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Langmuir 1998, 14, 2490-2496
Molecular-Hydrodynamic Description of Langmuir-Blodgett Deposition Jordan G. Petrov* and Peter G. Petrov Institute of Biophysics, Bulgarian Academy of Sciences, 1 Acad. G. Bonchev Str., Block 21, 1113 Sofa, Bulgaria Received September 8, 1997. In Final Form: February 23, 1998 The molecular-hydrodynamic description of Langmuir-Blodgett deposition presented in this study relates the velocity of deposition to the activation free energy of adsorption occurring along the three-phase contact line, the density of the hydrophilic headgroups (considered as adsorption centers), and the viscosity, density, and surface tension of the liquid phase. It introduces rigorously defined dynamic contact angles that can be easily determined in the experiment. It is shown that the pure molecular-kinetic and hydrodynamic theories do not adequately describe the experimental relationship between velocity of deposition and dynamic contact angles while the molecular-hydrodynamic one fits the experimental data up to the maximal velocity, Umax, at which a liquid film is entrained during the upstroke stage of deposition. Comparison between this theory and experiment points out that the hydrodynamic deformation of the fluid interface is negligible when deposition is performed from dilute aqueous subsolutions but can become important for viscous subsolutions and deposition at high surface pressure. Analysis of energy dissipation in the three-phase zone and in the bulk of the moving meniscus shows that the nonhydrodynamic dissipation in the three-phase contact zone prevails in the whole velocity range up to Umax. This dissipation depends on the nature of the hydrophilic headgroups, and their modification via counterion adsorption significantly changes its value. It is shown that substitution of the carboxylic groups by barium-carboxylate ones decreases this dissipation and causes an increase of Umax, probably due to facilitated dehydration of the heads.
Introduction Although deposition of Langmuir-Blodgett multilayers has been intensively studied during the last two decades, relatively little attention has been paid to the dynamics and kinetics of the process. The first attempt in this direction followed the idea of Langmuir1 that transfer of an insoluble monolayer from a liquid onto a solid substrate resembles adsorption of surfactants at the solid-liquid interface. Both processes are governed by the interaction between the functional groups of the amphiphilic compound and the solid surface and the affinity of these groups toward water. Thus the kinetics of Langmuir-Blodgett deposition was considered as kinetics of adsorption occurring along the three-phase contact line (Figure 1). The quantitative description2 was based on the molecularkinetic theory of Blake and Haynes3 that relates the contact line velocity (which is also velocity of deposition4) to the activation barriers of adsorption and surface density of adsorption centers on the solid substrate. A good agreement between molecular-kinetic theory and experimental data was found for deposition of methyl arachidate from 1 × 10-2 M HCl subsolutions2 and for arachidic acid deposited from 3 × 10-4 M CdCl2 and LaCl3 subsolutions.5 At the same time, deviations from this theory were found for deposition of arachidic acid from 1 × 10-2 M HCl and 1 × 10-4 M BaCl2 subsolutions.2,6 Careful observation of the contact line motion led to the conclusion that these deviations were due to a viscous friction in the receding liquid that became significant at higher velocities of the solid substrate. Incorporation of this friction in the (1) Langmuir, I. Trans Faraday Soc. 1920, 15, 62. (2) Petrov, J. G.; Radoev, B. P. Colloid Polym. Sci. 1981, 259, 753. (3) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (4) Langmuir, I. Science 1938, 87, 493. (5) Buhaenko, M. R.; Richardson, R. M. Thin Solid Films 1988, 159, 231. (6) Petrov, J. G. Z. Phys. Chem. Leipzig 1985, 266, 706.
