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NMR diffusion studies of spherical molecules: Tetramethylsilane and buckyballs
Journal of Molecular Liquids 214 (2016) 157–161
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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
NMR diffusion studies of spherical molecules: Tetramethylsilane and buckyballs Amninder S. Virk, Allan M. Torres, Scott A. Willis, William S. Price ⁎ Nanoscale Organisation and Dynamics Group, School of Science and Health, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia
a r t i c l e
i n f o
Article history: Received 1 August 2015 Received in revised form 16 October 2015 Accepted 23 November 2015 Available online xxxx Keywords: Diffusion Obstruction Spherical molecules C60 Tetramethylsilane
a b s t r a c t The structural properties and hydrodynamic size of a molecule in solution at infinite dilution are connected to its diffusion coefficient through the Stokes–Einstein–Sutherland equation. In this study, buckyballs (C60) and mixtures of tetramethylsilane and CDCl3 which closely approximate spherical molecules in solution were investigated using nuclear magnetic resonance (NMR) self-diffusion experiments. It was found that the change in diffusion coefficient of mixtures of TMS and CDCl3 was only correlated to the viscosity of the solution. The C60 PGSE NMR data was then analysed using various models of obstruction. It was found that the decrease in C60 diffusion with increasing concentration can be explained on the basis of aggregation alone, and thus C60 self-obstruction must be negligible under these experimental conditions. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Self-diffusion measurements provide information on the size, shape and interactions of the diffusing molecules [1,2]. Specifically, the diffusion coefficient of a molecule depends on the friction factor which is related to the hydrodynamic size/shape of the molecule and the viscosity of the solvent. In the absence of any external forces acting on the diffusing particle, the infinite-dilution diffusion coefficient of a molecule is related to its effective hydrodynamic radius via the Stokes–Einstein–Sutherland equation [3–5], D0 ¼
kb T ; fr
ð1Þ
where k b is the Boltzmann constant, T (K) is temperature and fr (kg s − 1 ) is the friction coefficient of the diffusing molecule. For a spherical molecule with an effective hydrodynamic radius rH (m) in a solvent of viscosity η (Pa s), fr = cπηrH where the constant c characterizes the interaction between the solute and the solvent with c = 4 or 6 denoting the ‘slip’ and ‘stick’ conditions, respectively. The stick condition is more applicable to large solutes or solutes interacting strongly with solvent molecules, whereas the slip condition is applicable to small solutes or less interaction between solute and solvent molecules [6,7]. Many solvents, such as water, dimethyl-sulfoxide, ethanol and methanol, have complex solvent behaviour involving at least transient ⁎ Corresponding author. E-mail address: [email protected] (W.S. Price).
http://dx.doi.org/10.1016/j.molliq.2015.11.029 0167-7322/© 2015 Elsevier B.V. All rights reserved.
self-association due to the presence of hydrogen bonds [8–10]. Thus, the presence of H-bonding interactions complicates the interpretation of the diffusion data. Simplistically, from Eq. (1) the hydrodynamic radii of both the internal reference and probe molecules can be related to each other by (e.g., [7,11]), r probe ¼
Dref r ; Dprobe ref
ð2Þ
where Dref, rref and Dprobe, rprobe are the diffusion coefficient and hydrodynamic radius of the reference and probe molecules, respectively. Both the internal reference and probe molecules, being in the same solution, should experience similar viscosity effects from the solvent molecules. In addition, many diffusing systems are known to undergo selfassociation with increasing concentration and it can be difficult to determine the size of the different aggregates and their degree of size polydispersity from diffusion data alone [12–14]. In fact, apart from at very low concentration all measured diffusion coefficients include effects of both aggregation and obstruction. Most of the theory used to analyse the experimental diffusion in liquids has been derived on the simple assumption that diffusion is occurring at infinite dilution (i.e., Eq. (1)). Clearly, this is inadequate as almost always these are crowded systems with multiple collisions occurring between solute molecules within the timescale of the measurement. Thus, a diffusion measurement will simultaneously measure true self-diffusion, obstruction and molecular association. The diffusion path and the (time-dependent) coordinates of obstructing particles are very complex due to the presence of obstructing particles of varying shapes and sizes. The
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to achieve the desired concentrations and dispensed into 5 mm NMR tubes (Wilmad Lab glass). Then, straight capillary tubes with openings at both ends (1.5 mm, Wilmad Lab glass) were inserted the tubes. Due to the volatility of TMS (boiling point 299 K) and CDCl3, the samples were frozen immediately after preparation using liquid nitrogen and were flame sealed to prevent evaporation. The C60 (99.9%, sublimed, Sigma) solutions were prepared by mixing the desired amount of C60 with 1-chloronapthalene (85%, technical grade, Sigma). C60 has a maximum solubility of 69 mM in 1chloronapthalene [45]. A concentrated solution was made by adding C60 to 1-chloronapthalene and sonicating. A series of dilutions were performed to achieve the desired concentration solutions. No visible precipitation of the C60 was observed in any of the samples.
