J. Chem. Phys.

Role of Internal Motions and Molecular Geometry on the NMR Relaxation ofHydrocarbons

P. M. Singer, D. Asthagiri, Z. Chen, Arjun V. Parambathu, G. J. Hirasaki, and W. G. ChapmanDepartment of Chemical and Biomolecular Engineering,

Rice University, 6100 Main St., Houston, TX 77005, USA(Dated: January 25, 2018)

The role of internal motions and molecular geometry on 1H NMR relaxation times T1,2 in hydro-carbons is investigated using MD (molecular dynamics) simulations of the autocorrelation functionsfor intramolecular GR(t) and intermolecular GT (t) 1H-1H dipole-dipole interactions arising fromrotational (R) and translational (T ) diffusion, respectively. We show that molecules with increasedmolecular symmetry such as neopentane, benzene, and isooctane show better agreement with tra-ditional hard-sphere models than their corresponding straight-chain n-alkane, and furthermore thatspherically-symmetric neopentane agrees well with the Stokes-Einstein theory. The influence ofinternal motions on the dynamics and T1,2 relaxation of n-alkanes are investigated by simulatingrigid n-alkanes and comparing with flexible (i.e. non-rigid) n-alkanes. Internal motions cause therotational and translational correlation-times τR,T to get significantly shorter and the relaxationtimes T1,2 to get significantly longer, especially for longer-chain n-alkanes. Site-by-site simulationsof 1H’s along the chains indicate significant variations in τR,T and T1,2 across the chain, especiallyfor longer-chain n-alkanes. The extent of the stretched (i.e. multi-exponential) decay in the au-tocorrelation functions GR,T (t) are quantified using inverse Laplace transforms, for both rigid andflexible molecules, and on a site-by-site bases. Comparison of T1,2 measurements with the site-by-site simulations indicate that cross-relaxation (partially) averages-out the variations in τR,T and T1,2

across the chain of long-chain n-alkanes. This work also has implications on the role of nano-poreconfinement on the NMR relaxation of fluids in the organic-matter pores of kerogen and bitumen.

Keywords: Intramolecular relaxation, Intermolecular relaxation, BPP theory, Rigid molecules, Cross-relaxation

I. INTRODUCTION

Recent studies have shown that MD (molecular dy-namics) simulations can successfully predict 1H NMR(nuclear magnetic resonance) relaxation times T1,2 anddiffusion coefficients of bulk hydrocarbons and water,without any adjustable parameters in the interpreta-tion of the simulation data [1]. Besides validatingthe forcefields used in the simulations, the MD simu-lations reveal new insight about the NMR relaxationfrom intramolecular versus intermolecular 1H-1H dipole-dipole interactions in fluids, which are not easily ac-cessible experimentally [2]. For instance, the simula-tions quantify the relative strength of the two relaxationmechanisms, indicating that intramolecular increasinglydominates over intermolecular relaxation with increasingmolecular chain-length (i.e. increasing carbon number).This validates the common practice of only consideringintramolecular dipole-dipole interactions for simulationsof macromolecules such as proteins [3] or polymers in theshort-time regime [4].

However, as also reported in [1], the functionalforms of the simulated autocorrelation functions forintramolecular GR(t) (i.e. rotational) and intermolecularGT (t) (i.e. translational) 1H-1H dipole-dipole interac-tions show significant deviations from the traditionalhard-sphere models by Bloembergen, Purcell, Pound(BPP) [5] and Torrey [6], respectively. In the caseof intramolecular interactions, the BPP model predictsa single-exponential decay for GR(t) with rotational-

correlation time τR however, the simulations clearly indi-cate an increasingly “stretched” (i.e. multi-exponential)decay with increasing chain-length. For intermolecularinteractions, the Torrey model predicts a specific func-tional form for GT (t) (and an associated translational-correlation time τT ), however, the simulations also indi-cate a “stretched” decay at large chain-lengths. Anotherprediction from the hard-sphere models [5, 6] is that theratio of translational-diffusion correlation-time τD (whereτD = 5

2τT ) [7] to rotational-correlation time τR should beτD/τR = 9 [8], which we show in this report is the case forspherically-symmetric neopentane. However, the simula-tions clearly show that τD/τR � 9 at large chain-lengths[1]. These findings indicate that the Stokes-Einstein ra-dius for rotational and translational motion are compara-ble for short-chain and spherical alkanes, but increasinglydiverge with increasing chain length, indicating limita-tions of ascribing a single “radius” for chain molecules.

While the molecular dynamics of long-chain n-alkanesare expected to deviate from hard-sphere predictions, theunderlying theory for the deviations remain elusive anddifficult to verify experimentally. Simplified models forthe autocorrelation function of non-spherical and non-rigid molecules have been successfully developed in thepast, such as those by Woessner and others which de-scribe spin-relaxation processes in two-proton systemsundergoing anisotropic reorientation [9–11] and internalmotions [12], as well as three-proton systems [13]. Theseanisotropic reorientation models have been successfulfor highly symmetric molecules such as benzene, where

arX

iv:1

801.

0780

3v1

[ph

ysic

s.ch

em-p

h] 2

3 Ja

n 20

18

2

in-plane and perpendicular-to-plane rotational diffusion-coefficients (and rotational correlation-times) can be de-termined from NMR measurements and MD simulations[14, 15]. In the case of intramolecular (i.e. rotational) re-laxation for molecules of lower symmetry, such theoriesresult in a multi-exponential decay for the autocorrela-tion function GR(t), with a corresponding distribution incorrelation times. Generalizations of such models for theautocorrelation function later arose which take a “modelfree” or heuristic approach to the internal motions, suchas the Lipari-Szabo model [16, 17] often used in poly-mers [18] and proteins [19]. However, as the internalmotions of the molecule become more complex, one isinvariably forced to develop even more heuristic modelsof the autocorrelation function and its associated spec-tral density [20, 21]. Such heuristic approaches include avariety of distribution functions for the underlying cor-relation times, including the Cole-Davidson distributionfunction [4, 22], the generalized gamma function [18], theKohlrausch-Williams-Watts [4] functions, and the Singer-Hirasaki function [23], each chosen to fit the observa-tions, but without theoretical justification. In the case ofbulk water (which exhibits hydrogen-bonding), GR(t) isstretched to a similar extent as n-pentane [1], which maybe a result of large, discrete angular-jumps [24] superim-posed on a continuous-time rotational-diffusion process[25].

In this report, we study the influence of internal mo-tions and molecular geometry on the molecular dynamicsof hydrocarbons by simulating the autocorrelation func-tions GR,T (t) of rigid n-alkanes (i.e. without internalmotions), compared with flexible n-alkanes (i.e. with in-ternal motions). While such rigid n-alkanes do not ex-ist in nature, they present an ideal testing ground forsimulating the influence of internal motions on correla-tion times and relaxation times. We also report site-by-site simulations GR,T (t) for the 1H’s across the chain,thereby quantifying the underlying distribution in cor-relation times τR,T and relaxation times T1,2 across themolecule. The simulations of both the rigid moleculesand the site-by-site 1H’s reveal key insights about thefunctional forms of GR,T (t) as a function of chain length,without invoking any heuristic models. The site-by-sitesimulations are also compared with T1,2 measurementsin the case of n-decane and n-heptadecane, which showthat cross-relaxation [26, 27] partially (in the case of n-heptadecane) averages out the underlying variations inτR,T and T1,2. Such comparisons between measurementand site-by-site simulation could in principle be used asa new method for determining the cross-relaxation ratesat low magnetic-fields, provided the proper theoreticalframework is developed.

Besides addressing the fundamental science of molec-ular dynamics of bulk fluids, the present work alsoopens up new opportunities for investigating the effectsof nanometer confinement on fluids, such as the light(i.e. low-viscosity) hydrocarbons confined in the organic-matter pores of the kerogen and bitumen typically found

in organic-rich shale [28, 29]. There is increasing evi-dence that the NMR surface relaxation of the light hy-drocarbons confined in such organic nano-pores is domi-nated by intramolecular [30] and intermolecular [31] 1H-1H dipole-dipole interactions, as opposed to surface para-magnetism. Provided the molecular dynamics of the bulkfluid is well understood, the MD simulations of 1H-1Hdipole-dipole interactions can then in principle be usedto characterize the complex NMR response of fluids con-fined in organic nano-pores [30–50], which as of yet isnot well understood. The effects of nano-confinement onfluids in organic shale can also affect the phase behav-ior and partitioning of components between the matrixand production fractures as a result of the strong in-teraction between the fluid molecules and pore surface.It is clear that the fundamental understanding of suchcomplex systems would significantly improve by integrat-ing NMR measurements, MD simulations, and molecularDFT (density functional theory) [51] techniques.

The rest of the article is organized as follows. SectionII presents the methodology including: the hydrocarbonsinvestigated and labeled in Section II A, the MD simula-tion background in Section II B, and the autocorrelationfunction and NMR relaxation background in Section II C.Section III presents the results and discussions includ-ing: the symmetric molecules and hard-sphere modelsin Section III A, the internal motions in flexible versusrigid molecules in Section III B, the site-by-site simu-lations and distribution in correlation times in SectionIII C, including the cross-relaxation effects and compar-ison with measurements in Section III C 1. Conclusionsare presented in Section IV.

II. METHODOLOGY

This section introduces the hydrocarbons under inves-tigation, and presents a brief review of the MD simu-lations, the definition and properties of the autocorrela-tion functions, and, the derivation of the NMR relaxationtimes. Further information about the methodology canbe found in [1].

