DOI: 10.1021/la102757t 15933Langmuir 2010, 26(20), 15933–15937 Published on Web 09/28/2010

pubs.acs.org/Langmuir

© 2010 American Chemical Society

Chemical Nano-Heterogeneities Detection by Contact Angle Hysteresis:

Theoretical Feasibility

Eyal Bittoun and Abraham Marmur*

Department of Chemical Engineering Technion, Israel Institute of Technology, 32000 Haifa, Israel

Received July 9, 2010. Revised Manuscript Received September 9, 2010

The theoretical feasibility of detecting chemical nanoheterogeneities on solid surfaces by measurement of contactangle hysteresis (CAH) was studied, using simplified models of cylindrical (2D) and axisymmetric (3D) drops oncorresponding models of chemically heterogeneous, smooth solid surfaces. This feasibility depends on the ratio betweenthe external energy input to the drop and the energies needed to deform its liquid-gas interface and move the contactline across energy barriers. A ubiquitous source of external energy is building vibrations, since most contact-anglemeasurements are done in buildings. The energy barriers that oppose the motion of the contact line were numericallycalculated for various parameters of the two systems. The variations of the liquid-gas interfacial energy are discussed interms of orders of magnitude. By comparing these energies, it is concluded that under regular (“barely perceptible”)building vibrations CAH measurements may detect chemical heterogeneities at the few nanometers scale.

Introduction

Real solid surfaces are usually rough and chemically hetero-geneous to some extent. As is well-known,1-4 these nonuniformitieslead to the phenomenon of contact angle hysteresis (CAH). Therange ofCAH (the difference between the advancing and recedingapparent contact angles) plays an important role in many applica-tions, e.g., coating, antibiofouling,5-7 andmicrofluidics.8,9The recentinterest in nonwettable surfaces (usually referred to as “super-hydrophobic”) has also emphasized the necessity of characteriz-ing surfaces in terms of theirCAH.6,10,11 In addition, aswill be dis-cussed in the present paper, CAH may serve as a sensitive meansfor detecting surface heterogeneities.12,13

The phenomenon of CAH stems from the multiplicity of meta-stable equilibrium states of a solid-liquid-fluid wetting systemthat is associated mainly with chemical heterogeneity or roughness

of the solid surface.14-22 Each metastable equilibrium state (a localminimum in the curve of the Gibbs energy versus the geometricCA) is represented by its corresponding apparent contact angle(CA) within the hysteresis range.14,15 The reason for the multi-plicity of minima points in the Gibbs energy is that at multiplelocations along the solid surface the local geometric CA equalsthe local Young CA.16,17 The lowest minimum point refers to theglobal minimum in the Gibbs energy and is represented by themost stable CA. In-between two successive local minima, theremust exist a local maximum. The height of this maximum definesthe energy barrier that has to be overcome in order to move fromone localminimum to the next. Themagnitude of the energy barrierbetween two successive minima increases as the system approachesthe global minimum.14,16

For a given wetting system, the theoretical and effective (experi-mentallymeasured) CAHrangemay be different. The highest CAforwhich there exists a localminimum in theGibbs energy definesthe theoretical advancing CA. Similarly, the lowest CA for whichthere exists a local minimum defines the theoretical receding CA.Practically, however, the wetting system is always exposed to someexternal energy source, such as vibrations.This energy sourcemayenable the overcoming of some of the energy barriers in theGibbsenergy as proposed by Johnson and Dettre.2,21 Since the energybarriers are largest near the global minimum and lowest near theadvancing and receding CAs, the external energy source affectsthe effective CAH range: the effective advancing CA must belower than the theoretical one, and the effective recedingCAmustbe higher than in theory. Therefore, if the external energy source issufficiently strong, the effective CAH may be eliminated and thesurface may appear to be chemically homogeneous despite itsactual heterogeneity.

