Temperature dependence of the spin relaxation time of Fe3O4 and hemozoin superparamagnetic nanocrystals

Chemical Physics 493 (2017) 120–132

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Temperature dependence of the spin relaxation time of Fe3O4 and hemozoin superparamagnetic nanocrystals I. Khmelinskii a, V. Makarov b,⇑ a b

University of the Algarve, FCT, DQF, 8005-139 Faro, Portugal University of Puerto Rico, Rio Piedras Campus, PO Box 23343, San Juan, PR 00931-3343, USA

a r t i c l e

i n f o

Article history: Received 9 February 2017 In final form 29 June 2017 Available online 30 June 2017

a b s t r a c t We report experimental temperature and concentration dependences of the natural spin relaxation time of superparamagnetic Fe3O4 and hemozoin nanocrystals. We recorded the 1H NMR spectrum of 0.5% benzene dissolved in CS2 in function of superparamagnetic particle concentration and temperature, interpreting the 7.261 ± 0.002 ppm benzene line broadening. Our model for the line broadening includes natural, hyperfine magnetic dipole-dipole, and contact hyperfine contributions. The latter arises due to exchange interaction between benzene molecules and suspended nanoparticles. Estimated frequency of fluctuation in the 1 cm3 sample volume is in the 107 Hz scale. Estimated natural electron spin-lattice relaxation frequencies of the superparamagnetic nanocrystals using frequency of fluctuations, and developed theoretical model applied to analysis of experimental data are in good agreement between each other. Thus the presently developed approach may be used to study fluctuations and natural spin-lattice relaxation frequencies in different media. Published by Elsevier B.V.

1. Introduction Recent rapid development of nanoscience and nanothechnology generated high interest in superparamagnetic nanosystems, smaller than the magnetic domain in the respective ferromagnetic material [1,2]. The evaluation of the total spin relaxation times in such systems, apart from being a fundamental problem [3] is important for the development of thermotherapy-related medical applications [4,5]. The Fe3O4 (magnetite) nanoparticles smaller than 3 nm are superparamagnetic [6]. Similarly, hemozoin is also superparamagnetic, even as nanocrystals with their larger dimension of about 200–300 nm [7]. Therefore, it is very important and interesting to compare the spin relaxation properties of these two systems. While the superparamagnetic properties of Fe3O4 have been studied quite extensively [8–10], the information available for hemozoin is clearly insufficient. Hemozoin is the primary product of heme detoxification in malaria parasites [11] that convert about 30% of the heme into hemozoin [12]. Thus, many malaria treatments are directed at disruption of crystallization of hemozoin [11–20]. Natural hemozoin crystals are prismatic with the maximum dimension in the 50–1000 nm range [16,21,22]. Hemozoin is produced upon ⇑ Corresponding author. E-mail address: [email protected] (V. Makarov). http://dx.doi.org/10.1016/j.chemphys.2017.06.015 0301-0104/Published by Elsevier B.V.

polymerization of the b-hematin dimer [23–25]. It was possible to artificially produce b-hematin polymeric crystals in the 50 nm–20 lm range [22]. Erythrocytes infested with parasites may also be separated in strongly non-homogeneous magnetic fields with high field gradients [26–31]. Therefore, the magnetic properties of hemozoin are inconsistent with mere paramagnetism, as the paramagnetics have very low magnetic permeabilities. It is well known that the magnetic properties of different systems depend on the particle size. Thus, the nanoparticles of ferromagnetic material smaller than the respective domain are superparamagnetic. On the other hand, such a nanoparticle contains N paramagnetic centers, each with the spin si, all coupled to each other by the exchange interaction that may be presented as follows: N X ^ exch ¼  J ^si  ^sj V ij i¼1 j–i

where Jij is the exchange integral between the paramagnetic centers i and j, and si is the spin operator of the ith paramagnetic center. Provided the exchange interaction is much larger than kBT, the energy gap between the closest electronic states is also much larger than kBT, therefore only the lowest electronic state is populated. Taking into account that the electronic motion has the highest degree of correlation in the state with the maximum spin angular

I. Khmelinskii, V. Makarov / Chemical Physics 493 (2017) 120–132

momentum, the lowest electronic state has the maximum spin angular momentum (Hund’s rule). Therefore, what we have is a superparamagnetic nanoparticle with the magnetic moment gslBNR|si|. This spin-induced magnetic moment may be coupled to the lattice of the nanoparticle by an interaction of the local orbital magnetic moments (spin-orbit interaction) or vibrational magnetic moments (spin-vibrational interaction) with the lattice. Such interactions may significantly reduce the spin relaxation time. We thus infer that the measurements of the spin relaxation time of superparamagnetic nanoparticles will probe the spin-orbital and spin-vibrational interactions, helping to elucidate the detailed spin-relaxation mechanism in such systems. We already reported [7] direct measurements of the magnetic permeability of hemozoin nanocrystals, concluding that they are superparamagnetic. However, earlier reports found paramagnetic Fe3+ centers with the electronic spin S = 5/2 in hemozoin nanocrystals [32,33]. This paramagnetism was explained by the EPR response of isolated Fe3+ paramagnetic centers present in the defects of hemozoin nanocrystals [7]. However, there are no known spin relaxation time measurements in such systems. On the other hand, while the magnetic structure and properties of the Fe3O4 nanocrystals have been studied quite extensively [34–37], their spin relaxation times are only known in a limited range of the experimental conditions [38]. Presently we report experimental measurements of the 1H NMR linewidth of benzene in presence of artificial hemozoin nanocrystals with the average size of 17.5 ± 3.5 nm and of Fe3O4 nanocrystals with the average size of 3.7 ± 1.3 nm in a range of temperatures. We developed theoretical tools to analyze the NMR line broadening. Note that 1H NMR was discussed very extensively [39–41]. Our present theoretical model includes the parameters describing the relaxation properties of the 1H nuclear spin and those of the electronic spin of superparamagnetic nanoparticles interacting with the benzene protons. A detailed analysis of the experimental data produced electronic spin relaxation rates for both hemozoin and Fe3O4 superparamagnetic nanocrystals. The numerical analysis employed a homemade FORTRAN code.

121

Fig. 1. TEM image of commercial hemozoin nanocrystals.

used without any additional treatment. Benzene and carbon disulfide (99.9%, Sigma-Aldrich) were used as received. 2.2. Experimental

2. Experimental methods and materials

NMR spectra of samples containing 0.5% v/v benzene, 99.5% of CS2 and a certain amount of nanocrystals were recorded using a Bruker FOURIER-300 MHz NMR equipped with a 16-position SampleXpress Lite sample changer, 5 mm DUL 1H and 13C EasyProbe with z-PFG, operating with TOPSPIN 3.1 software under Windows 7 Professional. The sample temperature was controlled between 30 °C and 120 °C using ER 4131VT temperature control system (Bruker Inc.). Digital signal recording provided for averaging the signal over multiple measurement cycles.

2.1. Materials

2.3. Sample preparation

Commercial hemozoin (InvivoGen, France; 93–95%) was used in direct measurements of hemozoin electronic spin relaxation time after 30 min ultrasonification of hemozoin suspension in CS2. The latter procedure was performed to obtain homogeneous distribution of single hemozoin nanocrystals in the suspension. The geometry of hemozoin nanocrystals was evaluated using an H-9500 TEM, with a typical image shown in Fig. 1. The sample for TEM was prepared using ultrasonicated hemozoin embedded into EMbed 812 EMS epoxy resin and LKB Ultratome (LKB-Produkter, Bromma, Sweden). It follows from Fig. 1 that hemozoin nanocrystals are prismatic, 17.5 ± 3.5 nm long, with a square cross-section 3.1 ± 0.3 nm on each side. The crystal structure of hemozoin was reported earlier [42,43]. Taking into account the nanocrystal structure, we infer the existence of strong anisotropy of the g-factor, provided the total spin angular momentum is strongly coupled to the crystal frame of reference via spin-orbital and spin-vibrational interactions, as will be discussed below. Commercial iron (II,III) oxide nanopowder (Sigma Aldrich; Special order) with the average nanocrystal size of 3.7 ± 1.3 nm was

