Compressibility of Nanoconfined Fluids

Compressibility of Nanoconfined Fluids:Relating Atomistic Modelingto Ultrasonic Experiments

Gennady Gor

Department of Chemical and Materials EngineeringNew Jersey Institute of Technology

Newark, NJ, USA

E-mail: [email protected]: http://porousmaterials.net

G. Gor (NJIT) Compressibility of Confined Fluids 8/06/2020 1 / 22

Acknowledgments

Max Maximov4th year Ph.D.student

Chris DobrzanskiPh.D. (2020)Currently: NJIT

Nick CorrenteBS (2019)Currently: Ph.D.student at Rutgers

Prof. Boris GurevichGeophysicsCurtin University,Australia


Motivation & Potential Industrial Needs

ExxonMobil Research & Engineering CompanyIndustrial need: exploration and development of unconventionalhydrocarbons (shale gas, shale oil)One of the key difference with conventional hydrocarbonsNanoporous 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.


Wave Propagation in an Elastic Medium

Seismic waves – characterization of geological formationsSonic/utrasonic waves – characterization of rock samples

Longitudinal waves↔ longitudinal modulus M

M = ρv2M (1)

Transverse waves↔ shear modulus G

G = ρv2G (2)

Image from https://chrisplouffe.comG. Gor (NJIT) Compressibility of Confined Fluids 8/06/2020 4 / 22

Properties of Fluid-Saturated Porous Media

K =M − 4

3G (3)

K = f (Ks,K0,Kf) (4)


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

φKf

+ (1−φ)Ks− K0

K2s

, (5)

Experimentally measured quantity: the longitudinal modulus M

M =M0 +(Ks −K0)

2Kf

φK2s + [(1− φ)Ks −K0]Kf

. (6)

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

Gassmann, F. Uber die Elastizitat poroser Medien Viertel. Naturforsch. Ges. Zurich, 1951, 96, 1-23Biot, M. A. J. Acoust. Soc. Am., 1956, 28, 168-178


Experiments on Saturated Nanoporous Media

Gas adsorption + Ultrasound on Nanoporous Vycor glass

PulseModulator

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


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 1.0p/p0

0.94

0.96

0.98

1.00

v/v

0

Longitudinal

Transverse

0.0 0.2 0.4 0.6 0.8 1.0p/p0

−0.01

0.00

0.01

0.02

0.03

0.04

∆G/G

0

Volumetric

0.0 0.2 0.4 0.6 0.8 1.0p/p0

−0.01

0.00

0.01

0.02

0.03

0.04

∆M/M

0

Ultrasonic

Volumetric

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


Experimental Data: Other Fluids

Argon at 87 K

0.0 0.2 0.4 0.6 0.8 1.0Relative Pressure p/p0

16.8

17.0

17.2

17.4

17.6

17.8

18.0

Lon

gitu

dina

lMod

ulus

M(G

Pa)

expt. adsorptionexpt. desorption

6.2

6.4

6.6

6.8

7.0

7.2

7.4

Shea

rM

odul

usG

(GPa

)

Hexane at 298 K

0.0 0.2 0.4 0.6 0.8 1.0p/p0

0.00

0.02

0.04

0.06

∆M/M

0Shear modulus does not appreciably changeLongitudinal modulus changes at capillary condensation andcontinues 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.


Experimental Data vs Gassmann Equation

K0,Ks,Kbulkf , φ→ K orM

0.0 0.2 0.4 0.6 0.8 1.0p/p0

�0.01

0.00

0.01

0.02

0.03

0.04�

M/M

0

Ultrasonic

Volumetric

Bulk



Argon at 87 K


16.8

17.0

17.2

17.4

17.6

17.8

18.0

Lon

gitu

dina

lMod

ulus

M(G

Pa)

expt. adsorptionexpt. desorptionKf(0), Ks(EMT)

Kf(0), Ks(AD)

Hexane at 298 K


19.0

19.2

19.4

19.6

19.8

20.0

20.2

Lon

gitu

dina

lMod

ulus

M(G

Pa)

expt. adsorptionexpt. desorptionKf(0), Ks(EMT)

Kf(0), Ks(AD)

The modulus changes with the vapor pressure pEven at p = p0 the modulus of saturated sample differs from themodulus 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.


Calculating Fluid Modulus from Molecular Simulation

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

Isothermal modulus and isothermal compressibility: KTf = 1/βT

Isothermal compressibility (definition):

βT ≡ − 1

V

(∂V

∂P

)T,N

Fluctuations of number of particles in the grand canonicalensemble (µ, V , T )

βT =V 〈δN2〉kBT 〈N〉2

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.


