J. Phys. Chem. B 2008, 112, 11333–11340
11333
Influence of Ion Transfer Kinetics on the Composition of Langmuir-Blodgett Films V. I. Kovalchuk,† M. P. Bondarenko,† E. K. Zholkovskiy,† and D. Vollhardt*,‡ Institute of Biocolloid Chemistry, 03142, KieV, Ukraine, and Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam/Golm, Germany ReceiVed: April 18, 2008; ReVised Manuscript ReceiVed: June 6, 2008
The effect of ion transfer kinetics on the ionic composition of Langmuir-Blodgett (LB) films formed by charged monolayers is analyzed. The dynamic regimes of the LB deposition are considered by taking into account the competitive adsorption of several counterions having different diffusivities, valences, binding constants, and bulk concentrations. It is shown that the composition of deposited films should change with the deposition rate. At lower deposition rates, the ion with higher binding constant is more represented within the deposited monolayer in comparison to the higher deposition rates. At low deposition rates, the ratio of counterion amounts within the LB films is the same as that within the floating monolayer excluding the ions within the diffuse layer. At high deposition rates, the ratio of the counterion amounts is the same as that within the floating monolayer when the potential-determining counterions within the diffuse layer are taken into account. Introduction During the past decade, a substantial interest has been focused on obtaining coatings whose local physicochemical properties change from point to point and form an ordered planar nanostructure (nanopattern).1–4 One of the intensively studied methods of the nanopatterning utilizes the self-assembling behavior that is observed during the Langmuir-Blodgett (LB) process.5–12 The main advantage of this method is related to the possibility of rapid patterning a large area with lateral resolution in the nanometer range. Recently, increased interest in patterned substrates, prepared by using the LB technique, is motivated by the perspective to use them for a variety of applications, for instance, in microfluidics,13 microlithography,14 and template-assisted electrodeposition of nanowires or nanoparticles,15 or in directed deposition of functional molecules and nanoparticles.16–18 Patterns with desired morphology that can be effectively controlled at micro- and nanoscales can be formed during the transfer of a Langmuir monolayer to a solid substrate as consequence of wetting instabilities observed under certain process conditions. Well-known examples are the patterns produced by transfer of phospholipid monolayers onto oxidized silicon substrates.19,20 In such a case, striped patterns are formed by the phospolipid layers deposited in more or less condensed two-dimensional phases and are a result of local variations of the monolayer molecular packing density near the contact line due to interaction with the substrate. Another type of the striped patterns have been found using fatty acid (arachidic acid) monolayers for the LB deposition process under special experimental conditions (subphase pH of ∼5.7 and CdCl2 concentrations in the range of 0.2-0.5 mM).21,22 In the latter experiments, the obtained stripes are composed of either a fatty acid or its cadmium salt and run in parallel to the meniscus during the LB film transfer. The skeletonization of the LB films in cyclohexane or benzene allows one to remove * To whom correspondence should be addressed. † Institute of Biocolloid Chemistry. ‡ Max Planck Institute of Colloids and Interfaces.
the fraction of the protonized fatty acid and to obtain the grooves having a depth in the range of the double layer thickness of cadmium arachidate. As shown previously,23 the stripe patterns formed by fatty acid monolayers are the result of the concentration polarization effect taking place in the subphase during the film transfer. For obtaining nanopatterns with a required structure, it is an important advantage to use charged (e.g., fatty acid) instead of electroneutral (e.g., DPPC) monolayers. In the case of the use of charged monolayers, the monolayer composition can be controlled by regulating the amounts of the different counterions within the patterned structures. Such a regulation becomes possible due to the concentration polarization effect discussed in our previous studies.23–27 Accordingly, when several sorts of counterions having different binding constants are present in the solution, competitive adsorption takes place in the system. Under dynamic conditions, electrochemical potential gradients of the ions are created in the meniscus region. The gradients increase as the monolayer transfer rate increases. Consequently, the local changes of counterion concentrations produced lead to a shift of their adsorption equilibria and, hence, to changes in the monolayer composition with the deposition rate. Thus, composition, structure, and morphology of the LB film can be regulated by changing the deposition rate and the subphase composition. Thus, the objective of the present paper is to study how deposition rate and subphase composition affect the ionic composition of the LB film formed by charged monolayers. To this end, in consideration of the dynamic regime of the LB deposition, we analyze the competitive adsorption of several counterions having different diffusivities, valences, binding constants, and bulk concentrations. The analysis is performed on the basis of a mathematical model developed in our previous studies24–27 for describing the balance of the individual ions within both the subphase and the monolayer in close vicinity to the three-phase contact line. This model describes adequately the existing experimental data on the dependence of the maximum monolayer deposition rate on the subphase composi-