Samples for NMR measurements were prepared as follows: 0.5 ml benzene was dissolved in 99.5 ml of CS2, 1 ml of this solution was transferred to a 1.5 ml glass vial, 6 mg of commercial hemozoin was added to the vial, the mixture was sonicated for 20 min, 0.2 ml of the sample was transferred to the NMR test tube, and 0.2 ml of benzene solution was added to the remaining sample (0.8 ml). The last two steps were repeated seven more times, with the total of eight samples thus prepared at increasing dilutions; the control sample contained 0.5% (v/v) C6H6 in CS2. The Fe3O4 samples were prepared in the same way. The number density of hemozoin and Fe3O4 nanocrystals in the most concentrated samples was estimated taking into account the densities of these materials [44,45] at 3.75  1010 cm3 and 5.47  1013 cm3, respectively, and will be used in our analysis. 2.4. Magnetization measurements Magnetic properties were measured using a 7400 series vibrating sample magnetometer (VSM) from Lake Shore Cryotronics Inc. (2 T maximum magnetic field; 300 pole gap, 84 Hz sample vibration

I. Khmelinskii, V. Makarov / Chemical Physics 493 (2017) 120–132

frequency). The external magnetic field varied in the 1.5 to +1.5 T range. The sample temperature could be set between 154 °C and 254 °C. Digital signal recording provided for averaging the signal over multiple field cycles. 3. Experimental results and data analysis 3.1. Experimental data First, we recorded magnetization curves of the studied materials. Fig. 2 shows the respective plots for both hemozoin and Fe3O4 samples, obtained at +20 °C. Fig. 2 shows that both plots exhibit no hysteresis at l = 3823 for hemozoin and l = 5440 for Fe3O4, respectively. We therefore conclude that both nanostructured materials are superparamagnetic. The 1H NMR line of C6H6 at the maximum concentration of Fe3O4 and hemozoin superparamagnetic particles is shown in Fig. 3. These spectra were recorded at T = 300 K. We see that the linewidth increases from 8.132 Hz for benzene – CS2 mixture to 128.131 Hz for benzene – CS2 – Fe3O4 mixture and 113.135 Hz for benzene – CS2 – hemozoin mixture. The NMR linewidth temperature dependence was recorded for different concentrations of added Fe3O4 and hemozoin. Fig. 4 shows selected results at 283, 300 and 323 K. We see in Fig. 4 that the NMR line becomes broader at higher temperatures, with the linewidths at different concentrations of Fe3O4 and hemozoin at different temperatures listed in Table 1. Fig. 5 shows the data of Table 1 in function of concentration at 323 K and in function of temperature for the two superparamagnetics, and at 5.47  1013 cm3 Fe3O4 and 3.75  1010 cm3 hemozoin, respectively. We note in Table 1 and Fig. 5 that the relaxation effects induced by Fe3O4 and hemozoin nanoparticles are quite similar. Therefore, we conclude that the properties of the Fe3O4 and hemozoin nanomaterials are quite similar as well. The magnetic permeability of Fe3O4 nanocrystals measured here is about 5440 (+20 °C), while the value reported earlier [46] is about 6783. This difference is probably due to different nanocrystal size. Earlier [7] we reported the magnetic permeability of hemozoin l = 4585 at 20 °C and l = 3843 at +20 °C. Hemozoin permeability measured here at +20 °C is 3823, i.e. it is in a good agreement with data reported earlier [7]. We therefore conclude that the two nanocrystalline systems have very similar superparamagnetic properties. As we shall see below, the observed relaxation properties of the benzene NMR line may be explained by the superparamagnetism of Fe3O4 and hemozoin. Note that superparamagnetic properties were also reported earlier in other biological systems, such as red pulp macrophages and ferritin, the protein that stores iron oxide [47–49].

Magnetization (Gs)

3.0

2.0 1.5

(1)

0.5

-500 -1000 -0.5

0.0

0.5

1.0

a

0.0 -8

-6

-4

-2

0

Chemical Shift (ppm) Fig. 3. 1H NMR spectra of benzene recorded in CS2 at 300 K: (a) 0.5% benzene; (b) 0.5% benzene +5.47  1013 cm3 Fe3O4 nanocrystals; (c) 0.5% benzene +3.75  1010 cm3 hemozoin nanocrystals. The spectra were shifted vertically for visual separation.

3.2. Theoretical background As we already noted, the size of Fe3O4 nanoparticles is 3.0 ± 0.2 nm and size of hemozoin nanocrystals is 17.5 ± 3.5 nm. Such small crystals may agglomerate, producing larger aggregates [6,21,22]. However, here we shall only consider individual nanocrystals. Since we need to interpret the broadening of the 1H NMR line of benzene in function of superparamagnetic particle concentration and sample temperature, we need to understand the mechanism of natural relaxation of nuclear and electronic spins, the mechanism of their interaction, and the mechanism determining the NMR linewidth in presence of superparamagnetic nanocrystals. To find the spin-Hamiltonian of an isolated superparamagnetic nanoparticle, we have to take into account the exchange interaction:

X ^ exch ¼  J ð^si  ^sj Þ H ij

ð1Þ

i;j

This interaction is much larger than kBT. Here, Jij is the exchange integral between i-th and j-th paramagnetic centers, kB is the Boltzmann constant and T is the temperature. Thus, the electronic ground state of the superparamagnetic system has the maximum spin angular momentum Ns, where N is the number of paramagnetic centers in a nanocrystal and s is the spin angular momentum of a single paramagnetic center. Thus, the spin-Hamiltonian of the superparamagnetic crystal may be written as follows [40,41]:

X X hf X ^ SP ¼  ½^S  T^ hf  ^Ii   ^ H ai ð^S  ^Ii Þ  ½^Ii  T^ nn ij  Ij  i i

i;j–i

ð2Þ

i

0

-1.0

b

1.0

^  ½^S  T^ SO  ^L  ½^S  T^ SV  ^l  lB ½^S  T^ ig  H X n ^ ^ ^  ln ½Ii  T i  H

(2)

500

c

2.5

i

1000

-1.5

3.5 Normalized Intensity

122

1.5

Magnetic Field (T) Fig. 2. Magnetization measurements of hemozoin and Fe3O4 at +20 °C: (1) hemozoin; (2) Fe3O4.

where the first term describes anisotropic electron spin-nuclear spin dipole-dipole interactions, with Thf ij being its tensor, the second term describes isotropic hyperfine interactions, with ahf ij,k being the Fermi contact interaction constant, the third term describes nuclear spin dipole – nuclear spin dipole interaction, with Tnn ij being its tensor, the fourth term describes the total spin – lattice orbital interaction, with TSO being its tensor, the fifth term describes total spin – lattice vibrational angular momentum interaction, with TSV being its tensor, the sixth term describes the Zeeman interactions of the electronic spin, with Tgi being the g-factor tensor, and the last term describes Zeeman interactions of the nuclear spin, with Tni

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3.5

3.5

Fe3O4

3.0 Normalized Intensity

Normalized Intensity

3.0 (c)

2.5 2.0 1.5

(b)

1.0 0.5

(a)

0.0

Hemozoin (c)

2.5 2.0 1.5

(b)

1.0 (a)

0.5 0.0

-8

-6

-4

-2

0

-8

-6

Chemical Shift (ppm)

-4

-2

0

Chemical Shift (ppm)

Fig. 4. Temperature dependences of NMR spectra of 0.5% benzene in CS2 with added 5.47  1013 cm3 Fe3O4 nanocrystals or 3.75  1010 cm3 hemozoin nanocrystals: (a) T = 283 K, (b) T = 300 K, and (c) T = 323 K. The spectra were shifted vertically for visual separation.

Table 1 Benzene 1H NMR linewidth (Hz) in function of temperature and superparamagnetic particle number density. The standard deviation is ±0.093 Hz for all data. Fe3O4 particle number density  1013, cm3 Temperature, K

5.47

4.38

3.50

2.80

2.24

1.79

1.43

1.15

0.92

0

283 290 300 315 323

119.13 123.28 128.13 131.11 134.91

107.43 111.06 115.28 118.18 121.59

96.87 100.11 103.89 106.52 109.56

87.48 90.38 93.75 96.14 98.86

79.08 81.67 84.69 86.85 89.29

71.53 73.84 76.54 78.51 80.69

64.77 66.84 69.25 71.04 72.99

58.90 60.76 62.92 64.56 66.31

53.51 55.18 57.11 58.61 60.17

7.897 7.998 8.132 8.198 8.213

Temperature, K 283 290 300 315 323

Hemozoin particle number density  1010, cm3 3.75 3.00 2.40 1.92 101.13 91.33 82.47 74.60 107.56 97.05 87.65 79.24 113.14 102.05 92.14 83.27 118.72 107.05 96.62 87.28 122.13 110.10 99.35 89.73

