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Dynamics of Spontaneous Spreading on Energetically Adjustable Surfaces in a Partial Wetting Regime S. Semal, M. Voue´, and J. De Coninck* Centre de Recherche en Mode´ lisation Mole´ culaire, Universite´ de Mons-Hainaut, 20, Place du Parc, B-7000 Mons, Belgium Received February 18, 1999. In Final Form: July 6, 1999 Using molecular-kinetic theory, the effects of the substrate surface energy on the time relaxation of the contact angle of squalane droplets are investigated. The studied surfaces are mixed Langmuir-Blodgett films composed of two kinds of molecules of equivalent length but with different end groups: n-tricosanoic and ω-tricosenoic acids. The two-dimensional miscibility of these chemical compounds over the complete range of relative concentration is discussed on the basis of various criteria. We show that the critical surface tension and the friction coefficient of the squalane molecules on these surfaces, calculated from the relaxation of the contact angle, are linear functions of the relative concentration of the ω-tricosenoic acid in the monolayers.
1. Introduction Wetting is a macroscopic property which is very sensitive to the molecular details of the considered systems.1,2 With the development of organic surface chemistry techniques,3 a variety of closely packed monolayer surfaces, single or binary, have been prepared, allowing controlled investigation of the effects of the surface morphology and of the chemical structure on the static contact angles and on the hysteresis.4-7 Macroscopic wetting theories8-10 predict the onset of a contact angle hysteresis when the size of surface irregularities is sufficiently large (e.g., 0.1 µm for patchwise heterogeneity8). However the hysteresis phenomenon has also been reported for microscopically heterogeneous2,5 and mixed6,7 self-assembled monolayer (SAM) surfaces. From these studies, and from the ones in refs 11-13, it is clear that there is a strong dependence of distribution and size of the heterogeneities on the contact angle hysteresis. Moreover, both roughness and chemical heterogeneities will drastically change the dynamics, of spreading. Hence there is a need to control the nature, the distribution and the size of the heterogeneities, and to understand the dynamic behavior of the contact angle on such heterogeneous substrates. Despite ingenious experimentation, it remains difficult to engineer surfaces of controlled structure or chemistry. (1) Whitesides, G. M.; Laibinis, P. E. Langmuir 1990, 6, 87. (2) Ulman, A.; Evans, S. D.; Shnidman, Y.; Sharma, R.; Eilers, J. E. Adv. Colloid Interface Sci. 1992 , 39, 175. (3) Ulman, A. An introduction to ultrathin organic films: from Langmuir-Blodgett to Self-Assembly; Academic Press: San Diego, 1991. (4) Ulman, A.; Evans, S. D.; Sharma, R. Thin Solid Films, 1992, 210/211, 810. (5) Folkers, J. P.; Laibinis, P. E.; Whitesides, G. M. Langmuir, 1992, 8, 1330. (6) Bain, C. D.; Evall, J.; Whitesides, G. M. J. Am. Chem. Soc., 1989, 111, 7155. (7) Bain, C. D.; Whitesides, G. M. J. Am. Chem. Soc. 1989, 111, 7164. (8) Johnson, R. E.; Dettre, R. H. In Surface and Colloid Science; Matjevic, E., Ed.; Wiley-Interscience: New York, 1969; Vol. 2, p 85. (9) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972, 38, 341. (10) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (11) Andrieu, C.; Sykes, C.; Brochard, F. Langmuir 1994, 10, 2077. (12) Drelich, J.; Wilbur, J. L.; Whitesides, G. M. Langmuir 1996, 12, 1913. (13) Wiegand, G.; Jaworek, T.; Sackmann, E. J. Colloid Interface Sci. 1993, 196, 29.
