Contact Angle Hysteresis on a Heterogeneous Surface: Solution in the Limit of a Weakly Distorted Contact Line.

Transcription

1 EUROPHYSICS LETTERS Europhys. Lett., 28 (6), pp (1994) 20 November 1994 Contact Angle Hysteresis on a Heterogeneous Surface: Solution in the Limit of a Weakly Distorted Contact Line. J. CRASSOUS and E. CHARLAIX Laboratoire de Physique, Ecole Nomle Supe'rieure de Lyon 46 Alle'e d'ltalie, Lyon Ce'dex 7, France (received 8 June 1994; accepted in final form 11 October 1994) PACS Fluid surfaces and interfaces with fluids (inc. surface tension, capillarity, wetting and related phenomena). PACS Solid-fluid interface processes. Abstract. - In the case of a weakly distorted contact line, we construct a one-variable equation giving the metastable configurations of a fluid interface in contact with a heterogeneous solid surface. In this limit it is possible to study the contact angle hysteresis created by semi-concentrated assemblies of individual defects. The results exhibit many collective effects, some of which can be described by a convenient renormalization of the hysteresis threshold and of the defect density. They are consistent with available experimental data. It is now established that the hysteresis of the static contact angle 8 between a liquid-gas interface and a solid wall is affected by the non-uniformity of the solid surface[l]. In a number of practical situations, this contact angle e is not uniquely determined by the Young-Dupr6 equation ylv cos 8 + yb = ysv, but may lie anywhere in an interval (Or, ea), depending on the history of the system. At a microscopic level, the non-uniformity of the solid surface allows many metastable configurations for the fluid interface, and the energy barrier between them is the source of hysteresis. The correlation between surface heterogeneity and contact angle hysteresis has been the focus of much theoretical [2-51 and experimental work [6-81. Joanny and de Gennes have solved analytically the anchoring of the contact line on individual defects [2]. The heterogeneity of the solid surface being described by a local fluctuation of the solid interfacial energies h(x, y) = ysv(x, y) - ysl (2, y) - ylv cos 80, they have shown that smooth defects create hysteresis only if they are strong enough, and that the hysteresis created by a dilute assembly grows like the number of defects. However, the theoretical description is difficult for concentrated defects because of the long-range character of the capillary interactions. Pomeau and Vannimenus[3] derived a non-linear equation giving the metastable configurations of the contact line; in the case of weak but concentrated heterogeneity they have shown that the hysteresis grows like the square of the magnitude of the heterogeneity. Di Meglio [6] has measured the hysteresis and its noise for a random assembly of defects, as a function of their density. Above a critical density he observes that defects interact, and hysteresis grows less than linearly with the density of defects. The noise of the hysteresis shows a regime of avalanches. This work proposes a numerical calculation of the contact line hysteresis on a random solid surface, in the limit where the contact line is weakly distorted around its average position. 8 Les Editions de Physique

2 416 EUROPHYSICS LETTERS Fig Liquid-gas interface on a rough solid surface. The system we consider (fig. 1) is a solid surface dipped into a liquid bath at a fvted angle eo equal to the contact angle in the absence of heterogeneity, ie. h(x, y) = 0 [91. The size of heterogeneities is assumed large enough so that van der Waals interactions and thermal fluctuations are negligible. Far away from the contact line (compared to the capillary length K -I), the gravity maintains the liquid-gas interface in a fvted horizontal plane. The capillary length also provides a macroscopic cut-off for the contact line perturbations: for instance, the perturbation induced by a local force applied on the contact line heals on a distance of the order of K [lo]. Thus periodic boundary conditions of wavelength K -' can be assumed in the z-direction parallel to the unperturbed contact line, without restricting the generality of the problem. Our approach follows the one of ref. [3,4]: the total (interfacial and gravitational) energy F of the system is calculated and minimized with respect to all Fourier components of the contact line, with the assumption that the components of order higher than one are small. The locally stable configurations of the contact line are then given by a one-variable equation involving the surface heterogeneity h(x, y), which is solved numerically. As an application we consider random Gaussian defects: in the limit of weak contact line distortion it is possible to address the case of semi-concentrated assemblies, in which we observe collective effects. If the contact line configuration may be written ps a function ~ (x) in the (x, y)-plane we characterize it by its Fourier components: U, = K ~(x) exp [- Bixq~x] dx, with U, = 0 being the unperturbed position. In the following we assume the condition of small slope: q I U, I << 1 and KU,, << 1. For each contact line configuration, the height C(x, x) of the fluid interface above its unperturbed position is then uniquely determined by the Laplace equation AC = rc2 C, with the boundary condition <(E, - ~ (x) cos 0,) = ~ (x) sin Bo. The sum of the liquid-gas interfacial energy and of the gravitational energy is then, to leading order in a, [3,9], The energy of the solid interface is F, = - r1 0 i r&x, y) + ylvcosoo)dxdy. In the approximation of weak distortions of the contact line, this expression can be linearized with respect to all components U, such that q f 0. A straightforward calculation leads to

