Molecular Dynamics Simulations of NMR Relaxation and Diffusion of Heptane Confined in a Polymer Matrix
Arjun Valiya Parambathu, Philip M. Singer, George J. Hirasaki, Walter G. Chapman, D. Asthagiri
MMolecular Dynamics Simulations of NMR Relaxation and Diffusion of HeptaneConfined in a Polymer Matrix
Arjun Valiya Parambathu, Philip M. Singer, George J. Hirasaki, Walter G. Chapman, and Dilip Asthagiri ∗ Department of Chemical and Biomolecular Engineering,Rice University, 6100 Main St., Houston, Texas 77005, USA (Dated: January 22, 2020)The mechanism behind the NMR surface relaxation and the large T / T ratio of light hydro-carbons confined in the nano-pores of kerogen remains poorly understood, and consequently hasengendered much debate. Towards bringing a molecular-scale resolution to this problem, we presentmolecular dynamics (MD) simulations of H NMR relaxation and diffusion of heptane in a polymermatrix, where the high-viscosity polymer is a model for kerogen and bitumen that provides an or-ganic “surface” for heptane. We calculate the autocorrelation function G ( t ) for H- H dipole-dipoleinteractions of heptane in the polymer matrix and use this to generate the NMR frequency ( f )dependence of T and T relaxation times as a function of φ C . We find that increasing molecularconfinement increases the correlation time of the heptane molecule, which decreases the surfacerelaxation times for heptane in the polymer matrix. For weak confinement ( φ C >
50 vol%), we findthat T S /T S (cid:39)
1. Under strong confinement ( φ C (cid:46)
50 vol%), we find that the ratio T S /T S (cid:38) φ C , and that the dispersion relation T S ∝ f is consistent with previ-ously reported measurements of polymers and bitumen. Such frequency dependence in bitumen hasbeen previously attributed to paramagnetism, but our studies suggests that H- H dipole-dipoleinteractions enhanced by organic nano-pore confinement dominates the NMR response in saturatedorganic-rich shales, without the need to invoke paramagnetism.
INTRODUCTION
The traditional interpretation of H NMR relaxationtimes T and T (and implicitly the autocorrelation func-tion G ( t ) for H- H dipole-dipole interactions) rely on theBloembergen, Purcell, and Pound (BPP) [1] theory forintra-molecular relaxation by rotational diffusion, Tor-rey [2] and Hwang and Freed [3] for inter-molecular re-laxation by translational diffusion, and, Hubbard [4] andBloom [5] for spin-rotation relaxation. These early pi-oneering theories were nevertheless built on strong as-sumptions regarding the molecular structure and inter-action, such as the assumption of rigid molecules withoutinternal motions. In this regard, theories for the effectsof internal motions on T and T of ellipsoidal moleculeswere developed by Woessner [6], and later a more generalphenomenological model was developed by Lipari and Sz-abo [7, 8].Until recently, petro-physicists have been forced to relyon these classical theories to interpret T and T of com-plex fluids such as crude-oils and bitumen. One of thebiggest conundrums has been that at high viscosity, thelog-mean T LM becomes independent of viscosity overtemperature ( η/T ), and (roughly) proportional to theNMR frequency f , i.e. T LM ∝ f , for heavy crude-oilsand bitumen [9–14]. To address this, a new phenomeno-logical model [15, 16] was developed which uses the Lipariand Szabo model, plus it takes into account the multi-component nature of complex reservoir fluids by modify-ing the exponent of the frequency dependence in the BPPmodel. This new phenomenological model successfullyaccounted for the viscosity and frequency dependenceof T LM for heavy crude-oils, bitumen, and polymer- heptane mixes, without invoking paramagnetism.Another conundrum has been that at high viscosity,the log-mean T LM of heavy crude-oils and bitumen hasa viscosity dependence of T LM ∝ ( η/T ) − . [12–14],which was also observed for pure polymers [16]. Severalmodels have been proposed to account for this behavior[14, 17], yet no consensus has yet been reached. Overall,it is clear that the viscosity and frequency dependence of T LM and T LM present significant deviations from theclassical theories of NMR relaxation of bulk fluids, andthat further investigations are required.A better theoretical understanding of the observed re-laxation T , in fluids can be achieved if one can split thedifferent relaxation mechanisms such as intra-molecular,inter-molecular, and spin-rotation relaxation. One wayto do this experimentally is to perform H or H NMRon partially deuterated molecules (i.e. partially replace H with H), such as in the case of glycerol [18], polymermelts [19], or methane [5]. This technique assumes thatdeuterating does not alter the local molecular dynamicsdue to changes in the rotational and vibrational modesof the molecule. In the case of methane, the symmetry ofthe molecule is altered by deuteration, which complicatesthe interpretation.Molecular dynamics (MD) simulations have alreadyplayed a helpful, guiding role in this regard. We havealready shown that MD simulations of H- H dipole-dipole relaxation can naturally separate intra-molecularfrom inter-molecular T , for liquid-state n -alkanes andwater [20–22], as well as H spin-rotation relaxation formethane [23]. Our MD simulations have also shown theeffects of internal motions and molecular geometry on T , of various hydrocarbons, as well as the differences a r X i v : . [ phy s i c s . c h e m - ph ] J a n in T , between methyl and methylene H’s across the n -alkane chain [21, 22]. MD simulations have shed light onthe limitations of the classical theories, and they signifi-cantly advanced our understanding of T , of bulk fluids,without any free parameters or models in the interpreta-tion of the MD