Figure 1. Schematic presentation of a steady dynamic meniscus during Langmuir-Blodgett deposition.
force balance of the process produced the first molecularhydrodynamic description of the LB deposition.2 It approximated the part of the meniscus close to the contact line as a thin liquid film and combined its hydrodynamics with the molecular-kinetic theory. Unfortunately, the solution did not give the profile of the fluid interface and introduced an unknown numerical constant that should be determined via comparison of theory and experiment. The next theoretical paper on this subject was published by de Gennes7 who proposed an entirely hydrodynamic description of the process. It predicted the existence of a maximum deposition velocity above which monolayer transfer during the upstroke stage should occur atop of an entrained macroscopic film of the liquid phase. This velocity was related to the solid-air and solid-liquid interfacial energies through the static contact angle, thus explaining the experimental observation that monolayers whose hydrophilic heads strongly adhere to the solid surface or to the heads of the last deposited monolayer exhibit “dry” deposition up to much higher threshold velocities.8,9 (7) de Gennes, P. G. Colloid Polym. Sci. 1986, 264, 463. (8) Petrov, J. G.; Kuhn H.; Mo¨bius, D. J. Colloid Interface Sci. 1980, 73, 66.
S0743-7463(97)01011-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/11/1998
Molecular-Hydrodynamic Description of LB Deposition
In 1992 the authors of the present paper published a molecular-hydrodynamic theory of wetting dynamics of pure liquids10 that was extended recently by one of us for liquids covered with a condensed insoluble monolayer.11 Both papers utilized the idea of ref 2 to use the BlakeHaynes relationship as a dynamic boundary condition at the moving contact line. They applied the creeping flow approximation12,13 that holds at larger dynamic contact angles and gave solutions for the dynamic profile of the fluid interface that did not contain empirical constants. All above-mentioned theoretical considerations relate contact line velocity to dynamic contact angles and material properties of the system. However, some of them use different definitions of dynamic contact angle. On the other hand, experimental studies often apply different methods for their determination without paying special attention to the correspondence between theoretical definition and experimental determination of this quantity. Several studies with pure liquids14-17 have shown that this problem might be important when the moving meniscus is hydrodynamically deformed. Such a situation might appear at velocities of LB deposition close to the threshold speed of liquid film entrainment. Therefore, when introducing viscous friction in the dynamics of LB deposition, one should also consider the viscous deformation of the moving meniscus and its effect on the definition of dynamic contact angle. This paper completes the former molecular-hydrodynamic descriptions of the LB deposition2,11 by proposing a solution that enables both rigorous definition and experimental determination of the dynamic contact angle. Theoretical expressions for contact line velocity versus quasi-static and extrapolated contact angles (see definitions in the next section) relating kinetics of the process to interfacial and bulk properties of the system are obtained on the basis of the hydrodynamic lubrication approximation and the Blake-Haynes molecular-kinetic theory. The effect of the viscous deformation of the fluid interface on dynamic contact angle and its relationship to experimental data are analyzed. The nonhydrodynamic energy dissipation in the three-phase contact zone is compared with the viscous dissipation in the receding meniscus. The modification of the headgroups via counterion adsorption and its significance for the nonhydrodynamic dissipation are demonstrated on two LB systems previously experimentally studied. Theoretical Definitions and Experimental Determination of Dynamic Contact Angles in Langmuir-Blodgett Systems Description of the kinetics of LB deposition requires the velocity of the three-phase contact line to be related to the material properties of the system. This relationship includes the steady dynamic contact angle called by Langmuir4 a “zipper angle” in order to stress the significance of the interactions along the wetting perimeter. According to him, these interactions are responsible for the elementary act of the deposition process closing the molecular “zip” of the two last monolayers and expelling water between them.4 (9) Petrov, J. G. Colloids Surf. 1986, 17, 283. (10) Petrov, P. G.; Petrov, J. G. Langmuir 1992, 8, 1762. (11) Petrov, P. G. J. Chem. Soc., Faraday Trans. 1997, 93, 295. (12) Voinov, O. V. Mech. Zhid. Gaza 1976, 5, 76 (in Russian); Fluid Dynam. 1976, 11/5, 714 (English Translation). (13) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (14) Huh, C.; Mason, S. G. J. Fluid Mech. 1977, 81, 401. (15) Kafka, F. Y.; Dussan, V. E. B. J. Fluid Mech. 1979, 95, 539. (16) Petrov, J. G.; Sedev, R. V. Colloids Surf. 1993, 74, 233. (17) Petrov, P. G.; Petrov, J. G. Langmuir 1995, 11, 3261.
