Determination of Molecular Self-Diffusion Coefficients Using Pulsed

Laboratory Experiment pubs.acs.org/jchemeduc

Determination of Molecular Self-Diffusion Coefficients Using PulsedField-Gradient NMR: An Experiment for Undergraduate Physical Chemistry Laboratory Jennifer Harmon,†,# Cierra Coffman,†,# Spring Villarrial,†,# Steven Chabolla,†,# Kurt A. Heisel,† and Viswanathan V. Krishnan*,†,‡ †

Department of Chemistry, California State University, Fresno California 93740, United States Department of Pathology and Laboratory Medicine, School of Medicine, University of California, Davis California 95616, United States

‡

S Supporting Information *

ABSTRACT: NMR spectroscopy has become one of the primary tools that chemists utilize to characterize a range of chemical species in the solution phase, from small organic molecules to medium-sized proteins. A discussion of NMR spectroscopy is an essential component of physical and biophysical chemistry lecture courses, and a number of instructional laboratory exercises have been described. The latter includes experiments to understand restricted rotations, measure relaxation times, and run two-dimensional NMR experiments. This note describes how NMR spectroscopy can be used to measure the translational diffusion coefficients using pulsed-fieldgradients (PFG). Though the principle of the diffusion coefficient measurements is based on one of the earliest pulse-sequences proposed, the advent of standard availability of PFG in commercial NMR spectrometers has made the implementation of this experiment straightforward. In addition to learning the basic operation of an NMR spectrometer, the specific goals of the experiment may include understanding the effect of temperature, solvent viscosity, and concentration on molecular motions as well as the analysis of a mixture. Complete details on how to implement the experiment and perform data analysis are provided in the Supporting Information. KEYWORDS: Upper-Division Undergraduate, Laboratory Instruction, Physical Chemistry, Hands-On Learning/Manipulatives, Biophysical Chemistry, NMR Spectroscopy, Noncovalent Interactions, Solutions/Solvents, Transport Properties

H

shielded gradient coils in commercial spectrometers.7,8 A PFG along the z axis is standard in all the major commercial spectrometers as they are often utilized for optimizing the z shims using automated shimming routines. In this way, classical experimental methods using PFG9 have been applied to measure the diffusion coefficients. This technique can be utilized in the analysis of small molecules and polymers in solution10 as well as proteins and DNA oligomers to understand the biomolecular association processes.11−14 PFG-based diffusion coefficient measurements can be incorporated into the undergraduate physical chemistry laboratory curriculum to enable the students to learn about molecular diffusion properties and NMR spectroscopy simultaneously. Maki and Loening described in detail how NMR-based diffusion experiments can be adopted in undergraduate chemistry curriculum.15 A concise description of Maki and Loening’s approach is provided that can be easily adopted in an undergraduate physical chemistry laboratory. The learning objectives of this experiment in an undergraduate physical or biophysical chemistry laboratory are to (a) understand the concept of Brownian motion in solution by measuring the translational diffusion coefficient of solutes; (b) explore the nature of the Stokes−Einstein equation on how molecular

igh-resolution solution-state nuclear magnetic resonance (NMR) has become one of the standard spectroscopic techniques in undergraduate chemistry curricula. In addition to students using NMR spectroscopy in organic chemistry laboratory courses, this technique has recently taken a central role in laboratory experiments for physical and biophysical chemistry courses.1,2 Understanding the process of molecular diffusion is critical in physical chemistry because, in all molecular chemical processes, a formal upper limit on the rates of chemical and biochemical reactions is set by the diffusion process. In spite of its importance, there are a limited number of experimental methods for measuring diffusion coefficients in solution. One of the first physical chemistry experiments concerning diffusion reported as a laboratory experiment was by Linder et al.,3 which was based on the free boundary method devised earlier by Polson.4 Fate and Lynn5 proposed a method using UV−vis spectroscopy and Williams et al. introduced a method based on capillary electrophoresis.6 A pulsed field gradient (PFG) in an NMR experiment is a short, timed pulse with spatial-dependent magnetic field intensity. PFGs can be applied along the x, y, or z axis. PFGs play an integral role in magnetic resonance imaging (MRI) and spatial selection in magnetic resonance spectroscopy (MRS). The use of pulsed magnetic field gradients in high-resolution NMR is a major advancement due to the availability of self© 2012 American Chemical Society and Division of Chemical Education, Inc.

