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Local Surface Potentials in the Three-Phase Coexistence Region of a Langmuir Monolayer P. Heinig,† S. Wurlitzer,‡ P. Steffen,† F. Kremer,‡ and Th. M. Fischer*,† Max Planck Institute of Colloids and Interfaces, Am Mu¨ hlenberg 1, D-14476 Golm, Germany, and Fakulta¨ t fu¨ r Physik und Geowissenschaften, Universita¨ t Leipzig, Linne´ strasse 5, 04103 Leipzig, Germany Received June 5, 2000. In Final Form: September 14, 2000 The relative surface potentials of Langmuir monolayer phases at the air/water interface have been determined in the three-phase coexistence region gas/liquid expanded/liquid condensed of methyloctadecanoate. The monolayer is assumed to be in hydrostatic equilibrium. A force balance applied to the morphology of the monolayer observed with fluorescence microscopy directly yields the relative surface potentials. This enables the determination of surface potential variations on a micron scale.
1. Introduction Langmuir monolayers at the air/water-interface are a unique system to study pattern formation of coexisting phases. The asymmetric arrangement of the molecules at the interface (hydrophilic headgroups immersed in the water subphase and hydrophobic tails pointing toward the air) leads to macroscopic electric dipole moments perpendicular to the interface. A relation between the molecular dipole moments and the surface potential is given by Demchak and Fort.1 The electrostatic interactions between the dipole moments compete with interactions due to the line tension between the coexisting phases. If two phases coexist, many patterns, the emulsification of the minority phase2 and shape and fingering instabilities,3 arising from this competition can be determined, as explained by McConnell.4 The variety of phenomena further increases if three phases coexist. Partial or complete wetting of the interface between two phases by the third phase may occur5 depending on the surface potentials of the phases, the line tensions between the phases, and the size of the emulsificated droplets. Increasing the number of coexisting phases also increases the number of parameters (the surface potentials and the line tensions), which determines the pattern observed in the experiment. If the three-phase coexistence were a usual triple-point coexistence, then a lever rule would hold; that is, the properties of the coexisting phases would not change as the relative areas of the phases are varied. Applying the lever rule, the surface potential determined by the macroscopic Kelvin method is valid in the entire three-phase region, if the domains can be considered to mainly consist of 2d-bulk phase (neglecting effects near the phase boundaries). Near the boundary between two phases it has been shown by Pandit and Fisher6 that the bulk properties may be modified. Hence, the lever rule * To whom correspondence should be addressed. E-mail:
[email protected]. † Max Planck Institute of Colloids and Interfaces. ‡ Universita ¨ t Leipzig. (1) Demchak, R. J.; Fort, T. J. Colloid Interface Sci. 1973, 46, 191. (2) McConnell, H. M.; Moy, V. T. J. Phys. Chem. 1988, 92, 4520. (3) Lee, K. Y. C.; McConnell, H. M. J. Phys. Chem. 1993, 97, 9532. (4) McConnell, H. M. Annu. Rev. Phys. Chem. 1991, 42, 171. (5) Khattari, Z.; Steffen, P.; Heinig, P.; Wurlitzer, S.; Lo¨sche, M.; Fischer, Th. M. In preparation. (6) Pandit, R.; Fisher, M. E. Phys. Rev. Lett. 1983, 51, 1772.