3 J. CRASSOUS et al.: CONTACT ANGLE HYSTERESIS ON A HETEROGENEOUS SURFACE: ETC. 417 where the functions I, ( y) are the Fourier components of the normalized surface hetero- geneity along a line y = const: I, ( y) = h(x, y) exp [- 2ixq~xl dx. The minimization of the total energy F = F, + Fcap + Fg may now be conducted in two steps. First the value of the average position of the contact line is chosen, and the minimization of F with respect to all other Fourier components of the contact line is performed. Since the expressions (1) and (2) have been linearized with respect to those components, the solution is uniquely determined, The spectral coefficients are then reinjected into the free energy F, which now depends only on the mean position of the contact line: 0 i' The minimization of F with respect to Q yields the locally stable configurations of the contact line: with the additional condition d2f/dc$ > 0. The 1.h.s. of eq. (4) is also the change in contact angle cos 6 - cos eo. For a given surface heterogeneity h(z, y), all the stable solutions can be computed numerically. The lowest and the highest possible solutions give the advancing and the receeding contact angle. We first look at the hysteresis generated by a single defect of Gaussian shape h(z, y) = = /A, exp [ - (x2 + y2)/02], isolated on a surface IC -' x IC -'. The unperturbed contact angle is fixed to Bo = 0.5. The defect size U is fixed, and we vary the length K -' of the system between IC -' = 300 and K -' = 240u. For each value of the heterogeneity parameter ho /y, we compute the r.h.s. f( y) of eq. (4) with a resolution Ay = IC -' /4000 and with a number of 100 Fourier modes. We check that decreasing Ay, or increasing the number of modes, does not change the result. We start the numerical experiment with the unperturbed value Bo of the contact angle, and receed the solid surface (i.e. drag it out of the liquid bath) by successive steps of length Ay. At each step, we compute the mean position a. of the contact line, and deduce from it the contact angle. Averaging on every solid position gives the average receeding contact angle. The symmetric procedure is used to get the advancing contact angle. At the end of the simulation, we compute the average hysteresis H = (cos8,) - (cose,) of the contact angle. Figure 2 shows the square root of the hysteresis vs. the strength of the defect for various ratios K -' /U between 30 and 240. As expected, there is no hysteresis until the strength exceeds a threshold value h,. We find a weak dependence of the threshold value on the system size: h, - l /va. This dependence is slightly different from the one calculated by Joanny and de Gennes [2]: h, - l / In (IC-'/U). This comes from the difference in the constraint imposed to the system: in [2] the contact line is pinned at its two extremities and the fluid interface is free, whereas here the contact line is free and the external constraint is provided by gravity. Above the threshold value, H1l2 varies linearly with the defect strength h, as shown in [2]. The slope dh'/2/dh varies as the inverse of the system size, with a logarithmic factor ln(mc). So the hysteresis for an isolated defect can be written as H - (UK)' with h, - In( 1/2.50~)-~/~. ln(1/2.5o~)(h - he)',

4 418 EUROPHYSICS LETTERS h=h,ly Fig Square root of the hysteresis H generated by an isolated defect of sue U, us. the defect strength h = ho/y, for different system sues: K - /U ~ = 30,60,120,240. There is no hysteresis below a threshold value he. For h > he, H 1/2 grows linearly with h as H112 = S(h - he). Inset: variation of h;' (x) and (S/KU)' (0) with the system size ln(ic-'/u). We consider now the hysteresis generated by a system of identical Gaussian defects randomly localized on the solid surface. The macroscopic cut-off length K -', the unperturbed contact angle eo, and the shape of the defects are fixed. We study the variation of the hysteresis with: 1) the amplitude of defects, and 2) the density of defects. In order to keep the mean value between cose, and cose, equal to coseo, the defects are chosen wetting and non-wetting in equal quantity. The total number of defects on a K-' X K - square ~ is N. We first check the validity of the two approximations of small slopes and of small distortions. The perturbation function -is: h(x, y) = hox exp [- (x- - (y- yi)2/~21, i where (xi, yi) is the coordinate of the centre of the defect number i, and =? 1 according to its wetting or non-wetting character. Reintroducing this perturbation function in the expression of q(x>, we find for the distortion 1 q(x> - ho U0 = -- - exp [- ( ~OXIC)~] ylv 2xsin2eO P+O Ipl. exp [- (3- yi)2/g2l Ei COS(~~~K(X - xi>). (5) i The number of defects crossing the contact line is of the order of NG-K. The sum in the above expression, with randomly distributed values of xi, is of order of m. The condition for weak distortions, q(x> - uo c G-, then becomes 5 < 3.5 sin20,. Y It is easy to check that this condition is stronger than the small-slopes condition. In the following numerical study, the mean contact angle will be fured to eo = 0.5, the defect width to G- = 1/120~, and the strength ho will vary between 0 and 0.3~~". The above condition is then equivalent to a density of defects lower than 1000 defects on a K - x ~ K-' square. For a fixed density of defects, we compute the defects positions on a surface of width K -'