Published: March 22, 2012 780

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diffusion is related to other physical parameters such as solvent viscosity, sample temperature, and solvation radius of the molecule; (c) learn the basics of optimizing a NMR experiment, the effect of pulsed-field gradients, and observe how a modified spin−echo pulse sequence is used to measure the diffusion coefficient; and (d) combine the above-mentioned key aspects to estimate the molecular radius of a set of molecules dissolved in the same sample. As the number of laboratory exercises pertaining to mass transfer in undergraduate chemistry curriculum is limited,16 this experiment introduces the concept using modern technology, that is, NMR spectroscopy.

(S vs q2Δ′) or a linear fit (−ln S vs q2Δ′) can be used to estimate D.

THEORETICAL BACKGROUND Several excellent review articles provide the theoretical basis needed for an understanding of PFG-based diffusion coefficient measurements.15,17 Only an overview is provided here. For a spherical object of radius R moving through a liquid with viscosity η, at a velocity v, the Stokes’ law predicts that

Samples are prepared in standard 5 mm NMR tubes. For each sample, caffeine (11 mg) and 2-ethoxyethanol (50 μL) were dissolved in 600 μL of D2O (∼99%). A one-dimensional NMR experiment was first performed at room temperature to identify the distinct resonances for each compound (caffeine, δ = 7.59 ppm; 2-ethoxyethanol, δ = 0.09 ppm). In addition, the residual H2O peak (δ ∼ 4.7 ppm) was also used.

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EXPERIMENTAL PROCEDURES All NMR experiments and analyses were performed by the students (first four authors) for their final project assignments in a physical chemistry laboratory. Caffeine and 2-ethoxyethanol samples were purchased from Sigma-Aldrich and were used as received. Chemical shifts are expressed in δ with units of ppm.

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F = −6πηRv

Samples

(1)

Diffusion Measurements

Using the pulse sequence reported by Morris and co-workers,20 self-diffusion coefficients were measured on a Varian VNMRS (νH = 400 MHz) NMR spectrometer, equipped with a doubleresonance pulsed-field-gradient probe with an actively shielded z gradient coil and a gradient amplifier. A detailed description of the pulse sequence and implementation for Varian and Bruker instruments is available from the authors of the original work.21 The experimental parameters for the pulse sequence are Δ (diffusion delay) = 0.2 s, δ (pulse length of the z gradient) = 2 ms, and the constants α = 0.2 (imbalance factor) and τ = 1 ms (time between the midpoints of the individual gradient pulses in one diffusion-encoding period).20 These parameters provided an actual diffusion delay (Δ′) of 0.19833 s (eq 3). The gradient strength was calibrated using the known diffusion constant of water, 2.30 × 10−9 m2 s−1 at 298 K.22 Diffusion data at each temperature was collected from a gradient strength of approximately 1 to 20 G cm−1 as defined by the sequence. NMR experiments were done at five temperatures between 15 and 35 °C. The chemical shift difference between the methyl and hydroxyl groups in methanol was used for temperature calibration.23 Pulse widths of 90° were determined at each of the temperatures. The cooling rate between the temperatures was approximately 1 °C/min and more than 10 min was allowed for temperature stabilization between the experiments. The acquisition time for each transient was 0.328 s over a spectral width of 8,000 Hz. Each FID (free induction decay) was signal-averaged over 16 scans with a recycling delay of 10 s. Each experiment was repeated three times to obtain the experimental standard deviations.

