Surface Adhesion (Adsorption) in LBM

Key Papers

• Martys, N. and H. Chen, 1996, PRE 53, 743-750

• Raiskinmäki, P., A. Koponen, J. Merikoski, and J. Timonen, 2000, Comp. Materials Sci. 18, 7 – 12

Key Books

• Adamson, A. W., and A.P. Gast, Physical Chemistry of Surfaces, New York, John Wiley & Sons, Inc., 1997.

• Israelachvili, J. N., Intermolecular and Surface Forces, 2nd ed. Academic Press, London, 1992.

Wetting

http://www.hdm-stuttgart.de/projekte/printing-inks/b_sel42.jpg

Wetting

http://psii.kist.re.kr/Teams/psii/research/Con_4.jpg

Geometrically-controlled Superhydrophobic surfaces

http://www.nature.com/nmat/journal/v1/n1/images/nmat715-f1.jpg

LBM Adhesive Force Formula

• s is a ‘switch’ that takes on value 1 if the site at x + eat is a solid and is 0 otherwise

• We seem to have flexibility in the choice of the pre-sum factor; the papers cited use or

a

aaaadsads tswtGt eexxxF )(),(),(

Computation of

• // Compute psi, Eq. (61).• for( j=0; j<LY; j++)• for( i=0; i<LX; i++)• if( !is_solid_node[j][i])• {• psi[j][i] = 4.*exp( -200. / ( rho[j][i]));• }

Sforce•

// Compute interaction force, Eq. (66).

• for( j=0; j<LY; j++)• {• jp = ( j<LY-1)?( j+1):( 0 );• jn = ( j>0 )?( j-1):( LY-1);• for( i=0; i<LX; i++)• {• ip = ( i<LX-1)?( i+1):( 0 );• in = ( i>0 )?( i-1):( LX-1);• if( !is_solid_node[j][i]) • {• sum_x=0.;• sum_y=0.;

• if( is_solid_node[j ][ip]) // neighbor 1

• { sum_x = sum_x + WM*ex[1];• sum_y = sum_y + WM*ey[1]; }

• if( is_solid_node[jp][i ]) // neighbor 2• { sum_x = sum_x + WM*ex[2];• sum_y = sum_y + WM*ey[2]; }

• if( is_solid_node[j ][in]) // neighbor 3• { sum_x = sum_x + WM*ex[3];• sum_y = sum_y + WM*ey[3]; }

Sforce• if( is_solid_node[jn][i ]) // neighbor 4• { sum_x = sum_x + WM*ex[4];• sum_y = sum_y + WM*ey[4]; }

• if( is_solid_node[jp][ip]) // neighbor 5• { sum_x = sum_x + WD*ex[5];• sum_y = sum_y + WD*ey[5]; }

• if( is_solid_node[jp][in]) // neighbor 6• { sum_x = sum_x + WD*ex[6];• sum_y = sum_y + WD*ey[6]; }

• if( is_solid_node[jn][in]) // neighbor 7• { sum_x = sum_x + WD*ex[7];• sum_y = sum_y + WD*ey[7]; }• if( is_solid_node[jn][ip]) // neighbor 8

• { sum_x = sum_x + WD*ex[8];• sum_y = sum_y + WD*ey[8]; }

• sforce_x[j][i] = -Gads * psi[j][i] * sum_x;• sforce_y[j][i] = -Gads * psi[j][i] * sum_y;• }• }• }

Contact Angles in SCMP LBM

Interplay between these forces will determine wetting

a

aaa twtGt eexxxF )(),(),(

a

aaaadsads tswtGt eexxxF )(),(),(

• Cohesive force:

• Adhesive force:

Young’s Equation?

