Interpretation of NMR Relaxation in Bitumen and Organic Shale using

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Interpretation of NMR Relaxation in Bitumen and Organic Shale using Polymer-Heptane Mixes Philip M. Singer, Zeliang Chen, Lawrence B. Alemany, George J Hirasaki, Kairan Zhu, Harry Xie, and Tuan D. Vo Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b03603 • Publication Date (Web): 10 Jan 2018 Downloaded from http://pubs.acs.org on January 10, 2018

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Interpretation of NMR Relaxation in Bitumen and Organic Shale using Polymer-Heptane Mixes Philip M. Singer,∗,† Zeliang Chen,† Lawrence B. Alemany,‡ George J. Hirasaki,† Kairan Zhu,¶ Z. Harry Xie,§ and Tuan D. Vo§ Department of Chemical and Biomolecular Engineering, Rice University, 6100 Main St., Houston, TX 77005, USA, Department of Chemistry, Rice University, 6100 Main St., Houston, TX 77005, USA, School of Electronic Engineering, Xi’an Shiyou University, No. 18, 2nd East Dianzi Rd., Xi’an, Shaanxi, 71006, P.R. China, and Core Laboratories, 6316 Windfern Rd., Houston, TX 77040, USA E-mail: [email protected]

Abstract One of the much debated mysteries in 1 H NMR relaxation measurements of bitumen and heavy crude-oils is the departure from expected theoretical trends at high viscosities, where traditional theories of 1 H-1 H dipole-dipole interactions predict an increase in T1 with increasing viscosity. However, previous experiments on bitumen and heavy crude-oils clearly show that T1LM (i.e. log-mean of the T1 distribution) becomes independent of viscosity at high viscosities; in other words, T1LM versus viscosity approaches a plateau. We report 1 H NMR data at ambient conditions on a set ∗

To whom correspondence should be addressed Rice University, Chemical and Biomolecular Engineering ‡ Rice University, Chemistry ¶ Xi’an Shiyou University § Core Laboratories †

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of pure polymers and polymer-heptane mixes spanning a wide range of viscosities (η = 0.39 cP ↔ 334,000 cP) and NMR frequencies (ω0 /2π = f0 = 2.3 MHz ↔ 400 MHz), and find that at high viscosities (i.e. in the slow-motion regime) T1LM plateaus to a value T1LM > ∝ ω0 independent of viscosity, similar to bitumen. More specifically, on a frequency-normalized scale, we find that T1LM > ×2.3/f0 ' 3 ms (i.e. normalized relative to 2.3 MHz), in good agreement with bitumen and previously reported polymers. Our findings suggest that in the high-viscosity limit, T1LM > and T2LM > for polymers, bitumen and heavy crude-oils can be explained by 1 H-1 H dipole-dipole interactions, without the need to invoke surface paramagnetism. In light of this, we propose a new relaxation model to account for the viscosity and frequency dependences of T1LM and T2LM , solely based on 1 H-1 H dipole-dipole interactions. We also determine the surface relaxation components T1S and T2S of heptane in the polymer-heptane mixes, where the polymer acts as the “surface” for heptane. We report ratios up to T1S /T2S ' 4 and dispersion T1S (ω0 ) for heptane in the mix, similar to previously reported data for hydrocarbons confined in organic matter such as bitumen and kerogen. These findings imply that 1 H-1 H dipole-dipole interactions enhanced by nano-pore confinement dominate T1S and T2S relaxation in saturated organic-rich shales.

Keywords Heavy crude-oil, Viscosity, Surface relaxation, Surface relaxivity, Nano-pore confinement

1

Introduction

NMR is a powerful technique for measuring the viscosity (and composition) of crude oils and mineral oils, both in the lab and down-hole. Research in this field is rich, and dates back almost 30 years. 1–40 Nevertheless, there is much debate about the 1 H NMR relaxation mechanism in crude oils, especially crude oils which contain asphaltenes. While the NMR signal from the asphaltenes themselves are not observable at low field (. 20 MHz) due to 2


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experimental limitations, the general belief is that the remaining observable NMR signal from the maltenes is affected by paramagnetic sites on the surface of the asphaltenes. The maltenes diffuse in and out of the asphaltene aggregates, during which time they come into contact with the paramagnetic sites on the asphaltene surface. 32,34–36,38 Another explanation for the 1 H NMR relaxation mechanism in crude oils and mineral oils is the 1 H-1 H dipole-dipole interactions, 1–5,9,13,25,29,30 without the need to invoke surface paramagnetism from asphaltenes. In the case of bitumen and heavy (i.e. high-viscosity) crudeoils, we postulate that the 1 H-1 H dipole-dipole interactions of the maltenes are greatly enhanced while inside the confined transient “pores” of the asphaltene macro-aggregates. NMR relaxation from 1 H-1 H dipole-dipole interactions are well understood for single-component bulk fluids, and is often modeled using the traditional BPP (Bloembergen, Purcell, and Pound) 41 and Torrey 42 theories for hard-spheres. While BPP theory works well for lowviscosity (i.e. light) crude-oils in the fast-motion regime, it is inadequate for bitumen and heavy crude-oils in the slow-motion regime. While the measured T2LM of crude oils roughly follows the BPP model at high viscosity, the measured T1LM shows a significant deviation at high viscosity. More specifically, the BPP model predicts that T1LM should increase with viscosity in the slow-motion regime; however, measurements clearly show that T1LM plateaus. 2,4,25,30 This conundrum lead to alternative theories that the T1LM plateau versus viscosity was due to surface paramagnetism from asphaltenes. 35 In this report, we maintain that the NMR relaxation in bitumen and heavy crude-oils is due to 1 H-1 H dipole-dipole interactions. To show this, we study pure polymers and polymerheptane mixes that do not contain paramagnetic impurities, or at least negligible amounts thereof. The study of NMR relaxation and diffusion in polymers using field-cycle relaxometry, double-quantum spectroscopy, and MD (molecular dynamics) simulations is a rich area of research that also dates back 30 years. 43–60 Polymer-alkane mixes (or “blends”) have previously been shown to be good systems for studying diffusion by NMR. 61 In this study, we compare the NMR relaxation of polymers (and polymer-heptane mixes) to bitumen and

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heavy crude-oils, and we find the same frequency-normalized T1LM > ×2.3/f0 ' 3 ms plateau at high viscosity, which confirms that paramagnetism is not the dominant interaction at high viscosity for either polymers, bitumen or heavy crude-oils. Our plateau also agrees with previously reported data at high frequency and viscosity (i.e. low temperature) for poly(butadiene) and tristyrene polymers, 48 where T1R> ×2.3/f0 ' 3 ms was also shown (using the units in this report). We then proceed to fit our polymer and polymer-heptane data by modifying the BPP model for 1 H-1 H dipole-dipole interactions to incorporate internal motions of the non-rigid polymer-branches, and a distribution in correlation times inherent in these multi-component systems. Our new model yields two distinct correlation times; a fast τL (' 10’s ps) correlation time for the local motion of the non-rigid branches, and a slower τR (' 10’s µs) correlationtime for the rotational motion of the entire polymer. We then compare our new model to previously published bitumen and heavy crude-oil data. Another (more recent) conundrum is the origin of the NMR surface relaxation for hydrocarbons confined in organic-rich shales, where large ratios T1S /T2S & 4 for the hydrocarbons were previously reported, 62–82 as well as dispersion T1S (ω0 ) 70,83 for the saturating hydrocarbons. In order to shed light on this, we analyze the isolated heptane signal in the polymer-heptane mixes, and we interpret the heptane relaxation as surface relaxation T1S and T2S , where the polymer is considered as the “surface” for heptane. We determine the surface-relaxivity parameter for heptane in the organic transient “pores” created by the polymers, which by analogy gives us insight into the surface-relaxation mechanism for hydrocarbons in bitumen and kerogen. Our premise behind this analogy is that the organic matter in bitumen and kerogen is essentially made up of cross-linked polymers, where cross-linking turns the liquid polymers into highly-viscous bitumen or solid kerogen (with increasing crosslinkage). We show that heptane absorbed in polymers exhibits ratios up to T1S /T2S ' 4 and dispersion T1S (ω0 ), consistent with what was previously found for hydrocarbon-saturated organic-shale. 70,83 Our findings imply that 1 H-1 H dipole-dipole interactions dominate in sat-

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urated organic-shale, 75,82 without the need to invoke surface paramagnetism. The rest of this manuscript is organized as follows: Section 2 presents the methodology for experiments and the nomenclature for the relaxation models; Section 3 presents the relaxation versus viscosity data, along with fits from the new model, and comparison with previously published bitumen and heavy crude-oil data; Section 4 presents the surface relaxation of heptane in the polymer-heptane mix, along with a model for pore fluid versus absorbed heptane in the mix; followed by the conclusions in Section 5.

