Molecular velocity auto-correlations in glycerol/water mixtures studied by NMR MGSE method
Janez Stepišnik, Carlos Mattea, Siegfried Stapf, Aleš Mohorič
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Molecular velocity auto-correlations in glycerol/water mixturesstudied by NMR MGSE method ⋆ Janez Stepišnik a , ∗ ,1 , Carlos Mattea c , Siegfried Stapf c and Aleš Mohorič a,b a University of Ljubljana, Faculty of Mathematics and Physics, Physics Department, Jadranska 19, 1000 Ljubljana, Slovenia c Dept.Technical Physics II, TU Ilmenau, 98684 Ilmenau, Germany, EU b Institute Jožef Stefan, Jamova 39, 1000 Ljubljana, Slovenia, EU
A R T I C L E I N F O
Keywords :molecular dynamicsglycerol/water mixtureNMRgradient spin-echoself-diffusionmolecular velocity auto-correlationshear rate viscosity thickeningprotein folding
A B S T R A C T
Molecular dynamics in binary mixtures of water and glycerol was studied by measuring the spec-trum of water velocity auto-correlation in the frequency range from .
05 − 10 kHz by using theNMR method of modulated gradient spin echo. The method shows that the diversity of diffusionsignature in the short spin trajectories provides information about heterogeneity of molecularmotion due to the motion in the micro-vortexes of hydrodynamic fluctuation, which is especiallypronounced for the mixtures with low glycerol content. As concentration of glycerol increasesabove vol % , a new feature of spectrum appears due to interaction of water molecules withthe clusters formed around hydrophilic glycerol molecules. New spectrum exposes a rate thick-ening of molecular friction, according to Einstein-Smoluchowski-Kubo formula, which inhibitsrapid molecular motions and creates the conditions for a slow process of spontaneously foldingof disordered poly-peptides into biologically active protein molecules when immersed in such amixture.
1. Introduction
Earliest models of liquids as totally disordered structures have been replaced by models of systems with a longrange-disordered and short range-ordered systems where molecules can associate in clustes [25, 28] due to intermolec-ular interactions [76, 58], but there is still lack of understanding of molecular dynamic that plays an important role inbiological systems. It refers primarily to liquids with hydrogen bonding such as water, alcohol, glycerol and mixturesthereof. The molecular mechanisms of these liquids, which spontaneously folds a disordered poly-peptide into theunique structure of a biologically active protein molecule, when imersed in them, are still not understood. Particularlypronounced in maintaining the structure of biologically active macromolecules and promoting protein self-assemblyare glycerol-water (G/W) mixtures [15]. Despite considerable research efforts [31, 11] these questions remain amongthe key unresolved issues in soft condensed matter physics, physical chemistry, materials science and biophysics.The G/W mixtures have already been a subject of extensive research involving thermodynamic measurements [44],broadband dielectric measurement [32, 56], NMR [17, 50, 16], infrared (IR) [15], and Raman spectroscopy [48], etc.These studies reveal changes in the aqueous structure beyond the first neighbor level, but general properties of the G/Wmixtures and how they affect the molecular dynamics are not revealed. They also show that macrophages are formed,because the glycerol is a small molecule of trihydric alcohol with large affinity to form 3D hydrogen-bonded networkpervading the bulk of mixtures [13, 54, 51]. Hydrogen bond energy between glycerol molecules ( . eV) and betweenwater ( . eV ) are lower than the binding energy between water and glycerol molecules ( . eV), leading to themolecular clusterization [19] confirmed by the dielectric measurements with almost tri-fold increase of the activationenthalpy of . mol % G/W mixture compared to the pure water at 𝐻 = 16 . kJ/mol [5].In liquids the thermal molecular motion is impeded by interactions with its neighbors. Velocity auto-correlationfunction (VAF) is a quantity containing information about the underlying processes of molecular interaction and dy-namics [36]. The VAF, which is associated with a number of physical properties, such as thermal and mass diffusion, ⋆ This document is the results of the research program funded by the Slovenian research agency, ARRS, under the program P1-0060, “Experi-mental biophysics of complex systems and imaging in bio-medicine”.All authors participated in the measurements, the first author and partly the last one participated in the analysis and interpretation of the dataand drafting the manuscript. We acknowledge also the contribution of prof. I. Serša and dr. F. Bajd from Josef Stefan Institute to assist in thepreparation of experiments on the
