Viscous Calibration Liquids for Self-Diffusion Measurements - Journal

Article pubs.acs.org/jced

Viscous Calibration Liquids for Self-Diffusion Measurements Kenneth R. Harris* School of Physical, Environmental and Mathematical Sciences, The University of New South Wales, PO Box 7916, Canberra BC, ACT 2610, Australia

Batchimeg Ganbold and William S. Price Nanoscale Organisation and Dynamics Group, University of Western Sydney, Penrith, NSW 2751, Australia ABSTRACT: Self-diffusion measurements made by steady or pulsed field gradient spin−echo NMR are not absolute and the magnetic field gradients employed must normally be determined by calibration with liquids with known selfdiffusion coefficients. The primary calibrant is water, with selfdiffusion coefficient values having been extrapolated from the tracer diffusion of HDO and of HTO in ordinary water by Mills,1 with a relative standard uncertainty of 0.2 %. This and other liquids presently used for calibration all have low viscosities. Current work on ionic liquids, which are generally quite viscous, suggests there may be problems with the pulsed field gradient (PGSE) techniques usually employed as results dependent on the time interval between gradient pulses have been reported by Hayamizu et al.2 In this work, self-diffusion coefficients, obtained by a steady gradient (SG) technique, are reported for the viscous molecular liquids squalane, ethylhexyl benzoate, and bis(ethylhexyl) phthalate (DEHP), and it is suggested that these substances may be suitable secondary reference materials for the calibration of spin−echo NMR apparatus when self-diffusion in viscous liquids is to be measured. New PGSE measurements for squalane and DEHP are in good agreement with the SG results. We also report on systematic errors found in the secondary calibration data of Holz et al.3 for cyclohexane, n-dodecane, dimethyl sulfoxide, and pentan-1-ol (though not for 1,4-dioxane) and suggest toluene in their place as a more convenient low-viscosity calibrant that is also suitable for low temperature work.

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INTRODUCTION

D12 = −

The spin−echo NMR technique is now a well-established method for the measurement of the self-diffusion coefficients of pure liquids and intradiffusion coefficients4,5 in solutions (DS).6−9 These are a measure of molecular “Brownian” motion and can be expressed in terms of Einstein-Kubo velocity autocorrelation functions describing the randomization of the molecular velocity (v) of particle α of a particular component (i) due to intermolecular collisions: Dsi =

1 3

∫0

×

(1)

where the average is taken over the ensemble of individual i molecules indicated by the index α.10 The technique is widely used in physical, biological, and medical chemistry. On the other hand, classical measurements of interdiffusion coefficients D12 require a concentration gradient, and the corresponding Kubo relation for a binary system contains a cross-correlation function for the molecular velocities of the two separate components 1 and 2, again averaged over the total ensemble of different molecules, α and β: © XXXX American Chemical Society

∫0

∞

⟨v1α(0)·v2β(t )⟩ dt

(2)

where xi, Mi and f i indicate mole fraction, molar mass and the mole-fraction-scale activity coefficient, respectively:11 the activity coefficient factor incorporates positional correlations or solution structure. Radio-tracer methods for the measurement of self- and intradiffusion take advantage of the ease with which very small concentrations of radio-actively labeled substances can be accurately determined, so the necessary gradients are very small, with measurements being made at essentially infinite dilution of the radio-tracer containing species.12 Under these circumstances, the measurement of a tracer diffusion coefficient approximates the self- or intra-

∞

⟨viα(0)·viα(t )⟩ dt

∂ ln f2 ⎞ ⎛ c 2VNA ⎞ (x1M1 + x 2M 2)2 ⎛ ⎟ ⎟ ⎜ ⎜1 + M1M 2x 2 ∂ ln x 2 ⎠T , p⎝ 3 ⎠ ⎝

Special Issue: Memorial Issue in Honor of Anthony R. H. Goodwin Received: March 15, 2015 Accepted: July 27, 2015

A

DOI: 10.1021/acs.jced.5b00246 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

PGSE measurements in this work. Particular care must be taken in such calibrations.28 The temperature range for H2O was extended to 90 °C by Easteal et al.29 in 1989, again using radio-tracer measurements and with the assumption of the same value of the isotope effect correction used by Mills. Reliable measurements of the selfdiffusion coefficient of water now extend above and below the normal melting and boiling points, and a critical summary is available.30 A range of other secondary calibrants have been suggested for various circumstances and values recommended for use.3,20,31−33 Nevertheless, there is a need for calibrants with much lower diffusion coefficients than are listed in these sources (the lowest calibrant DS given is that of pentan-1-ol at 5 °C, 143 × 10−12·m2·s−1 3) as more and more experimental work deals with systems with DS in the range of 10−11·m2·s−1. Examples are confined systems such as biological cells, polymer solutions and very viscous systems such as ionic liquids. The latter seem to present some particular experimental difficulties. For PGSE experiments, the Stejskal-Tanner working equation for the dependence of the observed signal, A, on the gradient, g, the width of the gradient pulse (assumed to be rectangular), δ, and the time interval between the gradient pulses, Δ, is, in simplified form and with the neglect of background gradients and cross-terms and the assumption of a constant 90°−180° rf pulse interval to remove the effect of spin−spin relaxation (T2):6