10.1021/jp8033984 CCC: $40.75 2008 American Chemical Society Published on Web 08/15/2008
11334 J. Phys. Chem. B, Vol. 112, No. 36, 2008
Figure 1. Electrical double layers overlap and convective flow in vicinity of the three-phase contact line (lD is the Debye length, θ the contact angle).24
tion and explains the observed very large meniscus relaxation times after stopping the substrate upward motion. Description of the System Ionizable amphiphilic monolayers spread at the air-water interface bear a net electric charge due to dissociation of ionizable groups of the molecules forming the monolayer.28,29 The interfacial charge density depends on the subphase composition because of counterion binding. Under equilibrium conditions, the number of different counterions bound at the monolayer depends on their concentration in the bulk phase and the association (binding) constant but also on the presence of other ions. During the transfer to the substrate surface, the ionized monolayers should bind some amounts of counterions from the subphase to neutralize the surface charges. Therefore, during the deposition process, the monolayer composition can change to a certain extent. However, it is usually assumed that, within the experimental error, the composition of the floating monolayer coincide with that of the deposited LB film. The latter assumption is valid for a sufficiently small degree of headgroup ionization in the monolayer. Sometimes, a variation of the monolayer composition during the deposition is taken in an approximate manner into account. Ahn and Franses proposed to assume that the ratio of amounts of the additionally bound counterions coincides with the ratio of their amounts within the initial diffuse double layer.30 Later, the same authors used another assumption. The ratio between the amounts of the additionally bound counterions was supposed to be proportional to the product of their relative concentrations in the diffuse layer and the binding constants.31 It was established that the respective corrections for the composition changes during the monolayer transfer are relatively small for solutions containing Pb2+ or Cd2+ ions but become significant for Ca2+ and Ba2+ ions.30 For an infinitely slow deposition process, the electrical double layers formed at the floating monolayer and the substrate surface remain in equilibrium state. In close vicinity to the contact line, the diffuse double layers overlap with each other (Figure 1), and therefore, the local charge and composition of the monolayer are different from those far from the meniscus.32 Having solved the respective electrostatic problem with accounting for the counterion adsorption equilibrium, we obtained the distributions of ion concentration and electric potential within the meniscus
Kovalchuk et al. region. In immediate vicinity to the contact line, the monolayer is in an electroneutral state, and its composition corresponds to that of the deposited LB film. Accordingly, the solution of the equilibrium problem made it possible to obtain the exact ionic composition of the deposited LB film for an infinitely slow process.32 However, at finite rates of the process, the ion distributions near the contact line should differ from that under equilibrium conditions. Although the ion transfer kinetics is usually assumed to be very fast, it turns out to be insufficient for holding the equilibrium distributions of ions near the contact line. Our previous studies have shown that, during the deposition of a charged monolayer, concentration and electric potential profiles develop within the meniscus region.23–27 This effect is similar to the concentration polarization in selective membranes, disperse systems, or electrode surfaces. The deviations from the equilibrium distributions increase with the deposition rate. Consequently, due to the shift of the adsorption equilibria, the redistribution of the ions near the contact line produces local changes in charge and composition. The latter affects the monolayer adhesion to the substrate and leads to meniscus instability. A microheterogeneous stripelike structure can be formed within the deposited LB film under these conditions. Thus, for finite rates of the process, the monolayer composition becomes dependent on the deposition rate. In particular, when two or more counterions are present in the subphase, their amounts within the deposited film can be altered by changing the deposition rate. The concentration and electric potential profiles discussed above develop in the meniscus region because of the misbalances of the partial ion fluxes transferred during the deposition process through each of the cross-sections of the meniscus region.23–27 The deposited film removes counterions from the subphase in amounts required to hold the film electroneutrality. Under steady-state conditions, to provide electroneutrality of the deposited film, all the counterions distributed within the diffuse layers should follow the monolayer and move along the interface with the exactly same velocity. However, because of the hydrodynamic back flow produced by water expelled from