simulations.Aside from bulk fluids, another major conundrum inpetro-physics is the origin of the NMR surface relaxation T S and T S of fluids confined in organic-rich shales [24–57]. For instance, Ozen and Sigal [24] first reported thatlight hydrocarbons exhibit large ratios T S /T S (cid:38) T S /T S (cid:39) H- Hdipole-dipole relaxation is thought to be the dominantmechanism [31, 36, 38, 42, 46, 49, 50], while in othercases surface-paramagnetism is thought to be the domi-nant mechanism [32, 33, 48, 56].In order to shed light on this subject, we present MDsimulations of T and T of heptane in polymer-heptanemixtures, where the high-viscosity polymer is a model forkerogen and bitumen that provides an organic “surface”and organic transient “pores” for heptane. Our premisebehind this analogy is that the organic matter in kero-gen and bitumen is essentially made up of cross-linkedpolymers (plus aromatics), where cross-linking turns theliquid polymers into highly-viscous bitumen or solid kero-gen when more cross-links are present. We simulate thesurface-relaxivity parameters ρ and ρ for heptane, andthe surface-relaxation ratio T S /T S , as a function ofNMR frequency f and heptane concentration φ C in thepolymer-heptane mix. Our MD simulations of T , T anddiffusion are also compared to previously reported mea-surements of similar systems [16].The rest of this manuscript is organized as follows: sec-tion presents the methodology for the polymer-heptanemixtures, MD simulations of diffusion and relaxation,auto-correlation and NMR relaxation, and effects of dis-solved oxygen on measurements; followed by the resultsin section for diffusion of heptane in the mix, intra andinter-molecular relaxation, total relaxation, surface re-laxation and relaxivity of heptane in the polymer matrix;followed by the conclusions in section . METHODOLOGYPolymer-heptane mixtures
An illustration of the polymer-heptane mix is shownin Fig. 1, where n -heptane is used as the representa-tive alkane, and a 16-mer oligomer of poly(isobutene) ofmolecular mass M w = 912 g/mol is used for the polymermatrix. Poly(isobutene) is based on a Brookfield vis- cosity standard used in previously reported NMR mea-surements of polymer-heptane mixes, where the empiri-cal relation η (cid:39) A M αw was reported with α (cid:39) . A (cid:39) . × − at ambient temperature [16]. Theempirical viscosity relation implies that the simulatedpoly(isobutene) 16-mer has a viscosity of η (cid:39) M w and η for the simulationsare lower than the measured polymer where M w = 9436g/mol and η (cid:39)
333 400 cP at ambient temperature andpressure [16].Fig. 1(a) shows the “dissolved” regime correspondingto simulations with low heptane volume fractions φ C <
50 vol%, where heptane molecules rarely contact otherheptane molecules. Fig. 1(b) shows the “pore fluid”regime corresponding to simulations with high heptanevolume fractions φ C >
50 vol% where heptane moleculesfill a more conventional pore.
Simulation details
Both n -heptane and the polymer are modeled us-ing using the CHARMM General Force Field [58, 59],which is known to accurately describe the thermophyscialproperties as well as the NMR relaxation and diffu-sion properties[20] of hydrocarbons. To construct thesimulation system, we first created the structure of n -heptane using Avogadro[60, 61] and poly(isobutene) us-ing the PRO-DRG server [62, 63]. We then created N copies of heptane and M copies of poly(isobutene), andpack them separately at a low density (0.1 g/cm ) usingPACKMOL[64, 65]. The initial numbers are chosen con-sidering ideal mixing and using the experimental densityof polymer (0.89 g/cm ) [16] and heptane (0.68 g/cm )[66] NIST at 298.15 K. The numbers are chosen in sucha way that 100% polymer corresponds to 40 molecules.The two boxes are then combined to form the initial sim-ulation box. We use the NAMD[67, 68] code to performthe simulations. The equations of motion are integratedusing the Verlet algorithm with a time step of 1 fs. Toremove possible steric clashes, we minimize the systemenergy using 1000 steps of conjugate gradient minimiza-tion. This starting system is necessarily at a much lowerpressure due to the low density. We then compress thesystem to atmospheric pressure using Langevin dynam-ics, where the temperature of 298.15 K is controlled usinga Langevin thermostat and the pressure of 1 atm. is con-trolled using a Langevin barostat. Compressing from alow density state also ensures we have a well mixed sys-tem. This is critical because relying on diffusive motionto ensure mixing is not recommended for systems withlow diffusivity, such as the polymer-alkane melt.We find that after about 1 ns, all the systems studiedhere achieve a constant density and temperature. Weequilibrate this system at constant temperature ( N V T ensemble) for 1 ns. The temperature during this phase
FIG. 1. Illustration of a cross-section of (locally) cylindricaltransient “pores” in a poly(isobutene) matrix (black) filledwith n -heptane (red), where only carbon atoms are shown.x(a) “Dissolved” regime corresponds to simulations with lowheptane volume fractions φ C <
50 vol% where heptanemolecules rarely contact other heptane molecules. (b) “Porefluid” regime corresponds to simulations with high heptanevolume fractions φ C >
50 vol% where heptane molecules filla more conventional pore. was controlled by reassigning velocities (obtained from aMaxwell-Boltzmann distribution) every 250 steps. Thesubsequent production run was carried out for 10 ns atconstant NVE. Frames were archived every 100 steps foranalysis. The Lennard-Jones interactions were smoothlyswitched to zero between from 13˚A and 14 ˚A. We use theparticle mesh Ewald procedure to describe electrostaticinteractions, with a grid spacing of 0.5 ˚A.