CAHhas the potential to serve as a sensitivemeasure of chemicalheterogeneity. For example, it has been experimentally demon-strated that it is possible to identify uniformly spread, nanometric“holes” in a monolayer by CAH measurements.13 Therefore, animportant general question is: what is the order of magnitude ofthe scale of chemical heterogeneity that can be detected by CAH

*Corresponding author. Fax: 972-4-829-3088. E-mail: marmur@technion.ac.il.(1) Shuttleworth, R.; Bailey, G. L. J. Disc. Faraday Soc. 1948, No. 3, 16-22.(2) Johnson, R. E.; Detrre, R. H. In Contact Angle, Wettability, and Adhesion;

Advances in Chemistry Series; Fowkes, F. M., Ed.; American Chemical Society:Washington, DC, 1964; Vol. 43, pp 112-135.(3) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972, 38, 341–358.(4) Joanny, J. F.; de Gennes, P.-G. J. Chem. Phys. 1984, 81, 552–562.(5) Scardino, A. J.; Zhang, H.; Cookson, D. J.; Lamb, R. N.; de Nys, R.

Biofouling 2009, 25, 757–767.(6) Marmur, A. Biofouling 2006, 22, 107–115.(7) Schmidt, D. L.; Brady, R. F., Jr.; Lam, K.; Schmidt, D. C.; Chaudhury,

M. K. Langmuir 2004, 20, 2830–2836.(8) Chen, J. Z.; Troian, S. M.; Darhuber, A. A.; Wagner, S. J. App. Phys. 2005,

97, 140906–140915.(9) Fang, G.; Li, W.; Wang, X.; Qiao, G. Langmuir 2008, 24, 11651–11660.(10) Li, W.; Amirfazli, A. J. Colloid Interface Sci. 2005, 292, 195–201.(11) Patankar, N. A. Langmuir 2004, 20, 8209–8213.(12) Wolff, V.; Perwuelz, A.; Achari, A. E.; Caze, C.; Carlier, E. J. Mater. Sci.

1999, 34, 3821–3829.(13) Bittoun, E.; Marmur, A.; Ostblom, M.; Ederth, T.; Liedberg, B. Langmuir

2009, 25, 12374–12379.(14) Marmur, A. Adv. Coll. Inter. Sci. 1994, 50, 121–141.(15) Marmur, A. Soft Matter 2006, 2, 12–17.(16) Marmur, A. J. Colloid Interface Sci. 1994, 168, 40–46.(17) Marmur, A. Coll. Surf. A 1998, 136, 209–215.(18) Fang, C.; Drelich, J. Langmuir 2004, 20, 6679–6684.(19) de Gennes, P.-G., Brochard-Wyart, F., Qu�er�e, D. Capillarity and Wetting

Phenomena; Springer-Verlag: New York, 2003; p 84.(20) Prevost, A.; Rolley, E.; Guthmann, C. Phys. Rev. Lett. 1999, 83, 348–351.(21) Johnson, R. E., Jr.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744–1750. (22) Marmur, A.; Bittoun, E. Langmuir 2009, 25, 1277–1281.

15934 DOI: 10.1021/la102757t Langmuir 2010, 26(20), 15933–15937

Article Bittoun and Marmur

measurements under regular laboratory conditions? This questionwas partially answered by Fang and Drelich,18 who theoreticallyshowed that the critical strip width, below which CAH cannot bedetected on a surfacemade of periodic strips of two different typesof chemistries is of the order ofmagnitude of 10 nm. The problemwas also analyzedwith regard to thermal vibrations by deGenneset al.19 and Pr�evost et al.,20 who concluded that an energy of theorder ofmagnitude of 20kbT (where kb is theBoltzmann constant)is required toovercomehysteresis causedby surfacedefects of about20 nm in size. Since almost all CAmeasurements are performed inlaboratories within buildings, the objective of the present paper isto answer this question with regard to liquid drops on chemicallyheterogeneous, smooth surfaces under typical, natural buildingvibrations. It will be shown that under regular conditions ofbarely perceptible building vibrations CAH measurements candetect chemical heterogeneity at the nanoscale.

