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Compressibility of Nanoconfined Fluids: Relating Atomistic Modeling to Ultrasonic Experiments

Gennady Gor

Department of Chemical and Materials Engineering New Jersey Institute of Technology Newark, NJ, USA

E-mail: gor@njit.edu Web: http://porousmaterials.net

• G. Gor (NJIT)

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Acknowledgments

Max Maximov 4th year Ph.D. student Chris Dobrzanski Ph.D. (2020) Currently: NJIT Nick Corrente BS (2019) Currently: Ph.D. student at Rutgers

• Prof. Boris Gurevich

Geophysics Curtin University, Australia

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Motivation & Potential Industrial Needs

ExxonMobil Research & Engineering Company Industrial need: exploration and development of unconventional hydrocarbons (shale gas, shale oil) One of the key difference with conventional hydrocarbons Nanoporous system with hydrocarbons in adsorbed state

Loucks, R. G.; Reed, R. M.; Ruppel, S. C. & Jarvie, D. M. J. Sediment. Res., 2009, 79, 848-861.

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Wave Propagation in an Elastic Medium

Seismic waves – characterization of geological formations Sonic/utrasonic waves – characterization of rock samples Longitudinal waves ↔ longitudinal modulus M M = ρv2

M

(1) Transverse waves ↔ shear modulus G G = ρv2

G

(2)

Image from https://chrisplouffe.com

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Properties of Fluid-Saturated Porous Media

K = M − 4 3G (3) K = f (Ks, K0, Kf) (4)

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Properties of Composite and its Constituents

Fluid does not affect the shear modulus Gf = 0 ⇒ G = G0 Gassmann’s equation (low frequency limit of Biot’s theory): K = K0 +

• 1 − K0

Ks

2

φ Kf + (1−φ) Ks

− K0

K2

s

, (5) Experimentally measured quantity: the longitudinal modulus M M = M0 + (Ks − K0)2Kf φK2

s + [(1 − φ)Ks − K0] Kf

. (6) Derived for “classical” macroporous media (Gassmann, 1951) Does it work for nanoporous media?

Gassmann, F. ¨ Uber die Elastizit¨ at por¨

• ser Medien Viertel. Naturforsch. Ges. Z¨

urich, 1951, 96, 1-23 Biot, M. A. J. Acoust. Soc. Am., 1956, 28, 168-178

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Experiments on Saturated Nanoporous Media

Gas adsorption + Ultrasound on Nanoporous Vycor glass

Pulse Modulator Receiver

Oscilloscope

t T p m

Schematic for the experimental setup from: Warner, K. L.; Beamish, J. R. J. Appl. Phys. 1988, 63, 4372-4376. Dobrzanski, C. D.; Gurevich, B.; Gor, G. Y. Appl. Phys. Rev., 2020, submitted

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Experimental Data: Density and Velocities (N2 at 77 K)

0.00 0.25 0.50 0.75 1.00

p/p0

0.000 0.001 0.002 0.003 0.004 0.005

n/m [mol/g]

Volumetric

0.0 0.2 0.4 0.6 0.8 1.0

p/p0

0.94 0.96 0.98 1.00

v/v0

Longitudinal Transverse

0.0 0.2 0.4 0.6 0.8 1.0

p/p0

−0.01 0.00 0.01 0.02 0.03 0.04

∆G/G0

Volumetric

0.0 0.2 0.4 0.6 0.8 1.0

p/p0

−0.01 0.00 0.01 0.02 0.03 0.04

∆M/M0

Ultrasonic Volumetric

Warner, K. L.; Beamish, J. R. J. Appl. Phys. 1988, 63, 4372-4376.

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Experimental Data: Other Fluids

Argon at 87 K

0.0 0.2 0.4 0.6 0.8 1.0 Relative Pressure p/p0 16.8 17.0 17.2 17.4 17.6 17.8 18.0 Longitudinal Modulus M (GPa)

• expt. adsorption
• expt. desorption

6.2 6.4 6.6 6.8 7.0 7.2 7.4 Shear Modulus G (GPa)

Hexane at 298 K

0.0 0.2 0.4 0.6 0.8 1.0

p/p0

0.00 0.02 0.04 0.06

∆M/M0

Shear modulus does not appreciably change Longitudinal modulus changes at capillary condensation and continues to change beyond it

Page, J.H., Liu, J., Abeles, B., Herbolzheimer, E., Deckman, H.W. and Weitz, D.A., Phys. Rev. E, 1995, 52(3), 2763. Schappert, K. and Pelster, R., EPL (Europhysics Letters), 2014, 105(5), 56001.

