Jumping Sundogs, Cat's Eye and Ferrofluids

Article Jumping Sundogs, Cat’s Eye and Ferroﬂuids

Alberto Tufaile 1,* , Michael Snyder 2 , Timm A. Vanderelli 3 and Adriana Pedrosa Biscaia Tufaile 1

1 Soft Matter Lab, School of Arts, Sciences and Humanities, University of São Paulo, São Paulo 03828-000, Brazil; [email protected] 2 Technical Space Science Center, 235 Martindale Drive, Morehead State University, Morehead, KY 40531, USA; [email protected] 3 Ferrocell USA, 739 Route 259, Ligonier, PA 15658, USA; [email protected] * Correspondence: [email protected]

Received: 20 April 2020; Accepted: 2 July 2020; Published: 8 July 2020

Abstract: We have explored some features of the complex fluids present in Earth’s atmosphere by the observation of some optical phenomena and compared them to the optical phenomena observed in gems and magnetic materials. The main feature of a complex fluid is that it contains polyatomic structures such as polymer molecules or colloidal grains. This paper includes some setups using tabletop experiments, which are intended to show concretely the principles discussed, giving a sense of how well the idealizations treated apply to the atmospheric systems. We have explored sundogs, light pillars, and the halo formation, which involve the existence of a certain structure in the atmospheric medium, resembling the structures observed in some types of gems and ferrofluids.

Keywords: sundogs; crown flash; ferrofluids; complex fluids; polarization; cat’s eye effect

1. Introduction The observation of atmospheric optical phenomena has caused a sense of wonder in many people throughout history, such as the sundogs, crown flashes, parhelic circles, light pillars, or the miracle of the sun. All of these phenomena are rare, and we can hardly predict their occurrence. However, the technological development that allows for the availability of cameras and camcorders, associated with the dissemination of the recordings of these phenomena on social networks, allows us to try to give a plausible explanation for these rare events. For example, the crown flash in Figure1 in meteorology is an atmospheric optical phenomenon that consists of a luminous column that moves near dense clouds during the day [1,2]. These images of this atypical event were obtained from a YouTube user’s video (QuadM13) observed at Greenwood, Indiana, USA [3], on the same day that another YouTube user’s video from Shelbyville, Indiana, USA (Chris Mercer) captured another video of this jumping sundog on June 12, 2015 at 6:39 pm CDT [4], with a distance of almost 30 km between the two. Clouds can be an example of a complex fluid [5] that changes constantly due to atmospheric conditions, in which a gas phase contains structures such as tiny ice crystals in motion. The existence of such structure for this kind of system distinguishes a cloud from a simple fluid, with properties which allow self-organization in the presence of an electric field or air flow. In the case of a crown flash or jumping sundogs, one possible explanation for this phenomenon to occur is that sun light is interacting through ice crystals above the clouds in the presence of an oscillating electrical field, while in the case of light pillars, the spatial distribution of particles inside this complex fluid creates the observable light pattern.

Condens. Matter 2020, 5, 45; doi:10.3390/condmat5030045 www.mdpi.com/journal/condensedmatter Condens. Matter 2020, 5, 45 2 of 12 Condens. Matter 2020, 5, x FOR PEER REVIEW 3 of 12

Figure 1. 1.The The solar solar halo halo is shown is shown in (a) andin (a the) and laser the halo laser obtained halo using obtained ferrofluid using subjected ferroflu toid a subjectedmagnetic field to a is magnetic shown in (fieldb). The is shown jumping in sundog (b). The phenomenon jumping sundog is shown phenomenon in this sequence is ofshown three picturesin this sequence in (c) in states of three 1, 2 and pictures 3, and thein ( dic)ff erentin states values 1, of2 and the magnetic 3, and t fieldhe different can create values a sequence of the of magneticlight patterns, field as can in (d create). For di aff sequenceerent configurations of light patterns of the magnetic, as in (d fields). For 1, different 2 and 3 of configurations these patterns in (ofd), the we canmagnetic emulate field somes 1, patterns 2 and observed 3 of the inse thepatterns crown flash.in (d), we can emulate some patterns observed in the crown flash. Thus, these properties in such a natural environment, which cause so much astonishment in people,Thus, are these the resultproperties of the in emergence such a natural of light environment patterns in, which a complex cause system so much on astonishment a scale of huge in proportions.people, are the This result feeling of of the wonder emergence can lead of light to a sense patterns of transcendence, in a complex systeminducing on in a many scale people of huge a proportions.deep feeling ofThis connection feeling of withwonder nature, can butlead sometimes to a sense of leads transcendence, to classification inducing of these in many phenomena people as a deepmiracles, feeling omens, of connection conspiracy with theories nature, or but even sometimes an extraterrestrial leads to classif manifestation,ication of these as we phenomena can see in the as miracles,comments omens, made byconspiracy several users theories on the or videoseven a mentionedn extraterrestrial above. manifestation In this way, it, isas interesting we can see to in carry the commentsout relatively made simple by andseveral accessible users on experiments the videos to mentioned demonstrate above some. In of this the mainway, conceptsit is interesting present to in carrythese phenomena.out relatively simple and accessible experiments to demonstrate some of the main concepts presentThe in use these of analogous phenomena. systems can facilitate the presentation and interpretation of these phenomena. See, forThe example, use of the analogous formation systems of the solar can halo facilitate in Figure the 1 presentationa and the formation and interpretation of the laser halo of using these phenomena.ferrofluids in See Figure, for 1example,b. Even consideringthe formation that of the solar first casehalois in mainly Figure related1a and tothe geometric formation optics, of the whilelaser halo the second using caseferrofluids is associated in Figure with 1b. light Even diffraction, considering both systemsthat the show first howcase lightis mainly can be related projected to geometricaround the op lighttics, sourcewhile the due second to its interactioncase is associated with particles with light suspended diffraction in, both a fluid. systems Another show case how is lightthe oscillating can be projected light flash around in Figure the 1lightc, which source can due beassociated to its interaction with di withfferent particles states of suspend the staticed lightin a patternsfluid. Another in Figure case1d, is ifthe we oscillating consider thatlight a flash characteristic in Figure light 1c, which pattern can can be respond associated to changes with different in the statesstate of of some the static type oflight field patterns that controls in Figure the movement1d, if we consider of particles that that a characteristic are interacting light with pattern light, as can in respondthe case ofto Figurechanges1d, in showing the state some of some of these type possibleof field that field controls states for the the movement same light of sourceparticle placeds that are in interactingdifferent positions, with light, simultaneously. as in the case of An Figure important 1d, showing result that some can of be these obtained possible with field all states of this for is the same light source placed in different positions, simultaneously. An important result that can be obtained with all of this is the existence of conditions that allow the emergence of a self-organization that manifests itself through the observation of an optical phenomenon on an atmospheric scale.

