# A Simple Elimination of the Thermal Convection Effect in NMR Diffusiometry Experiments

^{*}

## Abstract

**:**

_{2}O solvent were used as model molecules measured at T = 319 K in order to show that thermal convection sometimes remains hidden in experiments. In this paper, we demonstrate that the increase in apparent diffusion coefficient with increasing diffusion time is a definite indicator of thermal convection. Extrapolation to zero diffusion time can also be used to obtain the real diffusion coefficient, likewise applying the less sensitive pulse sequences designed for flow compensation or the expensive hardware, e.g., sapphire or Shigemi NMR tubes, to decrease the temperature gradient. Further, we show experiments illustrating the effect of a long diffusion time in which the periodic changes of the echo intensity with gradient strength appear as predicted by theories.

## 1. Introduction

_{c}[5,6]. This number is determined by such geometrical parameters as the height of the liquid level (the lower, the better), the radius of the NMR tube (the narrower, the better), material constants as the coefficient of the kinematic viscosity (the larger, the better), the thermal expansion of the liquid and the thermal diffusion [4,5,6]. The estimated R

_{c}numbers for a 5 mm standard glass NMR tube are 67.4 and 215.8 in the case of insulating and conducting walls, respectively [3]. Of course, these numbers are approximate and still affected by some other factors. According to Morris et al. in chloroform, only 0.3 K cm

^{−1}is enough to cause convection, while for water, about 6 K cm

^{−1}is an approximate limit; therefore, in experiments in aqueous solutions, it is rarely taken into account [6].

**B**

_{0}) [7]. Carr and Purcell presented not only the effect of self-diffusion on transverse relaxation experiments but warned of the effect of convection and showed the difference in intensities between the odd-numbered and the even-numbered echoes [8]. The advent of PGSE (Pulse Field Gradient Spin-Echo) NMR introduction of the time-dependent field gradient brought new dynamism into NMR diffusion experiments [9]. In the case of self-diffusion, the so-called Stejskal–Tanner equation is used to evaluate the apparent diffusion coefficient, D

_{app}[9].

^{2}Δ << D and δ << Δ. Model calculations in the paper by Morris et al. show that at higher diffusion time, the uncertainty of D

_{app}obtained by fitting the echo decay by Equation (1) increases. In the case of an extreme long Δ negative echoes may appear in PGSE experiments [6].

_{app}with increasing Δ was experienced in almost all cases. By intentionally applying higher diffusion times than the optimal ones we could even experimentally demonstrate a negative and periodic change in intensity of the stimulated echo vs. g

^{2}curves, as the above-mentioned considerations and model calculations show [6,22]. We also confirm that the extrapolation of D

_{app}to Δ = 0 gives real values of diffusion coefficients. We show that by applying the standard PGSE pulse sequences without convection compensation, the problem can be recognized and avoided by measuring the diffusion time dependence.

## 2. Results and Discussion

^{1}H NMR spectra of sodium decyl-sulfate (NaDS) as a function of concentration.

_{2}groups. From this, we can conclude that the chemical shifts, in this case, are not good parameters to check the presence of micelles in the solutions in spite of our expectations.

_{2}O solutions of different concentrations at 319 K. The measured self-diffusion coefficients show significant but different dependences on the diffusion time.

^{−1}concentration, the D

_{app}values extrapolated to Δ = 0 are constant. At concentrations higher than 0.029 mol kg

^{−1}, the extrapolated diffusion coefficients start to decrease, probably because of the appearance of micelles in an increasing concentration ratio. Figure 3 shows the dependence of the intercept on the concentration. The estimated critical micelle formation concentration (cmc) is approximately 0.035 mol kg

^{−1}, which is in good agreement with the literature data [32].