mathematical complexity of calculating obstruction factors of different sized obstructing particles involves solving the diffusion equation under appropriate boundary conditions. The simple models derived to account for the presence of obstructing particles assume the same shape (mostly spheres) for all obstructing particles and are only valid at very low concentrations [15–23]. The theoretical modelling to account for the combined effects of aggregation and obstruction effects in liquids is still in its infancy. At present, models consider particles as hard spheres [24–26]. The importance of developing cogent models of aggregation that include the effects of obstruction has been highlighted in a number of recent studies [27,28]. TMS is an inert and nonpolar solvent which is widely used as a standard internal chemical shift reference [11]. Since TMS will not be influenced by the complications of Hbonding and because TMS has a pseudo-spherical structure, it is an appropriate choice for our investigation. In this work, the diffusion of two pseudo-spherical molecules, buckyballs (C60) and tetramethylsilane (TMS) [29–31], have been investigated. The physical and chemical properties of C60 are of great interest due to potential applications of C60 in biological and chemical systems. A number of studies have used dynamic light scattering (DLS) or fluorescence spectroscopic techniques to investigate the aggregation properties of C60 in different solvents [32–36]. Nath et al. [33] have indicated the presence of aggregates at high C60 concentrations (i.e., N100 μM) in benzonitrile solutions. Similarly, Chen et al. [32] reported the aggregation and deposition behaviours of C60 nanoparticles in toluene solutions with electrolytes and Ying et al. [35,36] have studied the aggregation behaviour of C60 in benzene solutions. However, Huo et al. [37,38] have shown that solvent properties of the mixtures have a significant impact on the structure of resultant C60 aggregates. All these studies have used the DLS technique to determine the average size distribution near the solubility limit of C60 in the respective solvents. If the radii are measurably different, the DLS technique is, in principle, capable of distinguishing between sizes of different aggregates. However, it also depends on concentration fluctuations, as it measures the mutual diffusion coefficient. NMR diffusion spectroscopy is a unique tool for studying the binding of biomolecules, ranging from selfassociating systems to ligand–protein associations. Kato et al. [39] and Haselmeier et al. [40] have reported the diffusion coefficient of C60 in benzene and carbon disulphide, respectively using 13C NMR. In both studies, C60 was 13C enriched (30% [39] and 5% [40]). However, C60 aggregation and obstruction properties have never been investigated using NMR diffusion measurements. Concomitant with the improved technical abilities of modern NMR spectrometers to measure diffusion, there has been increased awareness of the experimental limitations and the need for more realistic models to analyse diffusion data of associated systems [6,14,41,42]. A detailed description of the aggregation and obstruction models used to analyse diffusion data of associated/aggregated systems can be found elsewhere [43,44]. In this work, the self-diffusion properties of TMS dissolved in CDCl3 were studied over the entire composition range using 1H NMR and the C60 diffusion coefficient was measured at concentrations ranging from 1.65 to 35 mM in 1-chloronapthalene using 13C NMR. This work is organized as follows: In the Results and discussion section, the diffusion coefficient of TMS was measured using 1H NMR at 298 K. Then, NMR self-diffusion measurements of TMS and CDCl3 mixtures are discussed in relation to the hydrodynamic radii (rH) derived from Eq. (1) [3–5]. Then the NMR diffusion results of C60 are discussed in relation to aggregation and obstruction models. Concluding remarks are presented in the final section.
H (TMS) and 2H (CDCl3) pulsed gradient stimulated echo (PGSTE) NMR diffusion experiments were performed at 400 MHz and 61 MHz, respectively at 298 K with a 5 mm BBO probe on a 400 MHz Bruker Avance NMR spectrometer. 13C pulsed gradient spin-echo (PGSE) NMR diffusion experiments (C60 and 1-chloronapthalene) were performed at 125 MHz and 298 K with a Micro5 probe and 5 mm 13C/1H rf insert on a 500 MHz Bruker Avance II NMR spectrometer. A standard methanol sample (99.97% MeOH + 0.03% HCl) was used for NMR temperature calibration. The Micro5 probe has a triple axes gradient set with maximum gradient strengths along the x, y and z-axes of 2.92 T m− 1, 2.91 T m−1 and 2.95 T m− 1 at 298 K, respectively. The gradient strength along each direction was calibrated by measuring the known diffusion coefficient of residual water (HDO, 1.9 × 10−9 m2 s−1) in D2O at 298 K [46]. 1 H and 2H NMR diffusion experiments were performed with typical acquisition parameters gradient pulse duration (δ) = 1.5 ms, timescale of the diffusion measurement (Δ) = 0.2 s with the gradient amplitude (g) varied from 0.005–0.253 T m−1 in increments of 0.017 T m−1 to give 16 data points for each echo attenuation curve. Each spectrum was averaged over at least 8 scans depending on the signal-to-noise ratio. The acquisition parameters for the 13C diffusion experiments were δ = 5.0 ms, Δ = 0.2 s with g varying from 0.03 to 0.64 T m−1 in increments of 0.09 T m−1. The values δ, g and Δ were selected so that the echo signal was attenuated by at least 80% with the largest value of g. Recycle delays were set to at least five times the spin–lattice relaxation time of the measured species. The 13C spin–lattice relaxation time of C60 (34 mM in 1-chloronapthalene) was determined to be 28 s using the inversion recovery pulse sequence. Unwanted convection effects can easily result from temperature gradients when measurements are performed away from ambient temperature [6,47–49]. Since TMS is very volatile and the diffusion measurements were performed at 298 K (i.e., 1° below boiling), capillaries were added to help minimize convection effects. Undesirable convection effects can also be minimized by using the double-stimulated echo pulse sequence (DSTE) [6,47–49]. Experiments were performed using a sample with capillaries and measurements were made with PGSE, PGSTE and DSTE to eliminate undesirable convection effects. The NMR diffusion data were analysed by non-linear least squares regression of the appropriate attenuation expression using OriginPro 9 (OriginLab) software [6]. The error values quoted are those obtained from data fitting. However, including factors like temperature fluctuations and other instrumental errors, the true error is likely to be on the order of 1% [50].