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existenceCondens. Matter of conditions 2020, 5, x FOR that PEER allow REVIEW the emergence of a self-organization that manifests itself through3 of the 12 observation of an optical phenomenon on an atmospheric scale. InIn this this way, way, this this work work presents presents some some plausible plausible explanations explanations for some for of some these of events these and events connects and themconnect to others them interesting to other interesting systems that systems are based that in are the based same in concepts the same but concepts in diﬀerent but scales,in different using scales, some principlesusing some of principle physicss and of physics material and science. material In thescience. next In section, the next we section present, we our present method; our after method that,; weafter explore that, we the explore phenomenon the phenomenon of light pillars of light and pillars compared and compared them with them the with light the patterns light patterns in gems. in Ingems. Section In 4S,ection we investigate 4, we investigate the polarization the polarization patterns patterns of ferroﬂuid of ferrofluid and related and patterns related in patterns crystals. in Finally,crystals. we Finally, present we our present conclusions our conclusion in Sections in5. Section 5.

2.2. Materials and Methods InIn orderorder toto simulatesimulate somesome opticaloptical phenomena,phenomena, wewe havehave usedused materialsmaterials withwith magneto-opticalmagneto-optical properties.properties. We havehave usedused aa thinthin filmfilm of ferrofluidferrofluid subjected to an external magnetic field field (Ferrolens) (Ferrolens).. WhenWhen light light interacts interacts with with this this thin thin film film of of ferrofluid, ferrofluid, it it undergoes undergoes di differentfferent e effects,ffects, and and we we cancan observeobserve some some analogies analogies between between the atmosphericthe atmospheric phenomena phenomena and the and magneto-optical the magneto light-optical patterns, light whichpatterns, intend which to show intend some to principles show some involved, principles giving involved, a sense of giving how well a sense the idealizations of how well treated the applyidealizations to the atmospheric treated apply systems to the atmospheric [1,6,7]. systems [1,6,7]. InIn addition addition to to this, this, we we also also have have explored explored the optical the optical properties properties of gems, of such gems, as such the light as the patterns light presentpatterns in present the cat’s in eye the e ffcat’sect and eye light effect polarization and light patterns polarization obtained patterns from obtained gemology, from comparing gemology, the formationcomparing of the patterns formati inon crystals of patterns with thein crystals light polarization with the light patterns polarization in ferrofluids patterns in Figurein ferrofluids2. in Figure 2.

FigureFigure 2. 2.Light Light polarization polarization using using ferrofluid: ferrofluid (a) Diagram: (a) Diagram of the of experimental the experimental apparatus apparatus showing inshowing (i) the thin in of (i) film the of thin ferrofluid of film between of ferrofluid two crossed between polarizers two with crossed no magnetic polarizers field and with in (ii), no themagnetic ferrofluid field subjected and in to (ii) an, externalthe ferrofluid magnetic subjected field; (b) Lightto an pattern external in themagnetic ferrofluid field obtained; (b) Light with lightpattern polarization. in the ferrofluid Depending obtained on the configuration with light ofpolarization. the magnetic Depending field, different on patterns the configuration are observed. of the magnetic field, different patterns are observed. We also have used magnetostatic equations to find the solutions related to the light patterns obtainedWe also from have the thinused films magnetostatic of ferrofluid, equations using treatments to find the based solutions on the related idea ofto magneticthe light charges,patterns enablingobtained usfrom to use the the thin representation films of ferrofluid, of magnetic using scalar treatments potentials, based because on the when idea of an magnetic external magnetic charges, fieldenabling is applied us to to use the the ferrofluid, representation the magnetic of magnetic moments scalar of thepotentials nanoparticles, because in when the ferrofluid an exter arenal orientedmagnetic along field the is externalapplied magneticto the ferrofluid, field. Although the magnetic the magnetic moments charges of the are nanoparticles a useful abstraction, in the withferrofluid no physical are oriented existence, along they the are external analogous magnetic to the field. case of Although electric charges the magnetic in electrostatic, charges are making a useful the analysisabstraction of the, with physical no physical system simpler; existence to, deal they with are scalar analogous potentials to the ismuch case more of electric intuitive charges than the in magnetostaticelectrostatic, making case of Amperianthe analysis currents, of the whichphysical leads system to the simpler vector potential.; to deal with In this scalar way, potentials considering is themuch magnetic more intuitive field BM thanproduced the magnetostatic by a magnetic case field ofρ MAmperian: currents, which leads to the vector potential. In this way, considering the magnetic field BM produced by a magnetic field M: → BM = ρM, ∇·∇ ∙ ⃗ = , (1) → (1) BM⃗= 0, ∇∇ ×× = 0,

where M is a fictitious magnetic charge. Following this analogy, we introduce the magnetic scalar potential M, defined by the relation:

⃗ = −∇ (2)

