Langmuir 1985, 1 , 219-230 219

It should be noted, however, that a t low temperatures N20* may be stable enough to desorb, as has been dem- onstrated recently by the observation of NzO IR chemi- luminescence in the reaction of N atoms with NO over a Pt foil via the most likely reaction 17b.27 This process may also be responsible for the production of N20 in the catalytic oxidation of NH3 on a supported Pt catalyst at low temperatures (<570 K), which, interestingly, produces mainly Nz and N20 instead of NO.4

Concluding Remarks Free radical products NH and OH have been detected

during the catalytic oxidation of NH3 over a polycrystalline Pt surface in the range of temperature from 800 to 1400 K employing the sensitive technique of laser-induced fluorescence (LIF). NH production as evidence by LIF measurement in the gas phase was found to be severely inhibited by the presence of 02. This is consistent with the result of a previous SIMS study that NH+ signal was strongly quenched by the addition of a small amount of OP5 The activation energy for NH desorption under 02-inhibited conditions was, however, found to be essen- tially the same as that determined with pure NH3 or NH3 + H2 mixtures, 66 f 3 kcal/mol.14

OH production, on the other hand, was strongly en- hanced by the continuing addition of O2 with no apparent inhibition effect, even at high Oz pressures. This finding contrasts to the strong inhibition of OH production by NH, when it is used in excess of 02. The activation energy for

(27) Halpern, B. L.; Murphy, E. J.; Fern, J. B. J. Catal. 1981, 7,434.

OH desorption under these conditions was found to de- pend on the 02/NH3 ratio, peaking at a value of 42 kcal/mol near the stoichiometric ratios for N2 and NO formation (02/NH3 = 1) with declining values as the ratio was either increased or decreased. This variation in the OH desorption energy with the oxidant/fuel ratio appears to be consistent with the site competition effect previously noted by Gland and Korchak for N2 and NO formation.6

On the basis of the mechanism for NH, decomposition established previously14 and the results of this study and earlier works7 on the catalytic oxidation of NH3 on Pt, we conclude that the key reaction steps leading to NH, N2, OH, and NO formation can be summarized as follows:

N2 N2 H2

decomposition IN. IN* lH* oxidation

/ q P

- 2 H N H ~ * NH* w N' + H*

NO* NO* OH*

An extensive search for the presence of triatomic tran- sient and radical products, which can be readily and un- equivocally detected by LIF, NH2, HNO, and NO2 under all conditions studied proved to be of no avail. These polyatomic species are believed to be thermodynamically unstable a t the high temperatures used in this work and thus are likely to be of no importance to the chemistry of the catalytic oxidation of NH, under these conditions.

Contact Angle Hysteresis on Heterogeneous Surfaces

Leonard W. Schwartz and Stephen Garoff* Corporate Research Science Laboratories, Exxon Research and Engineering Company,

Clinton Township, Annandale, New Jersey 08801

Received November 7, 1984

The problem of capillary rise onto a vertical plate whose surface is partially coated with a low surface energy material is considered. Various energy-minimiiation techniques have been applied to several doubly periodic coverage patterns. The total energy contains contributions from free-surface dilation, gravity, and the wall heterogeniety. Hysteresis, the difference between advancing and receding contact angles, is associated with the existence of multiple minima in the total energy functional, Le., a metastable range. We find that hysteresis is a strong function of the details of the "patch" structure rather than only fractional coverage. On the patch scale, the progression of the three-phase line consists of alternating sticking, stretching, and jumping events. At small fractional coverage, the energy associated with interface distortion is sufficient to reduce hysteresis; thus a "cleanliness" criterion may be formulated.

I. Introduction Wetting is a commonly observed phenomenon in the

natural and technological world. The relative wettability of a solid surface by a pair of immiscible fluids is governed by the balance of the pairwise interfacial energies of the three phases and is characterized by the contact angle, i.e., the angle formed by the intersection of the interfaces bounding the three phases. For an ideal surface, a unique relation between interfacial energy densities and the con- tact angle was derived by Young.' Difficulties in exper-

(1) Young, T. Philos. Tram R. SOC. London 1805,95,65.

imentally verifying Young's relation have occurred because a given system produced different contact angles de- pending on how the experiment was performed. In par- ticular, careful experiments have shown two relatively reproducible values of the contact angle-one as the three-phase line advances across the surface and one as it recedes. Traditionally, this difference in advancing and receding contact angle has been denoted as contact angle hysteresis. Experimentally, the contact angle (or equiva- lently capillary rise height) has been examined macro- scopically by using a low (roughly 20X-5OX) magnification scheme. Particularly in the case treated in this paper where the heterogeneity of the surface is on a scale much

0743-7463/85/2401-0219$01.50/0 0 1985 American Chemical Society

220 Langmuir, Vol. 1, No. 2, 1985 Schwartz and Garoff

pattern of "grease patches" on an otherwise uniform sur- face, the energy is calculated in dimensionless terms. Measured in units of ax2, where u is surface tension and X the patchwork wavelength, the total energy is repre- sented as a function of the contact line shape, the Bond number, and the geometric details and intensity of the wall wettability. The Bond number is the ratio of a charac- teristic length of the periodic surface energy heterogeneity to the characteristic length over which the liquid/vapor surface is distorted by the presence of the wall. Since we wish to examine cases where the heterogeneity does not cause gross distortion of the liquid/vapor surface, we treat only the case of very small Bond number, i.e., where the scale of heterogeneity on the surface is much smaller than the capillary rise on the plate. The term wettability rep- resents the point function equal to the cosine of the local equilibrium contact angle.

Section 111 contains a description of two approximate solutions to the energy-minimization problem. The first approximation, called the horizontal-average method, ig- nores contributions to the energy from deformation of the three-phase line. This method, which only examines the horizontal-averaged variation in the heterogeneity, predicts hysteresis which depends both on the coverage and the distribution of the heterogeneity. However, it also predicts very large hysteresis at small coverages, and no hysteresis for any case where the horizontally averaged wettability is constant. The second method considers, to a first order of approximation, the energy content of contact line de- formation and the vertical gradient of the wettability. This more accurate method agrees with the first method in most situations; however, it improves on the first method by predicting finite hysteresis even where the horizontally averaged wettability is constant. In this case, hysteresis varies as the square of the intensity of the variation of the wettability. Both techniques discussed in this section predict that the free energy curve will be the same for the advancing and receding casea; contact angle hysteresis then is a manifestation of the system being caught in different local minima for these two motions of the system.