Theory

When a drop on a solid surface vibrates (e.g., due to buildingvibrations), the gained kinetic energy may be consumed by a fewprocesses, mainly oscillatory variations in the liquid-gas inter-facial area andmotionof the contact line. These processes are verycomplex and may occur either independently or simultaneously.Therefore, a relatively simple analysis is desirable in order toidentify the orders of magnitude of the energy consumed by eachprocess. In addition, energy dissipation takes place, due to eitherviscous flowof liquid inside the dropormotionof the contact line.The latter is assumed here to be included in the energy needed toovercome the energy barriers between metastable states, anddissipation due to internal flow is assumed negligible. First, theenergy barriers that oppose the motion of the contact line areestimated using simple models of a drop on a chemically hetero-geneous but smooth solid surface. Then, the variations in liquid-gas interfacial area are discussed at an order of magnitude level.These estimates enable the discussion of the feasibility of chemicalheterogeneity detection by CAH.

Two simplified models of wetting systems are studied in thepresent paper: a cylindrical (two-dimensional) liquid drop and anaxisymmetric drop, both on a solid surface in a gaseous atmo-sphere (Figure 1a). The solid is smooth but chemically hetero-geneous. In the cylindrical system, the chemical heterogeneity ischaracterized by a periodic variation of the Young CA along thelength of the solid surface (Figure 1b). The axisymmetric system ischaracterized by a periodic radial variation of the Young CA(Figure 1c). It is of interest to analyze both systems, in order tofind out whether the results are effectively independent of theparticular system. The periodic variations in the local Young CAare assumed sinusoidal as a zeroth-order approximation of thechemical heterogeneity

cos θYðxÞ ¼ cos θ0 þj cos2πx

λ

� �ð1Þ

where θY is the local Young CA, θ0 is the average CA, j is theamplitude of heterogeneity, x is the coordinate along the solidsurface, and λ is the wavelength of the chemical heterogeneity.

The Gibbs energy for a cylindrical drop (per unit depth) is

G2D ¼ σlA2Dl þ 2

Z r2D

0

½σslðxÞ- σsðxÞ� dx ð2aÞ

and for an axisymmetric drop

G ¼ σlAl þ 2π

Z r

0

½σslðxÞ- σsðxÞ�x dx ð2bÞ

In these equations, the superscript 2D stands for cylindrical drop,Al is the interfacial area of the liquid-air interface, σl and σs arethe surface tensions of the liquid and solid, respectively, σsl is the

solid-liquid interfacial tensions, and r is the base half-lengthor base radius of the drop. The first term in the right-hand-sideof these equations represents the surface energy of the liquid-airinterface and the second one stands for the change in surfaceenergy due to wetting (or dewetting) of the solid surface. In theabsence of gravity, which is assumed here for simplicity, the shapeof the drop must be circular in the 2D case and spherical foraxisymmetric systems. Therefore, Al is given by

A2Dl ¼ 2r2Dθ

sin θð3aÞ

Al ¼ 2πr21- cos θ

sin2 θ

� �ð3bÞ

where θ is the geometric CA, which, at equilibrium, becomesidentical with the local Young CA. The base half-length or radiusof the drop relates to the drop volume, V, (or volume per unitdepth, V2D) by

r2D ¼ sin θ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2V 2D

2θ- sin 2θ

sð4aÞ

r ¼ 3V

π

� �1=3sin θð2- 3 cos θþ cos3 θÞ- 1=3 ð4bÞ

Figure 1. Schematic description of the model systems: (a) A sideviewof two-dimensional and axisymmetricwetting systems; (b) thesinusoidal variationsof the localYoungCAalong the surface in thetwo-dimensional model; (c) the sinusoidal variations of the localYoung CA along the surface in the axisymmetric model.