2 2.1

Methodology Experimental

The Brookfield viscosity standards used in this study are listed in Table 1. The composition of the standards ranged from high molecular-weight polymers at high viscosity, to petroleum distillates at low viscosity. All the polymer samples were clear transparent liquids, while the petroleum distillates had a light-brown color. In the case of the polymers, the average molecular weight Mw and poly-dispersivity index Mw /Mn listed in Table 1 were measured using gel permeation chromatography (GPC) based on a polystyrene calibration, using an Agilent Technologies 1200 module with tetrahydrofuran (THF) at 1 mL/min. The data in Table 1 indicate that the polymers are highly dispersive, up to Mw /Mn ' 3.11 in the case of B360000 poly(isobutene). The large poly-dispersivity is designed to minimize the shear-rate dependence of viscosity, thereby making a good viscosity “standard”, and a good comparator with NMR measurements (which are measured at zero shear-rate). By comparison, monodispersive poly(butadiene) with Mw /Mn . 1.08 is often used in polymer-NMR research, 48,49 however it is shear thinning. In the case of the three poly(isobutene) polymers in Table 1, the viscosity fit well to the functional form η = A Mwα , with α ' 2.4 and A ' 1.07 × 10−4 . The exponent α ' 2.4 is less than for mono-dispersive poly(isoprene) where α ' 3.9, 84 which is most likely a result of the large poly-dispersivity of the poly(isobutene) polymers. 5


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Table 1: Brand name, composition, viscosity η at 25 o C, density ρ at 25 o C, average molecular weight Mw , and poly-dispersivity index Mw /Mn , for the Brookfield viscosity standards used in this study. The polymer-heptane mixes were made using B360000 poly(isobutene). Brand name

Composition

η (25 o C) ρ (25 o C)

Mw

(cP)

(g/cm3 )

(g/mol)

Mw /Mn

B360000

Poly(isobutene)

333,400

0.885

9,436

3.11

B73000

Poly(isobutene)

68,070

0.888

4,368

2.53

B10200

Poly(isobutene)

10,650

0.875

2,256

2.13

B1060

Poly(1-decene)

1,043

0.841

4,204

1.49

B200

Petroleum distillate

196

0.873

B29

Petroleum distillate

29

0.857

The polymer-heptane mixes were prepared using the Brookfield B360000 poly(isobutene) polymer and n-heptane (n-C7 H16 , or “C7” for short). The mixtures were prepared with heptane volume-fractions φC7 ranging from 0 vol% to 100 vol%. The volume fractions were computed using the mass of polymer and heptane used, together with the measured densities of the polymer (ρpoly = 0.885 g/cm3 ) and heptane (ρC7 = 0.684 g/cm3 ). The desired amounts of polymer and heptane were placed in a 1.0 inch diameter glass vial, and sealed with a Teflon spacer in the cap, which resulted in negligible heptane loss over time. The samples were mixed by random re-orientation of the glass vial once a day, and the completeness of mixing was monitored with NMR T2 at a resonance frequency of f0 = 2.3 MHz. The T2 distributions were stable after ' 4 weeks of mixing, indicating that mixing was complete after that time. The viscosity measurements were made using a Brookfield AMETEK viscometer, using a selection of cylindrical spindles to cover the viscosity range under investigation. The viscosities did not depend on shear-rate (within experimental uncertainties), thereby making them suitable viscosity “standards” for comparing with NMR measurements (which are measured at zero shear-rate). The viscosity measurements were made at both ambient temperature (' 25 o C), and at an equilibrated temperature of 30 o C using a circulation

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heat-bath. For sample loading, the glass vial was opened and the polymer-heptane mix was placed inside the viscometer probe. The viscosity data was acquired within minutes of opening the glass vial, and repeated to ensure reproducibility. Electron paramagnetic resonance (EPR) measures the concentration of paramagnetic ions plus the (weight equivalent) concentration of free radicals (i.e. unpaired valence electrons), which both contribute to NMR paramagnetic relaxation. The 1 GHz EPR apparatus used here was designed for measuring paramagnetism in the asphaltenes of crude oils. The EPR data on the Brookfield viscosity standards indicated < 100 ppm paramagnetic impurities, i.e. the signal was below the detection limit of the apparatus. This is at least an order of magnitude less than previously reported EPR data which indicate paramagnetic concentrations of ' 5,000 ppm for organic shales 83 and asphaltenes. 34,35 One can then infer a paramagnetic concentration of ' 1,000 ppm for bitumen from EPR, given that the asphaltene concentration in Athabasca bitumen (for instance) is ' 20 wt%. 85 This paramagnetic concentration is consistent with elemental analysis from ICP-MS (inductively coupled plasma mass spectrometry) for Athabasca bitumen, 85 which also predicts ' 1,000 ppm (after summing over transition metals).

2.1.1

NMR Relaxation Distributions

The NMR measurements at ω0 /2π = f0 = 2.3 MHz were made with a GeoSpec2 from Oxford Instruments, with a 29 mm diameter probe. The f0 = 22 MHz measurements were made with a special spectrometer from MR Cores at Core Laboratories, with a 30 mm diameter probe. The polymer-heptane mix samples were measured at an equilibrated temperature of 30 o C, while the pure polymer samples were measured at ambient conditions (' 25 o C). The NMR probe sizes allowed for measurements without opening the glass vials. 2D T1 -T2 correlation data were acquired, 86 however in the present case the 2D correlation maps (not shown) did not provide more information than the separate 1D distributions. The T2 data were encoded using a CPMG (Carr-Purcell-Meiboom-Gill) sequence with an echo spacing

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Figure 1: (a) T1 distributions and (b) T2 distributions of pure Brookfield viscosity standards (Table 1) at different NMR frequencies, at ambient (' 25 o C). Viscosities at 25 o C are also listed. At 400 MHz, only the average value hT2 i = 1/π∆f (represented as a delta function) is displayed. of TE = 0.1 ms. The T1 data were encoded using an inversion-recovery sequence, and the amplitude was obtained from the first echo. The NMR measurements at f0 = 400 MHz were made with a Bruker Avance spectrometer. For sample loading, the glass vial was opened and the polymer-heptane mix was placed in a 5 mm NMR tube, with a Teflon vortex plug directly above the mix to limit heptane evaporation. The NMR data were acquired within minutes of opening the glass vial. The Teflon vortex plug ensured minimal heptane loss within that time. The T1 data were encoded using an inversion-recovery sequence, and the amplitude was obtained from the first point in the FID (free induction decay). The T1 and T2 distributions of the pure viscosity standards are shown in Fig. 1, while that of the polymer-heptane mixes are shown in Fig. 2. The T1 and T2 distributions were determined using a 1D inverse Laplace transform. 87,88 The SNR (signal-to-noise ratio) for all the data were large (SNR > 500). The measured hydrogen index HI of all the data in Figs.

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Figure 2: (a) T1 distributions and (b) T2 distributions of (poly(isobutene) B360000) polymerheptane mixes at different NMR frequencies, at 30 o C for 2.3 MHz and 22 MHz, and at ambient (' 25 o C) for 400 MHz. Viscosities at 30 o C are listed, from pure polymer (top) to pure heptane (bottom). At 400 MHz, only the average value hT2 i = 1/π∆f (represented as a delta function) is displayed. 1 and 2 were in the range HI = 1.0 ↔ 1.2 v/v, indicating that all the signal was captured by NMR. In the case of the highest viscosity standard B360000, the measured value HIpoly = 1.17 v/v was consistent with known HI correlations 5 which predict HIpoly = 1.14 v/v, using ρpoly = 0.885 g/cm3 and a hydrogen:carbon ratio of R = 2. The fluid viscosities are compared with the log-mean (i.e. geometric-mean) values of the T1 and T2 distributions, defined as T1LM and T2LM , 8 respectively, where T1LM,2LM = 9


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exphln T1,2 i. By contrast, at f0 = 400 MHz (where a CPMG echo train was not an option on our spectrometer) the average value hT2 i was determined from the relation hT2 i = 1/π∆f , where ∆f is the full-width at half-maximum of the spectrum. The pure standards in Fig. 1 show large distributions in relaxation times, especially at lower frequencies, while the polymer-heptane mixes in Fig. 2 often show two distinct peaks; one fast-relaxing peak for the polymer in the mix, and one slow-relaxing peak for the heptane in the mix.