In contrast, the substrates investigated in computer simulations can be at the same time both heterogeneous and perfectly controlled at the microscopic level. A few attempts have been made to describe the dynamics of wetting of heterogeneous substrates (refs 14-16 and references therein). Recently, Molecular Dynamics (MD) simulations, combined with the molecular-kinetic theory of wetting,17 were used to study the dynamics of relaxation of the contact angle of sessile drops in the partial wetting regime.18 Another attempt for a better understanding of the wetting of heterogeneous surfaces by liquids, using the molecular-kinetic theory of wetting, is also reported in ref 19. In the latter study, the dynamics of wetting of squalane droplets on partially octadectltrichlorosilane (OTS)-grafted silicon wafers was investigated. The main disadvantage of such substrates is that their surfaces are characterized by the coexistence of two or three phases on the silica (SiO2) surface:20 the “liquid expanded” and “liquid condensed” ones. Nevertheless, the coverage of the surface was unambiguously characterized by the OTS fraction fOTS measured using spectroscopic ellipsometry. Time relaxation experiments of the contact angle of squalane droplets were also performed, and their associated static contact angles appeared to follow a Cassie-like law.21 To overcome the difficulties related to the coexistence of the SiO2 and the liquid-expanded and liquid-condensed phases at the surface of the partially OTS-grafted substrates, we have chosen to use the Langmuir-Blodgett (L-B) technique. It allows direct variation of the monolayer composition and makes it experimentally easier to (14) Topolski, K.; Urban, D.; Brandon, S.; De Coninck, J. Phys. Rev. E. 1997, 56, 3353. (15) Urban, D.; Topolski, K.; De Coninck, J. Phys. Rev. Lett. 1996, 76, 4388. (16) Collet, P.; De Coninck, J.; Dunlop, F.; Reignard, A. Phys. Rev. Lett. 1997, 79, 3704. (17) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. Blake, T. D. In Wettability; Berg, J.-C., Ed.; Marcel Dekker: New York, 1993; p 251. (18) Ada˜o, M. H.; de Ruijter, M. J.; Voue´, M.; De Coninck, J. Phys. Rev. E. 1999, 59, 746. (19) Semal, S.; Voue´, M.; de Ruijter, M. J.; Dehuit, J.; De Coninck, J. J. Phys. Chem. B 1999, 103, 4854. (20) Davidovits, J. V. De´termination des conditions d’obtention de films monomole´culaires organise´s: application aux silanes autoassemble´s sur silice. Ph.D. Thesis, Universite´ de Paris VI, 1998. (21) Cassie, A. B. D. Discuss. Faraday. Soc. 1952, 57, 5041.
10.1021/la9901783 CCC: $18.00 © 1999 American Chemical Society Published on Web 09/11/1999
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control the chemical homogeneity.22 The domain shape and size in the spread monolayers can be influenced by means of the surface pressure, the compression speed, the subphase temperature, the counterion content, and its concentration.23,24 We covered silica surfaces with monolayers consisting of a mixture of two kinds of fatty acid molecules of the same length but different functional tail groups, methyl and vinyl, respectively. Investigations of the miscibility of binary monolayers on the basis of two-dimensional thermodynamic analysis are well developed,25-27 and it is possible to control the quality and the structure of the covered surfaces. The resulting surface coverage exhibits at least three advantages for the wetting experiments: (i) the multilayers perfectly cover the surfaces with closely packed molecules, (ii) they exhibit a very low roughness (root mean square roughness e0.4), and (iii) the surface energy can be tuned as a function of the relative concentration of the components. The paper is organized as follows. Section 2 is devoted to the presentation of the experimental setup and materials. The results and their analysis are presented in section 3. Concluding remarks are given in section 4. 2. Experimental Setup and Materials 2.1. Mixed L-B Monolayers on Glass. Unless otherwise specified, all the chemicals were purchased from Sigma Chemical (>99.9% purity). The fatty acids used in this study were n-tricosanoic and ω-tricosenoic acids (Interchim, >99.9% purity). The structure of the molecules of these two components is similar. Both are amphiphilic molecules and exhibit a carboxylic head and an alkyl chain of 23 carbon atoms. The n-tricosanoic acid is methyl-terminated (-CH3) while the end group of the ω-tricosenoic acid is a vinyl one (-CHdCH2). These amphiphilic materials, either pure or using the premixed spreading technique,27 were dissolved in chloroform at a concentration of 1.5 × 10-3 M. The aqueous subphase was a 10-3 M CdCl2 (Fluka, p.a.) solution at pH 5.6. MilliQ purified water (Millipore Corp.) with a resistivity >18 MΩ m was used throughout the experiments. The deposition experiments were done using a KSV-3000 balance. It consists of a PTFE trough with two barriers which allow symmetrical compression of the spread monolayer. The trough