential to the case:

_{m}= constant/x. Hence the vector associated with light lines D and the magnetic scalar potential V is

#### 3.2. Light Patterns and Chirality

_{1,2}) in a fluid of dipoles with magnetic moments μ

_{1}and μ

_{2}is given by

_{d}(i,j)/k

_{b}T, with k

_{b}T representing the thermal interaction. The dipolar interaction influences the organization of the rod structures at ambient temperatures, where l is close to 1. The magnetic perturbations twist the structure of nanoneedles, and this twist could be the source of this magnetochiral effect.

## 4. Results

#### Exploring Fixed Points and Trajectories with Magneto-Optics

**V**of Figure 15 can be represented by:

**i**and

**j**are unit vectors in the plane x-y plane, with x = θ and y = dθ/dt. This function represents a circulation about the z axis perpendicular to this plane. Another useful function which represents a radial flow from some central point is the vector function

**V**′:

**W**of Figure 16 representing the limit cycle (1) corresponds to a not uniform magnetic field represented by the vector potential

**R**

^{2}to represent the behavior of the system around some points, with the form

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Tufaile, A.; Sartorelli, J.C. Hénon-like attractor in air bubble formation. Phys. Lett. A
**2000**, 275, 211–217. [Google Scholar] [CrossRef] - Tufaile, A.; Vanderelli, T.A.; Tufaile, A.P.B. Observing the Jumping Laser Dogs. J. Appl. Math. Phys.
**2016**, 4, 1977–1988. [Google Scholar] [CrossRef] - Snyder, M.; Frederick, J. Photonic Dipole Contours of Ferrofluid Hele-Shaw Cell. Available online: http://www2.warwick.ac.uk/go/reinventionjournal/issues/volume2issue1/snyder (accessed on 24 February 2019).
- Scherer, C.; Neto, A.M.F. Ferrofluids, properties and applications. Braz. J. Phys.
**2005**, 35, 718–727. [Google Scholar] [CrossRef] - Philip, J.; Laskar, J.M. Optical properties and applications of ferrofluids–A review. J. Nanofluids
**2012**, 1, 3–20. [Google Scholar] [CrossRef] - Chiolerio, A.; Quadrelli, M.B. Smart Fluid Systems: The Advent of Autonomous Liquid Robotics. Adv. Sci.
**2017**, 4, 1700036. [Google Scholar] [CrossRef] [PubMed][Green Version] - Grossman, J.; McNeil, S. Nanotechnology in cancer medicine. Phys. Today
**2012**, 65, 38–42. [Google Scholar] [CrossRef] - Yang, Z.; Wang, H.; Guo, P.; Ding, Y.; Lei, C.; Luo, Y. A multi-region magnetoimpedance-based bio-analytical system for ultrasensitive simultaneous determination of cardiac biomarkers myoglobin and C-reactive rprotein. Sensors
**2018**, 18, 1765. [Google Scholar] [CrossRef] - Lin, J.F.; Tsao, Y.M.; Lee, M.Z. Measurement of birefringence and dichroism in magnetic fluids doped with nonmagnetic polystyrene microsphere. J. Jpn. Soc. Exp. Mech.
**2013**, 13, 13–17. [Google Scholar] - Tufaile, A.; Vanderelli, T.A.; Tufaile, A.P.B. Light polarization using ferrofluids and magnetic fields. J. Adv. Condens. Matter. Phys.
**2017**, 2017, 2583717. [Google Scholar] [CrossRef] - Rablau, C.; Vaishnava, P.; Sudakar, C. Magnetic-field-induced optical anisotropy in ferrofluids: A time-dependent light-scattering investigation. Phys. Rev. E
**2008**, 78, 051502. [Google Scholar] [CrossRef] [PubMed] - Singh, G.; Chan, H.; Baskin, A.; Gelman, E.; Repnin, N.; Kral, P.; Klajn, R. Self-Assembly of magnetite nanocubes into helical superstructures. Science
**2014**, 345, 1149–1153. [Google Scholar] [CrossRef] [PubMed] - Fowles, G.R. Introduction to modern optics, 2nd ed.; Dover: New York, NY, USA, 2013; pp. 108–110. ISBN 0-486-65957-7. [Google Scholar]
- Hubert, A.; Shäfer, R. Magnetic Domains: The Analysis of Magnetic Microstructures; Springer: Berlin, Germany, 1998; pp. 15–16. ISBN 978-3-540-64108-7. [Google Scholar]
- Witten, T.A.; Pincus, P.A. Structured Fluids, Polymers, Colloids, Surfactants, 1st ed.; Oxford University Press: Cary, NC, USA, 2004; ISBN 019958382X. [Google Scholar]

**Figure 1.**Magnetic contours observed in (

**a**) using red light, and in (

**b**) with green light for two distinct magnetic field configurations using the device Ferrolens.

**Figure 2.**Image of the magnetic contours obtained for a single cylindrical magnet in two orthogonal planes using the Ferrolens. In (

**a**) the polar configuration is presented and in (

**b**) the dipolar configuration, using LED ring lights providing radial illumination around the magnet. The diagram of the magnetic field generated by this magnet is presented in (

**c**), with the previous images positioned in each respective orthogonal plane.

**Figure 3.**Illustrating a magneto-optic phenomenon in ferrofluids: approximation function for the optical response (the retardance in degrees) of the ferrofluid for different liquid concentrations, as a function of the magnetic field intensity. Besides the birefringence (retardance), the magnetic-optic phenomenon in ferrofluids is related to Faraday rotation, the Kerr effect, and linear dichroism.