will not hold for the properties of the phases within the boundary region. The long-range dipolar interactions inhibit the formation of extended one-phase regions.4 Instead finite droplets of the minority phase will be immersed in the majority phase and most regions of each phase are close to a phase boundary. The validity of the lever rule will not hold exactly. Therefore, techniques are required, which are able to measure the surface potential of the individual phases in circumstances where all phases cover a non-negligible amount of the entire area. These have to be local techniques, which work on the scale of the observed patterns. McConnell, Rice, and Benvegnu7 have measured thermal fluctuations of the position of trapped domains in the two-phase coexistence region and thereby determined the local surface potentials. In the present work a related method is presented, where the surface potential of one phase can be determined from a force balance of dipolar forces, if the surface potentials of the other phases are known. The results obtained are compared with macroscopic Kelvin probe measurements of the surface potential. 2. Theory and Experiments 2.1. Macroscopic Measurement. The monolayer was spread to an area of 170 Å2 per molecule. The compression of the monolayer was performed at a rate of 0.01-0.1 Å2/molecule‚s, and the surface pressure and surface potential (using a Kelvin probe Nima KP1) were recorded simultaneously. Figure 1 shows the surface pressure and surface potential as a function of the area per molecule. The three-phase coexistence region of the liquid condensed (LC), liquid expanded (LE), and gaseous (G) phases is located in the range between 25 and 39 Å2/molecule. The surface pressure vanishes in this region and starts to increase below 25 Å2/molecule, where only the LE and LC phases remain. In the three-phase coexistence region the surface potential is a linear function of the area, in accordance with a lever rule. Above 39 Å2/molecule a law according to VG ∝ A-1 applies, which is observed if only the density of the molecular dipoles and not the molecular dipole strength is changed during compression. At three areas per molecule (A ) 23.75, 25.1, and 30.9 Å2/molecule) the area fractions φLC ) 85, 83, and 49%, φLE ) 11, 7, and (7) McConnell, H. M.; Rice, P. A.; Benvegnu, D. J. J. Phys. Chem. 1990, 94, 8965.
10.1021/la000790q CCC: $19.00 © 2000 American Chemical Society Published on Web 11/21/2000
Three-Phase Coexistence Region of a Langmuir Monolayer
Figure 1. Surface potential and surface pressure of a methyl octadecanoate Langmuir monolayer at the air/water interface measured on compression of the monolayer. Within the threephase coexistence region, liquid condensed/liquid expanded/ gaseous (25-39 Å2/molecule), the surface potential is a linear function of the area in accordance with a lever rule. The data of this range are taken to calculate the surface potentials of the three coexisting phases.
3%, and φG ) 0, 10, and 48% have been determined respectively by analyzing 10 different fluorescence microscope images. The lever rule reads
V ) φLCVLC + φLEVLE + φGVG
(1)
With the values at the three areas per molecule, eq 1 is a system of three linear equations, which can be resolved for the three unknowns VLC, VLE, and VG. A similar equation holds for the specific areas per molecule of the three phases. One finds VLC ) (505 ( 10) mV , VLE ) (230 ( 100) mV, and VG ) (95 ( 50) mV, respectively, and ALC ) 23 Å2/molecule, ALE ) 29 Å2/molecule, and AG ) 39 Å2/ molecule. The agreement of the values of AG and ALC with the edges in the surface potential curve and respectively the pressure area isotherms gives confidence in these values. 2.2. Micron Scale Measurement. The monolayer was spread and compressed to an area of 30 Å2/molecule at a temperature of 25°. At this temperature the gaseous, liquid expanded, and liquid condensed phases coexist and they are visualized using polarized fluoresence microscopy. Figure 2 shows a fluorescence microscopy image of the three-phase coexistence region. The gaseous phase (black) is the majority phase covering φG ≈ 50% of the area of the trough. Circular liquid condensed droplets (gray, φLC ≈ 40%) are immersed into the gas phase. They are connected by a network of LE phase (bright, φLE ≈ 10%). The monolayer is at rest, and none of the LC domains are moving (|ν| < 1 µm/s). Calculating the hydrodynamic force acting on a moving domain of typical diameter of some microns,8 one may conclude that the net force acting on an individual domain is smaller than F ≈ 10 fΝ. The forces arising due to electrostatic interactions between two domains are in the range of some F ≈ (1-10) fΝ. If both forces are of the same order of magnitude, they give rise to typical fluctuating Brownian motion of the domains around their equilibrium position. In Figure 2 these fluctuations are too small to be observed. It is justified to assume that the electrostatic force on the average domain position FD arising from the sum over all electrostatic forces from the surrounding domains balances; that is, on (8) Wurlitzer, S.; Lautz, C.; Ibn-Elhaj, M.; Fischer, Th. M. In preparation.