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where ρM is a fictitious magnetic charge. Following this analogy, we introduce the magnetic scalar potential φM, defined by the relation: Condens. Matter 2020, 5, x FOR PEER REVIEW → 3 of 12 BM = φM (2) −∇ To simulateTo simulate the magnetic the magnetic field field around around the themagnets, magnets, we wehave have used used the the computer computer code code Pic2mag Pic2mag [8]. [8 ]. DueDue toto thethe interaction interaction between between the the magnetic magnetic field field and and the the ferrofluid, ferrofluid, some some ordered ordered structures structures are formed,are formed, and and consequently, consequently, there there is a chainis a chain formation formation of magnetic of magnetic particles particles along along the magnetic the magnetic field, similarfield, similar to the caseto the where case therewhere are there iron are fillings iron infillings the presence in the presence of a magnetic of a magnetic field. When field. light When interacts light withinteracts these with structures, these structures, it is scattered, it is formingscattered, light forming patterns. light The patterns. same principleThe same occurs principle when occurs light interactswhen light with interacts the ice with crystals the inice the crystals atmosphere in the atmosphere [6,7]. [6,7]. WeWe havehave used used di ffdifferenterent types types of setups of setups to explore to ex theplore properties the properties of the light of patternsthe light in patterns ferrofluids in andferrofluids compare and them compare with atmosphericthem with atmospheric patterns. We patterns. can observe We can patterns observe in patterns the plane in ofthe Figure plane1 dof directlyFigure 1d in thedirectly thin film in the of ferrofluid thin film [ 1of,9 –ferrofluid11], in which [1,9–11], the source in which of lights the consists source ofof LEDs.lights Theconsists second of caseLEDs. is The laser second transmission case is throughlaser transmission the ferrofluid throug [1,10h ]the subjected ferrofluid to a[1,10] magnetic subjected field to in a Figure magnetic1b. Thefield third in Figure case is 1b. the The light third polarization case is the presented light polarization in Figure2b [presented10,11]. We in were Figure inspired 2b [10,11]. to compare We were the lightinspired patterns to compare of gems the and light ferrofluids patterns by of a demonstrationgems and ferrofluids at the Geo by Gallery,a demonstration Smithsonian at Naturalthe Geo MuseumGallery, Smithsonian of Natural History Natural in Washington,Museum of D.C.,Natural USA History [12], which in Washington, consisted of D.C., grids USA of lines [12], with which one whiteconsisted LED of as grids the source of lines of with light. one white LED as the source of light.

3.3. Light Pillars and Chatoyancy Initially,Initially, wewe willwill exploreexplore thethe interestinginteresting casecase ofof lightlight pillars,pillars, inin whichwhich columnscolumns ofof lightlight emanateemanate fromfrom aa lightlight source,source, becausebecause thethe iceice crystalscrystals inin thethe atmosphereatmosphere reﬂectreflect thethe sourcesource ofof lightlight towardstowards thethe viewer,viewer, asas can can be be seen seen in in Figure Figure3a,b 3a,b [ 6 ].[6]. The The same same principle principle of theseof these bands bands of lightof light can can befound be found at the at phenomenonthe phenomenon known known as chatoyancy, as chatoyancy, where where we can we see ca narrown see narrow bands ofbands light of within light gemstones,within gemstones, like the onelike inthe Figure one in3c. Figure 3c.

FigureFigure 3. 3.Pillars Pillars and and chatoyancy: chatoyancy: (a) Two (a) Two views views of light of pillars light formed pillars byformed the interaction by the interaction of street lights of withstreet di fflightserent colorswith different and the ice colors crystals and floating the ice in thecrystals atmosphere; floating (b )in the the light atmosphere; pillars are observed (b) the mainlylight pillars at night, are while observed during mainly the day, at there night, are wh sunile pillars; during (c) athe gem day, with there a light are stripe sun knowpillars; as cat’s(c) a eye.gem In with both a cases, light there stripe is a structuredknow as cat’s material eye. reflecting In both the cases, light tothere the viewer.is a structured material reflecting the light to the viewer. We can find gemstones with single bands patterns like the slits in cat’s eye, but on other gemstones, the patternsWe can form find stars gemstones with multiple with single rays. Thesebands beautiful patterns optical like the eff ectsslits are in createdcat’s eye, in many but on cases other by lightgemstones, reflected the in patterns beams of form needle-like stars with crystals, multiple or light rays. reflected These beautiful in tiny hollow optical tubes effects within are created a crystal. in manyYet, cases what by islight the reflected physical in concept beams which of needle-like gives the crystals, connection or light between reflected these in twotiny phenomena?hollow tubes Whatwithin are a thecrystal. properties in the atmosphere behind these light pillars which make possible the emergence of theseYet, patterns? what is the The physical answer concept of the first which question gives isthe the connection law of reflection, between while these the two answer phenomena? for the secondWhat are question the properties suggests the in existencethe atmosphere of a flat hexagonalbehind these plate light assembly pillars of icewhich crystals make filling possible the space the betweenemergence the of light these sources patterns? and the The viewer. answer The of existence the first ofquestion these floating is the icelaw plates of reflection, depends onwhile several the atmosphericanswer for the factors second that question vary during suggests the year, the exis suchtence as the of temperaturea flat hexagonal and theplate humidity, assembly but ofcan ice remaincrystals present filling the for space a period between of time, the which light sources allows theand observation the viewer. ofThe the existence phenomenon. of these For floating example, ice plates depends on several atmospheric factors that vary during the year, such as the temperature and the humidity, but can remain present for a period of time, which allows the observation of the phenomenon. For example, the formation of ice needles occurs mainly within the temperature range from −10oC to 5oC, and for the water vapor density around 0.15 g/m3, outside of the region of water saturation, according to the Nakaya diagram [12] of the shapes of ice crystals. In the case of the