diffusion coefficient. In the case of the diaphragm cell, the technique can yield a precision of ± (0.1 to 0.2) % in DS.13 However, the different mass of the labeled atoms can produce isotope effects, the magnitude of which can vary with the position of the labeling within the molecule, and the number of such labeled atoms. In some cases, such as benzene, the isotope effect is so small as to be almost negligible:14,15 for water, due to its extensive intermolecular H-bonding, labeling with tritium produces an isotope effect of about 3 % for tracer diffusion1 as both the mass and moment of inertia of the molecule are affected. (The difference between the selfdiffusion coefficients of pure H2O and D2O is of course much larger, 23 % at 25 °C). Isotope effects for other substances lie between these extremes,16,17 and in the case of intradiffusion, the isotope effect may depend on the nature of the solvent.18,19 With the NMR technique, where the signal attenuation due to diffusion of the resonant nuclei in a magnetic field gradient is determined following perturbation of the equilibrium distribution of nuclear magnetic moments by radio frequency (rf) pulses, labeling is not usually required, and isotope effects can generally be avoided. NMR spin−echo techniques lend themselves to use over wide ranges of temperature and pressure, with the appropriate experimental equipment, there being no manual sampling as is normally required in radio-tracer analysis. With continuing improvements to NMR equipment and techniques, the precision of spin−echo measurements is now approaching that of radiotracer measurements, though great care needs to be taken with temperature control and thermometer calibration.20 There are two major variants: steady field gradient spin−echo NMR (SG), where the gradient is held constant during the pulse sequence and echo measurement, generally used on (older) low field instruments; pulsed field gradient spin−echo NMR (PGSE), now standard on most high-field instruments, and its variant, where a PGSE sequence is incorporated into a magnetic resonance imaging (MRI) sequence, most often employed in biological or medical work, including measurements in vivo. While in principle it is possible to determine the amplitude of the magnetic field gradient from the shape of the echo21,22 or from the dependence of the Larmor frequency of a sample in a narrow capillary as a function of position,23 and hence to make absolute measurements, most workers choose to make relative measurements, calibrating their apparatus with samples of known DS. Using a calibrant also has the advantage of reducing the systematic effects of background magnetic gradients and other nonideal behavior,24 at least to first order. However, questions then arise as to what is the appropriate calibrant to use. At present, the most common calibrant is water. Mills1 derived values for both H2O and D2O self-diffusion coefficients in the temperature range (1 to 45) °C from his own radiotracer diffusion measurements for HTO/H2O and DTO/D2O and the infinite dilution interdiffusion values for HDO/H2O and HDO/D2O obtained from the Rayleigh interferometric measurements of Longsworth25 of the composition dependence of the mutual diffusion coefficient for H2O−D2O mixtures. He assumed a linear isotope effect in extrapolating from the tracer diffusion of the deuterium and tritium labeled isotopomers to the self-diffusion of unlabeled H2O. Longsworth’s limiting interdiffusion coefficients are sometimes employed directly by using the impurity proton signal for residual water in D2O samples.26,27 This has been done for the

A = A′exp[−γ 2 g 2δ 2DS(Δ − δ /3)]

(3)

where A′ is the signal height at zero gradient and γ is the magnetogyric ratio of the resonant nucleus. At long Δ, the effect of any convection present can lead to erroneously high DS, despite an apparently good fit to eq 3.34,35 For short Δ, Hayamizu and her colleagues have reported anomalously high DS in the particular case of ionic liquids, generally at low temperatures where the viscosity is high.2,36−38 The reason for this latter effect is, as yet, unknown. This work extends the range of calibrant DS by an order of magnitude, down to 12·10−12 m2·s−1, by using the viscous esters 2-ethylhexyl benzoate (EHB) and bis(2-ethylhexyl) phthalate (DEHP), and the hydrocarbon squalane (2,6,10,15,19,23hexamethyltetracosane), to provide reference materials for work on viscous systems and those with small self- or intradiffusion coefficients. In addition, results for SG and PGSE measurements for DEHP and squalane are compared, and the latter technique is used to investigate whether there are any anomalies at high viscosity and low temperature with molecular liquids as has been observed for ionic liquids. As toluene has a very large liquid range we have repeated and extended our earlier self-diffusion measurements39 for this substance to encompass the range (−54 to 70) °C so that it may be a reference material for low temperature studies. We have also attempted to extend the range of temperatures available for some of the other molecular liquids suggested as secondary calibrants. In doing so, we have found discrepancies in the results of Holz et al.3 for some liquids studied by this group, which are inconsistent with both our results and some from the literature. New values are therefore recommended for cyclohexane, n-dodecane, dimethyl sulfoxide (DMSO), and pentan-1-ol, but we have, in the process, confirmed the measurements of Holz et al.3 for 1,2-dioxane. B



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Article

EXPERIMENTAL SECTION Steady gradient measurements were made as described previously40 using a computer-programmable Bruker Minispec PC100 NMR spectrometer, employing both fixed gradient and variable pulse spacing and fixed pulse spacing with variable gradient methods. Both positive and negative gradients were used with each method to eliminate systematic errors. Echo voltages were measured with an ADC card in the computer. The diode detector of the PC100 was calibrated frequently using a TTi TGR1040 rf signal generator. A quadrupole coil machined from Macor ceramic was used to produce the magnetic field gradient. The rf coil is wound on the same former. The gradient current was generated by a Phillips PM programmable power supply. The sample was contained in a flat-bottomed 5 mm NMR tube, centered within the coil: its length was less than half that of the quadrupole coil, about (35 to 40) % to ensure maximum field gradient homogeneity. Where convection might be a problem at higher temperatures (e.g., cyclohexane and toluene), Teflon inserts were placed in the NMR tube to hinder this. The sample and coil former were in a thermostated container whose temperature was controlled by Julabo model HC or FC50 circulators. The container was filled with 3 M FC75, a totally fluorinated substance, for 1H measurements and methylcyclohexane for 19F measurements. The temperature was determined from a 4-wire measurement of the resistance of a small Pt resistor within the container, which had been calibrated with a 100 Ω Leeds and Northrup Pt resistance thermometer, model 8930, in turn calibrated by the National Measurement Laboratory, Sydney, NSW. Resistances were read with Keithley 5 1/2 and Agilent 34401A 6 1/2 digit multimeters. The standard uncertainty of the temperature was 0.02 K. Diffusion coefficients were obtained by regressing the equation13,41 ⎛ 2 2τ ⎞ A = A″exp⎜ − γ 2DSτ 3[g 2 + 2g ·g 0] − ⎟ T2 ⎠ ⎝ 3

