Nuclear spin–lattice relaxation in nanofluids with paramagnetic impurities

Journal of Magnetic Resonance 261 (2015) 175–180

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Nuclear spin–lattice relaxation in nanofluids with paramagnetic impurities Gregory B. Furman ⇑, Shaul D. Goren, Victor M. Meerovich, Vladimir L. Sokolovsky Physics Department, Ben Gurion University of the Negev, Beer Sheva 84105, Israel

a r t i c l e

i n f o

Article history: Received 3 September 2015 Available online 2 November 2015 Keywords: Nanosize cavity Paramagnetic impurities Spin lattice relaxation

a b s t r a c t We study the spin–lattice relaxation of the nuclear spins in a liquid or a gas entrapped in nanosized ellipsoidal cavities with paramagnetic impurities. Two cases are considered where the major axes of cavities are in orientational order and isotropically disordered. The evolution equation and analytical expression for spin lattice relaxation time are obtained which give the dependence of the relaxation time on the structural parameters of a nanocavity and the characteristics of a gas or a liquid confined in nanocavities. For the case of orientationally ordered cavities, the relaxation process is exponential. When the nanocavities are isotropically disordered, the time dependence of the magnetization is significantly non-exponential. As shown for this case, the relaxation process is characterized by two time constants. The measurements of the relaxation time, along with the information about the cavity size, allow determining the shape and orientation of the nanocavity and concentration of the paramagnetic impurities. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction The use of nuclear magnetic resonance (NMR) spectroscopy [1,2] in the study of structure and dynamic properties of molecules (both organic and inorganic) in solutions [4], liquid [5–7], liquid crystals [3], and solids, soft matter systems [8–10], such as nanostructures and nanoporous materials [11–13], and compound environments including membranes [14] is rapidly extended. Multifarious NMR techniques and methods, cryoporometry technique [13], spin locking [15], spin dynamics in a local field [16,17], multiple-quantum NMR [18–22] and multiple-pulse NMR [23] spin-relaxation experiments [10,24] were used to identify and quantify finite size effects. A major role in the structure investigation plays dipole–dipole interactions, due to the fact that the dipole–dipole coupling constant depends on the structural parameters [1,2]. Spin–lattice relaxation NMR methods applying to nanoscale objects can give important information about their shape, orientation, and sizes [25–30]. Recently, using continuous spin locking [15] and pulsed spin locking methods [23], we performed the study of the spin lattice relaxation due to direct interaction of the nuclear spins with the lattice. The relaxation times obtained in this study lie within a wide range from few to hundreds of seconds. Among developments of NMR for use in study of matter, the inclusion of contrast agents which cause proton relaxation ⇑ Corresponding author. E-mail address: [email protected] (G.B. Furman). http://dx.doi.org/10.1016/j.jmr.2015.10.013 1090-7807/Ó 2015 Elsevier Inc. All rights reserved.

enhancement and improve identifying structure [8]. These agents usually contain paramagnetic impurities (PI), being in stable free radicals such as nitroxides or transition metal/lanthanide cations. The large magnetic moment of PI has the effect of enhancing the local magnetic fields which fluctuate and resulting in assisting the relaxation process. Another use of PI is the dynamic nuclear polarization to prepare polarized targets for nuclear and particle physics [31]. In view of these applications, use of PI for study of nanoobjects is also important. It was revealed that in a bulk containing nuclear spins (I) coupled with spins (S) of paramagnetic impurities (PI’s), the heteronuclear dipole–dipole interactions (DDI’s) play the dominant role in the nuclear spin–lattice relaxation [32–36]. Existence of PI’s decreases the nuclear spin–lattice relaxation time several times in comparison of that in the sample without PI. In this paper we consider the spin–lattice relaxation in liquids or gases containing PI’s and entrapped in nanosized cavities (Fig. 1). We investigate two cases. In the first case, the material is in orientational order where the cavities, their long axes, a, (Fig. 1a), are oriented along a common direction. In the second case, the material is isotropically disordered, the long axes are directed arbitrary, as schematically illustrated in Fig. 1b. This paper is organized as follows. In Section 2 we describe the procedure of averaging of DDI’s between nuclear and PI spins in coordinate spaces inside nanosized cavities. In Section 3, we consider evolution of the nuclear spin system due to the spin interaction with PI. We show that DDI’s between nuclear and PI spins are

