S S symmetry

Review Spin Symmetry Breaking: Superparamagnetic and Spin Glass-Like Behavior Observed in Rod-Like Liquid Crystalline Organic Compounds Contacting Nitroxide Radical Spins

Shuichi Sato 1,*, Yoshiaki Uchida 2,* and Rui Tamura 3,*

1 Department of Physics, Osaka Dental University, Hirakata, Osaka 573-1121, Japan 2 Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan 3 Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan * Correspondence: [email protected] (S.S.); [email protected] (Y.U.); [email protected] (R.T.); Tel.: +81-90-2997-3881 (S.S.); +81-6-6850-6256 (Y.U.); +81-77-577-1337 (R.T.)


Received: 24 October 2020; Accepted: 16 November 2020; Published: 20 November 2020


Abstract: Liquid crystalline (LC) organic radicals were expected to show a novel non-linear magnetic response to external magnetic and electric fields due to their coherent collective molecular motion. We have found that a series of chiral and achiral all-organic LC radicals having one or two five-membered cyclic nitroxide radical (PROXYL) units in the core position and, thereby, with a negative dielectric anisotropy exhibit spin glass (SG)-like superparamagnetic features, such as a magnetic hysteresis (referred to as ‘positive magneto-LC effect’), and thermal and impurity effects during a heating and cooling cycle in weak magnetic fields. Furthermore, for the first time, a nonlinear magneto-electric (ME) effect has been detected with respect to one of the LC radicals showing a ferroelectric (chiral Smectic C) phase. The mechanism of the positive magneto-LC effect is proposed and discussed by comparison of our experimental results with the well-known magnetic properties of SG materials and on the basis of the experimental results of a nonlinear ME effect. A recent theoretical study by means of molecular dynamic simulation and density functional theory calculations suggesting the high possibility of conservation of the memory of spin-spin interactions between magnetic moments owing to the ceaseless molecular contacts in the LC and isotropic states is briefly mentioned as well.

Keywords: spin symmetry breaking; magnetic liquid crystals; magneto-LC effect; nitroxide radicals; superparamagnetic domain; spin glass state

1. Introduction Following the birth and establishment of ‘Einsteinian general theory of relativity’ and ‘quantum theory and mechanics’ at the beginning of the 20th century [1], the ‘complexity theory’ developed rapidly since the 1970s is recognized as one of the paradigm shifts or innovations in science in the same century [2–4]. Currently, a concept of the nonlinear and nonequilibrium (or out of equilibrium) complexity theory is known to govern a variety of dynamic behaviors observed in both natural and social sciences [5–9]. In the nonequilibrium complexity system, symmetry breaking takes place easily during transition from a chaotic or dissipative state to another one. The fluctuation in a nonequilibrium state induces a phase transition to trigger the symmetry breaking, and, eventually, the nonlinear amplification of fluctuation leads to dissymmetric circumstances [5–10]. The most dramatic and important event is the birth of the universe by the cosmic inflation followed by the Big

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SymmetryBang starting2020, 12 from, 1910 a quantum fluctuation, according to the ‘uncertainty principle of energy and 2time’ of 29 as a hypothetical explanation [1]. After the selection of ‘matter’ against ‘anti-matter’ resulting from Bangthe CP starting (charge from conjugation a quantum parity fluctuation, symmetry) according violation to the in ‘uncertaintythe universe, principle the important of energy and and familiar time’ asevents a hypothetical on the earth explanation include [ 1].many After body the selection interact ofions ‘matter’ of multiple against ‘anti-matter’elements responsible resulting fromfor the CPnonlinearity (charge conjugation [11]. The intermolecular parity symmetry) interactions violation ca inn create the universe, nonequilibrium the important objects and such familiar as cells events [12], onbubbles the earth [13], include and metastable many body crystals interactions [14], ofwhich multiple may elements have strong responsible links to for the the origin nonlinearity of selected [11]. Thechirality intermolecular of life. Thus, interactions symmetry canbreaking create has nonequilibrium been playing a objects primordial such role as cells in physics, [12], bubbles chemistry, [13], andlife science, metastable and crystals so forth. [14 ], which may have strong links to the origin of selected chirality of life. Thus, symmetryLiquid breaking crystals, has which been are playing defined a primordial as a therma rolel mesophase in physics, chemistry,between crystalline life science, and and isotropic so forth. phases,Liquid and, crystals, hence, can which be regard are defineded as ashigh-temperature a thermal mesophase polymorphs between of crystallinecrystals, are and unique isotropic soft phases,materials and, that hence, combine can anisotropy be regarded and as fluidity. high-temperature From a different polymorphs point of view, of crystals, liquid arecrystalline unique (LC) soft materialsphases are that considered combine to anisotropy be a sort of and complexity fluidity. Fromsystem a diconsistingfferent point of none of view,quilibrium liquid crystallinedynamic states (LC) phasesdue to the are consideredmolecular motion to be a sortand ofthe complexity coherent collecti systemve consisting properties of of nonequilibrium molecules in the dynamic LC state states [8]. dueTherefore, to the molecularthey are so motion sensitive and to the external coherent stimuli, collective such properties as electric of or molecules magnetic in field, the LCheat, state light, [8]. Therefore,temperature, they pressure, are so sensitiveand added to chiral external dopants, stimuli, that such the asLC electric superstructure or magnetic can field,easily heat,be altered light, temperature,and the change pressure, can be and controlled added chiral [15,16]. dopants, In this that context, the LC it superstructure seems challenging can easily and bepromising altered and to thedevelop change organic can be soft controlled materials [15 that,16 can]. In show this context,a novel complexity it seems challenging phenomenon and in promising response to to developvarious organicexternal softstimuli materials by making that can use show of the a LC novel environment. complexity phenomenon in response to various external stimuliIn bythis making connection, use of magnetic the LC environment. LC compounds had attracted great interest as soft materials to enhanceIn this the connection, effect of magnetic magnetic fields LC compounds on the electr hadic attracted and optical great properties interest as softof liquid materials crystals. to enhance They thewere eff ectanticipated of magnetic to fieldsexhibit on unique the electric magnetic and optical interactions properties and, of liquid thereby, crystals. unconventional They were anticipated magneto- toelectric exhibit [17–20] unique or magneticmagneto-optical interactions properties and, thereby,in the LC unconventional state [21–23]. However, magneto-electric there had [17 been–20] no or magneto-opticalprominent study propertieson these interesting in the LC subjects state [21 in–23 th].e However,20th century there because had been the majority no prominent of magnetic study onliquid these crystals interesting examined subjects were in of the a highly 20th century viscous because transition the (d-block majority or of f-block) magnetic metal-containing liquid crystals examinedmetallomesogens were of (Figure a highly 1) viscous [24,25], transition which were (d-block not suitable or f-block) for investigation metal-containing of the metallomesogens swift molecular (Figuremotion 1and)[ 24 reorientation,25], which in were the notLC suitablephases at for moderate investigation temperatures of the swift in weak molecular magnetic motion fields. and At reorientationpresent, no appreciable in the LC phasesintermolecular at moderate ferromagn temperaturesetic or superparamagnetic in weak magnetic interaction fields. At (in present, other nowords, appreciable spin symmetry intermolecular breaking) ferromagnetic has been orobserved superparamagnetic in the LC state interaction of d-block (in other or f-block words, spinmetallomesogens. symmetry breaking) has been observed in the LC state of d-block or f-block metallomesogens.

Figure 1. Representative metallomesogens and their LC transition temperatures. Figure 1. Representative metallomesogens and their LC transition temperatures. Only a few all-organic LC radical compounds were prepared until 2003 because it was believed that theOnly geometry a few all-organic and bulkiness LC radical of the radical-stabilizingcompounds were substituentsprepared until were 2003 detrimental because it to was the believed stability ofthat liquid the crystals,geometry which and requiresbulkiness molecular of the radical-st linearityabilizing or planarity substituents [25–27]. Althoughwere detrimental several rod-like to the organicstability LCof liquid compounds crystals, with which a stable requires cyclic molecular nitroxide unitlinearity as the or spin planarity source [25–27]. were prepared Although (Figure several2), theirrod-like molecular organic structures LC compounds were limited with a to stable those cyclic containing nitroxide a nitroxyl unit as group the spin in the source terminal were alkyl prepared chain, away(Figure from 2), their the rigidmolecular core, structures and, thereby, were allowed limited theto those free containing rotation of a the nitrox nitroxylyl group moiety in the inside terminal the molecule,alkyl chain, resulting away from in a verythe rigid small core, dielectric and, thereby, anisotropy allowed (∆ε) ofthe the free whole rotation molecule of the [25 nitroxyl–27]. moiety inside the molecule, resulting in a very small dielectric anisotropy (∆ε) of the whole molecule [25–27].

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Figure 2. Rod-like LC organic mono-radicals synthesized before 2004 and their LC transition Figure 2. Rod-like LC organic mono-radicals synthesized before 2004 and their LC transition temperatures [25–27]. temperatures [25–27]. With this situation in mind, since 2004, we have designed and synthesized a series of With this situation in mind, since 2004, we have designed and synthesized a series of chiral and chiral and achiral all-organic LC nitroxide radicals such as 1–5 having one or two PROXYL achiral all-organic LC nitroxide radicals such as 1–5 having one or two PROXYL (2,2,5,5- (2,2,5,5-tetrassubstituted-1-pyrrolidinyloxyl) units in the core position and, thereby, with a negative tetrassubstituted-1-pyrrolidinyloxyl) units in the core position and, thereby, with a negative dielectric dielectric anisotropy (∆ε < 0) (Figure3)[ 28–44]. Their LC phases and magnetic properties have been anisotropy (∆ε < 0) (Figure 3) [28–44]. Their LC phases and magnetic properties have been fully fully characterized. Consequently, we could observe a magnetic hysteretic behavior between the characterized. Consequently, we could observe a magnetic hysteretic behavior between the heating heating and cooling processes for all of these LC compounds by a SQUID magnetic susceptibility and cooling processes for all of these LC compounds by a SQUID magnetic susceptibility measurement. The molar magnetic susceptibility (χM) always increased at the crystal-to-LC phase measurement. The molar magnetic susceptibility (χM) always increased at the crystal-to-LC phase transition and this χM increase was always preserved during the cooling process. This unique magnetic transition and this χM increase was always preserved during the cooling process. This unique hysteretic behavior was referred to as a ‘positive magneto-LC effect’ [34–36]. Measurement of the magnetic hysteretic behavior was referred to as a ‘positive magneto-LC effect’ [34–36]. Measurement magnetic field (H) dependence of molar magnetization (M) indicates that the observed magnetic of the magnetic field (H) dependence of molar magnetization (M) indicates that the observed hysteresis corresponds to the emergence of superparamagnetic behavior, that is, the partial formation magnetic hysteresis corresponds to the emergence of superparamagnetic behavior, that is, the partial of ferromagnetic domains in the LC phases. These experimental results were contrary to the general formation of ferromagnetic domains in the LC phases. These experimental results were contrary to knowledge that the possibility of a ferromagnetic LC material had been considered unrealistic due the general knowledge that the possibility of a ferromagnetic LC material had been considered to the inaccessibility of long-range spin-spin interactions between rotating molecules in the LC state unrealistic due to the inaccessibility of long-range spin-spin interactions between rotating molecules at ambient or higher temperatures [26]. However, since the positive magneto-LC effect is observed in the LC state at ambient or higher temperatures [26]. However, since the positive magneto-LC effect only for specific PROXYL radicals showing an LC phase (Figure3), it is suggested that the breaking is observed only for specific PROXYL radicals showing an LC phase (Figure 3), it is suggested that of time reversal symmetry of radical spins is likely to have a strong connection with the partially the breaking of time reversal symmetry of radical spins is likely to have a strong connection with the broken space symmetry of LC phases of these compounds. Thus, the observed magnetic behavior partially broken space symmetry of LC phases of these compounds. Thus, the observed magnetic reminds us of a spin glass state. In fact, for example, considerable enhancement of a positive behavior reminds us of a spin glass state. In fact, for example, considerable enhancement of a positive magneto-LC effect or superparamagnetic behavior was noted by adding a small amount (5 to 20 mol%) magneto-LC effect or superparamagnetic behavior was noted by adding a small amount (5 to 20 of racemic cis-diastereomers (R*,R*)-3 [a 1:1 mixture of (R,R)-3 and (S,S)-3] to meso (R,S)-3 as an organic mol%) of racemic cis-diastereomers (R*,R*)-3 [a 1:1 mixture of (R,R)-3 and (S,S)-3] to meso (R,S)-3 as impurity [42]. Such thermal and impurity effects are known to be commonly observed for metallic an organic impurity [42]. Such thermal and impurity effects are known to be commonly observed for spin glass materials. metallic spin glass materials.