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Condens. Matter 2020, 5, x FOR PEER REVIEW 3 of 12 the formation of ice needles occurs mainly within the temperature range from 10 ◦C to 5 ◦C, and for 3 − theformation water vapor of plates, density we have around found 0.15 in g/ mthe, outsidediagram of previously the region cited of water that saturation, the temperature according is around to the Nakaya−15 oC and diagram the water [12] ofvapor the shapesdensity of is ice 0.03 crystals. g/m3, inside In the the case region of the formationof water saturation. of plates, weThere have is foundalso a indynamic the diagram balance previously between citedthe formation that the temperature and falling isof around these ice15 plates◦C and filling the waterthe space vapor between density the is 3 − 0.03light g source/m , inside and thethe region observer of water from saturation.the ground There level, is and also go a dynamic upwards, balance creating between a complex the formation fluid in andmotion. falling of these ice plates filling the space between the light source and the observer from the groundOne level, simple and experiment go upwards, showing creating this a complexoptical effect fluid of in the motion. light pillars and chatoyancy [13] can be seen Onein Figure simple 4a, experiment when light showing reflects off this parallel optical ewiffectres ofon the the light screen, pillars creating and chatoyancy a single band [13 ]of can light. be seenIn Figure in Figure 4b, we4a, whencan see light a six-rayed reflectso star,ff parallel since the wires source on the of screen,light iscreating reflecting a singlein wires band aligned of light. in Inthree Figure directions.4b, we can In Figure see a six-rayed 4c,d, we star,can see since the the two source cases of from light a isdifferent reflecting perspective. in wires aligned in three directions. In Figure4c,d, we can see the two cases from a di fferent perspective.

Figure 4. 4.The The chatoyance chatoyance at theat Hallthe Hall of Gems of Gems in the Smithsonianin the Smithsonian Museum: Museum: (a) A single (a) LED A single is the sourceLED is of the light source behind of a gridlight of behind parallel a wires, grid formingof parallel a light wires, stripe forming due to the a reflectionlight stripe of lightdue into each the wire;reflection (b) Changing of light the in positioneach wire; of the (b) viewer, Changing the image the position of the light of sourcethe viewer, is displaced, the image but the of light the stripelight remainssource is crossing displaced, it; (c) Thebut samethe elightffect observedstripe remains previously, crossing but now it; we(c) have The threesame crossing effect grids;observed (d) Again, previously, the displacement but now we of the have viewer three moves crossing the pattern grids; in(d the) Again, grid. the displacement of the viewer moves the pattern in the grid. From the point of view of geometric optics, this effect is explained by the law of reflection in FigureFrom5 in thethe casepoint of of chatoyancy, view of geometric because theoptics, reflection this effect still occursis explained if the light by the source law andof reflection the viewer in areFigure at the 5 in same the sidecase ofof thechatoyancy, grid. In thebecause specific the case reflection of chatoyancy, still occurs there if the are microscopiclight source acicularand the inclusionsviewer are in at some the same elbaite side crystals of the forming grid. In the the grid. specific case of chatoyancy, there are microscopic acicularIt is inclusions interesting in to some note thatelbaite the crystals effect is forming still present the grid. if the dimensions of the grid are changed, so that the physical properties of light have to be represented in the realm of physical optics. The light scattering will follow similar patterns. This transition is valid in the case of gems or in the case of the ice crystals. There are some important results that we can obtain from this diagram, as for example, from the point of view of the observer, the image of the source of light will be in the direction of the streak of light, no matter the position of the observer, if the grid is normal to the direction of the light propagation. In the case of oblique incidence of light on the grid elements, the source of light will still be in the light streak, but this light streak will not be necessarily be a straight line, but a curve like a parabola or a circle. For triangular or hexagonal structures, the light patterns may have the crossing of some lines as in the case of Figure4c,d, as discussed in some of our previous papers [ 6,7]. This effect, called asterism, can be observed in the sapphire “Star of Asia” [14] at the Smithsonian Museum, because sapphires are a gem variety of the mineral corundum. This pattern of a six-rayed star forms because the titanium atoms are trapped within the growing corundum crystal, forming needlelike crystals of the mineral rutile, oriented in three directions.

Figure 5. The cat’s eye effect and light pillars: Essentially, this diagram presents the light rays travelling from the source of light to the observer, subjected to the existence of the

Condens. Matter 2020, 5, x FOR PEER REVIEW 3 of 12 formation of plates, we have found in the diagram previously cited that the temperature is around −15 oC and the water vapor density is 0.03 g/m3, inside the region of water saturation. There is also a dynamic balance between the formation and falling of these ice plates filling the space between the light source and the observer from the ground level, and go upwards, creating a complex fluid in motion. One simple experiment showing this optical effect of the light pillars and chatoyancy [13] can be seen in Figure 4a, when light reflects off parallel wires on the screen, creating a single band of light. In Figure 4b, we can see a six-rayed star, since the source of light is reflecting in wires aligned in three directions. In Figure 4c,d, we can see the two cases from a different perspective.

Figure 4. The chatoyance at the Hall of Gems in the Smithsonian Museum: (a) A single LED is the source of light behind a grid of parallel wires, forming a light stripe due to the reflection of light in each wire; (b) Changing the position of the viewer, the image of the light source is displaced, but the light stripe remains crossing it; (c) The same effect observed previously, but now we have three crossing grids; (d) Again, the displacement of the viewer moves the pattern in the grid.

From the point of view of geometric optics, this effect is explained by the law of reflection in Figure 5 in the case of chatoyancy, because the reflection still occurs if the light source and the viewer are at the same side of the grid. In the specific case of chatoyancy, there are microscopic acicularCondens. Matterinclusions2020, 5 ,in 45 some elbaite crystals forming the grid. 6 of 12 Condens. Matter 2020, 5, x FOR PEER REVIEW 3 of 12

grid for two different positions of the viewer—one at the top of the diagram (1), and another in the middle of the diagram (2).