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Figure 2. Illustration of different definitions of dynamic contact angles adopted in theoretical investigations of the LB deposition (the shadowed areas indicate the region of viscous deformation). Some experimental techniques are also illustrated. (a) Definition of Θqs used in ref 2 and in this study. (b) Definition of Θd adopted in ref 7. (c) Definition of Θext used in this study. (d) Experimental determination of the dynamic capillary height, Ld, and Θapp. (e) Tensiometric determination of Θd for a quasi-static meniscus.
To compare theoretical predictions with experimental results, one should define and measure the same “zipper angle”. In our first attempt to describe the process,2 the final equation utilized the quasi-static dynamic angle that is the slope of the tangent to the fluid interface at the boundary between the hydrodynamically deformed and the quasi-static part of the moving meniscus (Figure 2a). In the hydrodynamic description of de Gennes7 this is the constant final slope of the liquid wedge at the contact line (Figure 2b). As shown for pure liquids,12-17 one could also define an extrapolated dynamic angle that serves as a boundary condition of the quasi-static part of the dynamic meniscus. This angle is easily determined via extrapolation of the hydrodynamically undisturbed meniscus profile to the solid wall (Figure 2c). The experimental determinations of dynamic contact angles in LB systems reported in the literature often applied different techniques. In refs 2, 6, 18, and 19 dynamic capillary heights, Ld, were measured (Figure 2d) and apparent dynamic contact angles, Θapp, were evaluated by means of the static relationship:
(
Θapp ) arcsin 1 -
Fg 2 L 2γ d
)
(1)
Here g is gravity acceleration, F is density, and γ is the surface tension of the liquid. The slope of the dynamic fluid interface possibly close to the contact line was often determined either goniometrically2,5,20 or optically by utilizing reflection of a laser beam focused on this region.21 Both methods give an average slope and could probably be used for large angles, but when viscous deformation is present, they neglect the strong change of curvature in the vicinity of the wetting perimeter (see Figure 2a,c). (18) Gaines, G. L. J. Colloid Interface Sci. 1977, 59, 438. (19) Spratte, K.; Riegler, H. Macromol. Chem., Macromol. Symp. 1991, 46, 113. (20) Neuman, R. D. J. Colloid Interface Sci. 1978, 63, 106. (21) Graff, K.; Riegler, H. Colloids Surf., in press.
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In some investigations22,23 the force exerted on the solid substrate during dipping or withdrawal through the monolayer was measured. Except for the buoyancy component this force contains a surface component γ cos Θd, which acts at the real contact line if the moving meniscus has a quasi-static shape (Figure 2e). In this case one can evaluate the steady dynamic angle Θd if surface tension is known and buoyancy accurately accounted for. When a significant viscous deformation occurs (see Figure 2a,c) the surface force component is ascribed to an effective quasi-static meniscus with the same volume and weight as the real one, thus yielding an effective (apparent) dynamic contact angle. To additionally underline the effect of the viscous deformation on dynamic contact angle definitions, one might point to the different limiting values that the quasistatic, extrapolated, apparent, and goniometric dynamic angles attain at the threshold velocity of liquid film entrainment. Earlier publications2,7,16 and our recent investigations24showed that the minimum value of Θext is 0°, while Θqs and Θd might have nonzero minimum values. When the LB-deposition kinetics follows the molecularkinetic theory that ignores viscous friction, the “zipper” contact angle is the boundary condition of a dynamic meniscus that has a quasi-static shape up to the moving contact line. Under such conditions all theoretical definitions coincide and the experimental methods give the same results because the hydrodynamic deformation of the fluid interface is negligible. Hydrodynamic Boundary Conditions of the LB Deposition Langmuir-Blodgett multilayers are built up via transfer of condensed (closely packed) monolayers from the airwater interface onto a solid substrate that is alternatively dipped and withdrawn through it. During the process of deposition, surface pressure is automatically maintained