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O Me O O O H2n+1CnO Me OCmH2m+1 Me O N Me N O O O Me O N trans-1 O O Me OC16H33 In the case of m = n =13: O 2 O (S,S,S,S)- Racemate C 71.9 SmC 86.1 N 90.3 I C 98.3 N* 105.1 I C 67.2 SmC* 86.0 N* 90.8 C H O (S,S) 16 33 OC16H33 OC16H33

OC16H33 OC16H33 O O HN O O Me C16H33O C16H33O N N H OC16H33 N O N C H O N 16 33 H N Me C16H33O H N OC H Me Me 16 33 O O O C16H33O C16H33O (R,S)-3 (R*,R*)-3

C 83.3 Colh 99.96 I non-LC

C12H25O O Me C H O 12 25 Me N OH O O C12H25O trans-4 trans-5

Racemate Colh 38.6 I Racemate I 117.5 N 91.3 C (monotropic) C 113.4 N* 135.3 I (R,R) Colh 46.9 I (S,S)

FigureFigure 3. Representative3. Representative PROXYL-based PROXYL-based organic organic radicals radi synthesizedcals synthesized since since 2004 and2004 their and LC their transition LC temperatures.transition temperatures Rod-like monoradicals. Rod-like monoradicalstrans-1 [28 trans–40]- and1 [28–40] biradical and biradical (S,S,S,S)- (2S,[S41,S],,S)- discotic2 [41], discotic diradical mesodiradical (R,S)- 3,mesoand ( theR,S)- non-LC3, and the racemic non-LC diastereomers racemic diastereomers (R*,R*)-3 [ 42(R*],,R* discotic)-3 [42], monoradical discotic monoradicaltrans-4 [44 ], andtrans rod-like-4 [44], hydrogen-bonded and rod-like hydrogen-bonded monoradical monoradicaltrans-5 [43]. trans The-5 LC [43]. temperatures The LC temperatures refer to the refer heating to process,the heating except process, for monotropic except for racemice monotropictrans- 5racemice. These compounds trans-5. These except compounds (R*,R*)-3 showedexcept (R a*, positiveR*)-3 magneto-LCshowed a positive effect. The magneto-LC individual effect. magnetic The individual properties magnetic for 1–4 are properties discussed for in 1 Section–4 are discussed4. in Section 4. Here, it should be emphasized that the positive magneto-LC effect (χM increase) (i) proved to be independentHere, it of should the magnetic be emphasized field-induced that the molecular positive magneto-LC reorientation effect because (χM increase) of the too (i) small proved molecular to be magneticindependent anisotropy of the ( ∆magneticχ)[45] and field-induced (ii) did not resultmolecular from reorientation the contamination because by of extrinsic the too magnetic small metalmolecular or metal magnetic ion impurities, anisotropy as ( confirmed∆χ) [45] and by (ii) the did quantitative not result from analysis the contamination using inductively by extrinsic coupled plasmamagnetic atomic metal emission or metal spectroscopy ion impurities, (ICP-AES) as confirmed as well by asthe the quantitative observation analysis of thermal using andinductively impurity effectscoupled [42]. plasma atomic emission spectroscopy (ICP-AES) as well as the observation of thermal and impurityThus far, effects there [42]. have been two theoretical studies to contribute to the elucidation of the mechanism of positiveThus magneto-LC far, there have effect. been By two the quantitativetheoretical studies analysis to of contribute angular dependenceto the elucidation of g-values of the and mechanism of positive magneto-LC effect. By the quantitative analysis of angular dependence of g- ∆Hpp (line-width) of EPR spectra and DFT calculations of spin density distribution in the interacting moleculesvalues and based ∆Hpp on (line-width) the crystal of structure EPR spectra of anand analogous DFT calculations compound, of spin Vorobiev density distribution et al. revealed in the that interacting molecules based on the crystal structure of an analogous compound, Vorobiev et al. an intermolecular spin polarization mechanism operating between neighboring radical molecules revealed that an intermolecular spin polarization mechanism operating between neighboring radical rather than the direct through-space interactions between the paramagnetic centers contributes to molecules rather than the direct through-space interactions between the paramagnetic centers the occurrence of the positive magneto-LC effect [46]. Furthermore, quite recently, by means of contributes to the occurrence of the positive magneto-LC effect [46]. Furthermore, quite recently, by molecular dynamic (MD) simulation and density functional theory (DFT) calculations, Uchida et al. means of molecular dynamic (MD) simulation and density functional theory (DFT) calculations, hasUchida revealed et al. that has the revealed positive that magneto-LC the positive e ffmagnetect cano-LC originate effect fromcan originate the conservation from the conservation of the memory of of spin-spinthe memory interactions of spin-spin between interactions magnetic between moments, magn owingetic moments, to the ceaseless owing to molecularthe ceaseless contacts molecular in the LCcontacts and isotropic in the LC states. and isotropic They incorporated states. They theincorp molecularorated the mobility molecular effects mobility [44] intoeffects the [44] quantitative into the Thouless-Anderson-Palmerquantitative Thouless-Anderson-Palmer approach, which approa was proposedch, which to was solve proposed the Edwards-Anderson to solve the Edwards- model for spin-glassesAnderson withmodel inhomogeneous for spin-glasses magneticwith inhomogeneous interactions magnetic [47]. interactions [47]. InIn addition, addition, it it is is noteworthy noteworthy thatthat thethe nonlinearnonlinear magneto-electric (ME) (ME) effect effect was, was, for for the the first first time,time, detected detected for for organic organic LC LC materials, materials, i.e.,i.e., ((SS,,SS)-)-1 (m(m = nn == 13)13) showing showing a aferroelectric ferroelectric LC LC [chiral [chiral SmCSmC (SmC*)] (SmC*)] phase phase [ 37[37–39].–39]. ByBy applying an an external external DC DC electric electric field field (between (between +25+ V25 and V and−25 V)25 to V) − tothe ( (SS,,SS)-)-11 confinedconfined in in a arubbed rubbed surface-stabilized surface-stabilized liquid-crystal liquid-crystal thin thin(4 μm(4 thick)µm thick) sandwich sandwich cell, the cell, thehysteresis hysteresis loops loops were were observed observed for forthe theexperimental experimental EPR EPRparameters parameters such suchas the as g-value, the g-value, ∆Hpp, and∆Hpp , andχrelχ rel(relative(relative paramagnetic paramagnetic susceptibility). susceptibility). These These experimental experimental results results suggest suggest that that the the interactions interactions betweenbetween the the electric electric polarization polarization and the the magnetic magnetic moment moment in inthe the magnetic magnetic LC LCphase phase are likely are likely to to contribute to the enhancement of a positive magneto-LC effect in weak magnetic fields. In fact, Symmetry 2020, 12, 1910 5 of 29 Symmetry 2020, 12, x FOR PEER REVIEW 5 of 29 thecontribute highly enantiomerically-enriched to the enhancement of a compoundspositive magneto- (S,S)-1LCalways effect exhibited in weak a magnetic larger positive fields. magneto-LC In fact, the ehighlyffect than enantiomerically-enriched the corresponding racemic compounds ones trans (S-1,S[)-351, 36always]. exhibited a larger positive magneto-LC effectDue than to the corresponding latest comprehensive racemic review ones trans as well-1 [35,36]. as database-like article regarding the preparation, characterization,Due to the latest and magneticcomprehensive properties review of a as wide well array as database-like of all-organic article LC radicals regarding synthesized, the preparation, thus, far,characterization, an excellent one and entitled magnetic “Liquid properties crystalline of a wide derivatives array of ofall-organic heterocyclic LC radicals radicals” synthesized, by Kaszy´nskiis thus, stronglyfar, an excellent recommended one entitled to refer “Liquid to Reference crystallin [27e]. derivatives of heterocyclic radicals” by Kaszyński is stronglyIn Section recommended2, the fundamentals to refer to Reference of super-para-magnetism [27]. are briefly mentioned. In Section3, the typicalIn Section properties 2, the fundamentals of canonical of spin super-para-magnetism glasses composed are of dilutedbriefly mentioned. magnetic alloysIn Section and 3, thethe non-canonicaltypical properties spin glass-likeof canonical behavior spin ofglasses the other composed systems of are diluted summarized. magnetic In Section alloys4 ,and the selectedthe non- importantcanonical spin experimental glass-like resultsbehavior featuring of the other the magnetic systems are properties summarized. of positive In Section magneto-LC 4, the selected effect observedimportant for experimental LC nitroxide radicalresults compoundsfeaturing the are magnetic summarized. properties In Section of 5positive, the mechanism magneto-LC of positive effect magneto-LCobserved for e ffLCect isnitroxide proposed radical and discussed compounds on the ar basise summarized. of the magnetism In Sectio ofn the 5, spinthe glassmechanism state and of thepositive ME e ffmagneto-LCect. effect is proposed and discussed on the basis of the magnetism of the spin glass state and the ME effect. 2. Super-Para-Magnetism 2. Super-Para-MagnetismSuper-para-magnetism is a paramagnetic behavior of magnetic domains or clusters. The collective magneticSuper-para-magnetism spins in a domain is area paramagnetic aligned parallel behavior to the of external magnetic magnetic domains field, or andclusters. there The is nocollective remanent magnetic magnetization spins in a domain of the sample,are aligned similarly parallel to the paramagnets. external magnetic However, field, and the there spins is areno ferromagnetically-orderedremanent magnetization of inside the sample, each domain. similarly To to behaveparamagnets. like paramagnets, However, the domains spins are of ferromagnetically-orderedferromagnetically-ordered momentsinside each should domain. readily respondTo behave to the like applied paramagnets, field (Figure domains4). Such aof situationferromagnetically-ordered can be realized in domains moments with should a rotational readily degree respond of freedomto the applied or domains field with(Figure a su 4).ffi cientlySuch a lowsituation energy can barrier be realized of magnetic in domains anisotropy with (i.e., a coercive rotational force). degree of freedom or domains with a sufficientlyThe net low magnetization energy barrier of super-para-magnets of magnetic anisotropy are expressed (i.e., coercive by the force). equation below. The net magnetization of super-para-magnets are expressed by! the equation below. µH kBT M = NµL(µ) = Nµ coth 𝜇𝐻 𝑘𝑇 (1) 𝑀=𝑁𝜇𝐿𝜇 =𝑁𝜇×coth× kBT −−µH (1) 𝑘𝑇 𝜇𝐻 N denotes a number of domains, µ does the magnetization summed in a domain, and kB is the 𝑁 denotes a number of domains, 𝜇 does the magnetization summed in a domain, and 𝑘 is the ( ) BoltzmannBoltzmann factor.factor.L 𝐿µ𝜇is is called called Langevin Langevin function. function. In In the the case case that that there there is is distribution distribution in inµ 𝜇,, the the net net ( ) magnetizationmagnetization shouldshould be be estimated estimated by by integration integration over over the the distribution distribution function functionP 𝑃µ𝜇[48 [48].].

Figure 4. Temperature variation in the magnetic field dependence of Langevin function L(µ) shown in Figure 4. Temperature variation in the magnetic field dependence of Langevin function 𝐿𝜇 shown Equation (1) for µ = 1000 µB, where µB is the Bohr magneton. in Equation (1) for 𝜇 = 1000 𝜇 , where 𝜇 is the Bohr magneton. Here, it is worth noting that the size effect of the domain is a very important factor, since the Here, it is worth noting that the size effect of the domain is a very important factor, since the anisotropic energy in a domain is expressed by the product of the anisotropic coefficient (K) and anisotropic energy in a domain is expressed by the product of the anisotropic coefficient (K) and volume size (V) of the domain. The V value, which satisfies KV < T, makes the magnetic moments volume size (V) of the domain. The 𝑉 value, which satisfies 𝐾𝑉 < 𝑇, makes the magnetic moments paramagnetic. As the V becomes larger, the anisotropic energy of a domain also increases so that it paramagnetic. As the 𝑉 becomes larger, the anisotropic energy of a domain also increases so that it turns to the coercive force at KV > T. turns to the coercive force at 𝐾𝑉 >𝑇.

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3. Spin Glass 3. Spin Glass

3.1. What is Is a a Spin Spin Glass?—G Glass?—Generaleneral Experimental Features Spin glass glass (SG) (SG) is isa frozen a frozen state state of spins of spins withou withoutt long-range long-range periodicity periodicity [49]. In [the49]. SG, In although the SG, t althoughhe long-time the long-time average averageof spins of at spins each at site each ha sites a has non-zero a non-zero value value as asseen seen in in a a conventional, magnetically-ordered state, state, both both th thee magnitude magnitude and and direction direction are are random random at each at each site site(Figure (Figure 5). The5). Thename name SG comes SG comes from from the glass the glass transition transition in which in which the atomic the atomic positions positions take take random random values. values. If the SG state is expressed in a thermodynamic way with a spin (Si) of the i-th site, the local If the SG state is expressed in a thermodynamic way with a spin (Si) of the i-th site, the local spontaneous magnetization (mi) has a non-zero value (Equation (2)), even though the site averaged spontaneous magnetization (mi) has a non-zero value (Equation (2)), even though the site averaged value (m)) isis zerozero (Equation(Equation (3)).(3)). m𝑚i ==S〈i𝑆 〉,00 (2) h i (2) N 1 X m = 1 mi = 0 (3) 𝑚=N 𝑚 =0 (3) 𝑁 i Here, N denotes the total number of spins. To distinguish the SG state from a paramagnetic state or a longHere, range 𝑁 denotes ordered the state, totalthe number combination of spins. of Tom distinand theguish second the SG order state moment from a paramagnetic (q) with respect state to or a long range ordered state, the combination of 𝑚 and the second order moment (q) with respect the thermal average of Si is often used (Equation (4)). to the thermal average of 𝑆 is often used (Equation (4)). N X 1 2 q = 1 Si , 0 (4) 𝑞=N h〈𝑆i〉 0 (4) 𝑁 i

Figure 5. An image of the SG state compared to conventional long-range magnetic ordering. Figure 5. An image of the SG state compared to conventional long-range magnetic ordering. Historically, the first SG state was found in dilute magnetic alloys such as AuFe and CuMn [50,51]. Historically, the first SG state was found in dilute magnetic alloys such as AuFe and CuMn Such systems are called canonical SGs, which are the most studied and established system in the SG [50,51]. Such systems are called canonical SGs, which are the most studied and established system in theory. Canonical SGs consist of a random distribution of local magnetic impurities in nonmagnetic the SG theory. Canonical SGs consist of a random distribution of local magnetic impurities in metal hosts. The features in which canonical SGs show experimentally can be summarized as follows. nonmagnetic metal hosts. The features in which canonical SGs show experimentally can be 1.summarizedHysteresis: as follows. The emergence of hysteresis of magnetic susceptibility (χ) under a different 1. Hysteresis:temperature The control emergence process (Figureof hysteresis6[51]). of magnetic susceptibility (χ) under a different 2. temperatureImpurity eff controlect: Non-linear process (Figure or non-monotonic 6 [51]). increase of χ with the increment of an amount of 2. Impuritythe magnetic effect: impurity Non-linear (Figure or non-monotonic6[51]). increase of χ with the increment of an amount of 3. theLong-time magnetic scale impurity dynamics. (Figure 6 [51]). 4.3. Long-timeAC magnetic scale susceptibility dynamics. sensitive to the frequency of the external magnetic fields near the spin 4. ACfreezing magnetic temperature. susceptibility sensitive to the frequency of the external magnetic fields near the spin 5. freezingA broad temperature. peak in magnetic specific heat around the spin freezing temperature. 5. A broad peak in magnetic specific heat around the spin freezing temperature.