the three-phase contact zone, the counterions in the diffuse layers move slower than the monolayer, on average. Moreover, the counterions in the central part of a thin liquid film, formed between the substrate surface and the monolayer, move back, i.e., toward the bulk solution (Figure 1). Hence, the resulting convective flux of counterions directed toward the contact line is insufficient to maintain electroneutrality. Accordingly, during an initial transitional period, a deficit of counterions is created near the contact line. As a consequence, electrochemical potential gradients of counterions are formed within the meniscus region. These gradients drive the electrodiffusion ion fluxes directed toward the contact line.23–27 Under steady-state conditions, such electrodiffusion ion fluxes provide both the necessary balances of the partial ion fluxes and electroneutrality of the deposited film. In contrast to potential-determining counterions, both the coions and the indifferent counterions are not bound at the surfaces. Consequently, they are not removed from the subphase with the deposited film. At the same time, their convective fluxes though each cross-section of the meniscus are not zero because they are nonuniformly distributed near the moving charged interfaces. Therefore, to provide the steady-state balances of the respective ions, electrodiffusion fluxes of coions and indifferent counterions are formed to be directed toward and outward the contact line, respectively.23–27
Ion Transfer Kinetics and the Composition of LB Films
J. Phys. Chem. B, Vol. 112, No. 36, 2008 11335
As mentioned above, the electrodiffusion ion fluxes appear due to the difference in the electrochemical potential of the ions between the solution at the contact line and the bulk solution. This difference is produced during the initial transitional period after starting the substrate upward motion. After stopping the substrate withdrawal, the distribution of electrochemical potentials should gradually disappear. A very long meniscus relaxation that is observed after stopping the deposition process33–35 corresponds to the slow diffusion processes and, thus, supports the qualitative picture discussed above. Consider now the most illustrative case of small contact angles. For such a case, the electrodiffusion ion fluxes are directed along a thin liquid film formed between the substrate surface and the monolayer. The local equilibrium in the film cross-section establishes much faster than along it, and therefore, the ion electrochemical potential can be considered as a function of only one coordinate directed along the film. For each of the ions, its electrodiffusion flux and electrochemical potential gradient depend on the rate of the ion removal with the deposited film. Also, both the flux and the gradient depend on the convective transfer rate from the bulk solution toward the contact line where the ion is adsorbed at the monolayer. A rough estimation yields
∆µiel ≈ RD[Jidep - (JiS + JiC)] ∆µeli
(1)
where is the difference between the ith ion electrochemical potentials attributed to the bulk solution and the immediate vicinity of contact line, JDep is the ith ion flux produced by its i removal together with the deposited film, and JSi + JCi is the flux of the ith ion transferred toward the contact line due to surface and bulk convection, respectively. To the coefficient RD, we refer to a partial diffusion resistance calculated per unit length of the three-phase contact line. Both the rate of the ion removal with the deposited film and the ion convective fluxes increase with the monolayer transfer rate U, and for relatively small transfer rates, the increase is proportional to the transfer rate.24,25 Accordingly, the differences between the ion electrochemical potentials attributed to the contact line and the bulk are monotonously increasing functions of U. As discussed above, the removal of the potential-determining counterions with the deposited film is faster than the convective transfer from the bulk solution, i.e., the difference in the square brackets in eq 1 is positive. Therefore, their electrochemical potential decreases near the contact line. In contrast, the transfer of the indifferent counterions with the monolayer is absent (the first term in the square brackets in eq 1 is zero), and their electrochemical potential increases near the contact line. The situation becomes more complex when two (or more) potential-determining counterions with different equilibrium binding constants are present in the solution (we consider here a typical situation when the kinetics of counterion binding with ionized groups is high and the local chemical equilibrium holds at the interfaces during the monolayer transfer). At equal bulk concentrations, the counterion with the higher binding constant is preferably adsorbed at the monolayer. Therefore, its flux together with the deposited film is larger, and its electrochemical potential near the contact line is smaller. The counterion with lower binding constant is less adsorbed at the monolayer, and its flux with the deposited film is smaller. Consequently, this counterion is slower removed than it is supplied by convective transfer from the bulk solution, i.e., the difference in the square brackets in eq 1 is negative. Hence, similar to the case of indifferent counterions, electrochemical potential of the ion having a lower binding constant increases near the contact line.