Calculating Fluid Modulus from Molecular Simulation

Lennard-Jones nitrogen, spherical silicapores, T = 77 K, LJ solid-fluid interactions,integrated spherical potentialMonte Carlo in the grand canonicalensemble (GCMC)109 equilibration moves, then 3-5 series of5× 109 moves

Interaction σ, nm ε/kB, K ns, nm−2 rcut, σff

N2-N2 0.36154 101.5 - 5.0SiO2-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, 216Rasmussen, C. J.; Vishnyakov, A.; Thommes, M.; Smarsly, B. M.; Kleitz, F.; Neimark, A. V. Langmuir 2010, 26, 10147-10157


Results: Adsorption Isotherms

0 0.2 0.4 0.6 0.8 1p/p0

0

0.2

0.4

0.6

0.8n∗ 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


Results: Modulus Isotherms

0.2 0.4 0.6 0.8 1.0p/p0

0.2

0.4

0.6

0.8

1.0

KT f

(GP

a)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


Results: Modulus vs Laplace Pressure

−30 −20 −10 0PL (MPa)

0.2

0.4

0.6

0.8

1.0K

T f(G

Pa)

2 nm

3 nm

4 nm

5 nm

6 nm

7 nm

8 nm

Laplace pressure: PL =RgT

Vllog

(p

p0

)(7)


Results: Comparison with Experiment

0.0 0.2 0.4 0.6 0.8 1.0p/p0

−0.01

0.00

0.01

0.02

0.03

0.04∆M/M

0Ultrasonic

Volumetric

Theory (GCMC 8 nm)

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



Argon at 87 K


16.8

17.0

17.2

17.4

17.6

17.8

18.0

Lon

gitu

dina

lMod

ulus

M(G

Pa)

expt. adsorptionexpt. desorptionKf(Pf), Ks(EMT)

Kf(Pf), Ks(AD)

Hexane at 298 K


19.0

19.2

19.4

19.6

19.8

20.0

20.2

Lon

gitu

dina

lMod

ulus

M(G

Pa)

expt. adsorptionexpt. desorptionKf(Pf), Ks(EMT)

Kf(Pf), Ks(AD)

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

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


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

1.2K

T f(G

Pa)

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 poresizeModulus in small pores is higher than in bulk by a factor of three


Physical Mechanism

Density enhancement

0 1 2 3 4 5Radial coordinate r∗

0.0

0.5

1.0

1.5

2.0

Loc

ald

ensi

tyn∗

Modulus-Pressure Equation

Solvation pressure:Pf = Psl + PL

Laplace pressure:PL = RgT/Vl log (p/p0) .

Solvation pressure (solid-fluidinteractions): 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, 194506Gor, G. Y.; Siderius, D. W.; Shen, V. K. & Bernstein, N. J. Chem. Phys., 2016, 145, 164505


Conclusions

Analysis of experimental data suggests that the moduli of confinedfluids differ from the bulkIsothermal compressibility (or modulus) of a confined fluid can becalculated using GCMC simulationsThe modulus is affected by the “solvation” pressure in the poreThe modulus changes logarithmically with the vapor pressureThe modulus is a linear function of the reciprocal pore size 1/dext

The results are consistent with the ultrasonic data on porousglasses 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


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 Sizeand 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 SimpleFluid” 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 AdsorptionExperiments” Langmuir 2018, 34(51), 15650-15657DOI: 10.1021/acs.langmuir.8b02909

Corrente, N. J.; Dobrzanski, C. D.; Gor, G. Y.* “Compressibility of Supercritical Methane in Nanopores: a MolecularSimulations Study” Energy & Fuels, 2020, 34(2), 1506-1513DOI: 10.1021/acs.energyfuels.9b03592

Dobrzanski, C. D.; Corrente, N. J.; Gor, G. Y.* “Compressibility of Simple Fluid in Cylindrical Confinement: MolecularSimulation and Equation of State Modeling” Ind. Eng. Chem. Res., 2020, 59(17), 8393-8402.DOI: 10.1021/acs.iecr.0c00693


http://dx.doi.org/10.1021/la503877q

http://dx.doi.org/10.1063/1.4935430

http://dx.doi.org/10.1063/1.4965916

http://dx.doi.org/10.1063/1.5008490

http://dx.doi.org/10.1002/2017GL075321

http://dx.doi.org/10.1021/acs.langmuir.8b02909

http://dx.doi.org/10.1021/acs.energyfuels.9b03592

http://dx.doi.org/10.1021/acs.iecr.0c00693

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Compressibility of Nanoconfined Fluids

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