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Figure 6. Temperature dependence of magnetic susceptibility for the SG alloy CuMn containing the Figure 6. Temperature dependence of magnetic susceptibility for the SG alloy CuMn containing the magnetic impurity Mn of 1.08- and 2.02-at.%. After zero field cooling (ZFC, H < 0.05 G), the initial magnetic impurity Mn of 1.08- and 2.02-at.%. After zero field cooling (ZFC, H < 0.05 G), the initial susceptibilities (b,d) increased with rising temperature at a field of 5.90 G. The susceptibilities (a,c) susceptibilities (b,d) increased with rising temperature at a field of 5.90 G. The susceptibilities (a,c) were obtained at the same magnetic field, which was applied above TSG before cooling the samples. were obtained at the same magnetic field, which was applied above TSG before cooling the samples. Data cited from Ref. [51]. Data cited from Ref. [51]. The features 1–3 observed for LC nitroxide radicals are described in Section4. For feature 4, The features 1–3 observed for LC nitroxide radicals are described in Section 4. For feature 4, we we do not discuss it because of the difficulty in the measurement of AC magnetic susceptibility for LC do not discuss it because of the difficulty in the measurement of AC magnetic susceptibility for LC nitroxide radicals at high temperatures. Likewise, we cannot discuss the feature 5 concerning the LC nitroxide radicals at high temperatures. Likewise, we cannot discuss the feature 5 concerning the LC nitroxide radicals at high temperatures because the contribution by molecular motion is dominant in nitroxide radicals at high temperatures because the contribution by molecular motion is dominant in the specific heat data for these compounds. the specific heat data for these compounds. 3.2. Theoretical Key Ingredients in Canonical SGs 3.2. Theoretical Key Ingredients in Canonical SGs The key ingredients in canonical SGs are considered to be randomness, frustration, and competing interactions,The key which ingredients are fully in equipped canonical in dilute SGs magneticare considered alloys [ 52to]. be A magneticrandomness, moment frustration, interacts withand othercompeting moments interactions, by means which of Ruderman-Kittel-Kasuya-Yoshida are fully equipped in dilute magnetic (RKKY) alloys interactions [52]. A magnetic via the conductive moment electron.interacts with The RKKYother moments interactions by means are expressed of Ruderman-Kittel-Kasuya-Yoshida as shown in Equation (5). (RKKY) interactions via the conductive electron. The RKKY interactions are expressed as shown in Equation (5).   cos 2kFrij cos2𝑘𝑟 Jij (5) 𝐽 ~ 3 (5) ∼ r𝑟ij where k𝑘F denotes denotes the the Fermi Fermi energy energy of the of hostingthe hosting metal metal and r ijandis the 𝑟 distance is the distance between twobetween magnetic two magnetic moments at 𝑖 and 𝑗 sites. Since cos2𝑘 𝑟 is an oscillating function of 𝑟 , 𝐽 can have moments at i and j sites. Since cos 2kFrij is an oscillating function of rij, Jij can have various plus and various plus and minus values as a function of 𝑟 . minus values as a function of rij. The random distribution of magnetic impurities then results in randomrandom magneticmagnetic interactionsinteractions between each impurity site. Under such a network of irregular interactions, some of these local sites are likely to undergo competition of interactions and be unable to settle down in a certain spin state, state, which satisfiessatisfies the energyenergy minimumminimum forfor eacheach interaction.interaction. Such aa situationsituation isis calledcalled frustration.frustration. The combination of of randomness randomness and and frustration frustration gives gives many many stable stable states states in the in the local local energy, energy, so sothat that it would it would bring bring the manifold the manifold metastable metastable states states in the in whole the whole system. system. Accordingly, Accordingly, the free the energy free energy(F) of the (F )SG of thestates SG has states many has local many minima local minima (Figure (Figure 7) and7 )this and is this called is called a multi-valley a multi-valley structure. structure. This Thischaracteristic characteristic structure structure inherent inherent in SGs in results SGs results in various in various features features listed listed in Section in Section 3.1. 3.1.

Symmetry 2020, 12, 1910 8 of 29 Symmetry 2020, 12, x FOR PEER REVIEW 8 of 29

FigureFigure 7. Schematic7. Schematic for for thethe temperature temperature variation variation of free of freeenergies energies (F) in order (F) in parameter order parameter space. The space. The horizontalhorizontal axes axes denote denote the the set setof magnetic of magnetic order orderparameters parameters {M} of spins. {M} of (a) spins. Conventional (a) Conventional long range long rangesecond-order second-order transition transition at T at= TTN. =(bT) NSG.( transitionb) SG transition at T = TSG at. T = TSG.

3.3. Character3.3. Character of the of the Magnetism Magnetism

3.3.1.3.3.1. Hysteresis Hysteresis TheThe hysteresis hysteresis results results from from the the multiple multiple metastablemetastable states states permitted permitted in the in thematerial, material, whereas whereas an an ordinaryordinary long long range range ordered ordered state state hashas a unique unique ground ground state. state. As shown As shown in Figu inre Figure 6, the 6hysteresis, the hysteresis in in magneticmagnetic susceptibilitysusceptibility is is most most noticeable noticeable when when the the material material is cooled is cooled below below the SG the transition SG transition temperature (𝑇) from the paramagnetic phase. If the sample is cooled in the absence of external temperature (TSG) from the paramagnetic phase. If the sample is cooled in the absence of external magnetic field [this is called ‘zero field cooling’ (ZFC)], the net magnetization of the sample generally magnetic field [this is called ‘zero field cooling’ (ZFC)], the net magnetization of the sample generally does not have a large value. This is observed as a cusp near the spin freezing temperature. On the does not have a large value. This is observed as a cusp near the spin freezing temperature. On the other hand, if the material is cooled in the presence of a non-zero external field [this process is called other‘field hand, cooling’ if the (FC)], material some is of cooled the spins, in thebut presencenot all of them, of a non-zerowould align external with the field magnetic [this field process leading is called ‘fieldto cooling’ a ferromagnetic (FC)], some behavior. of the Thus, spins, in butthe notSG phases all of them,, the magnetization would align after with the the FC magnetic process is field larger leading to a ferromagneticthan that after the behavior. ZFC process. Thus, in the SG phases, the magnetization after the FC process is larger than that after the ZFC process. 3.3.2. Impurity Effect 3.3.2. ImpurityIn canonical Effect SGs, the concentration (x) of magnetic impurities in nonmagnetic metal hosts has considerableIn canonical effects SGs, on the its concentration magnetic property. (x) of In magnetic a wider range impurities of x values, in nonmagnetic each SG state takes metal one hosts of has considerablevarious ground effects states. on its Figure magnetic 8 shows property. a schematic In a widerfor the rangerelationship of x values, between each the SGx values state and takes the one of resulting magnetic properties. At the low limit of x value, dilute magnetic impurities can be regarded various ground states. Figure8 shows a schematic for the relationship between the x values and the resulting magnetic properties. At the low limit of x value, dilute magnetic impurities can be regarded as isolated spins, so that the Kondo effect would work between the conducting electron spins and the isolated spins (Figure8a). A pair of conducting electron spin and isolated spin adopts a singlet Symmetry 2020, 12, x FOR PEER REVIEW 9 of 29 asSymmetry isolated 2020 spins,, 12, x soFOR that PEER the REVIEW Kondo effect would work between the conducting electron spins and9 theof 29 Symmetryisolated 2020spins, 12 ,(Figure 1910 8a). A pair of conducting electron spin and isolated spin adopts a singlet9 of 29 groundas isolated state spins, to vanish so that the the susceptibility Kondo effect at wouldlow temperatures. work between With the the conducting increasing electron x value, spins the average and the impurity-to-impurityisolated spins (Figure distance 8a). A becomespair of conducting shorter enou electrongh to induce spin andRKKY isolated interactions, spin adopts forming a ansinglet SG groundstateground (Figure state state to 8b).to vanish vanish When the the the susceptibility susceptibility x value exceeds at at low low ~10 temperatures. temperatures. at.%, some With Withof the the the impurities increasing increasing xexist xvalue, value, in thethe the average nearestaverage impurity-to-impurityneighborimpurity-to-impurity lattice points distance distanceso that such becomes becomes locally shorter shorter high-concentrated enough enough to to induce induce spin RKKYclusters RKKY interactions, interactions,result in the forming coexistenceforming an an SG SGof statethestate SG (Figure(Figure state and8 b).8b). ferro-magnetic When When the the xx value valueor anti-ferro-magneti exceedsexceeds ~10~10 at.%,at.%,c domains somesome ofof (Figure thethe impuritiesimpurities 8c). These existexist clusters ininthe the arenearest nearest most neighborlikelyneighbor to behave lattice lattice pointslike points super-para-magnets, so so that that such such locally locally and high-concentrated high-concentrated also exhibit a hysteresis spin spin clusters clusters in the result result temperature in in the the coexistence coexistence dependent of of theofthe magnetic SG SG state state andsusceptibility, and ferro-magnetic ferro-magnetic as a orsort or anti-ferro-magnetic anti-ferro-magnetiof SG behavior ofc domains thedomains clusters. (Figure (Figure Such8 c).8c). behavior These These clusters clustersis often are arecalled most most a likely‘clusterlikely to to glass.’ behave behave A like likefurther super-para-magnets, super-para-magnets, increase in the and x and valu also alsoe exhibit makes exhibit a the hysteresis a hysteresis ferro-magnetic in thein the temperature temperatureor anti-ferro-magnetic dependent dependent of magneticdomainsof magnetic larger, susceptibility, susceptibility, finally, asgiving a sortas ana of sort almost SG of behavior SG ordered behavior of state the clusters.of (Figure the clusters. Such8d). behavior Such behavior is often calledis often a ‘clustercalled a glass.’‘cluster A glass.’ further A increase further inincrease the x value in the makes x valu thee makes ferro-magnetic the ferro-magnetic or anti-ferro-magnetic or anti-ferro-magnetic domains larger,domains finally, larger, giving finally, an almost giving orderedan almost state ordered (Figure state8d). (Figure 8d).

Figure 8. Schematic for the effect of the concentration (x) of magnetic impurity on the magnetic properties of alloy systems. See text for details. The grey circles represent the lattice points. FigureFigure 8. 8. SchematicSchematic forfor thethe eeffectffect ofof thethe concentrationconcentration (x(x)) ofof magnetic magnetic impurity impurity on on the the magnetic magnetic 3.3.3.properties propertiesLong Time of of alloyScale alloy systems. systems.Dynamics See See text text for for details. details. The The grey grey circles circles represent represent the the lattice lattice points. points. 3.3.3. Long Time Scale Dynamics 3.3.3.Long Long time Time scale Scale dynamics Dynamics is another characteristic aspect of canonical SGs. Since there are many metastableLong time states, scale hopping dynamics over is the another energetic characteristic barrier to other aspect states of canonical is possible. SGs. Such Since metastable there are manystates Long time scale dynamics is another characteristic aspect of canonical SGs. Since there are many metastableare realized states, by a rapid hopping change over in the external energetic stimuli barrier such to other as temperature states is possible. or magnetic Such metastablefield around states the metastable states, hopping over the energetic barrier to other states is possible. Such metastable states areSG realizedphase transition. by a rapid Rapid change cooling in external from stimulithe temperature such as temperature above 𝑇 or tomagnetic that below field 𝑇 around compels the the SG systemare realized to take by aa rapidcertain change metastable in external state stimuliwithou tsuch immediate as temperature relaxing or to magnetic the lowest field energy. around The the phase transition. Rapid cooling from the temperature above TSG to that below TSG compels the system SG phase transition. Rapid cooling from the temperature above 𝑇 to that below 𝑇 compels the torelaxation take a certain to the metastablelowest energy state state without is called immediate the ‘aging relaxing phenomenon,’ to the lowest which energy. takes a Thetypical relaxation time of system to take2 a certain4 metastable state without immediate relaxing to the lowest energy. The tothe the order lowest of energy10 ~10 states or ismore. called Thus, the ‘aging the aging phenomenon,’ phenomenon which can takes be followed a typical by time statistical of the order DC relaxation to the lowest energy state is called the ‘aging phenomenon,’ which takes a typical time of ofmagnetization 102~104 s or measurements. more. Thus, the For aging example, phenomenon after keep caning be the followed SG state by at statistical a high temperature DC magnetization region the order of 102~104 s or more. Thus, the aging phenomenon can be followed by statistical DC measurements.in the SG phase For under example, the ZFC after conditions, keeping the the SG ma stategnetization at a high at temperature the same temperature region in the gradually SG phase magnetization measurements. For example, after keeping the SG state at a high temperature region underincreases the with ZFC conditions,the elapse of the time magnetization by applying at thean external same temperature magnetic field gradually (Figure increases 9) [53]. with After the a in the SG phase under the ZFC conditions, the magnetization at the same temperature gradually elapsesufficiently of time long by time, applying the anmagnetization external magnetic gets cl fieldose to (Figure the value,9)[ 53 which]. After is a observed sufficiently after long the time, FC increases with the elapse of time by applying an external magnetic field (Figure 9) [53]. After a theprocess. magnetization gets close to the value, which is observed after the FC process. sufficiently long time, the magnetization gets close to the value, which is observed after the FC process.

