ffc application note: diffusion

Fast field cycling (FFC) NMR relaxometry is a low-field magnetic resonance technique which measures the dependence of the spin-lattice relaxation rate R1 (= 1/T1 ) on the magnetic field over a wide range of field strengths with just one instrument. The important information extracted from the relaxation dispersion curves (NMRD profiles) concerns molecular motions (molecular dynamics) described by means of spectral density J(ω)∝ R1.

Fast field cycling NMR relaxometry is a versatile technique that can be applied to determine diffusion coefficients for molecular and ionic systems in liquids as well as in the solid state [1-5,13]. The simplicity of the method relies on the fact that it allows calculating the diffusion coefficient D inde-pendently from the sample (i.e. diffusive model) be-ing tested. At low frequencies the value of R1 depends only on the translational/diffusive motion of the molecules as the rotational contribution can be ne-glected [1-5,13]. The value of the diffusion coefficient D can be de-termined from the linear dependence of R1 on the square root of the magnetic field frequency.

Fast Field Cycling as method to calculate the diffusion coefficient

M. Pasina, G. Ferrantea, D. Krukb a Stelar Srl, Via E. Fermi, 27035 Mede (PV), Italyb Faculty of Mathematics an d Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna 54, PL-10710 Olsztyn, Poland

Introduction

Application of FFC NMR to calculate the diffusion coefficient

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How to calculate the diffusion coefficient: theory

How to calculate the diffusion coefficient: a practical example

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Using the information from an NMRD profile it is possible to calculate the value of the diffusion coef- ficient D in a straightforward way by plotting the rela- xation rate R1 versus the square root of the resonance frequency and then fitting the linear dependence at low fields (Fig. 3). The relaxation rate R1 can be written as the sum of 2 contribution [1, 4, 6, 7, 13]:

R1(ν) ≅ R1(intra)(ν)+ R1(inter)(ν) [1]but at low frequencies:

R1(ν) ≅ R1(0)- A √ν [2]where R1(0) includes the intramolecular contribution and cit is not relevant for the diffusion coefficient. It results:

R1(ν) = R1(inter) (ν) ≅ - A √ν [3] Thus, from the slope A of the linear best-fit curve, one can easily determine the diffusion coefficient D, using the relationship [6, 7, 13]:

A = NSpin ( γ2 ћ)2 ( ) ( )3/2 [4]

where γ is the gyromagnetic constant of the nucleus being measured and NSpin stands for the spin density (number of spins per unit of volume). The spin density can be obtained from the following relation:

NSpin = [5]

where n is the number of spins (hydrogen atoms) per molecule, NAvog is the Avogadro number, ρ is the den-sity of the liquid and MMol is the molecular mass [13]. Equation [4] still holds for nuclei other than hydrogen.

μ0

n NAvog ρMMol

π√2+84π D30

fig. 1: Plot of relaxation data for rape oil at different tempera-tures presented as a function of the square root of Larmor frequency. The solid lines at low fields are the best-fit lines to the experimental points. The slope of these best-fit lines is indicated with the letter A in eq. [3] and can be used to obtain the diffusion coefficient D(T). (From [6]).

fig. 4: Plot of relaxation data for 1-ethyl-3- methylimidazolium thiocyanate (EMIM-SCN) at different tem-peratures presented as a function of the square root of Larmor frequency. The solid lines at low fields are the best-fit lines to the experimental points. (From [7]).

Let’s refer to a sample of 1-ethyl-3- methylimidazoli-um thiocyanate (EMIM-SCN) [7], a ionic liquid that has been measured at different temperatures and the corresponding R1 values are reported as a function of the square root of the frequency in Fig. 2. Knowing the chemical formula for EMIM-SCN (C7H11N3S), we can obtain NSpin=4.36*10-2 Å-3 by substituting in eq. [5] the values:

n=11;NAvog=6,022*1023; ρ=1.11 gr/cm3; MMol=(7*12)+(11*1)+(3*14)+(1*32)=169 u.m.a. Looking at Fig. 4, the diffusion coefficient D, can be obtained easily from eq. [4]:

D = ( )2/3 [6]

where C=( γ2ћ)2( ) ( )3/2 = = 9,95*10-49m5s-3/2

is a constant (that depends only on the nucleus consid-ered and A = 1.018 s-1/2, the angular coefficient of the best-fit line, can be obtained from the experimental data (Fig. 4). For example at 220K the value of D is: 1.25*10-13 m2/s.