By consideration of two counterions with equal charges (e.g., Pb2+ and Cd2+), then, one having lower binding constants (Cd2+ in this example) accumulates in the solution near the contact line and replaces more ions with higher binding constant there. The latter is the result of the faster removal of the counterions having a higher binding constant. This effect becomes stronger for higher deposition rates. However, with accumulation of the less surface active ion near the contact line, its relative adsorption at the monolayer increases. Accordingly, with increasing the deposition rate, the removal of the counterions having smaller binding constant becomes more intensive. As a result, for the less surface active ion, the difference in the square brackets of eq 1 also becomes positive, i.e., at a certain deposition rate sufficiently high, this counterion begins to behave as a potential-determining (not like indifferent) one. Accordingly, its electrochemical potential near the contact line begins to decrease. After that, the electrochemical potentials of both the highly and lowly active counterions decrease simultaneously with increasing deposition rate. In such a regime, the composition of the deposited monolayer does not change. Thus, according to the above discussion, the composition of the deposited monolayer should change with the deposition rate. At lower deposition rates, the ion with higher surface activity is more represented within the deposited monolayer in comparison to higher deposition rates. At low deposition rates the concentration polarization is insignificant and, under steady state conditions, the ratio of counterion amounts should be the same as that within the floating monolayer excluding the ions within the diffuse layer. At high deposition rates, the ratio of counterion amounts within the deposited monolayer should be the same as that within the floating monolayer to take into account the potential-determining counterions within the diffuse layer. Next, we will address the above-described effects, quantitatively. We will analyze the system with subphases containing the following potential-determining counterions: (i) one monovalent and one bivalent counterion, e.g., hydrogen and cadmium ions; (ii) two bivalent counterions, e.g., cadmium and lead ions; (iii) one monovalent and two bivalent counterions. Mathematical Model In consideration of a stationary regime, the electrodiffusion flux of the ith ion through each cross-section of the thin liquid film near the contact line can be written in the form given in ref 24 as
JiED ) 2U
∑ νik(Xikdep - Xik) - JiC
(2)
k
where U is the velocity of the monolayer and the substrate withdrawal motion, Xik and XDep are the interfacial molar ik concentrations of the kth interfacial complex containing the ith ion within the floating and the deposited monolayer, respectively, and νik are stoichiometric coefficients. For derivation of eq 2, two assumptions were used: (i) the liquid film is symmetrical with an equal charge density and equal composition at both interfaces; (ii) the transfer ratio for the monolayer is close to unity. The convective and electrodiffusion ionic fluxes through the film cross section per unit length of the three-phase contact line are defined as24
JiC )
∫0h V(y)Ci(x, y)dy
JiED ) -Di
∫0
h
Ci(x, y) ∂µiel dy RT ∂x
(3) (4)
where the X axis is directed along the thin liquid film and the Y axis normal to the film. In eqs 3 and 4, h is the local film
11336 J. Phys. Chem. B, Vol. 112, No. 36, 2008
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thickness, µeli ) ziFφ + RT ln Ci, and Ci(x,y) are the local electrochemical potential and molar concentration of the Ith ion, respectively (the standard chemical potential is omitted), Di, and zi are the ith ion diffusion coefficient and valence, φ is the electric potential (defined with respect to the subphase bulk), R is the gas constant, T is the absolute temperature, F is the Faraday constant, and V(y) is the velocity distribution within the film cross section which, in the lubrication approximation, is given as36
y 1 1 V(y) ) U - + 6 2 h 2
(
(
2
))
(5)
According to eq 2, the electrodiffusion flux is given by the difference of two terms: term
2UΣ νikXikdep ) Jidep k
representing the rate of the ith ion removal with the deposited film, and term 2U∑k νik Xik+JCi ) JSi + JCi , representing the rate of the ith ion transfer toward the contact line due to the surface and bulk convection, respectively. Since we assume that, for sufficiently slow depositions, the equilibrium across the thin film is established much faster than along it, all the ion electrochemical potentials are set approximately constant within the film cross-section and dependent on the coordinate x only.24–27 Then, from eq 4 with account for eq 2 one obtains
JiED ) -
Di ∂µiel RT ∂x
∫0
h
∂µiel ∂µiel Ci(x, y)dy ) -λD and ∂x ∂x 1 )2U νik(Xikdep - Xik) - JiC (6) λD k
[
]
∑
where λD ) Di/RT ∫0h Ci(x,y) dy. The set of eq 6 can be integrated numerically.26,27 As a rough estimation we can write
[
∆µiel ≈ RD 2U
∑ νik(Xikdep - Xik) - JiC]
(7)
k
where RD ≈ L/λD ) L(Di/RT ∫0hef Ci(x,y) dy Ci(x,y) dy)-1, L is the diffusion length, and hef is the effective film thickness. The resistance RD depends on the diffusivity of ions, their distribution within the film and the diffusion length. Clearly, eq 1 is another form of eq 7. The composition of a floating monolayer is described by the set of interfacial molar concentrations of the complexes of the ionized groups with the counterions, Xik. By assumption that the respective chemical reactions are sufficiently fast, we can use the equilibrium conditions24–27
XRH ) KHXR-CHS +
whereCSH+
(9)
S XR2M ) KM2(XR-)2CM 2+
(10)
S XRM+ ) KM1XR-CM 2+