Figure 9. The relaxation of magnetization divided by the external field after waiting for tw at zero field. Figure 9. The relaxation of magnetization divided by the external field after waiting for tw at zero field. The data are obtained at the temperature of 0.91TSG and under the external field of 0.8 G. The horizontal The data are obtained at the temperature of 0.91TSG and under the external field of 0.8 G. The axisFigure denotes 9. The the relaxation elapse of of timemagnetization in the measurement. divided by the Reprint external with field permission after waiting [53]: for Copyright tw at zero 2020 field.

AIPThe Publishing. data are obtained at the temperature of 0.91TSG and under the external field of 0.8 G. The

Symmetry 2020, 12, 1910 10 of 29

3.4. Non-Canonical SG-Like Behavior Besides canonical SG alloy systems, some of the magnetic features summarized in Section 3.1 are observed in a variety of magnetic systems, which are often called ‘SG-like behavior’ [52]. The features can show up even if some of the key ingredients (i.e., randomness, frustration, and competing interactions) are weak or missed in the systems. In general, neither the diluted conditions of spins nor RKKY interactions are needed for the realization of SG-like behavior. The SG-like behavior is reported in many non-diluted magnets with exchange/super-exchange interactions between the nearest neighbor spins. It seems that the SG-like behavior tends to arise if only there exist some factors of randomness and/or inhomogeneity in the interaction network. They have been often found in atom-doped/substituted systems, grain boundary, systems with site-mixing or random local distortion, and more [52].

4. Positive Magneto-LC Effect Here, the unique magnetism of LC nitroxide radicals, which is regarded as SG-like behavior, is described. Although the solid-state picture of usual SG-like magnets is not straightforwardly applied to the liquid crystal systems, we actually observed the SG-like features 1–3 (hysteresis, impurity effect, and long-time scale dynamics) listed in Section 3.1. In Section 4.1, our discovery of the positive magneto-LC effect, which refers to a magnetic hysteresis (i.e., thermal effect) associated with the emergence of superparamagnetic behavior in the LC state, is demonstrated. The substantial enhancement of superparamagnetic behavior by impurity effect or controlling the magnetic inhomogeneity is mentioned in Section 4.2. An example of long-time scale dynamics is presented in Section 4.1.3. In Section 4.3, the ME effect observed for the first time in the ferroelectric LC phase of the LC nitroxide radical is explained in detail.

4.1. Magnetic Hysteresis by a Thermal Effect

4.1.1. Monoradical trans-1 In 2008, for the first time, we observed a magnetic hysteresis for nitroxide monoradical (S,S)-1 (m = n = 13) showing enantiotropic (reversible) SmC* and N* phases (Figure3) by SQUID magnetic suscetibiliy measurement (Figure 10a) [34,35]. In the first heating process from the initial crystalline phase, the molar magnetic susceptibility (χM) increased at the crystal–to-SmC* phase transition. An additional χM increase was noted at the SmC*-to-N* and N*-to-isotropic phase transitions. Interestingly, the overall 12% χM increase at the phase transitions in the heating process was preserved during the cooling one from the isotropic phase and even in the second heating one, and the χM in the first cooling and second heating processes never followed the curve drawn in the first heating process. Such magnetic hysteretic behavior, an unusual χM increase at the crystal–to-LC phase transition in the first heating process, was referred to as ‘positive magneto-LC effect,’ which was observed for all of the LC nitroxide radical compounds 1~5 with full reproducibility. Therefore, the χM in the first cooling process are expressed by the equation below.

χM = χpara + χTIM + χdia (6) where χpara represents the paramagnetic susceptibility of radical spins, which follows the Curie-Weiss law (Equation (7)). The C and θ are Curie constant and Weiss temperature, respectively. For the mono-radical nitroxide compounds examined, the values of C were close to 0.375 corresponding to the S = 1/2 system. C χ = (7) para T θ − SymmetrySymmetry2020 2020, ,12 12,, 1910 x FOR PEER REVIEW 1111 of of 29 29

Figure 10. Magnetic hysteresis observed for (S,S)-1 (m = n = 13) by a thermal effect. Temperature Figure 10. Magnetic hysteresis observed for (S,S)-1 (m = n = 13) by a thermal effect. Temperature (T) (T) dependence of (a) molar magnetic susceptibility (χM) measured at 0.05 T and (b) EPR line-width dependence of (a) molar magnetic susceptibility (χM) measured at 0.05 T and (b) EPR line-width (Hpp) (Hpp) measured at around 0.34 T in the first heating and cooling processes, and magnetic field (H) measured at around 0.34 T in the first heating and cooling processes, and magnetic field (H) dependence of molar magnetization (M) measured at (c) 73 and (d) 77 ◦C in the first heating process. dependence of molar magnetization (M) measured at (c)− −73 and (d) 77 °C in the first heating process. In panels a and b, open and filled circles represent the first heating and cooling processes, respectively. In panels a and b, open and filled circles represent the first heating and cooling processes, respectively. Data cited from Ref. [35]. Data cited from Ref. [35].

The χTIM indicates the temperature-independent magnetic susceptibility corresponding to the χM The χTIM indicates the temperature-independent magnetic susceptibility corresponding to the χM increase at the crystal-to-LC phase transition, while the χdia represents the temperature-independent diamagneticincrease at the susceptibility, crystal-to-LC which phase arises transition, from the while orbital the currentχdia represents and can the be temperature-independent calculated using Pascal’s diamagnetic susceptibility, which arises from the orbital current and can be calculated using Pascal’s constant for each atom and bond. Since the sum of χTIM and χdia can be experimentally obtained as constant for each atom and bond. Since the sum 1of χTIM and χdia can be experimentally obtained as the temperature-independent component by χM-T− plots, according to Equations (6) and (7), the χTIM − valuethe temperature-independent can be derived. component by χM-T 1 plots, according to Equations (6) and (7), the χTIM valueWe can observed be derived. a nonlinear relationship between the applied magnetic field (H) and the observed magnetizationWe observed (M) ina nonlinear the chiral andrelationship achiral LC between phases the of trans applied-1 (m magnetic= n = 13) field by SQUID (H) and magnetometry the observed (Figuremagnetization 10d). Unlike (M) in the the Brillouin chiral and function, achiral whichLC phases ordinary of trans paramagnetic-1 (m = n = 13) radical by SQUID compounds magnetometry display, a(Figure steep increase 10d). Unlike was observed the Brillouin at low function, magnetic which fields ordinary (~0.1 T) paramagnetic and an oblique radical linear compounds response was display, noted ata highersteep increase magnetic was fields. observed This resultat low implied magnetic the fie emergencelds (~0.1 T) of and superparamagnetic an oblique linear behavior response in was the LCnoted phase. at higher magnetic fields. This result implied the emergence of superparamagnetic behavior in the LCSuch phase. spin glass (SG)-like hysteretic or superparamagnetic behavior was also seen for the other LC nitroxideSuch radicals spin glass listed (SG)-like in Figure hysteretic3. In contrast, or superpar non-LCamagnetictrans-1 analogues behavior bearingwas also much seen shorterfor the alkylother chainsLC nitroxide failed to radicals show the listed magnetic in Figure hysteresis 3. In contrast, because non-LC of their trans common-1 analogues paramagnetic bearing properties. much shorter It is worthalkyl chains noting failed that, with to show respect the tomagnetic LC trans hysteres-1 and theis LCbecause derivatives, of their enantiomerically-enriched common paramagnetic properties. samples alwaysIt is worth exhibited noting a largerthat, with positive respect magneto-LC-e to LC transff-ect1 and (or the magnetic LC derivatives, hysteresis) enantiomerically-enriched at the crystal-to-LC phase transitionsamples always than the exhibited corresponding a larger racemic positive samples magneto-LC-effect [35,36]. Thus, (or the magn positiveetic hysteresis) magneto-LC at the eff ectcrystal- is a to-LC phase transition than the corresponding racemic samples [35,36]. Thus, the positive magneto-

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LC effect is a unique property inherent in these LC nitroxide radicals, which suggested some uniquerelationship property between inherent the spatial in these symmetry LC nitroxide of LC radicals, phases which and the suggested positive somemagneto-LC relationship effect. between the spatialHere, symmetrywe should of stress LC phases that such and a the magnetic positive hy magneto-LCsteresis or superparamagnetic effect. behavior did not resultHere, from we theshould magnetic stress field-induced that such amolecular magnetic reorientation hysteresis or in superparamagnetic the LC and isotropic behavior phases. didThe notcontribution result from of theoverall magnetic magnetic field-induced anisotropy molecular(∆χ) is, at reorientationmost, ~10−5 emu/mol in the LC [26], and while isotropic the observed phases. The contribution of overall magnetic anisotropy (∆χ) is, at most, ~10 5 emu/mol [26], while the observed χM increases at the crystal-to-LC phase transition points amount to− much more than 10−4 emu/mol in 4 χvariousM increases LC nitroxide at the crystal-to-LC radicals. It phaseis also transition important points to examine amount the to possibility much more of than contamination 10− emu/mol by inferromagnetically-ordered various LC nitroxide radicals. metal or It ismetal also ion important spins for to such examine nonlinear the possibility magnetization of contamination curves [54]. byConsequently, ferromagnetically-ordered we could exclude metal this or metal possibi ionlity spins by forthe such quantitative nonlinear analysis magnetization using curvesinductively [54]. Consequently,coupled plasma–atomic we could excludeemission this spectroscopy possibility by(ICP-AES), the quantitative verifying analysis 0.5~1.3 using ppm inductively for the iron coupled content plasma–atomicand 0.3~0.6 ppm emission for the nickel spectroscopy content for (ICP-AES), LC nitrox verifyingide radicals 0.5~1.3 [42]. These ppm forvalues the ironare less content than one- and 0.3~0.6 ppm for the nickel content for LC nitroxide radicals [42]. These values are less than one-tenth tenth of the observed 𝜒 increase for the individual compounds. Furthermore, these two ofpossibilities the observed (molecularχM increase reorientation for the individual and contamination) compounds. were Furthermore, ruled out theseby observing two possibilities a large (molecular‘impurity effect’ reorientation explained and in Section contamination) 4.2. were ruled out by observing a large ‘impurity effect’ explainedThe measurements in Section 4.2. of the AC magnetic susceptibilities are necessary to investigate the long time- scaleThe spin measurements dynamics of super-para-magnetic of the AC magnetic componen susceptibilitiests in LC arenitroxide necessary radicals. to investigate However, we the could long time-scalenot obtain spinthese dynamics data due to of the super-para-magnetic difficulties with the components measurement in LCat high nitroxide temperatures. radicals. This However, is our wefuture could subject. not obtain these data due to the difficulties with the measurement at high temperatures. This is our future subject. 4.1.2. Biradical (S,S,S,S)-2 4.1.2. Biradical (S,S,S,S)-2 In order to investigate the relationship between the positive magnet-LC effect and the number In order to investigate the relationship between the positive magnet-LC effect and the number of of spins, we prepared the biradical (S,S,S,S)-2 (>99% ee) showing an enantiotropic chiral nematic (N*) spins, we prepared the biradical (S,S,S,S)-2 (>99% ee) showing an enantiotropic chiral nematic (N*) phase (Figure 3) in which the two radical spins do not interact. The χM increase at the crystal-to-LC phase (Figure3) in which the two radical spins do not interact. The χM increase at the crystal-to-LC phase transition in the first heating process was 0.8 × 10 3 emu/mol (38%) (Figure 11), which was phase transition in the first heating process was 0.8 10− emu/mol (38%) (Figure 11), which was slightly larger than those of LC mono-radicals such× as (S,S)-1. This χM increase was maintained slightly larger than those of LC mono-radicals such as (S,S)-1. This χ increase was maintained during during the cooling process, similarly to the case of (S,S)-1. M the cooling process, similarly to the case of (S,S)-1.

Figure 11. Temperature dependence of χM for (S,S,S,S)-2 by SQUID magnetometry at a field of 0.05 TFigure in the 11. temperature Temperature range dependence of (a) 170 of to χM+ for120 (SC,S, andS,S)- (2b )by 100SQUID to 120 magnetometryC (magnification). at a field Open of 0.05 and T − ◦ − ◦ filledin the circles temperature represent range the of first (a) heating −170 to and+120 cooling °C and processes,(b) −100 to respectively. 120 °C (magnifica The solidtion). line Open indicates and filled the Curie-Weisscircles represent fitting the curve. firstThe heating LC transition and cooling temperatures processes, are resp shownectively. by arrows The solid in the line lower indicates and upper the sidesCurie-Weiss in panel fitting b. (Reprint curve. with The permission LC transition [41], temp Copyrighteratures 2020 are The shown Royal by Society arrows of in Chemistry). the lower and upper sides in panel b. (Reprint with permission [41], Copyright 2020 The Royal Society of Chemistry). 4.1.3. Diradical (R,S)-3 The non-π-delocalized achiral meso diradical (R,S)-3 (Figure3) showed an enantiotropic hexagonal columnar (Colh) phase in which each disc was composed of a hydrogen-bonded molecular trimer (Figure 12)[42]. This compound displayed both thermal and impurity effects. Here, only the thermal effect is stated and the impurity effect is described in Section 4.2.

Symmetry 2020, 12, x FOR PEER REVIEW 13 of 29

4.1.3. Diradical (R,S)-3 The non-π-delocalized achiral meso diradical (R,S)-3 (Figure 3) showed an enantiotropic hexagonal columnar (Colh) phase in which each disc was composed of a hydrogen-bonded molecular Symmetry 2020, 12, 1910 13 of 29 trimer (Figure 12) [42]. This compound displayed both thermal and impurity effects. Here, only the thermal effect is stated and the impurity effect is described in Section 4.2.