A. Hydrocarbons investigated and labeled

The hydrocarbons simulated in this report are shownin Fig. 1. Neopentane (C5H12) refers to the isomer2,2-dimethylpropane, and is labeled “oC5” for short; itis spherically symmetric, and all 1H’s are topologicallyequivalent. Benzene (C6H6) is labeled as “aC6” forshort; it is planar symmetric, and all 1H’s are equiva-lent. Cyclohexane (C6H12) is labeled as “cC6” for short;it contains two inequivalent 1H’s sites, namely axial andequatorial. Isooctane (C8H18) refers to the isomer 2,2,4-trimethylpentane, and is labeled “iC8” for short; it ismore spherical than the normal isomer n-octane (or “nC8for short), with many inequivalent 1H’s. The other n-

3

#2

#1

#3

#4

#5

#6

#7

#8

#9

#10

benzene (aC6) cyclohexane (cC6)neopentane (oC5)

n-heptadecane (nC17)

#2

#1

#3

#4

#5

#6

n-decane (nC10) isooctane (iC8)

FIG. 1. Ball-and-stick illustrations of the hydrocarbons sim-ulated in this report, including (in order of increasing mo-lar mass): (1) neopentane C5H12 (2,2-dimethylpropane) or“oC5” for short, (2) benzene C6H6 or “aC6” for short, (3)cyclohexane C6H12 or “cC6” for short, (4) isooctane C8H18

(2,2,4-trimethylpentane) or “iC8” for short, (5) n-decane n-C10H22 or “nC10” for short, and (6) n-heptadecane n-C17H36

or “nC17” for short. n-decane and n-heptadecane also showlabels for the inequivalent 1H’s along the chain. The othern-alkanes (not shown) are labeled nC# for carbon numberC#.

alkanes (not shown) are labeled nC# for carbon numberC#.

The site-by-site simulations are conducted on n-decane (n-C10H22) labeled as “nC10” for short, and n-heptadecane (n-C17H36) labeled as “nC17” for short. Inthe case of n-decane, sites #2 ↔ #6 have a degener-acy of 4, while #1 has a degeneracy of 2. In the caseof n-heptadecane, sites #2 ↔ #9 have a degeneracy of4, while #1 and #10 have a degeneracy of 2. Thesedegeneracies are used to compute the weighted averageof the autocorrelation functions. The site-by-site simula-tions are also conducted on rigid n-decane, with the samelabels as flexible n-decane.

B. Molecular dynamics simulation

The MD simulations of all the flexible hydrocarbonswere performed using NAMD [52] version 2.11. The bulkalkanes were modeled using the CHARMM General Forcefield (CGenFF) [53]. The protocol for setting-up the ini-tial simulation configuration was exactly as before [1].We use the n-alkane simulation trajectory from the ear-lier work for the analysis noted below. For the cyclicalkanes and isooctane, as before, we created the initialsimulation system by packing N copies of the moleculeinto a cube of volume L3 using the Packmol program

[54]. The volume was chosen such that the number den-sity N/V corresponds to the experimentally determinednumber density at 293.15 K. The simulation approach forthese systems using NAMD was as before [1].

All the rigid body simulations are performed withinLAMMPS [55]. For the rigid n-alkanes, we took the ini-tial configuration from our earlier work, i.e. the config-uration obtained after the Packmol packing procedure.At this stage, by construction all the n-alkanes are in thefully extended (all-trans) configuration. Then using theCHARMM-to-LAMMPS tool from the Enhanced MonteCarlo package [56], we prepared the molecular config-uration and forcefield information into a LAMMPS in-put data file. To remove potential steric overlaps fromthe packing procedure, within LAMMPS we first run 30steps of constant energy molecular dynamics using thenve/limit 0.1 option. (The limit option enforces themaximum distance a particle can move and allows thesimulation to proceed despite possible overlaps.) Ini-tial velocities for the MD simulation are obtained with aGaussian distribution and are adjusted to give a temper-ature of 293.15 K. We find that 30 steps is more thansufficient to remove any potential steric overlaps andalso avoid distorting the geometry of the alkanes fromtheir original all-trans configuration. After this dynamicsstep, we run rigid body molecular dynamics simulationsusing the rigid/nve/small molecule option withinLAMMPS. The system is thermostated using a Langevinthermostat with a damping coefficient of 5 ps−1. The cut-off within LAMMPS was exactly as it was in our earlierNAMD runs for the flexible molecules. Specifically, theLennard-Jones interactions were terminated at 14.00 Aby smoothly switching to zero starting at 13.00 A. Thetime step for integrating the equations of motion is 1 fsand the equilibration phase was over 1 ns. In the subse-quent production phase, we remove the thermostat. Theproduction phase lasted at least 1 ns and configurationsare saved every 100 steps to obtain at least 104 framesfor analysis. As before [1], the average temperature inthe rigid body simulations is within ±3 K of the targettemperature of 293.15 K (20oC).

C. Autocorrelation function and NMR relaxation

The theory of NMR relaxation in liquids is well known[5–8, 57, 58], and the expressions for the autocorrela-tion functions for isotropic intramolecular GR(t) andintermolecular GT (t) 1H-1H dipole-dipole interactionsare derived in Ref. [1]. Cross-correlation effects are ne-glected, which is generally justified for isotropic systems[27]. We use the convention based on the text by Mc-

4

Connell [57], with GR,T (t) in units of s−2:

GR,T (t) =3

16

(µ0

4π

)2~2γ4

1

NR,T×

NR,T∑i6=j

⟨(3 cos2θij(t+ τ)− 1)

r3ij(t+ τ)

(3 cos2θij(τ)− 1)

r3ij(τ)

⟩τ

. (1)

The correlation times τR,T are derived from the followingexpressions [7]:

τR,T =1

GR,T (0)

∫ ∞0

GR,T (t) dt. (2)

In other words, τR,T is defined as the normalized areaunder the autocorrelation functions GR,T (t). The areasin this report are determined using Simpson’s rule, asopposed to the trapezoidal rule used previously [1]. Allflexible n-alkane data in Sections III A and III B are takenfrom [1], and interpreted using Simpson’s rule.

Besides the correlation time, the other parameter ofinterest is the “second-moment” ∆ω2

R,T given by the fol-

lowing expression [7]:

GR,T (0) =1

3∆ω2

R,T . (3)

The dipolar strength ∆ωR,T of the dipole-dipole interac-tion is derived from Eqs. 1 and 3 to yield the followingexpression:

∆ωR,T =√√√√ 9

16

(µ0

4π

)2~2γ4

1

NR,T

NR,T∑i 6=j

⟨(3 cos2θij(τ)− 1)2

r6ij(τ)

⟩τ

. (4)

The simulated ∆ωR,T are shown in Fig. 2, in units ofkHz. The data indicate that ∆ωR ' 2∆ωT in all cases,except the case of benzene where ∆ωR ' ∆ωT . Thecase of benzene can be understood from Eq. 4 since (a)the 1H coordination number (i.e. the number of nearestneighbors) nH is smaller and (b) the nearest neighbor1H-1H distance rH is larger, therefore ∆ωR ∝

√nH/r

3H is

smaller. In all cases, ∆ωT is roughly the same given thatthe 1H spin density is roughly the same. Note that allcases reported here are in the motional averaging regime∆ωR,T τR,T � 1, which is typical of low-viscosity liquids.

The spectral densities JR,T (ω) (in units of s−1) of thelocal magnetic-field fluctuations can be determined fromthe Fourier transform of the autocorrelation function:

JR,T (ω) = 2

∫ ∞0

GR,T (t) cos (ωt) dt, (5)

where GR,T (t) is real and even. In the fast motionregime applicable here (i.e. ω0 τR,T � 1, where ω0 is theLarmor frequency for 1H), the spectral density followsJR,T (ω0) = JR,T (0), and the following can be derived:

1

T1,R,T=

1

T2,R,T= 5JR,T (0) =

10

3∆ω2

R,T τR,T . (6)

FIG. 2. MD simulation results for the dipolar strength of (a)intramolecular ∆ωR and (b) intermolecular ∆ωT interactions,using Eq. 4 and reported in units of kHz. The hydrocarbonshort-hand labels are defined in Fig. 1, and are plotted in or-der of increasing carbon number C#. nC# refers to n-alkanesof carbon number C#. “Rigid” refers to rigid hydrocarbons,while all others are flexible. The flexible nC# data are takenfrom [1].

In other words, T1,R = T2,R and T1,T = T2,T , which ischaracteristic for low viscosity fluids in the fast motionregime. The total relaxation rates are then equal to thesum of intramolecular and intermolecular rates:

1

T1=

1

T2=

10

3∆ω2

RτR +10

3∆ω2

T τT , (7)

which is the final expression used to predict the NMRrelaxation time T1 = T2 from simulation results.

In the case of symmetric molecules such as neopentane,benzene, and methane (CH4), there is an additional con-tribution to 1H NMR relaxation from the spin-rotationinteraction [59]. In the case of methane, relaxation fromthe spin-rotation interaction dominates at low densities(i.e., in the gas phase), whereas relaxation from the 1H-1H dipole-dipole interaction dominates at high densities(i.e. in the liquid phase) [60]. Simulations of the spin-

5

rotation interaction for these molecules will be reportedelsewhere.

In order to quantify the departure of GR(t) fromsingle-exponential decay, we fit GR(t) to a sum of multi-exponential decays and determine the underlying dis-tribution in correlation times τ . More specifically, weperform an inversion of the following Laplace transform[61, 62]:

GR,T (t) =

∫PR,T (τ) exp

(− tτ

)dτ,

GR,T (0) =

∫PR,T (τ) dτ,

(8)

where PR,T (τ) (in units of s−3) is the probability distri-bution function derived from the inversion. Also listed isthe integral of PR,T (τ), which is equal to GR,T (0).

In the case of the BBP hard-sphere model, PR(τ) is adelta-function at τR, i.e. PR(τ) = GR(0) δ(τ−τR). How-ever, as shown by the simulations below, GR(t) is alwaysstretched (i.e. multi-exponential) to some degree, there-fore PR(τ) has a finite distribution. The decompositionof GR,T (t) into a sum of exponential decays is commonpractice in heuristic models of complex molecules [20, 21],where the more complex the molecule dynamics, the moreterms are required. This justifies our general approach ofdecomposing GR,T (t) into a “model free” sum of expo-nential decays in Eq. 8, for the purposes of quantifyingthe departure from hard-sphere models.