2. Materials and methods
3. Results and discussion
2.1. Materials
3.1. TMS diffusion
TMS (99.9%, NMR grade, Sigma) and CDCl3 (99.8%, Cambridge Isotope Laboratories) were used as supplied. They were mixed together
A phase change was observed in the TMS diffusion spectra acquired with the magnetic gradient applied along the z-axis using PGSE and
2.2. Experimental details 1
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Table 1 Comparisons of the self-diffusion coefficient of TMS measured by different groups using steady or pulsed gradient NMR diffusion measurements at 298 K. Experimental studies
D (×10−9 m2 s−1)
Douglass et al. [51] Kessler et al. [51] [52] Kessler et al. [51] [53] Parkhurst et al. [51] [54] Brüsewitz et al. [51]. [55] This work (DSTE and capillaries)
3.6 (4.0)a 3.9 (4.11)b 4.13c 3.89d 4.1e 4.45 ± 0.01
a
The first value is from Table 1 and the second value is from Figs. 3 and 4 of Ref. [51]. The first value is from Table 1 using Eq. 7 and the second value is from Fig. 6 of Ref. [52]. c According to Ref. [53] the error is between 5 and 10%. d Extrapolated value. e Estimated from Fig. 3 in Ref. [55]. b
PGSTE sequence. The origin of the phase change was unclear and its presence hampered tests to determine the presence of convection effects [6]. However, it was found that such a phase change did not occur when the gradient was applied along the x or y axes (possibly because convective motion is primarily along the axis of the NMR tube and the regions at the top and bottom of the sample were outside the rf coils). Nonetheless, in the present study, by inserting straight capillary tubes with openings at both ends into the 5 mm NMR tube and using the DSTE based diffusion sequence with gradient along the z-axis eliminated the phase change. Since, the diffusion coefficient of TMS measured using the PGSE and PGSTE sequence with g along z-axis and sample with capillaries did not change significantly with Δ, it was concluded that convection effects were removed by the introduction of the capillaries. The measured diffusion coefficient of pure TMS at 298 K was found to be (4.45 ± 0.01) × 10−9 m2 s−1. This value was found to be slightly higher than some of the values reported in the literature (Table 1). However, it should be noted that all of the literature values were obtained on less sophisticated NMR spectrometers and had little indication as to the magnitude of the errors in the reported diffusion coefficient values. Further, as will be seen in Section 3.2 and from Fig. 1B, the slip condition predicts the newly measured diffusion coefficient accurately. 3.2. Diffusion of TMS and CDCl3 mixtures The diffusion coefficient of both TMS and CDCl3 were found in this study to increase with increasing TMS concentration. The diffusion coefficient of CDCl3 determined using 1H and 2H NMR diffusion was in agreement with the published experimental value of (2.45 ± 0.01) × 10−9 m2 s−1 [56]. The hydrodynamic radius of TMS and CDCl3 was calculated using slip (i.e., c = 4) and stick (i.e., c = 6) conditions with ηTMS = 0.239 mPa s [57] and ηCDCl3 = 0.542 mPa s at 298 K
Fig. 2. NMR measured diffusion coefficients of 1-chloronapthalene molecules via 13C (■) in C60 solutions as a function of C60 concentration at 298 K. The error bars for the measurements are also included; however, the errors are much smaller than the symbols.
[58]. The hydrodynamic radii of TMS and CDCl3 are 2.05 Å and 1.65 Å under stick conditions, and 3.08 Å and 2.47 Å under slip conditions, respectively. As TMS is not involved in H-bonding, the increase in diffusion coefficient with increasing TMS concentration is more likely due to a change in solution viscosity. The change in solution viscosity of the TMS and CDCl3 mixture with TMS concentration is shown in Fig. 1A. The viscosity of the solutions, ηsolution, assuming ideal solution behaviour is given by [59], ηsolution ¼
1 xTMS =ηTMS þ xCDC13 =ηCDC13
ð3Þ
where xTMS and xCDCl3 are the mole fractions of solvents TMS and CDCl3, respectively. The calculated hydrodynamic radii and viscosities of the solutions were then used to predict the diffusion coefficients (Fig. 1B). Both the predicted (i.e., slip conditions) and experimental diffusion coefficients closely agreed with each other (Fig. 1B). However, there was a slight deviation in DTMS from the predicted value at high concentrations. This deviation is probably due to experimental error introduced in the sample preparation (i.e., difficulty in preparing mixtures with highly volatile solvents). Hence, diffusion data indicated the diffusive path of TMS molecules was obstructed by the CDCl3 molecules (i.e., reduction of the diffusion coefficient of TMS with increasing CDCl3 concentration) and that no aggregation of CDCl3 or TMS was observed over the entire composition range (i.e., results are predictable with a change in viscosity only).
Fig. 1. (A) Predicted viscosity values (━) of TMS and CDCl3 solutions with TMS concentration at 298 K. (B) Predicted (i.e., using slip conditions and prediction solution viscosities) and experimental diffusion coefficients of TMS and CDCl3 with respect to TMS concentration at 298 K. The experimental diffusion coefficients were determined using 1H and 2H NMR for TMS and CDCl3, respectively. The error bars for the measurements are also included; however, the errors are much smaller than the symbols. The solid lines are guides for the eyes.
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results in Fig. 3 clearly show that none of the obstruction models are able to describe the experimental data and the measured diffusion coefficient is much lower than that predicted by the obstruction models. The poor fitting from the obstruction models to the experimental C60 diffusion data indicates that C60 molecules are undergoing aggregation in the presence of obstruction from other C60 molecules. Hence, aggregation models need to be taken into account to explain the decreased C60 coefficients. The experimentally determined C60 diffusion coefficients were then analysed by fitting isodesmic association model. As this model describes the experimental data it implies that only aggregation needs to be taken into account to explain the decreased C60 diffusion coefficients. 5. Conclusions Fig. 3. The measured diffusion coefficients of C60 (●) in 1-chloronapthalene at 298 K as a function of C60 concentration. Clearly, none of the current obstructions models (i.e., Han and Herzfeld (━) [15] overlapped with Lekkerkerker and Dhont ( ) [22], Tokuyama and Oppenheim's short interactions model ( ),Tokuyama and Oppenheim's long interactions model ( ) [21], and Jönsson et al. ( ) [16]), fit the experimental data. The measured diffusion coefficient is mostly lower than the obstruction models indicating obstruction have a minor effect on C60 diffusion coefficients with concentration. Also shown is the fitting of the isodesmic aggregation model ( ) [23] to the experimental data.