(a) (b) 30.6 Å

C16H33O a O C16H33O O Me OC16H33 N N H a 51.8 Å C16H33O H N N OC H Me 16 33 C16H33O O O O OC H C H O 16 33 16 33 O Me OC16H33 N N H C H O N 16 33 H N OC16H33 Me O = C16H33O O O OC16H33 C16H33O c 4.4 Å O Me H OC16H33 N N H C16H33O N N OC H Me 16 33 O O OC16H33 68.6 Å

(c)

O C16H33O H H O C H O 16 33 Me C16H33O N N H OC16H33 H N Me N OC16H33 OC16H33 O O O C16H33O H H O = C H O 16 33 Me C H O N N 16 33 H OC16H33 H N Me N OC16H33 OC16H33 O O

FigureFigure 12. 12.The The Col Colh hLC LC structure structure ofof ((RR,,S)-)-3.(. (aa)) Hexagonal Hexagonal lattice lattice and and the the columnar columnar structure, structure, (b) ( bone) one ofof possible possible trimer trimer structuresstructures formed formed by by hydrogen hydrogen bonds bonds in one in onedisc disclayer layerand (c and) intermolecular (c) intermolecular spin spinpolarization polarization via viaCH/O CH /interactionsO interactions between between two two (R, (SR)-,3S )-molecules3 molecules in inthe the neighboring neighboring disc disc layers layers expectedexpected in in the the magnetically magnetically inhomogeneous inhomogeneous LC domains. Data Data cited cited from from Ref. Ref. [42]. [42 ].

MeasurementMeasurement ofof thethe temperaturetemperature dependence dependence of of χMχ Mfor forthe thepure pure (R,S)- (R3, Sindicated)-3 indicated that (i) that (i)intramolecular intramolecular anti-ferro-magnetic anti-ferro-magnetic interactions interactions with witha singlet a singlet ground ground state between state between the two thespins two spinsdominated dominated below below 100 K 100 and K and(ii) the (ii) Curie-Weiss the Curie-Weiss fitting fitting could could be implemented be implemented between between 100 and 100 250 and 250K Kto to give give the the average average θ θvaluevalue of of nearly nearly zero zero (Figure (Figure 13). 13 ).Although Although the the pure pure (R, (SR)-,S3)- showed3 showed neither neither the χM increase nor the superparamagnetic M-H behavior in the first heating process at 0.05 T, distinct the χM increase nor the superparamagnetic M-H behavior in the first heating process at 0.05 T, distinct χM increase and superparamagnetic M-H behavior were observed in the first cooling from the χM increase and superparamagnetic M-H behavior were observed in the first cooling from the isotropic phaseisotropic and inphase the secondand in the heating second run heating at 0.05 run T. at 0.05 T.

Symmetry 2020, 12, x FOR PEER REVIEW 14 of 29

Symmetry 2020, 12, 1910 14 of 29

(a) (b) 0.02 0.02 o:1st heating o:1st heating st st 0.015 :1 cooling 0.015 :1 cooling

0.01 0.01

0.005 0.005

0 0

-0.005 -0.005 0 100 200 300 400 0 20406080100 Temperature / K Temperature / K (c) (d) 20 6 150 K in the 1st heating 150 K in the 1st heating 15 4 10 2 5

0 0 -5 -2 -10 -4 -15

-20 -6 -15000 -10000 -5000 0 5000 10000 15000 -15000 -10000 -5000 0 5000 10000 15000 H /Oe H /Oe (e) (f) 20 6 150 K in the 2nd heating 150 K in the 2nd heating 15 4 10 2 5

0 0

-5 -2 -10 -4 -15

-20 -6 -15000 -10000 -5000 0 5000 10000 15000 -15000 -10000 -5000 0 5000 10000 15000 H /Oe H /Oe (g) (h) 20 6 363 K in the 2nd heating 363 K in the 2nd heating 15 4 10 2 5

0 0

-5 -2 -10 -4 -15

-20 -6 -15000 -10000 -5000 0 5000 10000 15000 -15000 -10000 -5000 0 5000 10000 15000 H /Oe H /Oe

Figure 13. The temperature dependence of χM (a,b) at 0.05 T and the H dependence of M (c–h) for Figure 13. The temperature dependence of χM (a,b) at 0.05 T and the H dependence of M (c–h) for pure (R,S)-3 (a) in the first heating and cooling processes (b) by magnification (between 2 and 100 K) of pure (R,S)-3 (a) in the first heating and cooling processes (b) by magnification (between 2 and 100K) panel a, (c) in the solid at 150 K by the first heating from 2 K, (d) after subtraction of the oblique linear of panel a, (c) in the solid at 150 K by the first heating from 2 K, (d) after subtraction of the oblique base line in panel c, (e) in the solid at 150 K by the second heating after the first heating up to 400 K linear base line in panel c, (e) in the solid at 150 K by the second heating after the first heating up to followed by cooling to 2 K at 0.05 T, (f) after subtraction of the oblique linear base line in panel e, (g) in 400 K followed by cooling to 2 K at 0.05 T, (f) after subtraction of the oblique linear base line in panel the Colh phase at 363 k by the second heating, and (h) after subtraction of the oblique linear base line e, (g) in the Colh phase at 363 k by the second heating, and (h) after subtraction of the oblique linear in panel g. Open and filled circles in panels a and b represent the first heating and cooling processes. base line in panel g. Open and filled circles in panels a and b represent the first heating and cooling Data cited from Ref. [42]. processes. Data cited from Ref. [42].

Symmetry 2020, 12, 1910 15 of 29 Symmetry 2020, 12, x FOR PEER REVIEW 15 of 29

Next, to learn whether a helical columnar structure can be formed in the Colh phase of achiral (R,S)-3, the temperature-resolved second harmonic ge generationneration (TR-SHG) micr microscopyoscopy was performed with an excitation wavelength of 1200 nm and SHG em emissionission at 600 nm in the presence of a magnetic fieldfield (<0.5 (<0.5 T) T) (Figure (Figure 14).14). We We observed observed a a weak weak SHG SHG emission emission from from the surface of the initial crystal phase at room temperature. On On the other hand, when the solid sample was heated to 90 °C◦C and the resulting Col h phasephase was was annealed annealed at at the same temperature, the SHG intensity increased with an elapse of time over 10 min. This This observation observation indicates indicates a a gradual gradual growth growth of of domains domains with an an inversion inversion symmetry brokenbroken state state in in a magnetica magnetic field. field. The growthThe growth of this of domain this domain structure structure with broken with inversion broken inversionsymmetry symmetry is assumed is to assumed be responsible to be responsible for the generation for the generation of a positive of magneto-LC a positive magneto-LC effect, since effect, this is sincequite this analogous is quite to analogous the long-time to the relaxation long-time in relaxati a magneticon in fielda magnetic observed field for observed the SG state. for the SG state.

Figure 14. TR-SHGTR-SHG images images for for (R (R,S,)-S)-1.1 (.(a)a No) No SHG SHG signal signal was was detected detected for forsolid solid I at I26.5 at 26.5 °C. Then,◦C. Then, the –1 solidthe solid I was I was heated heated at a at rate a rate of of10 10°C◦ minC min–1 andand the the resulting resulting LC LC phase phase was was annealed annealed at at 90 90 °C.◦C. The The SHG images were were recorded at ( b) 0, (c) 5,5, andand ((d)) 1010 minmin afterafter thethe beginningbeginning ofof annealing.annealing. The observed SHG intensity was increased with the elapse of time. Data cited from Ref. [[42].42]. 4.1.4. Monoradical trans-4 4.1.4. Monoradical trans-4 The non-π-delocalized racemic trans-4 and (R,R)-4 (Figure3), which formed intermolecular The non-π-delocalized racemic trans-4 and (R,R)-4 (Figure 3), which formed intermolecular hydrogen bonds between the NO and OH groups in the condensed phase displayed an enantiotropic hydrogen bonds between the NO and OH groups in the condensed phase displayed an enantiotropic Colh phase at room temperature and formed LC glasses at lower temperatures. The SQUID Colh phase at room temperature and formed LC glasses at lower temperatures. The SQUID magnetometry revealed that the positive magneto-LC effect was gradually increased through the magnetometry revealed that the positive magneto-LC effect was gradually increased through the LC LC glass-to-LC-to-Iso phase transition sequence in the first heating process at both 0.05 and 0.5 T glass-to-LC-to-Iso phase transition sequence in the first heating process at both 0.05 and 0.5 T (Figure (Figure 15)[44]. This result suggests that molecular mobility is one of the origins of the positive 15) [44]. This result suggests that molecular mobility is one of the origins of the positive magneto-LC magneto-LC effect, which is consistent with the conclusions by means of MD simulation and DFT effect, which is consistent with the conclusions by means of MD simulation and DFT calculations so calculations so that the positive magneto-LC effect can originate from the conservation of the memory that the positive magneto-LC effect can originate from the conservation of the memory of spin-spin of spin-spin interactions between magnetic moments owing to the ceaseless molecular contacts interactions between magnetic moments owing to the ceaseless molecular contacts (Figure 16) [47]. (Figure 16)[47].

Symmetry 2020, 12, x FOR PEER REVIEW 16 of 29 Symmetry 2020, 12, 1910 16 of 29 Symmetry 2020, 12, x FOR PEER REVIEW 16 of 29

Figure 15. Temperature dependences of the χparaT plot for (a) racemic trans-4 and (b) (R,R)-4 at 0.5 T Figure 15. Temperature dependences of the χparaparaT plot for (a) racemic trans-4 and (b)(R,R)-4 at 0.5 T by byFigure SQUID 15. magnetometry,Temperature dependences and of (c) EPR of the∆H ppχ forT racemic plot for trans (a) racemic-4 at around trans -0.334 and T. The(b) ( circlesR,R)-4 denoteat 0.5 T SQUID magnetometry, and of (c) EPR ∆H for racemic trans-4 at around 0.33 T. The circles denote the theby SQUIDexperimental magnetometry, data in the and heat ofing (c) process,EPRpp ∆H ppand for the racemic horizontal trans- 4so atlid around lines indicates 0.33 T. The the circles Curie-Weiss denote experimental data in the heating process, and the horizontal solid lines indicates the Curie-Weiss fitting fittingthe experimental curves for thedata LC in glassthe heat state.ing The process, vertical and broken the horizontal lines denote solid the lines estimated indicates glass the Curie-Weissstate-to-LC curves for the LC glass state. The vertical broken lines denote the estimated glass state-to-LC phase phasefitting transitioncurves for temperatures,the LC glass state. and Thethe solidvertical lines broken on the lines circles denote to thethe estimatedright of theglass broken state-to-LC lines transition temperatures, and the solid lines on the circles to the right of the broken lines represent the representphase transition the fitting temperatures, curves for LC and and theisotropic solid phlinesases. on (Reprint the circles with to permission the right [44]:of the Copyright broken 2020,lines fitting curves for LC and isotropic phases. (Reprint with permission [44]: Copyright 2020, American Americanrepresent theChemical fitting Society).curves for LC and isotropic phases. (Reprint with permission [44]: Copyright 2020, ChemicalAmerican Society). Chemical Society).

FigureFigure 16.16. High-frequency exchange of interacting pairs detectabledetectable as an increase of the coordination number.number.Figure 16. (a ))High-frequency MoleculesMolecules interactinteract exchange withwith eacheach of interacting otherother inin fluids.fluids. pairs Because detectable LC-NR as an molecules increase of are the bulky, coordination eacheach ofof themthemnumber. usually (a) Molecules interacts interacts with withinteract only only with one one each other other other molecule. molecule. in fluids. In Inthis Because this case, case, LC-NRthe the coordination coordination molecules number are number bulky, z is each z1. is(b 1.of– (dbthem)– Generallyd) Generallyusually speaking, interacts speaking, withz valuesz onlyvalues for one one, for otherone, two, molecule.two, and andthree-dimens In three-dimensionally this case,ionally the coordination interacting interacting numberNR crystals NR zcrystals is 1.are (b 2– are(db)), Generally 24 ((bc),), and 4 (c ),6speaking, ( andd), respectively. 6 (d ),z values respectively. (fore) Schematic one, (e two,) Schematic illustrationand three-dimens illustration of the attenuationionally of the interacting attenuation of incremental NR of crystals incremental magnetic are 2 magneticfield(b), 4( (ηc)), for fieldand a 6 (certain η(d) for), respectively. a certainpair of pairmolecules. (e) of Schematic molecules. The illustrationhorizontal The horizontal ofaxis the denotes attenuation axis denotes the oftime the incremental timeelapsed elapsed sincemagnetic since the theinteractionfield interaction (η) for occurs. a occurs.certain After After pair a molecule aof molecule molecules. interacts interacts The with horizontal with another another axis molecule, molecule, denotes it it themoves moves time and and elapsed the interactioninteraction since the disappears. The effect of the exchange magnetic field disappears after time constant τ .(f) Each disappears.interaction occurs.The effect After of thea molecule exchange interacts magnetic with field another disappears molecule, after ittime moves constant and the spinτspin interaction. (f) Each moleculemoleculedisappears. changes The effect the partner of the ofexchange the interaction magnetic many field times disappears before disappearanceafter time constant of the τ eeffectspinffect. (f of)of Each thethe exchangeexchangemolecule magneticmagneticchanges the field.field. partner Each moleculeof the interaction interacts many withwith manytimes moleculesbefore disappearance as if the coordination of the effect number of the substantiallysubstantiallyexchange magnetic increasedincreased field. (zˆ ).( 𝑧Each (Reprint̂ ). (Reprint molecule with with permission interacts permission wi [47th]: many Copyright[47] :molecules Copyright 2020, as American 2020,if the coordinationAmerican Chemical Chemical Society). number substantially increased (𝑧̂ ). (Reprint with permission [47]: Copyright 2020, American Chemical 4.2. ImpuritySociety). Effect—Diradical (R,S)-3 Society). 4.2. ImpurityIt was the Effect—Diradical discovery of an(R,S)- impurity3 effect that was required to demonstrate the similarity of the4.2. positiveImpurity magneto-LCEffect—Diradical effect (R,S)- to the3 emergence of superparamagnetic domains in the SG state. It was the discovery of an impurity effect that was required to demonstrate the similarity of the We implemented the χM-T measurement for two samples with different ratios (80:20 and 95:5) of (R,S)-3 positiveIt was magneto-LC the discovery effect of anto theimpurity emergence effect thatof superparamagnetic was required to demonstrate domains in the the similarity SG state. of We the positive magneto-LC effect to the emergence of superparamagnetic domains in the SG state. We