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Experimental Data vs Gassmann Equation

K0, Ks, Kbulk

f

, φ → K or M 0.0 0.2 0.4 0.6 0.8 1.0

p/p0

−0.01 0.00 0.01 0.02 0.03 0.04

∆M/M0

Ultrasonic Volumetric Bulk

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Experimental Data vs Gassmann Equation

Argon at 87 K

0.0 0.2 0.4 0.6 0.8 1.0 Relative Pressure p/p0 16.8 17.0 17.2 17.4 17.6 17.8 18.0 Longitudinal Modulus M (GPa)

• expt. adsorption
• expt. desorption

Kf(0), Ks(EMT) Kf(0), Ks(AD)

Hexane at 298 K

0.0 0.2 0.4 0.6 0.8 1.0 Relative Pressure p/p0 19.0 19.2 19.4 19.6 19.8 20.0 20.2 Longitudinal Modulus M (GPa)

• expt. adsorption
• expt. desorption

Kf(0), Ks(EMT) Kf(0), Ks(AD)

The modulus changes with the vapor pressure p Even at p = p0 the modulus of saturated sample differs from the modulus of the bulk-fluid saturated sample

Page, J.H., Liu, J., Abeles, B., Herbolzheimer, E., Deckman, H.W. and Weitz, D.A., Phys. Rev. E, 1995, 52(3), 2763. Schappert, K. and Pelster, R., EPL (Europhysics Letters), 2014, 105(5), 56001.

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Calculating Fluid Modulus from Molecular Simulation

Bulk (adiabatic) modulus and isothermal modulus: Kf = γKT

f

Isothermal modulus and isothermal compressibility: KT

f = 1/βT

Isothermal compressibility (definition): βT ≡ − 1 V ∂V ∂P

• T,N

Fluctuations of number of particles in the grand canonical ensemble (µ, V , T) βT = V δN2 kBTN2

Bratko, D.; Curtis, R.; Blanch, D.; and Prausnitz, J. J. Chem. Phys. 2001, 115, 3873-3877. Coasne, B.; Czwartos, J.; Sliwinska-Bartkowiak, M.; Gubbins, K. E. J. Phys. Chem. B, 2009, 113, 13874. Strekalova, E.G., Mazza, M.G., Stanley, H.E. and Franzese, G., Phys. Rev. Lett., 2011, 106(14), 145701.

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Calculating Fluid Modulus from Molecular Simulation

Lennard-Jones nitrogen, spherical silica pores, T = 77 K, LJ solid-fluid interactions, integrated spherical potential Monte Carlo in the grand canonical ensemble (GCMC) 109 equilibration moves, then 3-5 series of 5 × 109 moves Interaction σ, nm ǫ/kB, K ns, nm−2 rcut, σff N2-N2 0.36154 101.5

• 5.0

SiO2-N2 0.317 147.3 15.3

• Allen, M. P

. & Tildesley, D. J. Computer simulation of liquids. 1987. New York: Oxford, 385. Norman, G. & Filinov, V. High Temp., 1969, 7, 216 Rasmussen, C. J.; Vishnyakov, A.; Thommes, M.; Smarsly, B. M.; Kleitz, F.; Neimark, A. V. Langmuir 2010, 26, 10147-10157

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Results: Adsorption Isotherms

0.2 0.4 0.6 0.8 1

p/p0

0.2 0.4 0.6 0.8

n∗

2 nm 3 nm 4 nm 5 nm 6 nm 7 nm 8 nm

Maximov, M. A.; Gor, G. Y. Langmuir, 2018, 34 (51), 15650-15657 & 2020, 36 (17), 4853-4854

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Results: Modulus Isotherms

0.2 0.4 0.6 0.8 1.0

p/p0

0.2 0.4 0.6 0.8 1.0

KT

f (GPa)

2 nm (Eq.8) 2 nm (Eq.10) 3 nm (Eq.8) 3 nm (Eq.10) 4 nm (Eq.8) 4 nm (Eq.10) 5 nm (Eq.8) 5 nm (Eq.10) 6 nm (Eq.8) 6 nm (Eq.10) 7 nm (Eq.8) 7 nm (Eq.10) 8 nm (Eq.8) 8 nm (Eq.10) Bulk

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Results: Modulus vs Laplace Pressure

−30 −20 −10

PL (MPa)

0.2 0.4 0.6 0.8 1.0

KT

f (GPa) 2 nm 3 nm 4 nm 5 nm 6 nm 7 nm 8 nm

Laplace pressure: PL = RgT Vl log p p0

• (7)
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Results: Comparison with Experiment

0.0 0.2 0.4 0.6 0.8 1.0

p/p0

−0.01 0.00 0.01 0.02 0.03 0.04

∆M/M0

Ultrasonic Volumetric Theory (GCMC 8 nm)

Maximov, M. A.; Gor, G. Y. Langmuir, 2018, 34 (51), 15650-15657 & 2020, 36 (17), 4853-4854

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Experimental Data vs Gassmann Equation