4. Diffusion of C60 solutions The identities of the C60 peaks were confirmed by comparison with spectra of the pure 1-chloronapthalene solvent. The diffusion coefficient of 1-chloronapthalene has been observed before using desorption measurements [60]. However, since the self-diffusion coefficient of pure 1-chloronapthalene has not previously been studied using PGSE NMR diffusion, this work therefore reports the first NMR diffusion results of this compound. It was observed that the diffusion coefficient of the 1-chloronapthalene in the C60 solutions was lower than that of the pure solvent (Fig. 2). The decrease in the diffusion coefficient therefore suggests the presence of obstruction in these solutions. As the concentration of C60 increases the excluded volume increases, hence the longer diffusive path for the same displacement leads to a decreased diffusion coefficient. The results of diffusion measurements performed on C60 in 1chloronapthalene at concentrations ranging from 0.02 to 0.33 mM are summarized in Fig. 3. The van der Waals radius of C60 has been reported in various studies [39,40,61]. To compare this literature with our work, the hydrodynamic radius of C60 was calculated using stick and slip conditions, and η = 3.020 mPa s at 298 K [62]. Extrapolation of the measured diffusion coefficients using a linear function gave the diffusion coefficient of C60 at infinite dilution. The hydrodynamic radii calculated from the extrapolated infinite dilution diffusion coefficient values were compared with literature in Table 2. The diffusion coefficient of C60 also decreased with concentration (Fig. 3). The decrease in the obtained C60 diffusion coefficients can be explained by analysing the data using aggregation and obstruction models. The extrapolated infinite dilution diffusion coefficient was then used to predict the obstruction factor from the C60 as a function of concentration and compared with experimental data. The predicted Table 2 Comparison of the hydrodynamic radii of C60 reported in literature. NMR diffusion studies
Radius (Å) Stick
Slip
Kato et al. [39] Haselmeier et al. (C6H6) [40] Haselmeier et al. (CS2) [40] This work
4.10 3.68 3.37 3.75
6.15 5.52 5.06 5.63
In this work, the diffusion coefficient of pure TMS at 298 K was precisely determined. The diffusion coefficients of both TMS and CDCl3 were found to increase with TMS concentration. From the experimental data of TMS and CDCl3 mixtures, there was strong evidence that the diffusive path of TMS molecules was obstructed by the CDCl3 molecules. The increase in the diffusion coefficients of the solutions of TMS and CDCl3 is likely a result from the decrease in the viscosity of the solutions which can be related to the absence of hydrogen bonds between the molecules (i.e., low attractive forces). It can be concluded that TMS and CDCl3 was only dependent on viscosity (or obstruction) and this method needs to be further applied to other organic solvents/TMS mixtures to investigate the effects of aggregation and obstruction due to the presence of H-bonds. PGSE NMR C60 diffusion coefficients were used to predict the presence of aggregation and obstruction at all C60 concentrations. The prediction of experimental diffusion data via the obstruction models indicates that the obstruction has a very limited effect on C60 diffusion data. Hence, the decrease in diffusion coefficient of C60 was likely only due to aggregation. Acknowledgements The authors acknowledge the facilities, and the scientific and technical assistance of the National Imaging Facility, Western Sydney University Node. References [1] S.A. Willis, G.R. Dennis, G. Zheng, W.S. Price, Hydrodynamic size and scaling relations for linear and 4 arm star PVAC studied using PGSE NMR, J. Mol. Liq. 156 (2010) 45–51. [2] M.W. Germann, T. Turner, S.A. Allison, Translational diffusion constants of the amino acids: measurement by NMR and their use in modeling the transport of peptides, J. Phys. Chem. A 111 (2007) 1452–1455. [3] G.G. Stokes, On the effect of internal friction of fluids on the motion of pendulums, Camb. Phil. Soc. Trans. 9 (1856) 8–106. [4] W. Sutherland, XVIII. Ionization, Ionic Velocities, and Atomic Sizes, Philos Mag Ser 6, 3 1902, pp. 161–177. [5] Einstein, A., Investigations on the Theory of the Brownian Movement Edited with notes by Fürth, R., Translated by Cowper, A. D., (The Five Papers of Albert Einstein from 1905–1908), Dover Publications, Inc., New York, (1956). [6] W.S. Price, NMR Studies of Translational Motion: Principles and Applications, Cambridge University Press, Cambridge, 2009. [7] A. Macchioni, G. Ciancaleoni, C. Zuccaccia, D. Zuccaccia, Determining accurate molecular sizes in solution through NMR diffusion spectroscopy, Chem. Soc. Rev. 37 (2008) 479–489. [8] A. Ellis, F.M. Zehentbauer, J. Kiefer, Probing the balance of attraction and repulsion in binary mixtures of dimethyl sulfoxide and N-alcohols, Phys. Chem. Chem. Phys. 15 (2013) 1093–1096. [9] I.A. Borin, M.S. Skaf, Molecular association between water and dimethyl sulfoxide in solution: a molecular dynamics simulation study, J. Chem. Phys. 110 (1999) 6412–6420. [10] W.S. Price, H. Ide, Y. Arata, Solution dynamics in aqueous monohydric alcohol systems, J. Phys. Chem. A. 107 (2003) 4784–4789. [11] C.A. Crutchfield, D.J. Harris, Molecular mass estimation by PFG NMR spectroscopy, J. Magn. Reson. 185 (2007) 179–182.