where F is the frictional force. Using the relationship between the kinetic energy of the molecule and the frictional force, the Stokes−Einstein equation is written as

D=

kBT 6πηR

(2)

where kB is the Bolzmann constant (1.3806503 × 10−23 m2 kg s−2 K−1), T is the temperature (K), and D (m2 s−1) is the diffusion coefficient. During the measurement of diffusion, molecules undergo translational motion between two gradient pulses: the first pulse is to dephase the magnetization and the second pulse is to refocus it at some later point in the experiment (by a delay designated as Δ). If there is no molecular motion between the two gradient pulses, the signal intensity will be the same, except for the loss of signal due to relaxation events such as T1 and T2 processes (S ≈ S0, where S and S0 are the NMR signal intensity with and without the diffusion, respectively). As the molecules in the solution undergo thermal motion, the observed signal will be reduced (S < S0) because the refocusing gradient will not properly restore the transverse magnetization to its initial phase. Similar to any other random motion, a Gaussian distribution function can be used to describe the distribution of molecular displacements due to translational diffusion.18 Therefore, in an NMR experiment with dephasing and refocusing gradients, there will be a Gaussian distribution of phases due to translational diffusion between these two gradients pulses. Using modified Bloch equations to account for the diffusion process, Stejskal−Tanner derived the expression9

Data Analysis

2

S = S0 e−Dq Δ′

NMR data were processed using the VnmrJ software. Areas under the curves from three distinct resonances, one from each caffeine, 2-ethoxyethanol, and residual water, were measured. The natural log of the integrations of these peaks were recorded and averaged for each temperature. The averages were plotted against values of the inverse of the diffusion coefficient (in units of s/m2). A linear fit (−ln S vs q2Δ′) using standard worksheet routines such as Microsoft Excel were utilized to produce reliable results. However, it has been noted that a nonlinear fit yields more accurate results.24 For this application, both the methods provide comparable results.

(3)

where S and S0 are as defined before, q is the area of the gradient pulse (γgzδ), γ is the magnetogyric ratio of hydrogen atoms (2.676 × 108 rad−1 s−1 T−1), gz is the gradient amplitude (normally along the z axis), δ is the width of the gradient pulse, and Δ′ is the diffusion time corrected for the effects of finite gradient pulse widths and other pulse sequence dependent delays.19 To modify the above equations for alternative schemes, Δ′ needs to be defined for each experiment, whereas all other parameters remain the same. Either an exponential fit 781

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HAZARDS 2-Ethoxyethanol is toxic and may be harmful if swallowed, inhaled, or absorbed through the skin.

Using the viscosity (η) of D2O, derived using the empirical relation log η = 757.9[T−1 − (1/304.6)] and the Stokes− Einstein equation (eq 2), it is possible to estimate the Stokes’ molecular radius of the diffusing sphere. Average values of the Stokes’ radius based on the D values measured at different temperatures were caffeine (3.2 ± 0.9 Å), 2-ethoxyethanol (2.4 ± 0.4 Å), and residual water (1.0 ± 0.30 Å). There are several important assumptions implicit to this conversion, as it predicts the radius of spherical molecules; therefore, deviations from this geometry, such as prolate or oblate ellipsoids, could lead significant error (30−40%) in the estimated effective radius.25 For solutes similar to the size of the solvent, the 6 in the denominator of eq 2 is sometimes replaced by a value of 4.26,27 As Stokes’s law was developed to deal with macroscopic properties such as random displacement of colloidal particles, the estimated radius is only a rough estimate of the effective radius of the solute. Diffusion coefficients reflect the transport properties of the solute under solvated conditions and therefore the estimated R represents the radius of the solute that may contain bound solvent molecules (e.g., hydrogen-bonded). The measured R is a time-averaged radius that includes a solvation shell, often referred to as a hydrodynamic radius.