Diffusion coefficient
The simulated diffusion coefficient is obtained usingthe Einstein relation D sim = 16 δ (cid:104) ∆ r (cid:105) L δt (1)where (cid:104) ∆ r (cid:105) is the mean-square displacement of thecenter-of-mass of the molecule as a function of diffusionevolution time t . Following Yeh and Hummer [69] (seealso Ref. 20), we correct the simulated diffusion coeffi-cient for finite size effects using the relation D T = D sim + k B T πη ξL (2)where D T is the diffusion coefficient, L is the simulationboxlength, η is the shear viscosity, T is the tempera-ture of the system, k B is the Boltzmann constant, and ξ = 2 . ∼ ∼
2% at low fractions.
Auto-correlation and NMR relaxation
The autocorrelation function G ( t ) for fluctuating mag-netic H- H dipole-dipole interactions is central to thedevelopment of the NMR relaxation theory in liquids [1–3, 73–76]. The details of the derivation of G ( t ) can befound in Ref. 20, and only the essential elements areprovided below. For an isotropic system, G ( t ) is givenas: G R,T ( t ) = 316 (cid:16) µ π (cid:17) (cid:126) γ × N R,T N
R,T (cid:88) i (cid:54) = j (cid:42) (3 cos θ ij ( t + τ ) − r ij ( t + τ ) (3 cos θ ij ( τ ) − r ij ( τ ) (cid:43) τ (3)where t is the lag time of the autocorrelation, µ is thevacuum permeability, (cid:126) is the reduced Planck constant, γ/ π = 42 .
58 MHz/T is the nuclear gyro-magnetic ratiofor H (spin I = 1 / r ij is the magnitude of the vectorthat connects the pair ( i, j ) H- H dipoles, and θ ij is thepolar angle the vector forms with the external magneticfield. The subscript R refers to autocorrelation of intra-molecular interactions from rotational diffusion, and sub-script T refers to autocorrelation of inter-molecular inter-actions from translational diffusion. From G R,T ( t ), one FIG. 2. (a) MD simulations of the intra-molecular ( G R ( t ))and inter-molecular ( G T ( t )) auto-correlation functions forheptane in a polymer-heptane mix with heptane concentra-tion φ C = 5 vol%, compared with pure heptane ( φ C = 100vol%). (b) Corresponding probability distributions P R,T ( τ )determined from inverse Laplace transforms of G R,T ( t ) (Eq.8). can determine the spectral density function J R,T ( ω ) byFourier transform as such: J R,T ( ω ) = 2 (cid:90) ∞ G R,T ( t ) cos ( ωt ) dt, (4)for G R,T ( t ) in units of s − [74]. The expressions (whichdo not assume a molecular model) for T and T are then given by [74, 75]:1 T R, T = J R,T ( ω ) + 4 J R,T (2 ω ) , (5)1 T R, T = 32 J R,T (0) + 52 J R,T ( ω ) + J R,T (2 ω ) , (6)1 T , = 1 T R, R + 1 T T, T , (7)where J R,T ( ω ) are the spectral densities at the res-onance frequency ω = 2 πf . Note that the intra-molecular and inter-molecular rates add to give the totalrelaxation rate (Eq. 7).The autocorrelation function G R,T ( t ) was constructedusing fast Fourier transforms, for lag time ranging from0 ps to 1000 ps in steps of 0 . ≈ × per time framefor intra-molecular relaxation, and ≈ × per timeframe for inter-molecular relaxation.The results of the intra-molecular G R ( t ) and inter-molecular G T ( t ) are shown in Fig. 2(a) for heptane inthe polymer-heptane mix at φ C = 5 vol%, and for pureheptane (i.e. φ C = 100 vol%). In order to quantify thedeparture of G R,T ( t ) from single-exponential decay, wefit G R,T ( t ) to a sum of multi-exponential decays and de-termine the underlying probability distribution P R,T ( τ )in correlation times τ . More specifically, we perform aninversion of the following Laplace transform [77, 78]: G R,T ( t ) = (cid:90) ∞ P R,T ( τ ) exp ( − t/τ ) dτ, (8) τ R,T = 1 G R,T (0) (cid:90) ∞ P R,T ( τ ) τ dτ, (9)∆ ω R,T = 3 G R,T (0) , (cid:90) ∞ (10) J R,T ( ω ) = (cid:90) ∞ τ ωτ ) P R,T ( τ ) dτ, (11)where P R,T ( τ ) are the probability distribution functionsderived from the inversion, and are plotted in Fig. 2(b).Details of the inversion procedure can be found in [23]and in the supporting information in [21]. The P R,T ( τ ) inFig. 