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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
NMR diffusion studies of spherical molecules: Tetramethylsilane and buckyballs Amninder S. Virk, Allan M. Torres, Scott A. Willis, William S. Price ⁎ Nanoscale Organisation and Dynamics Group, School of Science and Health, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia
a r t i c l e
i n f o
Article history: Received 1 August 2015 Received in revised form 16 October 2015 Accepted 23 November 2015 Available online xxxx Keywords: Diffusion Obstruction Spherical molecules C60 Tetramethylsilane
a b s t r a c t The structural properties and hydrodynamic size of a molecule in solution at infinite dilution are connected to its diffusion coefficient through the Stokes–Einstein–Sutherland equation. In this study, buckyballs (C60) and mixtures of tetramethylsilane and CDCl3 which closely approximate spherical molecules in solution were investigated using nuclear magnetic resonance (NMR) self-diffusion experiments. It was found that the change in diffusion coefficient of mixtures of TMS and CDCl3 was only correlated to the viscosity of the solution. The C60 PGSE NMR data was then analysed using various models of obstruction. It was found that the decrease in C60 diffusion with increasing concentration can be explained on the basis of aggregation alone, and thus C60 self-obstruction must be negligible under these experimental conditions. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Self-diffusion measurements provide information on the size, shape and interactions of the diffusing molecules [1,2]. Specifically, the diffusion coefficient of a molecule depends on the friction factor which is related to the hydrodynamic size/shape of the molecule and the viscosity of the solvent. In the absence of any external forces acting on the diffusing particle, the infinite-dilution diffusion coefficient of a molecule is related to its effective hydrodynamic radius via the Stokes–Einstein–Sutherland equation [3–5], D0 ¼
kb T ; fr
ð1Þ
where k b is the Boltzmann constant, T (K) is temperature and fr (kg s − 1 ) is the friction coefficient of the diffusing molecule. For a spherical molecule with an effective hydrodynamic radius rH (m) in a solvent of viscosity η (Pa s), fr = cπηrH where the constant c characterizes the interaction between the solute and the solvent with c = 4 or 6 denoting the ‘slip’ and ‘stick’ conditions, respectively. The stick condition is more applicable to large solutes or solutes interacting strongly with solvent molecules, whereas the slip condition is applicable to small solutes or less interaction between solute and solvent molecules [6,7]. Many solvents, such as water, dimethyl-sulfoxide, ethanol and methanol, have complex solvent behaviour involving at least transient ⁎ Corresponding author. E-mail address: [email protected] (W.S. Price).
http://dx.doi.org/10.1016/j.molliq.2015.11.029 0167-7322/© 2015 Elsevier B.V. All rights reserved.
self-association due to the presence of hydrogen bonds [8–10]. Thus, the presence of H-bonding interactions complicates the interpretation of the diffusion data. Simplistically, from Eq. (1) the hydrodynamic radii of both the internal reference and probe molecules can be related to each other by (e.g., [7,11]), r probe ¼
Dref r ; Dprobe ref
ð2Þ
where Dref, rref and Dprobe, rprobe are the diffusion coefficient and hydrodynamic radius of the reference and probe molecules, respectively. Both the internal reference and probe molecules, being in the same solution, should experience similar viscosity effects from the solvent molecules. In addition, many diffusing systems are known to undergo selfassociation with increasing concentration and it can be difficult to determine the size of the different aggregates and their degree of size polydispersity from diffusion data alone [12–14]. In fact, apart from at very low concentration all measured diffusion coefficients include effects of both aggregation and obstruction. Most of the theory used to analyse the experimental diffusion in liquids has been derived on the simple assumption that diffusion is occurring at infinite dilution (i.e., Eq. (1)). Clearly, this is inadequate as almost always these are crowded systems with multiple collisions occurring between solute molecules within the timescale of the measurement. Thus, a diffusion measurement will simultaneously measure true self-diffusion, obstruction and molecular association. The diffusion path and the (time-dependent) coordinates of obstructing particles are very complex due to the presence of obstructing particles of varying shapes and sizes. The
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to achieve the desired concentrations and dispensed into 5 mm NMR tubes (Wilmad Lab glass). Then, straight capillary tubes with openings at both ends (1.5 mm, Wilmad Lab glass) were inserted the tubes. Due to the volatility of TMS (boiling point 299 K) and CDCl3, the samples were frozen immediately after preparation using liquid nitrogen and were flame sealed to prevent evaporation. The C60 (99.9%, sublimed, Sigma) solutions were prepared by mixing the desired amount of C60 with 1-chloronapthalene (85%, technical grade, Sigma). C60 has a maximum solubility of 69 mM in 1chloronapthalene [45]. A concentrated solution was made by adding C60 to 1-chloronapthalene and sonicating. A series of dilutions were performed to achieve the desired concentration solutions. No visible precipitation of the C60 was observed in any of the samples.