1.54 67.56 71.80 75.42 79.02 81.21

1.23 61.23 65.02 68.27 71.50 73.45

0.98 55.57 58.90 61.81 64.70 66.45

0.78 50.65 53.41 56.02 58.60 60.17

0.62 46.14 48.48 50.83 53.14 54.53

0 7.897 7.998 8.132 8.198 8.213

160

140

A (b)

100

B

(a)

130

(a)

120

Linewidth (Hz)

Linewidth (Hz)

140

80 60 40

120

(b)

110 100

20 0 0

2

4

6

90 280

290

Number density, cm-3

300

310

320

330

Temperature (K)

Fig. 5. 1H NMR linewidth in function of (A) nanocrystal concentration at 323 K, for both superparamagnetic nanomaterials; abscissa: [Fe3O4]  1013, [Hemozoin]  107; and (B) temperature for 5.47  1013 cm3 Fe3O4 and 3.75  1010 cm3 hemozoin. The lines show the calculated values produced by the presently developed models for (a) Fe3O4 and (b) hemozoin.

being the nuclear spin g-factor. Here, the subscript i enumerates the non-equivalent nuclei. The same Hamiltonian may be written as follows for a free benzene molecule:

^ BZ H

X ^ ¼  ½^Ii  T^ nn;B  ^Ij   ln ½^I  T^ n;B  H ij

ð3Þ

i;j–i

The latter Hamiltonian has two terms, already described. The Hamiltonians (2) and (3) determine the respective eigenstates and energies. Since we are only interested in the strong-field limit, the Hamiltonians (2) and (3) become:

^ ^ SP ¼ l ½^S  T^ g  H H B ^ BZ ¼ l ½^I  T^ n;B  H ^ H

ð4Þ

n

and the complete zero-order spin-Hamiltonian becomes:

^0 ¼ H ^ SP þ H ^ BZ ¼ l ½^S  T^ g  H ^  l ½^I  T^ n;B  H ^ H B n

ð5Þ

The latter may be simplified for isotropic systems:

^ 0 ¼ g l ð^S  HÞ ^  g l ð1  rn Þð^I  HÞ ^ H s B n n

ð6Þ

where rn is the chemical shift. The eigenfunctions and eigenvalues for this Hamiltonian are:

I. Khmelinskii, V. Makarov / Chemical Physics 493 (2017) 120–132

ð7Þ

where g n ; ln ; g s ; lB and rn are nuclear g-factor, nuclear magneton, electron spin g-factor, Bohr magneton and chemical shift, respectively. The Hamiltonian (6) has no relaxation terms that could produce the spectral linewidth in NMR of benzene protons or EPR of superparamagnetic nanocrystals. Generally, spin-lattice and spin-spin relaxation should be included, although for the natural relaxation, the spin relaxation is negligible, because the benzene concentration is low and the concentrations of the 13C and 33S atoms present in the solvent are quite low as well. We also neglected the intramolecular magnetic dipole-dipole mechanism of the nuclear spin-spin interaction, as the respective term

g 2n 2n r 3HH

l

is quite small

compared to perturbations induced by the natural charge fluctuations, with rHH the distance between the neighboring hydrogen protons.

The nuclear or electronic spin relaxation may be presented as follows [50]:

ð8Þ

where I = ½ and MI = + 1/2. Such transitions are of the magneticdipole type, and the perturbations inducing these transitions may be created by charge fluctuations in solutions, generating electric field fluctuations. Such electric field fluctuations induce magnetic field fluctuations, according to the Maxwell equation:

~fl ðtÞ ¼ rot½H

@~ E @t

ð9Þ

presented in Gaussian units. These fluctuations are random in direction and in time, with the typical time scale of 1010–1011 s [51]. The charge fluctuations cause instantaneous polarization of a molecule due to interactions with the closest molecular neighbors. Taking into account the typical molecular size and typical duration of fluctuations, the amplitude of the magnetic field fluctuations should be about 10–20 G in the molecular volume [51]. To simplify the analysis of the natural spin relaxation, we approximate the time profile of the fluctuating magnetic field by a rectangular pulse:

8 t <  s20 > < 0; Hfl ðtÞ ¼ Hfl;0 ;  s20 6 t 6 þ s20 > : 0; t > þ s20

ð10Þ

  Sin s20 x

px

ð11Þ

Thus, the spin-lattice relaxation probability for both nuclear and electronic spins may be presented as follows (see Appendix I):

P12;n ¼ p42



P12;S ¼ 629 p2

Hfl;0 H0



2

 2:5  107

 Hfl;0 2 SðS H0

þ 1Þ  53  107 SðS þ 1Þ

ð12Þ

where H0 is the applied magnetic field. The natural width of the NMR line of hydrogen protons in the benzene – carbon disulfide mixture may be presented as follows:

c12;n ¼ P12;n N  2:5  107 N

ð14Þ

we calculated the temperature dependences of the natural spinstate width in superparamagnetic nanoparticles of hemozoin and Fe3O4, with the results shown in Fig. 7.

We already considered the relaxation of free spin in superparamagnetic nanostructures. However, the electronic spin is also coupled by intrasystem interactions with the nanocrystal lattice. Using the structure of Fe3O4 lattice and a simplified structure of the Fe ion vicinity in hemozoin (see Fig. 8), we carried out ab initio analysis of the spin-orbit interaction energies in the two systems (see Appendix II). Both systems may be assigned to the C4v point group, with the C4 axis perpendicular to the plane of Fe ions in Fe3O4, and perpendicular to the porphyrin ring in hemozoin. In both systems we expect spin-orbit interaction, as the maximum local orbital moment should be directed along the main symmetry axis [53]. In the same time, the electronic spin should also quantize along the main axis. Using commercial Gaussian-2000 software package and home-made FORTRAN code, we made ab initio calculations of the spin-orbit coupling energy (see Appendix II). The calculated spin-orbit coupling energies are 6.23  108 and 4.71  108 cm1, for Fe3O4 and hemozoin lattices respectively. The thermal motion of nanocrystals creates fluctuations in the crystal axis direction, causing spin direction fluctuations, and the respective spin relaxation. The minimum relaxation rate is determined by the spin-orbit coupling energy, being 1.869  103 Hz for Fe3O4 and 1.483  103 Hz for hemozoin nanocrystals. The maximum relaxation rate is determined by the rotational diffusion rate of the respective nanocrystals. The estimates of the rotational diffusion rates are 3.187  104 Hz for Fe3O4 nanocrystals and

-7

where s0 is the average duration of the fluctuation. An inverse Fourier transform applied to the latter transfers the fluctuation function from the time domain to the energy (frequency) domain:

Hfl ðxÞ ¼ Hfl;0

5 3

c12;S ¼ P12;S N   107 SðS þ 1ÞN

3.4. Other factors affecting spin relaxation in superparamagnetic nanoparticles

3.3. Natural relaxation

jI; MI i ! jI; MI  1i jS; M S i ! jS; MS  1i

where N = N0Vs is the frequency of fluctuations in the Vs sample volume, N0 being the density of fluctuations inducing nuclear/electron spin-lattice relaxation. The temperature dependence of N obtained using the data of Table 1 and the relationship (13) is shown in Fig. 6. This dependence will be used to estimate the natural width of the spin states in superparamagnetic nanoparticles. The estimated electronic spin of these particles is 538 [7] and 202, for hemozoin and Fe3O4, respectively. Thus, assuming that N magnetic field fluctuations induce spin relaxation in superparamagnetic nanoparticles, and using the relationship

-1

jSMS ; IMI i ¼ jSM S ijIMI i ^  g l ð1  rn ÞðM I  HÞ ^ E ¼ g s lB ðM S  HÞ n n

Frequency of fluctuations×10 , s

124

ð13Þ

3.30 3.28 3.26 3.24 3.22 3.20 3.18 3.16 3.14 0

10

20

30

40

50

60

Temperature, ºC Fig. 6. Frequency of magnetic field fluctuations inducing proton spin relaxation in benzene.

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2.26

15.9

Exp.

15.5

Fe3O4×10 , Hz

(a)

15.4

12,S,

12,S,

15.8

-5

-5

Hemozoin×10 , Hz

16.0

15.7

Theor.

15.6

Exp. (b)

2.24 2.22

Theor.

2.20 2.18 2.16

15.3 15.2

2.14

10

20

30

40

50

o

Temperature, C

10

20

30

40

50

o

Temperature, C

Fig. 7. Calculated spin relaxation frequencies of superparamagnetic nanoparticles: (a) hemozoin and (b) Fe3O4. Circles – experimental data; squares – theory.

Fig. 8. Structure of the Fe3O4 lattice [52] and part of hemozoin elementary cell: the Fe ion is in the center, with the porphyrin ring around and the polypeptide above it.