2(b) indicate a set of ∼ τ values for both intra-molecular P R ( τ ) and inter-molecular P T ( τ ) interactions. The intra-molecular P R ( τ )has an additional mode at short τ (cid:39) − ps for boththe polymer and heptane, while it is absent for P T ( τ ) inboth cases. Similar observations at τ (cid:39) − ps werereported in the supporting information in [21] for liquid-state alkanes.The decomposition of G R,T ( t ) into a sum of exponen-tial decays in Eq. 8 is common practice in phenomeno-logical models of complex molecules [79, 80], where themore complex the molecular dynamics, the more expo-nential terms are required [6, 81]. Also defined are thecorrelation times τ R,T (Eq. 9), and the square-root of the
FIG. 3. (a) Correlation times for the rotational ( τ R ( t )) andtranslational ( τ T ( t )) motions as a function of heptane concen-tration φ C . (b) Square-root of second moment (i.e. strength)of intra-molecular (∆ ω R ) and inter-molecular (∆ ω T ) interac-tions as a function of φ C . second moments ∆ ω R,T (Eq. 10) i.e. strength of the in-teraction, which are plotted in Fig. 3. While ∆ ω R,T areindependent of φ C , τ R,T increases by ∼ φ C = 100 vol% → τ R,T clearly show that decreasing φ C dra-matically slows the molecular dynamics of heptane dueto increasing confinement in the polymer matrix. T , as a function of f , i.e. T , dispersion, can alsobe determined from the P R,T ( τ ) distributions. This isderived by using the Fourier transform (Eq. 4) of G R,T ( t )(Eq. 8), resulting in Eq. 11. Once J R,T ( ω ) is known, Eqs.5, 6 and 7 are used to determine T , ( ω ) from J R,T ( ω )at ω = ω . RESULTSDiffusion of heptane in the mix
Fig. 4 compares the simulated diffusion coefficientagainst NMR diffusion measurements in the polymer-heptane mixtures. We emphasize that the polymer ma-trix used experimentally is of considerably higher vis-cosity ( η (cid:39)
333 400 cP at ambient) than the one usedin simulations ( η (cid:39) D T relative to the value in the bulk D (= 3.43 × − m /s at ambient) is consistent with [82]: D T D = 1 T = φ m − C , (12)where T is the tortuosity, and T = φ − mC is Archie’sequation with cementation exponent m . The simulationsindicate that m (cid:39) .
68, which agrees well with NMRmeasurements where m (cid:39) .
44. By comparison, m = 2is predicted in a capillary bundle model [82]. It shouldbe noted however that in the present case the polymer isitself diffusing, and therefore the pore walls are not rigid.We repeated our calculations for the low volume frac-tions ( ≤ ≤ ∼
5. This serves as a cautionary note inthe challenges that remain in simulating systems withvery low diffusivity. However, the overall agreement withmeasurements is encouraging, which gives us confidencein our molecular models and simulation forcefield.Furthermore, Fig. 4 also shows that the results forheptane in the polymer-heptane mix are consistent withpreviously reported NMR measurements of water in im-mature kerogen isolates. This suggests that the high-viscosity polymer is a good model for translational dif-fusion of fluids in an immature kerogen matrix, which isreasonable given that immature kerogen has fewer cross-links than mature kerogen [83].