constant so that monolayer density does not change. Under these conditions the solid wall and the monolayer being transferred move at the same velocity U (see Figure 1). Except for the initial stages, this velocity is constant and the LB deposition occurs at a steady dynamic contact angle. This yields an important relationship; the velocity of the solid substrate, U, and the three-phase contact line velocity, V, have the same values. Thus, any theoretical or experimental relationship between U and Θd simultaneously shows how the intrinsic kinetic parameter of the LB deposition, V, depends on material properties of the system. Because of the close packing of the condensed monolayer at the air-water interface, the tangential component of the velocity of the liquid with respect to the monolayer is zero. Thus, when describing hydrodynamics of the LB deposition, one should apply the nonslip boundary condition at both the solid-water and the air-water interface. The maintenance of constant surface pressure during the process excludes Marangony effects so that the hydrodynamic solution should be performed at constant airwater interfacial tension. Solution of the hydrodynamic problem of LB deposition meets the usual difficulties that the nonslip boundary condition generates close to the contact line.7,13 To avoid them, we follow the approach of Voinov12 and exclude the (22) Peng, J. B.; Ketterson, J. B.; Dutta, P. Langmuir 1988, 4, 1198. (23) Aveyard, R.; Binks, B. P.; Fletcher, P. D. I.; Ye, X. Colloids Surf. A 1995, 94, 279. (24) Petrov, J. G. Book of Abstracts, 9th International Conference on Surface and Colloid Science, 7-12 July 1997, Sofia; Sofia University Press: Sofia, 1997; 223.B4, p 128.
Petrov and Petrov
vicinity of the contact line from the hydrodynamic consideration. The molecular events in this zone affect hydrodynamics in the receding meniscus as a dynamic boundary condition. Usually one assumes that the local slope of the fluid interface at the boundary to this zone, ΘC, does not depend on velocity, retaining the value of the minimum static receding angle. The alternative idea is that ΘC ) ΘC(U). Since different dynamic behaviors of the LB systems were found at similar static contact angles,6,9 we believe that the velocity dependent boundary condition is more appropriate for hydrodynamic description of the LB deposition. Following Petrov and Radoev,2 we assume that the dependence ΘC(U) could be presented by the molecular-kinetic theory of Blake and Haynes,
cos ΘC ) cos Θ0 +
U 2nkT arsh γ 2Kλ
( )
(2)
where K is the equilibrium frequency of adsorptiondesorption of an amphiphilic molecule at the contact line, λ is the mean distance between the adsorption centers, n is their surface concentration (n ≈ λ-2), k is the Boltzmann constant, T is Kelvin’s temperature, and Θ0 is the static contact angle. This assumption simplifies the reality because LB multilayers are successfully built up when the adsorption of the last layer atop the previous one is irreversible. However, for many systems this requirement is not fulfilled so that one introduces “drying” pauses after the upstroke deposition in order to facilitate the following downstroke deposition. Molecular-Hydrodynamic Theory of LB Deposition Our computational procedure utilizes the lubrication approximation for a wedge like flow following the approach of Voinov.12 The velocity of the liquid parallel to the solid wall vx for nonslip boundary conditions at both the solidliquid and liquid-gas interfaces is
vx(z)/U ) -1 - 6[(z/h)2 - (z/h)]
(3)
The coordinate x is parallel to the solid surface, and z is perpendicular to it. The balance between hydrodynamic and capillary pressure at the fluid interface is
p ) p0 + γK
(4)
where p0 is the pressure in the gas phase and K ≈ -Θ dΘ/dh is the local curvature, with Θ being the local slope of the fluid interface and h the local thickness of the dynamic meniscus. To find the shape of the fluid interface, we substitute the expression for the pressure in the liquid wedge
µU p ) p0 + 12 hΘ
(5)
in eq 4 and integrate the result. In this way one obtains the profile of the fluid interface in the hydrodynamically affected region of the moving meniscus:
Θ3(h) ) ΘC3 - 36(µU/γ) ln(h/hmin)
(6)
The parameter hmin is the minimum thickness (of molecular dimensions) below which the continuum hydrodynamics is inapplicable.12 The angle ΘC ) Θ(hmin) is defined as a local slope of the fluid interface at this particular meniscus thickness.