1.273  104 Hz for hemozoin nanocrystals. We expect that the relaxation rate of the electronic spin depends on the spin-orbit interaction coupling the nanocrystal axis with the electronic spin angular momentum. Thus, the spin relaxation rate should correspond to the minimum relaxation rate. Taking into account the data of Fig. 6a and b, we conclude that the relaxation mechanism discussed in the present subsection has negligible effect on the results obtained. 3.5. Nuclear spin relaxation induced by Fe3O4 and hemozoin nanocrystals In this subsection, we study the nuclear-spin lattice and nuclear spin-spin relaxation induced by the interaction of benzene protons with superparamagnetic nanocrystals. We shall consider the hyperfine nuclear spin-lattice mechanism and the nuclear spin – electron spin magnetic dipole-dipole mechanism of the nuclear spin-lattice relaxation, and nuclear spin – electronic spin dipoledipole mechanism of the nuclear spin – electronic spin relaxation. 3.5.1. Hyperfine nuclear spin-lattice relaxation mechanism This mechanism involves partial spin density transfer from a superparamagnetic nanoparticle to benzene molecules in contact with it. Such transfer occurs via contact exchange interaction between the two. We performed ab initio analysis of this process, using Gaussian-2000 commercial software package that employed the coupled clusters method with the 6-31G(d) basis set and ran on a ROCKS server system.

The C6H6-Fe3O4 complex was modeled as shown in Fig.7a, with the plane of the benzene molecule normal to the C4 axis of the Fe3O4 lattice, and the axis passing through the center of the molecule. The plane of the benzene molecule is also parallel to the Fe4 plane of the lattice, with the distance between the two and the angle of the molecular rotation about the C4 axis being optimized. The benzene-hemozoin complex was modeled using the elementary hemozoin fragment (Fig.7b), with the benzene plane normal to the C4 axis passing through the center of the molecule. The benzene molecule is on the side of the porphyrin ring opposite to the unstructured polypeptide coordinated to the Fe3+ ion. In ab initio calculations, this polypeptide was substituted by an H atom, the benzene molecule was parallel to the porphyrin ring, and the distance between the two and the rotation angle of the molecule about the C4 axis were optimized. The ab initio calculations produced the spin density on the benzene carbon atoms of about 1.1  1010 for Fe3O4 nanoparticles, and 1.4  1010 for hemozoin. Taking into account the McConnell rule [40], we estimated the frequency of the nuclear spin relaxation induced by the contact hyperfine Fermi interaction. The resulting values were about 7.6 and 9.7 Hz for Fe3O4 and hemozoin nanocrystals, respectively. These frequencies are quite low; therefore, we neglected this relaxation mechanism in our analysis.

3.5.2. Nuclear spin – electron spin magnetic dipole-dipole mechanism of nuclear spin-lattice and nuclear spin – electron spin relaxation As we found above, the contact exchange interaction should be negligible as regards nuclear spin relaxation. Here we consider

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for the systems considered, the nuclear spin state relaxation width may be presented as follows (Appendix AIII):

long-distance magnetic interactions. With good accuracy the spinHamiltonian of the Zeeman and nuclear spin – electron spin dipole-dipole interaction may be presented as follows (see Appendix III):

^ S ¼ g l ð^Sz  HÞ ^  g l ð1  rn Þð^Iz  HÞ ^ þg l g l H s B n n s B n n   2 1  3Cos ðhÞ ^ ^ 1 ^ ^  Sz Iz  ðSþ I þ ^S^Iþ Þ r3 ðtÞ 4

cnS ¼

ð15Þ

DðTÞ ¼ DBenz ðTÞ þ DSP ðTÞ  DBenz ðTÞ DBenz ðTÞ  2 pR

13Cos2 ðhÞ ^ ^ ^ ^ ^ ^ ðSz  Iz Þ s lB ðSz  HÞ  g n ln ð1  rn ÞðIz  HÞ þ g s lB g n ln r 3 ðtÞ

B n

n

rðtÞ ¼ q þ

4r ðtÞ

Here, assuming strong external magnetic fields, the eigenfunctions of the H0 may be presented as:

s

2 ðhÞ

lB ððMF þ12ÞHÞþ12g n ln ð1rn ÞðHÞg s lB gn ln 13Cos 3

2r ðtÞ

 je

MS þi

c12;SF ðnSP Þ

i

t

2

ð17Þ

1 SP ; SM S ¼ M F þ 2ij Benz ; IM I

e

i

1

1

¼  12ieh½gs lB ððMF þ2ÞHÞþ2gn ln ð1rn ÞðHÞþi

c12;SF 2

t

and

  w2 ðtÞ ¼ eSP ; SMS ¼ MF  12 eBenz ; IMI ¼ þ 12 l

r

l

l

e   eSP ; SMS ¼ M F  12 jeBenz ; IMI c12;n i 1 1 ¼ þ 12 eh½gs lB ððMF 2ÞHÞ2gn ln ð1rn ÞðHÞþi 2 t c

ðn Þ

ð18Þ

We fitted the superparamagnetic nanoparticle concentration dependences of the NMR linewidth using the relationship (19), an appropriate homemade FORTRAN code and the data of Table 1. The experimental concentration dependences and the fitting curves for hemozoin and Fe3O4 superparamagnetic nanoparticles are shown in Fig.9a and b, respectively. We note a good agreement between the experimental data and the fitting curves in Fig. 9. The values of the fitting parameters for the number density plots of the NMR linewidth at different temperatures are listed in Table 2.

c

where the i 12;SF2 SP ; i 12;n are the imaginary components of the 2 energy, proportional to the state width [54]. Note that we used the experimentally determined c12;n values at different temperatures (see Table 1) in the fitting procedure, while ð0Þ c12;SF ðnSP Þ ¼ c12;SF þ rS  ðnSP Þ is a function containing two fitting ð0Þ parameters, c12;SF and rS , and dependent on the number density ð0Þ nSP of superparamagnetic nanoparticles. We compared the c12;SF parameter to the estimated values c12;S (see Fig. 6a and b). Next,

ð0Þ

Fig. 7a and b show temperature dependences of the c12;SF parameter for hemozoin and Fe3O4 superparamagnetic nanoparticles, respectively. Note an acceptable agreement between the

140

140 (b)

(a)

120

NMR linewidth (Hz)

NMR linewidth (Hz)

ð21Þ

3.6. Experimental data analysis

c12;n 13Cos2 ðhÞ i 1 1  ½g s B ððM F 2ÞHÞ2g n n ð1 n ÞðHÞþg s B g n n 2r 3 ðtÞ M S þi 2 t h

l

pffiffiffiffiffiffiffiffiffiffiffiffi DðTÞt

where gCS2 is the dynamic viscosity coefficient of liquid CS2, DBenz(T) and DSP(T) are the temperature-dependent diffusion coefficients of benzene and superparamagnetic particles in the liquid CS2, Reff,Benz and Reff,SP are the effective radii of benzene and superparamagnetic nanoparticles. The relationship (19) was used to fit the experimental dependences of the width of the benzene proton line at different sample temperatures versus nSP, with the following fitting parameters: c12;SF ; q  Reff ;Benz and gCS2 . We also compared the values of the gCS2 fitting parameter to the earlier reported experimental data [56].

w1 ðtÞ ¼ jeSP ; SM S ¼ M F þ 12ijeBenz ; IMI ¼  12i i ½g

ð20Þ

kB T eff ;Benz gCS2

DðTÞ ¼ DBenz ðTÞ þ DSP ðTÞ  DBenz ðTÞ q ¼ Reff ;Benz þ Reff ;SP4  Reff ;SP

ð16Þ

eh

ð19Þ

and

^  g l ð1  rn Þð^Iz  HÞ ^  g s lB ð^Sz  HÞ n n 2 13Cos ðhÞ ^ ¼g l g l ð^Sþ^I þ ^S^Iþ Þ VðtÞ 3 s

ðg e le g n ln Þ2 SðS þ 1Þ 2 24h  Z  1 ehi ½ðgs lB þgn ln ð1rn ÞÞHþ2i ðc12;SF ðnSP Þþc12;n Þt 2   dt  nSP  pffiffiffiffiffiffiffiffiffiffiffiffi 3   0;0 ½q þ DðTÞt 

where nSP is the number density of superparamagnetic nanoparticles, and DX(T) – the temperature dependent diffusion coefficient of either benzene or superparamagnetic nanoparticles that was calculated using the following relationship [55]:

where r(t) is the nuclear magnetic dipole and electronic magnetic dipole placed into the origin of the referential and h the angle between r and the z axis of the referential (see Appendix III). The spin-Hamiltonian (15) may be rewritten as follows:

^ ^ ¼H ^0 þ V H ^ H0 ðtÞ ¼ g

5p3 qDðTÞ

100 80 60

120 100 80 60 40

40 0

1

2

3 -10

-3

Number density×10 , cm

4

0

1

2

3

4

5

6

Number density×10-13, cm-3

Fig. 9. Experimental concentration dependences and fitting curves for (a) hemozoin and (b) Fe3O4 superparamagnetic nanoparticles. Temperature values, bottom to top: T = 283, 290, 300, 315, and 323 K.