(11)
CSM2+
and are the values of the bulk molar concentrations of H+ and M2+ at the interface (M2+ is either Pb2+ or Cd2+). In the literature, there are different opinions whether the bivalent metal ions form a positive RM+ complex or a neutral R2M complex with acidic groups at the interface.30,37,38 Therefore, we consider here different possibilities. The total interfacial amount of fatty acid groups is given by
XR ) XR- + XRH + 2XR2Pb + XRPb+ + 2XR2Cd + XRCd+ (12) For monolayers in a close-packed condensed state, the surface concentration of fatty acid, XR, depends on the surface pressure and on the bulk concentrations only weakly and can be approximated by a constant value. The interfacial charge density is represented as
σ ) -F(XR- - XRPb+ - XRCd+)
(13)
It is convenient to represent the ion concentrations within a film cross-section in a form, similar to equilibrium conditions24,26,27 ˜
Ci(x, y) ) Ciqe(x)e-ziΨ(x,y)
(14)
˜ (x,y) are so-called quasiequilibrium conwhere Cqe i (x) and Ψ centrations and (dimensionless) quasiequilibrium electric potential, respectively. Cqe i (x) has meaning of the ith ion concentration in a thought (virtual) electroneutral solution being in thermodynamic equilibrium with a given point of the diffuse layer. Accordingly, Cqe i (x) satisfies the electroneutrality condition
∑ ziCiqe(x) ) 0
(15)
i
Because of the local equilibrium within the film cross section, the quasiequilibrium concentrations Cqe i (x) are functions of the coordinate x only. Accordingly, the quasiequilibrium electric ˜ /F is a potential distribution what would potential Ψ(x,y) ) RTΨ be under equilibrium conditions within the given film cross section being in thermodynamic equilibrium with an electroneutral solution where the ith ion concentration is Cqe i (x). The quasiequilibrium quantities (concentration Cqe and electric i potential Ψ) and the potential Φ ) φ - Ψ can be used as a new set of variables instead of Ci and φ. Note, one of Cqe i is eliminated while making use of condition of eq 15. Consequently, the number of independent variables remains the same. While accounting for eq 911) and (14), expressions (12) and (13) take the form ˜
νik Xik ) KikXRR-ikCiS
XR-
(8)
where Kik are the equilibrium constants, the interfacial molar concentration of the dissociated groups (R-), Rik and νik the stoichiometric coefficients of the chemical reaction, and CiS the bulk concentrations of the ions at the interface. For multivalent ions several complexes of different stoichiometry are possible with the interfacial ionized groups. We will consider a fatty acid monolayer (RH) in contact with the electrolyte solution containing two bivalent metal ions, e.g., Pb2+ and Cd2+, a monovalent anion, A-, and a certain amount of an inorganic acid, HA. For this system the chemical equilibrium conditions are specified as
˜
qe -2ΨS XR ) XR-(1 + KHCHqe+e-2ΨS + 2KPb2XR-CPb + 2+e ˜
˜
qe -2ΨS qe -2ΨS + 2KCd2XR-CCd + KPb1CPb 2+e 2+e ˜
qe -2ΨS ) KCd1CCd 2+e ˜
(16)
˜
qe -2ΨS qe -2ΨS - KCd1CCd ) (17) σ ) -FXR-(1 - KPb1CPb 2+e 2+e
˜ S is the normalized quasiequilibrium electric potential where Ψ at the interface. By substitution of the new variables into eq 6, one obtains a set of ordinary differential equations written with respect to the 24,26 As for the concentrations Cqe i (x) and the potential Φ(x). quasiequilibrium electric potential, Ψ, according to refs 24 and
Ion Transfer Kinetics and the Composition of LB Films 26, it satisfies the Poisson-Boltzmann equation written for the film cross section having the coordinate x. In such an equation, Cqe i (x) turns out to be at the same place as that occupied by the bulk concentration of the ith ion in the usual Poisson-Boltzmann equation. Consequently, using eq 16 and 17, one can determine the local surface charge density that will be a function of the coordinate x. The above-discussed problem can numerically be solved and the distributions Cqe i (x), Φ(x), and Ψ(x,y) can be obtained for addressing the meniscus region at a given velocity U, contact angle, θ, equilibrium constants Kik, surface concentration of the fatty acid XR, ion diffusion coefficients Di, and ion bulk concentration C∞i fixed at a certain distance x ) L from the contact line. The length scale parameter L is a characteristic distance of the order of 1 mm for which the meniscus curvature becomes significant.24,26,27 This should be a self-consistent solution for which the obtained interfacial concentrations Xik, while approaching the contact line, should approach the values attributed to the deposited monolayer Xdep ik . Next, the distributions of the ion concentrations, Ci(x,y), the electric potential φ(x,y), the electrochemical potentials µeli (x), and the convective and electrodiffusion ionic fluxes JCi (x) and JED i (x) can be calculated. Finally, the composition of the deposited monolayer as a function of the velocity U and the bulk concentrations C∞i can be found. The formulated model assumes a small slope of the meniscus profile (dh/dx < 1). The meniscus profile can be obtained from the consideration of the dynamic force balance which should also include van der Waals and electrostatic forces. For the sake of simplicity, we assume constant slope of the meniscus profile, which has been widely used in the works on wetting dynamics. It is, however, important to note that the effect of concentration polarization should be observed for any meniscus profile (and even in the case that the slope is not small) as follows directly from the consideration of the ion balance. The assumption of constant slope, which is also used in our previous studies, is a reasonable simplification which allows us to consider the most important aspects of the effects.