was mounted on an antivibration table and the experimental setup was placed in a vertical laminar flow cabinet (Ho¨lten, class 100). Surface pressure is measured at a precision of 0.1 mN/m by the Wilhelmy plate method. Before each experiment, the trough and barriers were cleaned with hexane/ ethanol mixture (1:1, v/v) and then with pure methanol. The aqueous subphase temperature was maintained at 20 °C. Drops of the solution containing the fatty acid molecules were spread onto the subphase after repeated surface compression and aspiration. After evaporation of the chloroform, the surface pressure/area isotherms Π(A) were obtained with a compression speed of 20 mm/min. Using the premixed spreading technique, we forced the homogeneous 3-D mixing of the two different kinds of fatty acid molecules. The isotherms were obtained after several cycles of compression and decompression of the Langmuir monolayer. No further modification of the monolayers was observed after three cycles, and the isotherms presented on Figure 1, for several relative concentrations in ω-tricosenoic acid (Cω), therefore correspond to the true equilibrium of the mixture. We will discuss in the next paragraph the way these isotherms lead us to conclude the complete 2-D miscibility of the mixture. (22) Angelova, A. Thin Solid Films 1994, 243, 394. (23) Lo¨sche, M.; Mo¨hwald, H. J. Colloid Interface Sci. 1989, 131, 56. (24) Helm, C. A.; Mo¨hwald, H. J. Phys. Chem. 1988, 92, 1262. (25) Gaines, G. L. Insoluble monolayers at liquid/gas interface; WileyInterscience: New York, 1966. (26) Goodrich, F. C. Proceeding of the 2nd International Congress on Surface Activity; Butterworths: London, 1957; Vol. 1, p 85. (27) Do¨rfler, H. D. Adv. Colloid Interface Sci. 1990, 31, 1.
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Figure 1. Compression isotherms Π(A) of the binary systems, made up of n-tricosanoic and ω-tricosenoic acid molecules, with different relative concentrations of ω-tricosenoic acid: 0%, 35%, 50%, 65%, 100%. The surface pressure Π of the monolayer, as measured by the KSV-3000 balance, is given as a function of A, the mean molecular area. Before the transfer of the Langmuir monolayers onto standard microscope glass slides (Resco Trade), these substrates were cleaned by immersion for 1 h in hot 1 M sulfochromic acid solution, followed by 12 h in 1 M NaOH solution. On the basis of the compression isotherms of the Langmuir monolayers (Figure 1) and in view of having a surface pressure suitable whatever Cω, the deposition of monolayers was performed at 35 mN/m, at which pressure the molecules are closely packed (in a “liquid-condensed” (LC) phase, as determined from the compression isotherm shapes) and oriented perpendicular to the substrate surface.25,28 The hydrophilic substrate was dipped and withdrawn vertically through the Langmuir monolayer at a speed of 3 mm/min for the first layer and 25 mm/min for all subsequent layers. We waited for 5 min after depositing of the first layer before depositing the subsequent ones. Since the hysteresis decreases as the number of layers of fatty acid molecules increases, from one to seven layers and then reaches a plateau value,29 we covered the glass surfaces with seven monolayers, so that they exhibit the lowest hysteresis. 2.2. Contact Angle Measurements. The liquid used to measure the wetting characteristics of the substrates was squalane (2,6,10,15,19,23-hexamethyltetracosane, C30H62), a nonpolar hydrocarbon (Sigma, 99%). Squalane has a surface tension (γLV) of 31.1 mN/m and a viscosity of 35.0 mPa s at a temperature of 21 °C.30 The substrate was placed on a movable stage in front of a microscope, and a squalane droplet (∼6 mm3) was deposited with a syringe. Within a few seconds, the droplet had spread and reached its static contact angle. The viscosity of the liquid was high enough to allow us to follow the details of the time relaxation of the contact angle. The microscope was connected to a black-and-white CCD camera, and the images (768 × 576 pixels - 256 Gy levels) of the droplet were captured on a video recorder (VCR) at 50 images per second. Pictures were then selected from the videotape and transferred to a computer (Data Translation DT3155 frame-grabber) for further analysis. All the experiments were conducted at room temperature (21 ( 1 °C). Each image was processed as follows: After the localization of the edge of the liquid/gas interface, the shape of sessile drops in the presence of gravity was described by the Laplace equation:31 (28) Petty, M. Langmuir-Blodgett FilmssAn Introduction; Cambridge University Press: Cambridge, 1996. (29) Semal, S.; Blake, T. D.; Geskin, V.; de Ruijter, M. J.; Castelein, G.; De Coninck, J. Langmuir, in press. (30) de Ruijter, M. J. A microscopic approach of partial wetting: Statics and dynamics. Ph.D. thesis, Universite´ de Mons-Hainaut, Mons, Belgium, 1998. (31) Rusanov, A. I.; Prokhorov, V. A. Interfacial Tensiometry; Elsevier: Amsterdam, 1996.