Intra- vs. inter relaxation of heptane
MD simulations can naturally separate intra-molecular T R, R from inter-molecular T T, T relaxation. In Fig.5 we show the ratios of T T /T R and T T /T R , whereratios greater than one indicate that intra-molecular re-laxation dominates over inter-molecular relaxation, whileratios less than one indicate the opposite. It is foundthat intra-molecular relaxation dominates over inter-molecular (i.e. T T, T /T R, R >
1) for φ C (cid:38)
70 vol%.Below φ C (cid:46)
70 vol%, inter-molecular relaxation dom-inates (i.e. T T, T /T R, R < f (cid:38)
400 MHz where the reverse is found for T T /T R > FIG. 4. Ratio of translational diffusion coefficient D T to bulkdiffusion coefficient D for heptane in the polymer matrix forboth measurements and simulations, as a function of heptaneconcentration φ C . Lines are best fits using a Archie modelEq. 12 for tortuosity. Also shown are measurements of re-stricted diffusion of water in immature Kerogen, see Appendixfor more detail. The dependence of T T /T R on φ C can be loosely de-scribed from the results in Fig. 3 alone. According to Eq.6, the 1 /T R, T ∝ ∆ ω R,T τ R,T , if one ignores the smallamount of dispersion in T R, T . At φ C = 100 vol%, ∆ ω R is a factor ∼ ω T , while τ R is a factor ∼ τ T , which leads to T T /T R (cid:39) φ C ,∆ ω R,T stay constant (Fig. 3(b)), however τ T becomeslarger than τ R (Fig. 3(a)), which leads to T T /T R < φ C (cid:46)
70 vol%. Note that inter-molecular relaxation consistsof contributions from both heptane-heptane interactionsand polymer-heptane interactions. However below φ C (cid:46)
50 vol%, i.e. in the dissolved region (Fig. 1(a)), hep-tane molecules do not contact other heptane molecules,therefore the inter-molecular relaxation is dominated bypolymer-heptane interactions.The dependence of T T /T R on φ C can be understoodusing P R,T ( τ ) in Fig. 2(b). According to Eq. 11, con-tributions from ωτ (cid:29) ωτ (cid:28)
1. Contributions with ωτ (cid:29) ωτ (cid:28)
1. For φ C =5 vol% at f = 400 MHz this implies that contributionsfrom P R,T ( τ ) with τ (cid:29)
400 ps are negligible. In otherwords, for φ C = 5 vol% in Fig. 2(b), the peak at τ (cid:39) ps (where the inter-molecular contribution is larger thanintra-molecular contribution) no longer contributes to re-laxation f = 400 MHz, i.e. it is dispersed out. Mean-while P R ( τ ) and P T ( τ ) have peaks at similar τ values for τ (cid:28)
400 ps, however P R ( τ ) is a factor ∼ FIG. 5. Ratios of inter-molecular ( T T, T ) to intra-molecular( T R, R ) relaxation times at f = (a) 2.3 MHz, (b) 22 MHz,and (c) 400 MHz, as a function of heptane concentration φ C . Ratios greater than one (dashed lines) indicate thatintra-molecular relaxation dominates over inter-molecular re-laxation, while ratios less than one indicate the opposite. plitude than P T ( τ ) in that region, therefore T T /T R > f (cid:46)
400 MHz, the peak at τ (cid:39) ps in Fig.2(b) is not dispersed out, therefore similar arguments for T T /T R hold as for T T /T R above, where T T /T R < φ C (cid:46)
70 vol% (i.e. inter-molecular relaxationdominates).
Total relaxation of heptane
The expression for the total relaxation is given in Eq.7, and is plotted in Fig. 6 as a function of φ C at f =2.3 MHz, 22 MHz, 400 MHz. T decreases monotonicallywith decreasing φ C , which as shown below is a resultof larger surface relaxation due to increased confinement(i.e. a larger surface to pore-volume ratio), and a largersurface relaxivity ρ in the dissolved region. Likewise, T decreases monotonically with decreasing φ C at f = 2.3MHz and 22 MHz. However at f = 400 MHz, T tendsto level off with decreasing φ C , which as shown belowis due to a constant surface-relaxivity ρ in the dissolvedregion.Also shown in Fig. 6 are the measurements comparedwith simulations, which shows that the overall trendsagree. A cross-plot of measurements versus simulationsis also shown in Fig. 7 for better comparison. We findgood agreement between the measurements and simula-tions except in two regions: (1) the region φ C (cid:39)
50 vol%where the measurements are overestimated compared tosimulations, and (2) the region φ C (cid:46)
10 vol% wherethe simulations are overestimated compared to measure-ments.The deviation in the region φ C (cid:46)
10 vol% is mostlikely due to the fact that the maximum auto-correlationtime for G R,T ( t ) in Fig. 2(a) is t max = 1000 ps, whichlimits the accuracy in the inverse Laplace transforms P R,T ( τ ) for φ C (cid:46)
10 vol%. The maximum value of τ max in P R,T ( τ ) is chosen to be a factor ∼
10 largerthan the longest acquisition time in the data [77], i.e. τ max = 10 t max = 10 ps in the present case. This leadsto inaccuracies in P R,T ( τ ) if there are contributions with τ > τ max , which is likely to be the case for φ C (cid:46) t max , however this iscomputationally expensive.The deviation in the region φ C (cid:39)
50 vol% is believedto be due to uncertainties in the oxygen concentration inthe measurements. This is explored in Appendix , andthe proposition only qualitatively improves the agree-ment with experiments.
