A Statistical Mechanical Model for Hydrogen A Statistical Mechanical Model for Hydrogen Adsorption in Porous Substrates for Use in Adsorption in Porous Substrates for Use in

PEMS FuelPEMS Fuel--Cell SystemsCell Systems

E.H.Chimowitz. C.Martin, M.FabrizioDepartment of Chemical Engineering

University of Rochester, Rochester, NY,USA

For presentation at the 7th World Congress of Chemical Engineering, Glasgow, Scotland, 10-14 July 2005

Questions that motivated the researchQuestions that motivated the research

Why the need to understand the effects of confinement on Why the need to understand the effects of confinement on critical phenomena critical phenomena --both equilibrium and transportboth equilibrium and transport--what are what are the relevant technologies that might benefit from a better the relevant technologies that might benefit from a better understanding of this problem?understanding of this problem?Why should confinement affect the critical temperature of a Why should confinement affect the critical temperature of a fluid fluid --is there experimental evidence to support this?is there experimental evidence to support this?What sort of structures are of most interestWhat sort of structures are of most interest--quenched random quenched random or periodicor periodic--are there significant differences anticipated here?are there significant differences anticipated here?What are the best potential applications? fuelWhat are the best potential applications? fuel--cells, hydrogen cells, hydrogen storage in porous materials, supercritical catalysis, proton storage in porous materials, supercritical catalysis, proton transport in transport in nanoporousnanoporous membranes like those used in fuelmembranes like those used in fuel--cells.cells.

THE HYDROGEN STORAGE THE HYDROGEN STORAGE ““PROBLEMPROBLEM””

I litre of gasoline contains the energy equivalentof 634.8.10 JoulesThe fuel cell reaction:

32 2 2 2

1H O H O, 235.2 10 J/mole H2

+ → ×

Therefore, the volume of hydrogen at standardconditions equivalent to a litre of gasoline is

33.3 10 litres×

0.4 0.6 0.8 1.0 1.2

310

312

314

316

318

320SF6 phase diagram in porous glass

bulk co-existence curve 24 nm porous glass (CPG-240)

T,K

ρ(kg/dm-3)

THE EFFECT OF CONFINEMENT ON THE PHASE BOUNDARYdata adapted from Thommes and Findenegg, Langmuir, 10, 4270 (1994)

SCHEMATIC OF RANDOM PIXELIZED MEDIA

0

WHAT SORT OF MOLECULAR CONFIGURATIONS DO WE HAVE?

EXAMPLES IN A 2d LATTICE

Central molecule Central molecule

PROBABILITY OF CONFIGURATIONS IN THEPORE STRUCTURE

NΩ

Central-molecule

-matrix particle

z-lattice co-ordination number

p-porosity also the probability of observing blocked site

N-number of blocked sites

( )! 1( )! !

z N NN

z p pz N N

−Ω = −− -a binomial random variable

THE HAMILTONIAN FOR THE CONFINEDLATTICE GAS

4 [ (1 ) (1 )]LG i j i j i i j j j i i iij ij i

H n n n n nε ε ε ε ε ε µ ε< > < >

− = ℑ + Γ − + − +∑ ∑ ∑

number density of molecule ( 0 or 1)jn −

, nearest neighbor and fluid-solid coupling parametersℑ Γ −

chemical potentialµ −

random variablejε − [0 if pore matrix, 1 if molecule]

“SUMMING” THE HAMILTONIAN

RIGOROUS STATISTICAL MECHANICS

[ ][ ]0,1 0,1

exp...

expi k

i LGi

n n LG

n Hn

Hβ

ρβ= =

≡ = ∑ ∑

*-Impossible to do analytically, how about amean-field approximation?