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Figure 2. Fluorescence microscopy image within the threephase coexistence region of methyl octadecanoate. A liquid condensed domain (marked by black stripes) is positioned within the electrostatic potential arising from the other liquid condensed (gray), liquid expanded (bright), and gaseous (dark) regions. The balance of forces from the liquid condensed and liquid expanded regions (eq 3) yields the ratio (VLE - VG)/(VLC - VG) ) 0.45 ( 0.2 of the surface potentials of the three phases.
average the monolayer is in hydrostatic equilibrium. Using McConnell’s model of phases with homogeneous dipole density,4 one finds
FD )
[
30 2Wair (V - VG) (VLC - VG) 4π W + air D (VLE - VG)
∫D∫LCrr5 +
]
∫D∫LErr5
) 0 (2)
Here 0 is the permittivity constant, air and W are the relative permittivities of air and water, and V is the surface potential where the subscipt D refers to the individual domain, LC to the LC phase, LE to the LE phase, and G to the gas phase. The integrals are taken over the area of the domain and the areas of the surrounding LC and LE phases. The vector r starts within the individual domain under study and ends within the surrounding LC and respectively LE phases. Since the domain is completely wetted with gas, the radius r in eq 2 will be larger than 10 µm and no divergencies occur in the integrals. The x-component of eq 2 can be resolved for the ratio of the surface potentials (VLE - VG)/(VLC - VG) to yield
VLE - VG )VLC - VG
∫LC ∫Dx/r5 ∫LE∫Dx/r5
(3)
A similar equation holds for the y-component. Equation 3 is the basis for the determination of the relative surface potentials of the LE and LC phases. Two experimental examples are given in the present work. Equation 3 is most useful for an arrangement of the phases, where the force balance is achieved mainly due to opposing forces coming from the different phases and not to a force balance of equally distributed areas of one phase around the domain. Equation 3 is applied to the LC domain marked in Figure 2 and to four other domains with similar texture. The integrals are performed numerically, and one finds (VLE - VG)/(VLC - VG) ) 0.45 ( 0.2. Using VLC ) 505 ( 10 mV and VG ) 95 ( 50 mV from section 2.1, one finds VLE ) 280 ( 100mV.
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Heinig et al.
Figure 3. Fluorescence microscopy image within the threephase coexistence region of methyl octadecanoate. In comparison to Figure 2 the domains are smaller and the forces between the domains are larger. A colloidal region of small LE droplets immersed in the gas phase is opposed by a liquid condensed phase. From the thickness D of the depletion layer between the phases, the average radius of the colloidal LE particles, and their average separation, the ratio (VLE - VG)/(VLC - VG) ) 0.25 ( 0.02 is calculated.
In Figure 2 the size and the distances between the domains are large (≈50 µm), which facilitates the accurate measurement of the distances, but the resulting forces are weak (fN). In Figure 3 an arrangement of the phases is shown, where the forces are 1 order of magnitude stronger (10fN) and the distances are short. A region of small LE droplets immersed in the gas phase is opposed by an extended LC area. At the LC/LE boundary a depletion layer D ) 8 µm is formed, where the density of LE droplets is zero. This depletion layer is a result of the force balance (eq 3). The repulsion of the first layer of LE droplets from the LC phase is balanced by the repulsion of the individual LE droplets within the colloidal structure. The individual LE droplets perform pronounced Brownian motion. This ensures that on average a thermodynamic equilibrium is achieved. For the evaluation of eq 3 the average colloidal LE structure is approximated by a monodisperse hexagonal lattice with a lattice constant a of the disks with radius R and the LC phase is approximated by a halfplane. Following on from this, the integral in the nominator of eq 3 can be evaluated analytically and one finds
()
fDepl
D ) R
(
)
R/D + 1 -1 ∫-∞∞∫0∞∫discry5 ) 4π 3 x2R/D + 1
(4)
where R is the radius of the disk and D the thickness of the depletion layer (Figure 4). The integral
d ) R
()
fhex
∫LE'∫discry5
Figure 4. Scheme of the arrangement of the phases experimentally observed in Figure 3 with illustration of the definition of the depletion layer D, the lattice constant a, the radius R, and the average separation d.
Figure 5. Double logarithmic plot of the scaling function fhex(x) from eq 5, calculated numerically for 0.05 < x < 25. The asymptotic behaviors for x f 0 and x f ∞ are the power laws of eq 6, which are added to the figure. The mean field approximation hfhex(x) of eq 8 is a rough estimate of fhex(x) revealing the right scaling behavior but deviating as much as a factor 3 from the correct result.