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I. Khmelinskii, V. Makarov / Chemical Physics 493 (2017) 120–132 Table 2 The values of fitting parameters. Fe3O4

cð0Þ 12;SF , MHz

rS , 109, sec1 cm3

2q, nm

gCS2 , cP

283 K 290 K 300 K 315 K 323 K

0.216 ± 0.007 0.218 ± 0.006 0.221 ± 0.006 0.223 ± 0.008 0.224 ± 0.007

0.461 ± 0.009 0.467 ± 0.009 0.475 ± 0.010 0.479 ± 0.009 0.480 ± 0.011

4.3 ± 0.6 4.5 ± 0.7 4.1 ± 0.6 4.8 ± 0.5 4.4 ± 0.6

0.417 ± 0.017 0.393 ± 0.013 0.369 ± 0.015 0.338 ± 0.016 0.323 ± 0.014

Hemozoin

cð0Þ 12;SF , MHz

rS , 105, sec1 cm3

2q, nm

gCS2 , cP

283 K 290 K 300 K 315 K 323 K

1.527 ± 0.017 1.538 ± 0.019 1.560 ± 0.016 1.578 ± 0.020 1.583 ± 0.018

3.022 ± 0.015 3.060 ± 0.017 3.112 ± 0.018 3.137 ± 0.019 3.143 ± 0.021

10.7 ± 0.7 10.6 ± 0.7 10.2 ± 0.6 10.9 ± 0.8 10.6 ± 0.7

0.419 ± 0.018 0.391 ± 0.014 0.374 ± 0.014 0.339 ± 0.018 0.321 ± 0.015

estimated c12;S values and those obtained by the fitting procedure. Therefore, our theoretic estimates of the c12;S parameter for the electronic spin relaxation provided acceptable results. Note, however, that the observed difference between the experimental and theoretical temperature dependences (see Fig. 7) may be explained by nanoparticle agglomeration effects, which were not included into our model. Indeed, artificial b-hematin polymer crystals synthesized from hemin chloride under acidic conditions are chemically identical to hemozoin crystals of biological origin, although having different shape and size, ranging from 50 nm to 20 lm [22]. The large size of these crystals was explained by agglomeration of smaller hemozoin nanocrystals, with the authors proposing [22] that such agglomeration occurs due to interactions between the magnetic moments of the hemozoin nanocrystals. Since the magnetic properties of the hemozoin agglomerates may be different from those of individual nanocrystals, we may expect some deviations of the theoretical temperature dependences from the experimental results. The same should be true for Fe3O4 nanoparticles. Note also that the rS fitting parameter values in Table 2 for Fe3O4 are lower by the factor of 1.9  104 than those obtained for hemozoin. This difference may be partly accounted for by the difference in the size of the respective particles, as the total electronic spin is proportional to the particle volume, and the respective interaction is proportional to the total spin squared, with the ratio estimated at 3.2  103. We attribute the remaining discrepancy to high anisotropy of the hemozoin nanocrystals, which additionally amplifies their interactions. Note that the dynamic viscosity value listed in the last column of Table 2 is a temperature dependent parameter, while it was earlier reported [56] as a constant value (0.344 cP) in the 4 °C–29 °C temperature range. This contradiction may be attributed to low accuracy of the respective measurements [56], and will be discussed in detail below.

4. Discussion In the current study we measured the natural relaxation rates of the electronic spin in the Fe3O4 and hemozoin superparamagnetic nanoparticles using the NMR techniques. The numerical ab initio analysis revealed that the contribution of the intra-lattice SO interaction in both Fe3O4 and hemozoin to the natural relaxation rate of electronic spin is negligible. Since the NMR method was used to measure natural electronic spin relaxation rates, the contribution of the hyperfine contact Fermi interaction was neglected, because our ab initio analysis found very small spin density induced by exchange interaction between superparamagnetic nanoparticles and benzene molecules. Therefore, here both intra-lattice SO interaction and inter-superparamagnetic – benzene molecule contact

exchange interaction are not discussed, as no data are available for either of our superparamagnetics. A detailed study of spin dynamics in 6 nm Fe3O4 powder using the selective excitation double Mössbauer (SEDM) method was employed to separate static disorder, collective excitations, and moment reversals in spin relaxation [36]. Superparamagnetic spin flips were observed by the appearance of an additional line in the SEDM spectrum, defining the energy transfer during relaxation, with frequencies of 2.5 MHz at 70 K to 9.7 MHz at 110 K. Although our measurements were carried out at higher temperature, we recorded spin relaxation rates for Fe3O4 nanoparticles that are lower by the factor of ca. 44, comparing T = 283 K in our experiments and T = 110 K in [36]. Such difference may be explained by different conditions of spin relaxation, as the low-temperature measurements were done in solids [36], while we worked in solutions. Unfortunately we were unable to find any reports of the natural spin relaxation rates in conditions similar to those of the present measurements. Note that the natural spin relaxation rate for superparamagnetic hemozoin nanoparticles is higher by the factor of 7–8 as compared to that of Fe3O4. A detailed previous study [57,58] was focused on the direct relationship between the size and magnetization of the particles and their nuclear magnetic resonance relaxation properties. The experimental relaxation data on magnetite particles in a wide range of sizes and magnetizations agreed quite well with the relaxation theories. Using the experimental curve of the transverse relaxation rate versus particle size, the authors predicted that the MRI contrast efficiency of any type of magnetic nanoparticles should correlate with particle size. They found that such prediction requires the knowledge of the size of the particles impermeable to water protons and the saturation magnetization of the corresponding volume. The authors predicted [57,58] that the T2 relaxation efficiency of single magnetic Fe3O4 nanocrystals correlates directly with the crystal size and magnetization. The parameter values of the theoretical model calculating T2 from the particle size were obtained by a single Langevin fit of the magnetization curve. The maximum available relaxation was described by the rS parameter, with rS = 1010–109 sec1 cm3 [57]. The values we obtained in the present work are not very different, being in reasonable agreement with the previously reported rS values. The rS parameter for hemozoin nanoparticles is larger than that for Fe3O4 nanoparticles by a factor of about 105, as we already discussed above, providing a reasonable explanation. Note that the theoretical model used in [57] takes into account the magnetic dipole-dipole interaction, while leaving aside the natural relaxation. In comparison, our model uses the same mechanism, with natural relaxation properly included. As we already noted, we also neglected any eventual aggregation effects in the suspensions of nanoparticles. Therefore, it is impossible to directly compare the results obtained here with those of the previous study.

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As we can see in the fourth column of Table 2, the average diameter correlates quite well with the geometrical parameters of both Fe3O4 and hemozoin nanoparticles. Namely, the Fe3O4 nanoparticles may be described as approximate spheres, while the hemozoin nanocrystals may be described as parallelepipeds sized 17.5  3.1  3.1 nm3. Thus, hemozoin nanocrystals may be approximated by an equivalent-volume sphere 6.8 nm in diameter, quite close to the values listed in Table 2 for hemozoin. Thus, our theoretical analysis of the experimental data provides an acceptable estimate of the average diameter of superparamagnetic nanoparticles. We found that the dynamic viscosity values listed in Table 2 are the same for Fe3O4 and hemozoin, being dependent on temperature, while previously a constant value of 0.344 cP was used in the 4 °C–29 °C temperature range [56]. In fact, the temperature dependence of the CS2 dynamic viscosity may be described by the following expression, producing values in cP [59]:

gCS2 ¼

100c k Tc

ð22Þ

with k = 3.33 and c = 2.02, therefore the data presented in Table 2 are quite reasonable. Fig. 10 shows the temperature dependence of the dynamic viscosity coefficient in the temperature range of Table 2. Fig. 9 shows that the values calculated according to eq. (22) data are in acceptable agreement with the results of Table 2. Fitting the data of Table 2 using Eq. (22) gives k = 3.39 and c = 2.01, in good agreement with the previously reported values [59]. We believe that these results constitute an experimental confirmation of the presently developed theoretical model. The magnetic properties of hemozoin are similar to those of ferritin, which was extensively studied earlier [60–62]; however, we are reporting the natural spin relaxation properties for such systems for the first time. The knowledge of the spin relaxation might be useful for developing heat treatment methods for malaria and cancer, based on properties of hemozoin and Fe3O4, respectively. We expect that in aqueous solutions the natural relaxation rates of the electronic spin in both Fe3O4 and hemozoin will be larger than those measured in CS2 solutions, caused by high polarity of water molecules leading to stronger charge fluctuations. Presently we made no attempt to perform similar measurements in aqueous solutions, as it would be very difficult to quantify nuclear spin-spin relaxation in such a system. As we already noted, the measured natural spin relaxation rates for Fe3O4 and hemozoin may provide a key for heat treatment of cancer and malaria. The electromagnetic field-induced heating

was extensively studied on Fe3O4 superparamagnetic nanoparticles [63], with no data available for hemozoin. We expect that radiofrequency (RF) electromagnetic field or microwaves (MW) will induce magnetic-dipole transitions between the spin states of superparamagnetic nanoparticles, separated one from the other by the Zeeman interaction with a constant external magnetic field. The electromagnetic field energy absorbed by the system will be dissipated as heat. It is well known that the saturation of transitions between different spin states, with the selection rules DS ¼ 1; DMS ¼ 1, takes place when the electromagnetic fieldinduced transition rate is higher than the spin relaxation rate. Therefore, heating is limited by the spin relaxation time, with shorter spin relaxation times providing for stronger heating. We shall discuss these issues in detail in a follow-up publication. In conclusion, we note that spin relaxation dynamics in superparamagnetic systems has been extensively discussed earlier [64–70], with some of the previously proposed approaches used in our modeling analysis. However, the presently proposed natural relaxation mechanism is our novel and original contribution, describing spin relaxation rates in an isolated superparamagnetic nanoparticle. 5. Conclusions We developed a theoretical model describing line broadening in H NMR spectra of benzene dissolved in CS2, in presence of superparamagnetic nanoparticles. This model includes parameters describing natural relaxation of nuclear and electronic spin, involving natural relaxation parameters, and the effective nanoparticle diameter and diffusion coefficient, determined by the dynamic viscosity of liquid CS2. We found that the most important interaction coupling the proton nuclear spin states with the superparamagnetic nanoparticle electronic spin states is the nuclear spin – electronic spin dipole-dipole interaction. All of the modeling parameters were evaluated by fitting the model curve to the experimental data. We report that the natural spin-relaxation NMR spectroscopy of a known system in the presence of superparamag1

ð0Þ

netic nanoparticles allows measuring of the c12;SF ¼ c12;SF þ rS nSP , 2q  2Reff ;Benz and gCS2 parameter values. The knowledge of the relaxation parameters allows selecting the optimal frequency of the electromagnetic field inducing transitions between the Zeeman spin-states separated by an external magnetic field, with the field strength that is determined by the electromagnetic field frequency. Acknowledgements The authors are grateful for the NASA EPSCoR grant PR NASA EPSCoR (NASA Cooperative Agreement NNX13AB22A) to V. M.

Viscosity ( , cP)

0.44 0.42

Appendix I

0.40

The transitions between different spin-states are of the magnetic-dipole type, we should therefore look at how variable magnetic fields arise. The natural relaxation of the spin-states is created by the electric charge fluctuations. These generate electric field fluctuations, which in turn induce magnetic field fluctuations, described by the Maxwell equation

0.38 0.36 0.34 0.32 0.30 280

~fl ðtÞ ¼ rot½H 290

300

310

320

330

Temperature (K) Fig. 10. Temperature dependence of the dynamic viscosity coefficient: (a) calculated values (circles), Eq. (22), and (b) the values obtained in the present study (squares).

@~ E @t

ðAI:1Þ

presented in the Gaussian system. These are random Markov fluctuation in direction and in time, with the typical time scale of 1010–1011 s [51]. For simplicity, we approximate the time dependence of the fluctuating magnetic field by a rectangular function

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I. Khmelinskii, V. Makarov / Chemical Physics 493 (2017) 120–132

8 t <  s20 > < 0; Hfl ðtÞ ¼ Hfl;0 ;  s20 6 t 6 þ s20 > : 0; t > þ s20

 ¼ e ~ jSM S i

ðAI:2Þ

where s0 is average duration of fluctuations. Applying the inverse Fourier transfor to (AI.2), we obtain

Hfl ðxÞ ¼

1

Z

p

1

1

¼ Hfl;0

Hfl ðtÞeixt dt ¼

  Sin 2s x

i

p

Z s=2 Hfl;0

 hSM S j^Sx jSM0S i þ heX~ j^Ly jeX~ ihSM S j^Sy jSM0S ig ðAI:3Þ

(a) Nuclear spin

l

n gn

2

ESO ¼ g L g S l2B fheX~ j^Lz jeX~ ihSM S j^Sz jSM0S i þ heX~ j^Lx jeX~ i

eixt dt

px

Note that both Fe3O4 and hemozoin elementary cells may be described as belonging to the C4v point group, and the electronic ground state orbital symmetry should be described by the A1 irreducible representation of the C4v point group (see Table AII.1). The components of the orbital angular momentum are described by the B2 (Lz) and E (Lx and Ly) irreducible representations. Therefore, we expect that in the zero order of the perturbation theory, our matrix element equals zero:

X

ðAI:4Þ

^I ¼ ^Ix þ ^Iy þ ^Iz where

^I ¼ 1 ð^Ix  i^Iy Þ 2 ^Ix ¼ ^Iþ þ ^I

ðAI:5Þ

^Iy ¼ ið^Iþ  ^I Þ

2

ðAI:6Þ

Here we took into account that the maximum probability is obtained with

 2 Sin 2s x ¼ 1

ðAI:7Þ

x12 ¼ ln gnhHfl;0 (b) Electronic spin

PMS ;MS 1 ðxÞ ¼ jAMS ;MS 1 ðxÞj2 ¼ 4ð

Thus, in the numerical analysis we first used Gaussian-2000 to calculate the ground and the first excited electronic states (with B2 symmetry) of the Fe3O4 and hemozoin elementary cells. The software was running on a ROCKS cluster server using the Coupled cluster method with 6-31G(d) basis. Next, homemade FORTRAN code was used to calculate the SO interaction energy using the second-order perturbation theory, with the ground-state wavefunction perturbed by diabatic interactions used in the form:

Þ

Sin2 ðsxÞ ½SðS þ 1Þ  M S ðM S  1Þ  p2 x22 dðxMS ;MS 1  xÞ  2 H ¼ p42 Hfl;00 ½SðS þ 1Þ  M S ðM S  1Þ

ðAI:8Þ

Taking into account the relationship m X 1 x2 ¼ mðm þ 1Þð2m þ 1Þ 6 x¼1

ðAI:9Þ

 2 29 Hfl;0 SðS þ 1Þ 6p2 H0

ðAII5Þ

where H is the system Hamiltonian, and QB2 is the normal coordinate with B2 symmetry. Thus, the SO energy was presented as follows:

ESO ¼ g L g S l2B

   @ H^  heX;A i  eA;B ~ ~ 1 @Q B 2 2

DEð0Þ ~ ~A X

heX;A j^Lz jeA;B iMS ~ ~ 1 2

ðAII6Þ

 D   E   e  @H^ e @Q B  A;B   ~ ~ X;A 1 2 2 SO j ¼ 2g g l2  ^  jE h e j L j e i ~ ~ z A;B ð0Þ L S B X;A 1 2  DE ~ ~   XA   ^ @H  heX;A i ~   ~ 1 j@Q B2 jeA;B 2 ¼ g L g S l2B  heX;A j^Lz jeA;B i Sþ1 ~ ~ ð0Þ 1 2  2Sþ1 DE ~ ~  XA

X

jM S j

MS 2Sþ1

ðAII7Þ

The FORTRAN code was used to calculate the matrix elements

we average the relationship (AI.8) over the MS values, obtaining:

PMS ;MS 1 ðxÞ 

WXð2Þ ~

   @ H^   E E heX;A i  eA;B ~ ~ @Q 1 2  B2  ¼ eX;A jSMS i þ jSM S i eA;B ~ ~ ð0Þ 1 2 DEX~ A~