Molecular-Hydrodynamic Description of LB Deposition
Langmuir, Vol. 14, No. 9, 1998 2493
The quasi-static dynamic contact angle is defined at the boundary hqs between the region of viscous deformation and the quasi-static region (see Figure 2a):
Θqs3 ) ΘC3 - 36(µU/γ) ln(hqs/hmin)
(7)
The thickness hqs can be determined by matching the profiles of the fluid interface in the hydrodynamic and quasi-static regions at h ) hqs. The differential equation for the hydrodynamically deformed fluid interface
µU 1 d3h + 12 )0 γ h2 dx3
( )
(8)
enables us to obtain an approximate expression for the local curvature d2h/dx2 at h ) hqs:
( ) d2h dx2
∝
h)hqs
( ) µU γ
2/3
1 hqs
(9)
The curvature of the quasi-static region at h ) hqs is
( ) d2h dx2
)
h)hqs
( ) 2Fg γ
1/2
(1 - sin Θqs)1/2
(10)
Equations 9 and 10 yield the maximal thickness of the viscous deformation of the meniscus:
hqs) β
(Fgγ ) (µUγ ) 1/2
2/3
(1 - sin Θqs)-1/2
(11)
Following Derjaguin,25 we assume that the numerical constant β ≈ 2. Equation 11 shows that hqs depends on the solid substrate velocity U through the capillary number µU/γ and the velocity dependence of Θqs given by eq 7. The set of eqs 7 and 11 defines the dependence of Θqs on U and material properties of the system. The corresponding expressions for the extrapolated dynamic contact angle, Θext, combine eqs 7, 11, and 12,
F[Θext(U)] ) F[Θqs (U)] where F means
F(ξ) ) ln
[
(1 - sin ξ)1/2
]
21/2 - (1 + sin ξ)1/2
(Fgγ )
1/2
hqs
(12)
- [2(1 + sin ξ)]1/2 (13)
To complete the solutions for Θqs and Θext, one has to specify the dynamic behavior of ΘC. As mentioned in the former section, we assume that ΘC ) ΘC(U) and that this dependence follows the Blake-Haynes equation (2). Thus, the sets of eqs 2, 7, 11 and 2, 7, 11, 12 give the molecularhydrodynamic description of the LB deposition in terms of quasi-static or extrapolated dynamic contact angles, respectively. They take into account both the viscous friction in the receding meniscus via eqs 7 and 11 and the adsorption interactions in the three-phase contact zone through eq 2. Comparison between Molecular-Hydrodynamic Theory and Experimental Studies of LB-Deposition Kinetics Figure 3 presents the experimental dependence of Θapp versus U for deposition of multilayers of arachidic acid at (25) Derjaguin, B. V. Uspekhi kolloidnoi khimii; Nauka: Moscow, 1973; p 30 (in Russian).
Figure 3. Experimental dependence of the apparent dynamic contact angle, Θapp, on solid substrate velocity, U, during deposition of arachidic acid from 1 × 10-2 M HCl subsolution at 30 mN/m. Data from ref 2. Curve 1 is plotted according to the molecular-kinetic theory (eq 2) with K ) 9.4 × 104 s-1 and λ ) 11.1 Å and curve 2 represents the hydrodynamic description (eqs 7 and 11) assuming that ΘC is a constant determined as a free parameter.