It is interesting to note that the effect is still present if the dimensions of the grid are changed, so that the physical properties of light have to be represented in the realm of physical optics. The light scattering will follow similar patterns. This transition is valid in the case of gems or in the case of the ice crystals. There are some important results that we can obtain from this diagram, as for example, from the point of view of the observer, the image of the source of light will be in the direction of the streak of light, no matter the position of the observer, if the grid is normal to the direction of the light propagation. In the case of oblique incidence of light on the grid elements, the source of light will still be in the light streak, but this light streak will not be necessarily be a straight line, but a curve like a parabola or a circle. For triangular or hexagonal structures, the light patterns may have the crossing of some lines as in the case of Figure 4c,d, as discussed in some of our previous papers [6,7]. This effect, called asterism, can be observed in the sapphire “Star of Asia” [14] at the Smithsonian FigureFigure 5.5.The The cat’s cat’s eye eye effect effect and lightand pillars:light pillars: Essentially, Esse thisntially, diagram this presents diagram the presents light rays the travelling light Museum, because sapphires are a gem variety of the mineral corundum. This pattern of a six-rayed raysfrom thetravelling source of from light the to the source observer, of subjectedlight to the to the observer, existence subjected of the grid forto the two existence different positions of the star forms because the titanium atoms are trapped within the growing corundum crystal, forming of the viewer—one at the top of the diagram (1), and another in the middle of the diagram (2). needlelike crystals of the mineral rutile, oriented in three directions. A similar eeffectffect cancan bebe observedobserved usingusing aa thinthin filmfilm of of ferrofluid ferrofluid subjected subjected to to a a magnetic magnetic field field and and a alaser laser beam, beam, as as is is shown shown in in Figure Figure6. With6. With no no magnetic magnetic field, field, the the ferrofluid ferrofluid has has the the brownish brownish aspect aspect of ofFigure Figure6a, 6a, and and in Figurein Figure6c, we6c, seewe see the the laser laser projected projected on aon screen a screen after after going going through through this this thin thin film film of offerrofluid. ferrofluid. When When a magnetic a magnetic field field of 300 of gauss 300 isgaus applied,s is applied, we can observewe can theobserve formation the formation of needlelike of needlelikestructures instructures Figure6b. in The Figure laser undergoes6b. The laser di ff ractionundergoes when diffraction it passes through when theit passes ferrofluid through subjected the ferrofluidto this magnetic subjected field, to as this we canmagnetic see in Figurefield, as6d. we The can diagram see in ofFigure Figure 6d.7a representsThe diagram the eof ffFigureect of the7a representslaser beam the associated effect of with the laser the vector beamp associatedhitting the with nanoneedle the vector aligned p hitting with the the nanoneedle magnetic field alignedBM, withcreating the themagnetic diffraction field DB:M, creating the diffraction D: → → D⃗= →p BM⃗ (3) 𝐷 =𝑝⃗ ××𝐵 (3)

Figure 6.6.Ferrofluid Ferrofluid and and light light diff diffraction:raction: (a) Image (a) Image of the of ferrofluid the ferrofluid with no with magnetic no magnetic field using field an usingoptical microscope;an optical (microscope;b) Image of the (b ferrofluid) Image withof the a magnetic ferrofluid field with of 600 a G magnetic using the samefieldmicrospore of 600 G usingof the previousthe same case; microspore (c) Light patternof the previous on a screen case; in the (c case) Light of a pattern laser passing on a screen through in the the ferrofluid case of witha laser no passing magnetic through field; (d) the Light ferrofluid diffraction with pattern no obtainedmagnetic with field; the ( ferrofluidd) Light systemdiffraction of (b ).pattern obtained with the ferrofluid system of (b).

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According to the Geometrical Theory of Diffraction [6,7], when a light beam hits a needlelike structure, there is a cone of diffracted rays with the cross section that obeys the equation:

𝐷⃗(𝑟) =𝐾𝑝⃗𝑟/𝑒 (4) where K is the diffraction coefficient, r is the distance between the ferrofluid and the screen, and k =2π/λ is the wave number of the incident field with wavelength λ. Considering the z coordinate of the polar system for the ferrofluid needle axis aligned with the magnetic field BM, a conical scattering is observed instead of the circle scattering of Figure 6b. The tilt between the incident vector p and the axis of the nanoneedle is φ, with the light being scattered along the surface of a cone with an apical angle of (180o−2φ), as is shown in Figure 6c. We have explored these effects and compared them to Condens.the parhelic Matter circle,2020, 5, a 45 type of halo observed during the day in some weather conditions [6]. 7 of 12

Figure 7.7.Light Light interacting interacting with with the the magnetic magnetic field field in the in ferrofluid: the ferrofluid: (a) The ( propagationa) The propagation vector →p vector 𝑝⃗ diffracts in the→ direction 𝐷⃗, due to the magnetic→ field𝐵⃗; (b) Diagram of the light diffracts in the direction D, due to the magnetic field BM;(b) Diagram of the light scattering in a nanoneedlescattering in aligned a nanoneedle with themagnetic aligned field;with (thec) Conical magnetic diff raction.field; (c) Conical diffraction.

