Fractal applications: Wettability and contact angle

Fractal Applications: Wettability and Contact Angle R A N D Y DOYLE H A Z L E T T 1 Mobil Research and Development Corporation, Dallas Research Laboratory, 13777 Midway Road, Dallas, Texas 75244 Received May 9, 1989; accepted November l, 1989 A theromodynamic expression is derived for the equilibrium contact angle on fractal surfaces in which the wetted surface area is a function of the contacting fluid. The correction for fractal surfaces can be decomposed into an areal contribution analogous to Wenzel's roughness ratio, which always enhances the natural wettability characteristics of the material, and a wettability alteration factor due to fluid adsorbate size differences. Reported fractal properties of reservoir materials suggest that for most fluid pairs, such porous media will appear perfectly wetted by one of the fluids. Fractional and mixed wettability states are not prohibited by the theory but exist as combinations of perfectly wetting and perfectly nonwetting domains. ¢ 1990AcademicPress,Inc. INTRODUCTION

Due to the increasing discoveries of fractal geometry in nature and physics ( 1 ), it is believed that the use of fractals may yield insight into many unsolved or unsatisfactorily explained phenomena in fluid flow through porous media. Wettability is one such recurring problem. Expressions are herein described which account for the fractal nature of reservoir rock and may improve our understanding and predictive capabilities. A number of authors have documented the fractal character of reservoir rock surfaces. The difference between the fractal and topological dimensions of these surfaces is a characterization of surface roughness. By definition, the area of a fractal ascribes to a power law dependence A ~ ffl-D/2

[1]

where ff is the fundamental unit of area measurement and D is the fractal dimension. Should fractal behavior extend to the molecular scale, then fluids of different molecular size would "see" different solid surfaces (2).

1 To w h o m correspondence should be addressed.

Thus, expressions which include surface area or wetted surface as a constant (D = 2) cannot fully describe processes which involve solidfluid interaction. Since area enters naturally in thermodynamic expressions, many mathematical formulations of thermodynamics require close scrutiny. One such relationship considered here concerns the equilibrium contact angle at the three-phase point of contact when the solid surface is fractal. For smooth, homogeneous surfaces, the Young and Dupr6 equation describes the proper relationship between the interfacial tensions, 3"s, and the measured contact angle between two fluids resting on a solid (3, 4) "~sl -- '~s2 + ~12 cOS 0 ~-~ 0,

[2]

where the subscripts s, 1, and 2 denote the solid, fluid phase 1, and fluid phase 2, respectively. Equation [2] is readily obtained from a force balance at the three-phase contact. In accordance with Gibbsian thermodynamics, the associated interfacial energies are assumed to be extensive surface excess quantities. For rough surfaces, the meaning of a contact angle as a property is not clear from a force balance conceptual view (5). At sharp comers, the contact angle can assume a range of values.

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Journalof ColloidandInterfaceScience,Vol. 137,No. 2, July 1990

0021-9797/90 $3.00 Copyright© 1990by AcademicPress,Inc. All rightsof reproductionin any formreserved.

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RANDY DOYLE HAZLETT

Wenzel (6) recognized this difficulty and approached the problem in terms of surface energies, rather than forces at an apparent and sometimes ambiguous line of contact. Wetting of a solid occurs only if there is a resulting decrease in free energy. The apparent equilibrium contact angle is then determined by an integrated surface-fluid-fluid interaction and not by conditions existing only at the point of contact. The surface free energy change when one fluid resting on a solid reversibly displaces another is given by