mathematical complexity of calculating obstruction factors of different sized obstructing particles involves solving the diffusion equation under appropriate boundary conditions. The simple models derived to account for the presence of obstructing particles assume the same shape (mostly spheres) for all obstructing particles and are only valid at very low concentrations [15–23]. The theoretical modelling to account for the combined effects of aggregation and obstruction effects in liquids is still in its infancy. At present, models consider particles as hard spheres [24–26]. The importance of developing cogent models of aggregation that include the effects of obstruction has been highlighted in a number of recent studies [27,28]. TMS is an inert and nonpolar solvent which is widely used as a standard internal chemical shift reference [11]. Since TMS will not be influenced by the complications of Hbonding and because TMS has a pseudo-spherical structure, it is an appropriate choice for our investigation. In this work, the diffusion of two pseudo-spherical molecules, buckyballs (C60) and tetramethylsilane (TMS) [29–31], have been investigated. The physical and chemical properties of C60 are of great interest due to potential applications of C60 in biological and chemical systems. A number of studies have used dynamic light scattering (DLS) or fluorescence spectroscopic techniques to investigate the aggregation properties of C60 in different solvents [32–36]. Nath et al. [33] have indicated the presence of aggregates at high C60 concentrations (i.e., N100 μM) in benzonitrile solutions. Similarly, Chen et al. [32] reported the aggregation and deposition behaviours of C60 nanoparticles in toluene solutions with electrolytes and Ying et al. [35,36] have studied the aggregation behaviour of C60 in benzene solutions. However, Huo et al. [37,38] have shown that solvent properties of the mixtures have a significant impact on the structure of resultant C60 aggregates. All these studies have used the DLS technique to determine the average size distribution near the solubility limit of C60 in the respective solvents. If the radii are measurably different, the DLS technique is, in principle, capable of distinguishing between sizes of different aggregates. However, it also depends on concentration fluctuations, as it measures the mutual diffusion coefficient. NMR diffusion spectroscopy is a unique tool for studying the binding of biomolecules, ranging from selfassociating systems to ligand–protein associations. Kato et al. [39] and Haselmeier et al. [40] have reported the diffusion coefficient of C60 in benzene and carbon disulphide, respectively using 13C NMR. In both studies, C60 was 13C enriched (30% [39] and 5% [40]). However, C60 aggregation and obstruction properties have never been investigated using NMR diffusion measurements. Concomitant with the improved technical abilities of modern NMR spectrometers to measure diffusion, there has been increased awareness of the experimental limitations and the need for more realistic models to analyse diffusion data of associated systems [6,14,41,42]. A detailed description of the aggregation and obstruction models used to analyse diffusion data of associated/aggregated systems can be found elsewhere [43,44]. In this work, the self-diffusion properties of TMS dissolved in CDCl3 were studied over the entire composition range using 1H NMR and the C60 diffusion coefficient was measured at concentrations ranging from 1.65 to 35 mM in 1-chloronapthalene using 13C NMR. This work is organized as follows: In the Results and discussion section, the diffusion coefficient of TMS was measured using 1H NMR at 298 K. Then, NMR self-diffusion measurements of TMS and CDCl3 mixtures are discussed in relation to the hydrodynamic radii (rH) derived from Eq. (1) [3–5]. Then the NMR diffusion results of C60 are discussed in relation to aggregation and obstruction models. Concluding remarks are presented in the final section.
H (TMS) and 2H (CDCl3) pulsed gradient stimulated echo (PGSTE) NMR diffusion experiments were performed at 400 MHz and 61 MHz, respectively at 298 K with a 5 mm BBO probe on a 400 MHz Bruker Avance NMR spectrometer. 13C pulsed gradient spin-echo (PGSE) NMR diffusion experiments (C60 and 1-chloronapthalene) were performed at 125 MHz and 298 K with a Micro5 probe and 5 mm 13C/1H rf insert on a 500 MHz Bruker Avance II NMR spectrometer. A standard methanol sample (99.97% MeOH + 0.03% HCl) was used for NMR temperature calibration. The Micro5 probe has a triple axes gradient set with maximum gradient strengths along the x, y and z-axes of 2.92 T m− 1, 2.91 T m−1 and 2.95 T m− 1 at 298 K, respectively. The gradient strength along each direction was calibrated by measuring the known diffusion coefficient of residual water (HDO, 1.9 × 10−9 m2 s−1) in D2O at 298 K [46]. 1 H and 2H NMR diffusion experiments were performed with typical acquisition parameters gradient pulse duration (δ) = 1.5 ms, timescale of the diffusion measurement (Δ) = 0.2 s with the gradient amplitude (g) varied from 0.005–0.253 T m−1 in increments of 0.017 T m−1 to give 16 data points for each echo attenuation curve. Each spectrum was averaged over at least 8 scans depending on the signal-to-noise ratio. The acquisition parameters for the 13C diffusion experiments were δ = 5.0 ms, Δ = 0.2 s with g varying from 0.03 to 0.64 T m−1 in increments of 0.09 T m−1. The values δ, g and Δ were selected so that the echo signal was attenuated by at least 80% with the largest value of g. Recycle delays were set to at least five times the spin–lattice relaxation time of the measured species. The 13C spin–lattice relaxation time of C60 (34 mM in 1-chloronapthalene) was determined to be 28 s using the inversion recovery pulse sequence. Unwanted convection effects can easily result from temperature gradients when measurements are performed away from ambient temperature [6,47–49]. Since TMS is very volatile and the diffusion measurements were performed at 298 K (i.e., 1° below boiling), capillaries were added to help minimize convection effects. Undesirable convection effects can also be minimized by using the double-stimulated echo pulse sequence (DSTE) [6,47–49]. Experiments were performed using a sample with capillaries and measurements were made with PGSE, PGSTE and DSTE to eliminate undesirable convection effects. The NMR diffusion data were analysed by non-linear least squares regression of the appropriate attenuation expression using OriginPro 9 (OriginLab) software [6]. The error values quoted are those obtained from data fitting. However, including factors like temperature fluctuations and other instrumental errors, the true error is likely to be on the order of 1% [50].