DOI: 10.1021/la102757t 15935Langmuir 2010, 26(20), 15933–15937

Bittoun and Marmur Article

Substitutionof eqs 1, 3a, and 4a into eq2a leads to the followingexpression for the dimensionless Gibbs energy for the 2D drop

G�2D � G2D

λσl¼ 2r�2D θ

sin θ- cos θ0

� �-jπsinð2πr�2DÞ ð5aÞ

where r*2D � r2D/λ. Similarly, substitution of eqs 1, 3b, and 4binto eq 2b yields the dimensionless Gibbs energy for the axisym-metric drop

G� � G

λ2σl

¼ πðr�Þ2 21- cos θ

sin2 θ

� �- cos θ0

" #

-j r� sinð2πr�Þþ 1

2π½cosð2πr�Þ-1�

� �ð5bÞ

where r* � r/λ.As iswell-known14,16,18,21,22 anddemonstratedagain inFigures 2

and 3, the Gibbs energy curve has multiple minima that represent

metastable states. The transition between one metastable state toanother, more stable one, involves overcoming an energy barrier,defined in a dimensionless form as

B�2D � G�2Dlmax -G�2Dlmin ð6aÞ

B� � G�lmax -G�lmin ð6bÞwhere G*1max is a local maximum that follows a local minimum,G*1min, in the direction of the global minimum.

Prior to or simultaneously with overcoming energy barriersthat enables motion of the contact line, the drop oscillates and itsliquid-gas interfacial area, on the average, increases. It is extremelydifficult to predict such oscillations, so only a simple order-of-magnitude estimate is presented here. If a is the amplitude ofbuilding vibrations, then the change in the drop surface area is ofthe order of magnitude of πa for the 2D case and πra for theaxisymmetric drop. These expressions can be verified by checking

Figure 2. Dimensionless Gibbs energies (a-c), and energy barriers (d-f) vs the geometric contact angle for two-dimensional drops. Upperrow:j=0.25,R=102, andθ0=(a) 50�, (b) 75�, and (c) 100�. Lower row: the values ofj are givenby the numbers in the figures, the values ofR are 1�102 (Δ) and 1�105 (o), and θ0=(d) 50�, (e) 75�, and (f) 100�.

Figure 3. DimensionlessGibbs energies (a-c), and energy barriers (d-f) vs the geometric contact angle for axisymmetric drops.Upper row:j=0.25,R=102, andθ0=(a) 50�, (b) 75�, and (c) 100�. Lower row: the valuesofjare givenby thenumbers in the figures, thevaluesofRare1�102 (Δ) and 1�105 (o), and θ0 = (d) 50�, (f) 75�, and (f) 100�.

15936 DOI: 10.1021/la102757t Langmuir 2010, 26(20), 15933–15937

Article Bittoun and Marmur

some simple geometries, such as half a circle that turns by oscilla-tions to half an ellipse (for the 2D case), or half a sphere that turnsinto half a spheroid (ellipsoid).

Thus, a heterogeneous surface would appear homogeneous byCAH measurement if

E2Dvib > σlΔA

2Dl þB2D

max ¼ ∼ðσlπa þ λσlB�2DmaxÞ ð7aÞor

Evib > σlΔAl þBmax ¼ ∼ðσlπra þ λ2σlB�maxÞ ð7bÞwhereEvib is the kinetic energy that the drop gains from the build-ing vibrations,Bmax is the highest energybarrier of the system, andΔAl is the increase in the liquid-gas interfacial area. Obviously,the present treatment of overcoming the energy barriers is only anorder of magnitude analysis. Specific characteristics of the vibra-tions aswell as amore sophisticated analysis is required inorder toget more accurate results.23,24

To estimate the kinetic energy of the drop it is assumed that themaximumvelocity of the buildingdetermines the kinetic energy ofthe drop as a whole. Thus

E2Dvib � FV2Dð2πaf Þ2

2ð8aÞ

and

Evib � FVð2πaf Þ22

ð8bÞ

where F is the density of the liquid and f is the frequency of thebuilding vibrations. The use of the maximum velocity makes thisa stringent assumption on the “safe” side, since it actually over-estimates the available energy.