esis effects which involve metastable states. Huh and Mason (15) derived a modified form of Wenzel's equation to model hysteresis effects which includes a mechanistic surface texture term to augment the conventional thermodynamic roughness factor. Although the importance of Eq. [5] is fully realized, there is evidence to suggest its lack of generality (16). The force balance approach fails for fractal surfaces since the surface can be nowhere differentiable. Figure 1 illustrates this point. Mandelbrot (1) discussed this matter in refZX/t7 = A A s ( ' Y s l - - "Ys2) + AA12~/12, [3] erence to Koch curves as mathematical monwhere AAs is the change in solid surface area sters. Analogous to Wenzel's treatment for contacted by fluid 1, and AA12 is the change rough surfaces, fractal surfaces can alter the in fluid-fluid interfacial area. Although long- apparent contact angle or wettability characrange forces are manifested through the surface teristics. tensions, the strict proportionality implied Not only can fractal surfaces be rough, but here is the equivalent of a square-well poten- the area wetted by two fluids of different motial. lecular size can be vastly different. Avnir et al. Wenzel (6) recognized that more surface (17) have published fractal dimensions of energy is associated with rougher surfaces per graphites, fume silica, faujasite, crushed glass, unit projected area of contact and applied a charcoals, and silica gel where the surfaces correction factor for this area difference. The were probed with adsorbates of different moroughness factor, r, was defined as the ratio of lecular cross sections including nitrogen, alactual surface area to geometric surface area. kanes, polycyclic aromatics, quaternary amThis resulted in a correction to the Young- monium salts, and polymers. Actually, the Dupr6 equation surface area determined by adsorptive techniques gives the spacing of adsorption sites. r ( q / s l - - q/s2) + "Y12c O s 0 = 0. [4] Equating surface tiling with surface area is a The apparent contact angle for rough surfaces conventional practice and may differ only by could then be expressed in terms of that for the choice of molecular occupational areas smooth surfaces as (18). The wetted surface and adsorptive surface may be slightly different; however, mulcos 0rough= r cos 0smooth. [5 ] The most astounding result of this work which has been confirmed through experimentation (6, 7) is that contact angles for nonwetting fluids are elevated, while those for wetting fluids are depressed. Thus, surface roughness enhances the inherent wetting behavior. Both surface roughness and heterogeneity contribute to the phenomenon of contact angle hysteresis (8-13). The results of Wenzel were later extended to heterogeneous surfaces (14). However, Eq. [ 5 ] has a thermodynamic basis and cannot predict complicated hysterJournal of Colloid and Interface Science, Vol. 137, No. 2, July 1990

FiG. 1. Contact angle on a fractal surface.

529

FRACTALS AND WETTABILITY

tilayer adsorption phenomena confirm that surface mobility is fully developed above monolayer coverage (19). A thermodynamic treatment for the equilibrium contact angle follows in which the area is a function of the contacting fluid to account for the fractal nature of a solid substrate.

the reference state and U(O-rc/2) is the unit step function. The upper set of signs in g(0) corresponds to 0 > ~r/2, while the lower are for 0 < ~r/2. Equation [ 6 ] can be normalized with the interfacial free energy of the reference state fluid-fluid boundary to give -- C,YslO. I-D~2

THEORY

Define a reference state for free energy calculations as that shown in Fig. 2 with mutually saturated fluids and fluid phase 2 completely wetting the solid. Surface free energy changes can be calculated assuming constant volume and neglecting gravitational forces for contact angles less than and greater than 90 ° . Replacing AAs in Eq. [ 3 ] with two separate expressions for area, one obtains

4"a-R 2"Y12

1

- c o s O ) g ( O ) - 1.

[9]

In the first two terms, a fractal surface is divided by a Euclidean surface. If a reference area, 0.R, is introduced such that 7rR2sin20 = Co" l-D~2,

[10]

and the relationship between R and R0 given in Eq. [8 ] is used, the free energy becomes

~ F = C"Ysl0.~ -D/2 -- C7s20l-D/2 + AAI2"Y12,

C,.Ys2o.l-D/2

4~'R2yI2

[6]

where 0.1 and a2 are the occupational areas of molecular species 1 and 2. Equation [6 ] also assumes that the fractal dimension remains unchanged at least over the interval [ ~ , a2]. The fluid-fluid interfaeial change under the constant volume criterion is given as

"Ysl/(0._.]I-D/2 1

2~ff = 1 g ( O)sin 20[( t4 k \')'12 / \ 0.R]

\'Y12 ] \ O'R]

+

1

]

(1-cos0)g(0)-l.

[111

AAI2 = 4rrRg[(1 - cos O ) g ( O ) / 2 - 1], [7] Solving for the m i n i m u m free energy state, letting the term in brackets be denoted by K,

with g(O) = ( R / R o ) 2 = [U(O - re/2)

T- ( 1 + cos 8)2(2 T- COS 0)/4] -2/3,

[8]

dO -

=~cosO-~+

1

where R0 is the natural radius associated with

+~(1--cos0)

:':':':':'i':':'i':':'I.-;'1'4 S~ PHAS2~E

/

REFERENCSTATE E ,

sin 20

] sin20

T[g(O)13/2 I121

for which - K = cos 0 is a solution. Equation [12] also has the trivial solution of cos 0 = 1. So one obtains COS 0fraeta1 =

[ ( 1 -- I~fI-D/2,[0.1,1-D/2 ] 1- F )t~RR) J cOS 0Euclid. . . . [13]

FIG. 2. Free energy reference state and nomenclature.

with f --- (o'2/61) and £ --- (Ts2/'Ysl). JournalofColloidandInterfaceScience,Vol. 137,No. 2, July 1990

530

RANDY DOYLE HAZLETT RESULTS AND DISCUSSION

4 [

.