The PR,T (τ) distributions were determined by usingthe discrete form of Eq. 8, using 150 logarithmically-spaced τ bins ranging from 10−2 ps ≤ τ ≤ 103 ps, anda fixed regularization parameter α = 10−1 [61, 62]. Theresulting PR,T (τ) distributions were then analyzed usingthe following :

µR,T =1

GR,T (0)

∫PR,T (τ) ln(τ) dτ,

σ2R,T =

1

GR,T (0)

∫PR,T (τ) (ln(τ)− µR,T )

2dτ.

(9)

σR,T is the standard deviation and µR,T is the mean ofthe variable ln(τ). A natural logarithm in τ is used asthe variable since the underlying PR,T (τ) distributionsare discrete and evenly spaced in ln(τ).

In the case of n-decane and n-heptadecane, the MDsimulation were compared with 1H T2 measurements.The n-decane (≥ 99% purity) and n-heptadecane (99%purity) were obtained from Sigma-Aldrich. The n-alkanes were de-oxygenated by bubbling high-purity N2

gas overnight and sealing the vial, thereby removingparamagnetic O2 in solution. The NMR measurementswere acquired at ambient conditions (' 25 oC) using aGeoSpec2 from Oxford Instruments in an 18 mm probe,at a Larmor frequency of ω0/2π = 2.3 MHz for 1H(where ω0 = γB0, for magnetic-field strength B0 andgyro-magnetic ratio γ/2π = 42.57 MHz/T for 1H). ACPMG (Carr-Purcell-Meiboom-Gill) echo train was usedwith an echo spacing of TE = 0.2 ms, and averaged up

to a signal-to-noise ratio of SNR ' 103. The 1/T2 dis-tributions were determined using a 1D inverse Laplacetransform [61, 62], with a regularization parameter ofα ' 10−3 corresponding to the noise floor. The widthof the 1/T2 distributions was found to be the same at aslightly elevated temperature of 30 oC, which for the caseof n-heptadecane rules out any potential influence fromthe nearby phase-transition at ' 20 oC.

III. RESULTS AND DISCUSSIONS

A. Symmetric molecules and hard-sphere models

The hard-sphere models developed by BPP [5] andTorrey [6] have been the building blocks for the interpre-tation of NMR relaxation in liquids. In this section, weshow how the case of the more spherical isomers neopen-tane and isooctane approach the BPP and Torrey the-ories, and in particular neopentane shows good agree-ment with the Stokes-Einstein theory. Such agreementvalidates the hard-sphere theories in a manner never re-ported before, and at the same time further validates ourMD simulation methodology [1].

The intramolecular dipole-dipole interaction [5] is tra-ditionally derived using the rotational diffusion equationof rank 2 for hard-spheres of radius RR, and is param-eterized by the rotational diffusion coefficient DR. Theparameter DR is simply related to rotational correlationtime τR, which is defined as the average time it takesthe molecule to rotate by 1 radian. The intermoleculardipole-dipole interaction [6] is traditionally derived usingthe Brownian motion model where the diffusion propa-gator is derived for hard-spheres of radius RT , and is pa-rameterized by the translational diffusion coefficient DT .The parameter DT is simply related to the translational-diffusion correlation-time τD, which is defined as the av-erage time it takes the molecule to diffuse by one hard-core diameter 2RT [57].

For hard-spheres, the diffusion coefficients (and corre-lation times) can be related to the bulk properties of thefluid, namely the viscosity η and absolute temperatureT , using the traditional Stokes-Einstein relation [8]:

τR =1

6DR=

4π

3kBR3R

η

T,

τD =2R2

T

DT=

12π

kBR3T

η

T=

5

2τT .

(10)

Note that in order to relate the Stokes-Einsteintranslational-diffusion correlation time τD to the NMRderived translational correlation time τT (Eq. 2), a fac-tor of 5

2 is required, i.e. τD = 52τT [7]. An expression for

the two autocorrelation functions are then derived:

GR(t) = GR(0) exp

(− t

τR

),

GT (t) = GT (0)

∫ ∞0

3J23/2(x)

xexp

(−x2 t

52τT

)dx,

(11)

6

FIG. 3. MD simulations of the autocorrelation function GR(t)for intramolecular interactions using Eq. 1, for selected hy-drocarbons defined in Fig. 1, split into (a) and (b) for clarity.The y-axis has been normalized by zero time value GR(0),and the x-axis has been normalized by correlation time τR(Eq. 2). The prediction from the BPP [5] hard-sphere model(Eq. 11) is shown by the black dotted line.

where both expressions reduce to GR,T (0) (Eq. 3) att = 0.

The BPP hard-sphere model for the intramolecularautocorrelation GR(t) is plotted in Fig. 3(a) as thedotted black lines. The hard-sphere model is a single-exponential decay (Eq. 11), which is a straight-line decayon the semilog-y plot. The intramolecular GR(t) appearto be more “stretched” (i.e. multi-exponential) in na-ture than the hard-sphere model, however neopentane(solid line) is clearly closer to a hard-sphere, relative tothe linear isomer n-pentane (dashed line). Similarly, theisooctane (solid line) is closer to the hard-sphere predic-tion, relative to the linear isomer n-octane (dashed line).These observations are quantified in Fig. 5, where thestandard deviation σR,T are plotted for the same datasetas Fig. 2. A more stretched (i.e. multi-exponential) de-cay in GR(t) yields a larger distribution PR(τ) in cor-

FIG. 4. MD simulations of the autocorrelation function GT (t)for intermolecular interactions using Eq. 1, for selected hy-drocarbons defined in Fig. 1, split into (a) and (b) for clarity.The y-axis has been normalized by zero time value GT (0), andthe x-axis has been normalized by correlation time τT (Eq.2). The prediction from the Torrey [6] (“TOR”) hard-spheremodel (Eq. 11) is shown by the black dotted line.

relation times τ (Eq. 8), which yields a larger σR,T(Eq. 9). The BPP hard-sphere model GR(t) in Eq. 11,on the other hand, predicts a single exponential decay,i.e. PR(τ) = GR(0) δ(τ − τR), corresponding to σR = 0(which is limited to σR ' 0.038 in our case due to regular-ization effects of the inverse Laplace transform [61, 62]).The PR,T (τ) distributions from the inverse Laplace trans-form (Eq. 8) used to compute σR,T are shown in Ap-pendix A, including the hard-sphere model. The data inFig. 5(a) indicate that σR drops by ' 27 % for neopen-tane compared to n-pentane; likewise, σR drops by '22 % for isooctane compared to n-octane. This quanti-fies the observation in Fig. 3 that the more spherical themolecule, the closer to single-exponential decay in GR(t).

Meanwhile, the Torrey hard-sphere model “TOR” forthe intermolecular autocorrelation GT (t) is plotted inFig. 4(a) as the dotted black lines. The effect of molec-

7

FIG. 5. MD simulation results for the standard deviation of(a) intramolecular σR and (b) intermolecular σT interactions,using Eq. 9 determined from the PR,T (τ) distributions (Eq.8) shown in Appendix A. Same hydrocarbon labels as Fig. 2.Also shown are the BPP model which predicts σR = 0 (or' 0.038 in our computation), and the Torrey (TOR) modelwhich predicts σT ' 1.25.

ular symmetry does not have a significant effect on thefunctional form of GT (t), and remains equally departedfrom the hard-sphere model. More specifically, the sameanalysis of GT (t) in Fig. 4(b) indicates roughly simi-lar σT values with molecular symmetry. Note howeverthat the Torrey hard-sphere model (TOR) has a multi-exponential distribution characterized by σT ' 1.25.This is due to the fact that GT (t) in Eq. 11 is not a singleexponential decay, even though it contains only one cor-relation time τT (see Apendix A for more details). Whatis remarkable is that the Torrey hard-sphere predictionσT ' 1.25 is consistent with neopentane, and even tosome extent n-pentane.

Another way to quantify deviations from the hard-sphere model is to compute the ratio τD/τR, which ac-cording to Eq. 11 should be τD/τR = 9 for hard-spheres[8] where the Stokes-Einstein radius for rotation RR

FIG. 6. MD simulation result for ratio τD/τR of translational-diffusion correlation-time (τD = 5

2τT ) to rotational-

correlation time (τR), along with predicted value τD/τR = 9[8] from Stokes-Einstein relation (Eq. 10). Same hydrocarbonlabels as Fig. 2.

equals the Stokes-Einstein radius for translation RT . Theratio plotted in Fig. 6 clearly shows that lower n-alkanestend towards the hard-spheres, while the higher n-alkanesincreasingly depart from hard-spheres. Similarly, the ra-tio for isooctane is found to be closer to a hard-spherethan n-octane, as expected. Furthermore, in the caseof neopentane, the ratio is found to be τD/τR = 8.76,which is very close to the hard-sphere prediction of 9.This is a remarkable finding which validates the hard-sphere models by BPP, Torrey, and Stokes-Einstein in amanner never reported before.

A similar trend is found for the planar-symmetricmolecule benzene, where Fig. 3(b) indicates that ben-zene (solid line) is closer to the BPP prediction (i.e.straighter) than n-hexane (dashed line). More specif-ically, as shown in Fig. 5(a), σR is found to be '10 % lower for benzene compared to n-hexane. Like-wise, τD/τR for benzene in Fig. 6 is closer to the hard-sphere prediction of 9 than n-hexane. This indicates thatplanar-symmetric molecules are more “spherical” thanlinear chains of the same carbon number, at least whenmolecular dynamics are concerned.

On the other hand, Fig. 3(b) indicates that cyclo-hexane (which contains 1H’s in both axial and equato-rial positions) is further away from the BPP prediction(i.e. more stretched) than n-hexane. More specifically,as shown in Fig. 5(a), σR is found to be ' 15 % higherfor cyclohexane compared to n-hexane. Likewise τD/τRfor cyclohexane in Fig. 6 is further away from the hard-sphere prediction of 9 than n-hexane. This indicates thatthe out of plane 1H sites in cyclohexane result in largerdeviations from hard-spheres than linear chains of thesame carbon number.

8

FIG. 7. MD simulations of the autocorrelation function GR(t)for intramolecular interactions, for (a) rigid and (b) flexiblehydrocarbons, from n-pentane (nC5) to n-decane (nC10).