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(
∆P ) γ
)
1 1 + R1 R2
Semal et al. (1)
which relates the surface tension γ of the liquid to the pressure difference ∆P across the liquid/gas interface, R1 and R2 being the two principle radii of curvature of the curved surface. The best Laplace curve was fitted32 through the profile of the sessile drops (about 1500 experimental points)30 by our computer program G-contact. Since the hysteresis of the contact angles of droplets deposited on solid surface is given by the difference of the advancing (θadv) and receding (θrec) static contact angles, we measured θadv and θrec. The procedure is the following. A drop is first deposited on the substrate. The needle of a syringe is introduced into the drop, and the liquid volume of the drop is increased (decreased) until the liquid front moves. Just before it moves, the advancing (receding) angle is measured.
3. Results 3.1. Structure of the Mixed Monolayers. It is of the prime importance to control the nature, distribution, and size of the chemical heterogeneities. This unfortunately cannot be done using the standard characterization techniques of the surfaces (AFM, ellipsometry, contact angle measurements, ...). Indeed, even if these tools allow control of the quality of the covering films and to distinguish them quantitatively, they do not provide useful information about the local, molecular-scale structure of the deposited L-B monolayers. The organization of the molecules in the L-B film can be inferred from analysis of the corresponding Langmuir monolayers, if we assume that the monolayers which are transferred onto the substrate by the L-B technique have the same organization as the Langmuir monolayers.28 Hence the limitation of the miscibility tests of monolayers lies in the fact that the mixed state can be analyzed only in an indirect manner, i.e., by using the phenomenological properties of the multicomponent monomolecular systems. Application of thermodynamic analysis of mixing and of two-dimensional phase rule for miscibility in multicomponent films is based on the analysis of the dependencies on the monolayer composition of the average areas per molecule, the free energy of mixing, and the surface pressures at which the phase transitions and the collapse of the monolayer occur.25-27 We analyzed, at different surface pressures, the variation of the mean molecular area in the binary monolayers as a function of the ω-tricosenoic acid molar fraction. Despite complete miscibility of the components, the relationship between the mean molecular area and the relative concentration is not linear.25-27 For our systems, the mean molecular area is almost constant at the surface pressure of the L-B deposition, and the experimental errors (although small) did not allow us to decide unambiguously between miscibility or 2-D segregation of the mixture. Other miscibility criterions were therefore required. The relationship between the relative concentration and the surface pressures corresponding to either a phase transition or collapse, also allows miscibility investigations.25,27 A linear variation of the surface pressure as a function of the relative concentration, for phase transition or collapse, indicates complete miscibility. Any deviation from linearity provides additional evidence of 2-D segregation. To determine precisely the surface pressures at which either collapse or phase transitions appear (both (32) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, 1992.
Figure 2. (a) Compression isotherm Π(A) of a pure tricosanoic acid monolayer. (b) Second derivative of the previous curve versus A, used to monitor the phase transitions.
Figure 3. Surface pressures Π at which either phase transitions or collapse occur, as a function of Cω, the relative concentration of ω-tricosenoic acid: (a) liquid expanded (LE) to liquid condensed (LC); (b) liquid condensed (LC) to solid (S); (c) collapse. Linear variations indicate complete two-dimensional miscibility of the chemical compounds.
from the “liquid-expanded” phase (LE) to the “liquidcondensed” phase (LC) and from the LC phase to the “solid” phase (S)), we have calculated the second derivative32 of the compression isotherm Π(A) (Figure 2). Figure 3 reports the variation of the surface pressure Π at which occur the LE to LC and the LC to S phase transitions and monolayer collapse. All these relationship are clearly linear, within the error bars, suggesting that the miscibility of the two components in the mixed monolayer, and therefore in the transferred one, is complete. Further more, this result is also supported by the results of a third test. This last criterion for miscibility consists of measuring the molecular interactions within the binary films, i.e., the excess free energy of mixing, ∆Gminex, at a given surface pressure Π, as a function of the relative
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Figure 4. Excess free energy of mixing ∆Gmixex vs Cω, the relative concentration of ω-tricosenoic acid at different surface pressures: Π ) 10 mN/m (4), Π ) 20 mN/m (O), and Π ) 40 mN/m (]). Negative values of ∆Gmixex have been associated to complete miscibility of the 2-D system.