We then discuss a more detailed theory in section IV. Here, the energy associated with deformation of the con- tact line is calculated in detail. This model predicts a sticking, stretching, and jumping of the three-phase line as the mean height of the fluid traverses the plate. In addition, the free energy curve and the motion of the three-phase line are different in the advancing and re- ceding cases. We find that local minima of the free energy disappear as the scale of the heterogeneity gets small and thus predict that hysteresis should disappear gradually as the scale of the heterogeneity decreases. Our calculations suggest that hysteresis may effectively vanish for mi- cron-scale contamination of an otherwise "clean" surface-a size much larger than previously predi~ted.~," The jumps of the three-phase line provide a dissipative mechanism which may be related to the hysteretic behavior of the system.

In the final section, we summarize our work and con- clude with remarks regarding the nature of the needed experimental work as well as the application of our theory to further wettability patterns and to solution classes that were not considered in the body of the paper. The Ap- pendix contains a closed-form solution for the horizon- tal-average method for two wettability patterns.

11. Formulation of the Mathematical Problem for Doubly Periodic Wall Wettability

Consider a vertical plate, the surface of which is com- posed of a material that can be characterized by an

finer than these macroscopic measurements can discern, such measurements do not fully characterize the detailed shape of the three-phase contact line or even the precise contact angle. However, to make connection to these measurements and macroscopic phenomena, we will adopt the traditional view and define hysteresis as being present or absent on the basis of such macroscopic measurements.

Experimentally and theoretically, explanations of con- tact angle hysteresis have focused on the nonideality of real surfaces, Le., physical roughness a d heterogeneity of surface energy densities across the ~ i i r f ace .~ ,~ Experi- mentally, the wettability on surfaces with controlled or partially characterized roughness and heterogeneity has been in~estigated.~ In these experiments, suitable pre- cautions must be used to separate kinetic effects from static hys te re~is .~?~ A detailed description of an experi- mental procedure to eliminate dynamic effects may be found in the Appendix to ref 12. Early theoretical ap- proaches attempted to find the average wetting properties of roughs and heterogeneous' surfaces. While these ap- proaches predict a modification of contact angles on nonideal surfaces, they do not explain hysteresis. Good8 proposed a relation between the contact angle hysteresis and the existence of metastable, free energy states of the system, each of which show a different contact angle. Depending on the previous history and motion of the system, it will fall into one of these metastable states and not escape in the time frame of any experiment. Thus, the observed contact angle will depend on the prior state of the system. Most discussions of contact angle hysteresis on nonideal surfaces focus on the calculations of these metastable states in the free energy of the system.

In this paper, we will examine the effects of a particular geometry of surface heterogeneity on contact angle be- havior: doubly periodic variation of the surface energies on the surface of a flat plate held vertically in a reservoir of fluid. Previous analyses have examined simpler geom- etries, which were essentially singly periodic variations.+12 In the present work, we formulate the free energy for the doubly periodic system and then, using various methods, examine this free energy for metastable states. Each technique employs a different degree of approximation and reveals different aspects of the contact angle hysteresis. In contrast to our earlier treatment of the simpler geom- etry,12 we will treat only solution classes that pass con- tinuously one into another. We will not consider the ability of the system to use internal vibrational energy and thus either escape from metastable states or transit discontin- uously to new solution classes.

In section 11, we develop an energy-minimization for- malism for the determination of contact line shape as a function of the global parameters of the three-phase sys- tem. The total energy has contributions from surface dilation, gravity, and the wall heterogeneity. For a periodic

(2) A review of these discussions can be found in: Neumann, A. W. Ado. Colloid Interface Sci. 1974,4, 105.

(3) Johnson, R. E.; Dettre, R. H. Sur. Colloid Sci. 1969, 2, 85. (4) (a) Bigelow, W. C.; Brockway, L. 0. J. Colloid Sei. 1956,11,60. (b)

Bartell, L. S.; Ruch, R. J. J. Phys. Chem. 1956,60,1231. (c ) Gaines, G. L. J. Colloid Sci. 1960,15,321. (d) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1965,69,1507. (e) Neumann, R. D. J. Microsc. (Oxford) 1975,105, 283. (f) Oliver, J. F.; Huh, C.; Mason, S. G. Colloids Surf. 1980, 1, 79.

(5 ) Dussan V., E. B. Ann. Reu. Fluid Mech. 1979, 11, 371. (6) Wenzel, R. N. J. Phys. Chem. 1949,53, 1466. (7) Cassie, A. B. D. Discuss. Faraday SOC. 1948, 3, 11. (8) Good, R. J. J. Am. Chem. SOC. 1952, 75, 5041. (9) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972,343,341. (IO) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744. (11) Boruvka, L.; Neumann, A. W. J. ColloidZnterface Sci. 1978,65,

(12) Schwartz, L. W.; Garoff, S. J. Colloid Interface Sci., in press. 315.

Contact Angle Hysteresis Langmuir, Val. I , No. 2, 1985 221

consider the total energy over one-half wavelength of the horizontal wall wettability variation, i.e.,

Figure 1. Capillary rise on a plate with a doubly periodic wettability.

equilibrium contact angle that may vary from point to point on thii surface. The plate is partially submerged in a liquid whose free surface is horizontal at great distances from the plate. Let the free surface of the liquid be de- noted hy the function z = f ( x y ) where x and y are hori- zontal coordinates, and the plate is located at x = 0. z is the vertical coordinate; thus the solid-liquid-vapor three-phase line is denoted by the curve z = j(0y). The wettability or equilibrium contact angle variation on the plate is assumed to be doubly periodic in y and z with wavelengths h and &,respectively. The physical situation is shown schematically in Figure 1.

According to the principles of capillary hydrostatics, the shape of the liquid surface at rest will be a minimum-en- ergy configuration. This total energy, per wavelength X, is composed of three parts. The increase in surface energy due to the change in liquid/vapor surface area referred to the horizontal configuration z = 0 is

where yLv (=a) is the liquid-vapor surface energy density, or surface tension, and subscripts on f denote partial differentiation. The energy of r a i s i i or lowering the liquid against gravity, relative to the level z = 0, is

When the surface is raised at the wall, the solid-liquid interfacial area is increased, and the solid-vapor area is reduced by the same amount. Denoting the respective interfacial energy densities by ysL(r,z) and ysv(r,z), the wall energy is

The balance of forces parallel to the wall at a point on the three-phase line gives Young’s equation’

YSV = YSL + YLV cos @E Here f& is the equilibrium contact angle, and we will denote the ‘wall wettability function” by

c(y,z) = cos 0,

a doubly periodic function of its arguments that may as- sume values in the interval [-1, 11. Substituting these expressions in eq 3, the wall energy becomes