MHz NMR device. ∗ Corresponding author
ORCID (s): (J. Stepišnik)
First Author et al.:
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Page 1 of 14 elocity autocorrelations spectra in G/W mixtures sound propagation, transverse-wave excitation, and can have either a single-particle or a collective nature, has a pro-found meaning in the statistical physics of fluids [31, 11, 55], but it is extremely difficult to measure. Some informationcan be obtained by neutron scattering [57, 39, 4] and light scattering [45], but a short time scale of these methods can-not extracts information of its long time properties. This leads to the conclusion that the computer simulation ofmolecular dynamics is the most suitable tool for the study of translation dynamics in molecular systems [24, 13]. Thecurrent understanding of molecular dynamics in G/W mixtures is derived from experimental results in combinationwith computer simulations. Simulations can reproduce some macroscopic physical properties, but they largely dependon the chosen models. Thus, conclusions derived from models regarding molecular structure and dynamics remainuncertain [75].A lot of effort has been devoted toward understanding of the molecular translation dynamics in water, glycerol andG/W mixtures by measuring the self-diffusion coefficient, 𝐷 . Well known are the studies of molecular dynamics inliquids by measuring 𝐷 in water by tracer technique [49], the measurements in the G/W mixtures by using the inter-ferometric micro-diffusion method [52] and by NMR methods [72, 16]. The results of tracer technique are commonlyused to calibrate the diffusion measurements done by other techniques, especially those obtained by the NMR gradientspin echo method [30, 10]. This method uses the magnetic field gradient ∇ | 𝐁 | = 𝐆 (MFG) to detect the translationdisplacement of molecules via uneven precession of their atomic nuclear spins. A variant of this method, the pulsedgradient spin-echo (PGSE), provides the signal attenuation proportional to the molecular mean squared displacement(MSD) in the interval between two consecutive MFG pulses [74, 62]. However, the PGSE measurements in water atdifferent pressures and temperatures [35, 81] show values of 𝐷 scattered beyond the experimental uncertainty [49].Differences are commonly assigned to inaccurately calibrated MFG or to the convection flows in liquids. However,differences may also be due to the failure to observe properly some of experimental parameters as shown in the follow-ing. In the measurement of G/W mixture by the stimulated PGSE technique [42], only 𝐷 of water is obtained, sinceits contribution to the NMR signal is well distinguished by glycerol.The study proves the validity of the Arrhenius be-havior and the Stockes-Einstein relationship between 𝐷 and viscosity 𝜂 in the range of around room temperatures, butwith deviations at temperatures close to the glass transition. The diffusion in G/W mixtures was measured also with thePGSE-FT NMR method by measuring the NMR hydrogen signal of 𝐶𝐻 , and 𝑂𝐻 to which both water and glycerolcontribute [16]. Since the proton exchange between hydroxy groups of glycerol and water is much faster than the timeintervals of spin-echo sequence, 𝐷 of water component was calculated by taking into account numbers of 𝑂𝐻 groupsbelonging to water and glycerol molecules. The mutual diffusion coefficient of G/W mixtures was also measured bythe Gouy interferometric technique, but these results return substantially higher 𝐷 than obtained by the interferometricmicro-diffusion method [52] and lower than measured by the holographic interferometry [71]. No evident reason forthese discrepancies has been found [16].According to the Einstein definition [21], 𝐷 is a time derivative of the molecular mean squared displacement (MSD)in the long time limit. When measured in a finite time interval, as in the case of the PGSE methods, 𝐷 can exhibit timedependence, because the initial velocity of labeled molecule, 𝐯 ( 𝑡 ) , may not be forgotten fast enough. Thus, 𝐷 obtainedby this method may be different from that obtained by tracer techniques. According to the Green-Kubo formula [36]the measurement in the finite time interval gives 𝐷 𝑧𝑧 ( 𝜏 ) = ∫ 𝜏 ⟨ 𝑣 𝑧 ( 𝑡 ) 𝑣 𝑧 (0) ⟩ 𝜏 𝑑𝑡 = 2 𝜋 ∫ ∞0 𝐷 𝑧𝑧 ( 𝜔 ) 𝜏 sin( 𝜏𝜔 ) 𝜔 𝑑𝜔, (1)which is the time-dependent self-diffusion coefficient, 𝐷 𝑧𝑧 ( 𝜏 ) for the motion along 𝑧 -dirrection in the diffusion interval 𝜏 . It is related to the power spectrum i.e. the velocity autocorrelation spectrum (VAS) 𝐷 𝑧𝑧 ( 𝜔 ) 𝜏 = ∫ 𝜏 ⟨ 𝑣 𝑧 ( 𝑡 ) 𝑣 𝑧 (0) ⟩ 𝜏 cos ( 𝜔𝑡 ) 𝑑𝑡, where ⟨ ... ⟩ 𝜏 indicates the ensamble average over