2.1.2

Heptane Signal in the Mix

The analysis presented in the Heptane Surface-Relaxation section requires the separation of heptane signal from the polymer signal. In Fig. 3 we compare the volume fraction φC7 by NMR by NMR using the following relations, respectively: mass against φC7

MC7 /ρC7 , MC7 /ρC7 + Mpoly /ρpoly AC7 /HIC7 = . AC7 /HIC7 + Apoly /HIpoly

φC7 = φNMR C7

(1)

Mpoly and MC7 are the mass of polymer and heptane used during sample preparation, respectively. ρpoly = 0.885 g/cm3 and ρC7 = 0.684 g/cm3 are the densities of polymer and heptane, respectively. Apoly and AC7 are the NMR signal amplitudes of polymer and heptane using a T1,cutoff in Fig. 2, respectively. HIpoly = 1.14 v/v and HIC7 = 0.98 v/v are the hydrogen indices of polymer and heptane, respectively. The results in Fig. 3 shows good agreement NMR between φC7 and φC7 , indicating a good separation of heptane signal by NMR relaxation. NMR For f0 = 400 MHz, φC7 could not be determined below φC7 < 30 vol% due to merging of

the polymer and heptane T1 signal. For f0 = 2.3 MHz and 22 MHz, similar results for φC7 were obtained using T2 data and a T2,cutoff instead. In order to better understand and characterize the various relaxation mechanisms for heptane in the polymer-heptane mixes, the relaxation components for heptane T1C7 and

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NMR in polymer-heptane mixes according Figure 3: Cross-plot of heptane volume-fraction φC7 to NMR data (y-axis) at different NMR frequencies, versus φC7 by mass (x-axis), both defined in Eq. 1.

T2C7 can be expressed as such: 1 T1C7 1 T2C7

1 1 1 + + , T1B T1S T1O2 1 1 1 = + + . T2B T2S T2O2 =

(2)

The bulk components for pure heptane are defined as T1B and T2B , and the surface components for heptane are defined as T1S and T2S (where the surface in this case is the polymer itself). NMR relaxation of low-viscosity hydrocarbons (i.e. with T1LM , T2LM > 500 ms) is known to be affected by dissolved oxygen (O2 ). 18 Molecular oxygen is paramagnetic, which results in additional relaxation terms (T1O2 and T2O2 ) whose rate is proportional to the concentration of dissolved oxygen. 39 For alkanes at ambient conditions, the concentration of dissolved oxygen is typically ' 400 ppm, 39 which can be removed (i.e. de-oxygenated) by the freeze-pump-thaw technique. Note that fluids down-hole do not contain dissolved oxygen, and therefore down-hole NMR logs are not affected by oxygen.

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The correlation between T1B , T2B and η/T (viscosity over absolute temperature) for alkane liquids in the fast-motion regime is given by the well-established relation: 9

T1B = T2B = 9.56

T ms (alkanes). η

(3)

This results in T1B = T2B = 7,320 ms for n-heptane at 25 o C (298 o K), where η = 0.39 cP. In the case of pure heptane (bottom most data in Fig. 2), the surface component does not exist, and only T1B , T2B and T1O2 , T2O2 exist. Given the known values for T1B , T2B from Eq. 3, the T1O2 , T2O2 components can then be determined for pure heptane at the various NMR frequencies as such: T1O2 = T2O2 = 2,490 ms (2.3 MHz), T1O2 = T2O2 = 2,620 ms (22 MHz),

(4)

T1O2 = 5,580 ms (400 MHz). The larger value for T1O2 at f0 = 400 MHz is expected since the fast-motion regime no longer applies, given the typical electron-spin correlation-time of ' 1 ns. 89 Assuming the same T1O2 and T2O2 values for heptane in the polymer-heptane mixes, the relaxation components from oxygen were removed (i.e. rates were subtracted) from the log-mean values T1LM and T2LM .

2.2

Relaxation Models

In this section, the nomenclature for the BPP (Bloembergen, Purcell, Pound) 41 and LS (Lipari, Szabo) 90,91 models are presented, followed by an extension to incorporate a distribution in correlation times. All of these elements will be used to model the relaxation versus viscosity data in the next section. All the models presented here are based on intramolecular 1 H-1 H dipole-dipole interactions, 92,93 without the need to invoke surface paramagnetism. Another contribution to the relaxation is the intermolecular 1 H-1 H dipole-dipole relaxation, which is determined by translation of the molecules. 42 According to MD (molecular dynamics) 12


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simulations, with increasing chain length, intramolecular relaxation increasingly dominates over intermolecular. 94 However, recent studies in polymers indicate that intermolecular relaxation may dominate at low ( 1 MHz) frequencies. 59 Given the frequency range here (≥ 2.3 MHz), for simplicity and for clarity, we only consider intramolecular relaxation in this report, but note that the overall conclusions from using our new model (Section 3) would not change significantly if we were to include intermolecular relaxation as well. The theory behind 1 H-1 H dipole-dipole interactions are well understood; in fact, recent MD simulations of T1B = T2B for bulk alkanes (including n-heptane) have shown good agreement with measurements, 94 without any adjustable parameters in the interpretation of the simulation data. For 1 H-1 H dipole-dipole interactions, the underlying expressions (which do not assume a molecular model) for T1 and T2 in an isotropic system are given by: 92,93 1 = J(ω0 ) + 4J(2ω0 ), T1 5 1 3 = J(0) + J(ω0 ) + J(2ω0 ), T2 2 2

(5)

where J(ω) is the spectral density of the 1 H-1 H dipole-dipole interaction at angular frequency ω. The measured T1 and T2 are given by J(ω) at the measured NMR frequency ω0 (= 2πf0 ) and 2ω0 , plus T2 are has an additional zero-frequency component J(0). 2.2.1

BPP Model

The traditional BPP model 41 for intramolecular 1 H-1 H dipole-dipole relaxation assumes the molecules are hard-spheres of radius RR , rotating with a correlation time of τR , where the subscript R refers to molecular rotation. Using the BPP model leads to the following

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expression for the intramolecular spectral-density JR (ω): 1 2τR , JR (ω) = ∆ωR2 3 1 + (ωτR )2 4π 3 η τR = R , 3kB R T NR 9 µ0 2 2 4 1 X 1 2 ∆ωR = . ~γ 6 20 4π NR i6=j rij

(6) (7) (8)

The relation between τR and η/T in Eq. 7 is the traditional Stokes-Einstein relation for rotational diffusion of a hard sphere of radius RR . ∆ωR2 is the strength of the intramolecular interaction, also known as the“second moment” of the 1 H-1 H dipole-dipole interaction. According to MD simulations, the square-root of the second moment in Eq. 8 for n-heptane is ∆ωR /2π = 20.0 kHz (see 94 for more details), and does not change significantly with increasing chain length. The only free parameter in the BPP model (Eqs. 5, 6, 7, 8) required to match the measured T1B = T2B = 7,320 ms to the measured viscosity η= 0.39 cP for bulk n-heptane is the Stokes-Einstein radius, RR = 1.85 ˚ A. The resulting BPP model is shown as gray lines in Fig. 5. One important consequence of the BPP model shown in Fig. 5 is the transition from the fast-motion (low-viscosity) regime where ω0 τR 1, to the slow-motion (high-viscosity) regime where ω0 τR 1. In the fast-motion regime T1 /T2 = 1, and T1 , T2 are both inversely proportional to η/T . The ratio T1 /T2 = 1 in the fast-motion regime (ω0 τR 1) is a consequence of the fact that Eq. 6 simplifies to JR (ω) = 31 ∆ωR2 2τR . In the slow-motion regime T1 /T2 1; furthermore, T1 becomes proportional to η/T instead and depends on frequency (i.e. is dispersive). The transition from fast-motion to slow-motion, and the