.

.

.

.

.

.

.

i

~ = 2 .7 s

Application of both fractal and Euclidean concepts requires some crossover point where an object ceases to be a fractal. Indeed, most fractal physical p h e n o m e n a have upper and lower limits between which they are fractal ( 1, 20, 21 ). The reference area, aR, is the crossover area and represents that scale which would yield the Euclidean area if t h e fractal nature and dimension held to this scale. For sedimentary rocks, C a 1-m2 should be on the order of the grain size (22). One can also define a lower limit of fractal behavior, aL, which can typically be on the order of atomic dimensions of the substrate (23). The value of D may, of course, change with scale as one advances from the molecular level to the macroscopic, depending upon the mechanisms responsible for the spatial distribution of matter. It is, therefore, important to evaluate the reference area with the appropriate choice o f D . The correct fractal dimension for use is that displayed at the onset of fractal behavior upon increasing a. The modification factor ofEq. [ 13 ] has been subdivided into two contributions. The second factor is a measure of the extent of the fractal nature of the material or the range of applicability of Euclidean geometry. This term is always greater than or equal to unity, as seen in Fig. 3, and therefore tends to enhance the natural wetting characteristics of the material.

35~

. . . . . . . .

~

. . . . . . . .

~

. . . . . . . .

~

. . . . . . . .

k

'& 30~,

.:° F,, 20

w

10

\\\

~'~.

\~x\D=2.75 \~ D=2.50 ~

- ~-~--Z.~ _-~-~

.........................................

o i0-~

............... I0-~

TU--T-TT"~'"U=~Y':~:., i0-~

i0-I

i0 o

'

(a)

F=2

p- 0=2 50',

'. D=2 25

"~

"~,,

_1

" - ~ z : Z i .................. o . . . . . . . .

10 q

i

. . . . . . .

I0 o AREA RArI0,

101

%/=~

2 (b)

t

.--:"L:.............

,< u.

i

/ /

D=2.25~'',///

J ~ o i--

G . 5 0 /// / / '/D=2.75

F = 1/2

. . . . . . . .

10-1

I

. . . . . . . .

10 o

101

AREA RA"rIO, 0"2/0 1

FIG. 4. Wettabilitycontact angle correction factors for surfaces of selected fractal dimension: (a) phase 1 preferentially wetting, P = 2; (b) phase 1 preferentiallynonwetting, £ = ½.

The first term can either elevate or depress the contact angle depending upon both the natural wetting tendencies and which fluid "sees" more solid surface, as apparent from Figs. 4a and 4b. It can be readily shown that should either surface probe area, al or ff2, fall outside the range of fractal behavior, the appropriate modification of Eq. [13] would require only a substitution of the limiting value, aL or ~rR, for that area. For example, should both surface probes of a fractal surface be able to follow the smallest irregularities in the surface, O (aL), then the correct form is

AREA RATIO, O-I/O R

FIG. 3. Areal or roughnesscorrectionfactorsfor surfaces of selected fractal dimension. J o u r n a l o f C o l l o i d a n d Interface Science,

Vol. 137, No. 2, July 1990

COS 0fractal = /~RR )

COS 0Euclld . . . .

[14]

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FRACTALS AND WETTABILITY

The correction factor in this special case corrects only for surface roughness. Figure 3 also gives the relationship between the extent of fractal behavior, fractal dimension, and the equilibrium contact angle in a Euclidean sense for trl, a2 < aL if one replaces the abscissa and ordinate with the area ratio (aL/~R) and [sec 0], respectively. The lines then represent the envelope of partial wetting behavior at constant fractal dimension. Again, the fractal dimension of Eq. [ 14 ] should be that displayed at the onset of fractal behavior, i.e., crL, and need not be restricted to the molecular level. The results can immediately be extended to uniformly heterogeneous solids by analogy with Cassie (14) to give COS 0apparent • ~ picos Oi,

[ 15 ]

i

where Pi represents the fraction of surface substrate i, and contact angles for each material are evaluated with the appropriate fractal expression. Similarly, an effective composite surface can be achieved if molecules of a fluid phase with smaller molar volume are retained within the portions of the substrate surface inaccessible to the fluid of larger molar volume. Equations [ 13 ] and [ 14 ] highlight the effects of the fractal nature of the substrate on the equilibrium contact angle. Physically, however, the cosine of the apparent contact angle is restricted to the interval [ - 1, 1]. The results can be recast in terms of a dimensionless spreading coefficient (24, 25) to quantify wettability when the contact angle is no longer a unique measure of wetting characteristics.