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~0 k~ Fig. 1. Oriented (a) and disoriented (b) nanocavities containing water molecules. h is the angle between the external magnetic field (H z) and the principal axis a of the nanocavity.

characterized by a single universal dipolar coupling constant which depends on the volume and shape of a nanocavity and its orientation relatively to the external magnetic field. In the last section we discuss and conclude the main results. 2. Average Hamiltonian in a nanocavity Let us consider N nuclear spins, I ¼ 1=2 coupling with n PI spins, S by DDIs, enclosed in an elongated nanosized cavity with the prin~0 is direccipal axes a, b, and c ¼ b. A high external magnetic field H ted along the z-axis at the angle h to a axis (Fig. 1). Assuming that the dominant nuclear relaxation mechanism is dipole–dipole interaction between the nuclear and PI spins [32–36], the Hamiltonian of this system can be written as

H ¼ xI Iz þ xS Sz þ HSS þ HIS

ð1Þ

where xI and xS are the Larmor frequencies of the nuclei and PI’s, P P Iz ¼ Nl¼1 Izl ; Sz ¼ nj¼1 Szj , Izl and Szj are the projection of the angular

momentum operator of the l-th nuclear spin (l ¼ 1; 2; . . . ; N) and the j-th spin of PI’s (j = 1, 2, . . . , n) on the z-axis, respectively. HSS is the dipole–dipole interaction Hamiltonian of PI’s and HIS describes the dipole–dipole interaction between the nuclei and PI spins. At xI  xS , the Hamiltonian HIS has a part which gives the dominant contribution to the relaxation process [33,37–40]:

HIS ¼

N X n   X Szj F jl Iþl þ F jl Il

l¼1 j¼1

ð2Þ

3 cI cS h sin 2hjl eiujl ; 4 r 3jl

estimated as [23] D  2  109

r ¼ 8:94  104

ð3Þ

cI and cS are the gyromagnetic ratios of nuclei and PI, respectively, r jl connecting r jl , hjl ; ujl are the spherical coordinates of the vector ~ the j-th paramagnetic impurity and the l-th nucleus in a coordinate system with the z axis along the applied external field. Here the

Ns m2

m2 s

by using the dynamic viscosity,

and the molecular van der Waals radius,

k ¼ 2  1010 m [41]. The estimated diffusion coefficient is close to the experimental value D  2:299  109 ms at 25 °C [42]. The typical NMR time scale which characterizes the flip–flop transition due to   cI ¼ 4:2577 kHz and PI interaction between the nuclear G  1    h c c 6 cS ¼ 2810:1 kHz ’ 6:34  10 s. Therespins is t NMR ¼ Ir3S G 2

jl

fore, Hamiltonian (2) can be averaged if the condition t mov  tNMR is fulfiled. [25,27–29,43]. Then, to obtain nonzero averaged Hamiltonian, the typical length l of the water-confined cavity should be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi much less than 2Dt NMR ’ 160 nm. After averaging, the evolution of the spin system is governed by Hamiltonian (1) with the spaceaveraged Hamiltonian

HIS ¼ FIy Sz ;

ð4Þ

with spacing-independent coupling constants

F¼

where Il ¼ Ixl  iIyl , Ixl and Iyl are the projections of the spin operator onto axes x and y, respectively,

F jl ¼ 

Greek indices indicate the positions of the nuclei and the Latin those of the impurities. For an adequate description of a spin system restricted by a cavity, Hamiltonian (2) should be averaged in coordinate space. Let us estimate the characteristic size l of a nanocavity containing water for spins of which Hamiltonian (2) can be presented by its nonzero averaged form due to restricted diffusion in a nanocavity. The diffusion coefficient for spherical particles moving through a liquid can be

p cI cS h 2

V

ð5Þ

G sin 2h;

where V is the volume of the nanocavity, Iy ¼

PN

l¼1 Iyl

is the projec-

tions of the total nuclear spin operator onto the y axis, and form factor [28]

8    < 23 þ 2 e12  1 1  1e arctan h e ; a6b    G¼ 2 1 1 : 3  2 2 þ 1 1  e arctan jej ; a P b j j j ej and

e2 ¼ 1  ba22 ,  43p < G < 23p.