30 mN/m from an aqueous subsolution of 1 × 10-2 M HCl (data from ref 2). Curve 1 gives the trend of eq 2, showing that molecular-kinetic theory rather well fits the data at low and intermediate velocities, giving reasonable values of K ) 9.4 × 104 s-1 and λ ) 11.1 Å. However, it overestimates the dynamic contact angles at velocities approaching the threshold value, Umax. The value of Umax predicted by that theory via extrapolation of curve 1 to Θapp ) 0° is also strongly overestimated. Curve 2 is plotted according to eqs 7 and 11 assuming that ΘC ) const and leaving ΘC as an additional free parameter. This set of equations represents a pure hydrodynamic description of the LB-deposition process. It can be seen that the fit is good at intermediate and high velocities but underestimates the values of the dynamic contact angles at low velocities. The value of ΘC ) 52° obtained from the fit is also considerably below the experimental static receding angle ΘR ) 80°. The observed discrepancies of the molecular-kinetic and hydrodynamic descriptions of the process are avoided by the molecular-hydrodynamic theory presented in the former section. Figure 4 illustrates this fact via comparison of the same experimental data with the theoretical velocity dependencies of Θqs (curve 1) and Θext (curve 2). Both theoretical curves follow the trend of the experimental Θapp(U) dependence in the entire velocity range from 0 to Umax. The values of the parameters obtained from the best fit of eqs 2, 7, and 11 giving the minimum χ2 value (curve 1) are reasonable: K ) 1.1 × 105 s-1, λ ) 10.8 Å, and hmin ) 10 Å. Curve 2 is plotted by introducing the same values of K, λ and hmin in eqs 2, 7, 11, and 12. Figure 5 compares another set of experimental data Θapp(U) with theoretical dependencies of Θqs(U) (curve 1) and Θext(U) (curve 2). Here deposition of arachidic acid at 30 mN/m from an aqueous subsolution of 1 × 10-4 M BaCl2 at pH 6.4 is considered (data from ref 6). As in Figure 4 both theoretical dependencies satisfactorily fit the experimental data in the whole velocity range. The parameters obtained from the best fit are K ) 8.9 × 104 s-1, λ ) 17.1 Å, and hmin ) 10 Å. The different values of K and λ are obviously due to the modification of the carboxylic groups of the second system because of the binding of Ba2+ counterions to them. The same value of the cutoff thickness, hmin, reflects the hydrodynamic
2494 Langmuir, Vol. 14, No. 9, 1998
Figure 4. Comparison of the molecular-hydrodynamic dependencies of the quasi-static, Θqs (curve 1) and the extrapolated, Θext (curve 2), dynamic contact angles on solid substrate velocity U with the experimental data from Figure 3. The values of the parameters obtained from the best fit are K ) 1.1 × 105 s-1, λ ) 10.8 Å, and hmin ) 10 Å.
Figure 5. Comparison of the molecular-hydrodynamic dependencies Θqs(U) (curve 1) and Θext(U) (curve 2) with the experimental data for Θapp(U) obtained for deposition of arachidic acid from 1 × 10-4 M BaCl2 subsolution at pH 6.4; data from ref 6. K ) 8.9 × 104 s-1, λ ) 17.1 Å, and hmin ) 10 Å.
similarity of these systems, which were deposited at the same surface pressure and viscosity and density of the subsolutions. As can be seen from Figures 4 and 5, both the Θqs(U) and the Θext(U) dependencies follow the experimental Θapp(U) data better than the molecular-kinetic and hydrodynamic theories separately. As expected, they deviate from each other at high velocities where viscous friction is more pronounced. However their differences of 3-4° are within the scatter of the experimental points. On the other hand previous investigations with pure liquids15-17 came to the conclusion that for a receding liquid the values of Θapp stay between those of Θqs and Θext, i.e., Θqs g Θapp g Θext. Therefore, for the systems considered here, the values of Θqs and Θext should coincide with Θapp if reproducibility of determination of Θapp is (2-3°. This conclusion could probably be extended for all cases of LB deposition from dilute aqueous solutions because of their similar hydrodynamic behaviors. Thus one can conclude that hydrodynamics plays an important role in the LB-deposition process and ignoring it at high velocities leads to disagreement between theory
Petrov and Petrov
Figure 6. Fit of the molecular-hydrodynamic dependence of Θqs(U) to the experimental data of Θapp(U) from Figure 4 using the approximation hqs ≈ (2γ/Fg)1/2. The fit yields values of K ) 1.0 × 105 s-1, λ ) 11.1 Å, and hmin ) 10 Å, which are almost the same as those found in Figure 4.