As a further simplification, we will asaume c(y?) to be an even function of each of its arguments, thus it suffices to

(5 )

For a configuration in static equilibrium with a specified function c(y,z), the minimization of G is the criterion for the determination of j (xy). Equation 5 may be made dimensionless by selecting h as reference length and oh2 as the unit of energy. In addition, we introduce a Bond number

B = pgh2/a

and note that B’/’ is the ratio of a characteristic wall di- mension, such as the average spacing of patches of hete- rogeneity to a capillary length (o/pg)’lZ, a characteristic length over which the liquid surface is distorted by the presence of the wall. The parameter range of greatest interest for capillary hysteresis corresponds to B << 1. In dimensionless units, eq 5 becomes

G = x - d x x”’dy [(l + f,2 + f,’)’/’ - 1 + (B/2)f] -

The determination of the liquid surface shape that min- i m i i G may be recast, using the formalism of variational calculus, as a nonlinear partial differential equation for f(x,y). This equation is closely related to the minimal surface equation, differing from it only by a linear term that comes from the bond number term in (6). The last term in (6) then yields the boundary condition that the angle between the normals to the surface and to the wall is given by cos-‘ c(r,z) along the unknown contact line z = j (0y). Because the full problem is of such complexity that only numerical solutions are feasible in general, a more realistic approach would be to minimize (6) directly where G can be approximated as a nonlinear function of a vector of surface ordinates, via a finite-element algorithm, for example.

111. Two Approximate Solutions In order that qualitative features of the hysteresis

phenomena not be obscured by mathematical complexity, the energy functional (6) is simplified by assuming that the inclination of the normal to the free surface from the vertical direction is not large; specifically, we assume

f,’, f,’ << 1 (7)

Sufficient conditions for (7) to hold are that (a) extreme values of pointwise contact angle 0, are to be avoided, i.e., values of 0, near 0 (perfect wetting) or 0, near ‘A (perfectly nonwetting) are excluded and (b) interfaces between wettability patches are sufficiently diffuse so that at no point will the tangent to the three-phase line on the wall approach the vertical.

If (7) is invoked, the energy is approximated as

As in the general case, the differential equation for the free surface is determined only by the first integral in (8) since the extrema1 surface for a problem with a variable

222 Langmuir, Vol. 1, No. 2, 1985

bounding curve must also be extremal for a restricted class of variations where the bounding curve is given. Thus, a necessary condition for G to be an extremum is that the Euler-Lagrange equation

(9)

must hold. Since the wall wettability is an even periodic function of y, we seek solutions for the free surface that also have this property. Further, the surface must be undisturbed at great distances from the wall; thus, an appropriate eigenfunction expansion for f is

f x x + f y y - B f = 0

Schwartz and Garoff

wettability variation in the direction parallel to the gross contact line motion is retained.

Assume now that the horizontal-averaged wettability is periodic in z with wavelength Av measured in units of the characteristic horizontal wavelength A. A necessary con- dition for the existence of local energy minima is dG/dao = 0. If E is assumed to be a continuous function of its argument, equation 13 immediately yields

B1/2ao = E(ao) (15) The determination of individual roots of (15) is a nontrivial task; it will, in general, require a solution of a transcen- dental equation. However, in the limit of small vertical wavelength, the limits of the metastable range may be found immediately. Within each wavelength, E lies in the range [E1, e,]. Define Ae = E2 - E l . If the heterogeneity is such that the vertical wavelength of ita variation is much less than the difference in capillary rise heights appropriate to fictitious homogeneous materials with wettabilities El and c2 respectively, i.e.,

Av << AE/B'/2 (16) then local minima will occur over a range of rise heights from

aol El/B112

to ao2 N c2/B112

Within the small slope approximation 7, the advancing contact angle is

COS OA = B1I2aol N E l (17a)

cos OR = B1l2ao2 N E , (1%)

Thus, the expected values of advancing and receding contact angles correspond to the maximum and minimum angles in the wettability pattern determined by the av- eraging in (13b). This conclusion could have been reached directly from (15); the somewhat more involved argument was employed to emphasize the role of the inequality (16). For example, with micron-scale heterogeneity, B1I2 = 0

while AE is order one for a perfectly periodic patchwork; thus the inequality is easily satisfied in the cases of greatest interest.

This procedure may be applied to periodic patch structures in order to predict advancing and receding an- gles as a function of fractional coverage when an initially heterogeneous surface of high energy or "clean" material is spotted with a lower energy substance or "grease". Several families of structures are shown in Figure 2 along with the postulated pattern of "growth" as the coverage fraction is increased. Calculated contact angles for the centered-square and centered-circular families are shown in Figures 3 and 4. For these and other simple patterns, the effective contact angle variation with coverage may be found in closed form. The detailed solution for these two patterns is given in the Appendix. Certain features of this calculated hysteresis are reminicent of reported experi- mental r e ~ u l t s . ~ While the predicted hysteresis is, of course, zero in the extreme cases of F = 0 or 1, observe that a small amount of impurity is adequate to produce a sig- nificant difference between the advancing and receding angles.

A limitation of this simple theory is apparent in Figure 3, where, for the case of centered squares at exactly 50% coverage, no hysteresis is predicted; if the coverage deviates

and the receding angle is

m -

f = a,e-(B+4n2u2)'~zx cos znay (10) n=O

where the free surface is assumed to be periodic in y with the same periodicity as c(y,z). By use of (lo), the first integral in (8) can be evaluated explicitly, and the ex- pression for G now becomes

G = (B1/2 /4)a ,2 + y8 m

an2(B + 4n27r2)1/2 - IL=l

s, '"dy f(o'y)c(y,z) dz (1 1)

Note, for the case of a homogeneous wall, a, = 0 for n > 0, (11) predicts that the capillary rise is given as

a0 = H = ( u / p g ) l / P cos 0

A simple procedure will now be outlined for estimating the variation of total energy G with mean rise height a* To the extent that the resulting function G(a,) exhibits multiple energy minima, i.e., a metastable range, for a particular wall wettability patchwork, the difference be- tween advancing and receding contact angles or, alterna- tively, the difference between static meniscus heights in slow drainage and inhibition experimenh may be pre- dicted. In this approximation, to be referred to subse- quently as the method of horizontal averaging (HA), we shall ignore the energy content associated with the contact line deformation. Thus, in (ll), we set a, = 0 for n 1 1; the resulting energy function is given by

The limit on the z integration is now constant, and the order of integration may be reversed to yield

where F(z) is the horizontal-average wettability, i.e.,

We see that the assumption of a straight three-phase line leads to only the horizontal-averaged wettability appearing in the free energy. It will become apparent, by comparison with more detailed calculations to be discussed below, that the error in (13) is formally of order (Ac),, where Ac is the difference between maximum and minimum values of cos OE in the wettability pattern. More intuitively, this av- eraging approach is related to the classical prediction of C a s ~ i e , ~ who estimated the effective contact angle Oc of a heterogeneous surface composed of two materials as an areal average

COS Oc = F COS 0, + (1 - F) COS 82 (14)

where F is the fractional coverage of material 1. Here, because we wish to explore the hysteresis effect, the

Contact Angle Hysteresis Langmuir, Vol. 1 , No. 2, 1985 223

Increasing Coverage +- Figure 2. Three families of doubly periodic wettabfity structures and proposed "growth" of the shaded or low-energy portions of the surfaces.

