Presentation outline

Effects of Geometrical Corrugation and Energetical Heterogeneity

of Graphene Pore Walls on Adsorption in Nanoporous Carbons.PSD Characterization

Jacek Jagiello

Micromeritics Corporation, Norcross GA, USA

•Relationship between assumed carbon pore model and calculated PSD

•Standard slit pore model based on Steele potential

•Artifacts resulting from this model

•Modifications of the model– finite pores– energetically heterogeneous pore walls– geometrically corrugated (rough) walls– incorporation of both effects

•Improvements in PSD analysis

Fluid Density Calculated by DFT for Ar on Graphite

0 1 2 3 4 5 6 7

Distance from Surface, z/s

p/p0=0.99

p/p0=0.7

p/p0=0.3

p/p0=0.001

How the PSD is calculated

0 5 10 15 20

Pore Size, Å

1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e+0

Relative Pressure

1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e+0

Relative Pressure

4-60 Å5.3 Å7.3 Å10.2 Å20.9 Å

0.0000001 0.00001 0.001 0.1 10

Relative Pressure, P/Po

Experimental Isotherm

Theoretical Isotherms (Kernel)

Linear Fredholm integral equation of the first kind

Outline of 2D-NLDFT calculations

•Model NLDFT isotherms and density profiles are calculated using Tarazona approach [1, 2].

•Attractive fluid-fluid interactions are modeled by Weeks-Chandler-Andersen potential [3].

•Pore walls are constructed by structureless graphene sheets.

•External solid-fluid interaction potential is determined by numerical integration of the 12-6 Lennard-Jones potential over the graphene geometries.

______________________________1. Tarazona, P.; Marini Bettolo Marconi, U.; Evans R. Mol Phys 1987; 60, 573.2. Lastoskie, C.; Gubbins, K.E.; Quirke N. J Phys Chem 1993; 97, 4786-96.3. Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971; 54,5237.

Carbon Slit Pore Model (History)

Rosalind E. Franklin Proceedings of The Royal Society of London Series A. Mathematical and Physical Sciences1951, 209, 196-218

Steele WA. The Interactions of Gases with Solid Surfaces, Pergamon, Oxford, 1974

N. A. Seaton, J. P. R. B. Walton and N. Quirke Carbon ,1989, 27, 853-861

C. Lastoskie, K.E. Gubbins, N. Quirke, Langmuir, 1993, 9, 2693.

J.P. Olivier, W.B. Conklin, M.V. Szombathely, in Characterization of Porous Solids (COPS-III) Proceedings,ed. by F. Rodriguez-Reinoso, J. Rouquerol, K.S.W. Sing, K.K. Unger (Amsterdam, 1994)

Relative pressure, p/p0

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

10 Å7 Å

Artifacts of Carbon Slit Pore Model(NLDFT and Molecular Simulations)

NLDFT adsorption N2 isotherms for uniform flat slit pores (kernel)

Source ofNumericalProblem

Ustinov, E.A., Do, D.D., Fenelonov, V.B. Carbon 2006, 44, 653-663.

Neimark, A.V., Lin, Y., Ravikovitch, P.I., Thommes, M. Carbon 2009, 47, 1617-1628.

Lueking, A.D.; Kim, H.-Y.; Jagiello,J., Bancroft, K., Johnson, J.K., Cole, M.W. J. Low Temp. Phys. 2009, 157, 410–428.

Kernels by GCMC and NLDFTAre qualitatively similar.

15 Å 10 Å

7 Å 4 Å

HRTEM Images of Activated Carbons

Atul Sharma, Takashi Kyotani, Akira Tomita, Carbon 38 (2000) 1977–1984

Skeletonized images ->

Bright Field STEM Image of UMC (Westvaco) Carbon

(Oak Ridge National Lab) Jagiello, J., Kenvin J., Olivier J.P., Lupini A.R., Contescu C.I., Ads. Sci & Tech. 29, 2011, 769-780.