**Figure 4.**Our source of the magnetic field in this experiment is based on permanent magnets, and we obtained directly the intensity of the magnetic field of permanent magnets using the gaussmeter. In (

**a**), we obtained the magnetic field intensity for the polar configuration, and in (

**c**) for the dipolar configuration, for the one-dimensional case. The magnetic field intensity obtained for the plane of the polar case is in (

**b**), and for the dipolar case is in (

**d**). In (

**e**) a diagram of some magnet configurations are represented.

**Figure 5.**Light polarization observed in the Ferrolens for the polar configuration (

**a**), using a cylindrical magnet with a length of 0.60 cm, and the diameter is 0.60 cm. In (

**b**) simulation of light polarization using hyperbolic polynomials.

**Figure 6.**Light polarization observed in the Ferrolens for the dipolar configuration (

**a**), using a neodymium cube magnet (0.5 × 0.5 × 0.5) cm. In (

**b**) simulation of light polarization using hyperbolic polynomials.

**Figure 7.**Magnetic contours observed in Ferrolens using a tetrapolar configuration (

**a**), using a cylindrical magnet with a length of 0.40 cm, and a diameter of 1.00 cm. Light polarization observed in the Ferrolens for the tetrapolar configuration (

**b**), using a cylindrical magnet with a length of 0.60 cm, and a diameter of 0.60 cm. In (

**c**) simulation of light polarization using hyperbolic polynomials.

**Figure 8.**Observing the needle-like structures of ferrofluids using an optical microscope (magnification of 2500 ×). These structures are formed by the nanoparticles in the presence of magnetic fields, ranging from 100 G to 500 G, creating these elongated structures.

**Figure 9.**Two light patterns obtained using a straight array of LEDs. In (

**a**) for the case of polar configuration, and in (

**b**) for dipolar configuration, using a cylindrical magnet with a length of 1.50 cm, and a diameter of 1.30 cm.

**Figure 10.**The pattern of curved lines from placing the straight line of LEDs far from the magnet in the dipolar configuration, using a cylindrical magnet with a length of 1.50 cm, and a diameter of 1.30 cm.

**Figure 11.**The general pattern of the orbits does not depend on the wavelength of the light source. The same pattern is observed for the dipolar configuration of Figure 9b for the color red in (

**a**) and the color blue in (

**b**), using a cylindrical magnet with a length of 1.50 cm, and a diameter of 1.30 cm.

**Figure 12.**Light diffraction phenomenon with collimated and uncollimated light observed simultaneously (

**a**). The laser beam (coherent and collimated source of light) passing through the Ferrolens, while the formation of an elliptical light pattern in the plane of the Ferrolens is caused by light scattering of non-collimated light. The formation of patterns is a two-step process. This can be observed simultaneously in this figure, in which the first step is the diffraction of the green laser light through the Ferrolens in the presence of a magnetic field, projecting a curved line in the screen, with an intense light spot obtained due to the direct projection of the laser. The next step is obtained by the light spot of the laser hitting the screen, changing it to an uncollimated light source. In (

**b**) a diagram illustrates this.

**Figure 13.**We have in this image a comparison between a simulation of magnetic potential and the experiment using Ferrolens with various light sources of different colors. In (

**a**), the top view of the magnetic potential of three magnets with their paired poles, in (

**b**) the same arrangement observed through the Ferrolens. The size of the neodymium cube magnets is 9.5 mm, and the ferrofluid layer is 6.0 mm above the magnets.

**Figure 14.**Magnetochiral effect. This set up has a bright incandescent lamp at 45 degrees incidence angle to the surface of the Ferrolens, and three identical cylinder magnets, in a south-north-south configuration. The north pole of a 12.7 mm diameter neodymium cylinder is directly under the Ferrolens, and we are presenting the light pattern of this central magnet in (

**a**) and its respective diagram of observation in (

**c**). For different angles of observation, this pattern suffers a distortion as shown in (

**b**) and in (

**d**).

**Figure 15.**Example of the analogy between the phase space of a pendulum and the diffracted lines obtained from our experiment, with two centers and one saddle point.

**Figure 16.**Trajectories observed in the polar configuration for the different points of view. The complete diagram of the trajectories is presented in (

**a**). In (

**b**) there is a limit cycle with the label 1, in (

**c**) the separatrix is represented by the green line in the diagram with the label 2, and open trajectories with labels 3 and 4 are presented in (

**d**) and (

**e**), respectively.

**Figure 17.**Example of the structure of an attractor. The attractor obtained with a tetrapolar configuration is presented in (

**a**). The trajectories scape from the white disc of light, passing close to the magnets, and they return to the white disc, closing the circuit, forming a ribbon of light. This configuration develops the curved structure inside the attractor, and in this part of the attractor, the ribbon of light gives half a twist, like the case of a Möbius strip. In (

**b**) we present a top view of this curved structure close to the magnets.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tufaile, A.; Vanderelli, T.A.; Snyder, M.; Tufaile, A.P.B.
Observing Dynamical Systems Using Magneto-Controlled Diffraction. *Condens. Matter* **2019**, *4*, 35.
https://doi.org/10.3390/condmat4020035

**AMA Style**

Tufaile A, Vanderelli TA, Snyder M, Tufaile APB.
Observing Dynamical Systems Using Magneto-Controlled Diffraction. *Condensed Matter*. 2019; 4(2):35.
https://doi.org/10.3390/condmat4020035

**Chicago/Turabian Style**

Tufaile, Alberto, Timm A. Vanderelli, Michael Snyder, and Adriana Pedrosa Biscaia Tufaile.
2019. "Observing Dynamical Systems Using Magneto-Controlled Diffraction" *Condensed Matter* 4, no. 2: 35.
https://doi.org/10.3390/condmat4020035