,........ , 1 9 1 .(............" " ..,,,,, ,

R ': I

F W 8: hitdidions for advancing and receding contacts angles for a centered-square wettability pattern (see Figure 2b) as a function of coverage of the low-energy material. (-) Receding angle by HA method; (. . .) receding angle by linear theory; (- - -) advancing angle for HA method; (- -) advancing angle for linear theory. c1 = 0.025; c2 = 0.175.

'"h

: 8 ,I3/

.01 0 g .4 .I .e 1.0

F

Figure 4. Predictions of HA method for advancing (- - -) and receding (-) contact angles for a centered-circle wettability pattern (see Figure 2c). c1 = 0.025; c2 = 0.175.

infinitesimally from this value, however, large hysteresis results. At exactly one-half coverage, the centered-square pattern produces a perfect checkerboard and the hori-

Figure 5. Two doubly periodic wettability patterns exhibiting constant horizontal-averaged wettability at all coverage fractions.

zontal-averaged contact angles are constant. In fact, fam- ilies of wettability patterns can be constructed for which zero hysteresis is predicted by the simple theory at any coverage fraction. Two such patterns are shown in Figure 5. While advancing and receding angles are expected to be different for these cases, the energy component re- sponsible for a metastable range with multiple energy minima results entirely from the variation in contact line shape as it moves through the wettability pattern.

A more involved theory is required in order to include the energy of contact line deformation. Returning to the general problem with doubly periodic wall wettability, eq 11, our procedure will be to optimize, or minimize, the total energy for each value of a,,. That is, we regard ao, the mean elevation of the three-phase line on the wall, as a parameter and find the optimum values of the higher order Fourier coefficients, thereby determining both the energy and the shape of the contact line. The function G(ao) will then be examined to determine the characteristics of the meta- stable region. In dimensionless units, the unit cell on the wall is taken to be a rectangle with relative dimensions 1 and l / k v in the y and z directions, respectively. For sim- plicity, the wettability representation is a double Fourier series of the form

m m

c(y ,z ) = C Ccij cos 2aiy cos 2akQz (18) i 4 j = O

Potential difficulties associated with the Gibbs phenom- enon13 can be anticipated when gradients of wettability become very large. However, the approximation in (7) already requires a degree of diffuseness a t "patch" boundaries, thus the Gibbs effects need not be considered as an additional limitation.

The problem of determining the optimum values of a , for given a. is nonlinear and will require a numerical attack using Newton's method or the equivalent. The problem may be made linear, however, by invoking the diffuseness of the wettability variation, specifically in the vertical direction. Using a local expansion about the mean level, the energy is truncated at second order in the an. The optimization can then be accomplished by solving a linear system.

The nonlinearity comes from the wall integral in ( l l ) , which may be written

where OD

fib) = C a n cos 2any n=l

We write I = Il + 12; the integral

Il = J1/2dy Ja0c(y,z) dz

(13) Kreyazig, E. 'Advanced Engineering Mathematics"; Wiley: New York, 1972; p 411.

224 Langmuir, Vol. 1, No. 2, 1985

may be evaluated exactly, using the representation (18) to obtain

Schwartz and Garoff

minima of G(ao). Denoting the periodic or oscillatory portion of the energy in (23) by G,, this equation may be rewritten as aOCoo - co, sin 27rkvjao

I , = - + 72c (20) 2 ,=I 27rkQ

The portion of the wall integral involving the deviation of the three-phase line from the horizontal

is treated by expanding the integrand about z = ao; thus

where m

&,ao) = E d i cos 27riy i=O

ac m

- (y,ao) = C e i cos 27riy az i = O

and m

di = Ccij cos 27rkvjao

ei = -27rk, c ci,j sin 27rkvjao

j =O

m

j=1

Through 0 ( f 1 2 ) , the integral I2 is evaluated as

I2 = C aidi + LeoCa: + l e , C ajal+j + 4 i=l 8 8 j = 1 1 m k-1 m

Collecting results from (ll), (20) , and (22), the energy is B'/2 1 " aOCOO

4 8 n=l 2 1 Coj 1 "

4*k, j = 1 J 4 n=l

G = - ao2 + - a,2(B + 4n27r2)lI2 - - -

- C -sin 2akvjao - - C and,- b-1

with the understanding that the last summation is taken as zero when the lower limit exceeds the upper. For G(ao) to be minimum, we require aGlaa, = 0, n = 1 ,2 , ..., which, after some manipulation, yields the linear system

( (B + 4n2a2)lI2 - e0 /2 )a , - y2C [ev-,, + ej+,]a, = m

1=1

d , n = 1, 2 , ... (24)

where vertical bars have been used to signify absolute value. For purposes of computation, the Fourier series for fib) will be truncated at some convenient value n = N , say. Note that G(ao) will achieve sharper minima as N is increased.