AccordingIn other words, to the the Geometrical light scattering Theory in of Ditheff ractionferrofluid [6,7 ],is whensimilar a lightto the beam one hits observed a needlelike in a structure,cylindrical there rod-like is a conestructure of di ffalongracted the rays direction with the of crossthe propagation, section that obeysgiving the rise equation: to the formation of curved light patterns [6,7]. We can observe this effect of light scattering hitting obliquely the → 1/2 ikr needlelike structures of ferrofluids in FigureD(r) 7.= ThK→isp rmechanism− e is like the one observed in the grids(4) of Figure 4 related with the cat’s eye effect, and similar to same case of light scattering in ice crystals wherein the caseK is theof the diff ractionjumping coe sundogsfficient, r[2].is theWhen distance the magn betweenetic field the ferrofluid in ferrofluids and the or screen,the electric and kfield= 2π in/λ isclouds the wave changes number their orientation, of the incident we have field the with motion wavelength of the lightλ. Considering pattern of the the jumpingz coordinate sundogs, of theor polarthe jumping system laserdogs for the ferrofluid [1] of Figure needle 8, axis in the aligned case withof ferrofluid. the magnetic The use field ofB ferrofluidsM, a conical in scattering this work is observedreinforces insteadsome interesting of the circle properties scattering present of Figure in our6b. system. The tilt betweenFor example, the incident the optical vector responsep and of the a axisthin offilm the containing nanoneedle ferrofluid is φ, with is not the linear, light being as wa scattereds already alongreported the in surface our previous of a cone work with [10], an apical such angleas the of birefringence (180 2φ), asshown is shown in Figure in Figure 3 6ofc. the We haveprevious explored reference. these eBesidesffects and the compared birefringence them or to ◦ − theretardance, parhelic the circle, magneto-optics a type of halo of observed ferrofluids during coul thed involve day in someFaraday weather rotation, conditions the Kerr [6]. effect, and linearIn dichroism. other words, If we the compare light scattering Figure in1d thewith ferrofluid Figure 6d, is similarwe can to see the that one the observed light source in a cylindrical has quite rod-likedifferent structure relative positions along the with direction respect of to the diffracted propagation, light. giving While risein the to thefirst formation case, the diffracted of curved light light patternsis around [6 the,7]. laser, We can in observethe second this case, effect the of diffr lightacted scattering light emanates hitting obliquely directly thefrom needlelike the laser structuresbeam. of ferrofluids in Figure7. This mechanism is like the one observed in the grids of Figure4 related with the cat’s eye effect, and similar to same case of light scattering in ice crystals in the case of the jumping sundogs [2]. When the magnetic field in ferrofluids or the electric field in clouds changes their orientation, we have the motion of the light pattern of the jumping sundogs, or the jumping laserdogs [1] of Figure8, in the case of ferrofluid. The use of ferrofluids in this work reinforces some interesting properties present in our system. For example, the optical response of a thin film containing ferrofluid is not linear, as was already reported in our previous work [10], such as the birefringence shown in Figure3 of the previous reference. Besides the birefringence or retardance, the magneto-optics of ferrofluids could involve Faraday rotation, the Kerr effect, and linear dichroism. If we compare Figure1d with Figure6d, we can see that the light source has quite di fferent relative positions with respect to diffracted light. While in the first case, the diffracted light is around the laser, in the second case, the diffracted light emanates directly from the laser beam. Because this magneto-optical system exhibits a variety of effects, depending on the magnitude of the magnetic field, the magnetic nanoparticles can assemble in very different ways. For example, in Figure9, we can see some interesting light patterns for two di fferent magnetic field configurations, using, as the source of light, a ring of LEDs with different colors. Condens. Matter 2020, 5, x FOR PEER REVIEW 3 of 12

Condens. Matter 2020, 5, 45 8 of 12 Condens. Matter 2020, 5, x FOR PEER REVIEW 3 of 12

Figure 8. Three images of light diffraction controlled by the magnetic field of 600 G: (a) Light scattering obliquely in the nanoneedle for different angles between light propagation and magnetic field orientation at 83°; (b) 64°; (c) 32°.

Because this magneto-optical system exhibits a variety of effects, depending on the magnitude of theFigure magnetic 8. 8.Three Three field, images imagesthe ofmagnetic light of light diff ractionnanoparticles diffraction controlled controll can by assemble theed magnetic by thein fieldvery magnetic of different 600 G: field (a )ways. Light of 600 scatteringFor G: example, (a) in FigureobliquelyLight scattering9, inwe the nanoneedlecan obliquely see forsome in di theff erentinteresting nanoneedle angles between li forght different lightpatterns propagation angles for betweentwo and magneticdifferent light field propagation magnetic orientation field configurations,atand 83 ◦magnetic;(b) 64 using,◦;( cfield) 32 as◦ .orientation the source atof 83°;light, (b a) 64°;ring (ofc) LEDs32°. with different colors.

Because this magneto-optical system exhibits a variety of effects, depending on the magnitude of the magnetic field, the magnetic nanoparticles can assemble in very different ways. For example, in Figure 9, we can see some interesting light patterns for two different magnetic field configurations, using, as the source of light, a ring of LEDs with different colors.

Figure 9.9.Magnetostatic Magnetostatic and and magneto-optics magneto-optics of ferroﬂuids: of ferrofluids: (a) Simulation (a) Simulation of isopotentials of isopotentials of three cylindricalof three cylindrical magnets using magnets Pic2Mag using software; Pic2Mag (b) Simulation software; of(b isopotentials) Simulation of of six isopotentials cylindrical magnets; of six (cylindricalc) Light patterns magnets; obtained (c) Light from apatterns Ferrolens obtained with three from magnets; a Ferrolens (d) Light with patterns three obtained magnets; from (d a) FerrolensLight patterns with six obtained magnets. from a Ferrolens with six magnets.