Figure 5. Influence of Cω on the hysteresis of contact angles (CAH) for pure water deposited onto the mixed L-B films. Due to the absence of physical roughness, CAH remains constant within the errors bars over the complete range of investigated concentrations.
concentration. The values of ∆Gminex were determined following the method of Goodrich,26 by integration of the experimental compression isotherms:
∆Gmixex (Π) )
∫0Π A12 dΠ′ - x1∫0Π A1 dΠ′ - x2∫0Π A2 dΠ′
(2)
where A12 is the mean molecular area in the twocomponent film, A1 and A2 are the molecular areas in the pure films, and x1 and x2 are the molar fractions of the two components in the monolayer. The integration can be done for different surface pressures. Positive values are typical for unfavorable interactions, i.e., demixing, between the two components, and have been interpreted33 as an indication of partial phase segregation. Negative values are therefore associated with complete mixing. Figure 4 shows that the obtained excess free energies of mixing are all negative. Thus, we have further evidence for complete miscibility in the 2-D system. 3.2. Contact Angle Measurements. Additional information concerning the physical and chemical heterogeneities of the mixed L-B films can often be obtained from contact angle hysteresis measurements.34 Figure 5 shows the hysteresis of the contact angles of water droplets on the different substrates, as a function of the relative concentration of the ω-tricosenoic acid (Cω). In all cases, hysteresis is small. Furthermore, although the molecules in the monolayers differ by the end group, either methyl or vinyl, the level of hysteresis remains essentially constant, within the error bars, so that this measurement cannot help us to distinguish the different surfaces (11° for water and 3° for n-hexadecane). Therefore, to distinguish unambiguously the covered substrates, we measured their critical surface tension γc34 as a function of the relative concentration of the ω-tricosenoic acid. The homologous series of alkanes was used. It can be seen that the critical surface tension is a linear function of the relative concentration and that values increase steadily from 20.5 ( 0.3 to 23.6 ( 0.3 mN/m as the concentration of the vinyl end groups increases (Figure 6). (33) Costin, I. S.; Barnes, G. T. J. Colloid Interface Sci. 1975, 51, 106. (34) Adamson, A. W. Physical chemistry of surfaces; Wiley-Interscience: New York, 1990.
Figure 6. Critical surface tension γc of the mixed L-B films vs Cω, the relative concentration of ω-tricosenoic acid. γc was determined from standard Zisman plots, using the homologous series of n-alkanes. γc satisfies linearly the following linear relationship: γc ) a + bCω with a ) 20.47 ( 0.05 and b ) 2.99 ( 0.08 (correlation coefficient R2 ) 0.998).
In Figure 7, we plot the cosine of the static contact angle of squalane droplets on the covered surfaces vs the relative concentration. This curve can be fitted by a straight line, asserting the validity of Cassie’s law21 for microscopic heterogeneities in agreement with MD predictions.18 3.3. Wetting Dynamics. To determine the effect of chemical heterogeneities on the dynamics of spreading, we have measured the relaxation of the dynamic advancing contact angle of droplets of squalane spreading spontaneously on the covered surfaces (Figure 8). Since we showed the complete miscibility of the components in the mixed L-B films, the heterogeneities which are present at the surface of the samples have microscopic (i.e., molecular) sizes. Therefore, we choose the molecular kinetic theory17,30 to interpret the experimental data. This theory relates the dynamic angle θ to the velocity v of the wetting line
v)
(
)
(cos θ0 - cos θ)γLV) ∂r ) 2Kw0 λ sinh ∂t 2kBnT
(3)
where Kw0 is the equilibrium jump frequency of the liquid molecules at the wetting line (when the wetting line is at
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Figure 7. Linear variation of cos θeq, the equilibrium contact angle of squalane droplets on the mixed L-B films as a function of Cω, the relative concentrations of ω-tricosenoic acid: cos θeq ) a′ + b′Cω with a′ ) 0.788 ( 0.002 and b′ ) 0.117 ( 0.004 (R2 ) 0.998).
Figure 9. Distributions of Kw0 and λ values as calculated using the bootstrap method for a 50% relative concentration (Cω). Analogous results were obtained for the other surfaces.