The coefficients ei and di appearing in (24) and defined in (21) may be observed to be strictly periodic in a,,. Thus, the shape of the three-phase line, determined as the so- lution of (24), will also be periodic in ao. Note also that the energy is composed of a portion that is parabolic in a. plus a remainder that is strictly periodic in ao. This latter observation greatly simplifies the search for the local

Note that coo has the same meaning as the Cassie areal average cos BC in eq 14. The function G,, is simply a (periodic) function of a. once the optimal values of a,, given by the solution of the system (24), are inserted. As before, the local minima of the energy are found at every other root of the equation

dG B1/2 coo dGOBc (25) - - - - a o - - + - = 0

dao 2 2 da,

Since G,'(ao) is purely periodic, it assumes its maximum and minimum values within one vertical wavelength. When this vertical wavelength &,, assumed to be the same order as the horizontal wavelength, is much smaller than the capillary length (u/pg)1/2, many cycles of the wall wettability will separate the advancing and receding rise heights. In this limit, the two rise heights may be iden- tified with the largest and smallest roots of (25). We again associate rise height with equilibrium contact angle ac- cording to

cos 6 = B1i2ao

From (25) therefore, the advancing and receding contact angles can be calculated as

cos {t) = coo + (::( 2)) (26)

As an example, we repeat the computation of the ad- vancing and receding angles vs. fractional coverage for the family of centered squares using this linear theory. The results are shown in Figure 3, where they may be compared with the predictions of the horizontal-averaging method. Note that the anomalous feature of the HA method, that zero hysteresis is predicted for exactly one-half coverage, is no longer present. The hysteresis a t this value, using the linear theory, is about 13% of the difference between the wettabilities of the two wall materials. For purposes of this calculation, the equilibrium contact angles of the two wall constituents were taken to be cos-, (0.1 f 0.075), and the Bond number is Also, the linear theory represents the wettability using the truncated double- Fourier series approximation to the discrete patches; the overshoots in the figure, predicted valves of cos 8, that are less than cos 4, for example, arise from this representation. Were we to consider the truncated Fourier series to be the actual wall wettability pattern, the horizontal-averaged results would have been very close to those from the linear theory, except in the vicinity of F = 0.5 where the contact line deformation is the major constituent of the energy variation.

It is useful at this point to comment further on the relationship between the horizontal-averaged and linear- ized theories. If Ac represents the magnitude of the de- viation in the wall wettability pattern from its areal average value coo, eq 21a shows that the coefficients d,, n L 1, are of order Ac. From (24) therefore, the a , is also of order Ac. The horizontal-averaging method retains the first, third, and fourth terms on the right in the energy ex- pression (23); that is, the energy is calculated only to be of order Ac. The other terms retained by the linear theory, giving the contact line deformation, are of order ( A c ) ~ . Because, for small patch wavelength, the vertical gradient

Contact Angle Hysteresis Langmuir, Vol. 1, No. 2, 1985 225

method, but with the effects of gravity neglected, by Bo- ruvka and Newmann." Our solution for this case is

t

Figure 6. The functions L and g used in the detailed energy- minimization procedure.

of wettability can be much larger than the maximum wettability difference, the last term in (23) must be re- garded to be of second rather than third order in Ac.

IV. A More Detailed Theory The previously described treatments havz two limita-

tions: the energy associated with contact line deformation is considered only to a first order of approximation, or not a t d, and the contact line shape and its associated energy is a function only of the mean rise height a. independent of whether the line is advancing or receding on the solid surface. In this section, we will take a more complete look at contact line shape and system energy. As the mean rise height is varied, the contact line deformation will be seen to alternate between continuous and discontinuous events; the process may be described as one of "sticking, stretch- ing, and jumping". The detailed motion will differ in the advancing and receding cases. In addition, the extent of the metastable range and the resulting hysteresis will be considerably reduced, compared to the previously dis- cussed approximations, on a surface charaderized by small, widely separated patches.

For simplicity, we will treat only members of the rec- tangular family of patches shown in Figure 2a. The contact angle for the continuous region is given by c2 = cos 02, while for the discrete patches we have c1 = cos O1. In all cases to be discussed c2 > cl. The total energy per one-half horizontal wavelength is approximated by

G = (B1/2/4)a2 + (7r/4) C nu; - m

n = l

J1'2dy JQoc(y,z) dz - l l / ' d y 0 l f ' O Y ) c ( y , z ) QO dz (27)

where B has been neglected, relative to 47r2, in the second term on the right. The equation of the contact line is, as before,

(28)

For a given value of the mean rise height ao, we seek to minimize G. A necessary condition is

m

f(0,y) = a. + Ea, cos 27rny n=l

From (28), the an, n = 1, 2, ..., are given by

an = ~ ~ 1 ' 2 c ( y , f ) 7rn o cos 27rny dy (29)

A fundamental solution is the contact line shape for the vertical stripe case, i.e., where 2bv = Xv. This particular problem has been treated, using a conformal mapping

a, = -[ 2 cl& b cos 27rny dy + c2J 1'2 cos 27rny dy ] 7rn

-Ac = ""s," cos 27rny dy = - sin 27rnb

7rn a2n2

where Ac = c2 - cl. Then

f(0,y) - a0 = C ancos 27rny 01

n=l

- AC 2 4

= -[L(27r[b +

where we have defined the function

(31) sin n6 L(6) = c -

n=1 n2 The function L may be expressed as

-6 L(6) = In 2 - 2Joi2 In sin 8 de

where L(0) is the Lobachevskiy f~nct i0n . l~ Our modified function L(O), shown in Figure 6, has the following prop- erties:

L(6 + 2nr) = L(6)

L(2a - 6) = -I (6)

(324

(32b)

(32c)

(32d)

L(6) = 0 at 6 = n7r, n = 0, 1, 2, ... L'(6) - 00 as 6 - 2na, n = 0, 1, 2, ...

L1(7r/3) = 0; ' L(a/3) = 1.014 ... (324

The vertical stripe profiles computed from (30) exhibit infinite slope and an infinite curvature discontinuity at the strip boundary y = b. These features are also exhibited by the minimal surface solution of Boruvka and New- mann." The peak-to-trough amplitude of the contact line is

(Ac/a2)[L(2nb) - L(21rb + T ) ]

which achieves its maximum value for b = This value is 2CAc/7r2 where C is Catalan's constant 1/12 - 1/32 +

The vertical stripe solution calculated above is one of the regimes traversed as the mean elevation of the contact line is varied monotonically. For rectangular patches that are sufficiently elongated in the vertical direction, the shape of the contact line will usually pass through four discrete regimes. The formulas appropriate to each of these regimes, for the contact line and the total energy, will be derived in turn and calculated results will be dis- played. Focusing on a single rectangular patch, these re- gimes are shown in Figure 7.

In regime 1, the mean elevation a. is close to, but is somewhat greater than, the ordinate of the bottom of the

1/52 - ... = 0.91596 ...

(14) Gradshteyn, I. S.; Ryzhik, I. M. "Table of Integrals, Series and Products"; Academic Press, New York, 1965; p 933.