2. Materials and methods
3. Results and discussion
2.1. Materials
3.1. TMS diffusion
TMS (99.9%, NMR grade, Sigma) and CDCl3 (99.8%, Cambridge Isotope Laboratories) were used as supplied. They were mixed together
A phase change was observed in the TMS diffusion spectra acquired with the magnetic gradient applied along the z-axis using PGSE and
2.2. Experimental details 1
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159
Table 1 Comparisons of the self-diffusion coefficient of TMS measured by different groups using steady or pulsed gradient NMR diffusion measurements at 298 K. Experimental studies
D (×10−9 m2 s−1)
Douglass et al. [51] Kessler et al. [51] [52] Kessler et al. [51] [53] Parkhurst et al. [51] [54] Brüsewitz et al. [51]. [55] This work (DSTE and capillaries)
3.6 (4.0)a 3.9 (4.11)b 4.13c 3.89d 4.1e 4.45 ± 0.01
a
The first value is from Table 1 and the second value is from Figs. 3 and 4 of Ref. [51]. The first value is from Table 1 using Eq. 7 and the second value is from Fig. 6 of Ref. [52]. c According to Ref. [53] the error is between 5 and 10%. d Extrapolated value. e Estimated from Fig. 3 in Ref. [55]. b
PGSTE sequence. The origin of the phase change was unclear and its presence hampered tests to determine the presence of convection effects [6]. However, it was found that such a phase change did not occur when the gradient was applied along the x or y axes (possibly because convective motion is primarily along the axis of the NMR tube and the regions at the top and bottom of the sample were outside the rf coils). Nonetheless, in the present study, by inserting straight capillary tubes with openings at both ends into the 5 mm NMR tube and using the DSTE based diffusion sequence with gradient along the z-axis eliminated the phase change. Since, the diffusion coefficient of TMS measured using the PGSE and PGSTE sequence with g along z-axis and sample with capillaries did not change significantly with Δ, it was concluded that convection effects were removed by the introduction of the capillaries. The measured diffusion coefficient of pure TMS at 298 K was found to be (4.45 ± 0.01) × 10−9 m2 s−1. This value was found to be slightly higher than some of the values reported in the literature (Table 1). However, it should be noted that all of the literature values were obtained on less sophisticated NMR spectrometers and had little indication as to the magnitude of the errors in the reported diffusion coefficient values. Further, as will be seen in Section 3.2 and from Fig. 1B, the slip condition predicts the newly measured diffusion coefficient accurately. 3.2. Diffusion of TMS and CDCl3 mixtures The diffusion coefficient of both TMS and CDCl3 were found in this study to increase with increasing TMS concentration. The diffusion coefficient of CDCl3 determined using 1H and 2H NMR diffusion was in agreement with the published experimental value of (2.45 ± 0.01) × 10−9 m2 s−1 [56]. The hydrodynamic radius of TMS and CDCl3 was calculated using slip (i.e., c = 4) and stick (i.e., c = 6) conditions with ηTMS = 0.239 mPa s [57] and ηCDCl3 = 0.542 mPa s at 298 K
Fig. 2. NMR measured diffusion coefficients of 1-chloronapthalene molecules via 13C (■) in C60 solutions as a function of C60 concentration at 298 K. The error bars for the measurements are also included; however, the errors are much smaller than the symbols.
[58]. The hydrodynamic radii of TMS and CDCl3 are 2.05 Å and 1.65 Å under stick conditions, and 3.08 Å and 2.47 Å under slip conditions, respectively. As TMS is not involved in H-bonding, the increase in diffusion coefficient with increasing TMS concentration is more likely due to a change in solution viscosity. The change in solution viscosity of the TMS and CDCl3 mixture with TMS concentration is shown in Fig. 1A. The viscosity of the solutions, ηsolution, assuming ideal solution behaviour is given by [59], ηsolution ¼
1 xTMS =ηTMS þ xCDC13 =ηCDC13
ð3Þ
where xTMS and xCDCl3 are the mole fractions of solvents TMS and CDCl3, respectively. The calculated hydrodynamic radii and viscosities of the solutions were then used to predict the diffusion coefficients (Fig. 1B). Both the predicted (i.e., slip conditions) and experimental diffusion coefficients closely agreed with each other (Fig. 1B). However, there was a slight deviation in DTMS from the predicted value at high concentrations. This deviation is probably due to experimental error introduced in the sample preparation (i.e., difficulty in preparing mixtures with highly volatile solvents). Hence, diffusion data indicated the diffusive path of TMS molecules was obstructed by the CDCl3 molecules (i.e., reduction of the diffusion coefficient of TMS with increasing CDCl3 concentration) and that no aggregation of CDCl3 or TMS was observed over the entire composition range (i.e., results are predictable with a change in viscosity only).
Fig. 1. (A) Predicted viscosity values (━) of TMS and CDCl3 solutions with TMS concentration at 298 K. (B) Predicted (i.e., using slip conditions and prediction solution viscosities) and experimental diffusion coefficients of TMS and CDCl3 with respect to TMS concentration at 298 K. The experimental diffusion coefficients were determined using 1H and 2H NMR for TMS and CDCl3, respectively. The error bars for the measurements are also included; however, the errors are much smaller than the symbols. The solid lines are guides for the eyes.
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results in Fig. 3 clearly show that none of the obstruction models are able to describe the experimental data and the measured diffusion coefficient is much lower than that predicted by the obstruction models. The poor fitting from the obstruction models to the experimental C60 diffusion data indicates that C60 molecules are undergoing aggregation in the presence of obstruction from other C60 molecules. Hence, aggregation models need to be taken into account to explain the decreased C60 coefficients. The experimentally determined C60 diffusion coefficients were then analysed by fitting isodesmic association model. As this model describes the experimental data it implies that only aggregation needs to be taken into account to explain the decreased C60 diffusion coefficients. 5. Conclusions Fig. 3. The measured diffusion coefficients of C60 (●) in 1-chloronapthalene at 298 K as a function of C60 concentration. Clearly, none of the current obstructions models (i.e., Han and Herzfeld (━) [15] overlapped with Lekkerkerker and Dhont ( ) [22], Tokuyama and Oppenheim's short interactions model ( ),Tokuyama and Oppenheim's long interactions model ( ) [21], and Jönsson et al. ( ) [16]), fit the experimental data. The measured diffusion coefficient is mostly lower than the obstruction models indicating obstruction have a minor effect on C60 diffusion coefficients with concentration. Also shown is the fitting of the isodesmic aggregation model ( ) [23] to the experimental data.