Symmetry 2020, 12, x FOR PEER REVIEW 17 of 29 Symmetry 2020, 12, 1910 17 of 29 Symmetry 2020, 12, x FOR PEER REVIEW 17 of 29 implemented the χM-T measurement for two samples with different ratios (80:20 and 95:5) of (R,S)-3 implemented the χM-T measurement for two samples with different ratios (80:20 and 95:5) of (R,S)-3 and racemic cis-diastereomers (R*,R*)-(R*,R*)-33 (Figure(Figure3 3))[ [42].42]. These mixedmixed samplessamples couldcould stillstill showshow anan ColColhh phase.and racemic However, cis-diastereomers pure (R*,R*)-33 and(R*,R*)-and thethe 3 sample (Figure withwith 3) [42]. aa ratio These of 50:5050: mixed50 displayed samples no could LC phase.still show As shown an Colh phase. However, pure (R*,R*)-3 and the sample with a ratio of 50:50 displayed no LC phase. As shown in Figure 1717,, appreciableappreciable andand substantialsubstantial χχMM increasesincreases werewere notednoted duringduring thethe crystal-to-Colcrystal-to-Colhh-to-Iso in Figure 17, appreciable and substantial χM increases were noted during the crystal-to-Colh-to-Iso phase transition sequence for the samplessamples withwith thethe ratiosratios ofof 95:595:5 andand 80:20,80:20, respectively.respectively. TheThe χχMM phase transition sequence3 for the samples with the ratios of3 95:5 and 80:20, respectively. The χM increase waswas 11×1010− emu emu/mol/mol for for the the former former sample sample and and5 10 5×10− emu/ mol emu/mol for the for latter the one. latter As one. shown As ×1×10 × 5×10 showninincrease Figure in was13Figure, the hysteresis13, the emu/molhysteresis was not for was observed the not former observed for sample pure for (R,S)- and pure3 within (R,S)-3 the within emu/mol experimental the for experimental the error latter limits one. error inAs limitstheshown first in in heating the Figure first process. 13, heating the hysteresis In process. other words, was In othernot the observed contentwords, for ofthe superparamagnetic purecontent (R,S)- of3 withinsuperparamagnetic domainsthe experimental increased domains error by increasedaddinglimits (R*,R*)-in bythe adding first3 to (R,S)-heating (R*,R*)-3 as process.3 an to organic (R,S)- In3 impurity. asother an organic words, Furthermore, impurity. the content this Furthe organic of rmore,superparamagnetic impurity this organic effect convinced impuritydomains effectusincreased that convinced the positiveby adding us magneto-LCthat (R*,R*)- the positive3 to eff (R,S)-ect magneto-LC is3 intrinsic as an organic in effect LC nitroxide impurity. is intrinsic radicals Furthe in LC listedrmore, nitroxide in this Figure radicalsorganic3 because impuritylisted the in Figureadditioneffect convinced3 because of diastereomers theus thataddition the would positive of diastereomers not changemagneto-LC the wo total uldeffect amountnot is change intrinsic of contaminating the in total LC nitroxideamount magnetic of radicals contaminating metal listed ions, in magneticifFigure any. 3 becausemetal ions, the ifaddition any. of diastereomers would not change the total amount of contaminating magnetic metal ions, if any.

Figure 17. The temperature dependence of χM for the two samples with a different ratio (circle for Figure 17. The temperature dependence of χ for the two samples with a different ratio (circle for 80:20Figure and 17. triangle The temperature for 95:5) of dependence(R,S)-3 and (ofR* ,χR*MM )-for3 at the 0.05 two T insamples the temperature with a different range betweenratio (circle 2 and for 80:20 and triangle for 95:5) of (R,S)-3 and (R*,R*)-3 at 0.05 T in the temperature range between 2 and 40080:20 K. andOpen triangle and filled for 95:5)legends of (representR,S)-3 and the (R* firs,R*t)- heating3 at 0.05 and T in cooling the temperature processes, rangerespectively. between 2 and 400400 K. K. Open Open and and filled filled legends legends represent represent the the first first heating heating and and cooling cooling processes, processes, respectively. respectively. As shown in Figure 1818,, the H dependence of the superparamagnetic component extracted from As shown in Figure 18, the H dependence of the superparamagnetic component extracted from the M-H plots for the mixed sample of (R,S)-3 and (R*,R*)-3 (80:20) was well fittedfitted by Langevin the M-H plots for the mixed sample of (R,S)-3 and (R*,R*)-3 (80:20) was well fitted by Langevin function (Equation (1)), indicating the positive magn magneto-LCeto-LC effect effect that originates from the generation function (Equation (1)), indicating the positive magneto-LC effect that originates from the generation of superparamagnetic domains in the LC phase. The amount of super-para-magneticsuper-para-magnetic domains was of superparamagnetic domains in the LC phase. The amount of super-para-magnetic domains was enhanced compared to the one in the pure samplesample ofof ((RR,,SS)-)-33 shownshown inin FigureFigure 1313 h.h. enhanced compared to the one in the pure sample of (R,S)-3 shown in Figure 13 h.

Figure 18. The curve fitting fitting by Langevin function (Equation (1)) for the superparamagnetic domains extractedFigure 18. from The thecurve M- Hfitting plots by at Langevin 363 K for thefunction mixed (Equation sample of (1)) (R, ,Sfor)-)-33 the andand superparamagnetic ((RR*,*,RR*)-*)-33 (80:20).(80:20). domains extracted from the M-H plots at 363 K for the mixed sample of (R,S)-3 and (R*,R*)-3 (80:20).

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Thus,Thus, thethe originorigin ofof suchsuch uniqueunique magneticmagnetic propertiesproperties observedobserved inin thethe LCLC andand isotropicisotropic phasesphases ofof achiralachiral ((RR,,SS)-)-33 inin thethe absenceabsence oror presencepresence ofof thethe racemicracemic diastereomersdiastereomers ((RR*,*,RR*)-*)-33 asas thethe impurityimpurity cancan bebe rationalizedrationalized mesoscopicallymesoscopically inin termsterms ofof (i)(i) thethe preferentialpreferential formationformation ofof SG-likeSG-like inhomogeneousinhomogeneous magneticmagnetic domainsdomains consistingconsisting of of super-para-magnetic super-para-magnetic domains domains surrounded surrounded by by para-magnetic para-magnetic spins spins in thein the LC LC phase, phase, (ii) (ii) the the gradual gradual growth growth of theof the size size and and/or/or number number of super-para-magnetic of super-para-magnetic domains domains by thermalby thermal processing, processing, and and (iii) (iii) preservation preservation of of the the overall overall increased increased super-para-magnetic super-para-magnetic domains domains inin thethe solidsolid phase by cooling cooling until until cryogenic cryogenic temperatures temperatures,, inin weak weak magnetic magnetic fields fields (<0.1 (< 0.1T) (Figure T) (Figure 19). 19).

Figure 19. R S 3 Figure 19. SchematicSchematic of of super-para-magnetic super-para-magnetic domains domains in inthe the LC LCand and isotropic isotropic phases phases of (R of,S)- ( 3,. The)- . The formation of super-para-magnetic domains surrounded by para-magnetic spins by heating or in formation of super-para-magnetic domains surrounded by para-magnetic spins by heating or in the the presence of impurity (R*,R*)-3, and additional growth of the size and/or number of the domains presence of impurity (R*,R*)-3, and additional growth of the size and/or number of the domains by by heating. heating. 4.3. Magneto-Electric (ME) Effect 4.3. Magneto-Electric (ME) Effect The magneto-electric (ME) effect had been observed only for inorganic multiferroic materials such The magneto-electric (ME) effect had been observed only for inorganic multiferroic materials as YMnO3, TbMnO3, and so forth, which showed both ferroelectric and magnetic order (ferromagnetism orsuch anti-ferromagnetism) as YMnO3, TbMnO at3 cryogenic, and so temperaturesforth, which [showed17–20]. both ferroelectric and magnetic order (ferromagnetismIn this context, or anti-ferromagnetism) it was expected that aat uniquecryogenic ME temperatures effect might [17–20]. occur in the second-harmonic- generationIn this(SHG)-active context, it was [55 expected] and ferroelectric that a unique LC (SmC*) ME effect phase might of the occur (S,S in)-1 the(m second-harmonic-= n = 13), which generation (SHG)-active [55] and ferroelectric LC (SmC*) 2phase of the (S,S)-1 (m = n = 13), which displayed both an excellent ferroelectricity [PS = 24 nC/cm , τ10-90 = 213 µs, θ = 29◦, η = 73.0 m Pa s displayed both an excellent ferroelectricity [PS = 24 nC/cm2, τ10-90 = 213 μs, θ = 29°, η = 73.0 m Pa s at at 74 ◦C by the triangle wave method] in a rubbed surface-stabilized liquid-crystal sandwich cell (474µ °Cm thick)by the [triangle29] and awave positive method] magneto-LC in a rubbed effect surface-stabilized (super-para-magnetic liquid-crystal interactions) sandwich in the bulk cell (4 SmC* μm phasethick) [[29]35]. Suchand a an positive assumption magneto-LC was the case.effect The (sup temperatureer-para-magnetic dependences interactions) of relative in the paramagnetic bulk SmC* phase [35]. Such an assumption was the case. The temperature dependences of relative paramagnetic susceptibility (χrel), g-value, and ∆Hpp were measured by EPR spectroscopy using the thin sandwich cellsusceptibility into which (χ therel), most g-value, appropriate and ∆Hpp sample were measured of (S,S)-1 (mby =EPRn = spectroscopy13, 65% ee) was using loaded the thin (Figure sandwich 20). cell into which the most appropriate sample of (S,S)-1 (m = n = 13, 65% ee) was loaded (Figure 20).

Symmetry 2020, 12, 1910 19 of 29 Symmetry 2020, 12, x FOR PEER REVIEW 19 of 29

Figure 20. (a) Experimental setup to monitor the variable-temperature or electric field dependent EPR Figure 20. (a) Experimental setup to monitor the variable-temperature or electric field dependent EPR spectra of (S,S)-1 (m = n = 13, 65% ee) confined in a long 4-µm thin sandwich cell. (b) Principal axes of spectra of (S,S)-1 (m = n = 13, 65% ee) confined in a long 4-μm thin sandwich cell. (b) Principal axes of inertia and g-values of the trans-1 molecule. (Reprint with permission [37]; Copyright 2020 the Royal inertia and g-values of the trans-1 molecule. (Reprint with permission [37]; Copyright 2020 the Royal Society of Chemistry). Society of Chemistry). Since the magnetization measurement of the sample in the LC cell by SQUID magnetometry was Since the magnetization measurement of the sample in the LC cell by SQUID magnetometry was technically difficult, χpara was derived from the Bloch equation (Equation (8)) [56] by using the EPR technically difficult, χpara was derived from the Bloch equation (Equation (8)) [56] by using the EPR parameters, such as g, ∆Hpp, and maximum peak height (I0m and I0m) as described earlier [35,36], parameters, such as g, ∆Hpp, and maximum peak height (I’m and –−I’m) as described earlier [35,36],   = 2 √ χparaχpara =2 µ2BμgIBgI0m’m∆∆HH2pp/(/√3 3hνhHνH1) 1 (8)(8) wherewhereµ BμBis is the the Bohr Bohr magneton, magneton,h is Planck’s h is Planck’s constant, constant,ν is the frequency ν is the of thefrequency absorbed of electromagnetic the absorbed wave,electromagnetic and H1 is the wave, amplitude and H of1 theis oscillatingthe amplitude magnetic of the field. oscillat The relativeing magnetic paramagnetic field. susceptibilityThe relative (paramagneticχrel,T) in the absence susceptibility of an electric (χrel,T) in field the is absence defined of as: an electric field is defined as:

χrel,T = χpara/χ0 (9) χrel,T = χpara/χ0 (9) where χ0 is the standard paramagnetic susceptibility at 30 °C in the heating process. First, the wheregenerationχ0 is theof standardpositive magneto-LC paramagnetic effect susceptibility and ferroelectric at 30 ◦C in switching the heating in process. the liquid First, crystal the generation cell was ofconfirmed. positive magneto-LCBy measuring e fftheect temperature and ferroelectric dependence switching of inχrel, theT under liquid the crystal conditions cell was in confirmed.which the Byapplied measuring magnetic the temperaturefield was parallel dependence (Configuration of χrel,T under A) and the perpendicular conditions in (Configuration which the applied B) to magnetic the cell fieldsurface was and parallel the rubbing (Configuration direction A) in andthe perpendicularabsence of an electric (Configuration field, considerable B) to the cell χrel, surfaceT increases and (ca. the rubbing40%) together direction with in large the absence ∆Hpp ones, of anwhich electric were field, noted considerable at the crystal-to-LCχrel,T increases phase (ca. transition 40%) together in both withcases large (Figure∆H 21).pp ones, The whichferroelectric were notedswitching at the at crystal-to-LC 25 V was verified phase by transition polarized in bothoptical cases microscopy (Figure 21 in). Thethe absence ferroelectric of a switchingmagnetic field. at 25 VThe was bright verified fan-shaped by polarized texture optical at −25 microscopy V and the indark the one absence at +25 of V a magneticwere distinctly field. observed The bright [37]. fan-shaped texture at 25 V and the dark one at +25 V were distinctly − observed [37].