Schwartz and Garojj 226 Langmuir, Vol. I , No. 2, 1985

2

I I

". b I + y

Figure 7. Schematic of representative three-phase lines in each of the four regimes traversed as the average rise height moves past a patch of low-energy material (shaded region).

patch zb. The contact line follows the boundary of the patch from y = 0 to y = y* < b and then is free to rise. Thus, we seek solutions of the form

j(Oy) = Z b 0 5 y < y* = Xa.*cosnkCy-y*) y * 5 y < %

"=a where a,* is the Fourier coefficient of the free portion of the contact curve and

2 s k = -

Under the assumption (eq 7) that the contact line is almost horizontal, the form of the surface away from the wall is approximately

j(x,y) = zbe-BBLi2r

1 - 2y*

o 5 y < y* Y* 5 Y < x = Xa,*e-(m42+BS/2= cos &(y -

Note that while the contact line is continuous and pos- sesses a continuous y derivative on the wall at y = y*, this approximation introduces a small vertical discontinuity, for y = y*, on the sheet away from the wall. Assuming that thii discontinuity decays rapidly, the total energy becomes

B1/2aO*2 + (1 - 2Y*) 4

+ 2

G =

Ac ai* k i=i 1

b)+ c,(b - y*) + - X T sin ik (b - y )

Here I(zb;O) is the wall integral J;/'dy s a c dz. Minimizing G with respect to ai*, i = 1, 2, ..., we obtain

-Ac(l - Zy*) sin ik(b - y*)

i 2 2 ai* = (33)

where we again neglect B relative to k2. a,* is related to Q, according to

a, = 2zby* + ~ , * ( 1 - 2y*)

Inserting the optimum values ai* from (33) in the ex- pressions for the contact line and the total energy, we find

(Ac)2(1 - Z Y * ) ~

4 2 g ( 4 b - ~ * l ) (35)

The relation between y* and a, is

When a, = zb, we have y* = b. y* decreases as aa is in- creased. As a practical matter, it is easier to vary y* and find the corresponding a, then to solve the transcendental equation. The function g that appears in (35) is defined as

(36)

and may be calculated without difficulty. It is shown also in Figure 6. Note that

g" = L(28)

sin2 no g(0) = X-

"=I n3

For a sufficiently tall patch, the transition value y* = 0 wil l he reached, as aa is increased, before j(b) = zt. Then the vertical stripe solution for the contact line becomes applicable. The contact line f(0y) is given by (30) and the corresponding energy is

Bl/2 G = ?a,' - I(tb;O) - clb(aa - z b ) -

cz(% - Ma, - z b ) - (Ac)'/4?r'g(Zrb) (37)

This solution characterizes region 2. Observe that the contact line and the energy vary continuously across the transition hetween regions 1 and 2.

The region 2 solution is valid until j (b) reaches the top of the patch corresponding to

Ac 2 2

a. = zt + - L(4nb)

For somewhat larger a,. the contact line shape consists of a segment lying on the patch, a crossover point at a point (y,d = Cy*,z,) where y* < b, and the remainder of the line lying in the c2 region. By a calculation analogous to that used in previous regimes, the contact line and energy can he determined as

Ac 2.T

f(0.Y) = a0 - ,iL[2a(y + y*)l+ - W d y * - y)ll (38)

B1/2 G = ao2 - I(zb;o) - clb(z, - z b ) - Cly*(ao - Z t ) -

CP(% - b)(Zt - z b ) - Cz(% - Y*)(ao - Z t ) - -g(2aY*) 4a3

(39) In this third region, a, is related to y* by

Ac a, = zt + - L(4rry*) 2 2

(40)

Contact Angle Hysteresis Langmuir, Vol. 1, No. 2, 1985 227

t

Figure 8. Calculated line shapes forao advancing. j indicates jump between reFimes 3 and 4. h, = h; b = 0.2, b, = 0.35; c? = s, = 0.5; B = l(r . Shaded region is patch of low-energy matend.

The transition between regions 2 and 3 is again continuous. Referring to the graph of L(B), it is clear that, provided

< b < 5/12, y* will decrease as a. is increased until Ac

a. - zt = - L(s/3) 2 2

Equation 40 cannot be satisfied for larger values of a* The contact line snaps a t this point to the straight configura- tion, region 4, with a corresponding discontinuous reduc- tion in the total energy. In region 4, for completeness, the contact line and energy equations are simply

f = a , , (41)

G = (B1/2/4)aoZ - I ( Z b ; O ) - clb(tt - Zb) - Cz(% - b)(zt - z b ) - (Cz/2)(00 - Zt) (42)

In Figure 8, we show a succession of calculated contact line shapes as a,, is increased for the case S = h = 1, b = 0.2, b, = 0.35. Note in particular the tendency of the line to “stick” on the “grease” patch with consequent stretchiig until that value of a. when it jumps to the straight con- figuration. The jump is indicated in the figure by “?. A somewhat different sequence of profiles is obtained if a. decreases continuously. Using the same formulas, we calculate the sequence of profdes shown in Figure 9. Note that region 3 is absent. The straight contact line moves down until it just touches the top of the patch. It then jumps spontaneously to a region 2 configuration with a corresponding stepwise decrease in energy. The jump is shown in the figure.

For a periodic wall patch distribution, because of the smallness of B, these two sequences of profiles have a universal character. While the magnitude of the energy is a strong function of ao, that portion of the energy as- sociated with the deformation of the contact line is a pe- riodic function of a* In addition, the shape of the contact line is independent of B for B << 1. For B = and the above patch geometry, the energy G is displayed as a function of a. for both the advancing and receding cases

V

Figure 9. Contact line shapes for a. receding. j indicates jump between regimes 4 and 2. Parameters as in Figure 8.

1 -.2L

Figure 10. Free energy vs. a. curves for HA (- - -) and detailed theory (-) for a, increasing. j indicates position of jumps of the three-phase line. All parameters as in Figure 8.

i J . Figure 11. Free energy vs. a. curve for detailed theory for a, decreasing. j indicates position of jump of the three-phase line. All parameters as in Figure 8.

in Figures 10 and 11. For a. increasing, the first energy minimum occlv~ at a,, a 1.05. For slowiise, this minimum will correspond to the static level in inbibition. The small energy discontinuity (symbol j) occurs once each period as shown. The receding or drainage case, Figure 11, shows a first minimum at a. = 5.05. For slow drainage, this

228 Langmuir, Vol. I , No. 2,1985 Schwartz and Caroff

Y

Figure 12. Contact line shapes for a,, increasing at low coverage fraction. AU parameters as in Figure 8 except b, = 0.022. j indicates jump from distorted to horizontal lines. (*) indicates first distorted c w e attained on drainage. Shaded region is patch of low-energy material.

minimum will correspond to the static level. The energy jumps are also marked and are seen to be somewhat larger than in the advancing case. The horizontal-averaging estimates for this case, a, = 1 in advance and a. = 5 for recede, are very close to the calculated results of the more accurate theory.