B. Internal motions in flexible versus rigidmolecules

As discussed above, neopentane is a spherical moleculewith relatively stiff (i.e. rigid) bonds, and is thereforeexpected to agree with the above mentioned BPP, Tor-rey, and Stokes-Einstein theory of hard spheres. We nowturn our attention to the effects of internal motions in thelong-chain n-alkanes, which we study here by simulatingcompletely rigid n-alkanes. While rigid molecules do notexist in nature, they provide an ideal testing ground forquantifying the effects of rigidity on the molecular dy-namics and the NMR relaxation. For convenience, alldata in this section and the previous section are listed inthe supplementary material.

The characterization of internal motions in hydrocar-bons originates from Woessner’s theories [12], who pos-tulated that internal motions would decrease the corre-lation times τR,T and therefore increase the relaxationtimes T1,2 in liquids. As here for n-decane, we find thatinternal motions increase τR by a factor ' 12, resulting

FIG. 8. MD simulations of the autocorrelation function GT (t)for intermolecular interactions, for (a) rigid and (b) flexiblehydrocarbons, from n-pentane (nC5) to n-decane (nC10).

in a factor ' 13 decrease in T1,2. Furthermore, it waspostulated in [12] that internal motions would cause theintramolecular contribution to relaxation to decrease rel-ative to the intermolecular contribution. As shown herefor n-alkanes, we find that internal motions decrease theintramolecular contribution relative to the intermolecularcontribution by a factor ' 2, on average. In other words,we can test the original theories using MD simulations,without invoking any heuristic models.

The effects of internal motions on intramolecular au-tocorrelation functions GR(t) for nC5↔nC10 are shownin Fig. 7, presented in such a way as to allow for directcomparison between (a) rigid molecules and (b) flexiblemolecules. The internal motions clearly decrease the cor-relation times τR, as shown when going from (a) rigid to(b) flexible molecules. Likewise, the intermolecular auto-correlation functions GT (t) for nC5↔nC10 are shown inFig. 8, and show a similar decrease in correlation timesτT in going from (a) rigid to (b) flexible. What is alsosignificant is that the stretched decay for intramolecularGR(t) in Fig. 7 persists for rigid molecules, indicat-

9

FIG. 9. MD simulation results for correlation times from(a) intramolecular τR and (b) intermolecular τT interactions,using Eq. 2. Same hydrocarbon labels as Fig. 2.

ing that molecular geometry plays a crucial role in thefunctional-form for GR(t). If internal motions were theonly cause of the stretched decay, then the functionalforms for rigid GR(t) (which do not have internal mo-tions) would be straighter (i.e. closer to single exponen-tial). In fact the opposite is found, namely the standarddeviation σR is a factor ' 2 larger for rigid moleculescompared to flexible molecules, as shown in Fig. 5(a).This suggests that internal motions have a tendency tonarrow the underlying distribution PR(τ) (Eq. 8) in cor-relation times τ .

Meanwhile, the functional forms for intermolecularGT (t) of rigid molecules in Fig. 8 continues to deviatefrom the Torrey model, again indicating that moleculargeometry also plays a crucial role in the functional-formfor GT (t). As shown in Fig. 5(b), the standard devia-tion σT is a factor ' 2 larger for rigid nC9 and nC10compared to flexible nC9 and nC10. This suggests thatinternal motions have a tendency to narrow the underly-ing distribution PT (τ) (Eq. 8) in correlation times, fornC9 and above.

FIG. 10. MD simulation results for (a) the ratio of relaxationtimes T1,2,T /T1,2,R using Eq. 6, where T1,2,T /T1,2,R � 1indicates that intramolecular dominates over intermolecular,and (b) the total relaxation time T1,2, using Eq. 7. Samehydrocarbon labels as Fig. 2.

In order to quantify these findings, the same analysisfor the dipolar strengths ∆ωR,T (Eq. 4) and correlationtimes τR,T (Eq. 2) is applied to the rigid molecules. Inthe case of ∆ωR,T , Fig. 2 indicates that rigidity doesnot significantly effect the dipolar strength. In the caseof ∆ωR this is expected since on average over time, thenearest neighbor 1H’s are the same distance apart forboth rigid and flexible, i.e. the equilibrium positions arethe same. It is evident however that the rigid ∆ωR simu-lation shows some scattering of the order ±10% at largechain-lengths. Likewise, ∆ωT remains roughly the samesince the density of rigid and flexible fluids are the same,implying that the 1H spin density is roughly the same.

The correlation times τR,T computed using Eq. 2 forrigid and flexible molecules are plotted in Fig. 9 for(a) intramolecular and (b) intermolecular interactions.The rotational correlation-time τR for flexible benzeneagrees well with previous estimates from NMR measure-ments and MD simulations [14, 15]. In the case of rigid

10

molecules, the maximum autocorrelation time t = 150ps did not fully capture the decay in GR,T (t), implyingthat τR,T is underestimated for rigid nC9 and rigid nC10,and T1,2 is correspondingly overestimated for rigid nC9and rigid nC10. Nevertheless, Fig. 9(a) shows that theestimated intramolecular τR is a factor ' 2 greater forrigid nC5 than flexible nC5, and a factor ' 12 greaterfor rigid nC10 than flexible nC10. Meanwhile, Fig. 9(b)shows that the estimated intermolecular τT is roughlythe same between rigid nC5 and flexible nC5, but a fac-tor ' 5 greater for rigid nC10 than flexible nC10. Thesefindings clearly imply that internal motion effects becomemore prominent with increasing chain-length.

The next step is to compute the corresponding re-laxation times from the ∆ωR,T and τR,T data us-ing Eq. 6. Fig. 10(a) shows the ratio in relax-ation times T1,2,T /T1,2,R, where T1,2,T /T1,2,R � 1 in-dicates that intramolecular relaxation dominates overintermolecular relaxation, while T1,2,T /T1,2,R � 1 indi-cates that intermolecular dominates over intramolecularinstead. The data indicates that internal motions de-crease the intramolecular contribution relative to theintermolecular contribution by a factor ' 2 on average,although some scattering exists in the T1,2,T /T1,2,R datafor rigid nC9 and rigid nC10. This is consistent withthe predictions in [12]. These results can also be viewedas the ratio τD/τR (where τD = 5

2τT ) in Fig. 6, whichshows that τD/τR for rigid molecules is on average bya factor ' 2 further away from the hard-sphere model.This indicates that rigid n-alkanes are less “spherical”than flexible n-alkanes, as might be expected.

Finally, Fig. 10(b) presents the total relaxation timeT1,2 computed using Eq. 7, showing that T1,2 is a fac-tor ' 1.4 shorter for rigid nC5 than flexible nC5, whileit is a factor ' 13 shorter for rigid nC10 than flexiblenC10. Again, these findings clearly imply that internalmotions effects become more prominent with increasingchain-length, in line with predictions from [12].

It should be noted in passing that for the flexible iso-mers discussed in Section III A, Fig. 10 shows that whileneopentane and isooctane show results consistent withtheir corresponding n-alkane, benzene and cyclohexaneclearly do not. Specifically, Fig. 10(a) shows that forall cases T1,2 relaxation is dominated by intramolecularinteractions, i.e. T1,2,T /T1,2,R � 1, except for benzeneand cyclohexane which show that T1,2 is dominated byintermolecular interactions instead, i.e. T1,2,T /T1,2,R �1. In the case of benzene, this is a result of a lower ∆ωR(Fig. 2(a)) compared to all the other n-alkanes. In thecase of cyclohexane, this is a result of a larger τT (Fig.9(b)) compared to n-hexane. These difference result inthe spread of T1,2 (Fig. 10(b)) for benzene and cyclohex-ane compared with n-hexane.

C. Site-by-site simulations and distribution incorrelation times

The above results show that internal motions are notthe primary cause of the stretched decay. Our next taskis therefore to determine whether the multi-exponentialdecay in GR(t) (and the deviations in GT (t) from theTorrey model) are a result of underlying variations inmolecular dynamics across the chain-length. This sce-nario was postulated previously, where the fast rotation(i.e. short τR) of the methyl groups act as relaxationsinks for the macro-molecule [27]. For convenience, alldata in this section are listed in the supplementary ma-terial.

In order to investigate variations across the chain, weperform the MD simulations for each 1H across the chain,and then compute the correlation times τR,T and relax-ation times T1,2 on a site-by-site basis across the chain.The 1H sites are labeled in Fig. 1 for n-decane and n-heptadecane, where the #1 1H is on the methyl end-group, and the largest number is in the middle of thechain (more details in Section II A). IntramolecularGR(t)for n-heptadecane in Fig. 11(a) shows a large distri-bution in correlation times, with the #1 and #2 1H’son the methyl clearly having the steepest decay (i.e.shortest τR), as expected from [27]. The average “Ave”is the weighted average over all sites, and constituteswhat was reported in the previous sections. Meanwhile,intermolecular GT (t) for n-heptadecane in Fig. 11(b)shows less variation between sites than intramolecular,which is intuitive since the distance of closest approachbetween molecules should be roughly independent of thelocation along the chain. The same site-by-site methodis used for n-decane. Both the intramolecular GR(t) forn-decane in Fig. 12(a) and the intermolecular GT (t) forn-decane in Fig. 12(b) show less variation between sitesthan n-heptadecane. This is intuitive since one expectsthere to be more variation in the molecular dynamicsacross longer chains.