( )(
∂θ 3V ∂r )∂t ∂t π
1/3
(1 - cos θ)2
)
(2 - 3 cos θ + cos3 θ)
4/3
(5)
Equations 3 and 5 are a linked pair of partial differential equations. The static contact angles values are those experimentally obtained. Thus these equations may be solved numerically with two adjustable parameters: the prefactor Kw0 λ and γLV/(2kBnT). To a first approximation, n and λ are related by n ) λ-2.17 This procedure was used to fit the experimental data in Figure 8 and to determine the Kw0 and λ values as a function of the relative concentration. As can be seen, the fits are of good quality. To estimate the associated errors, we have applied the bootstrap method.32 We use the original experimental data as the basis of a Monte Carlo simulation. From this original set, 37% (i.e., 1/e) randomly chosen points are replaced by duplicates from the original data according to Figure 8. Influence of Cω, the relative concentration of ω-tricosenoic acid on the relaxation of the advancing contact angles of squalane droplets on the mixed L-B films. Cω increases from bottom to top. The plain lines correspond to the best-fit values of the molecular-kinetic theory equations with respect to the effective jump frequency Keff0 and the distance between adsorption sites λ.
rest), λ is the distance between adsorption sites, kB is Boltzmann’s constant, n is the surface density of adsorption sites on the surface, and T is the absolute temperature. The initial shape of a liquid drop placed on the top of a substrate will be different from the static case. To minimize the free energy of the system, the drop will transform itself as a function of time. At constant volume, this dynamic behavior is characterized by the contact angle θ(t). For small drops, gravity can be neglected and the shape of the spreading drop will approximate a spherical cap. The geometry of a spherical cap is characterized by the following relationship
θduplicate ) θoriginal + ξ∆θmax
(6)
where ∆θmax is the maximum allowed variation fixed to 1° (i.e., corresponding to the maximum experimental deviation) and ξ is a random number normally distributed around a zero mean with a unit variance. Using this procedure, we replace the original data set by a new one for which the corresponding parameters Kw0 and λ can again be calculated. Typically, 100 successive simulations were considered, and the distribution of values was analyzed statistically for each parameter. Figure 9 shows the results obtained for a 50% relative concentration in ω-tricosenoic acid Cω. Analogous results are obtained for the other surfaces. The correlation between the Kw0 and λ shown on Figure 9 results from the applicability of the linearized form of eq 3
(4)
0 3 γ ∂r Kw λ γ ) (cos θ0 - cos θ) ) (cos θ0 - cos θ) ∂t kBT ζ (7)
where V is the drop volume. Taking advantage of the very low volatility of the squalane at 21 °C and the short time scales of the contact angle relaxation process, the droplet volume can be considered as constant, yielding the following equation for the time variation of the base radius:
since the order of magnitude of the sinh argument is significantly less than unity in these experiments. In eq 7, we introduced the friction coefficient, ζ ) kBT/Kw0λ3.30 This, in turn, would imply that the effective parameter in our experiments is the friction coefficient, ζ. The average of ζ and its standard deviation can then be plotted versus
r)
(
sin3 θ 3V π (2 - 3 cos θ + cos3 θ)
)
v)
1/3
Dynamics of Spontaneous Spreading
Figure 10. Average value of the friction coefficient, ζ, and its standard deviation versus Cω. As it can be seen, the linear variation of ζ can be described by the linear relation: ζ ) a + bCω with a ) 11.05 ( 0.32 and b ) 69.7 ( 2.32 (R2 ) 0.997).