= 0.5 and b = 0.2 hut consider b,, the vertical extent of the patch, to be reduced. Provided b, is greater than zero, the horizontal-averaging estimates of the first and last energy minimum remains unchanged; thus the predicted hysteresis remains constant until the patch vanishes. This paradoxical effect can be removed by using the nonlinear theory. The succession of contact line shapes, in advance, is given in Figure 12 for 2b, = 0.022. They consist of only three regimes. As before, region 1 has the contact line osculating the patch bottom and intersecting the vertical boundary y = b. Now, how- ever, as a. is increased, the line reaches the point b , z ) = (b,z,) before y* becomes zero. Thus region 2 is absent. It is replaced by a new configuration, regime la say, where the contact line osculates the bottom at y = yl* and in- terseds the top at y = yz*. For this configuration, we have

Let us now retain c2 =

= 0 0 < Y < Y1*

where, for convenience, we have taken z6 = 0 and k is as previously defined. Region la terminates when yl* reaches zero. At this point, one could obtain region 3 solutions as a, is further increased. However, here, because the patch height 2b, is small, the value of yz*, when yl* = 0, is less than 1/12; thus no regime 3 solutions are possible. The contact line must therefore snap to the straight line con- figurations as shown by the jump 6). The succession of profiles in drainage is, of course, somewhat different. As a. is continuously reduced, from a value above the patch, the contact line remains straight until a. reaches the top of the patch; a t this point it snaps to the region 1 config- uration corresponding to a. = 0.0218, i.e., to the marked profile in Figure 12. The snap is accompanied by a step- wise reduction in the energy.

Figure 13. Free energy vs. a. curve at low coverage fraction. AU parameters as in Figure 12. Insets a-c are expanded plots of the first three points where H A predicts local minima. HA (---); detailed theory (-).

The horizontal-averaged result for a patch of finite height is, as before, that the first energy minimum occurs at a. = 1 for B = lo-'. Figure 13 shows G vs. a, where the patch interaction areas in inhibition are circled. For clarity, these regions are each enlarged in the insets labeled a<. At the first patch interaction near a, = 1, the HA theory predicts an energy minimum while the nonlinear theory shows that the energy reduction due to the undu- lation of the contact line is sufficient to overcome the energy increase due to the wettahility change so that G continues to be monotonically decreasing. Note that the extent of the region over which the three-phase line in- teracts with the patch is increased by a fador of more than 3, consistent with the tendency of the contact line to stick to the patch as a, is increased. The jump reduction in G when the contact line finally breaks free is also visible in the fwes. Similarly, the second patch interaction, Figure 13b, also fails to produce a minimum. A true energy minimum finally occurs at the third interaction, which can be seen from Figure 13c to be a t a, = 3.03. The energy variation in drainage is not shown; because the patches are very small, the first drainage minimum will he at a. = 5 for both the horizontal-average and the detailed theories. In terms of contact angle hysteresis, the apparent angle is related to rise height by

cos 9 = B'I2ao

With cz - c1 = 0.5 and B = 0.01, the calculated advancing angle is cos-' (0.303) = 72.4", which may be compared with the horizontal-average result cos-' (0.1) = 84.3'. The re- ceding angle in both cases is cos-' (0.5) = 60'. Thus, the hysteresis or difference is 12.4' for the detailed theory and ahout twice as much using the HA results. It is clear that when the patch height is extremely small but finite all hysteresis will effectively vanish. (A similar conclusion regarding the "invisibility" of small patches has recently been reached by Joanny and de GenneP who approach this problem in the "dilute limit" of isolated "defects" on an otherwise homogeneous surface.) This is distinct from the contention that hysteresis only vanishes when the undulations of the contact line lie within the diffuse zone

(15) Joanny, J. F.; de Gemes, P. G., J. Chem. Phys. 1984.81,552.

Contact Angle Hysteresis Langmuir, Vol. 1 , No. 2, 1985 229

The unit-cell viewpoint used in this paper has allowed us to calculate hysteresis, over the full range of fractional coverage on “samples” that are large compared to patch dimensions. The assumption of perfectly periodic wetta- bility must, of course, yield to a statistical description for any natural heterogeneous material. In spite of its limi- tations, the horizontal-average method appears to repro- duce many of the features of the more detailed procedures. Because of its utter simplicity, it is the likely candidate for a generalization that considers the statistical nature of natural surface heterogeneity.

There is an obvious need for more intimate contact between theory and experiment in studies of hysteresis on patchy surfaces. At a minimum, theory indicates the im- portance of patch size, shape, and relative position in predicting wetting behavior. For micron-scale heterog- eneity, only the fractional coverage can now be determined with any degree of confidence. Consequently, we have begun to consider construction of models with well-defined patchwork and characteristic dimension of the order of 100 pm. At this scale the bond number is still small, of order

and the scale independence of the contact line shape used in this paper is valid. The patches are, however, sufficiently large that visual confirmation of the accuracy of the patchwork, and the contact line shape, can be ob- tained at low magnification. It is anticipated that con- struction of these models will be facilitated by the use of micro-fabrication technology borrowed from the electronics industry.

In section IV, a detailed theory was developed and used to calculate contact line progression for a simple rectan- gular patchwork. Physically plausible simulations of contact line “hang-up” at patch boundaries, subsequent stretching, and ultimate discontinuous snapping to lower energy configurations have been produced. Importantly, hysteresis is shown to diminish and, by inference, to vanish for micron-size patches at coverage fractions of about 1%. This result may be used as a rough measure of the level of cleanliness required to produce a nominally heteroge- neous surface. Note that while the small-slope approxi- mation in (7) used in the section IV solutions is violated locally at patch boundaries these nonuniform regions are exponentially small and thus produce only small error in the integrated energy.