ð6Þ

G.B. Furman et al. / Journal of Magnetic Resonance 261 (2015) 175–180

177

3. Evolution equation

function, hSz ðt ÞSz ie , which is mainly responsible for relaxation, can be described by the simple exponential correlation function [1,46]

We consider Hamiltonian (1) which describes the spin systems of the nuclei and PI’s, in which HIS is replaced by averaged Hamiltonian (4), HIS :

D E jt j hSz ðtÞSz ie ¼ S2z esc

H ¼ xI Iz þ xS Sz þ HSS þ HIS :

ð7Þ

The Hamiltonian contains the interactions of nuclear and PI spins causing rapid relaxation of the nuclear spins, while the direct connection of the nuclear spins with the lattice is weak and does not practically contribute in the relaxation. Under these conditions, using the method of the nonequilibrium density matrix [44,45], we formulate and solve the problem of calculating the rate of energy transfer from the nuclear spin system to the PI spin system. For the system under consideration, the form of the non-equilibrium density matrix is given by [45]



1 Z

q ¼ exp bI ðtÞxI Iz  bS xS Sz þ ðbI  bS Þ

Z

0

1

t dte K ðt Þ ;

ð8Þ

where Z is the partition function

 Z Z ¼ Tr exp bI ðt ÞxI Iz  bS xS Sz þ ðbI  bS Þ

0

1

t

dte K ðtÞ



ð9Þ

and bI and bS are the time-dependent inverse temperatures of the nuclear and PI’s spin systems, respectively. The quantity K ðtÞ, the so-called energy flux [45], is obtained from

K ðtÞ ¼ eiHt KeiHt ;

ð10Þ

 where K ¼ i H; xI Iz ¼ xI FSz Ix . The factor  is introduced in the time integral, and the limit  ! 0 should be taken after carrying out of all the calculations [45]. If the heat capacity of PI’s is large or if the spin–lattice relaxation time of PI’s is very short, it is reasonable to consider the relaxation process with a constant inverse spin temperature of PI’s system equal to the lattice temperature: bS ¼ b. Following to [45], the evolution equation for the nuclear magnetization M ¼ hxI Iz ie in the absence of spin diffusion is derived

dM ðbI  bÞ ¼ dt b

Z

Z

b

0

dk 0

1

t

dte hK 0 ðt  ikbÞK ie ;

ð11Þ

where the thermodynamic average h. . .ie corresponds to averaging using the quasi-equilibrium state operator qe in the hightemperature approximation

qe ¼

1 ð1  bI xI Iz  bxS Sz Þ Trð1Þ

ð12Þ

and

ð13Þ

Keeping in (13) only the zero-order terms with respect to b; for we obtain

dM x2I db ¼ n TrI2z I : N dt dt 2 ð2S þ 1Þ

dM dt

ð14Þ

Using (14), we obtain from Eq. (11) an equation for the inverse temperature of the nuclear spin system

dbI ¼ T 1 1 ðbI  bÞ; dt

ð15Þ

where

Z

0

1

t

dte hSz ðtÞSz ie cos xI t

ð16Þ

is the spin–lattice relaxation rate due to the interaction of the spins with PI’s. Assume that the fluctuating component of correlation

ð17Þ

where sc is the correlation time of the PI spins [1,46]. This time characterizes the material containing PI’s. After substitution of Eq. (17) into Eq. (16), integrating over time, and applying Eq. (5), one obtains finally