NSpin*CA

μ0 π√2+84π D30

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Diffusion experiments performed on very different materials such as ionic liquids [7, 9, 11], polymers [1, 10] and vegetable oils [6], confirm that the results from FFC NMR are always consistent with those from pulsed field-gradient (PFG) NMR methods (Fig. 3). Thus, FFC NMR relaxometry emerges as a comple-mentary method to field-gradient NMR diffusome- try [1, 4, 13] and has a few advantages compared to it:

• FFC is not limited by the strength of the field gradients

• FFC instrumentation allows to obtain low-field data that can be used not only for extracting the diffusion coefficient but also other kinds of dynamical parameters and information (ob-viously depending on the kind of sample and model being used)

• SMARTracer FFC relaxometer is less expensive than an NMR spectrometer equipped with gra-dients required for PFG measurements

• SMARTracer FFC relaxometer provides multi-nuclear capability

• SMARTracer FFC relaxometer provides good temperature control (-140 °C to +140 °C with a 0.1 °C resolution)

For studying the ionic conductivity in electrolytes in terms of ionic diffusion it is important to detect the transport properties of both cations and anions (that could be to some extent aggregated). In this context, the NMR method is extremely useful because it is very sensitive and selective allowing information to be ob-tained on the different ion dynamics. Apart from diffusive studies performed on 1H nucleus, the applicability of the method has been confirmed for nuclei other than 1H , such as 19F [7, 12, 13]. As shown in [7], 19F diffusion measurements can be easily per-formed to determine diffusion coefficient of molecules. In theory, this method is applicable to any NMR-active nuclei, such as 13C, 7Li and 31P [1], provided that NMR sensitivity is enough for detection. The FFC technique has also been applied to study the molecular dynamics of lithium ions in electrolytes [8] and the correspon- ding diffusion coefficients could be calculated.In ionic liquids, FFC NMR relaxometry can be used as an alternative to PFG NMR diffusometry, and the range (of interest) of 10-8> D (m2 s-1)>10-14 can be covered.

Furthermore, using FFC NMR relaxometry:

1. One can determine not only the self-diffu-sion coefficient (as in the gradient methods), but also the relative diffusion coefficient (for instance cation-anion) which provides infor-mation about whether the ionic dynamics are correlated;

2. The method favorably bridges the ranges cov-ered by field-gradient NMR diffusometry and quasi-elastic neutron scattering [4].

Contrary to the field gradient techniques, the FFC method is uniquely sensitive to the NMR signal of the spin nuclei belonging to a low-abundant fraction of molecules as long as their molecular dynamics is significantly differentiated from that of the highly abundant molecules. This is particularly in evidence for liquid molecules adsorbed at pore walls in porous materials where the NMR diffusometry applying field gradient NMR plays a minor role, whereas the FFC NMR relaxometry allows the diffusion of adsorbed molecules to be followed based on specific relaxation mechanisms [11].

Advantages of FFC NMR with respect to field gradient methods

Application to electrolytes

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fig. 3: This plot shows the good agreement between Diffusion coefficient D obtained from FCC NMR (full symbols) and PFG NMR (small open symbols) techniques for several liquids versus reciprocal temperature (From [1]) .