THE HAMILTONIAN SUM AT MEAN-FIELD

( )! 1( )! !

z N NN

z p pz N N

−Ω = −−

[ ][ ]0

exp ( )1exp ( ) 1

zN N

iNtotal N

yn

yβ µ

β µ=

Ω +=Ω + +∑

4 ( )N iy n z N N≡ ℑ − + Γ

( )0

1 11 tanh 2z

i N NNtotal

n yβ µ=

⎡ ⎤= Ω + +⎣ ⎦Ω ∑

NOTE: p enters into the equation through NΩ

THE CONFINED FLUID EQUATION OF STATE

See pg. 303 : Introduction to Critical Phenomenain Fluids, E.H.Chimowitz, Oxford University Press, 2005

''

'' '

(2 1) (1 ) tanh (2 )2

1tanh (2 ( 1) )2 2

zp z

zp z

µρ β ρ

µβ ρ

⎡ ⎤− = − +⎢ ⎥⎣ ⎦⎡ ⎤+ − + Γ +⎢ ⎥⎣ ⎦

NOTE: the dependence of density on p, z, chemical potential,temperature and fluid-solid coupling parameter

CRITICAL PROPERTIES AT THE LOW p LIMITPREDICTED BY THE THEORY

' '2 2 21 1 2 2tanh ( 1) sec

2 2 2 2c z z h pz z

ρ⎡ ⎤⎡ ⎤ ⎛ ⎞Γ − Γ −

= + − −⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

'' 2 ( 2)1 ( 1)sec 2cT z p z z h z

⎡ ⎤⎧ ⎫⎛ ⎞Γ −= − − −⎨ ⎬⎜ ⎟⎢ ⎥⎝ ⎠⎩ ⎭⎣ ⎦

'' 2 ( 2)2 tanh 2

2c z p z

zµ

⎡ ⎤Γ −= − −⎢ ⎥

⎣ ⎦

NOTE:Do these results conform to those given by other models?

HOW IS THE COMPARISON WITH OTHER KNOWN RESULTS?

Bulk 3d Lattice gas model in the mean-field approximation

1( 0) 2c pρ = = -yes

-yes' ( 0)cT p z= =' ( 0) 2c p zµ = = − -yes

THE MODEL ALSO AGREEES WITH THERANDOMLY SITE DILUTED ISING MODEL

CRITICAL TEMPERATURE AT VARIOUS VALUES OF pDe et al. AIChE J., 47, 463 (2001)

Note: Symmetry and higher critical temperatureat smaller p

-10 -5 0 5 10 15

0

1

2

3

4

5

6

7

8

p=0 p=0.05 p=0.1

T c'

Γ'

ADSORPTION ISOTHERMS FROM MEAN FIELD MODEL CALCULATIONS

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Γ=8p=0.1

T/Tc=1.00 T/Tc=1.06 T/Tc=1.11

<nR>

µR

THE EFFECTS OF ENERGY HETEROGENEITY

-15 -10 -5 0 5 10 15 20

3.0

3.5

4.0

4.5

5.0

5.5

p=0.1

p=0.1

σ/Γ0

'=0

σ/Γ0

'=0.3

T c'

Γ0

'

COMPARISON OF MODEL WITH GRAND CANONICALMONTE-CARLO COMPUTER SIMULATION RESULTS

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Γ=4p=0.01Tr=1.01

Mean Field Simulation

<nR>

µR

COMPARISON OF MODEL WITH GRANDCANONICAL MONTE-CARLO COMPUTERSIMULATION RESULTS

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Γ=8p=0.1

Mean Field (TR=1.00) Simulation (TR=1.00) Mean Field (TR=1.33) Simulation (TR=1.33)

<nR>

µR

Adsorption isotherm for ethylene-carbon sieve-modelpredictions and data from Nakahara et al., J.Chem.Eng.Data, 27,317, (1982)

0.0 0.2 0.4 0.6 0.8 1.01E-3

0.01

0.1

Model, p=0.001,Γ'=95.49 Experimental data 50 C

Log

P

Normalized density

Co-existence curve for ethylene-carbon sieve-modelpredictions

0.2 0.4 0.6 0.8250252254256258260262264266268270272274276278280282284

Critical point

Γ'=95.49

p=0.001Tc=282.29ρc=0.503=26.65 mg/gm adsorbent

T, K

Reduced density

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0Ethylene-activated carbon

Experimental data, T=301.4 K - Model, p=0.02, Γ'=60.88R

educ

ed d

esni

ty

Pressure (bar)

Adsorption isotherm for ethylene-carbon sieve-modelpredictions and data from Paul, B.K., Ind.Eng.Chem.Res., 26,928(1987)