the particle trajectories in the interval 𝜏 .Theories [38, 78, 26, 37] and simulations [2, 3, 77] predict a long-time asymptote of the VAF as the 𝑡 −3∕2 -long timetail in liquids. Recent studies by the NMR modulated gradient spin-echo (MGSE) method [69] have shown a deviationfrom this asymptotic properties in simple liquids such as water, ethanol and glycerol due to inter-molecular interactions.In these measurements the unusual heterogeneity of molecular motion is observed, when the measurement interval isvery short, which one cannot describe by a simple diffusion coefficient. In order to enlighten these phenomena andthe function of glycerol as a colligative solute we set out to study the molecular self-diffusion in G/W mixtures by theNMR MGSE method presented in the following. First Author et al.:
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Page 2 of 14elocity autocorrelations spectra in G/W mixtures
2. NMR gradient spin-echo method
In liquids rapid molecular motion on the time scale of pico- or nanoseconds completely nullifies the spin dipole-dipole and the first order quadrupole interactions, while the spin interactions with electrons in molecular orbitals andthe electron mediated spin-spin interactions cannot be ignored. They appear as the chemical shifts and 𝐽 -couplings inthe NMR spectrum. Fluctuation of these interactions can be characterized by correlation functions of relevant phys-ical quantities and affect spin relaxation. In the gradient spin echo method, the applied MFG is commonly strongenough that the effect of fluctuation of molecular translation velocity prevails in echo attenuation over all other in-teractions. NMR gradient spin echo sequences create the spatial spin phase discord described by its wave vector, 𝐪 ( 𝑡 ) = 𝛾 ∫ 𝑡 𝐆 ( 𝑡 ′ ) 𝑓 𝜋 ( 𝑡 ′ ) 𝑑𝑡 ′ , in which 𝜋 -RF pulses switch 𝑏 ( 𝑡 ′ ) in 𝑓 𝜋 ( 𝑡 ) = ∫ 𝑡 cos ( 𝑏 ( 𝑡 ′ )) 𝑑𝑡 ′ between ± 𝜋 [66]. Wheneverthe molecular displacements within the interval of phase modulation are shorter than | 𝐪 | , the decay of signal of thespin-echo peak at the time 𝜏 can be approximated by the cumulant series in the Gaussian approximation as [63, 8, 65] 𝐸 ( 𝜏 ) = ∑ 𝑖 𝐸 𝑜𝑖 e− 𝑖𝛼 𝑖 ( 𝜏 ) − 𝛽 𝑖 ( 𝜏 ) . (2)Here the sum goes over the sub-ensembles of spins with identical dynamical properties. The phase shift 𝛼 𝑖 ( 𝜏 ) = ∫ 𝜏 𝐪 ( 𝑡 ) ⋅ ⟨ 𝐯 𝑖 ( 𝑡 ) ⟩ 𝑑𝑡, (3)can be neglected, when the averaged velocity of spin bearing particle is zero, ⟨ 𝐯 𝑖 ( 𝑡 ) ⟩ = 0 . This it is not in the case ofcollective molecular motion or diffusion in non-homogeneous systems like porous media [65].Spin-echo attenuation is given by [8] 𝛽 𝑖 ( 𝜏 ) = 1 𝜋 ∫ ∞0 𝐪 ( 𝜔, 𝜏 ) 𝐃 𝑖 ( 𝜔, 𝜏 ) 𝐪 ∗ ( 𝜔, 𝜏 ) 𝑑𝜔, (4)where 𝐪 ( 𝜔, 𝜏 ) is the the spectrum of the spin phase discord 𝐪 ( 𝑡 ) [69] and where the VAS is 𝐃 𝑖 ( 𝜔, 𝜏 ) = ∫ ∞0 ⟨ 𝐯 𝑖 ( 𝑡 ) ⊗ 𝐯 𝑖 (0) ⟩ 𝜏 cos( 𝜔𝑡 ) 𝑑𝑡. (5)According to Eq.4, the measurement of liquid by the pulsed gradient spin echo (PGSE) sequence, which consists oftwo MFG pulses of width 𝛿 and separated for Δ , gives the spin echo attenuation in liquids [68] 𝛽 𝑖 (Δ , 𝛿 ) = 𝛾 𝐺 𝜋 ∫ ∞0 𝐷 ( 𝜔 ) ( 𝜔𝛿 ∕2) sin( 𝜔 Δ∕2) 𝜔 ) 𝑑𝜔, (6)from which 𝐷 ( 𝜔 ) and thus 𝐷 ( 𝑡 ) can be extracted with considerable difficulty by changing Δ and 𝛿 if they are in therange of VAF correlation time 𝜏 𝑐 . At the present state of the art, the MFG coil induction limits the width and the shapeof the MFG pulses to above ms, which is close to or slightly above values of 𝜏 𝑐 in some liquids [69]. Thus, neglectingthe dependence of the echo decay on 𝛿 and Δ according to Eq.6, when measuring the self- diffusion in water by PGSEmethod, gives an apparent self-diffusion coefficients that may differ from one another [49, 35, 81] and also deviatefrom those obtained from the theory and the simulations of molecular dynamics and water binary mixtures [41, 23].The determination of the long time asymptotic properties of VAF in dense systems is still a challenge that canbe tackled by a NMR method, which directly probes the VAS. Such method is the modulated gradient spin echo [8],where the sequence of RF-pulses and MFG modulates the spatial dispersion of the spin phase. The method is basicallya Carr-Purcell-Meiboom-Gill sequence (CPMG) consisting of initial 𝜋 ∕2 -RF-pulse and the train of 𝑁 𝜋 -RF pulsesseparated by time intervals 𝑇 [10, 47], which was used initially to reduce the effect of diffusion in the measurementof 𝑇 relaxation. Detailed analysis shows that the sequence imprints information about VAS, when applied in thecombination with MFG [63, 64]. In first applications of MGSE method, pulsed or oscillating MFG were used tomeasure water flow through porous material [7] and molecular restricted self-diffusion in porous media [67, 9, 73, 53].It was also demonstrated how the MGSE sequence improves the resolution of the diffusion-weighted MR images of thebrain and the MRI of the diffusion tensor of neurons [1]. As in the case of PGSE, the frequency range of MGSE withthe pulsed MFG is limited to below kHz due to the gradient coil self-inductance. With the development of the MGSEtechnique in constant MFG, the frequency induction limit was avoided. New technique enables higher frequency limit, First Author et al.:
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Page 3 of 14elocity autocorrelations spectra in G/W mixtures which is now determined by the power of the RF transmitter and the magnitude of the MFG, while the lowest limit isinversely proportional the spin relaxation time. The use of the MGSE techniques with fixed MFG was first proposedin reference [8] with concern about the side effects of using RF pulses in the presence of background MFG. Analysisof adverse interferences of both fields [66, 69] shows that at suitable experimental conditions the MGSE signal decaycan be described as 𝐸 ( 𝜏, 𝜔 𝑚 ) = ∑ 𝑖 𝐸 𝑜𝑖 e− 𝜏𝑇 𝑖 − 8 𝛾 𝐺 𝜋 𝜔 𝑚 𝐷 𝑧𝑧𝑖 ( 𝜔 𝑚 , 𝜏 ) 𝜏 . (7)Here, 𝜏 = 𝑁𝑇 and 𝐷 𝑧𝑧𝑖 ( 𝜔 𝑚 , 𝜏 ) denotes the component of VAS tensor of the 𝑖 -th spin sub-ensemble in the direction ofapplied MFG at the modulation frequency 𝜔 𝑚 = 𝜋 ∕ 𝑇 averaged over the interval 𝜏 , and where 𝑇 𝑖 is the spin relaxationtime. The advantage of the new MGSE technique was demonstrated by measuring the VAS of restricted diffusion inpores smaller than 0.1 𝜇 m [66], by measuring the VAS of granular dynamics in fluidized granular systems [40] and bythe discovery of a new low frequency mode of motion in polymer melts [70]. Instead of using the externally appliedMFG, this MGSE method also allows the exploitation of MFG generated by the susceptibility differences on interfacesin porous systems to obtain information about the pore morphology and distribution of internal MFG [66].
3. Experiments
NMR spectrometer with
MHz proton Larmor frequency equipped with the Maxwell gradient coils to generateMFG in steps to the maximum of . T/m was used to measure the VAS of in G/W mixtures at room temperatures bythe MGSE method. Only water echoes are traced, as the contribution of glycerol to signal decay can be neglected dueto its slow diffusion rate and short spin relaxation time. Results were checked by repeated measurements of the samesamples on the NMR-MOUSE [6] operating at , MHz proton Larmor frequency with fixed MFG of . T/m. Thehigh magnetic field of the
MHz spectrometer allows measurements with a high signal to noise ratio, but with thetop frequency limited to kHz due to the weak MFG. On the other hand, the large MFG of one-sided magnet of theNMR MOUSE allows measurements of very slow diffusion in the frequency range up to about kHz. However, itsfix MFG limits the measurements on G/W mixtures to below 1 kHz due to fast diffusion rate of water.Samples of pure glycerol (99.5 % -Sigma-Aldrich), distillate water and mixtures with several different volume frac-tions of glycerol were prepared in plastic bottles of ml volume. Tightly sealed with a plastic lid were kept forseveral days before used for the measurements. The samples were loaded into mm long and mm wide pyrex glassampules and closed with paraffin tape to be inserted in the head of the MHz NMR spectrometer. While the flatshelf of one-sided magnet of the NMR-MOUSE allows the measurements of mixtures in bottles without the loadingstep. In order to avoid a possible impact of restricted diffusion, containers with the diameter much larger than themolecular displacements in the time of ms long intervals of measurement were used. In addition, the effect of re-stricted diffusion is even further reduced by the initial 𝜋 ∕2 -RF pulse of the MGSE sequence applied in the backgroundof MFG, because it excites only a few mm narrow slice of sample.In the measurements on the MHz NMR device, the amplitudes of echo peaks are recorded in the time intervalof ms. Measurements are repeated with different 𝑇 and magnitude of MFG, 𝐺 , in a way to keep the product 𝑇 × 𝐺 = 𝑐𝑜𝑛𝑠𝑡 . Fig.1 shows that the number of recorded peaks increases with the shortening of 𝑇 . In this way theexponential decay of 𝑁 -echo peaks is effected only by the changes of 𝐷 ( 𝜔, 𝜏 ) at the frequency 𝜔 = 𝜋 ∕ 𝑇 , where 𝜏 denotes average over the interval 𝜏 = 𝑁𝑇 . The contribution of spin relaxation to the signal decay is determined byseparate measurements in zero MFG as shown in the inset picture in the Fig.1. The spin relaxation exhibits almosta clear mono-exponential decay, except in the initial interval of a few milliseconds, with the rapid decay believed tobelong to water bounded in clusters..