STEM Image - Atomic Resolution

(Oak Ridge National Lab) Guo J, Morris JR, Ihm Y, Contescu CI, Gallego NC, Duscher G, Pennycook SJ, Chisholm MF.Small 2012;8:3283-3288.

Annular dark-fi eld (ADF) STEM images of UMC (a,b) and PFAC (c,d) processed to remove high-frequency noise and probe tail effects. The in-plane carbon atoms are clearly resolved, and large areas of hexagonal lattice (marked in blue) with a few five- and seven-atom ring defects (marked in red) can be seen.

Finite slit shape pores

Width, H

x L, length

Strip pore

Partially closed strip pore (channel)

Disc pore

Effective pore width, w=H-3.4 Ǻ

Marini Bettolo Marconi, U.; van Swol, F. Phys. ReV. A 1989, 39, 4109–4116.

Monson, P. A. J. Chem. Phys. 2008, 128, 084701.

Kozak, E., Chmiel, G., Patrykiejew, A., Sokolowski, S. Phys. Lett. A 1994, 189, 94-98.

Wongkoblap, A, Do, D.D. J. Phys. Chem. B 2007, 111, 13949-13956

Jagiello, J., Olivier, J. P. J. Phys. Chem. C 2009, 113, 19382-19385.

Jagiello, J., Kenvin J., Olivier J.P., Lupini A.R., Contescu C.I., Ads. Sci & Tech. 29, 2011, 769-780

-∞ ∞

Density profiles across the pores H=27.6 Ǻ (w=24 Ǻ), L=30 Ǻ, p/p0=0.001

p/p0=0.001

-3 -2 -1 0 1 2 3

z/s ff

p/p0=0.001

-3 -2 -1 0 1 2 3

p/p0=0.001

-4 -3 -2 -1 0 1 2 3 4

z/s ff

Disc pore Strip pore Channel

p/p0=0.01

-3 -2 -1 0 1 2 3

z/s ff

p/p0=0.01

-3 -2 -1 0 1 2 3

p/p0=0.01

-4 -3 -2 -1 0 1 2 3 4

z/s ff

Disc pore Strip pore Channel

p/p0=0.05

-4 -3 -2 -1 0 1 2 3 4

z/s ff

Disc pore Strip pore Channelp/p0=0.05

-3 -2 -1 0 1 2 3

z/s ff

p/p0=0.05

-3 -2 -1 0 1 2 3

p/p0=0.1

-3 -2 -1 0 1 2 3

z/s ff

p/p0=0.1

-3 -2 -1 0 1 2 3

-4 -3 -2 -1 0 1 2 3 4

z/s ff

-3 -2 -1 0 1 2 3

z/s ff

p/p0=0.2

-4 -3 -2 -1 0 1 2 3 4

z/s ff

p/p0=0.2

-3 -2 -1 0 1 2 3

z/s ff

p/p0=0.3

-4 -3 -2 -1 0 1 2 3 4

z/s ff

p/p0=0.3

-3 -2 -1 0 1 2 3

p/p0=0.5

-4 -3 -2 -1 0 1 2 3 4

z/s ff

p/p0=0.5

-3 -2 -1 0 1 2 3

z/s ff

-3 -2 -1 0 1 2 3

z/s ff

p/p0=0.7

-4 -3 -2 -1 0 1 2 3 4

z/s ff

p/p0=0.7

-3 -2 -1 0 1 2 3

NLDFT Adsorption N2 Isotherms for Slit Pores

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

10 Å7 Å

Infinite Slit Pores

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

10 Å7 Å

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

10 Å7 Å

Finite Pores, L=30 Ǻ

Channel

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

Experimental

Finite pore model

Infinite pore model

PSD Analysis for UMC (Westvaco) Carbon using Infinite and Finite Pore Models (Mix)

Fits of N2 Adsorption Isotherm Calculated PSDs

UMC sample kindly provided by Dr. Frederic Baker

Fitting error=16.8

10-5<p/p0<10-2 Fitting error = 4.2

Pore width, Å

0 10 20 30 40 50

cm3 /Å

Infinite modelFinite model (Strip+Channel)Strip component

Simple energetically and geometrically heterogeneous carbon infinite slit pore model

The model is derived from the Steele potential where the geometrical

heterogeneity is introduced only to the surface layer.