Surface relaxation of heptane
The decrease in heptane relaxation times T , fromtheir bulk relaxation T B, B is due to interactions of hep- FIG. 6. (a) T and (b) T relaxation times for heptane in thepolymer matrix according to simulations (closed symbols) andmeasurements (open symbols) at f = 2.3 MHz, 22 MHz, 400MHz, as a function of heptane concentration φ C . tane with the polymer surfaces. Commonly, this prop-erty is analyzed as relaxation induced by the surface itselfand termed “surface relaxation”. As indicated in Fig. 1,if we assume the polymer matrix to form “pores” for theheptane molecules, we can interpret the polymer-heptaneinteractions as “surface” interactions, where the polymeris the confining surface.The surface relaxation is obtained using the followingrelation: 1 T , = 1 T S, S + 1 T B, B , (13)where T B, B (cid:39) η = 0.39 cP) [85, 86].Fig. 8 shows the resulting dispersion for T S , T S , andthe ratio T S /T S , for the various φ C mixtures.While T B, B for bulk heptane has minimal frequencydependence within the range shown (simulation notshown), T S for heptane in the polymer matrix clearly FIG. 7. Correlation cross-plot of measurements vs. simula-tions of T and T at f = 2.3 MHz, 22 MHz, 400 MHz, forvarious heptane volume fractions φ C . has a large amount of dispersion. More specifically, thesimulations show that T S is dispersive above f (cid:38) T S tends towards the func-tional form T S ∝ f (specifically T S × . /f (cid:39) φ C (cid:46)
50 vol% and high frequencies f (cid:38)
500 MHz. Thisfunctional form for T S dispersion is consistent with thepreviously reported measurements of polymers and bitu-men where T LM ∝ f (specifically T LM × . /f (cid:39) T LM ∝ f found for all bitumen and polymersin the slow-motion regime (i.e. ω τ R (cid:29)
1) is also foundfor T S of heptane at low volume fractions ( φ C (cid:46) T LM ∝ f is predicted at high-viscosities [1]. In the caseof bitumen and polymers, a phenomenological model wasproposed to account for T LM ∝ f dispersion at highviscosities [16]. The results in Fig. 8(a) indicate thatthe same phenomenological model may apply to T S forheptane under nano-confinement in an organic matrix.Fig. 8(b) shows that T S has much less dispersion, asexpected. The increase in T S from low to high f (i.e.from the fast- to slow-motion regime) is given by 10/3,independent of the details in J R,T ( ω ). The factor 10/3can be calculated by comparing Eq. 6 in the fast-motionregime ( ω τ (cid:28)
1) to the slow-motion regime ( ω τ (cid:29) T S /T S ratio is shown in Fig. 8(c),where T S /T S (cid:39)
20 at f = 400 MHz for φ C = 5 vol%. FIG. 8. Surface relaxations (a) T S , (b) T S , and (c) T S /T S ratio as a function of frequency f , for heptane volume frac-tion φ C of 5%, 10%, 15%, 20% to 90% in increments of 10%.Dashed blue line in (a) shows dispersion relation T S ∝ f (specifically T S × . /f = 7 ms). FIG. 9. Surface relaxivities (a) ρ , (b) ρ , and (c) T S /T S (= ρ /ρ ) ratio as a function of heptane volume fractions φ C forboth simulations (closed symbols) and measurements (opensymbols), at frequencies f = 2.3 MHz, 22 MHz, and 400MHz. Dashed vertical line shows dissolved heptane region φ C <
50 vol%, and pore fluid region φ C >
50 vol%.
Surface relaxivity of heptane
The surface-relaxivity parameter ρ , is given by thefollowing expression [87]:1 T S, S = ρ , SV p (14)where V p is the pore volume, S is the surface area ofthe pore, and ρ , are the surface-relaxivity parameters. S and V p incorporate the geometric factors related poregeometry, while ρ , incorporate the surface interactionsbetween heptane and the polymer surfaces. The surfaceto pore-volume ratio of the polymer matrix is related tothe surface to grain-volume ratio of the polymer as such: SV p = 1 − φ C φ C SV g = 4 d . (15) V g is the grain volume of the polymer, which MD sim-ulations have previously shown is S/V g ≈ .
859 ˚A − forbranched alkanes [88], independent of the chain length. d is the equivalent diameter of a cylindrical pore shownin Fig. 1, where we imagine heptane to be extended ina cylindrical pore at high confinement. The diameter ofthe extended heptane is around d = 4 . φ C = 50 vol% according to Eq. 15. It is fairto assume that below φ C <
50 vol%, heptane moleculesinteract mainly with the polymer surfaces, and can bethought of as being ab sorbed (i.e. dissolved) in the poly-mer matrix. Note that Eq. 14 is valid in the small-poreregime, otherwise known as the “fast-diffusion” regime,where ρ , d/D (cid:28) S/V g results in thefollowing expressions:1 ρ , = T S, S − φ C φ C SV g , (16) T S T S = ρ ρ . (17)The resulting ρ and ρ are plotted in Figs. 9(a) and(b), respectively, while the ratio is plotted in Fig. 9(c).Also shown in Fig. 9 is the separation between dissolvedand pore-fluid states at φ C = 50 vol%. The simulationsshow that the surface-relaxivities ρ and ρ are indepen-dent of φ C and f for φ C (cid:38)
50 vol%, as expected inconventional pores. However below φ C (cid:46) ρ and ρ increase with decreasing φ C , which weinterpret as the “dissolved” region where heptane is nolonger in contact with other heptane molecules due toincreased confinement in the polymer matrix. We alsofind that ρ decreases with increasing f , i.e. is disper-sive, in the dissolved region. The simulations also showthat T S /T S (cid:39) T S /T S (cid:38) ρ , ρ , and T S /T S , where the trends are consistent with simula-tions. T S /T S is found to be systematically lower forsimulations compared to measurements. This discrep-ancy is attributed to the fact that the simulated polymerin the mix has a lower molecular weight M w = 912 g/mol( η (cid:39) M w = 9436 g/mol ( η (cid:39)
333 400 cP at ambient temperatures). Nevertheless, bothsimulations and measurements indicate that T S /T S in-creases with increasing confinement and increasing fre-quency, indicating that H- H dipole-dipole relaxationenhanced by nano-pore confinement is the dominantsurface-relaxation mechanism in saturated organic-richshales.