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corresponding minimum in T1 , is given by the following relation: 93

ω0 τR = 0.615 (fast ↔ slow), η 2.3 cP/K, = 22.1 T f ↔s f0 2.3 ηf ↔s = 6,590 cP (25 o C), f0

(9) (10) (11)

which using Eq. 7 corresponds to a fast f ↔ s slow transition at (η/T )f ↔s (with f0 in units of MHz), or ηf ↔s at ambient conditions. As shown below, the measurements suggest that the slow-motion regime is best defined by η/T ×f0 /2.3 & 102 cP/K, which is close to Eq. 10. Furthermore, the corresponding fast-motion regime η/T ×f0 /2.3 . 102 cP/K is split into the ultra fast-motion regime η/T ×f0 /2.3 . 10−2 cP/K, and the intermediate regime 10−2 cP/K . η/T ×f0 /2.3 . 102 cP/K. Note that the transition point in viscosity is inversely proportional to f0 . As such, one common way to plot T1 and T2 data versus η/T is to multiply η/T by f0 /2.3 (for f0 in units of MHz), and divide T1 and T2 by f0 /2.3. 30 As shown in Fig. 6, this has the effect of collapsing the BPP data for different frequencies onto one universal curve, normalized to a chosen frequency of 2.3 MHz. Note that a similar frequency-normalization for T1 is used for measurements of the normalized susceptibility χ˜00(ω) = ω/T1(ω). 47–51,55 2.2.2

LS Model

Building on the BPP model for hard-spheres, a common refinement to account for the internal motions of non-rigid molecules using the LS (Lipari, Szabo) model. 90,91 In the LS “modelfree” approach, the spectral density JR (ω) in Eq. 6 is modified as such: 2τR 1 2τe 2 2 2 JLS (ω) = ∆ωR S + 1−S , 3 1 + (ωτR )2 1 + (ωτe )2 1 1 1 = + . τe τR τL

15


(12)

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For clarity the last term in JLS (ω) is written using τe (effective), where τe is defined in the next line as a function of the local correlation-time τL of the polymer branches, and rotational correlation-time τR of the entire polymer. The LS spectral-density JLS (ω) has two additional parameters compared to BPP, (1) the generalized order-parameter S 2 , and (2) the local correlation-time τL . The order parameter is a measure of the rigidity of the polymer molecule, where S 2 = 1 for completely rigid molecules with no internal motion of the polymer branches, and S 2 = 0 for completely non-rigid molecules with full internal motion of the polymer branches. Note that complete non-rigidity S 2 = 1 reverts JLS (ω) to the BPP model JR (ω). The second free-parameter is the local correlation-time τL , which characterizes the fast τL (' 10’s ps, see fit below) motions of the polymer branches, in conjunction with the much slower (independent) rotational correlation-time τR which characterizes the slow τR (' 10’s µs, using Eq. 7) rotation of the entire polymer molecule. More recently, the implementation of the LS model (with dynamic order parameter S 2 ) in polymers distinguishes the polymer dynamics (S 2 term in Eq. 12) from the independent glassy dynamics ((1 − S 2 ) term in Eq. 12), given that the polymers are above the glass transition temperature T > Tg . In the case of poly(butadiene), the dynamic order parameter is small S 2 . 0.15, 47–49 in agreement with the values reported here. Interestingly, an alternative interpretation of the free parameters S 2 and τL proposes that the fast rotation of the methyl groups act as relaxation sinks for the macro-molecule. 95 In such a scenario, τL (' 10’s ps) is the fast rotation of the methyl groups, while S 2 is the fraction of methyl 1 H (versus methylene 1 H) in the macro-molecule.

2.2.3

Distribution in Correlation Times

The LS model is a generalized form of the theory originally put worth by Woessner for describing spin relaxation processes in a two-proton system undergoing anisotropic reorientation. 96 In the special case of a methyl group where the 1 H-1 H vector is perpendicular to the axis of internal motion, then Eq. 12 is valid with 2 terms (i.e. 2 correlation times)

16


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Energy & Fuels

Figure 4: Auto-correlation function G(β)(t) as a function of β in the range 0.5 ≤ β ≤ 1, derived from the inverse Fourier transform (defined in Eq. 14) of J (β)(ω) (defined in Eq. 13). The x-axis is normalized by τ and the y-axis is normalized by G(1)(0). and S 2 = 1/4 exactly. The more general non-perpendicular case results in 3 terms (i.e. 3 correlation times) in the spectral density. The even more general case of internal motions on an ellipsoid (rather than on a sphere as in the methyl case above) results in 8 terms (i.e. 8 correlation times). 97 It is clear from Woessner’s theories that as the internal motions in the molecule become more complex, the distribution in correlation times becomes more pronounced. There are many models that account for distributions in correlation times and the corresponding terms in the spectral density. 98 In the case polymers above the glass transition temperature T > Tg , the Cole-Davidson (CD) distribution function has been successfully used for frequency-temperature superposition of susceptibility χ˜00(ω) = ω/T1(ω) data from NMR fieldcycling relaxometry. 57,58 Other previously used distribution functions for polymers include the generalized gamma (GG) 48 and Kohlrausch-Williams-Watts (KWW) 58 functions. In light of these techniques, we propose the following model to account for the distribution

17


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in correlation times in the spectral density: 1 2τ J (β)(ω) = ∆ωR2 , 3 1 + (ωτ )2β

(13)

where τ is the generalized correlation time for the underlying distribution. The consequence of changing the exponent β < 1 in the denominator of Eq. 14 is to introduce an underlying distribution in correlation times τ . Changing β < 1 in J (β)(ω) results in a corresponding distribution in the autocorrelation function G(β)(t), defined as the real part of the inverse Fourier transform, as such: 2 G (t) = 2π (β)

Z

∞

J (β)(ω) cos (ωt) dω

(14)

0

Fig. 4 shows G(β)(t) as a function of β in the range 0.5 ≤ β ≤ 1, which clearly shows that decreasing β < 1 results in an increasingly “stretched” exponential (i.e. multi-exponential) decay. Justification for this model can be seen by comparing Fig. 4 with MD simulations of n-alkanes in Fig. 2a of Ref. 94. The similarity is evident; namely, the longer the chain length of the n-alkane, the lower the β exponent should be, indicating an larger distribution in correlation times with increasing chain-length. MD simulations of the polymers in question are currently underway to elucidate their complex autocorrelation function. It is also informative to note that an analytic expression exists for G(β)(t) in the two extreme cases of G(1)(t) and G(1/2)(t), 99 namely:

G(1)(t) G(1)(0) = exp (−t/τ ) , 2 (t) G(1)(0) = sin(t/τ ) π

(1/2)

G

(15)

Z

∞

t/τ

sin(x) 2 dx + cos(t/τ ) x π

Z

∞

t/τ

cos(x) dx. x

(16)

The case of β = 1 for G(1)(t) is simply the BPP model, resulting in a single-exponential decay. The case of β = 1/2 for G(1/2)(t) is the highly stretched case, which will be used in the next

18


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Energy & Fuels

section to account for the observed viscosity and frequency dependence T1LM > ∝ ω0 at high viscosity for the polymers, bitumen and heavy crude-oils. Note that G(1/2)(t) starts to diverge below t/τ = 0.14, therefore the model is valid in the region t/τ & 0.14. Correspondingly, this implies that the model for J (β)(ω) is valid in the region ωτ . 7.1, which is fulfilled for the local correlation-time τ ' τL ' 10 ps at 400 MHz (i.e. ω0 τL ' 0.03). A similar range of validity exists for the proposed autocorrelation function for 1D diffusion G(1D)(t) ∝ t−1/2 of moderately viscous crude-oils (' 100 cP), which predicts a frequency 1/2

dependence of T1LM ∝ ω0 . 21,32,35,36,38 It is also interesting to note that G(1/2)(t) ∝ t−2 for t/τ 10 in Eq. 16, implying a steeper power-law decay for n-alkanes than either the Rouse or entangled regimes in polymers. 53,59 More rigorous analytical techniques and theories have also been used to interpret the autocorrelation function of entangled polymers, 53,54,60 including the more complex theory required to interpret T2 relaxation in entangled polymers. 2.2.4