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(1) Meier R., Kruk D., & Rössler E. A., Intermolecular Spin Re-laxation and Translation Diffusion in Liquids and Polymer Melts: Insight from Field‐Cycling 1H NMR Relaxometry, ChemPhysChem 14.13 (2013): 3071-3081;

(2) Kruk D., Meier R., & Rossler E. A., Translational and rota-tional diffusion of glycerol by means of field cycling 1H NMR relaxometry, The Journal of Physical Chemistry B, 115.5 (2011): 951-957;

(3) Flamig M., Becher M., Hofmann M., Körber T., Kresse B., Privalov A. F., Kruk D., Fujara F., & Rössler E. A. (2016). Perspectives of Deuteron Field-Cycling NMR Relaxometry for Probing Molecular Dynamics in Soft Matter, The Journal of Physical Chemistry B, 120(31), 7754-7766;

(4) Kimmich R., & Fatkullin N., (aug 2017), Self-diffusion studies by intra-and inter-molecular spin-lattice relaxometry using field-cycling: Liquids, plastic crystals, porous media, and polymer segments. Progress in Nuclear Magnetic Res-onance Spectroscopy, 101: 18-50;

(5) Kruk D., Meier R., & Rössler E. A., (2012), Nuclear mag-netic resonance relaxometry as a method of measuring trans-lational diffusion coefficients in liquids, Physical Review E, 85(2), 020201;

(6) Rachocki A. & Tritt-Goc J., (2014), Novel application of NMR relaxometry in studies of diffusion in virgin rape oil, Food chemistry, 152, 94-99;

(7) Kruk D., Meier R., Rachocki A., Korpała A., Singh R. K., & Rössler E. A., (2014), Determining diffusion coefficients of ionic liquids by means of field cycling nuclear magnetic resonance relaxometry, The Journal of chemical physics, 140(24), 244509;

references:

(8) Graf M., Kresse B., Privalov A. F., & Vogel M., (2013),- Combining 7Li NMR field-cycling relaxometry and stim-ulated-echo experiments: a powerful approach to lithium ion dynamics in solid-state electrolytes. Solid state nuclear magnetic resonance, 51, 25-30;

(9) Seyedlar A. O., Stapf S., & Mattea C. (2015). Dy-namics of the ionic liquid 1-butyl-3-methylimidazo- lium bis (trifluoromethylsulphonyl) imide studied by nu-clear magnetic resonance dispersion and diffusion. Physical Chemistry Chemical Physics, 17(3), 1653-1659;

(10) Kruk D., Herrmann A., & Rössler E. A. (2012), Field-cycling NMR relaxometry of viscous liquids and polymers. Progress in nuclear magnetic resonance spectroscopy, 63, 33-64;

(11) Rachocki A., Andrzejewska E., Dembna A., & Tritt-Goc J., (2015), Translational dynamics of ionic liquid imidazolium cations at solid/liquid interface in gel polymer electrolyte. Eu-ropean Polymer Journal, 71, 210-220;

(12) Petit D., Korb J. P., Levitz P., LeBideau J., & Brevet D., (2010), Multiscale dynamics of 1H and 19F in confined ion-ogels for lithium batteries, Comptes Rendus Chimie, 13(4), 409-411;

(13) Kruk D., Meier R., & Rössler E. A., (2012), Nuclear mag-netic resonance relaxometry as a method of measuring trans-lational diffusion coefficients in liquids, Physical Review E, 85(2), 020201.

The Stelar relaxometer works by fast electronic switch-ing of the magnetic field from an initial polarizing magnetic field (BPOL), where the equilibrium of nucle-ar magnetization is attained in about 4T1, to a field of interest (relaxation field; BRELAX) at which the nuclear spins relax to the new equilibrium state with a cha- racteristic relaxation time constant T1. After a delay time, τ, the BRELAX is switched to the field of acquisi-

FFC technique

fig. 4 (left): Fast Field Cycling NMR method.

fig.5 (right): Example of NMRD profile. A Gadolinium-based con-trast agent measured from 0.01MHz to 40MHz (from in-house data).

fig. 4 fig. 5

tion (BACQ) and the NMR signal is detected after a π/2 RF pulse (Fig. 4). The magnetic field dependence of 1/T1 is shown in the graphical form as a Nuclear Magnetic Reso-nance Dispersion (NMRD) profile (Fig. 5).Each point of the NMRD profile (i.e. a certain BRELAX) is obtained detecting the NMR signal using a number of different delay times τ .