CONCLUSION
We report on MD simulations of heptane confined ina polymer-heptane mix as a function of heptane volumefraction φ C in the mix. Our motivation for studying thissystem is that the high-viscosity polymer acts as a modelof kerogen and bitumen, where a decrease in φ C resultsin an increase in confinement of heptane in the transientorganic “nano-pores” of the polymer matrix. MD simu-lations of the restriction in translational diffusion coeffi-cient D T /D of heptane in the polymer-heptane mix in-dicates a power-law dependence D T /D (cid:39) φ m − C , with anArchie cementation exponent of m (cid:39) .
68. The simula-tions agree well with NMR measurements ( m (cid:39) .
44) onsimilar systems. Furthermore, these findings are consis-tent with previously reported measurements of water inimmature kerogen isolates, which indicate that the high-viscosity polymer is a good model for immature kerogen.We then report on MD simulations of H NMR T and T from H- H dipole-dipole interactions for hep-tane in a polymer-heptane mix, as a function of hep-tane volume fraction φ C and NMR frequency f . Thesimulations naturally separate the contributions fromintra-molecular T R, R (from rotational diffusion) versusinter-molecular relaxation T T, T (from translational dif-fusion). It is found that intra-molecular relaxation domi-nates over inter-molecular (i.e. T T, T /T R, R >
1) above φ C (cid:38)
70 vol%. Below φ C (cid:46)
70 vol%, inter-molecularrelaxation dominates (i.e. T T, T /T R, R < f (cid:38)
400 MHz where the reverse isfound for T relaxation (i.e. T T /T R > T and T arefound to monotonically decrease with decreasing φ C asa result of increasing confinement of heptane in the mix.The MD simulations are found to be consistent with T and T measurements at f = 2.3 MHz, 22 MHz, and 400 MHz. Good agreement is found between measurementsand simulation, except in the region around φ C (cid:39) T and T com-pared to simulations. We propose that the overestimatefrom the measurements is a result of a decrease in theconcentration of dissolved oxygen at φ C (cid:39)
50 vol%,which is qualitatively confirmed by independent MD sim-ulations of the solubility of oxygen in the mix.We use the MD simulation results to compute thesurface-relaxation components T S and T S of heptanein the polymer “pores”. The simulations show that T S is dispersive above f (cid:38)
10 MHz, and furthermore that T S tends towards the functional form T S ∝ f (specif-ically T S × . /f (cid:39) φ C (cid:46)
50 vol% andhigh frequencies f (cid:38)
500 MHz. Remarkably, this func-tional form of the dispersion is consistent with the pre-viously reported measurements of polymers and bitumenwhere T LM ∝ f (specifically T LM × . /f (cid:39) T LM ∝ f is predicted athigh viscosities [1]. In the case of bitumen and polymers,a phenomenological model was proposed to account for T LM ∝ f dispersion at high viscosities [16]. Our find-ings suggest that the same phenomenological model mayapply to the surface relaxation of light hydrocarbons inan organic nano-confined matrix.The simulations show that the surface-relaxivities ρ and ρ are independent of φ C and f for φ C (cid:38)
50 vol%,as expected in conventional pores. However below φ C (cid:46) ρ and ρ increase with decreasing φ C ,which we interpret as the “dissolved” region where hep-tane is no longer in contact with other heptane moleculesdue to increased confinement in the polymer matrix. Wealso find that ρ decreases with increasing f , i.e. it isdispersive, in the dissolved region. The simulations alsoshow that T S /T S (cid:39) T S /T S (cid:38) ρ , ρ , and T S /T S show consis-tent trends with simulations as a function of φ C and f , however T S /T S is found to be systematically lowerfor simulations compared to measurements. This discrep-ancy is attributed to the fact that the simulated polymerin the mix has a lower molecular weight compared tothe measured polymer in the mix. Nevertheless, bothsimulations and measurements indicate that T S /T S in-creases with increasing confinement and increasing fre-quency, indicating that H- H dipole-dipole relaxationenhanced by nano-pore confinement is the dominantsurface-relaxation mechanism in saturated organic-richshales.1
ACKNOWLEDGMENTS
We thank Chevron, the Rice University Consortiumon Processes in Porous Media, and the American Chemi-cal Society Petroleum Research Fund (No. ACS-PRF-58859-ND6) for funding this work. We gratefully ac-knowledge the National Energy Research Scientific Com-puting Center, which is supported by the Office of Sci-ence of the U.S. Department of Energy (No. DE-AC02-05CH11231), and the Texas Advanced Computing Center(TACC) at The University of Texas at Austin, for HPCtime and support.