PGSE measurements were performed on a Bruker Avance 400 MHz spectrometer. The samples were contained within a right-cylindrical Wilmad NMR tube containing 500 μL of sample. The sample temperature was calibrated using ethylene glycol for high temperatures (above 27 °C) and methanol for low temperatures (below 27 °C).42−44 A recycle delay greater than five times the longest spin−lattice relaxation time (T1) value was used for all measurements. Integrals of appropriate resonances were used in the analysis of the NMR diffusion and relaxation data. The pulsed gradient stimulated echo (PGSE) NMR diffusion sequence45 was used to measure the diffusion coefficients over the temperature range (10 to 70) °C, except for DEHP between (40 and 70) °C, where the double stimulated echo sequence was employed.46 The experimental parameters are given in Table 1. The magnetic field gradient Table 1. PGSE Experimental Parameters Δ/ms δ/ms gmax/T m−1 repetition delay/s

squalane

DEHP

HDO in D2O

25 to 100 3 0.55 3

150 to 250 5 0.55 7

100 1 0.55 80

was calibrated using the residual water signal in a D2O sample at 25 °C, based on the limiting interdiffusion coefficients of Longsworth for HDO in D2O.25 The analysis of the NMR diffusion measurements has been described in detail elsewhere.47 The errors given are those that arise from the data fitting; the relative standard uncertainty of each PGSE measurement is usually 1 %,48 but for squalane and DEHP, the materials studied here, the relative expanded uncertainty is judged to be 3 % and 4 %, respectively, again based on the scatter shown by the temperature plots, discussed below. All measurements were made at atmospheric pressure. The materials used are described in Table 2.

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RESULTS The measured self-diffusion coefficients are listed in Tables 3 and 4. The upper temperature limit for the SG NMR measurements is normally 90 °C, but this was reduced for toluene and 1,4-dioxane by the onset of convection, despite the use of Teflon inserts to hinder this, and for cyclohexane by its low boiling point, 81 °C. For a 5 mm i.d. NMR tube and sample length of 35 mm, the “critical” viscosity is about 0.35 mPa·s. The absence of convection was gauged by fits to the Andrade−Arrhenius equation for strong liquids and the Vogel− Fulcher−Tammann (VFT) equation for the fragile liquids, taken in conjunction with linear Stokes−Einstein−Sutherland (SES) plots [ln(D/T) versus ln ϕ, where ϕ is the fluidity or reciprocal viscosity] as described below. The results for cyclohexane fit an Andrade−Arrhenius equation49,50 [(10 to 75) °C]:

(4)

onto the echo height data, g (= kI) is the applied field gradient, g0 is the background gradient, τ is the 90° to 180° pulse spacing, and T2 is the spin−spin relaxation time. I is the measured gradient current, and k is the gradient coil constant. A″, go, and DS are obtained by least-squares regression. A″ normally agreed with the zero gradient value (at constant τ) within 1 %; go was normally constant over a set of experiments; k was obtained by calibration with water over the temperature interval (1 to 90) °C using the values of Mills,1 Easteal et al.,29 and of Holz et al.3 and with benzene over the range (25 to 50) °C using values tabulated by Tyrrell and Harris13 and derived from the work of several groups. The reported DS are obtained by averaging positive and negative fixed gradient-variable τ experiments to remove the effect of the T2 term in eq 1 when this is small, and then taking the mean of this value and that obtained for fixed τ with g being varied between the positive and negative limits. The three values generally agreed within ± 1 %. Where T2 was short enough to produce significant echo attenuation, only variable gradient-fixed τ experiments were carried out. This was the case for the more viscous liquids, EHB, DEHP, squalane and dodecane. The overall relative standard uncertainty is estimated at 1.5 %, based on the temperature fits below, and the relative expanded uncertainty, including the calibration, at 2.5 %.

ln(DS /10−9m 2·s−1) = (6.0044 ± 0.036) − (1679.5 ± 11)/(T /K)

(5)

with a relative standard uncertainty for the fit, urD, of 1.0 %.51 The high precision diaphragm cell radio-tracer measurements of Kulkarni et al.52 (estimated urD = 0.2 %), Harris et al.53 (urD = 0.2 %), Mills54 (urD = 0.3 %) and of Freer and Sherwood55 (urD = 0.3 %), all at 25 °C, are included in the fit. Figure 1 shows a deviation plot, including other literature data, obtained C



Article

Table 2. Sample Descriptions CAS No

source

cyclohexane

110-82-7

cyclohexane 1,4-dioxane dimethyl sulfoxide n-dodecane DEHP EHB

110-82-7 123-91-1 67-68-5

Ajax Chemical Co Ltd., Auburn, NSW Australia Sigma-Aldrich, Australia Sigma-Aldrich, Australia Sigma-Aldrich, Australia

pentan-1-ol squalane toluene

71-40-0 111-01-3 108-88-3

112-40-3 117-81-7 5444-75-7

specification or grade

BDH, UK Sigma-Aldrich, Australia ABCR GmbH, Karlsruhe, Germany Sigma-Aldrich, Australia Sigma-Aldrich, Australia Burdick and Jackson, Muskegan, MI, USA

mole fraction purity

Univar

n/a

anhydrous anhydrous ReagentPlus grade, D5879, lot 018 K0757 lot 93082300 D201154, lot 09428JC

0.995 by gc > 0.998 by gc 0.995 by gc, < 0.001 H2O 0.99 by gc 0.997 by gc 0.99 by gc

Fluka 85629, lot 1140335 High purity solvent

0.99 by gc 0.990 by gc 0.998 by gc

purification method distilled, dried over Na wire, 99.9 % by gc dried over mol. sieves

dried over mol. sieves dried over mol. sieves dried over mol. sieves dried over mol. sieves dried over mol. sieves dried over Na wire

Table 3. Self-Diffusion Coefficients of the Less-Viscous Liquids at a Pressure of 0.1 MPaa cyclohexane T °C Sample A 35.01 35.02 45.07 45.10 54.98 55.00 Sample B 10.32 10.38 15.16 15.17 24.97 25.00 34.87 34.88 44.53 44.58 49.89 59.87 59.90 69.64 75.01

a

DMSO 9

10 D 2 −1

m ·s

1.712 1.718 2.053 2.048 2.469 2.465 1.072 1.085 1.191 1.199 1.457 1.470 1.728 1.758 2.030 2.078 2.234 2.598 2.646 2.968 3.251

T °C 19.94 25.11 25.14 34.95 44.88 54.95 64.97 75.02 90.56

n-dodecane

1,4-dioxane 9

10 D 2 −1

m ·s

0.6553 0.7456 0.7447 0.9100 1.116 1.335 1.586 1.838 2.292

9

T

10 D 2 −1

°C

m ·s

15.16 15.18 19.96 24.98 25.02 25.03 25.05 29.97 34.77 34.91 44.70 45.09 54.79 54.80 64.94 64.97 64.98 74.72 74.82 85.10 85.34