Predicted adsorption isotherm for ethylene-carbon sieve-model with data from Paul, B.K., Ind.Eng.Chem.Res., 26,928(1987)

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0Ethylene-activated carbon

Experimental data, T=260.2 K - Model, p=0.02, Γ'=60.88R

educ

ed d

esni

ty

Pressure (bar)

Hydrogen in Ti-doped metal hydrides, see Bogdanovic,Schwickardi, J.Alloys and Compounds, 253,1(1997)

0.0 0.2 0.4 0.6 0.8 1.0

0.1

1

p=0.01,Γ'=1068.22 p=0.05,Γ'=667.27 p=0.1,Γ'=353.37 Experimental data 250 C

Pres

sure

(bar

)

Normalized density

Transport through porous media Transport through porous media near a critical pointnear a critical point

What are the consequences of being near What are the consequences of being near the critical pointthe critical point--is critical slowing down a is critical slowing down a significant issue?significant issue?How do we construct computer simulation How do we construct computer simulation algorithms to do these calculations ?algorithms to do these calculations ?--here here we use the FICKIAN ensemblewe use the FICKIAN ensemble..

Laser vs. Thermal Laser vs. Thermal OutgassingOutgassingof Hydrogen from Porous Glassof Hydrogen from Porous Glass

PhotoPhoto--induced induced outgassingoutgassing is much quicker than is much quicker than the equivalent the equivalent ““thermalthermal”” process process ((Rapp.D.B.,PhDRapp.D.B.,PhD thesis, Alfred University,2004)thesis, Alfred University,2004)Rate depends upon the nature of the metal Rate depends upon the nature of the metal dopantdopant--i.ei.e. 2.0 wt % Fe vs. Ni. 2.0 wt % Fe vs. NiSimilar effects seen with both Helium and ArgonSimilar effects seen with both Helium and ArgonMechanism not understood but thought to Mechanism not understood but thought to involve cage dynamics and surface hydrogen involve cage dynamics and surface hydrogen adsorptionadsorption

Laser vs. Thermal Laser vs. Thermal OutgassingOutgassingof Hydrogen from Porous Glassof Hydrogen from Porous Glass

Laser

Furnace 400 C

Hyd

roge

nC

once

ntra

tion

Time

Simulation of diffusion in the Fickian ensemble, De et al. J.Chem. Phys., 116, 3012 (2002)

Removable partitionPeriodic boundary conditions

Net diffusion direction

x

Concentration profile

Diffusion coefficient from simulation data

Therefore, at a time 0t t≥ , with t0 chosen in advance, we have to a good approximation

that,

2

2

4 Dk tLm eεπ

−∆ = (1)

So, for two values of t>t0 , namely, 1 2 and t t we get that,

22 1( )2

1

Dk t tm em

− −∆=

∆(2)

from which it follows that,

( )( )

1

22

2 1

ln mm

Dk t t

∆∆

=−

(3)

This simple equation is the basis for the proposed simulation method. We now use it to

provide error estimates for the simulation results.

COMPUTER SIMULATION OF FLUXES ASIS APPROACHED ALONG CO-EXISTENCE DIRECTION

cT

4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

Pure homogeneous 3d lattice fluid

Parti

cle

Flux

T

CONCLUSIONSCONCLUSIONS

MeanMean--field theory appears to work reasonably well for field theory appears to work reasonably well for predicting thermodynamic properties of confined predicting thermodynamic properties of confined fluids in fluids in ““regularregular”” systemssystemsThe conventional theoretical status with hydrogenThe conventional theoretical status with hydrogen--porous carbon or hydride models needs much more porous carbon or hydride models needs much more work work LaserLaser--induced transport interesting and currently an induced transport interesting and currently an open challenge (open challenge (Rapp,D.BRapp,D.B, PhD thesis, Photo, PhD thesis, Photo--Induced Hydrogen Induced Hydrogen OutgassingOutgassing of Glass, Alfred of Glass, Alfred University, Alfred NY, 2004)University, Alfred NY, 2004)

ACKNOWLEDGEMENTS

National Science Foundation and SUN Microsystems

Vikram Kumaran, Subhranil De

Profs. Yonathan Shapir, Steve Teitel