T 1 1 ¼

 2 1 c c h sc SðS þ 1Þn p I S G sin 2h : 12 V x2I s2c þ 1

ð18Þ

Expression (18) gives the dependence of the spin lattice relaxation rate on the correlation time sc , external magnetic field xI , the characteristics of gas or liquid inside the cavity: volume V, number of the PI spins n, and parameters of the cavity: shape G and orientation, angle h. Fig. 2 presents the obtained from (18) dependencies of T 1 on n and h (Fig. 2a), n and G (Fig. 2b), n and V (Fig. 2c), n and f ¼ xI sc (Fig. 2d). Fig. 2a and b shows that the relaxation time increases sharply at orientations of the cavity close to h ¼ 0 and h ¼ 90 and/or G ¼ 0. When G ¼ 0 (spherical cavity), average Hamiltonian (4) is zero and, for correct description of processes, one should consider the next approximations or other mechanisms of relaxation, such as the direct interaction of nuclear spins with a lattice [15,23]. In experiments, parameter f can vary in the wide range from 102 to 102 [8], however, the minimum of the relaxation time on f lie near the point f ¼ 1 (Fig. 2d). By utilizing this fact, we can calculate the minimum of T 1 from (18) and obtain

T1 ¼

24xI  2 :  SðS þ 1Þn p cI VcS h G sin 2h

ð19Þ

At f 1, the relaxation time increases proportionally to this parameter. The correlation time, sc is usually determined by the activation energy, E and takes the Arrhenius form sc ¼ s0 ebE [1]. After extracting sc from experimental data on T 1 by using of Eq. (18) and plotting ln sc vs b, a straight line should be obtained and its slope should give the activation energy E. Using Eq. (18) and knowing the activation energy, one can determine the temperature dependence of the relaxation time. In the case when all the cavities are oriented along a common direction (Fig. 1a), it follows from Eq. (15) that the inverse spin temperature, bI tends to the equilibrium exponentially and is described by a normalized relaxation function

Rðt Þ ¼

K 0 ðt  ikbÞ ¼ eiðtikbÞðHHIS Þ KeiðtikbÞðHHIS Þ :

2 T 1 1 ¼ F

e

ðbI ðt Þ  bÞ t ¼ e T1 : ðbI ð0Þ  bÞ

ð20Þ

In an isotropically disordered case (Fig. 1b), to compare the experimental data and theoretical results, the normalized relaxation function has to be averaged over the cavity orientation

hRðtÞih ¼

1 2

Z p

dh sin h RðtÞ

ð21Þ

0

The integral in (21) cannot be solved in quadratures, the results of the numerical calculation are shown in Figs. 3 and 4. Fig. 3 presents the averaged normalized relaxation function vs. n and V (Fig. 3a and b) and vs. of G and V (Fig. 3c and d) at t ¼ 0:2 s and at t ¼ 1 s. Fig. 4 shows the time dependence of the normalized relaxation function hRih and ln hRih at different stages of relaxation and for different f. This time dependence is substantially nonexponential. In the initial stage of relaxation, the magnetization decays with the characteristic relaxation time ’0.8 s for both f ¼ 0:05 and f ¼ 1 (Fig. 4a) while in the final stage, the characteristic time is ’7.5 s at f ¼ 0:05 (Fig. 4b) and ’5 s at f ¼ 1 (Fig. 4c). The difference in relaxation rates

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Fig. 2. T 1 as a function of: (a) n and h at V ¼ 20 nm3 , G ¼ 2 and f ¼ 0:05, (b) n and G at V ¼ 20 nm3 , f ¼ 0:05; h ¼ p=4, (c) n and V at G ¼ 2, f ¼ 0:05, h ¼ p=4, (d) n and f at V ¼ 20 nm3 , G ¼ 2, h ¼ p=4. Here S ¼ 1=2 and xI ¼ 500 MHz.

Fig. 3. Averaged normalized relaxation function, hRih : (a) versus n and V at t ¼ 0:2 s and G ¼ 2; (b) versus n and V at t ¼ 1 s, G ¼ 2; (c) versus G and V at t ¼ 0:2 s and n ¼ 6; (d) versus G and V at t ¼ 1 s and n ¼ 6. Here f ¼ 0:05, S ¼ 1=2, xI ¼ 500 MHz.