A more quantitative summary of the site-by-site cor-relation times τR,T (using Eq. 2) is given in Fig. 13,for both n-decane and n-heptadecane. In the caseof n-heptadecane, the 1H’s labeled #1 and #2 (bothon the methyl end-group) show a factor ' 4 shorterintramolecular τR than at the middle of the chain, anda factor ' 1.4 shorter intermolecular τT . In the caseof n-decane, #1 and #2 show a factor ' 2 shorterintramolecular τR than at the middle of the chain, anda factor ' 1.4 shorter intermolecular τT . Also shown inFig. 13 are results from the site-by-site simulation forrigid n-decane. It is interesting to note that for rigidn-decane, #1 and #2 only show a factor ' 1.4 shorterintramolecular τR than the chain middle, which is some-what less variation than for flexible n-decane. In otherwords, the internal motions have a tendency to enhancethe site-by-site variation of intramolecular τR across thechain. In the case of intermolecular τT , both rigid andflexible n-decane (as well as flexible n-heptadecane) show

11

FIG. 11. MD simulation results of the autocorrelation func-tion for (a) intramolecularGR(t) and (b) intermolecularGT (t)interactions using Eq. 1, for n-heptadecane (nC17) on a site-by-site basis with labels defined in Fig. 1. “Ave” indicatesweighted average over the site-by-site results.

the same factor ' 1.4 shorter τT for #1 and #2 ver-sus the chain middle, which is reasonable given that thedistance of closest approach between molecules shouldbe the same across the chain, regardless of whether themolecule is rigid or not.

Given the large variation in τR across the chain (Fig.13), one expects the average GR(t) to be a multi-exponential (i.e. stretched) decay, since the averageGR(t) is the weighted sum of decays over all sites. Indeedn-decane (Fig. 12) and n-heptadecane (Fig. 11) show astretched decay for the average. What is remarkable,however, is that GR(t) for both n-decane (Fig. 12) andn-heptadecane (Fig. 11) show a stretched decay at everysite. The extent of the stretched (i.e. multi-exponential)decay can be quantified on a site-by-site basis using σR,T(Eq. 9), in a similar fashion as Section III A. As shown in14(a), σR does not vary significantly across the chain, al-though the data may indicate a slightly lower value at thechain ends (#1 and #2), for both nC10 and nC17. The

FIG. 12. MD simulation results of the autocorrelation func-tion for (a) intramolecularGR(t) and (b) intermolecularGT (t)interactions using Eq. 1, for n-decane (nC10) on a site-by-sitebasis with labels defined in Fig. 1. “Ave” indicates weightedaverage over the site-by-site results.

fact that σR is roughly uniform across the chain impliesthat not only does each site show a stretched decay, butall the sites show roughly the same stretched functional-form as the average. Similar results are found for thecase of site-by-site GR(t) for rigid n-decane, where Fig.14(a) shows a roughly uniform σR across the rigid chain.All these findings imply that the overall molecular geom-etry is a crucial factor, perhaps even the dominant factor,behind the functional form in the decay of GR(t).

Similar conclusions can also be made for intermolecularGT (t) in Figs. 12(b) and 11(b), which clearly show thatthe site-by-site decays overlap with the average. Thesame can be inferred from the σT data in Fig. 14(b),with even more certainty since σT shows less scatteringthan σR .

It is also interesting to track the strength ofintramolecular versus intermolecular relaxation on a site-by-site basis across the chain. Fig. 15(a) presents theratio T1,2,T /T1,2,R across the chain sites, which shows

12

FIG. 13. MD simulation results for correlation times from(a) intramolecular τR and (b) intermolecular τT interactions,using Eq. 2, on a site-by-site basis with site labels defined inFig. 1. “Rigid” refers to rigid hydrocarbons, while all othersare flexible.

that the intramolecular relaxation dominates in the chainmiddle (i.e. T1,2,T /T1,2,R � 1), while intramolecular re-laxation is a factor of ' 2 stronger than intermolecularat the chain ends (i.e. #1 and #2). In the case of n-heptadecane, the chain middle shows a factor ' 3 largervalue of T1,2,T /T1,2,R than the chain ends. In the caseof n-decane, the chain middle shows only a factor '1.5 larger value of T1,2,T /T1,2,R than the chain ends,which is reasonable given the smaller variation in cor-relation times τR,T for n-decane (Fig. 13). Rigid n-decane shows almost no variation in T1,2,T /T1,2,R ' 7,and clearly intramolecular relaxation dominates acrossthe entire chain.

Finally, Fig. 15(b) presents the total relaxation timeT1,2 on a site-by-site basis. In the case of n-heptadecane,the chain middle shows a factor ' 3 shorter value of T1,2than the chain ends. In the case of rigid and flexiblen-decane, the chain middle shows only a factor ' 1.5shorter value of T1,2 than the chain ends, which is rea-

FIG. 14. MD simulation results for the standard deviation of(a) intramolecular σR and (b) intermolecular σT interactions,using Eq. 9 determined from the PR,T (τ) distributions (Eq.8). Same hydrocarbon labels as Fig. 13. Also shown arethe BPP model which predicts σR = 0 (or ' 0.038 in ourcomputation), and the Torrey (TOR) model which predictsσT ' 1.25.

sonable given the smaller variation in correlation timesτR,T in Fig. 13.

1. Cross-relaxation effects and comparison withmeasurements

Despite the site-by-site variation in correlation timesτR,T (Fig. 13) and subsequent variation in relaxationtimes T1,2 (Fig. 15(b)) across the chain, the measureddistribution in T1,2 values for liquids is never as large.This is a result of strong cross-relaxation effects, a.k.a.“spin diffusion” effects, which tend to average out (i.e.“wash out”) any such variations across the chain [27, 63–65].

The condition for either strong, intermediate, or weakcross-relaxation is determined by the relative strengthof the cross-relaxation rate σ1,2,ij between spin-pairs i

13

FIG. 15. MD simulation results for (a) the ratio of relaxationtimes T1,2,T /T1,2,R using Eq. 6, where T1,2,T /T1,2,R � 1indicates that intramolecular dominates over intermolecular,and (b) the total relaxation time T1,2, using Eq. 7. Samehydrocarbon labels as Fig. 13.

and j compared with the difference in individual relax-ation rates |1/T1,2,i − 1/T1,2,j | (see supplementary ma-terial for more details). More specifically, the differentcross-relaxation regimes can be defined as such [3, 27]:

σ1,2,ij �1

2|1/T1,2,i − 1/T1,2,j | (strong),

σ1,2,ij '1

2|1/T1,2,i − 1/T1,2,j | (intermediate),

σ1,2,ij �1

2|1/T1,2,i − 1/T1,2,j | (weak).

(12)

For strong cross-relaxation, which is generally the casefor low-viscosity liquids, one expects the measured T1,2to be single-valued and given by the average rate [10, 27]:

1

T1,2=

1

N

∑i

1

T1,2,i, (13)

where N is the number of 1H’s in the chain, and 1/T1,2,iis the relaxation rate of the i’th 1H in the chain shown in

FIG. 16. (a) MD simulation result for distribution in site-by-site values of 1/T1,2,i taken from Fig. 15(b), for (a) nC17and (b) nC10, with labels defined in Fig. 1 and with relativeheights according to degeneracy. “Ave” indicates weightedaverage over the site-by-site results Eq. 13, with arbitraryheight. Simulations results are compared with measured dis-tributions of 1/T2, with arbitrary height.

Fig. 15(b). One consequence of averaging the relaxationrates in Eq. 13 is that the average 1/T1,2 is equivalent tocomputing the weighted average GR,T (t) in Figs. 11 and12, and then using Eqs. 2−7.

The comparison between simulation and measure-ments are shown in Fig. 16. In the case of n-heptadecane,the simulation predicts an average value of 1/T1,2 ' 1.48s−1, which is close to the measured value of 1/T1,2 ' 1.45s−1. In the case of n-decane, the simulation predicts anaverage value of 1/T1,2 ' 0.308 s−1, which is also closeto the measured value of 1/T1,2 ' 0.328 s−1. Such agree-ment was previously reported in [1], which validates ourMD simulation methodology.

However, while the mean values agree, Fig. 16 clearlyshows that the widths of simulation versus measurementdo not agree. In the case of n-heptadecane (Fig. 16(a)),the simulation predicts a width of ∆1,2 ' 1.40 s−1 (de-

14

fined as the difference between maximum and minimumvalues in the distribution), which is a factor ' 3 timeslarger than the measured full-width at half-maximumW1,2 ' 0.42 s−1 of the distribution. The n-heptadecaneresults therefore indicate that cross-relaxation partiallyaverages out the site-by-site distribution in relaxation.In other words, in the case of weak cross-relaxationσ1,2,ij � 1

2 |1/T1,2,i − 1/T1,2,j |, the measured W1,2 would

agree with ∆1,2 ' 1.40 s−1 from simulation. In the caseof strong cross-relaxation σ1,2,ij � 1

2 |1/T1,2,i − 1/T1,2,j |,the measured W1,2 would tend towards zero (i.e. the1/T1,2 distribution would tend towards a delta function),which in practice will be limited by the experimental res-olution of ' 0.06/T1,2 (as determined on a water sam-ple). Fig. 16(a) indicates that σ1,2,ij is somewhere in-between the strong and weak cross-relaxation regime, i.e.in the intermediate regime, suggesting that σ1,2,ij couldin principle be calculated from the observed differencebetween simulation and measurements at low magnetic-fields (ω0/2π . 2.3 MHz).

In the case of n-decane (Fig. 16(b)), the simulationpredicts a width of ∆1,2 ' 0.11 s−1, which is much largerthan the measured W1,2 ' 0.02 s−1. Given that theexperimental resolution limit has been reached, the truemeasured width for n-decane must satisfy W1,2 < 0.02s−1. This implies that ∆1,2 is at least a factor & 6 timeslarger than W1,2, indicating that cross-relaxation is moreefficient for n-decane than for n-heptadecane.

IV. CONCLUSIONS

The traditional hard-sphere models developed byBloembergen, Purcell, Pound [5] and Torrey [6] continueto be the building blocks in describing NMR relaxationfrom rotational (R) and translational (T ) diffusion ofmolecules, respectively. The BPP hard-sphere model pre-dicts a single-exponential decay in the autocorrelationfunction GR(t), however GR(t) for the n-alkanes becomesincreasingly “stretched” (i.e. multi-exponential) with in-creasing chain-length [1]. Likewise, GT (t) shows greaterdeparture from the Torrey hard-sphere model with in-creasing chain-length [1]. We quantify the departureof GR(t) from the single-exponential model by fittingGR(t) to a sum of exponential decays using an inverseLaplace transform, and determine the standard devia-tion σR (i.e. the normalized standard deviation) of theunderlying distribution PR(τ) in correlation times τ . Wefind that hydrocarbons of greater molecular symmetrysuch as neopentane and isooctane have ' 25 % lower σRthan their corresponding linear n-alkanes. Furthermore,in the case of the spherically-symmetric neopentane, wefind that the ratio of translational-diffusion to rotationalcorrelation times is found to be τD/τR = 8.76 (whereτD = 5

2τT ), which agrees well with the Stokes-Einsteinprediction of τD/τR = 9.