4. Polarization Figure 9. Magnetostatic and magneto-optics of ferrofluids: (a) Simulation of isopotentials of three cylindrical magnets using Pic2Mag software; (b) Simulation of isopotentials of six Another interesting parallel between ferrofluids,ferrofluids, clouds and some gems is the patterns obtained cylindrical magnets; (c) Light patterns obtained from a Ferrolens with three magnets; (d) using light polarization, because because the the polarization polarization effects effects map map the the orientation orientation of of the the particles particles in in both both a Light patterns obtained from a Ferrolens with six magnets. acrystal crystal and and a complex a complex fluid fluid [15-19]. [15–19 Ferrofluids]. Ferrofluids are are liquids liquids where where birefringence birefringence can can be explored be explored by byan anexternal external magnetic magnetic field field using using a polariscope a polariscope that that examines examines the the sample sample placed placed between between a a pair of 4. Polarization polarizers. Placing the thin film film of ferrofluidferrofluid between two crossed polarizers, we can observe some eeffectsffectsAnother of light interesting polarizationpolarization parallel for didifferent betweenfferent configurationsconfigurations ferrofluids, clouds of thethe and magneticmagnetic some gems field.field. is For the thispatterns case, obtained we have usedusing a light white polarization, lightlight sourcesource because usingusing anan the LEDLED polarization panelpanel light light effects (12 (12 W), W), map illuminating illuminating the orientation over over theof the the complete complete particles surface surface in both of of a square of the thin film of ferrofluid (22 22 mm) homogeneously, with a methacrylate diffuser using acrystal square and of athe complex thin film fluid of [15-19].ferrofluid Ferrofluids ×(22 × 22 mm)are liquids homogeneously, where birefringence with a methacrylate can be explored diffuser by backlightingan external magnetic technique. field On theusing other a polariscope hand, the conoscopy that examines allows the exploring sample ofplaced the light between polarization a pair inof gemspolarizers. [17]. Placing the thin film of ferrofluid between two crossed polarizers, we can observe some effects of light polarization for different configurations of the magnetic field. For this case, we have used a white light source using an LED panel light (12 W), illuminating over the complete surface of a square of the thin film of ferrofluid (22 × 22 mm) homogeneously, with a methacrylate diffuser

Condens. Matter 2020, 5, x FOR PEER REVIEW 3 of 12 using backlighting technique. On the other hand, the conoscopy allows exploring of the light Condens. Matter 2020, 5, 45 9 of 12 polarization in gems [17]. The physical concept connecting both the physical systems is the presence of the birefringence. In theThe case physical of the uniaxial concept crystal, connecting the refractive both the physicalindices n systemsx and ny have is the the presence same value, of the represented birefringence. in Inthe the index case ellipsoidal of the uniaxial of Figure crystal, 10a, the with refractive the polari indiceszationn xmodesand n yaroundhave the the same optic value, axis illustrated represented in inFigure the index 10b, while ellipsoidal in a biaxial of Figure crystal, 10a, withthe three the polarization indices are modesdifferent. around In Figure the optic 10c, we axis present illustrated the incomparison Figure 10 b,between while in conoscopic a biaxial crystal, pattern the in three a quar indicestz crystal are diff erent.and the In Figurelight polarization10c, we present in the comparisonferrofluid using between orthoscopic conoscopic illumination. pattern in a quartzThe tw crystalo arms and of thethe lightcross polarization form the inisogyres, the ferrofluid while usingmelatope orthoscopic is the point illumination. where the Theisogyres two armscross ofeach the other cross in form the thecase isogyres, of gems while in Figure melatope 10c, with is the a pointsimilar where pattern the obtained isogyres in cross Figure each 10d other in the in case the of case the of ferrofluid gems in subjected Figure 10 c,to withan axially a similar magnetized pattern obtainedring. in Figure 10d in the case of the ferrofluid subjected to an axially magnetized ring.

Figure 10. 10.Polarization Polarization and and conoscopic conoscopic pattern: pattern: (a) Ellipsoid (a) Ellipsoid that shows that theshows orientation the orientation of refractive of indicesrefractive (nx, nindicesy and nz )( innx, a uniaxialny and crystalnz) in known a uniaxial as indicatrix; crystal (b ) Theknown polarization as indicatrix; modes around (b) The the opticpolarization axis; (c) Quartzmodes conoscopy around showingthe optic isogyres axis; and(c) Quartz melatope conoscopy at the center; showing (d) Light isogyres polarization and in amelatope thin film ofat ferrofluidthe center; subjected (d) Light to an polarization axially magnetized in a thin ring. film of ferrofluid subjected to an axially magnetized ring. While the ferrofluid system used orthoscopic illumination, with a plane wave, the light pattern obtainedWhile with the quartz ferrofluid is obtained system with used conoscopic orthoscopic illumination illumination, [17], with represented a plane bywave, a series the light of concentric pattern lightobtained cones. with To examinequartz is a diobtainedfferent magnetic with conoscopic field configuration, illumination we present[17], represented in Figure 11 bya thea series diagram of ofconcentric the intensity light of cones. a magnetic To examine field of a the different diametrically magnetic magnetized field configuration, ring, and in we Figure present 11b,c, in theFigure pattern 11a obtainedthe diagram with of a the Ferrolens. intensity of a magnetic field of the diametrically magnetized ring, and in Figure 11b,c,In the Figure pattern 11 ,obtained we present with thea Ferrolens. magnetic field intensity diagram of a magnet with diametric magnetization. The light polarization patterns of this magnet ring with the ferrofluid were obtained in Figure 11b,c; they were obtained by rotating the ring around its axis by an angle of 45 degrees. We have obtained a model of the light intensity and the components of the magnetic field in the case of light polarization [9] given by:

2 I(x, y, z) = B(x, y, z) , (5) (x2+y2)/w2(z) iϕ(z) B(x, y, z) = B0Hm(x)Hn(y)e− e , in which, I(x,y,z) is the light intensity directly observed from the polariscope with the sample of ferrofluid. B0 is the maximal intensity of the magnetic field, Hm(x) and Hn(y) are the spatial derivatives of the magnetic field functions in the x-axis and y-axis, respectively, in the case of the isogyres-like pattern of Figure 12a. Using Equation (1), we can obtain the simulation of the light intensity in