Throughout our calculations we have assumed that for a horizontally periodic patch structure the contact line will be periodic with the same wavelength. If the patch structure is slightly perturbed, or if we assume that the system possesses a small amount of extra energy, due to vibration for example, other configurations are possible. Such alternative configurations can be expected to be observable experimentally if they are energetically favor- able with respect to the originally “period one“ waveform. To investigate this possibility, a solution has been calcu- lated by the method of section IV for a wave of twice the wall wavelength. Its energy is then compared with the period one result a t the same value of mean rise height. It is not a t all difficult to find parameter values for which the longer waveform is of lower energy. Such a comparison is shown in Figure 15; the relevant parameters may be found in the figure caption. The subsequent progress of this longer wave, once formed, raises a number of inter- esting questions. It is possible that this period-doubling phenomenon may repeat, yielding waves that bypass greater and greater numbers of patches. It seems clear that these longer waves will be more capable of reducing hys- teresis for widely separated patches, since grease patches can now be overcome consecutively rather than concur-

I

-lI L

Figure 14. Movement along the L(6) curve as a. advances. Movement to point A is in regime 2. j, indicates a jump to regime 3. Movement from B to C is in regime 3; and j, indicates a jump to regime 4.

of the liquid-vapor interface. Suppose hysteresis vanishes by the above mechanism when 2b, = 0.01. For B = 0.01 and (a/pg)’I2 - 0.2 cm, the patch height is about 2 X lo4 cm or 2 X lo4 A. This may be compared with the lo-A result estimated by the diffuseness a rg~ment .~J l

The above computations show that the movement of the three-phase line consists of three classes: uniform trans- lation of either a deformed or straight line (regions 2 or 4), stretching of a partially pinned configuration (regions 1, la, and 31, and discontinuous snapping. In the cases shown, there was only one snap or jump per cycle in either advance or drainage. A more complicated patch geometry or unit cell can result in multiple jumps. This effect can even be seen with the present simple patchwork if b > 5/12.

Consider the 2-3 transition. From the I ( 6 ) graph and eq 40 it can be observed that the 2-3 transition must be discontinuous. For given b > 5/12, the contact line reaches the upper corner of the rectangular grease patch at a value of a. corresponding to point A in Figure 14. As a. is increased still further, no continuous deformation is pos- sible; rather the contact line must jump (jl) to the other value of y* with I constant (point B). Continuous de- formation then proceeds until the “universal” jump, j,, a t point C.

V. Further Remarks Several different approaches have been presented for

the computation of capillary hysteresis effects for the model problem of rise onto a vertical wall having a periodic wettability pattern. Each technique employs a degree of approximation in the minimization of the free energy; more computationally intensive schemes could be formulated to provide sharper estimates of local energy minima. As an example, the linear scheme developed in section I11 can be interpreted as the first “pass” in a Newton-iteration attack on the nonlinear problem of determination of the optimal contact line shape, with fured mean level, on a wall with differentiable contact angle variation. Our interest is, a t this stage, to identify basic mechanisms involved in three-phase line motion. Thus, while this particular nonlinear solution might have been produced without great difficulty, it would only add minimally to our overall un- derstanding. In addition, the requirement of smooth wettability variation is, to a degree, incompatible with the concept of discrete grease patches. The linear solution, as employed, is sufficient to resolve the paradoxical aspect of the simple horizontal-averaging method. Thus, energy components that are second-order in the wettability dif- ference have been shown to become important and lead to finite hysteresis, when the HA method predicts zero hysteresis for certain special patch structures.

230 Langmuir 1985,1, 230-232

f

.5 0 1

Figure 15. Comparison of period one (curve 1) and longer wavelength (curve 2) solutions for the three-phase line. Shaded regions are patches of low-energy material. G for curve 1 is -0.059; G for curve 2 is -0.063. a. is the average height of both curves.

rently. Such calculations are possible within the present scheme. In addition, the experiment discussed above should provide valuable insight.

Our modeling has described several features of capillary rise and contact angle hysteresis on a vertical plate with a doubly periodic heterogeneity of surface energies. First, hysteresis does not depend only on the coverage of the different materials composing the heterogeneous surface. Thus, experiments must examine both the coverage and the distribution of, for example, a surfactant on a high energy surface, in order to understand the hysteresis phenomenon on that surface. Hysteresis on heterogeneous surfaces can disappear for larger patches of heterogeneity than previous predictions have indi~ated.~Jl Even our predictions must be considered a lower limit on critical patch size since the ability of the internal energy of the system to overcome the barriers of metastable states has not been considered. Also, for periodic patterns with av- erage wettabilities that are translationally invariant along the direction of capillary rise, some residual hysteresis will be observed. Finally, our more detailed theory identifies a dissipative mechanism-the jumps of the three-phase line which may be related to the hysteretic behavior of the system.

Appendix The following are closed-form solutions for advancing

and receding contact angles according to the horizontal- average theory for two wettability patterns.

For the centered-square and centered-circular patterns shown in Figure 2, let the contact angle of the low-energy patches be c0s-l c1 and the high-energy angle be c o d c2. With the averaged wettability E given by eq 13b, the co- sines of the advancing and receding angles are determined as the minimum and maximum values of E, respectively. Using simple geometrical considerations, these angles may be calculated as functions of F, the coverage fraction of material 1.

For the centered-square pattern the results are COS Ba = ~2 - Ac(F/2)lI2 F C 7 2

= ~2 - AC F > 72 (A.la) cos 8, = c2 F < Y2

= c2 - Ac( 1 - ( Y)l2) F > '/z (A.lb)

The analogous results for the family of centered circles where Ac is defined as c2 - c l .

are COS ea = ~2 - RAc R I 1/3lI2

= ~2 - (4R2 - 1)l12Ac = ~2 - AC

1 / 3 l I 2 < R C 1/2lI2 1/2lI2 I R C 1

(A.2a) cos 9, = c2 R I f / z

~2 - (2R - 1)'I2Ac 7 2 C R I 1 (A.2b) Here R is proportional to the circle radius and is related to the coverage fraction F according to F = (7r/2)R2 0 I R I 1/2lI2

= (2R2 - 1)'l' + 1 - (2R2 - 1)lI2

2R2 sin-' ( 2R ) 1/2lI2 < R 5 1

(A.2c)

Oscillation of Electrical Potential across a Liquid Membrane Induced by Amine Vapor

Kenichi Yoshikawa* and Yasuhiro Matsubara College of General Education, University of Tokushima, Minami- josanjima, Tokushima 770,

Japan

Received November 9, 1984

Studies were made on the electrical potential across a liquid membrane consisting of an oil layer, 90% oleic acid and 10% 1-propanol containing tetraphenylphosphonium chloride, between aqueous solutions of 0.5 M NaCl and KC1. When the oil phase was exposed to amine vapor, the system showed periodic changes of electrical potential of 10-20 mV with an interval on the order of a few minutes. It is suggested that this system can serve as a model of biological olfactory transduction.

Introduction

and taste sensing, are among the most important problems in biological science. In spite of extensive studies on

0743-7463/85/2401-0230$01.50/0

chemoreception, its molecular mechanism is not well un- The mechanisms of sensing, such olfactory derstood. Recently we reported's2 that rhythmic oscilla-

(1) Yoshikawa, K.; Mataubara, Y. Biophys. Chem. 1983, 17, 183.

0 1985 American Chemical Society