By comparing the relaxation of rigid and flexible n-alkanes, we find a factor ' 12 increase in the rotational

correlation-time τR for rigid n-decane compared withflexible n-decane, together with a factor ' 5 increase inthe translational correlation-time τT , thereby revealingthe strong influence of internal motions on τR,T and T1,2for long-chain n-alkanes. The extent of the stretched (i.e.multi-exponential) decay of GR,T (t) is greater for rigid n-alkanes, namely the σR,T values are a factor ' 2 largerfor rigid compared to flexible n-alkanes, indicating thatinternal motions tend to narrow the underlying distribu-tion PR,T (τ) (Eq. 8) in correlation times τ .

We find that T1,2 relaxation from intramolecular inter-actions dominates over intermolecular interactions (i.e.T1,2,T /T1,2,R � 1) for all hydrocarbons investigated, ex-cept for benzene and cyclohexane where intermolecularinteractions dominate (i.e. T1,2,T /T1,2,R � 1). The rigidn-alkanes show a factor ' 2 larger ratio T1,2,T /T1,2,Rthan flexible n-alkanes, indicating that internal motionssomewhat diminish the influence of intramolecular inter-actions.

Site-by-site simulations of GR,T (t) for the 1H’s acrossthe chain indicate that τR decreases by a factor ' 4 to-wards the chain-ends of n-heptadecane, together witha factor ' 1.4 decrease in τT , thereby revealing varia-tions in τR,T and T1,2 across the chain for long-chainn-alkanes. Moreover, the simulations indicate that thestretched functional-forms of site-by-site GR,T (t) are ap-proximately the same as the chain averages, namely theσR,T values are roughly the same across the chain, indi-cating that the overall molecular geometry plays a crucial(if not dominant) role in the functional-form of the decayin GR,T (t).

Corresponding T1,2 measurements of n-heptadecane in-dicate a narrower distribution in T1,2 than site-by-sitesimulations, implying that cross-relaxation (partially)averages-out the variations in τR,T and T1,2 across thechain of long-chain n-alkanes. Such T1,2 comparisons be-tween site-by-site simulations and measurements at lowmagnetic-field (ω0/2π . 2.3 MHz) could in principle beused to compute the cross-relaxation rates σ1,2,ij for long-chain n-alkanes.

This work informs our on-going work in understandingthe NMR relaxation and diffusion of hydrocarbons (andother fluids) in nano-confined pores, such as the light hy-drocarbons found in the organic-matter pores of kerogenand bitumen [23, 30, 49] typically found in organic-shalereservoirs.

V. ACKNOWLEDGMENTS

This work was funded by the Rice University Consor-tium on Processes in Porous Media, and the AmericanChemical Society Petroleum Research Fund [ACS-PRF-58859-ND6]. We gratefully acknowledge the NationalEnergy Research Scientific Computing Center, which issupported by the Office of Science of the U.S. Depart-ment of Energy [DE-AC02-05CH11231], for HPC timeand support. We also gratefully acknowledge the Texas

15

Advanced Computing Center (TACC) at The University of Texas at Austin (URL: http://www.tacc.utexas.edu)for providing HPC resources.

[1] P. M. Singer, D. Asthagiri, W. G. Chapman, and G. J.Hirasaki, J. Magn. Reson. 277, 15 (2017).

[2] D. E. Woessner, J. Chem. Phys. 41 (1), 84 (1964).[3] J. Kowalewski and L. Maler, Nuclear Spin Relaxation in

Liquids: Theory, Experiments, and Applications (Taylor& Francis Group, 2006).

[4] P. Henritzi, A. Bormuth, and M. Vogel, Solid State Nucl.Magn. Reson. 54, 32 (2013).

[5] N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys.Rev. 73 (7), 679 (1948).

[6] H. C. Torrey, Phys. Rev. 92 (4), 962 (1953).[7] B. Cowan, Nuclear Magnetic Resonance and Relaxation

(Cambridge University Press, 1997).[8] A. Abragam, “Principles of nuclear magnetism,” (Oxford

University Press, International Series of Monographs onPhysics, 1961).

[9] D. E. Woessner, J. Chem. Phys. 36 (1), 1 (1962).[10] D. E. Woessner, J. Chem. Phys. 37 (3), 647 (1962).[11] W. T. Huntress, J. Chem. Phys. 48 (8), 3524 (1968).[12] D. E. Woessner, J. Chem. Phys. 42 (6), 1855 (1965).[13] P. S. Hubbard, J. Chem. Phys. 51 (4), 1647 (1969).[14] A. Laaksonen, P. Stilbs, and R. E. Wasylishen, J. Chem.

Phys. 108 (2), 455 (1998).[15] R. Witt, L. Sturz, A. Dolle, and F. Muller-Plathe, J.

Phys. Chem. A 104, 5716 (2000).[16] G. Lipari and A. Szabo, J. Amer. Chem. Soc. 104, 4546

(1982).[17] G. Lipari and A. Szabo, J. Amer. Chem. Soc. 104, 4559

(1982).[18] S. Kariyo, A. Brodin, C. Gainaru, A. Herrmann,

H. Schick, V. N. Novikov, and E. A. Rossler, Macro-molecules 41, 5313 (2008).

[19] D. Frueh, Prog. Nucl. Magn. Reson. Spect. 41, 305(2002).

[20] P. A. Beckmann, Phys. Rep. 171 (3), 85 (1988).[21] V. I. Bakhmutov, NMR Spectroscopy in Liquids and

Solids (CRC Press, Taylor & Francis Group, 2015).[22] A. Bormuth, M. Hofmann, P. Henritzi, M. Vogel, and

E. A. Rossler, Macromolecules 46, 7805 (2013).[23] P. M. Singer, Z. Chen, L. B. Alemany, G. J. Hirasaki,

K. Zhu, Z. H. Xie, and T. D. Vo, Energy Fuels , DOI:10.1021/acs.energyfuels.7b03603 (2018).

[24] C. Calero, J. Martı, and E. Guardia, J. Phys. Chem. B119, 1966 (2015).

[25] W. A. M. Madhavi, S. Weerasinghe, and K. I. Momot,J. Phys. Chem. B 121, 10893 (2017).

[26] I. D. Campbell and R. Freeman, J. Magn. Reson. 11, 143(1973).

[27] A. Kalk and H. J. C. Berendsen, J. Magn. Reson. 24, 343(1976).

[28] Q. R. Passey, K. Bohacs, W. L. Esch, R. Klimen-tidis, and S. Sinha, Soc. Petrol. Eng. SPE-131350-MS(2010).

[29] R. G. Loucks, R. M. Reed, S. C. Ruppel, andU. Hammes, AAPG Bull. 96, No. 6, 1071 (2012).

[30] P. M. Singer, Z. Chen, and G. J. Hirasaki, Petrophysics57 (6), 604 (2016).

[31] K. E. Washburn and Y. Cheng, J. Magn. Reson. 278, 18(2017).

[32] A. E. Ozen and R. Sigal, Petrophysics 54 (1), 11 (2013).[33] E. Rylander, P. M. Singer, T. Jiang, R. E. Lewis,

R. McLin, and S. M. Sinclair, Soc. Petrol. Eng. SPE-164554-MS (2013).

[34] T. Jiang, E. Rylander, P. M. Singer, R. E. Lewis, andS. M. Sinclair, Soc. Petrophys. Well Log Analysts , SP-WLA (2013).

[35] P. M. Singer, E. Rylander, T. Jiang, R. McLin,R. E. Lewis, and S. M. Sinclair, Soc. Core AnalystsSCA2013-18 (2013).

[36] K. E. Washburn and J. E. Birdwell, J. Magn. Reson. 233,17 (2013).

[37] R. Kausik, K. Fellah, E. Rylander, P. M. Singer,R. E. Lewis, and S. M. Sinclair, Soc. Core AnalystsSCA2014-73 (2014).

[38] H. Daigle, A. Johnson, J. P. Gips, and M. Sharma,Unconv. Resources Tech. Conf. URTEC-1905272-MS(2014).

[39] K. E. Washburn, Concepts Magn. Reson. A 43 (3), 57(2014).

[40] J.-P. Korb, B. Nicot, A. Louis-Joseph, S. Bubici, andG. Ferrante, J. Phys. Chem. C 118 (40), 23212 (2014).

[41] B. Nicot, N. Vorapalawut, B. Rousseau, L. F. Madariaga,G. Hamon, and J.-P. Korb, Petrophysics 57 (1), 19(2015).

[42] M. Lessenger, R. Merkel, R. Medina, S. Ramakrishna,S. Chen, R. Balliet, H. Xie, P. Bhattad, A. Carnerup,and M. Knackstedt, Soc. Petrophys. Well Log Analysts ,SPWLA (2015).

[43] J. E. Birdwell and K. E. Washburn, Energy Fuels 29,2234 (2015).

[44] M. Fleury and M. Romero-Sarmiento, J. Petrol. Sci. Eng.137, 55 (2016).

[45] R. Kausik, K. Fellah, E. Rylander, P. M. Singer, R. E.Lewis, and S. M. Sinclair, Petrophysics 57 (4), 339(2016).

[46] B. Sun, E. Yang, H. Wang, S. J. Seltzer, V. Montoya,J. Crowe, and T. Malizia, Soc. Petrophys. Well Log An-alysts , SPWLA (2016).

[47] C. Sondergeld, A. Tinni, C. Rai, and A. Besov, Soc.Petrophys. Well Log Analysts , SPWLA (2016).

[48] D. Yang and R. Kausik, Energy Fuels 30 (6), 4509(2016).