5 J. CRASSOUS et al.: CONTACT ANGLE HYSTERESIS ON A HETEROGENEOUS SURFACE: ETC h=h,jy Fig Square root of the normalized hysteresis H/N, for various densities of defects N. At high value of the defect strength h, (H/N)'l2 varies as (h- h,(~v))f(n)l/~. Inset: the variation off(n) and h,(n) with the defect density N. and of length 5 0 -' ~ in order to get good statistics. In order to avoid too strong variation of h(x, y), we impose non-overlapping conditions, i.e. defects centres are separated at least by Q. Calculations are done for concentrations of defects NIC~ ranging from 1 to 800. If the defects do not interact, we expect a hysteresis varying linearly with N, and quadratically with ho /y [2,31. Thus we plot in fig. 3 the normalized contact angle hysteresis W N us. the defect strength h, for different concentrations N of defects. We h d that W N varies linearly with h for every concentration. Outside a crossover region discussed below, the Fig Square root of the normalized hysteresis H/Nf(N) us. the normalized defect strength hff = = h - h,(n). Deviations from the master curve occw only in a small crossover region around the threshold h$ = 0. Inset: the detail of the crossover region for small N and large N.

6 420 EUROPHYSICS LETTERS hysteresis can be described with two parameters: H(N) = Nf(N)(h - h,(n))2. (6) For very weak density of defects (typically Nrc2 < 30), the system exhibits the.dilute>> behaviour: f(n) and the threshold h, (N) are essentially constant. The transition between isolated and interacting defects occurs at a density for which the contact line crosses an average number of defects of about 0.5. As soon as the defect density increases above 50, departure from the isolated-defect behaviour can be observed. The functionf(n) is no longer constant but exhibits a dependence of order N This corresponds to a hysteresis growing like No.'. This behaviour can be compared to the experiments of di Meglio [6] on assemblies of non-overlapping defects, which showed hysteresis varying as No.'. The threshold decreases even more abruptly with the defect density: the log-log representation of h, (N) in fig. 3 shows a N-0.5 decrease law. Using the two parameters f(n) and h, (NI, all curves collapse on the same master curve, except in a small crossover region (fig. 4). We now focus on this crossover region. At low density 1 Nrc2 30, the width of the transition rises with the number of defects. This reveals the cooperative nature of defects interaction near the threshold: in this range of density, there is a small but finite probability for the contact line to cross two or more defects, able to cooperate and anchor the contact line whereas an isolated defect is unable to do so. This weak pinning effect is discussed in [5]; finite hysteresis is expected to occur even at vanishing defect strength when the macroscopic size rc-l diverges. At higher densities (50 < Nrc2 < 800), the size of the crossover region decreases. The unperturbed area of the solid surface is progressively covered by defects, and the solid surface acts as homogeneous. In conclusion we summarize the main results obtained. In the case of a weakly distorted contact line, a one-variable equation giving the metastable configurations of a fluid interface in contact with a heterogeneous solid surface can be constructed. It is possible to study in this limit semi-concentrated assemblies of individual defects. The results exhibit many collective effects, some of which can be described by a convenient renormalization of the hysteresis threshold and of the defect density. The results are consistent with available experimental data. The weak distortion limit provides a simple and numerically tractable model for contact angle hysteresis. We plan to study contact angle noise and response to an oscillatory motion in future work. *** We have benefited from very helpful comments by J. F. JOANNY, M. 0. ROBBINS and S. Ron. REFERENCES [l] LEGER L. and JOANNY J. F., Rep. Pmg. Phys., 55 (1992) 431. [2] JOANNY J. F. and DE GENNES P. G., J. Chem. Phys., 11 (1984) 552. [3] POMEAU Y. and VANNIMENUS J., J. CoZloid Znte$bce Sci, 104 (1985) 477. [4] Cox R. G., J. Fluid Mech., 131 (1983) 1. [5] ROBBINS M. 0. and JOANNY J. F., Europhys. Lett., 3 (1987) 729. [6] DI MEGLIO J. M., Europhys. Lett., 17 (1992) 607. [71 NADKARNI G. D. and GAROFF S., Eumphys. Lett., 20 (1992) 523. [8] LAZARE S., GRANIER V., LUTGEN P. and FEYDER G., Rev. Phys. AppZ., 23 (1988) [9] JOANNY J. F. and ROBBINS M. O., J. Chem Phys., 92 (1990) [lo] RAPHAEL E. and JOANNY J. F., Europhys. Lett., 21 (1993) 4, 453.