Cω, as illustrated in Figure 10. These results show a linear increase of ζ as Cω increases. To interpret these results in terms of the jump frequency at equilibrium, we need to estimate λ. The value of λ may be computed from the molecular area of the Langmuir monolayer measured during deposition. We deposited the L-B layers at a constant pressure of 35 mN/m. Measuring the area occupied by the molecules on the subphase surface, we obtained a mean molecular area of about 19 Å2.29 Assuming that the cross section of the molecule is a square, this gives a value of λ about 4.5 Å (the presumed distance between the centers of two adjacent CH3 or CH2 groups). As previously mentioned, we observe that λ is constant for each relative concentration. Therefore, we get the variation of the inverse of the jump frequency versus Cω. The equilibrium jump frequency Kw0 refers usually to time relaxation of the contact angle of a pure liquid droplet on an ideal substrate, i.e., for which the effect of the surface roughness and/or chemical heterogeneity are negligible. When heterogeneous substrates are considered, the molecular-kinetic theory has to be considered as an effective theory. In that framework, Keff should be used instead of Kw0 and referred to as the effective jump frequency. The jump frequencies Keff vary between the limits for the pure monolayers: KCH2 ) 5.10 ( 1.39 105 Hz and KCH3 ) 40.13 ( 3.20 105 Hz, where KCH2 and KCH3 are the jump frequencies of the homogeneous LB films of ω-tricosenoic and n-tricosanoic acids, respectively. The effective jump frequency monotonically decreases as the relative concentration of ω-tricosenoic acid increases in the LB film (Figure 11), i.e., as the film becomes more lyophilic. This effect can unambiguously be attributed to the increase in the strength of the solid-liquid interactions, i.e., the driving term, when transforming a CH2 surface into a CH3 one; the more lyophilic the surface, the lower the tendency for the liquid molecules to vibrate on the solid surface, i.e., the lower their mobility. Thus, an increase in the interaction causes both the driving term for spreading and the frictional resistance to increase. Assuming a constant driving force F and the viscous nature of the friction of the liquid molecule on the solid substrates, we have shown19 that, on heterogeneous substrates and in the partial wetting regime, the effective jump frequency Keff is related to the individual frequencies Ki and the surface composition ci by
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Figure 11. Inverse of the effective jump frequency Keff vs Cω, the relative concentration of ω-tricosenoic acid. The predictions of the parameter-free model (eq 9) with KCH2 ) 5.10 × 105 Hz and KCH3 ) 40.13 × 105 Hz are represented by the solid line, while the best-fit predictions of the two-parameter model (giving KCH2 ) (5.76 ( 0.37) × 105 Hz and KCH3 ) (40.09 ( 0.77) × 105 Hz) are shown by the dashed line. Both theoretical curves correspond to the experimental data within the experimental error bars.
1 Keff
)
ci
∑i K
(8)
i
where the subscript i refers to all surface constituents. For LB monolayers with two chemical species, as considered in these investigations, eq 8 becomes
cCH3 cCH2 1 ) + Keff KCH3 KCH2
(9)
where the CH3 and CH2 subscripts denote the n-tricosanoic and ω-tricosenoic acids, respectively. In Figure 11, the solid line corresponds to the theoretical predictions of eq 9, with KCH2 and KCH3 equal to the value of the jump frequencies experimentally determined on the onecomponent LB monolayers, i.e., KCH2 ) 5.10 ( 1.39 105 Hz and KCH3 ) 40.13 ( 3.20 105 Hz. As can be seen, the agreement between the proposed parameter-free model given by eq 9 and the experimental result is quite remarkable. Equation 9 can also be considered as a twoparameter equation, with KCH2 and KCH3 as parameters, whose values can be adjusted to reproduce the experimental data. The least-squares32 best-fit solution, denoted using asterisks and corresponding to values of KCH2 ) 5.76 ( 0.37 105 Hz and KCH3 ) 40.09 ( 0.77 105 Hz, is also shown on Figure 11 (dashed line). The predictions of both the parameter-free and the twoparameter models coincide within the error bars and are in close agreement with the experiments. 4. Conclusions The present investigations provide experimental evidence that completely miscible, two-component Langmuir monolayers, of variable relative concentrations of ntricosanoic and ω-tricosenoic acid, behave as “ideal” heterogeneous substrates when transferred to glass substrates. By using molecules of almost equal length, we have minimized on such substrates the influence of the physical roughness to highlight the chemical heterogeneity. The critical surface tension of these substrates ranges
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from 20.5 ( 0.3 to 23.6 ( 0.3 mN/m and is a linear function of Cω, the relative concentration of the ω-tricosenoic acid. In addition, the time relaxation of the contact angle of branched alkanes droplets can be described by the molecular kinetic theory. The friction coefficient, which is the analogous of the inverse of the effective jump frequency, is a linear function of the concentration. On such well-defined surfaces, the dynamics of spreading of
Semal et al.
liquid droplets becomes a powerful tool to obtain useful information about their molecular structure. Acknowledgment. This research has been partially supported by the Ministe`re de la Re´gion Wallonne and the Fonds National pour la Recherche Scientifique. LA9901783