[49] Z. Chen, P. M. Singer, J. Kuang, M. Vargas, and G. J.Hirasaki, Petrophysics 58 (5), 470 (2017).

[50] A. Valori, S. V. den Berg, F. Ali, and W. Abdallah,Energy Fuels 31, 5913 (2017).

[51] J. Liu, L. Wang, S. Xi, D. Asthagiri, and W. G. Chap-man, Langmuir 33 (42), 11189 (2017).

[52] J. C. Phillips, R. Braun, W. Wang, E. Tajkhorshid,E. Villa, C. Chipot, R. Skeel, L. Kale, and K. Schul-ten, J. Comput. Chem. 26, 1781 (2005).

[53] K. Vanommeslaeghe, E. Hatcher, C. Acharya, S. Kundu,S. Zhong, J. Shim, E. Darian, O. Guvench, P. Lopes,I. Vorobyov, and A. D. M. Jr., J. Comput. Chem. 31,

16

671 (2010).[54] L. Martinez, R. Andrade, E. G. Birgin, and J. M. Mar-

tinez, J. Comput. Chem. 30, 2157 (2009).[55] S. Plimpton, J. Comput. Phys. 117, 1 (1995).[56] P. J. in’t Veld and G. C. Rutledge, Macromolecules 36,

7358 (2003).[57] J. McConnell, “The theory of nuclear magnetic relaxation

in liquids,” (Cambridge University Press, 1987).[58] R. Kimmich, NMR Tomography, Diffusometry and Re-

laxometry (Springer-Verlag, 1997).[59] P. S. Hubbard, Phys. Rev. 131 (3), 1155 (1963).[60] S.-W. Lo, G. J. Hirasaki, W. V. House, and

R. Kobayashi, Soc. Petrol. Eng. J. 7 (1), 24 (2002).[61] L. Venkataramanan, Y.-Q. Song, and M. D. Hurlimann,

IEEE Transactions on Signal Processing 50 (5), 1017(2002).

[62] Y.-Q. Song, L. Venkataramanan, M. D. Hurlimann,M. Flaum, P. Frulla, and C. Straley, J. Magn. Reson.154, 261 (2002).

[63] I. Solomon, Phys. Rev. 99 (2), 559 (1955).[64] H. T. Edzes and E. T. Samulski, Nature 265, 521 (1977).[65] J. B. Lambert and E. P. Mazzola, Nuclear Magnetic

Resonance Spectroscopy: An Introduction to Principles,Applications, and Experimental Methods (Prentice-Hall,2004).

APPENDIX A

As mentioned in Section III A, the probability distri-bution functions PR,T (τ) of correlation time τ are shownin Figs. 17 and 18. The PR,T (τ) are derived from theinverse Laplace transforms in Eq. 8 of the GR,T (t) sim-ulations in Figs. 3 and 4, respectively. The PR,T (τ) arethen used to derive the coefficients of variance σR,T us-ing Eq. 9. For instance, it is clear from Figs. 17 and18 that PR,T (τ) are narrower for neopentane, benzeneand isooctane (i.e. have lower σR,T ) compared to theircorresponding straight-chain n-alkane, respectively.

FIG. 17. Probability distribution function PR(τ) of rotationalcorrelation time τ derived from the inverse Laplace transform(Eq. 8) of the GR(t) simulations in Fig. 3(a) and (b), respec-tively. Also shown is the BBP prediction GR(t) from Eq. 11,generated with an arbitrarily chosen value of τR = 2 ps. They-axis has been divided by GR(0), which normalizes the areaof the distributions to unity (except for the BPP model).

As a consistency check of the inverse Laplace algorithmand the resulting PR,T (τ) distributions, it was verifiedthat the predicted τR,T defined as:

τR,T =1

GR,T (0)

∫PR,T (τ) τ dτ, (14)

gave similar values as τR,T from Eq. 2. Indeed, the τR,Tpredicted from Eq. 14 was within ± 5 % of Eq. 2, for allthe flexible molecules. In the case of the rigid n-alkaneshowever, Eq. 14 predicted (on average) a factor ' 1.5longer τR,T compared to Eq. 2. For the rigid n-alkanes,this discrepancy is a result of the incomplete decay in

17

FIG. 18. Probability distribution function PT (τ) of trans-lational correlation time τ derived from the inverse Laplacetransform (Eq. 8) of the GT (t) simulations in Fig. 4(a) and(b), respectively. Also shown is the Torrey (TOR) predictionGT (t) from Eq. 11, generated with an arbitrarily chosen valueof τT = 2 ps. The y-axis has been divided by GT (0), whichnormalizes the area of the distributions to unity.

GR,T (t) at t = 150 ps (Figs. 7 and 8), which tends tounderestimate τR,T according to Eq. 2.

For completeness, Fig. 19 shows the PR,T (τ) distribu-tions for the complete set of flexible n-alkanes and wa-ter, where the GR,T (t) were previously reported in Ref.[1]. The case of water shows two distinct peaks in rota-tional distribution PR(τ), which agrees well with previ-ous reports [24, 25]. According to Ref. [25] for water,the larger peak (' 79 % relative intensity) with longerrotational correlation-time (τ = 3.8 ps) is interpretedas Debye continuous-time rotational diffusion, while thesmaller peak (' 21 % relative intensity) with shorter ro-tational correlation-time (τ = 0.22 ps) is interpreted aslarge-amplitude discrete jumps.

FIG. 19. Probability distribution function for (a) rotationalPR(τ) and (b) translational PT (τ) correlation time τ derivedfrom the inverse Laplace transform (Eq. 8) of the GR,T (t)simulations taken from Ref. [1], including water (W). Alsoshown are the BBP and Torrey (TOR) predictions GR,T (t)from Eq. 11, generated with an arbitrarily chosen value ofτR,T = 2 ps. The y-axis has been divided by GR,T (0), whichnormalizes the area of the distributions to unity (except forthe BPP model).

1

SUPPLEMENTARY MATERIAL

Here we provide some background about cross-relaxation effects in liquids, and the definition of the T1and T2 cross-relaxation rates σ1,2,ij between spin-pairs iand j, respectively. Cross-relaxation plays an importantrole in 1H relaxation in liquids, and exact theories ofcross-relaxation exist for the simple case of two-spin sys-tems [3, 27, 63]. The case of n-heptadecane (n-C17H36)reported here consists of 36 1H spins for intramolecular

relaxation, and therefore presents a much more complexscenario than the two-spin case. However, we use thetwo-spin case to briefly explain the principles of cross-relaxation, and we neglect any possible cross-correlationeffects with other relaxation mechanisms.

1. Longitudinal cross-relaxation

The time evolution of longitudinal magnetization for atwo-spin system is solved using the well known Solomonequations [63]:

d

dt

[〈Iz〉〈Sz〉

]= −

[1/T1,I − σ1,IS σ1,IS

σ1,IS 1/T1,S − σ1,IS

][〈Iz〉 − Ieqz〈Sz〉 − Seqz

]. (15)

d

dt

[⟨I±⟩⟨

S±

⟩] = −

[1/T2,I − σ2,IS ∓ iδIS/2 σ2,IS

σ2,IS 1/T2,S − σ2,IS ± iδIS/2

][⟨I±⟩⟨

S±

⟩] . (16)

〈Iz〉 and 〈Sz〉 are the time-dependent expectation valuesof the operators for longitudinal (i.e. z) magnetization for1H spins I and S along the chain, respectively. Ieqz andSeqz are the equilibrium values of the longitudinal magne-tization for 1H spins I and S along the chain, respectively.1/T1,I and 1/T1,S are the longitudinal relaxation rates for1H spins I and S, respectively. σ1,IS is the longitudinalcross-relaxation rate between 1H spins I and S. Accord-ing to Ref. [27], in the case of σ1,IS � 1

2 |1/T1,I−1/T1,S |,spins I and S relax at their own distinct rates [27], aswould be predicted by the site-by-site simulations (e.g.#1, #2, etc.) in Fig. 16. In the opposite case ofσ1,IS � 1

2 |1/T1,I − 1/T1,S |, the two spins relax at the

same rate given by the average 1/T1 = 12 (1/T1,I+1/T1,S)

defined in Eq. 13, as would be predicted by the “Ave” inFig. 16.

When 1H spins I and S are not identical (i.e. whentheir Larmor frequencies are not exactly equal ωI 6= ωS),one can manipulate the two spins independently andmeasure T1,I , T1,S and σ1,IS using NOESY (nuclearOverhauser effect spectroscopy), and/or T2,I , T2,S andσ2,IS (see below) using ROESY (rotating frame Over-hauser effect spectroscopy), both at high magnetic-fields(typically ω0/2π & 100 MHz) [3].

2. Transverse cross-relaxation

A slightly more complicated expression exists for thetime evolution of transverse magnetization for a two-spinsystem [63] is given in Eq. 16, where

⟨I±⟩

and⟨S±

⟩are

the time-dependent expectation values of the operators

for transverse (i.e. x, y) magnetization for 1H spins I

and S along the chain, respectively, where step-up I+and step-down I− operators are used (I± = Ix ± iIy).1/T2,I and 1/T2,S are the transverse relaxation rates for1H spins I and S, respectively. σ2,IS is the transversecross-relaxation rate between 1H spins I and S. Theadditional term for transverse relaxation involves the fre-quency splitting δIS = |ωI−ωS | between the two 1H sites,which for example a methyl (spin I) versus a methylene(spin S) is δIS ' 0.4 ppm ·ω0 [65], or, δIS/2π ' 1 Hzat ω0/2π = 2.3 MHz. Eq. 16 for T2 is equivalent to Eq.15 for T1, provided δIS/2 . σ2,IS [3], which is the casehere given that the measured T1 and T2 distributions arefound to be roughly the same.

3. Fast-motion regime

Further insight can be obtained about the cross-relaxation rate σ1,2,IS , given that the present case is inthe fast-motion regime, i.e. ω0 τR,T � 1. In the fast-motion regime, the following equalities hold: T1,I = T2,I ,T1,S = T2,S , σ1,IS = σ2,IS , plus none of these quanti-ties are dispersive (i.e. none depend on ω0). One addi-tional equality holds in the fast-motion regime, namelyσ1,2,IS = 1

3 1/T1,2 [27], with the average defined as

1/T1,2 = 12 (1/T1,2,I + 1/T1,2,S) from Eq. 13.