0.9100 0.8974 1.031 1.102 1.124 1.091 1.126 1.238 1.304 1.324 1.555 1.571 1.791 1.813 2.153 2.092 2.105 2.365 2.426 2.660 2.658

T °C −9.25 −9.01 −6.65 −5.12 5.05 5.37 5.41 5.57 14.94 15.34 15.37 25.00 25.05 25.06 35.00 35.01 45.00 45.01 50.02 55.02 55.07 55.10 59.99 64.78 64.82 64.84 74.93 79.60 79.62 91.38 91.39

toluene 9

10 D 2 −1

m ·s

0.3878 0.3867 0.4205 0.4334 0.5604 0.5782 0.5785 0.5731 0.7043 0.7060 0.7099 0.8688 0.8406 0.8443 1.034 1.035 1.224 1.228 1.326 1.426 1.439 1.465 1.538 1.668 1.665 1.674 1.917 2.068 2.040 2.436 2.410

T

109D

°C

m2·s−1

2008 expts −19.88 −19.87 −14.99 −14.92 −4.83 0.39 4.92 5.00 14.90 24.85 25.03 25.05 25.06 25.06 25.06 34.92 50.01 50.03 2014 expts −34.06 −32.21 −24.58 −21.44 −9.64 0.28 19.93 49.54 59.78 60.01 70.30

1.007 1.015 1.131 1.128 1.376 1.517 1.674 1.668 1.984 2.301 2.267 2.324 2.278 2.280 2.299 2.711 3.316 3.302 0.712 0.753 0.898 0.977 1.236 1.497 2.125 3.151 3.556 3.562 4.095

The standard uncertainty for T is uT = 0.02 K and the relative expanded uncertainty for D at the 95 % confidence level is UrD = 2.5 % (k ≈ 2).

%) agree at lower temperatures but are higher above 40 °C. The PGSE NMR data of Iwahashi and Kasahara59 (off-scale, not shown; urD not given) at (25 to 40) °C lie some (6 to 10) % too high, whereas those of Polzin and Weiss,60 (urD = 1 %), at (22 to 80) °C, lie below the correlation above 60 °C, and, finally, those of Holz et al.,3 (urD = 1 %) at (15 to 55) °C, also deviate systematically below the correlation as the temperature is increased, from (−2 to −3) %.

by both NMR and radio-tracer measurements. The diaphragm cell and gel-sectioning tracer results of data of Freer and Sherwood,55 (urD = 2 %) at (8.5 to 35) °C, the PGSE NMR result at 25 °C of Kato et al.20 and the PGSE NMR results of Yoshida et al.,56 (urD = 3 %) at (30 to 100) °C, are in good agreement with the correlation. The (high pressure) diaphragm cell results of McCool and Woolf,57 (urD = 1 %), at (15 to 55) °C, and the SG NMR results of Jonas et al.58 (estimated urD = 3 D



Article

Table 4. Self-Diffusion Coefficients of the More-Viscous Liquids at a Pressure of 0.1 MPaa EHB

a

DEHP

pentan-1-ol

squalane

T

1012D

T

1012D

T

1012D

T

1012D

°C

m2·s−1

°C

m2·s−1

°C

m2·s−1

°C

m2·s−1

−14.73 −14.71 −14.69 −8.94 −8.94 0.27 0.35 0.52 5.25 5.28 5.30 5.35 15.26 15.31 20.22 20.23 24.91 24.94 24.99 25.12 25.14 34.77 34.92 35.11 35.15 39.90 39.91 44.84 44.91 54.66 54.72 59.62 59.65 60.15 64.66 64.66 74.34 74.37 79.03 79.18 79.26

22.56 21.84 22.10 32.98 33.14 55.66 56.11 56.68 73.19 72.45 72.71 73.35 110.1 111.3 134.3 134.8 163.3 163.8 164.7 160.5 160.4 225.4 225.6 222.2 221.0 253.9 254.4 291.4 292.2 374.0 373.7 417.8 418.3 421.7 472.5 471.3 584.5 582.1 637.8 635.9 635.7

30.03 39.74 49.55 49.69 49.73 49.88 49.90 60.01 60.05 70.50 70.55 79.71 79.73 89.57 89.60

17.14 29.09 45.01 45.29 45.33 45.69 45.70 67.44 67.79 97.37 97.95 129.8 130.9 168.7 171.6

−20.02 −19.96 −14.73 −14.68 −9.90 −9.87 −4.50 −4.49 5.25 5.28 15.16 15.18 25.01 25.02 25.04 25.06 25.06 25.12 25.12 34.94 34.96 44.93 44.94 49.95 49.95 50.00 50.00 54.71 54.71 64.56 64.59 74.06 74.10 79.20 79.21 79.40 91.37 91.44 91.58

52.71 53.17 66.15 66.12 80.57 81.12 101.3 101.9 150.9 152.2 216.6 216.7 300.7 300.2 298.8 302.1 302.1 302.1 302.1 412.9 413.6 554.2 556.4 638.5 638.5 642.5 642.5 730.4 735.9 956.3 949.4 1218 1218 1399 1396 1413 1844 1825 1835

10.15 15.13 19.98 20.04 25.04 25.29 29.85 30.03 39.84 49.54 49.57 50.01 59.88 60.34 69.66 75.05 79.67 80.00 89.45 89.67 89.73

13.45 18.11 21.95 22.62 28.75 29.14 35.67 36.85 57.00 81.48 82.35 81.23 117.1 114.3 150.4 173.6 195.9 198.2 244.4 249.9 260.2

The standard uncertainty for T is uT = 0.02 K and the relative expanded uncertainty for D at the 95 % confidence level is UrD = 2.5 % (k ≈ 2).