Objective:

Introduce minimal modifications to Steele potential that are necessary

to improve the model.

Carbon pore heterogeneity may be considered a combined effect of:

•Chemical composition (surface chemical groups)

•Variation of local density

•Variations of pore wall thickness

•Geometry (curvature, roughness)

External potential in heterogeneous pore

xxzx sfsfssf

2 )()(4),(ss

zzz sfsf

ssf sf

2,, 112 zzzxzxsf

The solid-fluid interaction potential of a gas molecule interacting with the graphitic surface:

For a uniform infinite graphitic surface:

zHxzHxzxV sfsfext 2

For a pore wall consisting of 3 graphitic surfaces:

sf, ssf –solid-fluid interaction parametersrs –solid surface density – distance between graphitic layers

The total external interaction potential in the pore Vext:

0 1 2 3 4 5 6

Surface energy distributionUnit cell in periodic boundary conditions

0 1 2 3 4 5 6

Multiple sites2 SitesUniform

Solid-fluid interaction parameter, sf

External potential at 1st adsorbed layer, Vext

Experimental and calculated adsorption isotherms on nongraphitized carbon black surface

Cabot BP280

Relative pressure

1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e+0

Carbon Black, BP 280Uniform surface

Multiple energy sites2 energy sites

N2 local densities in heterogeneous and uniform slit pores with w=24 Ǻ

p/p0=0.00001

-2 -1 0 1 2

p/p0=0.0001

-2 -1 0 1 2

p/p0=0.0005

-2 -1 0 1 2

p/p0=0.001

-2 -1 0 1 2

p/p0=0.01

-2 -1 0 1 2

p/p0=0.05

-2 -1 0 1 2

p/p0=0.1

-2 -1 0 1 2

p/p0=0.2

-2 -1 0 1 2

p/p0=0.3

-2 -1 0 1 2

p/p0=0.4

-2 -1 0 1 2

p/p0=0.5

-2 -1 0 1 2

NLDFT Adsorption N2 Isotherms for Uniform and Heterogeneous Slit Pores

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

10 Å7 Å

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

10 Å7 Å

Heterogeneous poresUniform pores

15 Å 10 Å

7 Å 4 Å

10 Å 7 Å

-2 -1 0 1 2

Effect of pore geometry on external wall potential

-4 -2 0 2 4

-2 -1 0 1 2

-16 -14 -12 -10 -8 -6

Vext/kT

Uniform flat wall Uniform cylindrical wall Curved (corrugated) wall

-2 -1 0 1 2

Pore geometry (roughness)Unit cell - periodic boundary conditions

Pore wall: 3 graphene layersSurface layer: z=0.3*sin(x)

External potential in flat slit and curved (rough) pores

zzz sfsf

ssf sf

2,, 11 zzzxzxsf

zHxzHxzxV sfsfext 2

The solid-fluid interaction potential of a gas molecule interacting with a graphene surface:

For a flat infinite graphene:

For a pore wall consisting of 3 graphene surfaces:

sf, ssf –solid-fluid interaction parametersrs –solid surface density – distance between graphitic layers

The total external interaction potential in the pore Vext:

For a curved surface is obtained by numerical integration.),( zx

10-6 10-5 10-4 10-3 10-2 10-1 100

A8 D4 D5 D8 BP 280Carbopack FStandard NLDFT

N2 Isotherms measured for standard reference carbon blacks, graphitized carbon black Carbopak F

and the standard NLDFT model

Experimental and calculated adsorption isotherms on nongraphitized carbon black surface