S = ( r f l-D/2 - 1) × (%All(ff_l_lll D / 2 _ 1. \~f12]\

[16]

fiR]

Positive values of S correspond to the condition of spreading. Katz and Thompson (22) have measured the fractal dimension of the pore-rock interface for a number of sandstones and report values ranging from D = 2.57 to 2.87. In addition, their research indicated that the fractal

behavior is apparent from linear dimensions of about 20 A, the suggested crystal nucleus size, to upper limits between 2.5 and 98 t~m. Krohn (26) has also reported fractal dimensions of sandstone, shale, and carbonate fracture surfaces between 2.27 and 2.89 with upper limits of fractal behavior of 2 to 50 #m. The work of Avnir et al. (27) supports these measurements where materials such as dolomite indicated fractal behavior down to probe occupational areas of 20 ~2. Although many c o m m o n reservoir fluids such as water ( 10.8 A2), nitrogen ( 16.2 A2), and methane ( 19.4 A2) may be able to follow the smallest surface irregularities, it is anticipated that the higher hydrocarbons cannot. Indeed, Winslow (28) ascertained that the fractal nature of portland cement pastes terminated at some molecular size between that of nitrogen and water vapor. The oil-wet character of many reservoirs can be ascribed to adsorption of asphaltenes (29, 30), which have reported occupational areas (31) in the range of 380-1100 A2. CONCLUSIONS

The modified Young-Dupr6 equation, and thus the equivalent Wenzel r ratio, was derived for a fractal surface from a thermodynamic basis. The correction for the contact angle could be decomposed into two factors: a roughness term which always enhances the inherent wettability characteristics of the substrate and a wettability alteration factor which allows for different surface probe sizes. Modifications are described for probe sizes which fall outside the range of fractal behavior. With reported differences in fractal and topological dimensions of reservoir rocks, Eq. [13] suggests that such porous media will appear to be either perfectly wetting (0 = 0 ° ) or perfectly nonwetting (0 = 180 ° ) for most fluid pairs. For example, a fractal dimension of 2.14 is sufficient to shift the contact angle for water on a surface of polytetrafluoroethylene from 108 o to 180 °, assuming that fractal behavior is exhibited from atomic dimensions to the micrometer range. This estimate used the data Journal of Colloid and Interface Science, Vol. 137, No. 2, July 1990

532

RANDY DOYLE HAZLETT

o f Melrose (32) to characterize the solid-fluid interfacial tensions and the assumption that % - %2 for a vapor phase and a low energy solid. If the same surface were fractal only to about 1000 A2, a fractal dimension o f 2.23 would suffice to p r o d u c e completely nonwetting behavior. These calculations offer a possible explanation to the historically difficult task o f obtaining reliable estimates o f surface free energies on solids. Even polished materials m a y yet display fractal behavior on the m o lecular level. Fractional and mixed wettability due to heterogeneity are not prohibited by the theory but are c o m b i n a t i o n s o f perfectly wetting and nonwetting domains. The intermediate wettability case is removed for surfaces with a fractal dimension differing significantly from the topological dimension and exists only as an effective contact angle in the case o f fractional wettability. These results suggest a new set o f wettability indices for reservoir materials, consisting o f the fraction o f each wetting state and some measure o f the spatial distribution o f those types. A n analysis for the wetting phase saturation and scaling behavior for fractal pore models as a function o f LaPlace pressure has recently been delineated (33). Such models m a y prove to be very useful in light o f this, yet another, example in which fractals cause " a p p a r e n t " behaviors which are limits o f classical ranges. APPENDIX: NOMENCLATURE A A12 As C D F ff f g K Pi

R

Area Fluid-fluid interfacial area Area o f solid-fluid contact Proportionality constant Fractal dimension H e l m h o l t z free energy Dimensionless Helmholtz free energy Ratio o f fluid occupational areas G e o m e t r i c constraint Parameter defined as bracketed term in Eq. [ l l l Fraction o f surface type i Radius o f drop curvature

lournal of Colloid and Interface Science, Vol. 137, No. 2, July 1990

Ro r S U r 7 7L2 "Ysi a ai

aL aR 0

D r o p radius o f fluid in reference state Ratio o f actual to projected surface area Dimensionless spreading coefficient Step function Ratio o f solid-fluid interfacial tensions Interfacial tension Fluid-fluid interfacial tension Solid-fluid phase i interfacial tension F u n d a m e n t a l unit o f area m e a s u r e m e n t Molecular occupational area o f fluid probe i Lower limit o f fractal behavior Reference area Contact angle ACKNOWLEDGMENTS