RESULTS The diffusion-dependent signal attenuation for three signals (caffeine, 2-ethoxyethanol, and residual water) from the same sample tube at room temperature (∼25 °C) is shown in Figure 1. The continuous lines show the corresponding nonlinear

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CONCLUSIONS The calculation of the diffusion coefficient of various compounds using pulsed-magnetic-field gradient (PFG) NMR is an excellent way to introduce students to a spectroscopic method that may be utilized in their current and future education as well as their research efforts. The complete experiment requires about 6 h and some portions can be performed in an automated manner. Analysis of the resultant data is straightforward enough that no detailed knowledge of NMR spectroscopy is necessary. The experiment and analysis can be completed within 2 or 3 laboratory periods. The value of the diffusion coefficient depends on the identity of the molecule of interest, the solvent, and the temperature. There are also several experimental factors that affect the accuracy of the measurement of the diffusion coefficient, including the uniformity of the gradient, other sources of motion, distortions of the baseline, and the uniformity of the temperature of the sample. Measurements of the diffusion coefficients can also be used to separate signals from different molecules. In this experiment, the diffusion constants of three different molecules were measured in the same sample. The basics of this experiment can be easily adopted to perform diffusion-ordered spectroscopy (DOSY) experiments where the indirect dimension is the diffusion coefficient.28 DOSY is particularly useful in situations where it is not convenient or possible to physically separate a mixture and is capable of separating the signals based on differences in the chemical shifts and molecular size.

Figure 1. Diffusion-dependent signal attenuation of caffeine (squares), 2-ethoxyethanol (circles), and the residual water (triangles) in D2O. Normalized area of the peaks for each molecule (S) is plotted vs q2Δ′, where q = γgzδ. Standard deviations were derived from three independent experimental measurements shown as the error bars. The continuous lines represent the exponential fit to the experimental data.

least-squares regression fit to the experimental data (eq 3, S vs q2Δ′, with gradient pulse area of q2 = γ2gz2δ2). The three molecules in this study (caffeine, M = 194.19 g/mol, δ = 7.59 ppm; 2-ethoxyethanol, M = 90.12 g/mol, δ = 0.09 ppm; and water, M = 18.02 g/mol, δ =4.72 ppm) are not reactive toward one another and possess distinct chemical shifts. This allows for the reliable measurement of their diffusion constants independent of each other. The temperature-dependent diffusion constants (measured at the five different temperatures) are presented in Table 1. Table 1. Diffusion Coefficients Measured Using PulsedField-Gradient NMR D/(10−9 m2 s−1) T/K 288.3 293.6 299.0 304.4 309.8 a

Water 1.42 1.89 1.79 2.59 3.31

± ± ± ± ±

a

0.01 0.07 0.09 0.06 0.17

2-Ethoxyethanol 0.51 0.67 0.84 1.00 1.54

± ± ± ± ±

0.02 0.02 0.02 0.02 0.14

Caffeine 0.39 0.45 0.59 0.84 1.31

± ± ± ± ±

0.01 0.04 0.02 0.03 0.12

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Residual water in D2O.

ASSOCIATED CONTENT

S Supporting Information *

In addition to providing a qualitative measure of transport properties, diffusion constants can provide structural characterization of a solute in a given solvent. The Stokes−Einstein equation (eq 2) relates the diffusion coefficient to an effective molecular radius, R. This can be determined because eq 2 relates the radius (and therefore the volume) of a spherical molecule to its diffusion coefficient: volume ∝ radius3 ∝ D−3.

Instructions for the students and notes for the instructor This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] or [email protected]. 782

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Notes #

Undergraduate B.S. Chemistry Honors students.

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ACKNOWLEDGMENTS The authors acknowledge S. Attar and A. Hasson for their critical reading of the manuscript. K.A.H. was supported by a graduate fellowship by National Science Foundation (NSF Award # 1059994). This work was in part supported by Research Infrastructure for Minority Institutions (RIMI) Grant P20 MD-002732 and student research training grant P20-CA138025.

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