Effect of dissolved oxygen on measurements
Solubility is determined by the excess chemical poten-tial. Here we predict the excess chemical potential of O in the alkane/polymer mixture using βµ exO = ln x − ln p − ln (cid:104) e − β ∆ U (cid:105) (18)which is the quasichemical organization of the poten-tial distribution theorem [89–91]. In the above equa-tion, ln x is the work required to move the solvent outof the inner shell defined around the oxygen molecule, − ln p is the work required to created the empty in-ner shell in the solvent (in the absence of the solute),and − ln (cid:104) e − β ∆ U (cid:105) is the contribution from the interac-tion of the oxygen with the rest of the solvent when theinner shell is empty. Exploratory calculations show thatthe interaction between O and alkane/polymer matrixis dominated by inner-shell exclusion (steric effects) andlong-range van der Waals interactions. To this end, wechoose an inner shell cavity that is large enough to accom-modate the solute but small enough such that ln x = 0.We make a conservative choice of 2.9 ˚A for the inner shellradius. Here O was modeled using the three site modelthat has both partial charge and dispersion contribution Ref .For computing p , we first define a cubic grid of size13 × ×
13 ˚A . The grid sites are separated by 3 ˚A. (Thesimulation boxes are all about 40 ˚A and hence the gridsits entirely within the simulation cell.) Using the gridsites as reference, we find the number of occurrences forwhich no carbon atom of the solvent is within 2.9 ˚A of thegrid site. All such sites are archived for further analysis.This calculation also directly provides the probability p of finding a cavity of size 2.9 ˚A in the hydrocarbon ma-trix. For the cavities archived from the study above, wecompute − ln (cid:104) e − β ∆ U (cid:105) by particle insertion [89–92]. Forthese calculations, based on the convergence of the freeenergy, we used only a smaller subset (up to 1000 frames)of the overall 5000 frames. Note that for each site, wealso consider three random orientations of the oxygenmolecule, further enhancing the statistical reliability. We obtain the solubility of oxygen in heptane to be676 ppm. The experiments suggest that the solubilityof oxygen in n -heptane is around 132 ppm (by weight)[93, 94]. To obtain this value (assuming oxygen partialpressure of 0.21 atm.), we need βµ exO ≈ .
0. In energyunits, the difference between our computed value and 5.0is about 0 . in the alkane/polymer mixture. Consid-ering the relative solubility also serves to minimize theerrors in the absolute solvation values, that are off by afactor of 5.The relation between the measured T meas1 , and the in-trinsic T , of interest is given by the following expression[95]: 1 T meas1 , = 1 T , + C O T , . (19)The measured T (= T ) for pure heptane at ambientconditions are T = 2490 ms, 2620 ms, and 5580 ms at f = 2.3 MHz, 22 MHz, and 400 MHz, respectively [16].It was previously shown that T is roughly constant forsolvents with the molecular weight of heptane or higher[95]. In other words, T (= T ) in Eq. 19 is assumedto be independent of φ C . As shown in Fig. 7, we findgood agreement between simulated T , and measured T , assuming C O = 1 for all φ C in Eq. 19. FIG. 10. MD simulations of concentration C O of dissolvedoxygen in the polymer-heptane mix, as a function of heptanevolume fraction φ C . C O is defined relative to pure heptane( φ C = 100 vol%), under ambient conditions. However, as shown in Fig. 10, C O decreases ataround φ C (cid:39)
50 vol%, and therefore the assumption2
FIG. 11. Correlation cross-plot of measurements vs. simula-tions of T and T at f = 2.3 MHz, 22 MHz, 400 MHz, forvarious heptane volume fractions φ C . Symbols use C O =1 in Eq. 19 to determine T , from measured T meas1 , , whileleftmost point of the individual vertical lines uses C O valuesfrom Fig. 10. that C O = 1 for all φ C may not be accurate. In orderto quantify this effect, Fig. 11 shows the measured T , using the simulated C O values as a function of φ C fromFig. 10, the results of which are shown as the leftmostpoint of the vertical lines in Fig. 11. Using the C O values from Fig. 10 improves the comparison betweenmeasurements and simulations in the region φ C (cid:39) φ C >
70 vol%. We note however that the simulated C O in Fig. 10 are qualitative and designed to capturethe overall trends in C O , namely that C O decreasesaround φ C (cid:39)
50 vol%, which coincides with the discrep-ancy between measurements and simulations of T , inthat region. This gives credibility, though not certainty,to the proposition that variations in C O with φ C arethe cause of the discrepancy between measurements andsimulations at φ C (cid:39)
50 vol%.3
Diffusivity of Water in Kerogen
FIG. 12. Diffusion- T ( D - T ) measurement at ambient ofwater-saturated isolated kerogen pellets from a Kimmeridgeoutcrop (same kerogen as used in [38, 44]). Right panel shows D projection (red). Upper panel shows the T projection from D - T (red), along with full T distribution (blue). The pro-jection from D - T (9.5 pu) shows less signal intensity thanthe full T (35.8 pu) due to limitations in the D - T measure-ment. The diffusion coefficient (taken at the peak of the D distribution) of the inter-granular water is detectable, whilethe diffusion coefficient for dissolved water is not detectable.Dashed horizontal line is the bulk D for water, while thedashed diagonal line is the bulk alkane line [85]. ∗ [email protected][1] Bloembergen, N.; Purcell, E. M.; Pound, R. V. Relax-ation effects in nuclear magnetic resonance absorption Phys. Rev.
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