Cross-Relaxation

Despite the distribution in correlation times and subsequent stretched decay of G(β)(t) in Fig. 4, the resulting T1 and T2 relaxation times will be single valued (i.e. mono-exponential). However, as our recent MD simulations have shown, for inequivalent 1 H’s along a long hydrocarbon chain, the methyl versus methylene 1 H (for instance) will have significantly different autocorrelation functions (i.e. different β and τ values). Nevertheless, these differences in autocorrelation function will be averaged out (i.e. “washed out”) due to cross-relaxation and associated “spin diffusion” effects. 95,100 The averaged autocorrelation function will once again result in single valued T1 and T2 relaxation times, provided cross-relaxation is efficient. However, except for the case of pure heptane, it is clear that the T1 and T2 data in Figs. 1 and 2 have finite distribution due to incomplete cross-relaxation. 95,100 In the case of the polymer-heptane mix, the degree of averaging (i.e. merging) of the polymer T1 and heptane T1 peaks is more pronounced at higher frequency and viscosity. Similar observations were previously reported, 95 where the degree of cross-relaxation (and spin diffusion)

19


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is expected to increase with increasing frequency and viscosity (i.e. molecular weight). In our case, imperfect cross-relaxation is most likely a result of chain-ends versus chain-centers, and/or polydispersity of the polymer melt, distributed over length-scales larger than the spin-diffusion length-scales. This then leaves the question of how to determine the mean of the T1,2 distributions in order to compare with the (single valued) viscosity. It is known that in the case of efficient crossrelaxation, the average autocorrelation function results in single values T1R,2R determined by the mean rate 1/T1R,2R = h1/T1,2 i. 95 In the case of incomplete cross-relaxation in crude oils, the constituent-viscosity-model shows that the log-mean T1LM,2LM = exphln T1,2 i is the best parameter for correlating T1,2 with viscosity. 8 For the data in Figs. 1 and 2, we find that T1LM,2LM ' T1R,2R at both low and high viscosity, while T1LM,2LM . T1R,2R . 3 T1LM,2LM at intermediate viscosity. Given that T1LM,2LM ' T1R,2R at high viscosity (i.e. the value of the T1 plateau is the same), the petrophysical convention of T1LM,2LM is used.

3

Relaxation versus Viscosity

The log-mean values T1LM and T2LM of the distributions from Figs. 1 and 2 are plotted in Fig. 5 versus η/T , where the oxygen component (Eq. 4) has been subtracted from the relaxation data. It is evident from Fig. 5 that T1LM plateaus as a function of η/T , and that the T1LM plateau value, defined as T1LM > , is proportional to frequency, i.e. T1LM > ∝ ω0 . Furthermore, T1LM > starts at lower η/T values with increasing ω0 , consistent with the shift in fast to slow-motion regimes defined in Eq. 10. This suggests that the T1LM plateau exists only in the slow-motion regime. It is also found that the pure standards (open symbols) and the polymer-heptane mix (solid symbols) plateau to the same T1LM > ∝ ω0 value, albeit along different paths. These findings are in stark contrast to the expectation from the BPP model (gray lines), which predicts that T1LM should increase with η/T in the slow-motion regime, defined as

20


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Figure 5: (a) T1LM and (b) T2LM of pure viscosity standards (open symbols) and polymerheptane mixes (solid symbols) at different NMR frequencies, versus viscosity over absolute temperature (η/T ). Gray lines are the BPP model for T1 (solid) and T2 (dashed) at 2.3 MHz, while colored (solid) lines are fits using new model (Eq. 17) at different NMR frequencies (same colors as data). Relaxation contributions from dissolved O2 (Eq. 4) have been subtracted from the data. the region where η/T ×f0 /2.3 & 102 cP/K (similar to Eq. 10). Meanwhile T2LM in Fig. 5 continues to decrease versus η/T , albeit with a shallower slope than predicted by BPP. For both T1LM and T2LM in the mix, departure from the BPP model is found after η/T & 10−2 cP/K, corresponding to heptane volume-fractions φC7 < 90 vol%. The BPP model has been successful for systems at constant composition (glycerol in the original publication 41 ), where η/T was varied by varying temperature, also in the case of certain polymers. 101 However, it is evident that the BPP model is inadequate at high viscosity in the present case where η/T 21


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Figure 6: Same data and fits as Fig. 5, except the x-axis has been multiplied by f0 /2.3 and the y-axis has been divided by f0 /2.3, with f0 in MHz. This plotting format is referred to as frequency-normalizing to 2.3 MHz. is varied by varying composition, at constant temperature. It should be pointed out that the BPP paper 41 also had some (although limited) T1 data where η/T was varied by varying composition (glycerol-water mixes), at constant temperature. The glycerol-water mix data showed a similarly shallow slope to the polymer-heptane mix in Fig. 5, and also showed no sign of increasing with increasing viscosity. The plateau in T1LM at high viscosity in Fig. 5 and the corresponding departure from the BPP model are consistent observations with previous bitumen and heavy crude-oil data. 30 In order to compare, we plot our polymer data on a frequency-normalized scale in Fig.

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Energy & Fuels

6. Similar frequency-normalization for T1 is also used for measurements of the normalized susceptibility χ˜00(ω) = ω/T1(ω). 47–51,55 Our data clearly shows that the plateau value T1LM > × 2.3/f0 ' 3 ms (frequency-normalized relative to 2.3 MHz) is independent of frequency (i.e. T1LM > ∝ ω0 ) in the slow-motion regime η/T × f0 /2.3 & 102 cP/K (i.e. the high-viscosity regime), which is the same as found for bitumen and heavy crude-oil. 30 This implies that T1LM >×2.3/f0 ' 3 ms is a “universal” value, independent of the concentration of paramagnetic impurities in the sample. If surface paramagnetism were the cause, the T1LM > would depend on the concentration of paramagnetics; however, this is clearly not the case. The logical conclusion is that T1 relaxation at high viscosity is due to 1 H-1 H dipole-dipole interactions, for pure-polymer/polymer-heptane mixes, bitumen, and heavy crude-oils. It is also the case that χ˜00(ω) = ω/T1(ω) shows a similar plateau at high frequency and viscosity (i.e. low temperature) for poly(butadiene) and tristyrene polymers 48 (Figs. 7 and 8), more specifically T1R> ×2.3/f ' 3 ms was also shown (using the units in this report). Meanwhile, T2LM in Fig. 6 continues to decrease versus η/T , albeit with a shallower slope than predicted by BPP, which is also consistent with previous bitumen and heavy crude-oil data.

3.1

New Model for Relaxation versus Viscosity

Given the clear departure from the traditional hard-sphere BPP model in Figs. 5 and 6, we now propose a new model to account for the η/T and ω0 dependences in T1LM and T2LM for our polymer-heptane mixes. The new model is based on the intramolecular 1 H-1 H dipoledipole interactions in Eq. 6; however, (1) we account for internal motions of the non-rigid polymer branches using the LS model in Eq. 12, and (2) we modify the LS model to account for a distribution in correlation times in Eq. 13. This results in the following new model: 2τe 2τR 1 2 2 2 , + 1−S JP (ω) = ∆ωR S 3 1 + ωτR 1 + ωτe 1 1 1 = + . τe τR τL 23


(17)

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Page 24 of 51

The motivation for introducing a distribution in correlation times was shown in Section 2.2.3 by using β < 1. Furthermore, the motivation behind using the particular value β = 1/2 in Eq. 17 is because it correctly leads to T1LM being independent of η/T in the slow-motion regime, i.e. JP (ω) is independent of τR in the limit ω0 τR 1. The subscript P in JP (ω) refers to recovering the P lateau defined as T1LM > , thereby distinguishing it from the JR (ω) in Eq. 6 and JLS (ω) in Eq. 12. Using Eq. 17 (along with Eqs. 5, 7 and 8) to fit the data in Figs. 5 and 6 results in the solid lines, respectively. Using the previously found values of ∆ωR /2π = 20.0 kHz and RR = 1.85 ˚ A from the BPP model, the remaining free parameters S 2 and τL are optimized by minimizing the L1 norm of the logarithms of the variables [ln(T1LM ), ln(T2LM ), ln(η/T )], using data at all frequencies, resulting in optimized values S 2 = 0.131 and τL = 39.4 ps. The fitting parameters can be summarized as such: ∆ωR /2π = 20.0 kHz, RR = 1.85 ˚ A, (18) 2