Averaging the SO energy over the absolute value of MS, we obtain:

lB g s Hfl;0 2  h

ðAII4Þ

X

heX~ j^Ly jeX~ i ) hA1 jEjA1 i ¼ E

and

P12 ðxÞ ¼ jA12 ðxÞj  2 Sin2 sx  2 l g H ð2 Þ 4 Hfl;0 ¼ 4 n nh fl;0 p2 x2 dðx12  xÞ ¼ p2 H0

ðAII3Þ

heX~ j^Lz jeX~ i ) hA1 jB2 jA1X i ¼ B2 he ~ j^Lx je ~ i ) hA1 jEjA1 i þ E

jhI; þM I jðHfl ðxÞ  ^IÞjI; M I ij2

P12 ðxÞ ¼ jA12 j ¼ h  2 Sin2 sx l g H ð2 Þ 2 ^ ¼ n nh fl;0 p2 x2 jhI; þM I jIjI; M I ij dðx12  xÞ

ðAII2Þ

Thus, the matrix element of the SO interaction may be presented as follows:

s=2

The latter is the Sinc function, describing the magnetic field fluctuations on the energy scale. Thus, the probabilities of the respective magnetic-dipole transitions for the nuclear and electronic spin states may be presented as follows:

2

wXSM ~ X ^ ðL  ^SÞ ¼ ^Lz  ^Sz þ ^Ly  ^Sx þ ^Ly  ^Sy

j @ H jeA;B i, heX;A j^ Lz jeA;B i, while DEX~ A~ is obtained from the outheX;A ~ ~ ~ ~ 1 @Q B 1 2 2 ^

ð0Þ

2

ðAI:10Þ

with S 1.

put of Gaussian-2000. The calculated values of the SO interaction energy were 1.869  103 Hz for Fe3O4 nanocrystals and 1.483  103 Hz for hemozoin nanoparticles, respectively. Appendix III

Appendix II III.1 Dipole-dipole interaction tensor We start with analyzing the structure of the spin-orbit (SO) interactions. In the first order of the perturbation theory, the SO interaction is determined by the following matrix element:

ESO ¼ g L g S l2B heX~ ; SM S j^L  ^SjeX~ ; SM0S i where gL is the orbital g-factor and

ðAII1Þ

Considering the nuclear spin of a benzene proton and superparamagnetic nanoparticles, the eigenfunctions and eigenvalues in the zero-order approximation may be presented as follows (we neglect the magnetic dipole-dipole interaction between different protons):

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Table AII.1 Table of the irreducible representations of the C4v point group. C4v

E

2C4

C2 (=C24)

2rv

2 rd

F

A1 A2 B1 B2 E

1 1 1 1 2

1 1 1 1 0

1 1 1 1 2

1 1 1 1 0

1 1 1 1 0

xy Xz; yz

wSMS ¼ jeSP ; SM S i ^2 ^S2 w SM S ¼ S jeSP ; SM S i ¼ SðS þ 1ÞjeSP ; SM S i ^Sz w ¼ ^Sz jeSP ; SM S i ¼ M S jeSP ; SM S i

In a diagonal tensor, we require

ðAIII:1Þ

xy ¼ xz ¼ yz ¼ 0

ðAIII:9Þ

This becomes possible by directing one of the axes of the Cartesian referential along the vector r that joins the two spins. Then the diagonal tensor components become:

SM S

and

wimi ¼ jeBenz ; IMI i ^I2 w ¼ ^I2 jeBenz ; IMI i ¼ IðI þ 1ÞjeBenz ; IMI i imi ^Iz w ¼ ^Iz jeBenz ; IMI i ¼ M I jeBenz ; IMI i

ðAIII:2Þ

dd 1 T dd xx ¼ T yy ¼ r 3 2 T dd zz ¼  r 3

imi

where eSP and eBenz are the set of quantum numbers of the electronic ground states of superparamagnetic nanoparticles and benzene. The zero-order energies of these states were introduced as phenomenological parameters ESP(0) and EBenz(0). The zero-order common spin states may be presented as follows:

jeSP ; eBenz ; FMF ; SM S ; IMI i ¼ jeSP ; SM S ijeBenz ; IMI i

ðAIII:3Þ

where F = S + i and MF = MS + MI are the total angular momentum and its projection on the z axis of the Cartesian referential. The magnetic dipole-dipole interaction may be presented as:

ðAIII:10Þ

This, the Trace of the diagonal tensor equals zero. III.2. Nuclear spin-lattice relaxation induced by superparamagnetic nanoparticles The Hamiltonian of the magnetic dipole-interaction in spherical coordinates may be written as:

^ dd ¼ g S lB g n ln ½C 1 þ C 2 þ C 3 þ C 4 þ C 5 þ C 6  H r3

ðAIII:11Þ

where

^ ^ ^ rÞð^I  ~ rÞ ^ dd ¼ g l g l ½ðS  IÞ  3 ðS  ~ H  S B n n 3 r ðtÞ r 5 ðtÞ

ðAIII:4Þ

C 1 ¼ ð1  3Cos2 ðhÞÞð^Sz  ^Iz Þ C 2 ¼  14 ð1  3Cos2 ðhÞÞð^Sþ  ^I þ ^S  ^Iþ Þ

and the total spin-Hamiltonian thus becomes:

C 3 ¼  32 SinðhÞCosðhÞei/ ð^Sz  ^Iþ þ ^Sþ  ^Iz Þ

^ S ¼ g l ð^S  HÞ ^  g l ð1  rn Þð^I  HÞ ^ H s B n n " # ð^S  ^IÞ ð^S  ~ rÞð^I  ~ rÞ þ g S lB g n ln 3 3 r ðtÞ r 5 ðtÞ

ðAIII:5Þ

C 4 ¼  32 SinðhÞCosðhÞeþi/ ð^Sz  ^I þ ^S  ^Iz Þ

ðAIII:12Þ

C 5 ¼  34 Sin ðhÞeþ2i/ ð^Sþ  ^Iþ Þ 2

C 6 ¼  34 Sin ðhÞeþ2i/ ð^S  ^I Þ 2

Taking into account the relationships

ð^S  ^IÞ ¼ ^Sx  ^Ix þ ^Sy  ^Iy þ ^Sz  ^Iz ð^S  ~ rÞ ¼ ^Sx  x þ ^Sy  y þ ^Sz  z

ðAIII:6Þ

ð^I  ~ rÞ ¼ ^Ix  x þ ^Iy  y þ ^Iz  z

The terms C3 – C4 may be neglected, because the matrix elements of these operators differ from zero only in the second order of the perturbation theory. Thus, the spin-Hamiltonian may finally be presented as follows:

we rewrite the magnetic dipole interaction:

n ^ dd ¼ g S lB g n ln ^Sx^Ix ðr 2  3x2 Þ þ ^Sy^Iy ðr 2  3y2 Þ þ ^Sz^Iz ðr 2 H 5 r  3z2 Þ  ð^Sx^Iy þ ^Sy^Ix Þ3xy  ð^Sy^Iz o þ ^Sz^Iy Þ3yzð^Sz^Ix þ ^Sx^Iz Þ3zx

ðAIII:7Þ

or else in the tensor form:

2 ðr2 3x2 Þ ^ dd H

6 ¼ g S lB g n ln ½^Sx ; ^Sy ; ^Sz 6 4 ¼ g S lB g n ln ½^S  T^ dd  ^I

r5

 3xy r5 ðr 2 3y2 Þ r5

 3zx r5

 3zy r5

32 3 ^I 76 x 7 74 ^Iy 5  3yz r5 5 ^Iz ðr 2 3z2 Þ 5  3xz r5

ðAIII:8Þ

r

Using the conventional relation between the spherical and the Cartesian referentials, with the electronic spin located in the center, and the nuclear spin on the sphere with the radius r, a new referential may be found, so that the tensor becomes diagonal (see Fig. AIII.1).

Fig. AIII.1. Schematic presentation of nuclear spin – electron spin interaction system in the spherical coordinate system.