G.B. Furman et al. / Journal of Magnetic Resonance 261 (2015) 175–180

ln R 0.0

system before the whole system reaches the equilibrium. The DDI, which are not averaged to zero [25], ensure the establishment of the quasi-equilibrium state described by Eq. (12). Comparison of the data of the spin–lattice relaxation obtained in the present study (T 1 0:1  10 s) and the experimental data for the spin–spin

θ

0.2 0.4 0.6 0.8 1.0 0.2

0.4

0.6

0.8

1.0

t, s

(a) ln R 1.2

θ

1.3 1.4 1.5 1.6 1.7 1.8 1.5

2.0

2.5

3.0

2.5

3.0

t, s

(b) ln R

179

θ

2.6 2.7 2.8 2.9 3.0

1.5

2.0

t, s

(c) Fig. 4. Time dependence of the averaged normalized function, hRih and ln hRih : (a) 0 < t < 0:1 s for both f ¼ 0:05 and f ¼ 1; (b) 1s < t < 3 s at f ¼ 0:05; (c) 1 s < t < 3 s at f ¼ 1. Here n ¼ 10, S ¼ 1=2, V ¼ 20 nm3 , G ¼ 2 and, xI ¼ 500 MHz.

at initial and final stages can be explained by the following. The relaxation depends on the cavity orientation and in the cavities oriented along and perpendicular to the external magnetic field occurs slower than the relaxation in the cavities with the other orientations.

relaxation time (T 2 103 s) [25] proves that the assumption of establishing a quasi-equilibrium state (12) in the spin system of the nanofluid is correct. Establishing of a quasi-equilibrium state is an important difference in the behavior of the fluid confined in   a nanovolume  106 nm3 from the bulk liquid. Commonly, the concentration of PI’s is much less than the concentration of the nuclear spins [31], but even Nn 1=100 still results to essential relaxation enhancement. Therefore, for the 600 molecules of H2 per a nanocavity with average volume of 45 nm3 [25] 6 PI’s is sufficient to enhance the relaxation. The spin–lattice relaxation time depends on the structural parameters of a nanocavity and the characteristics of a gas or a liquid, and it does not depend on the number of nuclear spins, N, as in the case when the relaxation is determined by the direct interaction of nuclear spins with a lattice [15,23]. We obtained that there are two orientations of the cavities, h ¼ 0 and 90°, where the spin relaxation through the interaction with PI’s is not effective (the spin–lattice relaxation time tends to infinity), while, in the case of the direct interaction of nuclear spins with a lattice, the spin lattice relaxation time goes to infinity near the magic angles: h ¼ 54:74 and 125.26° [15,23,25,27–29,43]. The dependence of the relaxation rate on the cavity orientation can be used to control and govern the relaxation processes in the nanofluids. It was shown that in an isotropically disordered matter, at the beginning of the relaxation process, the magnetization decays rapidly. At this stage the relaxation time in three order shorter than in the case of the direct interaction of nuclear spins with a lattice [15,23]. At the next stage, the relaxation rate decreases because it is only determined by the cavities oriented along and perpendicular to the external magnetic field with long relaxation time. The theoretical models we considered in this work touch upon various soft matter and biological systems. The most obvious connection comes from the physics of molecules dissolved in complex fluids. Our approach can be used to characterize complex fluids confined at the nanoscale. Our results are also relevant in the situation when nanoconfinement is provided by a vesicle, in particular, in biological systems. Finally, we would like to point out an advantage of the measurement of the relaxation time over the NMR cryoporometry technique [13,48,49]: the measurements of R (ordered) and hRih (disordered), along with the information about the cavity size, allow determining the shape and orientation of the cavity.

References 4. Discussion and conclusion We considered the spin–lattice relaxation process due to the interaction with PI. Using the method of the nonequilibrium state operator [45], we obtained the equation describing the relaxation of the nuclear magnetization and analytical expression for the spin–lattice relaxation time. An important part of the relaxation process in the considered system is the establishment of quasiequilibrium state (12). According to Bogolyubov’s theory [47] about hierarchy of the relaxation times, if the spin–spin relaxation time T 2 is shorter than the spin–lattice relaxation time T 1 ; T 2  T 1 , the quasi-equilibrium state is established in parts of the spin

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