(−2 to −3) % above 25 °C. The result of Mischler et al.61 at 25 °C (estimated urD = 2−3 %) lies somewhat higher. The results for 1,4-dioxane [(15 to 85) °C] were fitted to eq 7 with a relative standard uncertainty for the fit, urD, of 1.4 %. The plot is only slightly non-Andrade−Arrhenius, but the extra term was included as the viscosity is definitely non-Andrade− Arrhenius in the range (12 to 87)°C.62,63

The results for DMSO were fitted to an extended Arrhenius equation with a relative standard uncertainty for the fit, urD, of 0.6 %. ln(DS /10−9m 2·s−1) = (4.223 ± 0.44) − (710.2 ± 29)/(T /K) − (0.1904 ± 0.046)106 /(T /K)2

(6)

ln(DS /10−9m 2·s−1)

Figure 2 is the corresponding deviation plot. There is good agreement with the PGSE NMR datum of Connell et al.33 (urD = 1 %) at 25 °C, and those of Holz et al.17 at (15 to 35) °C (urD = 1 %), but again the later data of Holz et al.3 (urD = 1 %) at (25 to 55) °C deviate systematically below the correlation by

= (3.143 ± 0.81) − (1141 ± 516)/(T /K) − (236.1 ± 82)103/(T /K)2

(7)

Figure 3 is the corresponding deviation plot. In this particular case, there is very good agreement with the PGSE results of E



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PGSE data of Vogel and Weiss63 (smoothed AndradeArrhenius fit, (urD = 1 %) at (22 to 67) °C lie some (12 to 25) % and (6 to 23) % higher, respectively. Both these sets show fractional exponents greater than unity (1.09) in Stokes− Einstein−Sutherland plots against the fluidity, which is not expected for non-hydrogen-bonded liquids: this suggests these sets should be disregarded. The results for dodecane [(−9 to 91) °C] were also fitted to an extended Arrhenius equation with a relative standard uncertainty for the fit, urD, of 1.3 %. The need for the extra term is consistent with what has been found for the viscosity over a similar temperature range, dodecane being a slightly fragile liquid.66

Figure 1. Cyclohexane: deviation plot for the fit to the Andrade− Arrhenius equation, eq 5. Symbols: ●, this work, SG NMR, urD(fit) = 1.0 % (dashed lines); ◇, Kulkarni et al.52 and Harris et al.,53 14C tracer, diaphragm cell, urD = 0.2 %, at 25 °C; ○, Mills,54 14C tracer, diaphragm cell, urD = 0.2 %, at 25 °C; □, McCool and Woolf,57 14C tracer, (high pressure) diaphragm cell, urD = 2 %; ▽, Freer and Sherwood,55 14C and 3H tracers, diaphragm cell, urD = 0.5 %; △, Freer and Sherwood,55 14C and 3H tracers, gel sectioning, urD = 0.5 %; ■, Yoshida et al.,56 PGSE NMR, urD = 2 %; ▲, Jonas et al.,58 SG NMR, urD = 3 % (estimated); ▼, Polzin and Weiss,60 PGSE NMR, urD = 2 %; ◆, Holz et al.,3 PGSE NMR, urD = 1 %; ∗, Kato et al.,20 PGSE NMR, urD = 0.4 %.

ln(DS /10−9m 2·s−1) = (4.5102 ± 0.027) − (1040 ± 166)/(T /K) − (105.0 ± 25)103/(T /K)2

(8)

Figure 4 is the corresponding deviation plot. The PGSE NMR results of Brusewitz and Weiss,67 (urD = 1 %) at (5 to 55) °C,

Figure 4. Dodecane: deviation plot for the fit to eq 8. Symbols: ●, this work, SG NMR, urD(fit) = 1.3 % (dashed lines); ▼, Brüsewitz and Weiss,67 PGSE NMR, urD = 2 %; ◆, Holz et al.,3 PGSE NMR, urD = 1 %.

Figure 2. DMSO: deviation plot for the fit to eq 6. Symbols: ● this work, SG NMR, urD(fit) = 0.6 % (dashed lines); ◇, Holz et al.,17 PGSE NMR, urD = 1 %.◆, Holz et al.,3 PGSE NMR, urD = 1 %, ■, Mischler et al.,61 PGSE NMR, estimated urD = 2−3 %, ▲, Connell et al.,33 PGSE NMR, urD = 1 %.

are in excellent agreement, except for the high point at 4.4 °C. Again, the PGSE data of Holz et al.,3 (urD = 1 %) at (5 to 55) °C, lie below the correlation by approximately (−2.5 to −7) %. There are two other NMR studies,68,69 but regrettably neither includes numerical data or regression fits for direct comparison. The results for pentan-1-ol [(−20 to 91) °C] were fitted to the Andrade−Arrhenius eq 9 with a relative standard uncertainty for the fit, urD, of 0.9 %. ln(DS /10−12m 2·s−1) = (15.5821 ± 0.013) − (2943.72 ± 4.0)/(T /K)

(9)

Figure 5 is the corresponding deviation plot. Again, the data of Holz et al.3 (urD = 1 %) at (5 to 55) °C lie below the correlation by amounts similar to those for dodecane. The single point of Iwahashi et al.70 (urD not given) at 25 °C is in fair agreement with our correlation. The results for toluene [(−54 to 70) °C], including previous SG NMR and 14C tracer results from the Canberra laboratory,39 were fitted to eq 10 with a relative standard uncertainty for the fit, urD, of 1.6 %.

Figure 3. 1,4-Dioxane: deviation plot for the fit to eq 7. Symbols: ●, this work, SG NMR, urD(fit) = 1.4 % (dashed lines); ◇, Holz et al.,17 PGSE NMR, urD = 1 %; ◆, Holz et al.,3 PGSE NMR, urD = 1 %; ▲, Connell et al.,33 PGSE NMR, urD = 1 %.

Holz et al.,3 (urD = 1 %) at (15 to 55) °C, the earlier data of Holz et al.17 (urD = 1 %) at (15 to 35) °C and the single point of Connell et al.33 (urD = 1 %) at 25 °C. The SG data of Fratiello and Douglass64 (urD = 5 %65) at (25 to 59) °C, and the F



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Figure 7. EHB: deviation plot for the fit to VFT equation, eq 11 (dashed lines). Symbols: ●, this work, SG NMR, urD = 1.2 %; ▲, Walker et al.,74 NMR, urD = 10 %, (T1ρ method, ≤ 40 °C), 3 % (SG, ≥ 60 °C).