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∞ 2+ ∞ 2+ Figure 2. Diagrams of surface charge density σ(CCd , CPb ) and ∞ 2+ ∞ 2+ S surface potential φeq(CCd , CPb ) for a fatty acid monolayer under equilibrium conditions (both R2M and RM+ complexes): (a) lines of constant surface charge density (in percent to FXR) correspond to 0, 0.1, 0.25, 0.5, 0.75, 1.0, 1.25, and 1.5% (from top to bottom); (b) lines of constant surface potential (normalized by F/RT) correspond to 0, 0.01, 0.2, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 (from top to bottom); the dotted lines correspond to zero charge.
Results and Discussion The numerical results presented in this section are obtained for the following parameters, used in our previous studies:23–27 DH+ ) 9.34 × 10-9 m2/s, DCd2+ 7.2 × 10-10 m2/s, DCl- 2.04 × 10-9 m2/s, KH ) 6.54 × 104 dm3/mol, KCd1 ) 15.5 dm3/mol, and KCd2 ) 2.5 × 109 dm5/mol2, XR ) 8.3 × 10-6 mol/m2 (these parameters correspond to the conditions of the experiments dealing with the variation of the deposited monolayers composition with subphase pH32). For the lead ions we used the following parameters: DPb2+ 9.3 × 10-10 m2/s, KPb1 ) 120.2 dm3/mol, and KPb2 ) 1.09 × 1012 dm5/mol2. Similar to cadmium ions, the same constant KPb1 was chosen for short-chain fatty acids,37,39 and the constant KPb2 was taken from ref 30. Equilibrium Conditions. Figure 2 represents the equilibrium ∞ 2+ ∞ 2+ diagrams of surface charge density σ(CCd , CPb ) and surface S ∞ 2+ ∞ 2+ potential φeq(CCd , CPb ) for an individual fatty acid monolayer in contact with aqueous subphase containing Pb2+ and Cd2+ ions of different concentrations. Two types of complexes (R2M and RM+) are taken into account for bivalent metal ions. These diagrams are very similar to those for a fatty acid monolayer in contact with subphases containing H+ and Cd2+ ions, as reported in ref 26. At small Pb2+ concentrations, the surface charge density passes through a maximum with increasing cadmium concentration. The same is observed at high pH values in the H+/Cd2+ system.26 The increase of the surface charge density with increasing cadmium concentration is explained by the
monolayer conversion from the lead (or hydrogen) form into the cadmium form. Thus, at small concentrations, the Cd2+ ions behave as indifferent counterions and replace the more surface active Pb2+ ions within the diffuse layers. Such behavior is similar to that of sodium ions discussed in ref 27 dealing with the H+/Na+ system. In contrast, at higher concentrations, the Cd2+ ions behave as potential-determining counterions. Accordingly, the surface charge density decreases with increasing the cadmium concentration. In the presence of two types of complexes (R2M and RM+), the surface charge density is much lower than its maximum ∞ 2+ possible value, FXR. For CPb > 10-5 M, it is lower than 2% of FXR (Figure 2). In the presence of R2M complexes only, the charge may be several times higher but still much lower than FXR. Similar results are obtained also for the H+/Cd2+ system.26 In Figure 3 the composition diagrams are presented obtained for LB films under the assumption of infinitely slow deposition. The lines of constant composition correspond to constant fractions of the cadmium salt of the fatty acid within the dep + deposited LB films (PCd ) [2/XR](X Rdep 2Cd + X RCd ) ) const) for two particular systems: the H+/Cd2+ system (Figure 3a) and the Pb2+/Cd2+ system (Figure 3b). Two types of complexes (R2M and RM+) are assumed in both cases. Note that, because dep of electroneutrality of the deposited films, we have Xdep R- ) XRPb+ dep + + XRCd , and according to eq 12
11338 J. Phys. Chem. B, Vol. 112, No. 36, 2008 dep dep dep dep XR ) XRH + 2(XRdep + XRPb + + XR Cd + XRCd+) 2Pb 2
Kovalchuk et al.