Condens. Matter 2020, 5, x FOR PEER REVIEW 3 of 12

Condens. Matter 2020, 5, 45 10 of 12

Figure 12b for the case presented in Figure 12a. In Figure 12c we present the light pattern obtained for theCondens. dipolar Matter conﬁguration, 2020, 5, x FOR PEER with REVIEW its respective simulation in Figure 12d. 3 of 12

Figure 11. Biaxial polarization: (a) The magnetic field intensity diagram of a magnet with a diametrically magnetization, with a line indicating the minimum field axis; (b) Polarization pattern obtained with the ring magnet oriented obliquely; (c) Polarization pattern with the magnetic ring axis parallel to the horizontal direction.

ferrofluid. B0 is the maximal intensity of the magnetic field, Hm(x) and Hn(y) are the spatial Figurederivatives 11. 11.Biaxial Biaxialof the polarization: magnetic polarization: field (a) Thefunctions (a) magneticThe magnetic in the ﬁeld x-axis intensity field and intensity diagram y-axis, respectively, ofdiagram a magnet of with ina magnetthe a diametrically case withof the a magnetization,isogyres-likediametrically pattern withmagnetization,a of line Figure indicating 12 a.with Using the minimuma Equation line indicating ﬁeld (1), axis;we can (b )the Polarizationobtain mi thenimum simulation pattern field obtained ofaxis; the light with(b) theintensityPolarization ring magnet in Figure pattern oriented 12b forobtained obliquely; the case presentedwith (c) Polarization the inring Figu magnet patternre 12a. withIn oriented Figure the magnetic 12c obliquely; we present ring axis( thec) parallelPolarization light pattern to the horizontalobtainedpattern with for direction. the the dipolar magnetic configuration, ring axis withparallel its respective to the horizontal simulation direction. in Figure 12d.

In Figure 11, we present the magnetic field intensity diagram of a magnet with diametric magnetization. The light polarization patterns of this magnet ring with the ferrofluid were obtained in Figure 11b,c; they were obtained by rotating the ring around its axis by an angle of 45 degrees. We have obtained a model of the light intensity and the components of the magnetic field in the case of light polarization [9] given by: 𝐼(𝑥, 𝑦,) z = |𝐵(𝑥, 𝑦,| 𝑧) , ()/() () (5) 𝐵(𝑥, 𝑦,) 𝑧 =𝐵𝐻(𝑥)𝐻(𝑦)𝑒 𝑒 , in which, I(x,y,z) is the light intensity directly observed from the polariscope with the sample of ferrofluid. B0 is the maximal intensity of the magnetic field, Hm(x) and Hn(y) are the spatial derivatives of the magnetic field functions in the x-axis and y-axis, respectively, in the case of the isogyres-like pattern of Figure 12a. Using Equation (1), we can obtain the simulation of the light intensity in Figure 12b for the case presented in Figure 12a. In Figure 12c we present the light pattern obtained for the dipolar configuration, with its respective simulation in Figure 12d.

Figure 12. Simulation of the light polarization patterns: (a) Light pattern for the unipolar configuration with isogyres and melatope; (b) Simulation of the light intensity for the unipolar configuration of the previous case; (c) Light pattern for the dipolar configuration with isogyres and melatope in a different configuration; (d) Simulation of the light intensity for the polarization in the case of light dipolar configuration.

In this way, the thin film of ferrofluid possesses the ability to rotate the plane of polarization of light passing through them. For more information about optical properties of light scattering in structured fluids, ferrofluids and Ferrocell (Ferrolens), we recommend some references [20–22].

Condens. Matter 2020, 5, 45 11 of 12

5. Conclusions Stimulated by the report of sundogs and light pillars, we have explored some patterns in ferrofluids and crystals. The main idea involves the existence of certain structures in the atmospheric medium, resembling the patterns observed in some types of gems and ferrofluids, along with some properties of light polarization related to birefringence in ferrofluid and crystals. The light scattering in the ferrofluid is similar to the one observed in a cylindrical rod-like structure along the direction of the propagation, giving rise to the formation of curved light patterns. In the case of jumping sundogs, one possible explanation for this phenomenon is the motion of ice crystals above the clouds controlled by an oscillating electrical field, reflecting the sun light. One important aspect of this work is the versatility of the ferrofluid system, which is used as an analog computer to explore patterns, with the change of some optical parameters of the ferrofluid such as birefringence, ferrofluid concentration, and orientation of the magnetic field. We have explored some features of the complex fluids of ferrofluids subjected to a magnetic field and compared them to the optical phenomena present in some cases of isogyres observed using conoscopy in gems. One important feature of complex fluids is the presence of polyatomic structures such as polymer molecules or colloidal grains. In the case of the atmospheric optics, the thermodynamics related to the Nakaya diagram controls the shape of the ice crystals floating in the atmosphere, and consequently, enables the existence of light patterns, while in the case of the ferrofluid, the physical properties of the fluid along with the intensity and direction of the magnetic field define the type of light scattering. In order to illustrate what is happening in the light pillars, we have presented some tabletop experiments, which are intended to show concretely the principles discussed, giving a sense of how well the idealizations like chatoyancy can be applied to the atmospheric systems.

Author Contributions: All authors contributed to prepare the experiments, propose the mechanisms involved, and analyze the results. The preparation of this manuscript was done mainly by A.T. and A.P.B.T. Conceptualization was done A.T. and A.P.B.T.; methodology, A.T. and A.P.B.T.; software, A.T., A.P.B.T. and M.S. Editing was done by A.T., A.P.B.T., M.S. and T.A.V. All authors have read and agreed to the published version of the manuscript. Funding: This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos (INCT-FCx), and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) FAPESP/CNPq#573560/2008-0. Acknowledgments: A.T. and A.P.B.T. thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos (INCT-FCx), and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP). The authors also thank Tony Klein for the photo of the solar halo. Conflicts of Interest: The authors declare no conflict of interest.

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