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I. Khmelinskii, V. Makarov / Chemical Physics 493 (2017) 120–132

^ ¼ g l ð^Sz  HÞ ^  g l ð1  rn Þð^Iz  HÞ ^ þg l g l H s B n n s B n n

The time dependence of r(t) was obtained from

1  3Cos2 ðhÞ ^ ^ 1  3Cos2 ðhÞ ð Sz  I z Þ  g s l B g n l n 3 r ðtÞ 4r 3 ðtÞ  ð^Sþ^I þ ^S^Iþ Þ



rðtÞ ¼ q þ ðAIII:13Þ

The first three terms of the spin-Hamiltonian determine the zero-order energies of the electron-spin – nuclear-spin states, while the fourth describes the perturbation that mixes different electron-spin – nuclear-spin states. Thus, the latter spin-Hamiltonian may be rewritten as: ^ ¼H ^ 0 þ V^ H 2 ðhÞ ^ ^ ^ ^  g l ð1  rn Þð^Iz  HÞ ^ þ g l g l 13Cos H0 ðtÞ ¼ g s lB ð^Sz  HÞ ðSz  Iz Þ n n s B n n r3 ðtÞ ^  g l ð1  rn Þð^Iz  HÞ ^  g s lB ð^Sz  HÞ n n 2 ðhÞ ^ ^ ^ ðSþ I þ ^S^Iþ Þ VðtÞ ¼ g s lB g n ln 13Cos 4r3 ðtÞ

ðAIII:14Þ

Since the system is in a strong magnetic field, the eigenfunctions may be presented as follows:

 w1 ðtÞ ¼ eSP ; SMS ¼ M F þ 12 jeBenz ; IMI ¼  12i i ½g

eh

1 1 s B ððM F þ2ÞHÞþ2g n n ð1 n ÞðHÞg s B g n

l

l

r

l

2 ðhÞ

ln 13Cos 3

2r ðtÞ

MS þi

c12;SF 2

i

i

1

1

c12;SF 2

Here DBenz(T), DSP(T) are the temperature dependent diffusion constants of benzene and superparamagnetic nanoparticles, Reff, Benz, Reff,SP are their effective radii, and q is a fitting parameter, where for the temperature dependence of the diffusion coefficients we used the expression [55]:

kB T DðTÞ  DBenz ðTÞ  2  pReff ;Benz gCS2

ðAIII:15Þ and

here, gCS2 is the dynamic viscosity coefficient of liquid CS2, which we used as a fitting parameter. The values obtained by fitting were compared to the experimentally measured values of gCS2 in the temperature interval of 4 to 29 °C [56]. Note that the collision rate between the benzene molecule and the superparamagnetic nanoparticle as determined by diffusion is described by the rate constant

ðAIII:21Þ

cnS ¼ kdif  P12 f ðnSP Þ Z pqDðTÞ 2 ðg e le g n ln Þ ½SðS þ 1Þ  M S ðM S þ 1Þ  j ¼ 2 4h

0;0

i

l ððMF 1ÞHÞ1g l ð1rn ÞðHÞþg l g l

13Cos2 ðhÞ

c

MS þi

c12;n 2

t

ðAIII:16Þ

ð0Þ c12;SF ðnSP Þ ¼ c2;SF þ rS  ðnSP Þ; ð0Þ

ð0Þ

where c2;SF and rS  are fitting parameters. The obtained value of c2;SF was compared to the estimated values of c12;S (see Fig.6a and b). Using the time-dependent perturbation theory, the probability amplitude of the relaxation induced by magnetic dipole-dipole interaction may be written as:

¼

^ hw ðtÞjVðtÞjw ðtÞidt

1 2 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  4ih g S B g n n ½1  3Cos2 ðhÞ SðS þ 1Þ  M S ðM S þ 1Þ Rt i i  0 r31ðtÞ eh½ðgs lB þgn ln ð1rn ÞÞHþ2ðc12;SF c12;n Þt dt

l

l

ðAIII:17Þ Thus, the respective probability is given by:

P12 ðtÞ ¼ jAðtÞj2 ¼ h12 j

Rt 0

i

Thus, integrating by h and averaging over MS, the relaxation width may be written as:

cnS ¼

c

Rt

1 pffiffiffiffiffiffiffiffiffiffiffiffi 3 ½q þ DðTÞt ðAIII:22Þ

where the i 12;SF ; i 12;n are the imaginary energies that are propor2 2 tional to the state widths [54]. Note that the c12;n values determined experimentally for different temperatures (see Table 1) were used in the fitting procedure, while c12;SF ðnSP Þ is a fitting parameter dependent on nSP , the number density of the superparamagnetic nanoparticles. Therefore, we determined this parameter as:

a12 ðtÞ ¼ AðtÞ ¼  hi

1;p

 ½1  3Cos2 ðhÞeh½ðgs lB þgn ln ð1rn ÞÞHþ2ðc12;SF ðnSP Þþc12;n Þt dt  dhj2 nSP

  w2 ðtÞ ¼ eSP ; SMS ¼ MF  12 eBenz ; IM I ¼ þ 12 i ½g

ðAIII:20Þ

Thus, the nuclear spin relaxation rate may be written as:

t

s B n n 2 2 n n 2r 3 ðtÞ eh s B  1  eSP ; SMS ¼ MF  2 jeBenz ; IM I c12;n i 1 1 ¼ þ 12 eh½gs lB ððMF 2ÞHÞ2gn ln ð1rn ÞðHÞþi 2 t

ðAIII:19Þ

DðTÞ ¼ DBenz ðTÞ þ DSP ðTÞ  DBenz ðTÞ

q ¼ Reff ;Benz þ Reff ;SP4  Reff ;SP

kdif ðTÞ ¼ 4pqDðTÞ

t

 jeSP ; SMS ¼ M F þ 12ijeBenz ; IMI ¼  12ieh½gs lB ððMF þ2ÞHÞþ2gn ln ð1rn ÞðHÞþi

pffiffiffiffiffiffiffiffiffiffiffiffi DðTÞt

2 ^ hw1 ðtÞjVðtÞjw 1 ðs0; i0Þidtj

1 ¼ 16 ðg e le g n ln Þ2 j½1  3Cos2 ðhÞj2 ½SðS þ 1Þ  M S ðM S þ 1Þ h2 R t 1 i ½ðg l þg l ð1r ÞÞHþ i ðc þc Þt 2 n 12;n 2 12;SF j 0 r3 ðtÞ eh s B n n dtj

ðAIII:18Þ

5p3 qDðTÞ

ðg e le g n ln Þ2 SðS þ 1Þ 2 24h  Z  1 ehi ½ðgs lB þgn ln ð1rn ÞÞHþ2i ðc12;SF ðnSP Þþc12;n Þt 2    dt  nSP pffiffiffiffiffiffiffiffiffiffiffiffi 3   0;0 ½q þ DðTÞt 

ðAIII:23Þ

The latter relationship was used to fit the experimental dependences of the 1H NMR width of the benzene proton line for different sample temperatures versus the nSP, with the following fitting parameters: c12;SF ðnSP Þ, q  Reff ;Benz and gCS2 . References [1] B.D. Terris, T. Thomson, J. Phys. D: Appl. Phys. 38 (2005) R199. [2] S. Blügel, T. Brückel, C.M Schneider (Eds.), Magnetism goes Nano: Electron Correlations, Spin Transport, Molecular Magnetism. In: Lecture Manuscripts of the 36th Spring School of the Institute of Solid State Research, 14–25 February 2005. [3] H. Kronmüller, S. Parkin (Eds.), Handbook of Magnetism and Advanced Magnetic Materials, vol. 3, Wiley, 2007. [4] D.J. Sellmyer, R. Skomski (Eds.), Advanced Magnetic Nanostructures, Springer, 2005. [5] J. Daughton, Proc. IEEE 91 (2003) 681. [6] Yancong Zhang, Lianying Zhang, Xinfeng Song, Xiangling Gu, Hanwen Sun, Chunhua Fu, Fanzong Meng, Hindawi J. Nanomater., 2015, Article ID 417389, pp. 6, http://dx.doi.org/10.1155/2015/417389 [7] M. Inyushin, Yu. Kucheryavih, L. Kucheryavih, L. Rojas, I. Khmelinskii, V. Makarov, Sci. Rep. (2016). [8] Ihab M. Obaidat, Bashar Issa, Yousef Haik, Nanomaterialsm 5 (2015) 63–89, http://dx.doi.org/10.3390/nano5010063. [9] Bimali Sanjeevani Weerakoon, Toshiaki Osuga, Takehisa Konishi, Int. J. Chem. Mol. Nucl. Mater. Metall. Eng. 10 (2016) 50–51. [10] Ning-Ning Song, Hai-Tao Yang, Hao-Liang Liu, Xiao Ren, Hao-Feng Ding, XiangQun Zhang, Zhao-Hua Cheng, Sci. Rep. 1–5 (2013). [11] K.N. Olafson, M.A. Ketchum, J.D. Rimer, P.G. Vekilov, Proc. Natl. Acad. Sci. U.S.A. 112 (2015) 4946–4951. [12] P. Loria, S. Miller, M. Foley, L. Tilley, Biochem. J. 339 (1999) 363–370.

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