(18)
By assumption either concentration of one of the ions or pH is constant, from the diagrams in Figure 3 one obtains typical S-shaped curves describing the dependence of the film composition on the second ion concentration. Such S-shaped dependencies are usually observed in experiments, in particular for film composition vs pH dependencies.40–44 In the case of infinitely slow deposition, the ratio of the counterion amounts in the deposited monolayer is almost the same as that within the floating monolayer (without the ions within the diffuse layer),32 provided the surface charge density is as small as in the presence of Pb2+ and Cd2+ ions. However, the difference of monolayer compositions can be higher in the presence of ions with lower binding constant when the surface charge density in the floating monolayer is higher. Subphase Containing H+ and Cd2+ Ions. In Figure 4, the impact of the substrate speed on the LB film composition is shown for the subphase containing H+ and Cd2+ ions. Here, the fractions of the cadmium salt of fatty acid PCd is presented as a function of pH for two limiting cases: (a) infinitely slow deposition and (b) fast deposition for which the composition of the film becomes independent of the deposition rate. It is seen that, in the presence of two types of complexes (R2M and RM+), the effect of the deposition rate is negligible, whereas, in the presence of R2M complexes only, the effect is essential. This difference is explained by the fact that, in the presence of RM+ complexes, the quasiequilibrium potential ΨS at the contact line cannot increase above a certain limit, whereas in the presence of only R2M, the potential ΨS can increase without limit.32 With increasing the potential ΨS, the local concentration of the Cd2+ ions near the interface increases much more rapidly than the concentration of H+ ions. Accordingly, the amount of Cd2+ ions within the deposited film increases as well. The effect of the deposition rate becomes more substantial when the amount of cadmium ions in the deposited film is small. In this case, the deposited film removes mainly H+ ions, and with increasing deposition rate, Cd2+ ions are accumulated near the contact line (the difference in the square brackets in eq 1 is negative). At higher amounts of cadmium ions in the deposited film, they are removed faster and, thus, accumulated weaker near the contact line. It was demonstrated in our previous studies23–27 that the decrease of the quasiequilibrium counterion concentrations near the contact line to a critical value can lead to meniscus instability. Under certain conditions, the meniscus instability can result in a periodical change of the composition of the deposited LB film and the film acquires striped morphology which can be affected by both the substrate speed and the subphase composition. Subphase Containing Pb2+ and Cd2+ ions. Figure 5 shows the variation of profiles of quasiequilibrium concentrations of Cd2+ and Pb2+ ions in the meniscus region with the substrate speed (in fact, the profiles depend on the ratio of substrate speed and contact angle26,27). It is seen that, near the contact line (at small film thicknesses h), the concentration of lead ions is always smaller than in the bulk and decreases monotonously with increasing the velocity (the difference in the square brackets in eq 1 is always positive). In contrast, the concentration of cadmium ions initially increases with the substrate speed up to a certain value (while the difference in the square brackets in eq 1 is negative, curves 1-3 in Figure 6), and then it decreases (when the difference in the square brackets in eq 1 becomes positive, curves 4-6). The increase of the counterion concentration with the deposition rate is typical for indifferent counterions
Figure 3. Diagram of composition of LB film deposited at U ) 0 ∞ 2+ from the subphase containing (a) H+ and Cd2+ ions (CPb ) 0) and (b) Pb2+ and Cd2+ ions, (CH∞+) 0) (both R2M and RM+ complexes); lines of constant composition correspond to (a) PCd ) 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95 (from bottom to top) and (b) PCd ) 0.001, 0.005, 0.02, 0.09, 0.33, 0.67, and 0.91 (from top to bottom).
Figure 4. Composition of LB film deposited from the subphase ∞ ∞ 2+ 2+ ) 0): 1 and containing H+ and Cd2+ ions (CCd ) 0.25 mol/m3, CPb 1′, both R2M and RM+ complexes; 2 and 2′, only R2M complex; full lines (1 and 2), infinitely slow deposition; dashed lines (1′ and 2′), deposition at a high speed.
(cf. Figure 5a in ref 27). However, at higher deposition rate the cadmium ions behave already as potential-determining ions so that their concentration decreases. This phenomenon was already explained above. With increasing cadmium concentration and decreasing lead concentra-
Ion Transfer Kinetics and the Composition of LB Films
Figure 5. Profiles of quasiequilibrium concentrations of Cd2+ (full lines) and Pb2+ (dashed lines) in the meniscus region for different deposition rates: U/θ ) 0.05 mm/s (1), 0.1 mm/s (2), 0.25 mm/s (3), ∞ ∞ 2+ ) CPb2+ ) 0.25 0.5 mm/s (4), 2.5 mm/s (5), and 5 mm/s (6) (CCd mol/m3; both R2M and RM+ complexes).
J. Phys. Chem. B, Vol. 112, No. 36, 2008 11339
Figure 7. Composition of LB film deposited from the subphase containing H+, Pb2+, and Cd2+ ions: curve 1, PCd; curve 2, PCd + PH ∞ 2+ ∞ 2+ (PPb ) 1 - (PCd + PH)) (CCd ) 0.25 mol/m3, CPb ) 0.025 mol/m3, both R2M and RM+ complexes, U ) 0).