and experiment (see Figure 3). However, when diluted aqueous subsolutions are used, the viscous deformation of the moving meniscus yields rather small differences between Θqs and Θext within the usual experimental scatter of (2-3°. For such cases it is easier to compare the experimental data Θapp(U) with the theoretical Θqs(U) dependence because calculation of the latter avoids the use of eq 12. The above conclusion does not hold for LB deposition from glycerol26 or other subsolutions of high viscosity or at surface pressures close to the surface tension of pure water, i.e., at very low values of γ (e.g., deposition of phospholipids). In both cases the capillary number, µU/ γ, becomes large and the viscous deformation of the dynamic meniscus becomes significant.27 Equation 7 shows that hqs stays under logarithm so that its variation with velocity does not strongly affect the dependence of Θqs on U. To check this, we can use the approximation
hqs ≈ (2γ/Fg)1/2
(14)
which neglects the influence of µU/γ and Θqs(U) on hqs. Figure 6 shows the best fit of the Θqs(U) dependence obtained at this approximation to the data from Figure 4. It shows that substitution of eq 11 by eq 14 practically does not change the agreement between theory and experiment. The values of the parameters obtained, K ) 1.0 × 105 s-1, λ ) 11.1 Å, and hmin ) 10 Å, are very close to those from the best fit in Figure 4, K ) 1.1 × 105 s-1, λ ) 10.8 Å, and hmin ) 10 Å. Slightly worse agreement is found for the system from Figure 5; the lower approximation (eq 14) yields values of K ) 3.0 × 104 s-1, λ ) 19.9 Å, and hmin ) 10 Å, while those obtained with eq 11 are K ) 8.9 × 104 s-1, λ ) 17.1 Å, and hmin ) 10 Å. Dissipation of Energy in the Nonhydrodynamic Zone and in the Moving Meniscus The elementary act of the LB deposition takes place in the immediate vicinity of the three-phase contact line, i.e., at h < hmin, where the monolayer being transferred (26) Richard, J.; Barrauld, A.; Vandevyever, M.; Ruaudel-Teixier, A. Thin Solid Films 1988, 159, 207. (27) Chen, Q.; Rame, E.; Garoff, S. Phys. Fluids 1995, 7, 2631.
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Langmuir, Vol. 14, No. 9, 1998 2495
in this study. For the deposition of arachidic acid from 1 × 10-2 M HCl subsolutions (Figure 7a) the nonhydrodynamic dissipation at Umax is 32.2 erg/(cm‚s) while for 1 × 10-4 M BaCl2 subsolution ΦNH is only 17.5 erg/(cm‚s). This fact explains the sensitivity of the threshold velocity Umax to the nature of the hydrophilic headgroups observed in a series of experimental studies.2,6,8,9 The decrease of ΦNH in the second case seems to be related to the substitution of 40% of the carboxylic groups by barium carboxylate groups that occurs under the particular experimental conditions.28 Curiously, the value of ΦNH of the barium carboxylate system taken at the same velocity is 38% of the nonhydrodynamic dissipation of the pure arachidic acid system. Since the solubility of the divalent soaps in water is much less than the solubility of the corresponding fatty acids, we can speculate that the smaller value of ΦNH results from easier dehydration of the headgroups along the three-phase contact line when mixed arachidic acid-barium arachidate monolayers are deposited. This result quantifies our previous conclusions8,9 based on comparison of Umax for LB systems with different hydration of the headgroups by now relating the values of Umax to such a well-defined quantity as the nonhydrodynamic dissipation of energy at the moving three-phase contact line, ΦNH. Conclusions Figure 7. Dependencies of the energy dissipation in the nonhydrodynamic region, ΦNH (curves 1), and viscous dissipation, ΦV, in the bulk of receding meniscus (curves 2) on the solid substrate velocity U: (a) deposition of arachidic acid from 1 × 10-2 M HCl; K ) 1.1 × 105 s-1, λ ) 10.8 Å, hmin ) 10 Å; (b) deposition of arachidic acid from 1 × 10-4 M BaCl2 subsolution at pH 6.4, K ) 8.9 × 104 s-1, λ ) 17.1 Å, hmin ) 10 Å.