Argon at 87 K

0.0 0.2 0.4 0.6 0.8 1.0 Relative Pressure p/p0 16.8 17.0 17.2 17.4 17.6 17.8 18.0 Longitudinal Modulus M (GPa)

• expt. adsorption
• expt. desorption

Kf(Pf), Ks(EMT) Kf(Pf), Ks(AD)

Hexane at 298 K

0.0 0.2 0.4 0.6 0.8 1.0 Relative Pressure p/p0 19.0 19.2 19.4 19.6 19.8 20.0 20.2 Longitudinal Modulus M (GPa)

• expt. adsorption
• expt. desorption

Kf(Pf), Ks(EMT) Kf(Pf), Ks(AD)

Gassmann equation is applicable to nanoporous media The fluid modulus Kf has to be corrected for confined effects

Gor, G. Y. & Gurevich, B. Geophys. Res. Lett., 2018, 45, 146-155

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Results: Modulus vs Pore Size

0.0 0.1 0.2 0.3 0.4 0.5 0.6

1/dext (nm−1)

0.2 0.4 0.6 0.8 1.0 1.2

KT

f (GPa) Bulk GCMC

10.0 5.0 3.3 2.5 2.0 1.7

dext (nm)

Modulus of confined nitrogen is a linear function of reciprocal pore size Modulus in small pores is higher than in bulk by a factor of three

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Physical Mechanism

Density enhancement

1 2 3 4 5

Radial coordinate r∗

0.0 0.5 1.0 1.5 2.0

Local density n∗

Modulus-Pressure Equation Solvation pressure: Pf = Psl + PL Laplace pressure: PL = RgT/Vl log (p/p0) . Solvation pressure (solid-fluid interactions): Psl ∝ 1/dext Tait-Murnaghan Equation: K(Pf) = K(0) + αPf

Gor, G. Y.; Siderius, D. W.; Rasmussen, C. J.; Krekelberg, W. P .; Shen, V. K. & Bernstein, N. J. Chem. Phys., 2015, 143, 194506 Gor, G. Y.; Siderius, D. W.; Shen, V. K. & Bernstein, N. J. Chem. Phys., 2016, 145, 164505

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Conclusions

Analysis of experimental data suggests that the moduli of confined fluids differ from the bulk Isothermal compressibility (or modulus) of a confined fluid can be calculated using GCMC simulations The modulus is affected by the “solvation” pressure in the pore The modulus changes logarithmically with the vapor pressure The modulus is a linear function of the reciprocal pore size 1/dext The results are consistent with the ultrasonic data on porous glasses saturated with nitrogen (also argon and n-hexane) The modulus of supercritical methane is affected even stronger *

* Corrente, N. J.; Dobrzanski, C. D.; & Gor, G. Y. Energy & Fuels, 2020, 34 (2), 1506-1513

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References

Gor, G. Y.* “Adsorption Stress Changes the Elasticity of Liquid Argon Confined in a Nanopore”, Langmuir 2014, 30 (45),

• p. 13564-13569. DOI: 10.1021/la503877q

Gor, G. Y.*; Siderius, D. W.; Rasmussen, C. J.; Krekelberg, W. P .; Shen, V. K; Bernstein, N. “Relation Between Pore Size and the Compressibility of a Confined Fluid”, J. Chem. Phys., 2015, 143, 194506. DOI: 10.1063/1.4935430 Gor, G. Y.* Siderius, D. W.; Shen, V. K.; Bernstein, N. “Modulus-Pressure Equation for Confined Fluids” J. Chem. Phys. 2016, 145, 164505. DOI: 10.1063/1.4965916 Dobrzanski, C. D.; Maximov, M. A.; Gor, G. Y.* “Effect of Pore Geometry on the Compressibility of a Confined Simple Fluid” J. Chem. Phys. 2018, 148, 054503. DOI: 10.1063/1.5008490 Gor, G. Y.*; Gurevich, B. “Gassmann Theory Applies to Nanoporous Media” Geophys. Res. Lett., 2018, 45(1), 146-155. DOI: 10.1002/2017GL075321 Maximov, M. A.; Gor, G. Y.* “Molecular Simulations Shed Light on Potential Uses of Ultrasound in Nitrogen Adsorption Experiments” Langmuir 2018, 34(51), 15650-15657 DOI: 10.1021/acs.langmuir.8b02909 Corrente, N. J.; Dobrzanski, C. D.; Gor, G. Y.* “Compressibility of Supercritical Methane in Nanopores: a Molecular Simulations Study” Energy & Fuels, 2020, 34(2), 1506-1513 DOI: 10.1021/acs.energyfuels.9b03592 Dobrzanski, C. D.; Corrente, N. J.; Gor, G. Y.* “Compressibility of Simple Fluid in Cylindrical Confinement: Molecular Simulation and Equation of State Modeling” Ind. Eng. Chem. Res., 2020, 59(17), 8393-8402. DOI: 10.1021/acs.iecr.0c00693

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