I thank D. U. von Rosenberg for inspiring me to do this work and Mobil Research and Development Corporation for permission to publish it. REFERENCES 1. Mandelbrot, B. B., "The Fractal Geometry of Nature," Freeman, New York, 1983. 2. Pfeifer, P., and Avnir, D,, J. Chem. Phys. 79, 3558 (1983). 3. Adamson, A. W., and Ling, I., in "Contact Angle, Wettability, and Adhesion" (Advances in Chemistry, Series 43), p. 57. Amer. Chem. Soc., Washington, DC, 1964. 4. Johnson, R. E., J. Phys. Chem. 63, 1655 (1959). 5. Gibbs, J. W., "The Collected Works of J. Willard Gibbs," p. 326. Yale Univ. Press, New Haven, CN, 1948. 6. Wenzel, R. N., lnd Eng. Chem. 28, 988 (1936). 7. Dettre, R. H., and Johnson, R. E., in "Contact Angle, Wettability, and Adhesion" (Advances in Chemistry, Series 43 ), 19. 136. Amer. Chem. Soc., Washington, DC, 1964. 8. Cox, R. G., J. FluidMech. 131, 1 (1983). 9. Good, R. J., J. Amer. Chem. Soc. 74, 5041 (1952). 10. Joanny, J. F. and de Gennes, P. G., J. Chem. Phys. 81, 552 (1984). 11. Pomean, Y., and Vannimenus, J., J. Colloid Interface Sci. 104, 477 (1985). 12. Schwartz, L. W., and Garoff, S., Langmuir 1, 219 ( 1985). 13. Shuttleworth, R., and Bailey, G. L. J., Discuss. Faraday Soc. 3, 16 (1948). 14. Cassie, A. B. D., Discuss. Faraday Soc. 3, 11 (1948). 15. Huh, C., and Mason, S. G., J. Colloid Interface Sci. 60, 11 (1977). 16. Morrow, N. R., J. Canad. Pet. Technol. 14, 42 (1975).

FRACTALS AND WETTABILITY 17. Avnir, D., Farin, D., and Pfeifer, P., J. Chem. Phys. 79, 3566 (1983). 18. Adamson, A. W., "Physical Chemistry of Surfaces," p. 580. Interscience, New York, 1967. 19. Zimmerman, J. R., and Lasater, J. A., J. Phys. Chem. 62, 1157 (1958). 20. Argyrakis, P., and Kopelman, R., J, Chem. Phys. 83, 3099 (1985). 21. Orbach, R., Science 231, 814 (1986). 22. Katz, A. J., and Thompson, A. H., Phys. Rev. Lett. 54, 1325 (1985). 23. Wong, P., Phys. Today December, 24 (1988). 24. Cooper, W. A., and Nuttall, W. A., J. Agric. Sci. 7, 219(1915). 25. Harkins, W. D., and Feldman, A., J. Amer. Chem. Soc. 44, 2665 (1922). 26. Krohn, C. E., J. Geophys. Res. B 93, 3297 (1988).

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27. Avnir, D., Farin, D., and Pfeifer, P., Nature308, 261 (1984). 28. Winslow, D. N., Cem. Concr. Res. 15, 817 (1985). 29. Clementz, D. M., SPE/DOE Preprint 10683 presented at the Third Joint Symposium on Enhanced Oil Recovery, Tulsa, OK, April 4-7, 1982. 30. Collins, S. H., and Melrose, J. C., SPE Preprint 11800 presented at the International Symposium on Oilfield and Geothermal Chemistry, Denver, CO, June 1-3, 1983. 31. Dickie, J. P., Hailer, M. N., and Yen, T. F., J. Colloid Interface Sci. 29, 475 (1969). 32. Melrose, J. C., in "Contact Angle, Wettability, and Adhesion" (Advances in Chemistry, Series 43 ), p. 158. Amer. Chem. Soc., Washington, DC, 1964. 33. de Gennes, P. G., in "Physics of Disordered Materials" (D. Alder, H. Fritsche, and S. R. Ovshinsky, Eds.), p. 227. Plenum, New York, 1985.

Journalof ColloidandInterfaceScience,Vol.137,No. 2, July1990