4. Diffusion of C60 solutions The identities of the C60 peaks were confirmed by comparison with spectra of the pure 1-chloronapthalene solvent. The diffusion coefficient of 1-chloronapthalene has been observed before using desorption measurements [60]. However, since the self-diffusion coefficient of pure 1-chloronapthalene has not previously been studied using PGSE NMR diffusion, this work therefore reports the first NMR diffusion results of this compound. It was observed that the diffusion coefficient of the 1-chloronapthalene in the C60 solutions was lower than that of the pure solvent (Fig. 2). The decrease in the diffusion coefficient therefore suggests the presence of obstruction in these solutions. As the concentration of C60 increases the excluded volume increases, hence the longer diffusive path for the same displacement leads to a decreased diffusion coefficient. The results of diffusion measurements performed on C60 in 1chloronapthalene at concentrations ranging from 0.02 to 0.33 mM are summarized in Fig. 3. The van der Waals radius of C60 has been reported in various studies [39,40,61]. To compare this literature with our work, the hydrodynamic radius of C60 was calculated using stick and slip conditions, and η = 3.020 mPa s at 298 K [62]. Extrapolation of the measured diffusion coefficients using a linear function gave the diffusion coefficient of C60 at infinite dilution. The hydrodynamic radii calculated from the extrapolated infinite dilution diffusion coefficient values were compared with literature in Table 2. The diffusion coefficient of C60 also decreased with concentration (Fig. 3). The decrease in the obtained C60 diffusion coefficients can be explained by analysing the data using aggregation and obstruction models. The extrapolated infinite dilution diffusion coefficient was then used to predict the obstruction factor from the C60 as a function of concentration and compared with experimental data. The predicted Table 2 Comparison of the hydrodynamic radii of C60 reported in literature. NMR diffusion studies
Radius (Å) Stick
Slip
Kato et al. [39] Haselmeier et al. (C6H6) [40] Haselmeier et al. (CS2) [40] This work
4.10 3.68 3.37 3.75
6.15 5.52 5.06 5.63
In this work, the diffusion coefficient of pure TMS at 298 K was precisely determined. The diffusion coefficients of both TMS and CDCl3 were found to increase with TMS concentration. From the experimental data of TMS and CDCl3 mixtures, there was strong evidence that the diffusive path of TMS molecules was obstructed by the CDCl3 molecules. The increase in the diffusion coefficients of the solutions of TMS and CDCl3 is likely a result from the decrease in the viscosity of the solutions which can be related to the absence of hydrogen bonds between the molecules (i.e., low attractive forces). It can be concluded that TMS and CDCl3 was only dependent on viscosity (or obstruction) and this method needs to be further applied to other organic solvents/TMS mixtures to investigate the effects of aggregation and obstruction due to the presence of H-bonds. PGSE NMR C60 diffusion coefficients were used to predict the presence of aggregation and obstruction at all C60 concentrations. The prediction of experimental diffusion data via the obstruction models indicates that the obstruction has a very limited effect on C60 diffusion data. Hence, the decrease in diffusion coefficient of C60 was likely only due to aggregation. Acknowledgements The authors acknowledge the facilities, and the scientific and technical assistance of the National Imaging Facility, Western Sydney University Node. References [1] S.A. Willis, G.R. Dennis, G. Zheng, W.S. Price, Hydrodynamic size and scaling relations for linear and 4 arm star PVAC studied using PGSE NMR, J. Mol. Liq. 156 (2010) 45–51. [2] M.W. Germann, T. Turner, S.A. Allison, Translational diffusion constants of the amino acids: measurement by NMR and their use in modeling the transport of peptides, J. Phys. Chem. A 111 (2007) 1452–1455. [3] G.G. Stokes, On the effect of internal friction of fluids on the motion of pendulums, Camb. Phil. Soc. Trans. 9 (1856) 8–106. [4] W. Sutherland, XVIII. Ionization, Ionic Velocities, and Atomic Sizes, Philos Mag Ser 6, 3 1902, pp. 161–177. [5] Einstein, A., Investigations on the Theory of the Brownian Movement Edited with notes by Fürth, R., Translated by Cowper, A. D., (The Five Papers of Albert Einstein from 1905–1908), Dover Publications, Inc., New York, (1956). [6] W.S. Price, NMR Studies of Translational Motion: Principles and Applications, Cambridge University Press, Cambridge, 2009. [7] A. Macchioni, G. Ciancaleoni, C. Zuccaccia, D. Zuccaccia, Determining accurate molecular sizes in solution through NMR diffusion spectroscopy, Chem. Soc. Rev. 37 (2008) 479–489. [8] A. Ellis, F.M. Zehentbauer, J. Kiefer, Probing the balance of attraction and repulsion in binary mixtures of dimethyl sulfoxide and N-alcohols, Phys. Chem. Chem. Phys. 15 (2013) 1093–1096. [9] I.A. Borin, M.S. Skaf, Molecular association between water and dimethyl sulfoxide in solution: a molecular dynamics simulation study, J. Chem. Phys. 110 (1999) 6412–6420. [10] W.S. Price, H. Ide, Y. Arata, Solution dynamics in aqueous monohydric alcohol systems, J. Phys. Chem. A. 107 (2003) 4784–4789. [11] C.A. Crutchfield, D.J. Harris, Molecular mass estimation by PFG NMR spectroscopy, J. Magn. Reson. 185 (2007) 179–182.
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