4. Results and discussion
Data was analyzed in the following way. A fifth order polynomial is fitted to the series of echo peak amplitudeswith given 𝑇 as a function of time 𝜏 . The time derivative of this fit represents the diffusion spectrum if the the the spinrelaxation is accounted for by proper normalization of the echo decay. The MGSE measurements of G/W mixtureson the MHz NMR device give time dependencies of spin-echoes 𝐸 ( 𝜏 ) that deviate from the anticipated mono-exponential decays particularly in pure water and G/W mixtures with lower glycerol contents. The deviations are First Author et al.:
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Page 4 of 14elocity autocorrelations spectra in G/W mixtures
Figure 1:
MGSE spin echo decays in a mixture of vol % glycerol in water at different modulation frequencies ( 𝜔 = 𝜋 ∕ 𝑇 ).The inserted images show: A) the signal decay in zero MFG and B) MGSE sequence with RF pulses (blue) and MFG(green) giving spin echoes (black) whose peaks contain information about molecular motion. clearly visible in the 3D frequency/temporal plots 𝐷 ( 𝜔, 𝜏 ) of the time derivatives of curves obtained by fitting the echopeaks with the coefficient of determination of 𝑅 > . shown in Fig.2 and Fig.6. Instead of constant value alongthe echo time, 𝜏 , anticipated for the mono-exponential decays, discrepancies occur that cause the 3D plots to show acurved surface at short intervals 𝜏 and of higher frequencies. In the case of simple liquids [69], we interpreted similarcurved surface by the molecular diffusion diversity due to the motion in the vortexes of hydrodynamic fluctuations.Namely, during initial interval after the spin excitation, when the trajectories of the molecules are still short, the spinsmay observe local inhomogeneity caused by the distribution of internal MFG in the porous medium [69] or motionaldiversity due to fluxes or hydrodynamic oscillation in liquids [66]. By grouping the spin bearing particles into separatesub-ensembles corresponding to spins with the different spin-echo attenuation, we can describe the induction signalwith the distribution function 𝑃 ( 𝐷 ) as 𝐸 ( 𝜏 ) = ∫ 𝑃 ( 𝐷 ) 𝑒 − 𝑠𝐷𝜏 𝑑𝐷 with 𝑠 = 𝛾 𝐺 𝜋 𝜔 given in Eq.7. In the case of narrowdistribution, the signal attenuation can be approximated by log 𝐸 = 𝛽 ( 𝜏 ) ≈ − 𝜏 ∕ 𝑇 − 𝑠 ⟨ 𝐷 ⟩ 𝜏 + 𝑠 ⟨ Δ 𝐷 ⟩ 𝜏 + ... (8)Here ⟨ 𝐷 ⟩ is the mean diffusion coefficient and ⟨ Δ 𝐷 ⟩ is the variance of distribution. In the cases of a non-exponentialdecay the time derivative of echo attenuation does not give VAS, but some apparent one, 𝐷 𝑎𝑝𝑝 ( 𝜔, 𝜏 ) , which conveysinformation on the mean diffusion coefficient and its distribution. 3D frequency/temporal plots of 𝐷 𝑎𝑝𝑝 ( 𝜔, 𝜏 ) for purewater and for G/W mixtures with . , . and . volume fraction of glycerol are shown in Fig.2 and for the G/Wmixtures with . , . , . and . volume fraction of glycerol in Fig.5. In water and mixtures with the lowerglycerol content a curved surface is evident in the 3D plots of spectra in the range of short 𝜏 and at higher frequenciesdue to the distribution of the self-diffusion coefficients. The part of curved surface is clearly visible in the contourplot of the second derivative of 𝛽 ( 𝜏 ) of the mixture with vol % of glycerol content in Fig.3, which shows how ⟨ Δ 𝐷 ⟩ occurs at high modulation frequencies and disappears with increasing echo-time at 𝜏 > ms, when the trajectoriesof spins become long enough to span the whole extend of heterogeneity and to average off the diffusion diversity into ⟨ Δ 𝐷 ⟩ = 0 , what is also seen in Figs.2 and 6. Similar diversity was observed in the MGSE measurements of VAS First Author et al.:
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Page 5 of 14elocity autocorrelations spectra in G/W mixtures
Figure 2:
3D temporal-frequency plots of apparent VAS for the G/W mixtures with vol % , vol % , vol % , vol % ofglycerol. in water, ethanol and toluene with the explanation of the molecular self-diffusion in the vortexes of hydrodynamicfluctuations [69].By increasing the glycerol content, the curved surface of spectra is reduced, but in the mixtures with ≥ vol % ofglycerol a new spectral feature appears in the form of a low frequency ridge that levels at higher frequencies. In orderto enhance visibility of spectral changes and to avoid the part affected by hydrodynamic fluctuation, where ⟨ Δ 𝐷 ⟩ ≠ ,we present in Fig.4 the VAS at echo times 𝜏 > ms for the mixtures with . , . , . and . vol % of glycerol.The spectrum of mixture with vol % of glycerol has a form similar to that of pure water, only shifted up-wards byabout . The upwards shift indicates a disruption of hydrogen bonding in water caused by a low glycerol contentin water. A drastic change of spectra appears at the concentrations of and vol % of glycerol indicating new typeof inter-molecular interaction, which has a strong impact on the molecular dynamics. 