adsorbs onto the last deposited monolayer. This process gives rise to a nonhydrodynamic dissipation of energy that can be evaluated from the molecular-kinetic theory. Its value per unit contact line length can be written as:
ΦNH(U) ≈ 2nkTU arsh
U (2Kλ )
(15)
The viscous energy dissipation in the region hmin < h < hqs can be evaluated from the expression
ΦV(U) )
∫xx
qs
min
dx
∫0hdz [µ(∂vx/∂z)2]
(16)
Using eq 3 for the velocity profile and eq 6 for the shape of the fluid interface in the hydrodynamic region, we obtain
ΦV(U) )
γU [ΘC2(U) - Θqs2(U)] 2
(17)
Substitution of eq 2 for ΘC(U) and eqs 7 and 11 for Θqs(U) in eq 17 gives the velocity dependence of the viscous dissipation ΦV. Figure 7 compares ΦNH and ΦV for the two LB systems presented in Figures 4 and 5. The ΦV(U) dependence does not exhibit the unbounded increase at Umax shown in Figure 9 of ref 11. This substantial difference results from relaxing the approximation of the hydrodynamic region as a solid wedge in the present calculation and taking into account the variation of its local slope according to eq 6. It is rather interesting that the nonhydrodynamic dissipation prevails in the entire velocity range and that it significantly differs for the two LB systems considered
The molecular-hydrodynamic theory presented in this study relates the velocity of the LB deposition (that is also the velocity of the three-phase contact line) to dynamic contact angle and material properties of the system: activation free energy of adsorption and distance between adsorption centers on the solid surface, static wettability, viscosity, density, and surface tension of the liquid phase. In contrast to the pure molecular-kinetic and hydrodynamic theories, molecular-hydrodynamic theory fits the experimental dependence of contact line velocity on dynamic contact angle in the entire velocity range up to Umax, at which the liquid film is entrained during the upstroke stage of the LB deposition. The theory utilizes rigorously defined dynamic contact angles that can be precisely experimentally determined. In this sense it avoids the subjective comparison of the theoretical and experimental quantities. This fact becomes important when LB deposition is performed from subsolutions with high viscosity (glycerol) or at very high surface pressure, i.e., very low surface tension (phospholipids) when capillary numbers become large enough. Although the consideration of the viscous friction in the receding meniscus proves important, the hydrodynamic deformation of the liquid-air interface is small when dilute aqueous subsolutions are used. Under such conditions the theoretically defined quasi-static and extrapolated dynamic contact angles differ by only 3-4° and both agree within the experimental scatter with the apparent dynamic contact angle, which has been most often determined in the previous experimental studies. The comparison of the nonhydrodynamic dissipation in the three-phase contact zone and the viscous dissipation in the moving meniscus estimated on the basis of the molecular-hydrodynamic theory shows that the nonhydrodynamic dissipation prevails in the entire velocity range up to Umax. This fact explains the observations in many previous publications that Umax is sensitive to the (28) Petrov, J. G.; Kuleff, I.; Platikanov, D. J. Colloid Interface Sci. 1982, 88, 29.
2496 Langmuir, Vol. 14, No. 9, 1998
nature of the hydrophilic headgroups and their modification via polyvalent counterion adsorption. Dissipation analysis of two experimentally studied LB systems for which deposition of arachidic acid occurs from HCl and BaCl2 subsolutions shows that the nonhydrodynamic dissipation in the second system is proportional to the degree at which the carboxylic groups bind Ba2+ counterions. This fact and the well-known low solubility
Petrov and Petrov
of Ba soaps in water allow us to hypothesize that the nonhydrodynamic dissipation during the LB deposition is related to dehydration of the hydrophilic headgroups of the monolayer. Such a conclusion has been reported in some of our previous investigations, but there it was based on comparison of Umax in different LB systems. LA971011H