12

12cos

SS

Contact Angles in SCMP LBM

Assume uniform liquid or vapor surroundings:

a

aawG eF 2

a

aaa twtGt eexxxF )(),(),(

Contact Angles in LBM• Assume uniform surroundings:

a

aawG eF 2

a

aall wG eF 2

aaav

v wG eF 2

Liquid Vapor

Contact Angles in LBM• Assume uniform surroundings:

a

aawG eF

a

aaladsl

ads wG eF a

aavadsv

ads wG eF

Liquid surrounded by solid Vapor surrounded by solid

Contact Angles in LBM• Zero degree contact angle:

– Adhesive force equal to cohesive force for liquid

a

aaladsl

ads wG eF

Liquid surrounded by solid

a

aall wG eF 2

Liquid

lads GG

Contact Angles in LBM• 180 degree contact angle:

– Adhesive force on vapor equal to cohesive force for vapor

a

aavadsv

ads wG eF

Vapor surrounded by solid

a

aavv wG eF 2

Vapor

vads GG

Contact Angles in LBM• 90 degree contact angle:

– Adhesive force on vapor equal to cohesive force for ‘interface’ (= [l + v)

a

aaadsads wG eF

Interface surrounded by solid

a

aawG eF 2

Interface

GGads

Adsorption

• Asvl: Hamaker constant for interaction of solid with vapor through liquid

• : Disjoining pressure (P relative to flat, free interface)

3

6)(

svlAh

Adsorption

vap=85.7

vap=85.7857

vap=86.1285

Capillary Condensation

• Avll: Hamaker constant for interaction of liquid with liquid through vapor

• : Disjoining pressure (P relative to flat, free interface)

333 6266 hH

A

hH

A

h

A svlllvsvl

Capillary Condensation

vap=86.557

Adsorption/Capillary Condensation

0.0001

0.001

0.01

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

Saturation

-Dis

join

ing

Pre

ssu

re (

mu

ts

-2)

P

vap =

25.

560

mu

ts-2

Pva

p =

25.

574

mu

ts-2

Pva

p =

25.

632

mu

ts-2

Pvap

= 2

5.703 m

uts

-2

Te

mporal E

volutio

n

50 lu slit

28 lu slit

Hysteretic Wetting/Drying of Angular Pores

(Tuller, Or, and Dudley,1999 WRR) FrAwt

2

rpc

P

Arimb

2

n

i

i

i

F1 360

180

2tan

1

FF

Prd

2

A

FpS c

2

AS

Fpc

Saturation as a function of p at high tension

Drainage radius

Imbibition radius

Shape factor

Young-Laplace (zero contact angle)

Filled cross-sectional area

p as a function of saturation at high tension

Hysteretic Wetting and Drying

Hysteretic Wetting and Drying

0.1

1

10

0 0.2 0.4 0.6 0.8 1Saturation

-Dis

join

ing

Pre

ssu

re (

mu

ts-2

) 100 lu

Hysteritic Wetting and Drying

0.1

1

10

0 0.2 0.4 0.6 0.8 1

Saturation

-Dis

join

ing

Pre

ssu

re (

mu

ts

-2)

272 lu~2.2 cm

Invasion Percolation

Capillary Number

• v inlet/outlet velocity• viscosity of injected fluid• n porosity• interfacial tension between fluids• contact angle

cosn

vCam

Friedman, 1999. J. Adhesion Sci Technol. 13(12), 1495-1518.

Pore Selection and Impact of Ca on Pore Penetration

2,500 ts/movie step

r = 7.5

r = 6.5r = 5

v 10-3

Ca 2 x 10-4

v 10-4

Ca 2 x 10-5

Viscosity Ratio

displaced

injectedM

1.0

23

232

2

displaced

injected

displaced

injected

tc

tc

M

• For D2Q9 LBM:

-10

-8

-6

-4

-2

0

2

4

6

8

-10 -8 -6 -4 -2 0 2 4 6 8

log M

log

Ca Viscous

Fingering

CapillaryFingering

(Invasion Percolation)

StableDisplacement

Phase DiagramLenormand et al. 1988. J. Fluid Mech. 189, 165-187.

Air/Viscous Oil Glucose Soln./ Oil

Air/Viscous Oil

Frette et al., 1997. PRE 55(3) 2969-2975.

Viscosity-Matched Fluids

Monolayer of 0.7 mm beads

No Gravity Gravity

Drainage and gravity stabilization