Results and Discussion

Typical Gibbs energy and energy barrier curves are shown inFigures 2 and 3, for 2D and axisymmetric drops, respectively. Toenable convenient comparison of the various cases, the dimen-sionless Gibbs energy is represented by the difference between itsactual value and the value at the global minimum

ΔG�2D � G�2D -G�2Dgm ð9aÞand

ΔG� � G�-G�gm ð9bÞThe values of 50�, 75�, and 100� for the average CA, θ0, were

used for the calculations, to account for various degrees of hydro-philicity/hydrophobicity. All of the Gibbs energy curves given inFigures 2a-c and 3a-c were calculated for an amplitude of j=0.25 and a relative size of the drop of 102. The latter is defined asthe ratio of the diameter of the drop before touching the solidsurface to the wavelength of the surface heterogeneity22

R2D � ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V2D=π

pÞ

λð10aÞ

R � ð ffiffiffiffiffiffiffiffiffiffiffiffi6V=π3

p Þλ

ð10bÞ

The effects of the heterogeneity amplitude and the relative sizeof the dropon the dimensionless energy barriers are demonstratedin Figures 2d-f and 3d-f for 2D and axisymmetric drops,

respectively. The range of values for the amplitude,j, was limitedin such a way that the value of the local Young CA is alwayswithin the range of 0 e θY e 120�, as is the case in reality. Therelative sizes of the drops were 102 and 105. The maximum pointof each curve represents the highest energy barrier for moving tothe most stable CA. The highest and lowest CAs for each curverepresent the theoretical advancing and recedingCA, respectively.

As previously observed,16 the magnitudes of the dimensionlessenergy barriers for 2Ddrops,B*2D, on a given solid surface turn outto be almost independent of the drop volume. This is reemphasizedin Figure 2d-f, which show practically the same dimensionlessenergy barriers for drops that differ by 3 orders of magnitude intheir relative size. In contrast, the dimensionless energy barriersstrongly increasewith an increase in the amplitude of heterogeneity,which is associated with an increase of the theoretical CAH range.For axisymmetric drops, the magnitudes of the dimensionlessenergy barriers, B*, turn out to strongly depend on the relative sizeof the drop. The calculations done led to the conclusion that B* isactually linearly dependent on R for a given chemically hetero-geneous solid surface. Consequently, an additional dimensionlessform of the energy barrier for axisymmetric drops was defined as

B�� � B�=R ð11ÞThis form of dimensionless energy barriers is presented in

Figure 3d-f. As can be seen in this figure, B** is indeed practi-cally independent of the relative drop size, for a given solid surface.Similarly to the 2D case, the dimensionless energy barriers foraxisymmetric drops strongly increase with an increase in the ampli-tude of heterogeneity (theoretical CAH range).

To get an order of magnitude for the maximum dimensionlessenergy barriers, the following calculation was done. The maxi-mumdimensionless energy barrier for a given system,B*max

2D orBmax** ,

depends on the averageCA, θ0, and the heterogeneity amplitude,j.To findout the dependence of themaximumdimensionless energybarrier on these parameters it is necessary to screen all possiblecombinations ofθ0 andj.However, for eachθ0 amaximumvalueofj can be identified, based on the condition that 0e θYe 120�:it is the lowest of the values that leads to either a theoreticaladvancing CA of 120� or a theoretical receding CA of 0�. As seenin Figures 2d-f and 3d-f, the maximum dimensionless energybarrier increases with j. Therefore, when B*max

2D and Bmax** are

calculated as a function of θ0, assuming the maximum value ofj to apply for each θ0, their upper limits are elucidated. Theresults of these calculations are shown in Figure 4. As can be seen,0 <B*max

2D <0.47, and 0 < Bmax** <1.05.

It is useful at this stage to turn eq 7a,b into a dimensionlessequation, using eqs 8a,b, 10b, and 11, so that the various energiesare expressed as fractions of the available kinetic energy. Thus,a chemically heterogeneous surface would appear homogeneousby measurement of CAH if

1 > ∼ σl

Faðπrf Þ2 þ λσlB�2Dmax

Fπ3ðraf Þ2 !

ð12aÞ

and

1 > ∼ 3σl

4Faðπrf Þ2 þ 3ffiffiffi43

pλσlB

��max

4Fπ3ðraf Þ2 !