S = 0.131, τL = 39.4 ps. As shown in Fig. 7, the fit yields a strong correlation coefficient of R2 = 0.962 (computed from the logarithm of variables) against the measurements. The value of S 2 = 0.131 is consistent with previously reports in poly(butadiene) 45,47,49 in the high molecular-weight limit Mw > 4,000 g/mol. In the case of the polymer-heptane mixes (solid symbols), the new model in Eq. 17 with best fit values in Eq. 18 yields good agreement with T1LM and T2LM measurements across the entire η/T and f0 range. Note also how the fit correctly accounts for both the T1LM > ×2.3/f0 ' 3.5 ms at 2.3 MHz and 22 MHz, and the slightly lower T1LM > ×2.3/f0 ' 2 ms plateau at 400 MHz (Fig. 6). In the case of the three pure poly(isobutene) samples (open symbols, three most viscous samples, and same polymer as used in the mix), the fit is

24


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Energy & Fuels

Figure 7: Cross-plot of all measured T1LM and T2LM data (x-axis) from Fig. 5 (O2 subtracted), versus the fitted values (y-axis) using the new model Eq. 17 with parameters in Eq. 18. The correlation coefficient R2 (computed from the logarithm of variables) is also shown. similarly good. However, in the case of the pure poly(1-decene) and the two pure petroleum-distillates (open symbols, three least viscous samples), even though the T1LM > ×2.3/f0 ' 3 ms plateau is consistent, the approach to the plateau is different. The deviation from the fit can also be seen in Fig. 7, where the fits for these three samples overestimate T1LM and T2LM compared with measurements. One likely explanation for this discrepancy is the difference in molecular structure between samples. Strictly speaking, the BPP model should be plotted as a function of τR instead of η/T on the x-axis of Fig. 5. According to Eq. 7, using η/T instead of τR for the x-axis should result in the same universality, provided the Stokes-Einstein radius RR is the same for all samples. In fact, as pointed out previously, 20 it is remarkable that Eq. 3 even holds for the light alkanes, since RR is presumed to increase with increasing chain length. However, recent MD simulations of the light alkanes confirm that RR remains roughly constant with increasing chain length, 94 implying that NMR is indeed a local probe

25


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of the molecular dynamics (see also

13

Page 26 of 51

C T1 inversion-recovery spectra of n-C15 H32 and n-

3 C20 H42 102 ). Nevertheless, due to differences in molecular structure, it is plausible that RR

is up to a factor ' 4 larger for these three samples (i.e. RR is up to a factor ' 1.6 larger), resulting in the observed left-shift along the x-axis of Fig. 5 relative to the other samples and the fit. Finally, note that this explanation does not affect T1LM or T2LM since ∆ωR2 in Eq. 8 does not depend on RR ; in other words, this explanation does not result in an additional shift along the y-axis of Fig. 5, provided that intramolecular relaxation dominates over intermolecular (as is assumed throughout). It is informative to expand Eq. 17 (along with Eqs. 5, 7 and 8) in the slow-motion regime defined in Eq. 9. Defining T1LM > and T2LM > as the log-mean relaxation times in the slow-motion regime, results in the following leading-order terms: 1

2∆ωR2 S 2 = ω0

5 1 − S2 ω0 τL + ... , 1+ 3 S2

T1LM > 1 = ∆ωR2 S 2 τR + ... . T2LM >

(19)

Inputting the best-fit values from Eq. 18 into Eq. 19 results in the following expressions: 3.50 f0 ms, (1 + 0.0027f0 ) 2.3 T = 252 ms. η

T1LM > = T2LM >

(20)

The term in the denominator of Eq. 20 is of order unity (0.0027f0 ' 1) at f0 = 400 MHz, which correctly accounts for the slight difference in T1LM > plateau value between 2.3 MHz and 400 MHz on a frequency-normalized scale (Fig. 6). The proportionality constant in T2LM > of Eq. 20 (in the slow-motion regime) is ' 26 times higher than that of Eq. 3 (in the fast-motion regime). The factor ' 26 can be accounted for in the new model by a factor 10/3/S 2 ' 25. The factor 10/3 originates from the fast-motion to slow-motion transition in the BPP model 93 (which can be seen as the upward “kink” in the BPP T2 data with increasing viscosity), and the factor S 2 = 0.131 originates from our new model in Eq. 19. 26


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Energy & Fuels

3.2

Comparison with Bitumen and Heavy Crude-Oils

Fig. 8 compares the pure polymer and polymer-heptane mix data from Fig. 6 with previously published bitumen and crude-oil data. 2,4,25,30 As shown in Fig. 8, the pure polymers, polymer-heptane mixes, bitumen and heavy crude-oils all show a plateau consistent with T1LM > × 2.3/f0 ' 3 ms in the slow-motion regime η/T × f0 /2.3 & 102 cP/K (i.e. in the high-viscosity regime), at all frequencies investigated. The large difference in paramagnetic concentrations between the polymers (< 100 ppm) and bitumen (' 1,000 ppm 85 ) suggests that NMR relaxation in the high-viscosity regime of polymers, bitumen and heavy crude-oils can be explained by 1 H-1 H dipole-dipole interactions, without the need to invoke surface paramagnetism. As for the role of asphaltenes in bitumen and heavy crude-oils, we postulate that NMR relaxation from enhanced 1 H-1 H dipole-dipole interactions for maltenes in the confined transient “pores” of asphaltene macro-aggregates dominate over relaxation from surface paramagnetism in the asphaltenes. As also shown in Fig. 8, there is reasonable agreement between the polymer and crudeoil data in the ultra low-viscosity region of η/T × f0 /2.3 . 10−2 cP/K. The slightly lower T1LM and T2LM values for the lightest crude-oils is due to paramagnetic contributions from dissolved O2 , which is known to occur for low-viscosity crude-oils when T1LM , T2LM > 500 ms. 18 On the other hand, the polymer data in Figs. 5, 6 and 8 have the contribution from dissolved O2 subtracted according to Eq. 4; likewise, the BPP model does not contain a contribution from dissolved O2 . However, Fig. 8 shows a clear discrepancy between polymer and crude-oil data in the intermediate-viscosity region, defined as 10−2 cP/K . η/T×f0 /2.3 . 102 cP/K. More specifically, in the intermediate-viscosity region, T1LM (T2LM ) for polymers is up to ' 3 (10) times larger than for crude oils at a given viscosity, respectively. There are two possible explanations for this discrepancy. The first explanation is that surface paramagnetism in the asphaltenes 32,35,36,38 contributes to the relaxation for crude-oils in the intermediate-viscosity regime, while it does not for the polymer-heptane mixes. This will tend to shorten T1LM 27


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Figure 8: (a) T1LM and (b) T2LM of previously published bitumen and heavy crude-oil data taken from Refs. 2,4,25,30, all frequency-normalized to 2.3 MHz as in Fig. 6. Also shown are all the polymer data from Fig. 6 as gray symbols. and T2LM for crude-oils compared to polymers. As already demonstrated, surface paramagnetism does not contribute to relaxation in the slow-motion regime (i.e. high-viscosity regime) for bitumen and heavy crude-oils, where higher viscosities are generally (but not always) attributed to higher asphaltene concentrations. It may therefore seem somewhat counterintuitive that paramagnetism would dominate in the intermediate-viscosity regime where there are generally fewer paramagnetic relaxation sites, however this is a possibility. Another explanation for the discrepancy between polymer and crude-oil data in the intermediate-viscosity regime is the difference in molecular structure. The viscosity standards

28


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are saturated polymers, while the crude oils and bitumen can have significant poly-nuclear aromatic components; this difference in molecular structure can be expected to have different molecular motions. As discussed in Section 3.1, a difference in molecular structure can change the relation between the molecular correlation-time τR and the bulk viscosity η/T . More specifically, in the intermediate-viscosity regime, the Stokes-Einstein radius RR in Eq. 7 may be larger for crude oils than for polymers. This may also explain the T2LM < (defined as the log-mean T2 in the fast-motion regime) correlation with viscosity for crude oils 8 at 2 MHz:

T2LM < = 4.00

T ms (crude oils). η

(21)

The well-established pre-factor for alkanes in Eq. 3 is a factor 9.56/4.00 ' 2.4 larger than the well-established pre-factor for crude oils in Eq. 21. This can be accounted for if RR is a mere factor (9.56/4.00)1/3 ' 1.3 larger for crude oils than for alkanes. The impact of molecular structure on the NMR relaxation in crude oils compared to polymers is currently being investigated with MD simulations.