Symmetry 2020, 12, 1910 20 of 29 Symmetry 2020, 12, x FOR PEER REVIEW 20 of 29

Figure 21. Temperature dependence of (a and d) χ , (b and e) ∆H and g-value (c and f) for (S,S)-1 Figure 21. Temperature dependence of (a and d) χrel,rel,T, (b and e) ∆Hpppp and g-value (c and f) for (S,S)-1 (m = n = 13,13, 65% 65% ee) confined confined in aa thinthin rubbedrubbedsandwich sandwich cell cell by by EPR EPR spectroscopy spectroscopy at at a magnetica magnetic field field of ofaround around 0.33 0.33 T. T. The The magnetic magnetic field field was was applied applied ( a(–ac–)c) parallel parallel and and ( (dd––ff)) perpendicularperpendicular toto the rubbing direction. The The LC LC transition transition temperatures temperatures in the heating and cooling runs are shown in the lower and upper sidessides ofof the the panels, panels, respectively. respectively. (Reprint (Reprint with with permission permission [37]; Copyright[37]; Copyright 2020 the 2020 Royal the SocietyRoyal Societyof Chemistry). of Chemistry). The difference in temperature dependence of g-values along two direction shows that the magnetic The difference in temperature dependence of g-values along two direction shows that the susceptibility of the (S,S)-1 clearly has an anisotropy with respect to the molecular axis. Since the Curie magnetic susceptibility of the (S,S)-1 clearly has an anisotropy with respect to the molecular axis. constant revealed S = 1/2 nature of the system, the anisotropy stems from high-order perturbation term Since the Curie constant revealed S = 1/2 nature of the system, the anisotropy stems from high-order of exchange interactions with respect to spin-orbit interactions, such as Dzyaloshinkii-Moriya (DM) perturbation term of exchange interactions with respect to spin-orbit interactions, such as interactions. Moreover, the temperature dependence in the case of Configuration A (Figure 21c) shows Dzyaloshinkii-Moriya (DM) interactions. Moreover, the temperature dependence in the case of anomalies in the vicinity of ferroelectric LC (FLC) transition. That implies some changes in magnetism Configuration A (Figure 21c) shows anomalies in the vicinity of ferroelectric LC (FLC) transition. That implies some changes in magnetism in connection with ferroelectricity, while little anomaly was found in the case of configuration B (Figure 21f).

Symmetry 2020, 12, 1910 21 of 29 in connection with ferroelectricity, while little anomaly was found in the case of configuration B Symmetry 2020, 12, x FOR PEER REVIEW 21 of 29 (Figure 21f).

On theOn otherthe other hand, hand, the the relative relative paramagnetic paramagnetic susceptibilitysusceptibility (χ (χrel,rel,E) Ein) the in the presence presence of an of electric an electric fieldfield is defined is defined as: as: χrel,E = χpara/χ1 (10) χrel,E = χpara/χ1 (10) wherewhereχ1 is χ the1 is standard the standard paramagnetic paramagnetic susceptibility susceptibility at at the the initial initial potential potential of of+ 25+25 V V and and at at 75 75◦ °C.C. Next, the electricNext, the field electric dependences field dependences of χrel, ofE, χ∆rel,HEpp, ∆,H andpp, andg-value g-value were were plottedplotted for for the the (S (,S)-,S1)- (m1 (m= n = 13,n = 13, 65% ee),65% displayingee), displaying each each hysteresis hysteresis between between+ +2525 V and −2525 V V when when the the magnetic magnetic field field was wasapplied applied − only perpendicularlyonly perpendicularly to theto the electric electric field field and and parallel parallel to the the rubbing rubbing direction direction (Figure (Figure 22). 22 ).

FigureFigure 22. Electric 22. Electric field field dependence dependence of ofg-value, g-value,χ χrel,Erel,E,, and and ∆∆HHpppp forfor the the FLC FLC phase phase of (S of,S)- (S1, S(m-n)-1 (m-n = 13, = 13, 65% ee)65% confined ee) confined in ain thin a thin rubbed rubbed sandwich sandwich cellcell at 75 °C◦C by by EPR EPR spectroscopy spectroscopy at a at magnetic a magnetic field fieldof of aroundaround 0.33 T.0.33 The T. The magnetic magnetic field field was was applied applied (a (–ac–)c perpendicularly) perpendicularly to the the electric electric field field and and parallel parallel to the rubbingto the rubbing direction direction and (d –andf) parallel (d–f) parallel to the electricto the electric field and field perpendicular and perpendicular to the to rubbing the rubbing direction. Opendirection. and filled Open circles and represent filled circles the represent application the application of electric fieldsof electric from fields+25 from V to +2525 V V to and −25 fromV and 25 V − − to +25from V,respectively. −25 V to +25 V, (Reprint respectively. with (Reprint permission with [perm37]; Copyrightission [37]; Copyright 2020 the Royal 2020 the Society Royal ofSociety Chemistry). of Chemistry).

Symmetry 2020, 12, 1910 22 of 29

It is considered that the molecules with the spontaneous electric polarization is inversed by sweeping the external electric field. Therefore, it is noteworthy that the spin inversion accompanies the inversion of the molecule, which can be regarded as a nonlinear ME effect. Since the usual paramagnetic components cannot show such a nonlinear hysteresis loop, the hysteresis indicates the spin flipping of superparamagnetic domains. This result clearly shows the predominance of the electric field over a magnetic field in controlling the super-para-magnetic behavior, since the spin flipping by the electric field occurs even under a large magnetic field of 0.33 T with a constant direction used for this EPR measurement. The direction-of-magnetic field dependence of the g-value, χrel,E, and ∆Hpp indicates that the super-para-magnetic domains are directed almost perpendicularly to the spontaneous electric polarization. For both the nonlinear ME effect and the direction-of-magnetic field dependence, the high-order perturbation term of exchange interactions with respect to spin-orbit interactions is usually necessary for this S = 1/2 system. In summary of Section4, the unique magnetic properties observed for LC nitroxide radicals are characterized in connection with SG-like magnetic features (see Section 3.1) as follows:

1. The hysteresis observed by measurement of the temperature dependence of magnetic susceptibility (Figures 10a, 11 and 13) corresponds to the emergence of superparamagnetic domains in the LC and isotropic phases. 2. The superparamagnetic domains grow in the LC phases under external magnetic fields with the elapse of time in minutes (Figures 14 and 15). 3. The superparamagnetic components considerably grow with the increasing impurity content or inhomogeneity (Figure 17).

As shown in Figures 18 and 19, these results imply the formation of the super-para-magnetic domains, most likely due to the local inhomogeneity in LC phases. Therefore, this picture is analogous to the cluster glass (See Section 3.3.2), or super-para-magnetic system without magnetic interactions. A partially broken degree of freedom in LC phases is considered to affect the magnetic properties of the domains. The super-para-magnetic response to the external magnetic field is due to the domains’ rotational degree of freedom. Likewise, the factors of molecular mobility and intermolecular interactions resulting in a favorable molecular correlation in LC phases are likely to contribute to the long-time scale growth of superparamagnetic domains. To comprehend the microscopic state of the superparamagnetic domains, we have to investigate the dynamics and interactions between the domains. Elucidation of the microscopic mechanism for the formation of super-para-magnetic domains in the LC nitroxide radicals is a challenging subject. In this context, we discovered that the nonlinear ME effect occurs in the LC nitroxide radicals (Figure 22). Namely, the super-para-magnetic domains turned out to be controllable by external electric fields. This result demonstrates the ferroelectric aspects of super-para-magnetic domains. Accordingly, in Section5, we propose and discuss the mechanism of the positive magneto-LC effect on the basis of the nonlinear ME effect. Meanwhile, quite recently another mechanism has been suggested, in which the positive magneto-LC effect can be accounted for microscopically in terms of the molecular mobility and the resulting inhomogeneity of intermolecular interactions without assuming the formation of superparamagnetic domains by means of MD simulation and DFT calculations (Figure 16)[47]. Namely, the magnetic features seem to reflect the macroscopic inhomogeneity of the liquid crystal orientation field, even though the magnetic properties look like those resulting from the superparamagnetic domains. Since the macroscopic studies on this mechanism is under investigation, only the former mechanism based on the ME effect is discussed in Section5. In addition, it is fairly possible that both mechanisms operate complementarily. The latter mechanism originating from the molecular mobility is predominant in the LC and isotropic phases, while the formation of super-para-magnetic domains by the former mechanism results in the preservation of χTIM in the supercooled LC and solid phases during the cooling process. Symmetry 2020, 12, x 1910 FOR PEER REVIEW 23 of 29

5. Proposed Mechanism 5. Proposed Mechanism The existence of superparamagnetic domains in rod-like LC nitroxide radicals means the The existence of superparamagnetic domains in rod-like LC nitroxide radicals means the ferromagnetic spin arrangement inside the domains. However, ferromagnetic ordering through the ferromagnetic spin arrangement inside the domains. However, ferromagnetic ordering through direct exchange interactions between radical spins at ambient or higher temperatures in metal-free the direct exchange interactions between radical spins at ambient or higher temperatures in metal-free organic radicals had been believed to be unrealistic. Moreover, the molecular rotation of as fast as organic radicals had been believed to be unrealistic. Moreover, the molecular rotation of as fast as around 1010 s−1 in a rod-like LC phase may also make the average magnetic interactions much weaker around 1010 s 1 in a rod-like LC phase may also make the average magnetic interactions much weaker than in a crystalline− phase [25,26]. Contrary to such general believing, the positive magneto-LC effect than in a crystalline phase [25,26]. Contrary to such general believing, the positive magneto-LC effect has been observed in the rod-like LC nitroxide radicals with a specific molecular structure, but not in has been observed in the rod-like LC nitroxide radicals with a specific molecular structure, but not in the homologous non-LC nitroxide radicals. the homologous non-LC nitroxide radicals. Here, we propose the parasitic ferromagnetism [57] in terms of the nonlinear ME effect, that is, Here, we propose the parasitic ferromagnetism [57] in terms of the nonlinear ME effect, that is, the induction of ferromagnetism by the emergence of the domains with spontaneous electric the induction of ferromagnetism by the emergence of the domains with spontaneous electric polarization polarization (Figure 23). If there is a cross-correlation term between magnetization (M) and electric (Figure 23). If there is a cross-correlation term between magnetization (M) and electric polarization (P) polarization (P) in the free energy, it is plausible that the emergence of M can lower the free energy in the free energy, it is plausible that the emergence of M can lower the free energy of the system in the of the system in the presence of P. presence of P.

Figure 23. Two possible mechanisms for the emergence of ferromagnetism in LC nitroxide radicals Figure 23. Two possible mechanisms for the emergence of ferromagnetism in LC nitroxide radicals induced via DM interactions. (a) The induction of weak ferromagnetism (M) directed perpendicularly induced via DM interactions. (a) The induction of weak ferromagnetism (M) directed perpendicularly to the electric polarization (P). L denotes the antiferromagnetic vector S1 S2.(b) A helical magnetic to the electric polarization (P). L denotes the antiferromagnetic vector S1− S2. (b) A helical magnetic structure induced by helical molecular alignment in the LC phase. The following are the explanation of structure induced by helical molecular alignment in the LC phase. The following are the explanation individual symbols and signs in panel b—Spheroid: the rod-like LC nitroxide radical molecule, pi: the of individual symbols and signs in panel b—Spheroid: the rod-like LC nitroxide radical molecule, pi: electric polarization of the i-th molecule, pij: the summed electric polarization between the i-th and j-th the electric polarization of the i-th molecule, pij: the summed electric polarization between the i-th and molecules, and eij: translational vector directed to the screw axis. j-th molecules, and eij: translational vector directed to the screw axis. For the operation of this mechanism, the existence of spontaneous polarization (Ps) is necessary, For the operation of this mechanism, the existence of spontaneous polarization (Ps) is necessary, which emerges as the result of breaking the space inversion symmetry. As described in Section 4.3, which emerges as the result of breaking the space inversion symmetry. As described in Section 4.3, the Ps is already present in the LC phases composed of chiral nitroxide radical molecules. We could the Ps is already present in the LC phases composed of chiral nitroxide radical molecules. We could obtain the following three experimental results that indicated the correlation of super-para-magnetic obtain the following three experimental results that indicated the correlation of super-para-magnetic behavior and the breaking of space inversion symmetry. Firstly, the super-para-magnetic domains behavior and the breaking of space inversion symmetry. Firstly, the super-para-magnetic domains grow during the crystal-to-LC-to-Iso phase transition sequence in the heating process, in which the grow during the crystal-to-LC-to-Iso phase transition sequence in the heating process, in which the birth and growth of domains without space inversion symmetry was revealed by the gradual increase birth and growth of domains without space inversion symmetry was revealed by the gradual increase in the SHG signal (Figure 14). Second, the result that the super-para-magnetic domains augmented in the SHG signal (Figure 14). Second, the result that the super-para-magnetic domains augmented with the increasing ee value of the LC compounds (See Section 4.1.1) is likely to demonstrate the with the increasing ee value of the LC compounds (See Section 4.1.1) is likely to demonstrate the correlation of the positive magneto-LC effect and the breaking of space inversion symmetry. Third, correlation of the positive magneto-LC effect and the breaking of space inversion symmetry. Third, and more importantly, it was the external electric field that flipped the g-value of the EPR signal to draw and more importantly, it was the external electric field that flipped the g-value of the EPR signal to draw a ferromagnetic-like hysteresis loop (Figure 22a). These results suggest that the positive