ð12bÞ

For simplicity, the order-of-magnitude calculation of thevariations in the liquid-air interfacial area is done for a contactangle of 90�, i.e. for a 2D drop that is half a circle and anaxisymmetric drop that is half a sphere. It is worth noticing that,

(23) Daniel, S.; Chaudhury, M. K.; de Gennes., P.-G. Langmuir 2005, 21, 4240–4248.(24) Mettu, S.; Chaudhury, M. K. Langmuir 2010, 26, 8131–8140.

DOI: 10.1021/la102757t 15937Langmuir 2010, 26(20), 15933–15937

Bittoun and Marmur Article

in terms of order of magnitude, eq 12a,b yields the same result,sinceBmax* andBmax

** are of the same order ofmagnitude. Thus, theresults of the analysis for the 2D case and the axisymmetric caseare practically identical, and the conclusions are effectivelyindependent of the system geometry.

To enable an estimate of the orders of magnitude of the variousratios in eq 12a,b, it is necessary to know the typical amplitudes andfrequencies of building vibrations. For vibrations that are barelyperceptible by ahumanbeing, typical values are 10-6mand100Hz,respectively.25 For vibrations that are strongly perceptible byhuman beings, but do not cause damage to the building, typicalvalues are 10-3 m and 1 Hz, respectively.25 Substituting thesenumbers in, say, eq 12a, assuming the liquid to be water (surfacetension of 73 mN/m, and density of 103 kg/m3), eq 12a turns into

1 > ∼ 10-3

r2þ 102λ

r2

!for barely perceptible vibrations ð13Þ

and

1 > ∼ 10-2

r2þ λ

r2

!for strongly perceptible vibrations

ð14ÞIt is clearly seen that the first terms in these equations are >1

for any reasonable drop size that is used in CA measurement(usually smaller than about 10 mm). Thus, it is feasible, inprinciple, that all the kinetic energy will be consumed by thevibration-induced increase in liquid-gas interfacial area. How-ever, since motion of the contact line may occur simultaneouslywith the interfacial oscillations, it is essential to analyze the orderof magnitude of the second terms of the right-hand side of eqs 13and 14. First, these equations show that the ratio of themaximumenergy barrier to the available kinetic energy increases when thedrop size decreases. Thus, energy barriers are more easily over-come for larger drops then for smaller ones. For barely percep-tible building vibrations, and for a drop of the order ofmagnitudeof 10-3 m, this ratio is 0.1 for a 1 nm heterogeneous wavelength,and 1 for a 10 nm wavelength. Taking into account that at leastpart of the kinetic energy turns into liquid-air interfacial energy,it is, therefore, safe to conclude that CAH measurement underregular conditions of barely perceptible building vibrations maydetect chemical heterogeneities at the few nanometers scale. Thisobservation is indeed supported by some experimental evidence.13

Obviously, good isolation of the measurement system from thebuilding vibrations may enhance the detection sensitivity. It isalso interesting to notice that the situation changes when thevibrations are strongly perceptible, an observation that justifiesthe use of deliberately induced vibrations to measure the moststable CA.22,26-28

Acknowledgment. The authors acknowledge the supportfrom AMBIO (Advanced Nanostructured Surfaces for the Con-trol of Biofouling) project (NMP-CT-2005-011827) funded by theEuropean Commission’s sixth Framework Programme. Partialsupport from COST Action D43 is also acknowledged by theauthors.

Figure 4. Dimensionless energy barriers for two-dimensional(dashed line) and axisymmetric (solid line) drops vs the average ofthe average CA, θ0. The peak in each curve refers to highest energybarrier, B*max

2D or Bmax** , for 2D and axisymmetric drop, respectively.

(25) Merritt, F. S., Ricketts, J. T. Building design and construction handbook,6th ed.; McGraw-Hill: New York, 2001; pp 5.183-5.188.

(26) Decker, E. L.; Garoff, S. Langmuir 1996, 12, 2100–2110.(27) Andrieu, C.; Sykes, C.; Brochard, F. Langmuir 1994, 10, 2077–2080.(28) Meiron, T. S.; Marmur, A.; Saguy, I. S. J. Colloid Interface Sci. 2004, 274,

637–644.