One finds the asymptotic relations
d f R
()
fhex
{
xRRd
4π 3
(d)
Rπ2
4
for d/R f 0 (6) for d/R f ∞
where R is a Madelung constant defined as
(5) R)
1
x3
2 in the denominator of eq 3 is proportional to the force of the colloidal structure on a singular disk placed in the first layer of the structure and depends on the ratio d/R ) a/R - 2, where d is the separation of the LE droplets in the hexagonal lattice. The prime on the integral denotes the exclusion of the disk upon which the force is acting.
∑ k∈N,h∈Z
k (h2 + hk + k2)5/2
) 2.276
(7)
The function fhex(x) is evaluated numerically for 0.05 < x < 25 and plotted in Figure 5; the asymptotic equations (eq 6) are added as straight lines. An easy mean field approximation of fhex is to treat the hexatic lattice, where the first row of LE droplets is excluded (see Figure 4) as
Three-Phase Coexistence Region of a Langmuir Monolayer
a homogeneous phase of surface potential V h hex ) φhex LE VLE hex hex + (1 - φLE )VG, where φLE > φLE is the percentage of LE area within the hexagonal LE droplet region. One finds
hf hex(x) )
1 2π 1 fDepl x3x 2 x3 (x + 2)2
(
)
(8)
The mean field approximation is also shown in Figure 5. A comparison with the numerical result shows that hfhex(x) is a good estimate for fhex(x) in the whole range of x ) d/R. However, at large d/R it underestimates the actual value by a factor of about 3. Combining eqs 3-5, one obtains
VLE ) VG +
fDepl(D/R) (V - VG) fhex(d/R) LC
(9)
In the fluorescence image the number density of LE domains n within the colloidal region and φhex LE are determined by counting the domains and evaluating a histogram of gray values within a larger area of the colloidal LE region. They are related to R and d via
a)
x
R)
x
2 nx3
(10)
and
φhex LE πn
(11)
such that
d ) R
x
2π
x3φhex LE
-2
(12)
The thickness of the depletion layer D is taken from the image. Six different images have been evaluated. The results are listed in Table 1. Combining all data, one finds VLE ) 200 ( 70mV . The main error here is due to the uncertainty of VG (cf. section 2.1).
Langmuir, Vol. 16, No. 26, 2000 10257 Table 1 φhex LE 0.30 0.30 0.31 0.35 0.35 0.36 0.38
n
(1/µm2) 0.051 0.047 0.056 0.041 0.056 0.045 0.046
average
D (µm) 8.7 8.0 8.5 9.1 7.2 8.1 8.8
(VLE - VG)/(VLC - VG)
VLE (mV)
0.25 0.31 0.21 0.23 0.25 0.26 0.2
198 222 181 189 198 202 177
0.25 ( 0.02
200 ( 70
As noted in the introduction, the phase behavior near interfaces may be altered in comparison to the bulk phase behavior. This should be visible also in the surface potentials of the different phases, when changing the average size of the domains building up the texture of the monolayer. Also the values determined from Figures 2 and 3 are in accordance with each other and there is some tendency pointing toward a lowering of the LE potential near the LE/G interface; that is, large domains have a higher LE potential. Unfortunately the error of the measurements is too large to give a conclusive picture. A refinement of the measurements is needed to clarify this issue. 3. Conclusion A local technique for the measurement of surface potentials within coexisting Langmuir monolayer phases has been presented. The force balance between the phases yields the relative surface potential VLE - VG ≈ 145 mV of the liquid expanded phase with respect to the gaseous phase, being in good agreement with macroscopic Kelvin probe measurements. The local measurements point toward a decrease in the surface potential of the LE phase close to a LE/G boundary. Acknowledgment. We thank Prof. Mo¨hwald for buying the setup from the University of Leipzig. This work is supported by the German Science Foundation within the priority program wetting and structure formation at interfaces. Th.M.F. thanks the German Science Foundation for providing a Heisenberg fellowship. LA000790Q