Figure 5. Pentan-1-ol: deviation plot for the fit to the Andrade− Arrhenius equation, eq 9. Symbols: ●, this work, SG NMR, urD(fit) = 0.9 % (dashed lines); ◆, Holz et al.,3 PGSE NMR, urD = 1 %, ▲, Iwahashi et al.,70 PGSE NMR, urD not given.

ln(DS /10−9m 2·s−1) = (4.577 ± 0.17) − (901.6 ± 97)/(T /K) 3

2

− (64.4 ± 13) ·10 /(T /K)

method (urD = 10 %) was used because of the very high viscosity and short T2 relaxation times. The combined SG and PGSE results for DEHP [(−5 to 90) °C] were fitted to the VFT eq 12 with a relative standard uncertainty for the fit, urD, of 2.2 %:

(10)

(A VFT fit gives very similar residuals, but an unrealistically low value of the parameter, T0, 40.6 K.) Figure 6 is the

ln(DS /10−12m 2·s−1) = (10.234 ± 0.13) − (983.5 ± 36) /[T /K − (169.52 ± 2.4)]

Figure 6. Toluene: deviation plot for the fit to eq 10, urD(fit) = 2.2 % (dashed lines). Symbols: ■, 2008, ◆, 2014 this work, SG NMR; ●, Harris et al.,39 SG NMR, urD = 2 %; obscured, 25 °C, Harris et al.,39 14 C tracer, (high pressure) diaphragm cell, urD = 2 %; △, Pickup and Blum,71 PGSE NMR, urD = 10 %; ○, O’Reilly and Petersen,72 urD not given; □, Yemloul et al.,73 C7D8, mass effect normalized, (as in ref 16), urD = 3 %.

Figure 8. DEHP: deviation plot for the fit to VFT equation, eq 12. Symbols: ●, this work, SG NMR, urD(fit) = 1.2 %; ■, this work, PGSE, urD(fit) = 3.9 %; overall urD(fit) = 2.2 % (dashed lines); ▲, Hayamizu,75 PGSE NMR, urD eq 12 = 2.6 %.

corresponding deviation plot. The PGSE NMR results of Pickup and Blum71 (urD = 10 %) at (25 to 115) °C are displaced higher than those of this work. The points of O’Reilly and Petersen72 and of Yemloul et al.73 are also shown. The results for EHB [(−15 to 80) °C] were fitted to the VFT equation, eq 11, with a relative standard uncertainty for the fit, urD, of 1.4 %.

Figure 8 is the corresponding deviation plot. The PGSE data of Hayamizu75 for the same sample generally lie somewhat lower than our results. Her data can be fitted to

ln(DS /10−12m 2·s−1)

ln(DS /10−12m 2·s−1)

= (9.8256 ± 0.056) − (641.0 ± 15) /[T /K − (163.00 ± 1.5)]

(12)

= (11.404 ± 0.36) − (1350.9 ± 116) (11)

/[T /K − (146.57 ± 6.8)]

Figure 7 is the corresponding deviation plot. The SG NMR data of Walker et al.74 (urD = 3 %) show good agreement above 60 °C but poorer agreement below this temperature where the less precise radiofrequency (i.e., B1) field dependence of the rotating frame proton spin−lattice relaxation time, or T1ρ,

(13)

with a relative standard uncertainty for the fit, urD, of 2.6 %. The combined SG and PGSE results for squalane [(−3 to 90) °C] were fitted to the VFT eq 14 with a relative standard uncertainty for the fit, urD, of 2.1 %. G



Article

ln(DS /10−12m 2·s−1)

Table 5. Self-Diffusion Measurements for Squalane and DEHP at a Pressure of 0.1 MPa, Measured by PGSEa

= (10.303 ± 0.014) − (992.2 ± 43) /[T /K − (155.09 ± 3.3)]

1012D/m2s−1

(14)

T/°C

Δ/ms = 25

Δ/ms = 50

Δ/ms = 75

Δ/ms = 100

average

Squalane −2.75 −0.15 10.45 24.85 41.15 52.55 61.65 71.75 T/°C −5.15 −0.15 9.85 24.95 30.05 40.05 49.95 59.55 69.35

Figure 9. Squalane: deviation plot for the fit to VFT equation, eq 14. Symbols: ●, this work, SG NMR, urD(fit) = 2.2 %; ■, this work, PGSE, urD(fit) = 2.4 %; overall urD(fit) = 2.1 % (dashed lines); ▲, Hayamizu,76 PGSE NMR, urD eq 14 = 2.2 %; ◆, Mondello and Guest,77 PGSE NMR, urD(expt) = 10 %.

6.8 5.5 6.71 6.73 13.5 13.1 27.8 27.9 57.0 57.0 85.0 87.0 114 118 149 154 Δ/ms = 150 Δ/ms = 1.31 2.06 4.59 12.4 18.6 30.4 47.3 69.3 94.8

5.3 5.2 6.74 6.71 13.1 13.1 27.8 28.0 57.0 57.0 89.0 92.0 123 130 160 167 200 Δ/ms = 250

DEHP 1.32 2.05 4.53 12.4 18.6 30.5 47.3 69.4 94.8

1.31 2.05 4.60 12.4 18.6 30.5 47.3 69.3 94.9

5.32 6.72 13.2 27.9 57.0 88.3 118 160 average 1.31 2.05 4.57 12.4 18.6 30.5 47.3 69.3 94.8

a The standard uncertainty for T is uT = 0.05 K and the relative expanded uncertainty for D at the 95 % confidence level is (k ≈ 2) UrD = 3 % for squalane and UrD = 4 % for DEHP. Values in italics were excluded from the analysis as being outliers.

Figure 9 is the corresponding deviation plot. The PGSE data of Hayamizu76 for the same sample also lie somewhat lower than our results. Her data best fit the VFT equation ln(DS /10−12m 2·s−1) = (11.555 ± 0.026) − (1436.6 ± 92) /[T /K − (124.43 ± 5.7)]

(15)

with a relative standard uncertainty for the fit, urD, of 2.2 %. The PGSE results of Mondello and Grest,77 urD = 10 %, deviate from eq 14 by (+17 to −18) %.