4

Heptane Surface-Relaxation

Up to this point all the data has been analyzed in terms of the bulk properties of purepolymers or the polymer-heptane mixes (i.e. T1LM , T2LM , η/T ). We now turn our attention to the heptane NMR signal in the polymer-heptane mix, which as shown in Fig. 3 was successfully separated from the polymer signal. The T1 log-mean and T2 log-mean values of the heptane signal alone, defined as T1C7 and T2C7 respectively, are plotted in Fig. 9 (a). The oxygen contributions from Eq. 4 have again been subtracted from the data using Eq. 2. The data indicates no significant dispersion in T2C7 , as expected. However, below φC7 < 50 vol%, there is clearly dispersion in T1C7 , indicating a transition from the fast-motion to slowmotion regime for heptane below φC7 < 50 vol%. In the case of 22 MHz and 400 MHz, the dispersive region φC7 < 50 vol% is accompanied by ratios up to T1C7 /T2C7 ' 4 ratios, while 29


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in the case of 2.3 MHz the ratio T1C7 /T2C7 > 1 only becomes apparent at low φC7 = 5 vol%. The next step is to subtract the bulk-heptane contribution in Eq. 2, given the known value T1B = T2B = 7,320 ms from Eq. 3, and plot it in Fig. 9 (b). What remains are the surface-relaxation components T1S and T2S for heptane, where the surface is the polymer itself. With the surface components for heptane isolated, the surface-relaxivity parameters ρ1 and ρ2 for heptane are determined by the following relations in the fast-diffusion regime: 89,103 S 1 = ρ1 , T1S Vp 1 S = ρ2 . T2S Vp

(22)

Vp is the pore volume, i.e. the volume of heptane in the polymer matrix, and S/Vp is the surface to pore-volume ratio of the polymer “pore”. It should be noted that the above interpretation of ρ1,2 as surface-relaxivities is unusual in the sense that the pore walls are themselves moving. Nevertheless, this can also be said of the pores walls of kerogen and bitumen, which themselves are made of cross-linked polymers, and which have previously been interpreted as relaxation surfaces originating from intramolecular 75 and intermolecular 82 1 H-1 H dipole-dipole interactions. We can then determine Vp by using the following relations: S 1 − φC7 S = , Vp φC7 Vg S ≈ 0.859 ˚ A−1 (constant). Vg

30


(23) (24)

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Figure 9: (a) T1C7 and T2C7 of heptane signal in polymer-heptane mixes, versus heptane volume-fraction φC7 , at different frequencies. (b) Surface relaxation components T1S and T2S for heptane in mix, using Eq. 2. (c) Surface relaxivities ρ1 and ρ2 for heptane in the polymer, using Eq. 25. 31


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Vg is the grain volume of the polymer, and S/Vg is the surface to grain-volume ratio of the polymer. Molecular simulations indicate that S/Vg in Eq. 24 is roughly constant for the branched alkanes. 104 More specifically, a plot of smoothened S versus smoothened Vg for ˚−1 (see Fig. 1 in Ref. 104). branched alkanes indicates a constant slope of S/Vg ' 0.859 A Combining Eqs. 22 and 23 yields the following: 1 − φC7 S 1 = ρ1 , T1S φC7 Vg 1 1 − φC7 S = ρ2 . T2S φC7 Vg

(25)

Fig. 9 (c) plots the resulting values of ρ1 and ρ2 for heptane using Eq. 25. What is remarkable is that ρ1 and ρ2 are constant (ρ1 ' ρ2 ' 3 · 10−5 µm/s) above φC7 > 50 vol%. Furthermore below φC7 < 50 vol%, ρ1 and ρ2 increase with decreasing φC7 , and ρ1 is dispersive for 22 MHz and 400 MHz. We note that the results for T1C7,2C7 and T1S,2S as a function of φC7 in Fig. 9 are found to agree semi-quantitatively with independent MD simulations (not shown) of similar polymerheptane mixes. More specifically, the MD simulations show an increase in heptane correlation time, i.e. a slowing down of the molecular dynamics, due to the effects of nano-confinement in the transient “pores” created by the polymer. The MD simulations therefore confirm that 1 H-1 H dipole-dipole interactions enhanced by nano-confinement dominate the NMR response of hydrocarbons in organic nano-pores, without the need to invoke paramagnetism. This agreement between measurement and simulation helps justify our interpretation of surface-relaxivity for heptane in the polymer-heptane mixes using Eqs. 22−25.

4.1

Pore Fluid versus Absorbed Regimes

In order to enhance the interpretation of the surface relaxation for heptane, it is helpful from a petrophysics standpoint to have a toy model of the transient “pores” created by the polymers. The polymers are most likely only weakly aligned over a few monomers (i.e.

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Pore Fluid

…

…poly(isobutene) dp pore

…

…poly(isobutene) φC7

Absorbed

…

…poly(isobutene) dC7 n-heptane

…

…poly(isobutene)

Figure 10: Cross-section of (locally) cylindrical transient “pores” for the two regimes in the polymer-heptane mix (only carbon atoms are shown). (Top) “Pore fluid” regime (φC7 > 50 vol%), where dp is the cylindrical-pore diameter created by the polymers. (Bottom) “Absorbed” regime (φC7 < 50 vol%), where dC7 is the molecular diameter of n-heptane. < 10 ˚ A), beyond which the polymer orientation is perfectly random. 53,54,60 Nevertheless, over short length-scales, a cylindrical arrangement of the polymers is more likely than a spherical or planar one. Within this approximation, Fig. 10 illustrates a cross-section of a cylindrical pore model created by dissolution of heptane (the solute) in poly(isobutene) (the solvent), and a corresponding pore-diameter dp . Within the cylindrical pores lie the heptane molecules, where the diameter of the extended heptane-chain is roughly dC7 = 4.2 ˚ A (taken from 105 for n-butane). Assuming (local) cylindrical geometry yields the following: 4 1 − φC7 S S = = . Vp dp φC7 Vg

(26)

Using Eqs. 26 and 24, the values for dp corresponding to the specific values of φC7 used in

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Figure 11: Cylindrical-pore diameter dp for heptane in polymer-heptane mixes, versus φC7 at specific values in this study, using Eqs. 26 and 24. this study are plotted in Fig. 11. Also shown in Fig. 11 is the heptane molecular-diameter dC7 , where the equality dp ' dC7 corresponds to roughly φC7 ' 50 vol%. Fig. 11 indicates two regions: (1) the heptane “pore-fluid” region dp > dC7 where heptane molecules reside in a cylindrical pore created by the polymers. (2) The “absorbed” heptane region dp < dC7 where a heptane molecule is predominantly in contact with polymers, not other heptane molecules. The cylindrical-pore model does not strictly imply that dp < dC7 in the absorbed region, as this would be physically impossible. Instead, dp is an indicator of the separation Lp between two neighboring heptane molecules in the same cylindrical pore. Noting that in reality the empty volume (i.e. vacuum) in between the two heptane molecules would be filled in by the polymers, and assuming that the molecular diameter of the polymer is the same as dC7 for heptane, Lp can then be determined using the following expression φC7 ' LC7 / (9LC7 + 8Lp ), where LC7 ' 19 ˚ A is the approximate length of the heptane molecule. Equating this expression to φC7 as a function of dp (Eq. 26) leads to the following

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approximation in the limit dp S/Vg 1 (i.e. dp 1 ˚ A):

Lp '

LC7 1 2S/Vg dp

for dp 1 ˚ A .