Figure 8. Variation of composition of LB film with deposition rate and C∞H+ for the subphase containing C∞Cd2+ ) 0.25 mol/m3,C∞Pb2+ ) 0.025 mol/m3 (both R2M and RM+ complexes): curve 1, U/θ ) 0; curve 2, U/θ ) 0.5 mm/s.
Figure 6. Variation of composition of LB film with the deposition ∞ 2+ ∞ 2+ rate for the subphase compositions:(a) CCd ) CPb ) 0.25 mol/m3, ∞ 2+ ∞ 2+ 3 3 (b) CCd ) 0.25 mol/m , CPb ) 0.025 mol/m (both R2M and RM+ complexes).
tion within the subphase at the contact line, the ratio of the counterion amounts within the deposited monolayer changes as well. Such a change is seen in Figure 6, where the fraction of the cadmium salt of fatty acid as a function of the deposition rate is presented for two different bulk concentrations of lead ions. The increase of the fraction of cadmium ions results in their faster removal from the subphase with increasing deposition rate. Accordingly, their accumulation near the contact line reduces, and the increase of fraction of cadmium ions becomes slower and then disappears at higher deposition rate. By
comparison of the data of Figures 5 and 6a, one can see that, after the concentration of cadmium ions near the contact line begins to decrease (U/θ > 0.5 mm/s), the film compositions practically do not change with the deposition rate. In this regime the film is deposited with the composition, which coincides with the ionic composition of the floating monolayer taken with its diffuse layer. That means that, at higher deposition rates, the first assumption of Ahn and Franses becomes valid.30 Subphase Containing H+, Pb2+, and Cd2+ Ions. In Figure 7, the equilibrium composition of LB film is presented for a ternary system (with H+, Pb2+, and Cd2+ ions) varying with the concentration of hydrogen ions. Curve 1 represents the fraction of the cadmium salt of fatty acid, whereas curve 2 represents the sum of fractions in cadmium and hydrogen ions. The fraction of the lead ions can be quite easy obtained as the difference PPb ) 1 - (PCd + PH). With increasing hydrogen concentration, the two fractions in cadmium and lead forms decrease. Figure 8, shows the variation of LB film composition with both the deposition rate and hydrogen ion concentration. It is seen that the ratio of cadmium to lead amounts within the deposited monolayer can be increased either by increasing hydrogen ion concentration or by increasing deposition rate. Again, the increase of the cadmium amount with increasing
11340 J. Phys. Chem. B, Vol. 112, No. 36, 2008 deposition rate is the result of the accumulation of cadmium ions near the contact line. Conclusions It can be concluded that the kinetics of ions redistribution in close vicinity of the three-phase contact line is an important factor influencing the properties of the deposited LB film. Under dynamic conditions the local subphase composition near the contact line can change because of the concentration polarization effect. This results in a change of composition and structure of the deposited film with the monolayer transfer rate. In the presence of two potential-determining counterions, the counterions having a lower adsorption constant behave differently at low and high deposition rates. At small deposition rates, they accumulate in the subphase near the contact line and replace the more chemically active ions there, i.e., they act as indifferent counterions,27 whereas at high deposition rates, their behavior becomes similar to that of counterions having a higher interaction constant whose electrochemical potential and quasiequilibrium concentration decrease near the contact line. Thus, during the Langmuir wetting process, the kinetics of the particular counterion transfer defines the number of different counterions within deposited LB films and, therefore, affects composition and structure of these films. The concentration polarization effect yields an additional tool for control and modification the LB film properties. The obtained results can be applied in experimental studies intended for preparing nanostructured coatings with desired properties. By use of the proposed theory, one can inter-relate the composition of the deposited monolayer and the technological parameters of the process, namely deposition rate, subphase composition, monolayer type, etc. Acknowledgment. Financial assistance by the Bundesministerium fu¨r Bildung, Wissenschaft, Forschung and Technologie (BMBF), the Ukrainian Ministry of Education and Science (Common Project UKR 07/007), and National Ukrainian Academy of Sciences (Project 69/07-H) is gratefully acknowledged. The work was supported by COST D43 Action. References and Notes (1) Xia, Y.; Qin, D.; Yin, Y. Curr. Opin. Colloid Interface Sci. 2001, 6, 54. (2) Huang, Y.; Duan, X.; Wei, Q.; Lieber, C. M. Science 2001, 291, 30–633. (3) Yabu, H.; Shimomura, M. AdV. Funct. Mater. 2005, 15, 575. (4) Henzie, J.; Barton, J. E.; Stender, C. L.; Odom, T. W. Acc. Chem. Res. 2006, 39, 249. (5) Moraille, P.; Badia, A. Langmuir 2002, 18, 4414. (6) Howland, M. C.; Johal, M. S.; Parikh, A. N. Langmuir 2005, 21, 10468.
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