_{app}slightly decreases with increasing diffusion time, which indicates a kind of hindered diffusion. It may be explained that observing the motion of the molecule for a longer time makes the collisions between them more probable than in a shorter time. Therefore, the random walk conditions are not fulfilled [14,33]. The increasing trend of D

_{app}with Δ at larger concentrations is the probable indication of thermal convection, although the exchange process between the monomers and micelles may also be considered [16,34]. If the reason is thermal convection, then the slope is in connection with the convection velocity (v) according to Equation (2). Further analysis of the slope is out of the scope of this paper because there are very detailed considerations on it in the literature and where the convection rate is the parameter needed [6,22].

^{−1}. The cmc values for NaOS, NaDS and NaHDS are ~0.129 (extrapolated), 0.033 and 0.0006 mol kg

^{−1}, respectively [32]. It means that in the case of NaOS, practically only monomers, while in that of NaHDS, only micelles are present. The results are shown in Figure 4. The apparent diffusion coefficients of octyl- and decyl-sulfates show a conspicuous linear increase with the diffusion time, but for the hexadecyl-sulfate and the water, this dependence is negligible. These experimental results lead to the conclusion that the phenomenon causing the linear increase in D

_{app}as a function of diffusion time tends to be not of chemical origin but rather physical.

^{−1}solution of NaDS in a sapphire NMR tube (the thermal conductivity is 25 times that of borosilicate) and in a normal NMR tube but implementing a pulse sequence, which reduces the effect of thermal convection developed by Jerschow et al. [24,25]. The reduction of the effect of thermal convection is clearly seen in Figure 5.

_{app}values to zero diffusion time results in the real diffusion coefficient.

^{2}function. This effect of the cos term at a short diffusion time is not visible; the I/I

_{0}vs. g

^{2}function can be well fitted with Equation (1) in accordance with our experiences [6,22]. In Figure 6A, the results of the least square fit of Equation (1) on the echo decay data are seen at short diffusion times. One can see very good fit up to 50 ms (in this case δ = 4 ms). Only at large gradient values (see insert) do small deviations appear between the measured and calculated values. Applying a 120 ms diffusion time instead of the usual maximum of 60 ms resulted in us detecting the experimental periodic changes in peak intensities (Figure 6B). The amplitude is not large, but it is visible. Equation (3) can reproduce this periodicity (red line), but the fit is not very good, pointing out the rough approximations applied for the deduction of this equation [6,22,35]. The fits with Equation (1) are spectacularly worse than that with Equation (3) in this case (blue line). Further, we can observe (in Figure 6B) that at the lower gradient values, both equations fit well. It illustrates that the independent determination of the convection rate (v) and the diffusion coefficient (D

_{app}) is not possible only by parameter fitting procedures.

## 3. Materials and Methods

^{1}H NMR spectroscopy (Figure 1) [32]. The integrated intensity of the C-H protons was used to check the purity of the surfactants. Every measured sample was freshly prepared and checked before the diffusion experiment in order to avoid the effect of partial hydrolysis. Sodium-decyl-sulfate was mostly used. However, to verify some statements, we used the results obtained on other surfactants indicated above.

^{−1}solutions of three NaOSO

_{2}(CH

_{2})

_{n}CH

_{3}were prepared by weight (n are 9 NaDS, 7 NaOS and 15 NaHDS) and measured in deuterium oxide solution at 319 and at 298 K (laboratory temperature). A total of 500 uL solution was used as a standard volume in each case to reach good shim. Norell © type 5 mm NMR tubes were applied in all cases, but in one series of experiments, a sapphire tube was used.