■

DISCUSSION Viscous Liquids. The agreement between the SG and PGSE results for both DEHP and squalane is satisfactory given the difficult experimental conditions for low Ds measurements and is generally within the combined experimental uncertainties. It should be noted that the PGSE measurement requires Fourier transformation of the echo and the choice of a suitable resonance line in the spectrum (frequency domain) so there is more manipulation of the measured signal than with the SG technique, which is a simpler, time-domain experiment. Table 5 shows that the PGSE results appear to be independent of the gradient pulse interval, Δ, though there is one high value at the shortest Δ for squalane at −3 °C and some variation at (62 and 72) °C. The unpublished results of Hayamizu75,76 support this observation. Figure 10 shows linear Stokes−Einstein−Sutherland plots of the self-diffusion coefficients for the more viscous liquids, pentan-1-ol (viscosities), EHB, DEHP, and squalane, against their respective fluidities (ϕ, reciprocal viscosities) (viscosity data: pentan-1-ol,78,79 EHB, DEHP and squalane80). As is typical of liquids, the slopes are fractional.81 The fitted coefficients to

Figure 10. Stokes−Einstein−Sutherland plots of the self-diffusion coefficient versus fluidity for the viscous liquids pentan-1-ol, EHB, DEHP, and squalane. Symbols: green ▲, pentan-1-ol; black ●, EHB; blue ■, DEHP; red ◆, squalane.

ln[(1012DS /T )/m 2· s−1· K−1] = a + t ln[ϕ/(mPa ·s)−1] (16)

are given in Table 6. Figure 11 shows deviations from the fits for DEHP and squalane, including the data of Hayamizu,75,76 which show similar deviations as a function of temperature as in Figures 8 and 9. There appears to be a consistent difference between her results and ours between (5 and 40) °C. The SES viscosity correlation is very useful for comparing different data sets, independent of any fitting function assumed for the temperature dependence of self-diffusion data. It should also be useful for checking the extrapolation of eq 11, 12 and 14 to higher or lower temperatures where viscosity data are available. Unfortunately, the European Chemical Agency listed DEHP as a “Substance of Very High Concern (SVHC)” in October 2008, classifying it as toxic to (human) reproduction.82 H



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Table 6. Coefficients for Fits to the Stokes−Einstein−Sutherland Equation, eq 16 substance

a

t

u/ %

EHB DEHP squalane cyclohexaneb DMSOb 1,4-dioxaneb dodecaneb tolueneb 1-pentanolb

1.025 ± 0.005 0.779 ± 0.012 0.936 ± 0.011 −5.429 ± 0.003 −5.813 ± 0.003 −5.443 ± 0.004 −5.568 ± 0.002 −5.453 ± 0.004 −5.591 ± 0.002

0.9411 ± 0.0025 0.9580 ± 0.0038 0.9776 ± 0.0038 0.9244 ± 0.0077 1.067 ± 0.0072 0.860 ± 0.010 0.9047 ± 0.0045 0.9765 ± 0.0063 1.062 ± 0.0014

1.5 2.2 2.2 1.0 0.4 1.5 1.3 1.8 0.8

t (ref 64)a

0.93

0.88 to 0.91c 0.93

viscosity data ref 80 80 80 83 to 85 86 to 88 63 83, 89, 90 92 78, 79

a

Based on high-pressure self-diffusion and viscosity data in addition to atmospheric pressure results. u is the relative standard uncertainty for the least-squares regression. bFitted to 10−9D/m2 s−1 not 10−12D/m2 s−1 cHexane, 0.91; octane, 0.89; hexadecane, 0.88.

dodecane,83,89,90 and toluene91). The coefficients for eq 15 are again given in Table 6. The deviations of our data from those of Holz et al.3 are puzzling given the good agreement for 1,4dioxane and the common use of data for water for primary calibration. Nevertheless we think our data are to be preferred given the good agreement with radio-tracer diaphragm-cell data for both cyclohexane and toluene and the excellent agreement of values of t for cyclohexane, dodecane, and toluene with those obtained previously from high-pressure self-diffusion and viscosity data.81 Holz et al.3 employed a step-series of internal calibrations where results for the more viscous liquids were adjusted against those with lower viscosities in the sequence water; cyclohexane, 1−4-dioxane; dodecane, dimethyl sulfoxide; 1pentanol, tetradecane, with the underlined substances being used as reference materials for the next pair in the sequence. It is possible that a systematic error was introduced at some stage in this rather awkward procedure. Finally we recommend toluene as a low-viscosity calibrant for low-temperature work. This has a much lower freezing point than the substances studied by Holz et al.3 and does not have the long T1 relaxation time of 1,4-dioxane that can make measurements rather lengthy, though there are techniques that can obviate this.92

Figure 11. Deviations from Stokes−Einstein−Sutherland plots of the self-diffusion coefficient versus fluidity for DEHP, and squalane. Symbols: DEHP, ●, SG, ■, PGSE, this work; ▲, Hayamizu,75 PGSE; squalane, ○, SG, □, PGSE, this work; △, Hayamizu,76 PGSE. The dashed lines show the relative standard uncertainty of the SES fit to the combined SG and PGSE results of this work for DEHP, urD = 2.7 %: that for squalane is 2.2 %. SES plots of Hayamizu’s data appear to be nonlinear, becoming concave at lower temperatures.

Consequently this substance may not meet OH&S criteria in some countries or organizations for use as a reference or calibration material. However, the data for squalane cover the same range of viscosities, so this substance is a suitable alternative. The data for EHB are for slightly lower viscosities, but extend to lower temperatures than for squalane. Less-Viscous Liquids. Figure 12 shows Stokes−Einstein− Sutherland plots for the less-viscous liquids studied here (viscosity data: cyclohexane,83−85 DMSO,86−88 1,4-dioxane,63

■

CONCLUSIONS Self-diffusion measurements have been made for the viscous fluids ethylhexylbenzoate, bis(ethylhexyl)phthalate, and squalane using the steady-gradient spin−echo NMR technique supplemented for the latter two substances by pulsed-gradient measurements. Within the range of experimental conditions employed, the PGSE results are independent of the pulse gradient separation time, Δ. Squalane and ethylhexyl benzoate may be suitable secondary reference materials for the calibration of spin−echo NMR apparatus when low values of self-diffusion coefficients are to be measured, for example, for highly viscous liquids. The Stokes−Einstein−Sutherland relation is used to check the consistency of the self-diffusion coefficients with the best available viscosities from the literature. We also report systematic errors found in the secondary calibration data of Holz et al.3 for cyclohexane, n-dodecane, dimethyl sulfoxide, and pentan-1-ol and suggest toluene in their place as a more convenient low-viscosity calibrant that is also suitable for low temperature work.