Symmetry 2020, 12, 1910 24 of 29 aSymmetry ferromagnetic-like 2020, 12, x FOR hysteresisPEER REVIEW loop (Figure 22a). These results suggest that the positive magneto-LC24 of 29 effect is driven by collective alignment of LC molecules and the subsequent breaking of its space inversionmagneto-LC symmetry. effect is driven by collective alignment of LC molecules and the subsequent breaking of its space inversion symmetry. P As one of the origins of nonlinear ME effect, DM interactions, EDM Dij Si Sj, are known. As one of the origins of nonlinear ME effect, DM interactions, 𝐸~ ∑∼ij𝑫 ∙𝑺· ×𝑺× , are known. The Dzyaloshinkii vector vector (D (Dij)ij )can can be be present present when when the thespace space inversion inversion symmetry symmetry is broken. is broken. The Theexistence existence of DM of DM interactions interactions in LC in LC nitroxi nitroxidede radicals radicals were were indicated indicated by by the the anisotropic g-value-value (Figure 21c).21c). Next, Next, we we discuss discuss two two possibilities possibilities for forparasitic parasitic ferromagnetism ferromagnetism on the on basis the basis of DM of DMinteractions. interactions. The firstfirst possibility is thethe emergenceemergence ofof weakweak ferromagnetismferromagnetism accompanyingaccompanying the electric polarization (Figure 2222a).a). Such Such weak weak ferromagnet ferromagnetismism is is suggested when the contribution of the energy term, EPLMPLM~𝑷P ∙ (𝑳×𝑴L M),, exists exists in in the the free free energy energy of the syst systemem [58]. [58]. This This is is a a phenomenological phenomenological ∼ · × term resultingresulting from from DM DM interactions, interactions, where Pwheredenotes 𝑷 the denotes electric polarization,the electric L polarization,is antiferromagnetic 𝑳 is vector S1 S2, and M is the𝑺 remanent−𝑺 magnetization𝑴 perpendicular to L (i.e., weak ferromagnetism).𝑳 antiferromagnetic− vector , and is the remanent magnetization perpendicular to (i.e., IfweakP is ferromagnetism). present in the system, If 𝑷 the is present emergence in the of system,M leads the to aemergence finite EPLM ofvalue. 𝑴 leads Therefore, to a finite the E cantingPLM value. of anti-ferromagneticTherefore, the canting pairs of anti-ferromagnetic of S1 and S2 to the pairs direction of 𝑺 perpendicular and 𝑺 to the to direction the electric perpendicular polarization to canthe lowerelectric the polarization total energy can of lower the system the total by EenergyPLM (Figure of the 23 systema). As by expressed EPLM (Figure in L, 23a). large As anti-ferromagnetic expressed in 𝑳, correlationslarge anti-ferromagnetic between spins correlations by sufficient between intermolecular spins by sufficient contacts are intermolecular needed for this contacts mechanism. are needed It is highlyfor this plausible mechanism. that It the is largehighly molecular plausible mobility that the in large the LCmolecular and isotropic mobility phases in the is likelyLC and responsible isotropic forphases the inhomogeneousis likely responsible intermolecular for the inhomogeneou contact (Figures intermolecular 16)[47]. This contact picture (Figure is consistent 16) [47]. with This the resultspicture obtainedis consistent from with the experimentthe results obtained on the ME from effect the (Figure experiment 22). Both on the the MEg-value effect and (Figure the line 22). width Both (the∆H PPg-value) are switched and the byline the width external (∆H electricPP) are field switched (E) when by itthe is perpendicularexternal electric to thefield external (E) when magnetic it is fieldperpendicularH0 (Figure to 24 the). external magnetic field H0 (Figure 24).

Figure 24.24.Image Image of of domains domains with with electric electric polarization polarization and the and resulting the resulting weak ferromagnetism weak ferromagnetism proposed inproposed Figure 23ina. Figure The light 23a. blue-colored The light blue-colored background background denotes the denotes paramagnetic the paramagnetic LC region. Response LC region. of domainsResponse toof thedomains external to electricthe external field electric rather thanfield therather perpendicular than the perpendicu magneticlar field magnetic in theexperiment field in the onexperiment ME effect on (Figure ME effect 22). Note(Figure that 22). the Note correspondence that the correspondence of the direction of of the magnetic direction moments of magnetic with observedmoments gwith-value observed depends g-value on the depends internal magneticon the internal field atmagnetic resonant field spin at sites. resonant spin sites.

The secondsecond possibility possibility is theis the emergence emergence of helical of he magnetismlical magnetism by helical by alignmenthelical alignment of LC molecules of LC (Figure 23b). It is known that, in the helical magnetic structures, the spins Si and Sj can𝑺 induce𝑺 the molecules (Figure 23b). It is known that, in the helical magnetic structures, the spins and can electricinduce the polarization electric polarization (pij) to satisfy (pij)p to satisfyeij 𝒑Si ~ S𝒆j due×𝑺 to×𝑺 inverse due DM to inverse interactions, DM interactions, where eij is ij ∼ × × 𝒋 thewhere unit 𝒆 vector is the connecting unit vector the connectingi-th and j-th the sites i-th [59 and]. In j-th contrast, sites [59]. in the In case contrast, of LC nitroxidein the case radicals, of LC thenitroxide helical radicals, structure the formed helical bystructure local p ijformedfrom theby ilocal-th and 𝒑 j-thfrom molecule the i-th mightand j-th induce molecule the helicalmight S S M structureinduce the of helicali and structurej. This helical of 𝑺 magnetic and 𝑺 structure. This helical can have magnetic the net st magnetizationructure can havealong the thenet magnetization M along the direction of the screw axis. To the best of our knowledge, although there is no example for the helical electric polarization-induced helical magnetization, our proposed mechanism is likely to explain the unique magnetic properties observed in chiral helical LC phases (Figures 10 and 11) or achiral LC phases containing partial chiral helical domains (Figure 14).

Symmetry 2020, 12, 1910 25 of 29

Symmetrydirection 2020 of, the12, xscrew FOR PEER axis. REVIEW To the best of our knowledge, although there is no example for the25 helical of 29 electric polarization-induced helical magnetization, our proposed mechanism is likely to explain the As for the correlation of super-para-magnetism and electric polarization, similar magnetic uniqueSymmetry magnetic2020, 12, x FOR properties PEER REVIEW observed in chiral helical LC phases (Figures 10 and 11) or achiral25 of LC 29 properties can be found in relaxors [60]. A relaxor can be regarded as an electric version of SG because phases containing partial chiral helical domains (Figure 14). it has randomly directed ferroelectric domains called Polar Nano Regions (PNR) due to the inherent As for the correlationcorrelation ofof super-para-magnetismsuper-para-magnetism and electricelectric polarization,polarization, similar magnetic randomness and frustration in the system (Figure 25). Recently, the emergence of super-para- properties can can be be found found in inrelaxors relaxors [60]. [60 A]. relaxor A relaxor can be can regarded be regarded as an electric as an electric version versionof SG because of SG magnetism has been observed at high temperatures with respect to some relaxors (or the system with becauseit has randomly it has randomly directed ferroelectric directed ferroelectric domains called domains Polar called Nano Polar Regions Nano (PNR) Regions due to (PNR) the inherent due to relaxor-like behavior) containing magnetic metal ions [61,62] (Figure 26). For the mechanism of such therandomness inherent and randomness frustration and in frustration the system in (Figur the systeme 25). (FigureRecently, 25 ).the Recently, emergence the of emergence super-para- of super-para-magnetic relaxors, the presence of DM interactions are suggested, which will give weak super-para-magnetismmagnetism has been observed has been at observedhigh temperatures at high temperatures with respect withto some respect relaxors to some (or the relaxors system (or with the ferromagnetism perpendicular to the polarization of PNR [61]. In addition, the MD simulation systemrelaxor-like with behavior) relaxor-like containing behavior) magnetic containing metal magneticions [61,62] metal (Figure ions 26). [61 For,62 ]the (Figure mechanism 26). Forof such the showed that the addition of spherical apolar impurity to spheroidal polar particles produces PNR to mechanismsuper-para-magnetic of such super-para-magnetic relaxors, the presence relaxors, of DM interactions the presence are of suggested, DM interactions which will are suggested,give weak show the typical behavior of relaxors [63]. This seems analogous to the formation of ferroelectric whichferromagnetism will give weakperpendicular ferromagnetism to the polarization perpendicular of toPNR the polarization[61]. In additi ofon, PNR the [ 61MD]. Insimulation addition, domains in LC nitroxide radical samples by an impurity effect (See Section 4.2). theshowed MD simulationthat the addition showed of that spherical the addition apolar ofimpuri sphericalty to apolarspheroidal impurity polar to particles spheroidal produces polar particles PNR to producesshow the PNRtypical to showbehavior the typicalof relaxors behavior [63].of This relaxors seems [63 analogous]. This seems to the analogous formation to the of formationferroelectric of ferroelectricdomains in LC domains nitroxide in LC radical nitroxide samples radical by an samples impurity by aneffect impurity (See Section effect (See4.2). Section 4.2).

Figure 25. Image of PNR in relaxor systems. The electric polarization of each PNR is directed randomly. Figure 25.25. ImageImage of of PNR PNR in relaxorin relaxor systems. systems. The electricThe electric polarization polarization of each PNRof each is directed PNR is randomly. directed randomly.

Figure 26. (a) The hysteretic temperature dependence of magnetic susceptibility and (b) the Figure 26. (a) The hysteretic temperature dependence of magnetic susceptibility and (b) the super- super-para-magnetic behavior in the magnetization curves in ZnO Co relaxor-like nanocomposite thin para-magnetic behavior in the magnetization curves in ZnO−Co relaxor-like− nanocomposite thin films films [60]. The labels (a,b) in the original graphs were moved out of the frames by the authors of this [60]. The labels (a,b) in the original graphs were moved out of the frames by the authors of this paper. paper.Figure (Reprint26. (a) The with hysteretic permission temperature [62]; Copyright dependence 2020, American of magnetic Chemical susceptibility Society). and (b) the super- (Reprintpara-magnetic with permission behavior in [62]; the Copyrimagnetizationght 2020, curves American in ZnO Chemical−Co relaxor-like Society). nanocomposite thin films In[60]. some The labels of these (a,b) super-para-magnetic in the original graphs were relaxors, moved it out was of the revealed frames by that the the authors super-para-magnetic of this paper. domainsIn(Reprint some were withof identical these permission super-para-magnetic toPNR [62]; Copyri by estimatingght 2020, relaxors, American their sizesit was Chemical [61 revealed]. In Society). the that case the of LCsuper-para-magnetic nitroxide radicals, domainswe must were clarify identical the detailed to PNR relationship by estimating between their si thezes magnetic[61]. In the and case electric of LC propertiesnitroxide radicals, in order we to must Inclarify some theof thesedetailed super-para-magnetic relationship between relaxors, the magneticit was revealed and electric that the properties super-para-magnetic in order to uncoverdomains the were mechanism identical tofor PNR the birthby estimating and grow their of su siper-para-magneticzes [61]. In the case domains, of LC nitroxide which radicals,will lead weto themust elucidation clarify the of detailed the microscopic relationship mechanism between ofthe positive magnetic magneto-LC and electric effect properties that LC in nitroxideorder to radicalsuncover exhibited. the mechanism for the birth and grow of super-para-magnetic domains, which will lead to the elucidation of the microscopic mechanism of positive magneto-LC effect that LC nitroxide radicals exhibited.

Symmetry 2020, 12, 1910 26 of 29 uncover the mechanism for the birth and grow of super-para-magnetic domains, which will lead to the elucidation of the microscopic mechanism of positive magneto-LC effect that LC nitroxide radicals exhibited.

6. Conclusions and Prospects Since 2008, we have reported that a series of chiral and achiral all-organic LC nitroxide radicals having one or two PROXYL units in the core position display SG-like superparamagnetic features, such as a magnetic hysteresis (referred to as positive magneto-LC effect), and thermal and impurity effects during a heating and cooling cycle in weak magnetic fields. In general, the enantiomerically-enriched rod-like LC nitroxide radicals trans-1 and their derivatives always showed a distinct positive magneto-LC effect, irrespective of the molecular positive or negative dielectric anisotropy (∆ε), whereas the corresponding racemic samples exhibited the positive or negative magneto-LC effect, depending on the negative or positive sign of molecular ∆ε, respectively. Furthermore, for the first time, the magneto-electric (ME) effect was observed in the ferroelectric (SmC*) phase of S,S-enriched 1 (m = n = 13) at temperatures as high as 75 ◦C. Achiral meso diradical compound (R,S)-3 became SHG-active gradually in the discotic hexagonal columnar phase by heating in the presence of a magnetic field to form a chiral helical columnar structure and eventually showed a distinct positive magneto-LC effect. It is also noticeable that a very large impurity effect was observed for this compound when 20 mol% of racemic cis-diastereomers (R*,R*)-3 was added to the host (R,S)-3 as the impurity. By comparison of these experimental results with the well-known magnetic properties of SG materials and on the basis of the results of nonlinear ME effect, we suggest that the positive magneto-LC effect, i.e., partial formation of super-para-magnetic domains in the major paramagnetic spins in the LC phase, is most likely to originate from the emergence of weak ferromagnetism accompanying the electric polarization due to DM interactions and/or the emergence of helical magnetism by helical alignment of LC molecules. For the practical application of the positive magneto-LC effect to organic materials science, it is essential to enlarge the ratio of superparamagnetic domains to paramagnetic spins in the LC and isotropic phases. The utilization of the impurity effect to form SG-like inhomogeneous magnetic domains is quite promising because we can select the most suitable compounds among a variety of candidate magnetic and nonmagnetic organic compounds miscible with LC phases as the impurities. The electric field control of ferromagnetic domains is another interesting choice. However, this needs further studies to understand the detailed mechanism of the positive magneto-LC effect. These studies will offer novel access to the realization and application of ferromagnetic materials composed of organic radical spins. At the same time, it will be possible to measure the AC magnetic susceptibility at low or even higher temperatures for elucidating the mechanism of positive magneto-LC effect owing to a considerable increment of the super-para-magnetic region surrounded by the para-magnetic spins.

Author Contributions: Conceptualization, S.S. and Y.U.; methodology, Y.U. and R.T.; validation, S.S., Y.U. and R.T.; formal analysis, S.S. and Y.U.; investigation, Y.U. and R.T.; resources, R.T.; data curation, S.S. and Y.U.; writing—original draft preparation, S.S. and R.T.; writing—review and editing, R.T. and S.S.; supervision, R.T.; administration, R.T.; funding acquisition, R.T. and Y.U. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by JSPS KAENHI (Grant number 26248024). Conflicts of Interest: The authors declare no conflict of interest.

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