^{1}H–X inverse gradient probe head was used. The temperature was regulated by a Bruker BCU 4 cooler using dry air flow with a rate of 800 L/h. Under the standard Bruker TopSpin 2.1 software, a 2D pulse sequence (ledbpgp2s provided with the spectrometer) was used without modification for diffusion measurement. Stimulated echoes were recorded by applying LED (low eddy current delay) and bipolar gradient pulses, and two spoil gradients [36]. Usually, from 5% to 95% of the available gradient strength is used with 64 steps. The distances between consecutive gradient pulses were applied in the square mode. The gradient value was calibrated for D

_{2}O as D = 1.9 × 10

^{−9}m

^{2}s

^{−1}(298 K) [37]. In each experiment, δ and Δ in Equation (1) were kept constant, and g varied. The apparent diffusion coefficients were evaluated from the decay of individual peaks separately by using MestreNova 8.1© software, fitting Equation (1) to the experimental data. The average values of these apparent D values were used for further analysis. The decrease in stimulated echoes measured by using an extra-long diffusion time (Δ) was fitted by Equation (3) as well. In one series of experiments, we adapted a modified pulse sequence, named double-stimulated-echo pulse sequence, published by Jerschow and Müller dedicated to suppress “convection artifacts” in the PGSE experiments [24]. The real diffusion coefficients were obtained by linear regression from D

_{app}vs. (Δ − δ/3) curves, as illustrated in the Supplementary Materials.

## 4. Conclusions

_{2}O avoids thermal convection, we showed that it is not evident. Measuring the diffusion time dependence of the observed diffusion coefficients is always required.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

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**Figure 1.**

^{1}H NMR spectra of sodium decyl-sulfate (NaDS) below cmc (5 × 10

^{−3}mol kg

^{−1}, upper spectrum) and above cmc (5.6 × 10

^{−2}mol kg

^{−1}, lower spectrum). Peaks assignment: 4.71 ppm HDO, 4.22 ppm CH

_{2}(1), 1.84 ppm CH

_{2}(2), 1.4–1.6 ppm CH

_{2}(3–9) and 1.03 ppm CH

_{3}.

**Figure 2.**The apparent self-diffusion coefficients measured at different concentrations as a function of the diffusion time (Δ) at 319 K. Concentrations: ▲ 0.005 mol kg

^{−1}, ● 0.015 mol kg

^{−1}, ■ 0.029 mol kg

^{−1}, ○ 0.047 mol kg

^{−1}, □ 0.059 mol kg

^{−1}, Δ 0.074 mol kg

^{−1}.

**Figure 3.**Dependence of the intercept (D

_{app}) of lines in Figure 2. on the concentration of NaDS. The intersection of the lines gives the cmc.

**Figure 4.**The dependence of the apparent diffusion coefficient of molecules present on the diffusion time at 319 K. ■ NaHDS ● NaDS and ▲ NaOS while ○ HDO.

**Figure 5.**The apparent diffusion coefficient of NaDS as a function of diffusion time at 319 K. ● normal 5 mm NMR tube, ■ in a sapphire tube applying the standard pulse sequence without convection reduction and Δ with pulse sequence with convection compensation.

**Figure 6.**The decrease in the stimulated echo intensity with the increasing strength of the gradient square (0.07 mol kg

^{−1}NaDS, T = 319 K, δ = 4 ms). The inserts are the enlargements of the wrapped part of the curves. (A) The best fit of Equation (1) Δ = [circle = 20, diamond = 30, square = 40 and triangle = 50 ms] (B) Δ = 120 ms. The blue and red lines are the best fits of Equations (1) and (3), respectively.

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**MDPI and ACS Style**

Nyul, D.; Novák, L.; Kéri, M.; Bányai, I. A Simple Elimination of the Thermal Convection Effect in NMR Diffusiometry Experiments. *Molecules* **2022**, *27*, 6399.
https://doi.org/10.3390/molecules27196399

**AMA Style**

Nyul D, Novák L, Kéri M, Bányai I. A Simple Elimination of the Thermal Convection Effect in NMR Diffusiometry Experiments. *Molecules*. 2022; 27(19):6399.
https://doi.org/10.3390/molecules27196399

**Chicago/Turabian Style**

Nyul, Dávid, Levente Novák, Mónika Kéri, and István Bányai. 2022. "A Simple Elimination of the Thermal Convection Effect in NMR Diffusiometry Experiments" *Molecules* 27, no. 19: 6399.
https://doi.org/10.3390/molecules27196399