(27)

Given (for instance) that dp = 0.26 ˚ A at the lowest heptane concentration of φC7 = 5 vol%, Eq. 27 implies a separation Lp ' 40 ˚ A between heptane molecules at the lowest heptane concentration. Fig. 12 (a) shows the same ρ1 and ρ2 data as Fig. 9 (c), but plotted against cylindricalpore diameter dp taken from Fig. 11. As shown in Fig. 12 (a), both ρ1 and ρ2 are constant in the pore-fluid region dp > dC7 . Meanwhile, ρ1 and ρ2 increase with decreasing dp in the absorbed region dp < dC7 ; furthermore, ρ1 is clearly dispersive in the absorbed region, while ρ2 shows a hint of dispersion at φC7 = 5 vol% (although this may be within experimental uncertainties). It is therefore informative to combine Eqs. 22 and 26 to yield the following generalized form: 4 1 = ρ1 (ω0 , dp ) , T1S dp 1 4 = ρ2 (ω0 , dp ) . T2S dp

(28)

Eq. 28 explicitly shows the pore-diameter dp and frequency ω0 dependences of ρ1 and ρ2 , where ω0 in ρ2 is gray indicating a much weaker dependence. Fig. 12 (b) shows the ratio T1S /T2S (= ρ2 /ρ1 ) ratio versus pore-diameter dp , where T2S at 400 MHz is assumed to be equal to T2S at 2.3 MHz, which is reasonable given that ρ2 (ω0 , dp ) is much less dispersive than ρ1 (ω0 , dp ). The results in Fig. 12 (b) are consistent with previous results in saturated organicshale, where hydrocarbons saturated in organic-matter show a large ratio T1S /T2S & 4 (i.e. ρ2 /ρ1 & 4) (first reported in Ref. 62), and furthermore T1S (ω0 ) for the saturating hydrocarbons is dispersive. 70,83 While the analogy between the polymer-heptane mix and saturated organic-matter (i.e. kerogen) works well conceptually, there are some quantitative differences between the two 35


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Figure 12: (a) Same ρ1,2 data as Fig. 9 (c), but plotted against cylindrical pore-diameter dp taken from Fig. 11. Dashed line separates heptane pore-fluid region (dp > dC7 ) from absorbed heptane region (dp < dC7 ). (b) Plot of T1S /T2S (= ρ2 /ρ1 ) ratio versus dp . systems. The first difference is that the surface relaxivities for the pore-fluid region is ρ1,2 ' 3 · 10−5 µm/s, which is ' 5 orders of magnitude smaller than those found in kerogen ρ2 ' 3 µm/s. 63–65,75,80,106 It is well known that the values for ρ1,2 depend on the experimental method used to determine S/Vg . 89 According to Eq. 25, for given T1S and T2S values, using a larger value of S/Vg results in correspondingly smaller ρ1,2 . For instance, using thin-section imaging, SEM (scanning electron microscopy) imaging, MICP (mercury injection capillary pressure), or BET (Brunauer-Emmett-Teller) gas-adsorption yields increasingly large S/Vg values, and

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therefore decreasingly small ρ1,2 values. In the present case, the polymer molecules create transient “pores” which act as relaxation surfaces for heptane, therefore the large S/Vg of the polymer molecule itself 104 is the appropriate surface to grain-volume. The second quantitative difference in this analogy is that T1S /T2S ' 4 for heptane in kerogen at 2.3 MHz, whereas T1S /T2S > 1 for heptane in polymers occurs only at higher frequencies of 22 MHz and 400 MHz (with the exception of perhaps one point at 2.3 MHz at φC7 = 5 vol%, see Fig. 12 (b)). In other words, the dispersion effects in T1S of heptane are clearly apparent at 2.3 MHz in kerogen, but not clearly apparent until 22 MHz in the polymers used here. The most likely reason for this is that the heptane correlation-times τC7 are different between the two systems; therefore, the transition frequencies between fast and slow-motion are different. This is reasonable given that solid kerogen has more cross-links than the liquid polymer; therefore, one expects τC7 to be larger for kerogen than for the polymers used here. In the case of the polymers used here, we can estimate τC7 by the fact that the transition frequency satisfying ω0 τC7 = 0.615 (see Eq. 6) occurs somewhere inbetween 2.3 MHz and 22 MHz. Taking a log-mean results in a transition at f0 ' 7 MHz, corresponding to τC7 ' 14 ns for heptane in the polymers used here. In the case of kerogen, we can ascertain that the transition occurs somewhere below f0 < 2.3 MHz, 75,80 therefore τC7 > 43 ns for heptane in kerogen. To be more precise, previous T1S dispersion data in hydrocarbon-saturated organic-shale 70,83 indicate a transition frequency at ' 0.5 MHz, corresponding to τC7 ' 200 ns for hydrocarbons in kerogen.

5

Conclusions

We show that for pure polymers and polymer-heptane mixes in the slow-motion regime, defined as the high-viscosity regime η/T ×f0 /2.3 & 102 cP/K (frequency-normalized relative to 2.3 MHz), T1LM is independent of viscosity and plateaus to a value T1LM > ∝ ω0 . More specifically, on a frequency-normalized scale, T1LM >×2.3/f0 ' 3 ms is independent of viscosity.

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This finding agrees with previously reported bitumen and heavy crude-oil data, which also show T1LM >×2.3/f0 ' 3 ms at high viscosities. Likewise, in the high-viscosity limit, T2LM > for the polymers is found to be consistent with bitumen and heavy crude-oils. The concentration of paramagnetic impurities in the polymers under investigation are found to be at least an order of magnitude less than previously reported bitumen, which suggests that the NMR relaxation mechanism at high viscosities for polymers, bitumen and heavy crude-oils are all dominated by 1 H-1 H dipole-dipole interactions, and not surface paramagnetism. On the other hand, the discrepancies between the polymer and the crude-oil data in the intermediateviscosity region, defined as 10−2 cP/K . η/T ×f0 /2.3 . 102 cP/K, can be accounted for by surface paramagnetism in the crude oils and/or differences in molecular structure between the two. We introduce a new NMR relaxation model to account for the viscosity and frequency dependences of T1LM and T2LM in the polymer-heptane mixes, including the plateau T1LM > ∝ ω0 at high viscosities. The new model builds on the traditional hard-sphere BPP model of 1

H-1 H dipole-dipole interactions, and adds (1) the possibility of internal motions of the non-

rigid polymer-branches and (2) the distribution of molecular correlation-times inherent in the polymer-heptane mixes. With just two additional parameters, the new model fits T1LM and T2LM for pure polymer and polymer-heptane mixes across the entire viscosity range, with a strong correlation coefficient of R2 = 0.964 against the measurements. Separating the heptane signal in the polymer-heptane mixes allows for determination of the surface relaxation of heptane on the polymer surfaces. Assuming that the polymers create (local) cylindrical pores of diameter dp for the heptane to occupy allows for analyzing ρ1,2 versus pore-diameter dp . Two distinct regions are identified: (1) the “pore-fluid” region where dp is larger than the heptane molecule dC7 (i.e. dp > dC7 ), and the surface-relaxivity parameters ρ1,2 for heptane are equal and constant versus dp and frequency. (2) The “absorbed” region dp < dC7 where ρ1,2 increase with decreasing dp , and ρ1 is dispersive (i.e. frequency dependent). Analogies are made between heptane surface-relaxation in polymers

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and heptane surface-relaxation in bitumen and kerogen. Our findings suggest that the large ratios T1S /T2S ' 4 and dispersion T1S (ω0 ) previously reported in hydrocarbon-saturated organic-shales originate from 1 H-1 H dipole-dipole interactions enhanced by nano-pore confinement.

Acknowledgments We dedicate this manuscript to the memory of Prof. Riki Kobayashi, who was a pioneer in the investigation of NMR fluid properties and was a mentor to George J. Hirasaki in this area of research. We thank the Rice University Consortium on Processes in Porous Media for funding this work, Maura Puerto for help on sample preparation, Prof. Aydin Babakhani for the EPR measurements, Prof. Rafael Verduzco and Hao Mei for the GPC measurements, and, Prof. Walter G. Chapman, Dr. Dilip Asthagiri, Arjun V. Parambathu, Jinlu Liu, and Amin Haghmoradi for helpful discussions on molecular dynamics simulations and theory.

Supporting information The supporting information consists of a spreadsheet with: 1. Polymer-heptane mix and pure-polymer data shown in Figs. 5 and 6. 2. Previously reported bitumen and heavy crude-oil data shown in Fig. 8. 3. Heptane data in the polymer-heptane mix shown in Figs. 9 and 12.

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Interpretation of NMR Relaxation in Bitumen and Organic Shale using

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