3D presentations of spectra inFigs.2 and 5 clearly show how the increase of glycerol content lowers the spectrum level, reduces the spectral hump,i.e. the curved surface at short 𝜏 , which is attributed to the bulk water. The low frequency ridge of the new spectrum,which starts to appear at vol % of glycerol, remains almost unchanged with the increase of glycerol content.In order to ensure the validity of results obtained by the high frequency NMR device, the measurements of the sam-ples were repeated by using the . MHz NMR MOUSE device. Its strong and fixed MFG enables the measurementof samples up to about kHz, but with the lower S/N ratio due to lower Larmor frequency. It also does not permitsthe measurements of water and G/W mixtures below kHz, due to excessive spin echo attenuation. Fig.6 shows theresults of measurements on both devices, which match well in the overlapping frequency range. In references[25, 28, 76, 58] a formation of clusters in the G/W mixtures was explained by the energies of bondingbetween water molecules and between glycerol molecules, which are lower than between water and glycerol molecules.Thus, the new spectral features of G/W mixture can be attributed to the formation of clusters around the hydrofilic glyc-erol molecules, and this interaction with unbound water changes molecular dynamics. The disappearance of spectralhump with the increase of glycerol content indicates the depletion of the free water basin through the formation of newclusters. We believe that there are two contributions to the mixture VAS, when measuring proton signal by MGSEmethod: The contribution of free water that is not in contact with the clusters, and whose content decreases with the
First Author et al.:
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Figure 3:
The contour plot of the distribution variance of the diffusion coefficient, ⟨ Δ 𝐷 ⟩ , for the G/W mixture with vol % of glycerol showing the diversity of molecular motion observed only at short observation times and at high modulationfrequencies. increase of glycerol concentration, and the water that interacts with the clusters or even could exchange with the clus-tered water. The VAS of pure glycerol and water bounded in the clusters cannot be detected due to a slow diffusionrate [69] and the short relaxation time, and is only partially observed in Fig. 1 for the spin-echo decays in zero MFG.At room temperatures the self-diffusion coefficient of G/W mixtures does not deviate significantly from Arrhenius’slaw [18], allowing to consider the molecular diffusion as the motion of particles occasionally caught up in potentialwells created by their neighbors. Quite commonly the molecular dynamics is described by the Langevin equations,where the effect of the environment is taken into account through the fluctuating and frictional forces. The methodhas been widely used in the study of structural and thermal properties of matters in different phases [60]. However,the inter-molecular interaction and the coupling to other degrees of freedom are difficult to effectively include into thedescription. The description of interaction between water-water, water-glycerol and glycerol-glycerol molecules witha simple Lennard-Jones potential turned out to be unsuccessful even though the potential is modified by the bifurcationof the single minimum into two or more minima [61]. However, the results of MGSE measurements in ordinary fluids[69] showed that the results can be successfully explained by Langevin equations (LE), in which we consider thatmolecular motion at high temperatures alters molecular interactions to such an extent that they can be approximated byharmonic interaction [79]. This means that in mutual collisions, the molecules occasionally get caught in a harmonicwell, which can be described by a constant 𝑘 and with a minimum at 𝑎 ( 𝑡 ) between 𝑥 𝑖 and 𝑥 𝑗 locations of collidingmolecules. This allows to describe the molecular dynamics by the set of coupled LE 𝑚 𝑖 𝑑 𝑥 𝑖 𝑑𝑡 + 𝛾 𝑑𝑥 𝑖 𝑑𝑡 + 𝑘 𝑛 ∑ 𝑗 ≠ 𝑖 ( 𝑥 𝑖 − 𝑥 𝑗 − 𝑎 ( 𝑡 )) = 𝑓 𝑖 ( 𝑡 ) (9)in which the 𝑖 -th particle is coupled to 𝑛 of its closest neighbors. Here, 𝑚 𝑖 is the particle mass, 𝛾 is the frictioncoefficient, and 𝑓 𝑖 is the random force. The friction and the random force represent two consequences of the samephysical phenomenon and are interrelated [36] 𝛾 ( 𝜔 ) = 1 𝑘 𝐵 𝑇 ∫ ∞0 ⟨ 𝑓 ( 𝑡 ) 𝑓 (0) ⟩ exp ( 𝑖𝜔𝑡 ) 𝑑𝑡, (10)where 𝑘 𝐵 is the Boltzmann constant. Neglecting the fluctuation of 𝑎 ( 𝑡 ) and the inertial terms of LE at low frequencies , First Author et al.:
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Figure 4:
The dots show VAS of G/W mixtures measured at echo times 𝜏 > ms and the fitting curves correspond to theharmonic couplings of diffusing particles described with Eq.11 for vol % and with Eq.12 for vol % of glycerol content . when 𝜔 < 𝑘 ∕ 𝑚 𝑖 , and assuming that all particles are subjected to the same friction 𝛾 ( 𝜔 ) ≈ 𝛾 = 𝑘 𝐵 𝑇 ⟨| 𝑓 | ⟩ , the solutiongives the VAS of harmonically coupled water molecules, 𝐷 𝑤 ( 𝜔 ) , in the form 𝐷 𝑤 ( 𝜔 ) = 𝑘 𝐵 𝑇𝛾 𝑛 + 𝜏 𝑐 𝜔 𝑛 + 𝜏 𝑐 𝜔 . (11)where the correlation time is 𝜏 𝑐 = 𝛾 ∕ 𝑘 . At zero frequency, the Einstein diffusion coefficient, 𝐷 𝑤 (0) = 𝑘 𝐵 𝑇𝑛𝛾 , depends onthe number of coupled molecules, while at high frequencies, 𝐷 𝑤 (∞) = 𝑘 𝐵 𝑇𝛾 is the diffusion rate of molecules escapingthe inter-molecular capturing. This formula provides a good fit to the VAS of pure water, glycerol and ethanol [69]as well also to the results of our measurements for the VAS of G/W mixture with . vol % glycerol shown in Fig.4.However, 𝐷 𝑤 ( 𝜔 ) cannot fit to the VAS of G/W mixture with the glycerol concentrations equal to or greater than vol % as shown in the same figure. In the latter case, one must consider the interactions of water with the water clusteredaround glycerol. With a slight exaggeration that water molecules and clusters experience the same friction, 𝛾 , that thecoupling constant between small and large particles is the same 𝑘 , and given that the mass of clusters is large enoughthat the inertial term in their Langevin equations cannot be neglected at low frequencies, i.e. 𝜔 ≈ 𝛾 ∕ 𝑀 , the solutionfor the system of 𝑛 𝑤 light water molecules and 𝑛 𝑐 heavy clusters gives the VAS for the water molecules interactingwith clusters as 𝐷 𝑤𝑐 ( 𝜔 ) = 𝑘 𝐵 𝑇𝛾𝑛 𝑤 ⎛⎜⎜⎜⎝ ( 𝑛 𝑤 − 1 ) 𝜏 𝜔 ( 𝑛 𝑐 + 𝑛 𝑤 ) + 𝜏 𝜔 + 𝑛 𝑐 𝑛 𝑤 + 𝜏 𝜔 + ( 𝑛 𝑤 − 𝜔 𝜔 𝑜 ) ( 𝑛 𝑐 + 𝑛 𝑤 ) 𝑛 𝑐 + 𝜏 𝜔 + 𝑛 𝑐 𝜔 𝜏 𝜔 𝑜 + ( 𝑛 𝑤 − 𝜔 𝜔 𝑜 ) ⎞⎟⎟⎟⎠ , (12)and the VAS for the clusters interacting with water as 𝐷 𝑐𝑤 ( 𝜔 ) = 𝑘 𝐵 𝑇𝛾𝑛 𝑐 ⎛⎜⎜⎜⎝ ( 𝑛 𝑐 − 1 ) 𝜏 𝜔 ( 𝜔 𝜔 𝑜 − 𝑛 𝑐 − 𝑛 𝑤 ) + 𝜏 𝜔 + ( 𝑛 𝑐 + 𝑛 𝑤 ) 𝑛 𝑤 + 𝜏 𝜔 ( 𝑛 𝑐 + 𝑛 𝑤 ) 𝑛 𝑐 + 𝜏 𝜔 + 𝑛 𝑐 𝜔 𝜏 𝜔 𝑜 + ( 𝑛 𝑤 − 𝜔 𝜔 𝑜 ) ⎞⎟⎟⎟⎠ , (13)where 𝜔 𝑜 = √ 𝑘 ∕ 𝑀 . This may be an excessive simplification, but allows the calculation of results that serve at leastqualitatively to compare with our experimental results. Fig.4 shows a good fit of 𝐷 𝑤𝑐 ( 𝜔 ) to the data for the VASof mixtures with vol % of glycerol, if assuming that each water molecule interacts with one of adjacent aqueousmolecules and one cluster and that the interaction between clusters is neglected . First Author et al.:
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Page 8 of 14elocity autocorrelations spectra in G/W mixtures
Figure 5:
3D temporal-frequency plots of apparent VAS for the G/W mixtures with vol % , vol % , vol % , vol % ofglycerol Figure 6:
VAS of G/W mixtures obtained by MGSE method on the
MHz NMR set-up (frequencies below kHz) andon the . MHz NMR MOUSE (frequencies above kHz) match well in the overlapping region. As shown in references [14, 43] the self diffusion coefficient of G/W mixture decreases and the viscosity increaseswith decreasing temperature according the Stokes-Einstein formula (SE) in a wider range around room temperature.This formula is derived from the Einstein-Smoluchowski (ES) relation [20, 59] 𝐷 = 𝑘 𝐵 𝑇𝛾 (14) First Author et al.:
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Page 9 of 14elocity autocorrelations spectra in G/W mixtures
Figure 7:
The size of the hydrodynamic radius of the diffusing molecules, determined according to the SE formula, provesthat only the properties of water molecules in the G/W mixture were observed. by relating the fluid viscosity 𝜂 to the friction as 𝛾 = 𝑠𝜋𝑅𝜂 , which is derived from the Navier-Stokes equations forthe spherical objects of radius 𝑅 moving with small Reynolds numbers in a fluid. Here, the factor 𝑠 depends onthe boundary condition and is equal to 4 for the slip boundary condition and 6 for the stick boundaries. Thus, thehydrodynamic radius of a molecule 𝑅 may be derived directly from the diffusion coefficient using the SE relation, ifthe viscosity of solution is known. Fig.7 shows the dependence of water hydrodynamic radius on the glycerol contentobtained by using the diffusion coefficients derived from the MGSE measurement, shown in Fig.8, and the viscosityof G/W mixture from the reference [12] with the assumption of slip boundaries. The values are close to the commonlyaccepted radius of water, . nm [81] in the range of higher glycerol content, but slightly lower at low glycerol content,which can be attributed to the effect of hydrodynamic fluctuations.In the generalized LE, hydrodynamic interactions and the correlation between random forces at different time aretaken into account by introducing friction forces with a memory kernel, meaning that the friction acting on the particledepends on the velocity at an earlier time 𝛾 ( 𝑡 ) [36] giving the VAS in the form 𝐷 ( 𝜔 ) = 𝑅𝑒 𝑘 𝐵 𝑇𝑖𝜔𝑚 + 𝛾 ( 𝜔 ) . (15)which can be considered as a generalized ES equation, in the range of low frequencies, 𝜔 < 𝛾 ( 𝜔 )∕ 𝑚 , or a generalizedSE relation, if friction is expressed by a shear rate viscosity 𝜂 ( 𝜔 ) , 𝛾 ( 𝜔 ) = 𝑠𝜋𝑅𝜂 ( 𝜔 ) . Generalized SE relation was usedto obtain the viscoelastic module of a complex fluid from the microscopic motion of small particles [46] in order tounderstanding the swimming of microorganisms, and the sedimentation in fluids. While the generalized SE gives goodestimates for the motion of larger objects, its use for the molecular diffusion could lead to a systematic failure. Suchmodels tend to severely underestimate molecular radius 𝑅 from the diffusion coefficients or vice versa [22]. Since thehydrodynamic radius of small molecules is weakly dependent on the flexibility and density of the molecular structure,we still can assume the proportionality between the frequency dependent molecular friction and the shear rate viscosityin simple liquids. Thus, the inverse proportionality between the shear rate viscosity and the VAS, 𝜂 ( 𝜔 ) ≈ 𝐷 ( 𝜔 ) −1 ,reveals the shear rate thickening of viscosity of our mixtures, which appears at the glycerol contents equal to or largerthan vol % .Our interpretation with the simplified LE confirms that the thickening is a consequence of water interaction with thehydro-cluster formed around glycerol molecules. It corresponds to the common understanding that the shear thickeningviscosity is related to the presence of ”clumps” in liquids [34]. First Author et al.:
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Page 10 of 14elocity autocorrelations spectra in G/W mixtures
Figure 8:
The diffusion coefficient of G/W mixtures measured by our MGSE method, presented together with the resultsof PGSE measurements from the reference [16].
5. Conclusion
In Fig.8 the diffusion coefficient obtained by the PGSE method as reported in article [16] and obtained here byMGSE method are compared. In MGSE method the VAS at the decay time 𝜏 = 70 ms and at the modulation frequency 𝜈 = 3 kHz is taken as a self-diffusion coefficient in order to avoid the part of spectrum with the humped surfacebelonging to the diffusion diversity created by the hydrodynamic fluctuations and also to be outside the low frequencyspectral ridge, since its value cannot be exactly determined due to the low-frequency limit of the method. The results ofboth measurements match only in the cases of pure water and pure glycerol. The differences may be due to the differentinterpretation of spin-echo decay, given in Eq.6 for the PGSE and Eq.7 for the MGSE method. The article [16] doesnot mention the width of the MFG pulses in their experiments. This value is crucial for the correct analysis of thePGSE measurements. In addition, the differences may also appear because the proportions of glycerol, bounded andunbounded water in the hydroxyl NMR spectral line were not properly estimated in determination of water 𝐷 [16].Unlike in the PGSE method, in the processing of the MGSE spin-echo decay no calibration is required to match resultsof other methods, but one only needs to know the exact value of the MFG according to Eq.7.Study of molecular dynamics in G/W mixture by the NMR MGSE method unveils the low-frequency feature ofwater VAS with two indications: The small amounts of glycerol in water only partially weakens the hydrogen bondingnetwork in water and thus increases the diffusion rate, and that the glycerol content equal or higher than vol % brings about a new feature of VAS, which is the consequence of water interaction with the hydro-clusters formedaround hydrophilic glycerol molecules. These interactions strongly influence the translational molecular dynamics inliquid resulting in the shear rate thickening of viscosity of water in mixtures. The shear rate thickening of viscosityalters the dynamics of other molecules if immersed in such a liquid. Instead of using the SE relation, the effect ofa liquid on a submerged molecule one can treat it more correctly with the generalized EC equation, in which thefriction spectrum of friction embodies the interaction of a molecule in a fluid. With this in mind, instead of the shearrate viscosity thickening, we can talk about the rate thickening of the friction coefficient in liquids, which causes anenvironment where rapid molecular motions and collisions are strongly inhibited.It is well known that any protein exists as an unfolded polypeptide or random coil when translated from a sequenceof mRNA to a linear chain of amino acids. Protein folding is the physical process by which a protein chain acquiresits native 3-dimensional structure, a conformation that is biologically functional. A large number of experimentaland simulation studies tested whether folding reactions are diffusion-controlled, whether the solvent is the source ofthe reaction friction, and whether the friction-dependence of folding rates generally can provide insight into folding First Author et al.:
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Page 11 of 14elocity autocorrelations spectra in G/W mixtures dynamics [33, 29], but these some simple questions still remain unanswered. Simulations and theory provide someinsight into possible physical mechanisms of internal friction, but there are no experimental demonstrations of theseideas. An answer to this dilemma could be that in suitable solvents such as the G/W mixtures, the rate thickening ofthe friction coefficient dampens rapid molecular motion and collisions between molecules to create a condition forthe slow process of spontaneously folding of disordered poly-peptides into biologically active protein molecules whenimmersed in them [27, 80].As in the cases of MGSE measurements of pure liquids [69], the VAS of G/W mixtures with the low glycerol contentshow a similarity of the diffusion diversity explained by the molecular self-diffusion in the vortexes of hydrodynamicfluctuation, which disappears at higher glycerol concentrations. 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