Cabot BP280

Relative pressure

1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e+0

Carbon Black, BP 280 (Experimental)Flat surface model

Curved surface model

N2 local densities in curved and flat slit pores with w=24 Ǻ

p/p0=0.00001

-2 -1 0 1 2

p/p0=0.0001

-2 -1 0 1 2

p/p0=0.0005

-2 -1 0 1 2

p/p0=0.01

-2 -1 0 1 2

p/p0=0.05

-2 -1 0 1 2

p/p0=0.1

-2 -1 0 1 2

p/p0=0.2

-2 -1 0 1 2

p/p0=0.3

-2 -1 0 1 2

p/p0=0.4

-2 -1 0 1 2

p/p0=0.5

-2 -1 0 1 2

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

10 Å7 Å

NLDFT Adsorption N2 Isotherms for Flat Slit and Curved Pores

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

10 Å7 Å

Curved poresFlat slit pores

Source ofNumericalProblem

15 Å 10 Å

7 Å 4 Å

10 Å 7 Å

Pore width, Å

0 10 20 30 40 50

m3 /Å

Uniform slitHeterogeneous slit (EH)Corrugated slit (GC)

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

ExperimentalCorrugated slit (GC)Heterogeneous slit (EH)Uniform slit

PSD Analysis of PC76 Carbon using Flat Slit and Curved Pore Models

Fits of N2 Adsorption Isotherms Calculated PSDs

PC76 – PET derived carbon,

J. Jagiello, C.O. Ania, J.B. Parra, J.J. Pis, Carbon 45 (2007) 1066–1071.

New Model 2D- NLDFT Incorporating Energetical Heterogeneity and Geometrical Corrugation

Variation of the solid-fluid interaction parameter given by trigonometric series.

-18 -16 -14 -12 -10 -8 -6 -4 -2 0

-3 -2 -1 0 1 2 3

)/2sin()( xxsf

Vext/kT

New Model 2D- NLDFT Incorporating Energetical Heterogeneity and Geometrical Corrugation

10-6 10-5 10-4 10-3 10-2 10-1 100

Exp BP 280Standard NLDFTEH-GC 2D-NLDFT

Selected 2D-NLDFT isotherms for N2 adsorption calculated for carbon

slit pores and open surface compared with BP 280 data

10-6 10-5 10-4 10-3 10-2 10-1 100

Exp BP 280Open Surfacew=15 Åw=20 Åw=30 Å

N2 local densities in curved and energetically heterogeneous pores with w=24 Ǻ

p/p0 = 0.0001

-3 -2 -1 0 1 2 3

p/p0 = 0.001

-3 -2 -1 0 1 2 3

p/p0 = 0.01

-3 -2 -1 0 1 2 3

p/p0 = 0.1

-3 -2 -1 0 1 2 3

p/p0 = 0.2

-3 -2 -1 0 1 2 3

p/p0 = 0.3

-3 -2 -1 0 1 2 3

p/p0 = 0.5

-3 -2 -1 0 1 2 3

PSD Analysis of PC58 and PC76 Carbons using 2D-NLDFT Model for EH-GC Surface

Pore width, Å

0 10 20 30 40 50

Standard NLDFT2D-NLDFT, EH-GC

Pore width, Å

0 10 20 30 40 50P

3 /Å/g

Standard NLDFT2D-NLDFT, EH-GC surface

Conclusions

• We modify the standard carbon slit pore model by changing surfaces of pore walls from flat and homogeneous to:

a. Flat energetically heterogeneousb. geometrically corrugated (rough)c. geometrically corrugated and energetically heterogeneous

• These assumptions lead to the heterogeneity of the adsorption potential.

• As a result, filling of pores is gradual and the layering transition steps are significantly reduced or eliminated.

• The carbon PSD analysis from such models are free of known artifacts:– Fits are improved – Calculated PSDs do not show the typical gap at 10 Å.

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