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Figure 12. Stokes−Einstein−Sutherland plots of the self-diffusion coefficient versus fluidity for the nonviscous liquids cyclohexane, DMSO, dodecane, 1,4-dioxane, and toluene. Symbols: green ●, cyclohexane; black ■, DMSO; green ▲, dodecane; black ○, 1,4dioxane; blue ▼, toluene.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. I



Article

Notes

(16) Mills, R.; Harris, K. R. The Effect of Isotopic Substitution on Diffusion in Liquids. Chem. Soc. Rev. 1976, 5, 215−231. (17) Holz, M.; Mao, X.; Seiferling, D.; Sacco, A. Experimental study of dynamic isotope effects in molecular liquids: detection of translation-rotation coupling. J. Chem. Phys. 1996, 104, 669−679. (18) Aoyagi, K.; Albright, J. G. The Mutual Diffusion Study in the Systems Containing Benzene or Benzene- d6 at 25°. J. Chem. Phys. 1976, 64, 81−83. (19) Shankland, I. R.; Arora, P. S.; Dunlop, P. J. Isotope Effect in Diffusion of Perdeuteriobenzene and Benzene in a Series of Normal Hydrocarbons at 25°C. J. Phys. Chem. 1977, 81, 1518−1519. (20) Kato, H.; Saito, T.; Nabeshima, M.; Shimada, K.; Kinugasa, S. Assessment of Diffusion Coefficients of General Solvents by PFGNMR: Investigation of the Sources Error. J. Magn. Reson. 2006, 180, 266−273. (21) Murday, J. S. Measurement of Magnetic Field Gradient by Its Effect on the NMR Free Induction Decay. J. Magn. Reson. 1973, 10, 111−120. (22) Lamb, D. M.; Grandinetti, P. J.; Jonas, J. Fixed Field Gradient NMR Diffusion Measurements Using Bessel Function Fits to the Spinecho Signal. J. Magn. Reson. 1987, 72, 532−539. (23) Peereboom, P. W. E.; Luigjes, H.; Prins, K. O.; Trappeniers, N. J. NMR Spin-echo Study of Self-diffusion in Xenon and Ethane. Physica B+C 1986, 139−140, 134−136. (24) Yadav, N. N.; Torres, A. M.; Price, W. S. An Improved Approach to Calibrating High Magnetic Field Gradients for Pulsed Field Gradient Experiments. J. Magn. Reson. 2008, 194, 25−28. (25) Longsworth, L. G. The Mutual Diffusion of Light and Heavy Water. J. Phys. Chem. 1960, 64, 1914−1917. (26) Stilbs, P. Fourier Transform Pulsed-gradient Spin-echo Studies of Molecular Diffusion. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1−45. (27) The intra-diffusion coefficient of HDO in H2O-D2O mixtures has sometimes been confused with Longsworth’s measured interdiffusion coefficients. See, for example, Norwood, T. J. New NMR Methods for Measuring Diffusion. J. Magn. Reson., Ser. A 1993, 103, 258−267. (28) Price, W. S.; Stilbs, P.; Jönsson, B.; Södermann, O. Macroscopic Background Gradient and Radiation Damping Effects on High-field PGSE NMR Diffusion Measurements. J. Magn. Reson. 2001, 150, 49− 56. (29) Easteal, A. J.; Price, W. E.; Woolf, L. A. Diaphragm Cell for High-temperature Diffusion Measurements. Tracer Diffusion Coefficients for Water to 363 K. J. Chem. Soc., Faraday Trans. 1 1989, 85, 1091−1097. (30) Harris, K. R. Communication: The Fractional Stokes-Einstein Equation: Application to Water. J. Chem. Phys. 2010, 132, 231103. (31) Holz, M.; Weingärtner, H. Calibration in Accurate Spin-echo Self-diffusion Measurements Using 1H and Less-Common Nuclei. J. Magn. Reson. 1991, 92, 115−125. (32) Tofts, P. S.; Lloyd, D.; Clark, C. A.; Barker, G. J.; Parker, G. J. M.; McConville, P.; Baldock, C.; Pope, J. M. Test Liquids for Quantitative MRI Measurements of Self-diffusion Coefficient in Vivo. Magn. Reson. Med. 2000, 43, 368−374. (33) Connell, M. A.; Bowyer, P. J.; Adam Bone, P.; Davis, A. L.; Swanson, A. G.; Nilsson, M.; Morris, G. A. Improving the Accuracy of Pulsed Field Gradient NMR Diffusion Experiments: Correction for Gradient Non-uniformity. J. Magn. Reson. 2009, 198, 121−131. (34) Hayamizu, K.; Price, W. S. A New Type of Sample Tube for Reducing Convection Effects in PGSE-NMR Measurements of Selfdiffusion Coefficients of Liquid Samples. J. Magn. Reson. 2004, 167, 328−333. (35) Hedin, N.; Yu, T. Y.; Furó, I. Growth of C12E8 Micelles with Increasing Temperature. A Convection-Compensated PGSE NMR Study. Langmuir 2000, 16, 7548−7550. (36) Hayamizu, K.; Tsuzuki, S.; Seki, S.; Fujii, K.; Suenaga, M.; Umebayashi, Y. Studies on the Translational and Rotational Motions of Ionic Liquids Composed of N-methyl-N-propyl-pyrrolidinium (P13) cation and Bis(trifluoromethanesulfonyl)amide and Bis-

The authors declare no competing financial interests.

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ACKNOWLEDGMENTS The authors are grateful to Dr Kikuko Hayamizu, University of Tsukuba, Japan, (formerly at AIST Tsukuba), for graciously sharing both her data and her considerable experience in NMR spin−echo measurements. It is a pleasure to thank the late Dr Peter Dunlop, University of Adelaide, South Australia, for a sample of high purity cyclohexane and Dr Mitsuhiro Kanakubo, AIST, Sendai, Japan, for helpful discussions during this project.

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