One can then loosely apply the above two-spin casein the fast-motion regime to the more complex case ofn-heptadecane in Fig. 16(a); though this is a gross over-simplification, it can check whether our interpretationis at all reasonable. According to Fig. 16(a), the aver-

2

age 1/T1,2 ' 1.5 s−1, which loosely implies that σ1,2,IS '13 1/T1,2 ' 0.5 s−1. The simulations further indicate that

the spread is given by ∆1,2 ' |1/T1,I−1/T1,S | ' 1.4 s−1.Putting these two estimates together loosely indicatesthat σ1,2,IS ' 1

2 |1/T1,I − 1/T1,S | ' 0.5-0.7 s−1, implyingan intermediate cross-relaxation regime, which is quali-

tatively consistent with the findings in Fig. 16(a). Suchconsistency is motivation for a more thorough analysis ofσ1,2,ij for all spin-pairs, provided extensions of Eqs. 15and 16 are developed which (for instance) break down the36 1H spin system into magnetization modes [3]. This isthe first instance we are aware of where cross-relaxationeffects can be studied by comparing simulations withmeasurements at low magnetic-fields (ω0/2π . 2.3 MHz).

3

Nam

eL

ab

elR

igid

∆ωR/2π

∆ωT/2π

τ Rτ T

τ D/τ R

σR

σT

T1,2,T/T1,2,R

T1,2

(kH

z)(k

Hz)

(ps)

(ps)

(s)

neo

pen

tane

oC

521.6

29.1

10.5

51.9

38.7

60.3

91.3

51.6

118.1

7

ben

zene

aC

67.9

38.8

31.1

13.2

17.2

20.6

11.5

70.2

823.7

1

cycl

ohex

ane

cC6

17.8

610.5

51.4

37.4

012.9

00.8

61.8

70.5

65.9

3

isooct

ane

iC8

21.4

49.3

61.7

53.8

75.5

20.8

01.6

22.3

86.6

4

n-p

enta

ne

nC

520.1

79.6

60.7

22.2

77.9

20.5

31.4

41.3

815.1

0

n-h

exane

nC

620.0

59.7

21.1

02.8

46.4

50.6

81.5

21.6

510.6

9

n-h

epta

ne

nC

719.9

99.7

31.6

63.5

05.2

80.8

21.6

22.0

07.6

5

n-o

ctane

nC

819.9

39.7

32.4

54.3

04.3

81.0

21.7

22.4

05.5

0

n-n

onane

nC

919.8

79.7

63.3

45.1

93.8

81.2

11.8

42.6

74.1

9

n-d

ecane

nC

10

19.8

39.7

44.6

16.2

53.3

91.3

81.9

53.0

63.1

6

n-u

ndec

ane

nC

11

19.8

29.7

46.0

07.4

03.0

81.5

12.0

53.3

62.4

9

n-d

odec

ane

nC

12

19.8

09.7

38.2

210.5

73.2

21.5

92.1

53.2

21.8

0

n-t

ridec

ane

nC

13

19.7

69.7

010.7

412.3

72.8

81.8

22.2

33.6

01.4

2

n-t

etra

dec

ane

nC

14

19.7

19.6

612.3

413.6

12.7

61.9

02.2

93.7

71.2

5

n-p

enta

dec

ane

nC

15

19.7

59.6

916.2

416.3

12.5

12.0

72.4

14.1

30.9

7

n-h

exadec

ane

nC

16

19.6

69.6

918.9

018.1

92.4

12.2

62.4

84.2

80.8

4

n-h

epta

dec

ane

nC

17

19.7

99.5

922.3

520.0

02.2

42.0

82.5

34.7

60.7

2

n-p

enta

ne

nC

5R

igid

19.7

79.8

31.3

12.1

34.0

71.0

11.3

82.4

810.5

8

n-h

exane

nC

6R

igid

19.8

09.9

32.6

53.3

73.1

81.1

31.5

23.1

35.5

4

n-h

epta

ne

nC

7R

igid

19.5

310.0

56.7

75.8

12.1

41.4

01.6

84.4

02.4

0

n-o

ctane

nC

8R

igid

19.5

910.1

316.4

28.1

91.2

51.8

51.8

27.5

01.0

6

n-n

onane

nC

9R

igid

17.7

110.1

631.9

524.4

91.9

22.5

62.6

63.9

60.6

1

n-d

ecane

nC

10

Rig

id21.7

810.5

156.8

733.0

11.4

52.3

72.6

87.3

90.2

5

TA

BL

EI.

MD

sim

ula

tions

resu

lts,

incl

udin

g:

fluid

nam

e,fluid

lab

el(c

hem

ical

form

ula

giv

enin

Fig

.1

capti

on),

rigid

or

not,

intr

am

ole

cula

rdip

ola

rst

rength

∆ωR/2π

(Eq.

4),

inte

rmole

cula

rdip

ola

rst

rength

∆ωT/2π

(Eq.

4),

intr

am

ole

cula

rco

rrel

ati

on

tim

eτ R

(Eq.

2),

inte

rmole

cula

rco

rrel

ati

on

tim

eτ T

(Eq.

2),

rati

oof

transl

ati

onal-

diff

usi

on

toro

tati

onal

corr

elati

on

tim

esτ D/τ R

(wher

eτ D

=5 2τ T

),in

tram

ole

cula

rst

andard

dev

iati

onσR

(Eq.

9),

inte

rmole

cula

rst

andard

dev

iati

onσT

(Eq.

9),

rati

oof

inte

rmole

cula

rto

intr

am

ole

cula

rre

laxati

on

tim

esT1,2,T/T1,2,R

(Eq.

6),

and

tota

lre

laxati

on

tim

eT1,2

(Eq.

7).

4

Lab

elSit

eD

egen

.R

igid

∆ωR/2π

∆ωT/2π

τ Rτ T

τ D/τ R

σR

σT

T1,2,T/T1,2,R

T1,2

(kH

z)(k

Hz)

(ps)

(ps)

(s)

nC

17

#1

221.7

710.4

68.5

818.3

05.3

31.7

12.4

22.0

31.2

5

nC

17

#2

421.7

110.5

08.4

718.1

35.3

51.6

82.4

12.0

01.2

7

nC

17

#3

418.9

910.1

115.3

120.7

43.3

92.0

72.4

82.6

10.9

9

nC

17

#4

418.9

99.9

121.1

322.4

42.6

62.1

72.5

03.4

60.7

7

nC

17

#5

419.0

49.4

720.6

624.5

82.9

82.0

62.5

03.3

90.7

8

nC

17

#6

419.2

29.2

426.5

325.6

72.4

22.1

02.5

24.4

80.6

3

nC

17

#7

419.2

79.4

731.6

025.8

12.0

42.0

82.5

05.0

60.5

4

nC

17

#8

419.4

89.5

232.7

526.0

81.9

92.1

02.5

25.2

60.5

1

nC

17

#9

419.8

99.2

836.3

826.1

41.8

01.9

42.5

16.3

90.4

6

nC

17

#10

219.5

59.5

834.1

824.7

81.8

11.8

92.4

65.7

50.5

0

nC

10

#1

221.8

210.3

22.7

65.5

35.0

11.1

91.8

12.2

34.0

0

nC

10

#2

421.7

510.3

12.9

75.5

44.6

61.1

61.8

12.3

93.8

1

nC

10

#3

418.6

99.9

14.9

06.2

83.2

11.5

21.8

42.7

73.2

6

nC

10

#4

418.8

59.6

54.4

86.7

43.7

71.4

31.8

62.5

33.4

3

nC

10

#5

419.2

19.2

55.4

17.4

03.4

21.3

71.8

63.1

62.8

9

nC

10

#6

419.3

09.1

95.6

27.5

53.3

61.3

91.8

73.2

92.7

8

nC

10

#1

2R

igid

20.9

710.1

844.2

327.2

31.5

42.2

82.6

06.9

00.3

4

nC

10

#2

4R

igid

21.6

910.2

847.6

727.7

41.4

52.3

32.6

27.6

50.3

0

nC

10

#3

4R

igid

20.0

110.5

660.3

730.9

71.2

82.3

12.6

67.0

00.2

8

nC

10

#4

4R

igid

21.8

510.5

361.6

735.3

51.4

32.4

32.7

07.5

10.2

3

nC

10

#5

4R

igid

22.7

610.6

659.6

036.1

31.5

22.3

92.7

07.5

30.2

2

nC

10

#6

4R

igid

22.8

410.7

160.7

037.0

51.5

32.3

42.7

07.4

50.2

1

TA

BL

EII

.Sit

e-by-s

ite

MD

sim

ula

tions

resu

lts,

incl

udin

g:

fluid

lab

el(c

hem

ical

form

ula

giv

enin

Fig

.1

capti

on),

site

num

ber

(Fig

.1),

deg

ener

acy

,ri

gid

or

not,

intr

am

ole

cula

rdip

ola

rst

rength

∆ωR/2π

(Eq.

4),

inte

rmole

cula

rdip

ola

rst

rength

∆ωT/2π

(Eq.

4),

intr

am

ole

cula

rco

rrel

ati

on

tim

eτ R

(Eq.

2),

inte

rmole

cula

rco

rrel

ati

on

tim

eτ T

(Eq.

2),

rati

oof

transl

ati

onal-

diff

usi

on

toro

tati

onal

corr

elati

on

tim

esτ D/τ R

(wher

eτ D

=5 2τ T

),in

tram

ole

cula

rst

andard

dev

iati

onσR

(Eq.

9),

inte

rmole

cula

rst

andard

dev

iati

onσT

(Eq.

9),

rati

oof

inte

rmole

cula

rto

intr

am

ole

cula

rre

laxati

on

tim

esT1,2,T/T1,2,R

(Eq.

6),

and

tota

lre

laxati

on

tim

eT1,2

(Eq.

7).